Properties

Label 2151.4.a.a.1.2
Level $2151$
Weight $4$
Character 2151.1
Self dual yes
Analytic conductor $126.913$
Analytic rank $1$
Dimension $22$
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] = [2151,4,Mod(1,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, 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("2151.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2151.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: \(126.913108422\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 239)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 2151.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.58869 q^{2} +13.0561 q^{4} +15.1251 q^{5} -25.8403 q^{7} -23.2007 q^{8} +O(q^{10})\) \(q-4.58869 q^{2} +13.0561 q^{4} +15.1251 q^{5} -25.8403 q^{7} -23.2007 q^{8} -69.4042 q^{10} -0.294915 q^{11} +3.35456 q^{13} +118.573 q^{14} +2.01223 q^{16} -39.1710 q^{17} +11.2006 q^{19} +197.474 q^{20} +1.35327 q^{22} -44.6692 q^{23} +103.767 q^{25} -15.3930 q^{26} -337.372 q^{28} -75.9968 q^{29} -34.1402 q^{31} +176.372 q^{32} +179.744 q^{34} -390.836 q^{35} +114.610 q^{37} -51.3959 q^{38} -350.912 q^{40} +364.615 q^{41} +145.626 q^{43} -3.85043 q^{44} +204.973 q^{46} +322.485 q^{47} +324.721 q^{49} -476.157 q^{50} +43.7974 q^{52} +345.906 q^{53} -4.46061 q^{55} +599.513 q^{56} +348.726 q^{58} +701.310 q^{59} -357.131 q^{61} +156.659 q^{62} -825.414 q^{64} +50.7379 q^{65} -748.574 q^{67} -511.419 q^{68} +1793.42 q^{70} -730.542 q^{71} +41.7860 q^{73} -525.909 q^{74} +146.235 q^{76} +7.62069 q^{77} +450.983 q^{79} +30.4351 q^{80} -1673.11 q^{82} +826.811 q^{83} -592.464 q^{85} -668.230 q^{86} +6.84223 q^{88} -1294.94 q^{89} -86.6828 q^{91} -583.204 q^{92} -1479.78 q^{94} +169.409 q^{95} -533.174 q^{97} -1490.04 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.58869 −1.62235 −0.811173 0.584806i \(-0.801171\pi\)
−0.811173 + 0.584806i \(0.801171\pi\)
\(3\) 0 0
\(4\) 13.0561 1.63201
\(5\) 15.1251 1.35283 0.676413 0.736522i \(-0.263533\pi\)
0.676413 + 0.736522i \(0.263533\pi\)
\(6\) 0 0
\(7\) −25.8403 −1.39524 −0.697622 0.716466i \(-0.745759\pi\)
−0.697622 + 0.716466i \(0.745759\pi\)
\(8\) −23.2007 −1.02534
\(9\) 0 0
\(10\) −69.4042 −2.19475
\(11\) −0.294915 −0.00808365 −0.00404183 0.999992i \(-0.501287\pi\)
−0.00404183 + 0.999992i \(0.501287\pi\)
\(12\) 0 0
\(13\) 3.35456 0.0715683 0.0357841 0.999360i \(-0.488607\pi\)
0.0357841 + 0.999360i \(0.488607\pi\)
\(14\) 118.573 2.26357
\(15\) 0 0
\(16\) 2.01223 0.0314411
\(17\) −39.1710 −0.558845 −0.279422 0.960168i \(-0.590143\pi\)
−0.279422 + 0.960168i \(0.590143\pi\)
\(18\) 0 0
\(19\) 11.2006 0.135241 0.0676207 0.997711i \(-0.478459\pi\)
0.0676207 + 0.997711i \(0.478459\pi\)
\(20\) 197.474 2.20782
\(21\) 0 0
\(22\) 1.35327 0.0131145
\(23\) −44.6692 −0.404964 −0.202482 0.979286i \(-0.564901\pi\)
−0.202482 + 0.979286i \(0.564901\pi\)
\(24\) 0 0
\(25\) 103.767 0.830140
\(26\) −15.3930 −0.116109
\(27\) 0 0
\(28\) −337.372 −2.27705
\(29\) −75.9968 −0.486629 −0.243315 0.969947i \(-0.578235\pi\)
−0.243315 + 0.969947i \(0.578235\pi\)
\(30\) 0 0
\(31\) −34.1402 −0.197799 −0.0988993 0.995097i \(-0.531532\pi\)
−0.0988993 + 0.995097i \(0.531532\pi\)
\(32\) 176.372 0.974327
\(33\) 0 0
\(34\) 179.744 0.906640
\(35\) −390.836 −1.88752
\(36\) 0 0
\(37\) 114.610 0.509237 0.254618 0.967042i \(-0.418050\pi\)
0.254618 + 0.967042i \(0.418050\pi\)
\(38\) −51.3959 −0.219408
\(39\) 0 0
\(40\) −350.912 −1.38710
\(41\) 364.615 1.38886 0.694431 0.719559i \(-0.255656\pi\)
0.694431 + 0.719559i \(0.255656\pi\)
\(42\) 0 0
\(43\) 145.626 0.516458 0.258229 0.966084i \(-0.416861\pi\)
0.258229 + 0.966084i \(0.416861\pi\)
\(44\) −3.85043 −0.0131926
\(45\) 0 0
\(46\) 204.973 0.656992
\(47\) 322.485 1.00083 0.500417 0.865784i \(-0.333180\pi\)
0.500417 + 0.865784i \(0.333180\pi\)
\(48\) 0 0
\(49\) 324.721 0.946708
\(50\) −476.157 −1.34677
\(51\) 0 0
\(52\) 43.7974 0.116800
\(53\) 345.906 0.896488 0.448244 0.893911i \(-0.352050\pi\)
0.448244 + 0.893911i \(0.352050\pi\)
\(54\) 0 0
\(55\) −4.46061 −0.0109358
\(56\) 599.513 1.43059
\(57\) 0 0
\(58\) 348.726 0.789481
\(59\) 701.310 1.54750 0.773752 0.633488i \(-0.218378\pi\)
0.773752 + 0.633488i \(0.218378\pi\)
\(60\) 0 0
\(61\) −357.131 −0.749606 −0.374803 0.927104i \(-0.622290\pi\)
−0.374803 + 0.927104i \(0.622290\pi\)
\(62\) 156.659 0.320898
\(63\) 0 0
\(64\) −825.414 −1.61214
\(65\) 50.7379 0.0968195
\(66\) 0 0
\(67\) −748.574 −1.36497 −0.682484 0.730901i \(-0.739100\pi\)
−0.682484 + 0.730901i \(0.739100\pi\)
\(68\) −511.419 −0.912039
\(69\) 0 0
\(70\) 1793.42 3.06222
\(71\) −730.542 −1.22112 −0.610559 0.791971i \(-0.709055\pi\)
−0.610559 + 0.791971i \(0.709055\pi\)
\(72\) 0 0
\(73\) 41.7860 0.0669956 0.0334978 0.999439i \(-0.489335\pi\)
0.0334978 + 0.999439i \(0.489335\pi\)
\(74\) −525.909 −0.826158
\(75\) 0 0
\(76\) 146.235 0.220715
\(77\) 7.62069 0.0112787
\(78\) 0 0
\(79\) 450.983 0.642273 0.321136 0.947033i \(-0.395935\pi\)
0.321136 + 0.947033i \(0.395935\pi\)
\(80\) 30.4351 0.0425344
\(81\) 0 0
\(82\) −1673.11 −2.25322
\(83\) 826.811 1.09343 0.546713 0.837320i \(-0.315879\pi\)
0.546713 + 0.837320i \(0.315879\pi\)
\(84\) 0 0
\(85\) −592.464 −0.756020
\(86\) −668.230 −0.837874
\(87\) 0 0
\(88\) 6.84223 0.00828845
\(89\) −1294.94 −1.54229 −0.771143 0.636662i \(-0.780315\pi\)
−0.771143 + 0.636662i \(0.780315\pi\)
\(90\) 0 0
\(91\) −86.6828 −0.0998553
\(92\) −583.204 −0.660904
\(93\) 0 0
\(94\) −1479.78 −1.62370
\(95\) 169.409 0.182958
\(96\) 0 0
\(97\) −533.174 −0.558099 −0.279050 0.960277i \(-0.590019\pi\)
−0.279050 + 0.960277i \(0.590019\pi\)
\(98\) −1490.04 −1.53589
\(99\) 0 0
\(100\) 1354.79 1.35479
\(101\) −944.073 −0.930087 −0.465043 0.885288i \(-0.653961\pi\)
−0.465043 + 0.885288i \(0.653961\pi\)
\(102\) 0 0
\(103\) 174.838 0.167256 0.0836279 0.996497i \(-0.473349\pi\)
0.0836279 + 0.996497i \(0.473349\pi\)
\(104\) −77.8281 −0.0733815
\(105\) 0 0
\(106\) −1587.26 −1.45441
\(107\) 21.1974 0.0191517 0.00957585 0.999954i \(-0.496952\pi\)
0.00957585 + 0.999954i \(0.496952\pi\)
\(108\) 0 0
\(109\) −680.020 −0.597561 −0.298780 0.954322i \(-0.596580\pi\)
−0.298780 + 0.954322i \(0.596580\pi\)
\(110\) 20.4683 0.0177416
\(111\) 0 0
\(112\) −51.9966 −0.0438680
\(113\) 606.546 0.504948 0.252474 0.967604i \(-0.418756\pi\)
0.252474 + 0.967604i \(0.418756\pi\)
\(114\) 0 0
\(115\) −675.624 −0.547846
\(116\) −992.219 −0.794183
\(117\) 0 0
\(118\) −3218.09 −2.51059
\(119\) 1012.19 0.779725
\(120\) 0 0
\(121\) −1330.91 −0.999935
\(122\) 1638.76 1.21612
\(123\) 0 0
\(124\) −445.736 −0.322809
\(125\) −321.143 −0.229791
\(126\) 0 0
\(127\) −22.4357 −0.0156760 −0.00783798 0.999969i \(-0.502495\pi\)
−0.00783798 + 0.999969i \(0.502495\pi\)
\(128\) 2376.59 1.64112
\(129\) 0 0
\(130\) −232.821 −0.157075
\(131\) 778.644 0.519316 0.259658 0.965701i \(-0.416390\pi\)
0.259658 + 0.965701i \(0.416390\pi\)
\(132\) 0 0
\(133\) −289.426 −0.188695
\(134\) 3434.97 2.21445
\(135\) 0 0
\(136\) 908.794 0.573003
\(137\) −1685.66 −1.05121 −0.525605 0.850729i \(-0.676161\pi\)
−0.525605 + 0.850729i \(0.676161\pi\)
\(138\) 0 0
\(139\) −336.141 −0.205116 −0.102558 0.994727i \(-0.532703\pi\)
−0.102558 + 0.994727i \(0.532703\pi\)
\(140\) −5102.78 −3.08045
\(141\) 0 0
\(142\) 3352.23 1.98108
\(143\) −0.989310 −0.000578533 0
\(144\) 0 0
\(145\) −1149.46 −0.658325
\(146\) −191.743 −0.108690
\(147\) 0 0
\(148\) 1496.35 0.831078
\(149\) −1536.41 −0.844747 −0.422373 0.906422i \(-0.638803\pi\)
−0.422373 + 0.906422i \(0.638803\pi\)
\(150\) 0 0
\(151\) 3415.89 1.84093 0.920466 0.390822i \(-0.127809\pi\)
0.920466 + 0.390822i \(0.127809\pi\)
\(152\) −259.861 −0.138668
\(153\) 0 0
\(154\) −34.9690 −0.0182979
\(155\) −516.372 −0.267587
\(156\) 0 0
\(157\) 372.951 0.189584 0.0947921 0.995497i \(-0.469781\pi\)
0.0947921 + 0.995497i \(0.469781\pi\)
\(158\) −2069.42 −1.04199
\(159\) 0 0
\(160\) 2667.64 1.31810
\(161\) 1154.26 0.565024
\(162\) 0 0
\(163\) 2075.92 0.997536 0.498768 0.866735i \(-0.333786\pi\)
0.498768 + 0.866735i \(0.333786\pi\)
\(164\) 4760.44 2.26663
\(165\) 0 0
\(166\) −3793.98 −1.77392
\(167\) −182.621 −0.0846205 −0.0423102 0.999105i \(-0.513472\pi\)
−0.0423102 + 0.999105i \(0.513472\pi\)
\(168\) 0 0
\(169\) −2185.75 −0.994878
\(170\) 2718.63 1.22653
\(171\) 0 0
\(172\) 1901.30 0.842863
\(173\) 2283.44 1.00351 0.501753 0.865011i \(-0.332689\pi\)
0.501753 + 0.865011i \(0.332689\pi\)
\(174\) 0 0
\(175\) −2681.38 −1.15825
\(176\) −0.593437 −0.000254159 0
\(177\) 0 0
\(178\) 5942.08 2.50212
\(179\) −3077.36 −1.28499 −0.642493 0.766291i \(-0.722100\pi\)
−0.642493 + 0.766291i \(0.722100\pi\)
\(180\) 0 0
\(181\) −1134.36 −0.465837 −0.232918 0.972496i \(-0.574827\pi\)
−0.232918 + 0.972496i \(0.574827\pi\)
\(182\) 397.761 0.162000
\(183\) 0 0
\(184\) 1036.36 0.415224
\(185\) 1733.48 0.688909
\(186\) 0 0
\(187\) 11.5521 0.00451751
\(188\) 4210.38 1.63337
\(189\) 0 0
\(190\) −777.367 −0.296822
\(191\) 4440.56 1.68224 0.841120 0.540848i \(-0.181897\pi\)
0.841120 + 0.540848i \(0.181897\pi\)
\(192\) 0 0
\(193\) −1205.33 −0.449541 −0.224770 0.974412i \(-0.572163\pi\)
−0.224770 + 0.974412i \(0.572163\pi\)
\(194\) 2446.57 0.905430
\(195\) 0 0
\(196\) 4239.57 1.54503
\(197\) −3242.26 −1.17260 −0.586298 0.810095i \(-0.699415\pi\)
−0.586298 + 0.810095i \(0.699415\pi\)
\(198\) 0 0
\(199\) −663.638 −0.236402 −0.118201 0.992990i \(-0.537713\pi\)
−0.118201 + 0.992990i \(0.537713\pi\)
\(200\) −2407.48 −0.851171
\(201\) 0 0
\(202\) 4332.06 1.50892
\(203\) 1963.78 0.678967
\(204\) 0 0
\(205\) 5514.83 1.87889
\(206\) −802.279 −0.271347
\(207\) 0 0
\(208\) 6.75015 0.00225019
\(209\) −3.30321 −0.00109324
\(210\) 0 0
\(211\) 1279.83 0.417568 0.208784 0.977962i \(-0.433049\pi\)
0.208784 + 0.977962i \(0.433049\pi\)
\(212\) 4516.17 1.46308
\(213\) 0 0
\(214\) −97.2683 −0.0310707
\(215\) 2202.60 0.698678
\(216\) 0 0
\(217\) 882.193 0.275978
\(218\) 3120.40 0.969451
\(219\) 0 0
\(220\) −58.2379 −0.0178473
\(221\) −131.401 −0.0399956
\(222\) 0 0
\(223\) −4187.92 −1.25760 −0.628798 0.777569i \(-0.716453\pi\)
−0.628798 + 0.777569i \(0.716453\pi\)
\(224\) −4557.50 −1.35942
\(225\) 0 0
\(226\) −2783.25 −0.819200
\(227\) 4456.98 1.30317 0.651586 0.758575i \(-0.274104\pi\)
0.651586 + 0.758575i \(0.274104\pi\)
\(228\) 0 0
\(229\) −6368.11 −1.83763 −0.918813 0.394693i \(-0.870851\pi\)
−0.918813 + 0.394693i \(0.870851\pi\)
\(230\) 3100.23 0.888796
\(231\) 0 0
\(232\) 1763.18 0.498958
\(233\) 3358.02 0.944168 0.472084 0.881553i \(-0.343502\pi\)
0.472084 + 0.881553i \(0.343502\pi\)
\(234\) 0 0
\(235\) 4877.60 1.35396
\(236\) 9156.34 2.52554
\(237\) 0 0
\(238\) −4644.62 −1.26498
\(239\) −239.000 −0.0646846
\(240\) 0 0
\(241\) −5178.87 −1.38423 −0.692117 0.721785i \(-0.743322\pi\)
−0.692117 + 0.721785i \(0.743322\pi\)
\(242\) 6107.15 1.62224
\(243\) 0 0
\(244\) −4662.73 −1.22336
\(245\) 4911.42 1.28073
\(246\) 0 0
\(247\) 37.5730 0.00967900
\(248\) 792.076 0.202810
\(249\) 0 0
\(250\) 1473.63 0.372801
\(251\) 6252.08 1.57222 0.786111 0.618085i \(-0.212091\pi\)
0.786111 + 0.618085i \(0.212091\pi\)
\(252\) 0 0
\(253\) 13.1736 0.00327359
\(254\) 102.950 0.0254318
\(255\) 0 0
\(256\) −4302.13 −1.05032
\(257\) −5515.39 −1.33868 −0.669340 0.742957i \(-0.733423\pi\)
−0.669340 + 0.742957i \(0.733423\pi\)
\(258\) 0 0
\(259\) −2961.55 −0.710510
\(260\) 662.438 0.158010
\(261\) 0 0
\(262\) −3572.96 −0.842511
\(263\) −5472.00 −1.28296 −0.641480 0.767140i \(-0.721679\pi\)
−0.641480 + 0.767140i \(0.721679\pi\)
\(264\) 0 0
\(265\) 5231.85 1.21279
\(266\) 1328.09 0.306128
\(267\) 0 0
\(268\) −9773.42 −2.22764
\(269\) −1170.67 −0.265342 −0.132671 0.991160i \(-0.542355\pi\)
−0.132671 + 0.991160i \(0.542355\pi\)
\(270\) 0 0
\(271\) −1426.98 −0.319864 −0.159932 0.987128i \(-0.551127\pi\)
−0.159932 + 0.987128i \(0.551127\pi\)
\(272\) −78.8211 −0.0175707
\(273\) 0 0
\(274\) 7734.97 1.70543
\(275\) −30.6026 −0.00671056
\(276\) 0 0
\(277\) 4834.12 1.04857 0.524286 0.851542i \(-0.324332\pi\)
0.524286 + 0.851542i \(0.324332\pi\)
\(278\) 1542.45 0.332769
\(279\) 0 0
\(280\) 9067.66 1.93534
\(281\) 5865.40 1.24520 0.622598 0.782542i \(-0.286077\pi\)
0.622598 + 0.782542i \(0.286077\pi\)
\(282\) 0 0
\(283\) 7932.25 1.66616 0.833080 0.553153i \(-0.186576\pi\)
0.833080 + 0.553153i \(0.186576\pi\)
\(284\) −9538.00 −1.99287
\(285\) 0 0
\(286\) 4.53963 0.000938581 0
\(287\) −9421.77 −1.93780
\(288\) 0 0
\(289\) −3378.63 −0.687692
\(290\) 5274.49 1.06803
\(291\) 0 0
\(292\) 545.561 0.109337
\(293\) 9117.39 1.81790 0.908948 0.416910i \(-0.136887\pi\)
0.908948 + 0.416910i \(0.136887\pi\)
\(294\) 0 0
\(295\) 10607.4 2.09350
\(296\) −2659.03 −0.522138
\(297\) 0 0
\(298\) 7050.09 1.37047
\(299\) −149.845 −0.0289826
\(300\) 0 0
\(301\) −3763.01 −0.720585
\(302\) −15674.4 −2.98663
\(303\) 0 0
\(304\) 22.5381 0.00425214
\(305\) −5401.63 −1.01409
\(306\) 0 0
\(307\) −3145.60 −0.584784 −0.292392 0.956299i \(-0.594451\pi\)
−0.292392 + 0.956299i \(0.594451\pi\)
\(308\) 99.4961 0.0184069
\(309\) 0 0
\(310\) 2369.47 0.434119
\(311\) −8439.79 −1.53883 −0.769416 0.638748i \(-0.779452\pi\)
−0.769416 + 0.638748i \(0.779452\pi\)
\(312\) 0 0
\(313\) 4897.13 0.884351 0.442176 0.896929i \(-0.354207\pi\)
0.442176 + 0.896929i \(0.354207\pi\)
\(314\) −1711.36 −0.307571
\(315\) 0 0
\(316\) 5888.06 1.04819
\(317\) −4551.34 −0.806399 −0.403200 0.915112i \(-0.632102\pi\)
−0.403200 + 0.915112i \(0.632102\pi\)
\(318\) 0 0
\(319\) 22.4126 0.00393374
\(320\) −12484.4 −2.18094
\(321\) 0 0
\(322\) −5296.56 −0.916664
\(323\) −438.738 −0.0755790
\(324\) 0 0
\(325\) 348.094 0.0594117
\(326\) −9525.74 −1.61835
\(327\) 0 0
\(328\) −8459.33 −1.42405
\(329\) −8333.10 −1.39641
\(330\) 0 0
\(331\) −7408.96 −1.23031 −0.615156 0.788405i \(-0.710907\pi\)
−0.615156 + 0.788405i \(0.710907\pi\)
\(332\) 10794.9 1.78448
\(333\) 0 0
\(334\) 837.990 0.137284
\(335\) −11322.2 −1.84656
\(336\) 0 0
\(337\) −11206.3 −1.81141 −0.905707 0.423904i \(-0.860659\pi\)
−0.905707 + 0.423904i \(0.860659\pi\)
\(338\) 10029.7 1.61404
\(339\) 0 0
\(340\) −7735.24 −1.23383
\(341\) 10.0684 0.00159894
\(342\) 0 0
\(343\) 472.342 0.0743559
\(344\) −3378.61 −0.529542
\(345\) 0 0
\(346\) −10478.0 −1.62803
\(347\) −11731.2 −1.81489 −0.907444 0.420174i \(-0.861969\pi\)
−0.907444 + 0.420174i \(0.861969\pi\)
\(348\) 0 0
\(349\) 2945.57 0.451784 0.225892 0.974152i \(-0.427470\pi\)
0.225892 + 0.974152i \(0.427470\pi\)
\(350\) 12304.0 1.87908
\(351\) 0 0
\(352\) −52.0147 −0.00787612
\(353\) 4349.41 0.655795 0.327898 0.944713i \(-0.393660\pi\)
0.327898 + 0.944713i \(0.393660\pi\)
\(354\) 0 0
\(355\) −11049.5 −1.65196
\(356\) −16906.8 −2.51702
\(357\) 0 0
\(358\) 14121.0 2.08469
\(359\) −4682.08 −0.688331 −0.344166 0.938909i \(-0.611838\pi\)
−0.344166 + 0.938909i \(0.611838\pi\)
\(360\) 0 0
\(361\) −6733.55 −0.981710
\(362\) 5205.23 0.755748
\(363\) 0 0
\(364\) −1131.74 −0.162965
\(365\) 632.016 0.0906335
\(366\) 0 0
\(367\) 4364.51 0.620779 0.310389 0.950610i \(-0.399541\pi\)
0.310389 + 0.950610i \(0.399541\pi\)
\(368\) −89.8847 −0.0127325
\(369\) 0 0
\(370\) −7954.41 −1.11765
\(371\) −8938.32 −1.25082
\(372\) 0 0
\(373\) −9494.23 −1.31794 −0.658971 0.752168i \(-0.729008\pi\)
−0.658971 + 0.752168i \(0.729008\pi\)
\(374\) −53.0090 −0.00732896
\(375\) 0 0
\(376\) −7481.87 −1.02619
\(377\) −254.936 −0.0348272
\(378\) 0 0
\(379\) 3268.10 0.442931 0.221466 0.975168i \(-0.428916\pi\)
0.221466 + 0.975168i \(0.428916\pi\)
\(380\) 2211.82 0.298589
\(381\) 0 0
\(382\) −20376.4 −2.72918
\(383\) 1831.46 0.244342 0.122171 0.992509i \(-0.461014\pi\)
0.122171 + 0.992509i \(0.461014\pi\)
\(384\) 0 0
\(385\) 115.263 0.0152581
\(386\) 5530.87 0.729311
\(387\) 0 0
\(388\) −6961.15 −0.910822
\(389\) −15278.4 −1.99138 −0.995688 0.0927692i \(-0.970428\pi\)
−0.995688 + 0.0927692i \(0.970428\pi\)
\(390\) 0 0
\(391\) 1749.74 0.226312
\(392\) −7533.74 −0.970693
\(393\) 0 0
\(394\) 14877.7 1.90236
\(395\) 6821.15 0.868884
\(396\) 0 0
\(397\) 6226.03 0.787092 0.393546 0.919305i \(-0.371248\pi\)
0.393546 + 0.919305i \(0.371248\pi\)
\(398\) 3045.23 0.383526
\(399\) 0 0
\(400\) 208.804 0.0261005
\(401\) −3838.48 −0.478016 −0.239008 0.971018i \(-0.576822\pi\)
−0.239008 + 0.971018i \(0.576822\pi\)
\(402\) 0 0
\(403\) −114.525 −0.0141561
\(404\) −12325.9 −1.51791
\(405\) 0 0
\(406\) −9011.17 −1.10152
\(407\) −33.8002 −0.00411649
\(408\) 0 0
\(409\) −890.762 −0.107690 −0.0538452 0.998549i \(-0.517148\pi\)
−0.0538452 + 0.998549i \(0.517148\pi\)
\(410\) −25305.8 −3.04821
\(411\) 0 0
\(412\) 2282.70 0.272963
\(413\) −18122.0 −2.15915
\(414\) 0 0
\(415\) 12505.6 1.47922
\(416\) 591.651 0.0697309
\(417\) 0 0
\(418\) 15.1574 0.00177362
\(419\) −14865.7 −1.73326 −0.866632 0.498948i \(-0.833720\pi\)
−0.866632 + 0.498948i \(0.833720\pi\)
\(420\) 0 0
\(421\) 3211.47 0.371776 0.185888 0.982571i \(-0.440484\pi\)
0.185888 + 0.982571i \(0.440484\pi\)
\(422\) −5872.73 −0.677440
\(423\) 0 0
\(424\) −8025.26 −0.919201
\(425\) −4064.68 −0.463919
\(426\) 0 0
\(427\) 9228.38 1.04588
\(428\) 276.755 0.0312557
\(429\) 0 0
\(430\) −10107.0 −1.13350
\(431\) 7311.80 0.817162 0.408581 0.912722i \(-0.366024\pi\)
0.408581 + 0.912722i \(0.366024\pi\)
\(432\) 0 0
\(433\) −15748.5 −1.74786 −0.873931 0.486051i \(-0.838437\pi\)
−0.873931 + 0.486051i \(0.838437\pi\)
\(434\) −4048.11 −0.447731
\(435\) 0 0
\(436\) −8878.39 −0.975224
\(437\) −500.320 −0.0547679
\(438\) 0 0
\(439\) −9031.99 −0.981944 −0.490972 0.871175i \(-0.663358\pi\)
−0.490972 + 0.871175i \(0.663358\pi\)
\(440\) 103.489 0.0112128
\(441\) 0 0
\(442\) 602.961 0.0648867
\(443\) −8338.56 −0.894304 −0.447152 0.894458i \(-0.647562\pi\)
−0.447152 + 0.894458i \(0.647562\pi\)
\(444\) 0 0
\(445\) −19586.1 −2.08645
\(446\) 19217.1 2.04026
\(447\) 0 0
\(448\) 21328.9 2.24932
\(449\) −2254.14 −0.236926 −0.118463 0.992958i \(-0.537797\pi\)
−0.118463 + 0.992958i \(0.537797\pi\)
\(450\) 0 0
\(451\) −107.531 −0.0112271
\(452\) 7919.11 0.824078
\(453\) 0 0
\(454\) −20451.7 −2.11420
\(455\) −1311.08 −0.135087
\(456\) 0 0
\(457\) 3522.47 0.360556 0.180278 0.983616i \(-0.442300\pi\)
0.180278 + 0.983616i \(0.442300\pi\)
\(458\) 29221.3 2.98127
\(459\) 0 0
\(460\) −8820.99 −0.894089
\(461\) 12960.9 1.30943 0.654715 0.755876i \(-0.272789\pi\)
0.654715 + 0.755876i \(0.272789\pi\)
\(462\) 0 0
\(463\) 13150.7 1.32001 0.660003 0.751263i \(-0.270555\pi\)
0.660003 + 0.751263i \(0.270555\pi\)
\(464\) −152.923 −0.0153002
\(465\) 0 0
\(466\) −15408.9 −1.53177
\(467\) −4317.68 −0.427834 −0.213917 0.976852i \(-0.568622\pi\)
−0.213917 + 0.976852i \(0.568622\pi\)
\(468\) 0 0
\(469\) 19343.4 1.90446
\(470\) −22381.8 −2.19659
\(471\) 0 0
\(472\) −16270.9 −1.58671
\(473\) −42.9471 −0.00417487
\(474\) 0 0
\(475\) 1162.25 0.112269
\(476\) 13215.2 1.27252
\(477\) 0 0
\(478\) 1096.70 0.104941
\(479\) −4948.79 −0.472058 −0.236029 0.971746i \(-0.575846\pi\)
−0.236029 + 0.971746i \(0.575846\pi\)
\(480\) 0 0
\(481\) 384.466 0.0364452
\(482\) 23764.2 2.24571
\(483\) 0 0
\(484\) −17376.5 −1.63190
\(485\) −8064.29 −0.755011
\(486\) 0 0
\(487\) 9485.80 0.882633 0.441317 0.897351i \(-0.354512\pi\)
0.441317 + 0.897351i \(0.354512\pi\)
\(488\) 8285.69 0.768597
\(489\) 0 0
\(490\) −22537.0 −2.07779
\(491\) −11416.0 −1.04928 −0.524638 0.851325i \(-0.675799\pi\)
−0.524638 + 0.851325i \(0.675799\pi\)
\(492\) 0 0
\(493\) 2976.87 0.271950
\(494\) −172.411 −0.0157027
\(495\) 0 0
\(496\) −68.6979 −0.00621901
\(497\) 18877.4 1.70376
\(498\) 0 0
\(499\) 3525.64 0.316292 0.158146 0.987416i \(-0.449448\pi\)
0.158146 + 0.987416i \(0.449448\pi\)
\(500\) −4192.87 −0.375021
\(501\) 0 0
\(502\) −28688.9 −2.55069
\(503\) −1779.64 −0.157754 −0.0788770 0.996884i \(-0.525133\pi\)
−0.0788770 + 0.996884i \(0.525133\pi\)
\(504\) 0 0
\(505\) −14279.2 −1.25825
\(506\) −60.4496 −0.00531089
\(507\) 0 0
\(508\) −292.922 −0.0255833
\(509\) 19285.4 1.67939 0.839696 0.543056i \(-0.182733\pi\)
0.839696 + 0.543056i \(0.182733\pi\)
\(510\) 0 0
\(511\) −1079.76 −0.0934753
\(512\) 728.382 0.0628716
\(513\) 0 0
\(514\) 25308.4 2.17180
\(515\) 2644.44 0.226268
\(516\) 0 0
\(517\) −95.1056 −0.00809040
\(518\) 13589.6 1.15269
\(519\) 0 0
\(520\) −1177.15 −0.0992724
\(521\) 8435.69 0.709356 0.354678 0.934989i \(-0.384591\pi\)
0.354678 + 0.934989i \(0.384591\pi\)
\(522\) 0 0
\(523\) −1493.58 −0.124875 −0.0624375 0.998049i \(-0.519887\pi\)
−0.0624375 + 0.998049i \(0.519887\pi\)
\(524\) 10166.0 0.847528
\(525\) 0 0
\(526\) 25109.3 2.08140
\(527\) 1337.31 0.110539
\(528\) 0 0
\(529\) −10171.7 −0.836004
\(530\) −24007.3 −1.96757
\(531\) 0 0
\(532\) −3778.76 −0.307951
\(533\) 1223.12 0.0993985
\(534\) 0 0
\(535\) 320.612 0.0259089
\(536\) 17367.4 1.39955
\(537\) 0 0
\(538\) 5371.83 0.430476
\(539\) −95.7650 −0.00765286
\(540\) 0 0
\(541\) −4187.91 −0.332814 −0.166407 0.986057i \(-0.553217\pi\)
−0.166407 + 0.986057i \(0.553217\pi\)
\(542\) 6547.98 0.518930
\(543\) 0 0
\(544\) −6908.67 −0.544497
\(545\) −10285.4 −0.808396
\(546\) 0 0
\(547\) 4198.00 0.328142 0.164071 0.986449i \(-0.447537\pi\)
0.164071 + 0.986449i \(0.447537\pi\)
\(548\) −22008.1 −1.71558
\(549\) 0 0
\(550\) 140.426 0.0108869
\(551\) −851.207 −0.0658124
\(552\) 0 0
\(553\) −11653.5 −0.896128
\(554\) −22182.3 −1.70115
\(555\) 0 0
\(556\) −4388.68 −0.334751
\(557\) 3451.57 0.262563 0.131282 0.991345i \(-0.458091\pi\)
0.131282 + 0.991345i \(0.458091\pi\)
\(558\) 0 0
\(559\) 488.510 0.0369620
\(560\) −786.452 −0.0593458
\(561\) 0 0
\(562\) −26914.5 −2.02014
\(563\) 1630.16 0.122030 0.0610150 0.998137i \(-0.480566\pi\)
0.0610150 + 0.998137i \(0.480566\pi\)
\(564\) 0 0
\(565\) 9174.05 0.683107
\(566\) −36398.6 −2.70309
\(567\) 0 0
\(568\) 16949.1 1.25205
\(569\) 17097.3 1.25967 0.629837 0.776727i \(-0.283122\pi\)
0.629837 + 0.776727i \(0.283122\pi\)
\(570\) 0 0
\(571\) −9042.60 −0.662734 −0.331367 0.943502i \(-0.607510\pi\)
−0.331367 + 0.943502i \(0.607510\pi\)
\(572\) −12.9165 −0.000944171 0
\(573\) 0 0
\(574\) 43233.6 3.14379
\(575\) −4635.21 −0.336177
\(576\) 0 0
\(577\) −20921.9 −1.50952 −0.754759 0.656002i \(-0.772246\pi\)
−0.754759 + 0.656002i \(0.772246\pi\)
\(578\) 15503.5 1.11568
\(579\) 0 0
\(580\) −15007.4 −1.07439
\(581\) −21365.0 −1.52560
\(582\) 0 0
\(583\) −102.013 −0.00724690
\(584\) −969.464 −0.0686930
\(585\) 0 0
\(586\) −41836.8 −2.94926
\(587\) 6953.22 0.488910 0.244455 0.969661i \(-0.421391\pi\)
0.244455 + 0.969661i \(0.421391\pi\)
\(588\) 0 0
\(589\) −382.390 −0.0267506
\(590\) −48673.8 −3.39639
\(591\) 0 0
\(592\) 230.622 0.0160110
\(593\) −8046.53 −0.557220 −0.278610 0.960404i \(-0.589874\pi\)
−0.278610 + 0.960404i \(0.589874\pi\)
\(594\) 0 0
\(595\) 15309.4 1.05483
\(596\) −20059.4 −1.37863
\(597\) 0 0
\(598\) 687.594 0.0470198
\(599\) −15878.7 −1.08312 −0.541558 0.840663i \(-0.682166\pi\)
−0.541558 + 0.840663i \(0.682166\pi\)
\(600\) 0 0
\(601\) −4180.66 −0.283748 −0.141874 0.989885i \(-0.545313\pi\)
−0.141874 + 0.989885i \(0.545313\pi\)
\(602\) 17267.3 1.16904
\(603\) 0 0
\(604\) 44598.0 3.00442
\(605\) −20130.1 −1.35274
\(606\) 0 0
\(607\) 782.941 0.0523536 0.0261768 0.999657i \(-0.491667\pi\)
0.0261768 + 0.999657i \(0.491667\pi\)
\(608\) 1975.47 0.131769
\(609\) 0 0
\(610\) 24786.4 1.64520
\(611\) 1081.79 0.0716280
\(612\) 0 0
\(613\) −19886.3 −1.31028 −0.655138 0.755509i \(-0.727390\pi\)
−0.655138 + 0.755509i \(0.727390\pi\)
\(614\) 14434.2 0.948722
\(615\) 0 0
\(616\) −176.805 −0.0115644
\(617\) −12249.7 −0.799280 −0.399640 0.916672i \(-0.630865\pi\)
−0.399640 + 0.916672i \(0.630865\pi\)
\(618\) 0 0
\(619\) −25619.9 −1.66357 −0.831785 0.555097i \(-0.812681\pi\)
−0.831785 + 0.555097i \(0.812681\pi\)
\(620\) −6741.79 −0.436705
\(621\) 0 0
\(622\) 38727.6 2.49652
\(623\) 33461.7 2.15187
\(624\) 0 0
\(625\) −17828.2 −1.14101
\(626\) −22471.4 −1.43472
\(627\) 0 0
\(628\) 4869.27 0.309403
\(629\) −4489.39 −0.284584
\(630\) 0 0
\(631\) −19492.8 −1.22979 −0.614894 0.788610i \(-0.710801\pi\)
−0.614894 + 0.788610i \(0.710801\pi\)
\(632\) −10463.1 −0.658545
\(633\) 0 0
\(634\) 20884.7 1.30826
\(635\) −339.341 −0.0212068
\(636\) 0 0
\(637\) 1089.30 0.0677543
\(638\) −102.844 −0.00638189
\(639\) 0 0
\(640\) 35946.1 2.22015
\(641\) 15937.5 0.982047 0.491024 0.871146i \(-0.336623\pi\)
0.491024 + 0.871146i \(0.336623\pi\)
\(642\) 0 0
\(643\) −21204.1 −1.30048 −0.650239 0.759730i \(-0.725331\pi\)
−0.650239 + 0.759730i \(0.725331\pi\)
\(644\) 15070.2 0.922123
\(645\) 0 0
\(646\) 2013.23 0.122615
\(647\) −16437.7 −0.998817 −0.499408 0.866367i \(-0.666449\pi\)
−0.499408 + 0.866367i \(0.666449\pi\)
\(648\) 0 0
\(649\) −206.827 −0.0125095
\(650\) −1597.30 −0.0963863
\(651\) 0 0
\(652\) 27103.3 1.62799
\(653\) −8512.52 −0.510139 −0.255069 0.966923i \(-0.582098\pi\)
−0.255069 + 0.966923i \(0.582098\pi\)
\(654\) 0 0
\(655\) 11777.0 0.702545
\(656\) 733.690 0.0436674
\(657\) 0 0
\(658\) 38238.0 2.26546
\(659\) 587.889 0.0347510 0.0173755 0.999849i \(-0.494469\pi\)
0.0173755 + 0.999849i \(0.494469\pi\)
\(660\) 0 0
\(661\) 3835.60 0.225700 0.112850 0.993612i \(-0.464002\pi\)
0.112850 + 0.993612i \(0.464002\pi\)
\(662\) 33997.4 1.99599
\(663\) 0 0
\(664\) −19182.6 −1.12113
\(665\) −4377.59 −0.255271
\(666\) 0 0
\(667\) 3394.71 0.197067
\(668\) −2384.31 −0.138101
\(669\) 0 0
\(670\) 51954.2 2.99577
\(671\) 105.323 0.00605955
\(672\) 0 0
\(673\) −15030.8 −0.860912 −0.430456 0.902612i \(-0.641647\pi\)
−0.430456 + 0.902612i \(0.641647\pi\)
\(674\) 51422.3 2.93874
\(675\) 0 0
\(676\) −28537.2 −1.62365
\(677\) −20846.3 −1.18344 −0.591720 0.806144i \(-0.701551\pi\)
−0.591720 + 0.806144i \(0.701551\pi\)
\(678\) 0 0
\(679\) 13777.4 0.778685
\(680\) 13745.6 0.775174
\(681\) 0 0
\(682\) −46.2010 −0.00259403
\(683\) 11132.8 0.623697 0.311849 0.950132i \(-0.399052\pi\)
0.311849 + 0.950132i \(0.399052\pi\)
\(684\) 0 0
\(685\) −25495.7 −1.42210
\(686\) −2167.43 −0.120631
\(687\) 0 0
\(688\) 293.032 0.0162380
\(689\) 1160.36 0.0641601
\(690\) 0 0
\(691\) 22733.5 1.25156 0.625778 0.780001i \(-0.284782\pi\)
0.625778 + 0.780001i \(0.284782\pi\)
\(692\) 29812.7 1.63773
\(693\) 0 0
\(694\) 53831.0 2.94438
\(695\) −5084.16 −0.277486
\(696\) 0 0
\(697\) −14282.4 −0.776159
\(698\) −13516.3 −0.732950
\(699\) 0 0
\(700\) −35008.3 −1.89027
\(701\) 2040.80 0.109957 0.0549785 0.998488i \(-0.482491\pi\)
0.0549785 + 0.998488i \(0.482491\pi\)
\(702\) 0 0
\(703\) 1283.70 0.0688699
\(704\) 243.427 0.0130320
\(705\) 0 0
\(706\) −19958.1 −1.06393
\(707\) 24395.1 1.29770
\(708\) 0 0
\(709\) 23191.1 1.22844 0.614218 0.789136i \(-0.289471\pi\)
0.614218 + 0.789136i \(0.289471\pi\)
\(710\) 50702.7 2.68005
\(711\) 0 0
\(712\) 30043.5 1.58136
\(713\) 1525.01 0.0801013
\(714\) 0 0
\(715\) −14.9634 −0.000782655 0
\(716\) −40178.2 −2.09711
\(717\) 0 0
\(718\) 21484.6 1.11671
\(719\) −27756.3 −1.43969 −0.719845 0.694135i \(-0.755787\pi\)
−0.719845 + 0.694135i \(0.755787\pi\)
\(720\) 0 0
\(721\) −4517.88 −0.233363
\(722\) 30898.1 1.59267
\(723\) 0 0
\(724\) −14810.3 −0.760249
\(725\) −7885.99 −0.403970
\(726\) 0 0
\(727\) −14185.6 −0.723679 −0.361840 0.932240i \(-0.617851\pi\)
−0.361840 + 0.932240i \(0.617851\pi\)
\(728\) 2011.10 0.102385
\(729\) 0 0
\(730\) −2900.12 −0.147039
\(731\) −5704.30 −0.288620
\(732\) 0 0
\(733\) 14805.4 0.746042 0.373021 0.927823i \(-0.378322\pi\)
0.373021 + 0.927823i \(0.378322\pi\)
\(734\) −20027.4 −1.00712
\(735\) 0 0
\(736\) −7878.39 −0.394567
\(737\) 220.766 0.0110339
\(738\) 0 0
\(739\) −16831.9 −0.837853 −0.418926 0.908020i \(-0.637593\pi\)
−0.418926 + 0.908020i \(0.637593\pi\)
\(740\) 22632.4 1.12430
\(741\) 0 0
\(742\) 41015.2 2.02926
\(743\) 20561.8 1.01526 0.507631 0.861575i \(-0.330521\pi\)
0.507631 + 0.861575i \(0.330521\pi\)
\(744\) 0 0
\(745\) −23238.2 −1.14280
\(746\) 43566.1 2.13816
\(747\) 0 0
\(748\) 150.825 0.00737261
\(749\) −547.747 −0.0267213
\(750\) 0 0
\(751\) −21060.5 −1.02332 −0.511658 0.859189i \(-0.670968\pi\)
−0.511658 + 0.859189i \(0.670968\pi\)
\(752\) 648.914 0.0314674
\(753\) 0 0
\(754\) 1169.82 0.0565018
\(755\) 51665.5 2.49046
\(756\) 0 0
\(757\) −23440.4 −1.12544 −0.562719 0.826648i \(-0.690245\pi\)
−0.562719 + 0.826648i \(0.690245\pi\)
\(758\) −14996.3 −0.718588
\(759\) 0 0
\(760\) −3930.41 −0.187593
\(761\) −13575.3 −0.646656 −0.323328 0.946287i \(-0.604802\pi\)
−0.323328 + 0.946287i \(0.604802\pi\)
\(762\) 0 0
\(763\) 17571.9 0.833744
\(764\) 57976.3 2.74543
\(765\) 0 0
\(766\) −8403.99 −0.396408
\(767\) 2352.59 0.110752
\(768\) 0 0
\(769\) 13313.7 0.624323 0.312162 0.950029i \(-0.398947\pi\)
0.312162 + 0.950029i \(0.398947\pi\)
\(770\) −528.908 −0.0247539
\(771\) 0 0
\(772\) −15736.8 −0.733654
\(773\) 7378.75 0.343331 0.171666 0.985155i \(-0.445085\pi\)
0.171666 + 0.985155i \(0.445085\pi\)
\(774\) 0 0
\(775\) −3542.64 −0.164201
\(776\) 12370.0 0.572239
\(777\) 0 0
\(778\) 70107.8 3.23070
\(779\) 4083.90 0.187832
\(780\) 0 0
\(781\) 215.448 0.00987109
\(782\) −8029.00 −0.367156
\(783\) 0 0
\(784\) 653.413 0.0297655
\(785\) 5640.91 0.256475
\(786\) 0 0
\(787\) 32584.1 1.47585 0.737927 0.674880i \(-0.235805\pi\)
0.737927 + 0.674880i \(0.235805\pi\)
\(788\) −42331.1 −1.91369
\(789\) 0 0
\(790\) −31300.1 −1.40963
\(791\) −15673.3 −0.704525
\(792\) 0 0
\(793\) −1198.02 −0.0536480
\(794\) −28569.3 −1.27694
\(795\) 0 0
\(796\) −8664.49 −0.385810
\(797\) 30922.0 1.37430 0.687148 0.726517i \(-0.258862\pi\)
0.687148 + 0.726517i \(0.258862\pi\)
\(798\) 0 0
\(799\) −12632.1 −0.559311
\(800\) 18301.7 0.808827
\(801\) 0 0
\(802\) 17613.6 0.775507
\(803\) −12.3233 −0.000541569 0
\(804\) 0 0
\(805\) 17458.3 0.764379
\(806\) 525.521 0.0229661
\(807\) 0 0
\(808\) 21903.1 0.953650
\(809\) −39348.2 −1.71003 −0.855013 0.518607i \(-0.826451\pi\)
−0.855013 + 0.518607i \(0.826451\pi\)
\(810\) 0 0
\(811\) −20564.7 −0.890413 −0.445207 0.895428i \(-0.646870\pi\)
−0.445207 + 0.895428i \(0.646870\pi\)
\(812\) 25639.2 1.10808
\(813\) 0 0
\(814\) 155.098 0.00667838
\(815\) 31398.4 1.34949
\(816\) 0 0
\(817\) 1631.09 0.0698465
\(818\) 4087.43 0.174711
\(819\) 0 0
\(820\) 72002.0 3.06636
\(821\) −13125.1 −0.557939 −0.278969 0.960300i \(-0.589993\pi\)
−0.278969 + 0.960300i \(0.589993\pi\)
\(822\) 0 0
\(823\) −2653.20 −0.112375 −0.0561875 0.998420i \(-0.517894\pi\)
−0.0561875 + 0.998420i \(0.517894\pi\)
\(824\) −4056.37 −0.171493
\(825\) 0 0
\(826\) 83156.4 3.50288
\(827\) 16068.9 0.675660 0.337830 0.941207i \(-0.390307\pi\)
0.337830 + 0.941207i \(0.390307\pi\)
\(828\) 0 0
\(829\) −9738.24 −0.407989 −0.203995 0.978972i \(-0.565392\pi\)
−0.203995 + 0.978972i \(0.565392\pi\)
\(830\) −57384.2 −2.39980
\(831\) 0 0
\(832\) −2768.90 −0.115378
\(833\) −12719.6 −0.529063
\(834\) 0 0
\(835\) −2762.15 −0.114477
\(836\) −43.1270 −0.00178418
\(837\) 0 0
\(838\) 68214.1 2.81195
\(839\) −1386.67 −0.0570600 −0.0285300 0.999593i \(-0.509083\pi\)
−0.0285300 + 0.999593i \(0.509083\pi\)
\(840\) 0 0
\(841\) −18613.5 −0.763192
\(842\) −14736.5 −0.603149
\(843\) 0 0
\(844\) 16709.5 0.681475
\(845\) −33059.6 −1.34590
\(846\) 0 0
\(847\) 34391.2 1.39515
\(848\) 696.043 0.0281866
\(849\) 0 0
\(850\) 18651.5 0.752638
\(851\) −5119.53 −0.206222
\(852\) 0 0
\(853\) −15525.5 −0.623192 −0.311596 0.950215i \(-0.600863\pi\)
−0.311596 + 0.950215i \(0.600863\pi\)
\(854\) −42346.1 −1.69679
\(855\) 0 0
\(856\) −491.794 −0.0196369
\(857\) −7278.21 −0.290104 −0.145052 0.989424i \(-0.546335\pi\)
−0.145052 + 0.989424i \(0.546335\pi\)
\(858\) 0 0
\(859\) 23322.9 0.926388 0.463194 0.886257i \(-0.346703\pi\)
0.463194 + 0.886257i \(0.346703\pi\)
\(860\) 28757.2 1.14025
\(861\) 0 0
\(862\) −33551.6 −1.32572
\(863\) 30197.6 1.19112 0.595561 0.803310i \(-0.296930\pi\)
0.595561 + 0.803310i \(0.296930\pi\)
\(864\) 0 0
\(865\) 34537.1 1.35757
\(866\) 72264.9 2.83564
\(867\) 0 0
\(868\) 11518.0 0.450397
\(869\) −133.002 −0.00519191
\(870\) 0 0
\(871\) −2511.14 −0.0976884
\(872\) 15776.9 0.612700
\(873\) 0 0
\(874\) 2295.81 0.0888525
\(875\) 8298.44 0.320615
\(876\) 0 0
\(877\) −46507.4 −1.79070 −0.895350 0.445363i \(-0.853075\pi\)
−0.895350 + 0.445363i \(0.853075\pi\)
\(878\) 41445.0 1.59305
\(879\) 0 0
\(880\) −8.97577 −0.000343833 0
\(881\) 358.878 0.0137241 0.00686203 0.999976i \(-0.497816\pi\)
0.00686203 + 0.999976i \(0.497816\pi\)
\(882\) 0 0
\(883\) 5590.37 0.213059 0.106529 0.994310i \(-0.466026\pi\)
0.106529 + 0.994310i \(0.466026\pi\)
\(884\) −1715.59 −0.0652731
\(885\) 0 0
\(886\) 38263.0 1.45087
\(887\) −32199.9 −1.21890 −0.609451 0.792823i \(-0.708610\pi\)
−0.609451 + 0.792823i \(0.708610\pi\)
\(888\) 0 0
\(889\) 579.745 0.0218718
\(890\) 89874.4 3.38494
\(891\) 0 0
\(892\) −54677.7 −2.05241
\(893\) 3612.01 0.135354
\(894\) 0 0
\(895\) −46545.2 −1.73836
\(896\) −61411.8 −2.28976
\(897\) 0 0
\(898\) 10343.6 0.384375
\(899\) 2594.54 0.0962546
\(900\) 0 0
\(901\) −13549.5 −0.500998
\(902\) 493.424 0.0182142
\(903\) 0 0
\(904\) −14072.3 −0.517740
\(905\) −17157.3 −0.630196
\(906\) 0 0
\(907\) 2479.81 0.0907835 0.0453917 0.998969i \(-0.485546\pi\)
0.0453917 + 0.998969i \(0.485546\pi\)
\(908\) 58190.5 2.12679
\(909\) 0 0
\(910\) 6016.15 0.219158
\(911\) 18543.6 0.674399 0.337199 0.941433i \(-0.390520\pi\)
0.337199 + 0.941433i \(0.390520\pi\)
\(912\) 0 0
\(913\) −243.839 −0.00883887
\(914\) −16163.5 −0.584947
\(915\) 0 0
\(916\) −83142.4 −2.99902
\(917\) −20120.4 −0.724573
\(918\) 0 0
\(919\) −18668.1 −0.670081 −0.335040 0.942204i \(-0.608750\pi\)
−0.335040 + 0.942204i \(0.608750\pi\)
\(920\) 15674.9 0.561726
\(921\) 0 0
\(922\) −59473.3 −2.12435
\(923\) −2450.65 −0.0873933
\(924\) 0 0
\(925\) 11892.8 0.422738
\(926\) −60344.3 −2.14151
\(927\) 0 0
\(928\) −13403.7 −0.474136
\(929\) 5627.24 0.198734 0.0993669 0.995051i \(-0.468318\pi\)
0.0993669 + 0.995051i \(0.468318\pi\)
\(930\) 0 0
\(931\) 3637.06 0.128034
\(932\) 43842.5 1.54089
\(933\) 0 0
\(934\) 19812.5 0.694095
\(935\) 174.726 0.00611140
\(936\) 0 0
\(937\) 6982.60 0.243449 0.121724 0.992564i \(-0.461158\pi\)
0.121724 + 0.992564i \(0.461158\pi\)
\(938\) −88760.7 −3.08970
\(939\) 0 0
\(940\) 63682.3 2.20967
\(941\) 11915.5 0.412789 0.206395 0.978469i \(-0.433827\pi\)
0.206395 + 0.978469i \(0.433827\pi\)
\(942\) 0 0
\(943\) −16287.1 −0.562439
\(944\) 1411.20 0.0486552
\(945\) 0 0
\(946\) 197.071 0.00677308
\(947\) −56192.9 −1.92822 −0.964110 0.265503i \(-0.914462\pi\)
−0.964110 + 0.265503i \(0.914462\pi\)
\(948\) 0 0
\(949\) 140.174 0.00479476
\(950\) −5333.23 −0.182140
\(951\) 0 0
\(952\) −23483.5 −0.799480
\(953\) 16717.7 0.568246 0.284123 0.958788i \(-0.408298\pi\)
0.284123 + 0.958788i \(0.408298\pi\)
\(954\) 0 0
\(955\) 67163.8 2.27578
\(956\) −3120.40 −0.105566
\(957\) 0 0
\(958\) 22708.5 0.765842
\(959\) 43558.0 1.46670
\(960\) 0 0
\(961\) −28625.4 −0.960876
\(962\) −1764.19 −0.0591267
\(963\) 0 0
\(964\) −67615.7 −2.25908
\(965\) −18230.6 −0.608151
\(966\) 0 0
\(967\) 53927.3 1.79337 0.896683 0.442674i \(-0.145970\pi\)
0.896683 + 0.442674i \(0.145970\pi\)
\(968\) 30878.1 1.02527
\(969\) 0 0
\(970\) 37004.5 1.22489
\(971\) −21338.5 −0.705236 −0.352618 0.935767i \(-0.614708\pi\)
−0.352618 + 0.935767i \(0.614708\pi\)
\(972\) 0 0
\(973\) 8685.99 0.286187
\(974\) −43527.4 −1.43194
\(975\) 0 0
\(976\) −718.630 −0.0235684
\(977\) 41274.9 1.35159 0.675794 0.737091i \(-0.263801\pi\)
0.675794 + 0.737091i \(0.263801\pi\)
\(978\) 0 0
\(979\) 381.897 0.0124673
\(980\) 64123.8 2.09016
\(981\) 0 0
\(982\) 52384.2 1.70229
\(983\) −32718.2 −1.06160 −0.530798 0.847498i \(-0.678108\pi\)
−0.530798 + 0.847498i \(0.678108\pi\)
\(984\) 0 0
\(985\) −49039.4 −1.58632
\(986\) −13659.9 −0.441197
\(987\) 0 0
\(988\) 490.555 0.0157962
\(989\) −6504.97 −0.209147
\(990\) 0 0
\(991\) −11924.6 −0.382237 −0.191118 0.981567i \(-0.561211\pi\)
−0.191118 + 0.981567i \(0.561211\pi\)
\(992\) −6021.37 −0.192721
\(993\) 0 0
\(994\) −86622.6 −2.76409
\(995\) −10037.6 −0.319811
\(996\) 0 0
\(997\) 33617.3 1.06787 0.533936 0.845525i \(-0.320712\pi\)
0.533936 + 0.845525i \(0.320712\pi\)
\(998\) −16178.1 −0.513135
\(999\) 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 2151.4.a.a.1.2 22
3.2 odd 2 239.4.a.a.1.21 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
239.4.a.a.1.21 22 3.2 odd 2
2151.4.a.a.1.2 22 1.1 even 1 trivial