Properties

Label 304.4.a.e.1.1
Level $304$
Weight $4$
Character 304.1
Self dual yes
Analytic conductor $17.937$
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] = [304,4,Mod(1,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.9365806417\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 304.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.27492 q^{3} -6.27492 q^{5} +18.0997 q^{7} -16.2749 q^{9} +O(q^{10})\) \(q-3.27492 q^{3} -6.27492 q^{5} +18.0997 q^{7} -16.2749 q^{9} +33.9244 q^{11} -3.07558 q^{13} +20.5498 q^{15} -14.2508 q^{17} +19.0000 q^{19} -59.2749 q^{21} -114.072 q^{23} -85.6254 q^{25} +141.722 q^{27} -34.5257 q^{29} -107.698 q^{31} -111.100 q^{33} -113.574 q^{35} -181.698 q^{37} +10.0723 q^{39} -444.743 q^{41} -120.323 q^{43} +102.124 q^{45} -306.371 q^{47} -15.4020 q^{49} +46.6703 q^{51} +115.825 q^{53} -212.873 q^{55} -62.2234 q^{57} -161.680 q^{59} +274.571 q^{61} -294.571 q^{63} +19.2990 q^{65} -81.6254 q^{67} +373.577 q^{69} -773.492 q^{71} +148.794 q^{73} +280.416 q^{75} +614.021 q^{77} +557.341 q^{79} -24.7043 q^{81} -768.337 q^{83} +89.4228 q^{85} +113.069 q^{87} +457.884 q^{89} -55.6670 q^{91} +352.701 q^{93} -119.223 q^{95} -1500.82 q^{97} -552.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 5 q^{5} + 6 q^{7} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 5 q^{5} + 6 q^{7} - 25 q^{9} + 15 q^{11} - 59 q^{13} + 26 q^{15} - 104 q^{17} + 38 q^{19} - 111 q^{21} + 21 q^{23} - 209 q^{25} - 11 q^{27} - 137 q^{29} - 4 q^{31} - 192 q^{33} - 129 q^{35} - 152 q^{37} - 229 q^{39} - 210 q^{41} - 67 q^{43} + 91 q^{45} - 273 q^{47} - 212 q^{49} - 337 q^{51} + 209 q^{53} - 237 q^{55} + 19 q^{57} - 799 q^{59} + 149 q^{61} - 189 q^{63} - 52 q^{65} - 201 q^{67} + 951 q^{69} - 792 q^{71} - 246 q^{73} - 247 q^{75} + 843 q^{77} + 254 q^{79} - 442 q^{81} - 374 q^{83} - 25 q^{85} - 325 q^{87} - 564 q^{89} + 621 q^{91} + 796 q^{93} - 95 q^{95} - 178 q^{97} - 387 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.27492 −0.630258 −0.315129 0.949049i \(-0.602048\pi\)
−0.315129 + 0.949049i \(0.602048\pi\)
\(4\) 0 0
\(5\) −6.27492 −0.561246 −0.280623 0.959818i \(-0.590541\pi\)
−0.280623 + 0.959818i \(0.590541\pi\)
\(6\) 0 0
\(7\) 18.0997 0.977290 0.488645 0.872483i \(-0.337491\pi\)
0.488645 + 0.872483i \(0.337491\pi\)
\(8\) 0 0
\(9\) −16.2749 −0.602775
\(10\) 0 0
\(11\) 33.9244 0.929873 0.464936 0.885344i \(-0.346077\pi\)
0.464936 + 0.885344i \(0.346077\pi\)
\(12\) 0 0
\(13\) −3.07558 −0.0656163 −0.0328082 0.999462i \(-0.510445\pi\)
−0.0328082 + 0.999462i \(0.510445\pi\)
\(14\) 0 0
\(15\) 20.5498 0.353730
\(16\) 0 0
\(17\) −14.2508 −0.203314 −0.101657 0.994820i \(-0.532414\pi\)
−0.101657 + 0.994820i \(0.532414\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) −59.2749 −0.615945
\(22\) 0 0
\(23\) −114.072 −1.03416 −0.517081 0.855937i \(-0.672981\pi\)
−0.517081 + 0.855937i \(0.672981\pi\)
\(24\) 0 0
\(25\) −85.6254 −0.685003
\(26\) 0 0
\(27\) 141.722 1.01016
\(28\) 0 0
\(29\) −34.5257 −0.221078 −0.110539 0.993872i \(-0.535258\pi\)
−0.110539 + 0.993872i \(0.535258\pi\)
\(30\) 0 0
\(31\) −107.698 −0.623970 −0.311985 0.950087i \(-0.600994\pi\)
−0.311985 + 0.950087i \(0.600994\pi\)
\(32\) 0 0
\(33\) −111.100 −0.586060
\(34\) 0 0
\(35\) −113.574 −0.548500
\(36\) 0 0
\(37\) −181.698 −0.807322 −0.403661 0.914909i \(-0.632262\pi\)
−0.403661 + 0.914909i \(0.632262\pi\)
\(38\) 0 0
\(39\) 10.0723 0.0413552
\(40\) 0 0
\(41\) −444.743 −1.69408 −0.847038 0.531532i \(-0.821616\pi\)
−0.847038 + 0.531532i \(0.821616\pi\)
\(42\) 0 0
\(43\) −120.323 −0.426723 −0.213362 0.976973i \(-0.568441\pi\)
−0.213362 + 0.976973i \(0.568441\pi\)
\(44\) 0 0
\(45\) 102.124 0.338305
\(46\) 0 0
\(47\) −306.371 −0.950826 −0.475413 0.879763i \(-0.657701\pi\)
−0.475413 + 0.879763i \(0.657701\pi\)
\(48\) 0 0
\(49\) −15.4020 −0.0449038
\(50\) 0 0
\(51\) 46.6703 0.128140
\(52\) 0 0
\(53\) 115.825 0.300184 0.150092 0.988672i \(-0.452043\pi\)
0.150092 + 0.988672i \(0.452043\pi\)
\(54\) 0 0
\(55\) −212.873 −0.521887
\(56\) 0 0
\(57\) −62.2234 −0.144591
\(58\) 0 0
\(59\) −161.680 −0.356762 −0.178381 0.983961i \(-0.557086\pi\)
−0.178381 + 0.983961i \(0.557086\pi\)
\(60\) 0 0
\(61\) 274.571 0.576314 0.288157 0.957583i \(-0.406957\pi\)
0.288157 + 0.957583i \(0.406957\pi\)
\(62\) 0 0
\(63\) −294.571 −0.589086
\(64\) 0 0
\(65\) 19.2990 0.0368269
\(66\) 0 0
\(67\) −81.6254 −0.148838 −0.0744189 0.997227i \(-0.523710\pi\)
−0.0744189 + 0.997227i \(0.523710\pi\)
\(68\) 0 0
\(69\) 373.577 0.651789
\(70\) 0 0
\(71\) −773.492 −1.29291 −0.646455 0.762952i \(-0.723749\pi\)
−0.646455 + 0.762952i \(0.723749\pi\)
\(72\) 0 0
\(73\) 148.794 0.238562 0.119281 0.992861i \(-0.461941\pi\)
0.119281 + 0.992861i \(0.461941\pi\)
\(74\) 0 0
\(75\) 280.416 0.431729
\(76\) 0 0
\(77\) 614.021 0.908755
\(78\) 0 0
\(79\) 557.341 0.793743 0.396872 0.917874i \(-0.370096\pi\)
0.396872 + 0.917874i \(0.370096\pi\)
\(80\) 0 0
\(81\) −24.7043 −0.0338879
\(82\) 0 0
\(83\) −768.337 −1.01610 −0.508048 0.861329i \(-0.669633\pi\)
−0.508048 + 0.861329i \(0.669633\pi\)
\(84\) 0 0
\(85\) 89.4228 0.114109
\(86\) 0 0
\(87\) 113.069 0.139336
\(88\) 0 0
\(89\) 457.884 0.545344 0.272672 0.962107i \(-0.412093\pi\)
0.272672 + 0.962107i \(0.412093\pi\)
\(90\) 0 0
\(91\) −55.6670 −0.0641262
\(92\) 0 0
\(93\) 352.701 0.393262
\(94\) 0 0
\(95\) −119.223 −0.128759
\(96\) 0 0
\(97\) −1500.82 −1.57098 −0.785490 0.618874i \(-0.787589\pi\)
−0.785490 + 0.618874i \(0.787589\pi\)
\(98\) 0 0
\(99\) −552.117 −0.560504
\(100\) 0 0
\(101\) −1731.06 −1.70541 −0.852707 0.522389i \(-0.825041\pi\)
−0.852707 + 0.522389i \(0.825041\pi\)
\(102\) 0 0
\(103\) 1887.77 1.80590 0.902951 0.429743i \(-0.141396\pi\)
0.902951 + 0.429743i \(0.141396\pi\)
\(104\) 0 0
\(105\) 371.945 0.345697
\(106\) 0 0
\(107\) 411.468 0.371758 0.185879 0.982573i \(-0.440487\pi\)
0.185879 + 0.982573i \(0.440487\pi\)
\(108\) 0 0
\(109\) 1186.86 1.04294 0.521469 0.853270i \(-0.325384\pi\)
0.521469 + 0.853270i \(0.325384\pi\)
\(110\) 0 0
\(111\) 595.045 0.508821
\(112\) 0 0
\(113\) 509.478 0.424139 0.212069 0.977255i \(-0.431980\pi\)
0.212069 + 0.977255i \(0.431980\pi\)
\(114\) 0 0
\(115\) 715.794 0.580419
\(116\) 0 0
\(117\) 50.0548 0.0395519
\(118\) 0 0
\(119\) −257.935 −0.198697
\(120\) 0 0
\(121\) −180.134 −0.135337
\(122\) 0 0
\(123\) 1456.50 1.06771
\(124\) 0 0
\(125\) 1321.66 0.945701
\(126\) 0 0
\(127\) 1169.71 0.817284 0.408642 0.912695i \(-0.366002\pi\)
0.408642 + 0.912695i \(0.366002\pi\)
\(128\) 0 0
\(129\) 394.048 0.268946
\(130\) 0 0
\(131\) 1052.31 0.701838 0.350919 0.936406i \(-0.385869\pi\)
0.350919 + 0.936406i \(0.385869\pi\)
\(132\) 0 0
\(133\) 343.894 0.224206
\(134\) 0 0
\(135\) −889.292 −0.566949
\(136\) 0 0
\(137\) −2116.92 −1.32015 −0.660075 0.751200i \(-0.729476\pi\)
−0.660075 + 0.751200i \(0.729476\pi\)
\(138\) 0 0
\(139\) 1115.90 0.680929 0.340464 0.940257i \(-0.389416\pi\)
0.340464 + 0.940257i \(0.389416\pi\)
\(140\) 0 0
\(141\) 1003.34 0.599266
\(142\) 0 0
\(143\) −104.337 −0.0610148
\(144\) 0 0
\(145\) 216.646 0.124079
\(146\) 0 0
\(147\) 50.4402 0.0283010
\(148\) 0 0
\(149\) 472.693 0.259896 0.129948 0.991521i \(-0.458519\pi\)
0.129948 + 0.991521i \(0.458519\pi\)
\(150\) 0 0
\(151\) 1635.40 0.881372 0.440686 0.897661i \(-0.354735\pi\)
0.440686 + 0.897661i \(0.354735\pi\)
\(152\) 0 0
\(153\) 231.931 0.122552
\(154\) 0 0
\(155\) 675.794 0.350201
\(156\) 0 0
\(157\) −785.904 −0.399503 −0.199751 0.979847i \(-0.564013\pi\)
−0.199751 + 0.979847i \(0.564013\pi\)
\(158\) 0 0
\(159\) −379.316 −0.189193
\(160\) 0 0
\(161\) −2064.67 −1.01068
\(162\) 0 0
\(163\) −1224.95 −0.588623 −0.294311 0.955710i \(-0.595090\pi\)
−0.294311 + 0.955710i \(0.595090\pi\)
\(164\) 0 0
\(165\) 697.141 0.328923
\(166\) 0 0
\(167\) −1071.26 −0.496386 −0.248193 0.968711i \(-0.579837\pi\)
−0.248193 + 0.968711i \(0.579837\pi\)
\(168\) 0 0
\(169\) −2187.54 −0.995694
\(170\) 0 0
\(171\) −309.223 −0.138286
\(172\) 0 0
\(173\) 825.821 0.362925 0.181462 0.983398i \(-0.441917\pi\)
0.181462 + 0.983398i \(0.441917\pi\)
\(174\) 0 0
\(175\) −1549.79 −0.669447
\(176\) 0 0
\(177\) 529.489 0.224852
\(178\) 0 0
\(179\) 465.802 0.194501 0.0972506 0.995260i \(-0.468995\pi\)
0.0972506 + 0.995260i \(0.468995\pi\)
\(180\) 0 0
\(181\) −1214.74 −0.498847 −0.249423 0.968395i \(-0.580241\pi\)
−0.249423 + 0.968395i \(0.580241\pi\)
\(182\) 0 0
\(183\) −899.196 −0.363227
\(184\) 0 0
\(185\) 1140.14 0.453106
\(186\) 0 0
\(187\) −483.451 −0.189056
\(188\) 0 0
\(189\) 2565.12 0.987221
\(190\) 0 0
\(191\) −154.943 −0.0586980 −0.0293490 0.999569i \(-0.509343\pi\)
−0.0293490 + 0.999569i \(0.509343\pi\)
\(192\) 0 0
\(193\) −2776.64 −1.03558 −0.517790 0.855508i \(-0.673245\pi\)
−0.517790 + 0.855508i \(0.673245\pi\)
\(194\) 0 0
\(195\) −63.2026 −0.0232104
\(196\) 0 0
\(197\) 4865.09 1.75951 0.879754 0.475429i \(-0.157707\pi\)
0.879754 + 0.475429i \(0.157707\pi\)
\(198\) 0 0
\(199\) 3014.43 1.07381 0.536903 0.843644i \(-0.319594\pi\)
0.536903 + 0.843644i \(0.319594\pi\)
\(200\) 0 0
\(201\) 267.316 0.0938062
\(202\) 0 0
\(203\) −624.905 −0.216058
\(204\) 0 0
\(205\) 2790.72 0.950793
\(206\) 0 0
\(207\) 1856.52 0.623366
\(208\) 0 0
\(209\) 644.564 0.213327
\(210\) 0 0
\(211\) −3434.46 −1.12056 −0.560280 0.828303i \(-0.689307\pi\)
−0.560280 + 0.828303i \(0.689307\pi\)
\(212\) 0 0
\(213\) 2533.12 0.814867
\(214\) 0 0
\(215\) 755.017 0.239497
\(216\) 0 0
\(217\) −1949.29 −0.609800
\(218\) 0 0
\(219\) −487.288 −0.150356
\(220\) 0 0
\(221\) 43.8296 0.0133407
\(222\) 0 0
\(223\) 2247.73 0.674975 0.337488 0.941330i \(-0.390423\pi\)
0.337488 + 0.941330i \(0.390423\pi\)
\(224\) 0 0
\(225\) 1393.55 0.412903
\(226\) 0 0
\(227\) −1883.81 −0.550806 −0.275403 0.961329i \(-0.588811\pi\)
−0.275403 + 0.961329i \(0.588811\pi\)
\(228\) 0 0
\(229\) 3594.49 1.03725 0.518626 0.855001i \(-0.326444\pi\)
0.518626 + 0.855001i \(0.326444\pi\)
\(230\) 0 0
\(231\) −2010.87 −0.572750
\(232\) 0 0
\(233\) −5530.01 −1.55486 −0.777431 0.628968i \(-0.783478\pi\)
−0.777431 + 0.628968i \(0.783478\pi\)
\(234\) 0 0
\(235\) 1922.45 0.533647
\(236\) 0 0
\(237\) −1825.24 −0.500263
\(238\) 0 0
\(239\) −2587.82 −0.700387 −0.350193 0.936677i \(-0.613884\pi\)
−0.350193 + 0.936677i \(0.613884\pi\)
\(240\) 0 0
\(241\) −3576.45 −0.955932 −0.477966 0.878378i \(-0.658626\pi\)
−0.477966 + 0.878378i \(0.658626\pi\)
\(242\) 0 0
\(243\) −3745.58 −0.988804
\(244\) 0 0
\(245\) 96.6462 0.0252020
\(246\) 0 0
\(247\) −58.4360 −0.0150534
\(248\) 0 0
\(249\) 2516.24 0.640403
\(250\) 0 0
\(251\) −5966.87 −1.50050 −0.750251 0.661154i \(-0.770067\pi\)
−0.750251 + 0.661154i \(0.770067\pi\)
\(252\) 0 0
\(253\) −3869.84 −0.961638
\(254\) 0 0
\(255\) −292.852 −0.0719181
\(256\) 0 0
\(257\) 3841.01 0.932278 0.466139 0.884712i \(-0.345645\pi\)
0.466139 + 0.884712i \(0.345645\pi\)
\(258\) 0 0
\(259\) −3288.67 −0.788988
\(260\) 0 0
\(261\) 561.904 0.133260
\(262\) 0 0
\(263\) −3956.41 −0.927614 −0.463807 0.885936i \(-0.653517\pi\)
−0.463807 + 0.885936i \(0.653517\pi\)
\(264\) 0 0
\(265\) −726.791 −0.168477
\(266\) 0 0
\(267\) −1499.53 −0.343707
\(268\) 0 0
\(269\) 2158.66 0.489279 0.244639 0.969614i \(-0.421330\pi\)
0.244639 + 0.969614i \(0.421330\pi\)
\(270\) 0 0
\(271\) 2321.77 0.520435 0.260217 0.965550i \(-0.416206\pi\)
0.260217 + 0.965550i \(0.416206\pi\)
\(272\) 0 0
\(273\) 182.305 0.0404161
\(274\) 0 0
\(275\) −2904.79 −0.636966
\(276\) 0 0
\(277\) −6184.24 −1.34143 −0.670713 0.741717i \(-0.734012\pi\)
−0.670713 + 0.741717i \(0.734012\pi\)
\(278\) 0 0
\(279\) 1752.77 0.376113
\(280\) 0 0
\(281\) −7981.21 −1.69437 −0.847187 0.531294i \(-0.821706\pi\)
−0.847187 + 0.531294i \(0.821706\pi\)
\(282\) 0 0
\(283\) 4851.71 1.01910 0.509548 0.860442i \(-0.329813\pi\)
0.509548 + 0.860442i \(0.329813\pi\)
\(284\) 0 0
\(285\) 390.447 0.0811511
\(286\) 0 0
\(287\) −8049.69 −1.65560
\(288\) 0 0
\(289\) −4709.91 −0.958664
\(290\) 0 0
\(291\) 4915.06 0.990123
\(292\) 0 0
\(293\) −1269.75 −0.253172 −0.126586 0.991956i \(-0.540402\pi\)
−0.126586 + 0.991956i \(0.540402\pi\)
\(294\) 0 0
\(295\) 1014.53 0.200231
\(296\) 0 0
\(297\) 4807.83 0.939322
\(298\) 0 0
\(299\) 350.838 0.0678579
\(300\) 0 0
\(301\) −2177.81 −0.417032
\(302\) 0 0
\(303\) 5669.08 1.07485
\(304\) 0 0
\(305\) −1722.91 −0.323454
\(306\) 0 0
\(307\) 9699.35 1.80316 0.901581 0.432610i \(-0.142407\pi\)
0.901581 + 0.432610i \(0.142407\pi\)
\(308\) 0 0
\(309\) −6182.30 −1.13818
\(310\) 0 0
\(311\) 323.284 0.0589446 0.0294723 0.999566i \(-0.490617\pi\)
0.0294723 + 0.999566i \(0.490617\pi\)
\(312\) 0 0
\(313\) −3251.29 −0.587137 −0.293569 0.955938i \(-0.594843\pi\)
−0.293569 + 0.955938i \(0.594843\pi\)
\(314\) 0 0
\(315\) 1848.41 0.330622
\(316\) 0 0
\(317\) −7512.25 −1.33101 −0.665505 0.746393i \(-0.731784\pi\)
−0.665505 + 0.746393i \(0.731784\pi\)
\(318\) 0 0
\(319\) −1171.27 −0.205575
\(320\) 0 0
\(321\) −1347.52 −0.234303
\(322\) 0 0
\(323\) −270.766 −0.0466434
\(324\) 0 0
\(325\) 263.348 0.0449474
\(326\) 0 0
\(327\) −3886.86 −0.657320
\(328\) 0 0
\(329\) −5545.22 −0.929233
\(330\) 0 0
\(331\) −2857.14 −0.474449 −0.237225 0.971455i \(-0.576238\pi\)
−0.237225 + 0.971455i \(0.576238\pi\)
\(332\) 0 0
\(333\) 2957.11 0.486633
\(334\) 0 0
\(335\) 512.193 0.0835346
\(336\) 0 0
\(337\) 10635.9 1.71921 0.859606 0.510958i \(-0.170709\pi\)
0.859606 + 0.510958i \(0.170709\pi\)
\(338\) 0 0
\(339\) −1668.50 −0.267317
\(340\) 0 0
\(341\) −3653.58 −0.580213
\(342\) 0 0
\(343\) −6486.96 −1.02117
\(344\) 0 0
\(345\) −2344.17 −0.365814
\(346\) 0 0
\(347\) 8357.08 1.29289 0.646443 0.762962i \(-0.276256\pi\)
0.646443 + 0.762962i \(0.276256\pi\)
\(348\) 0 0
\(349\) 7741.28 1.18734 0.593669 0.804709i \(-0.297679\pi\)
0.593669 + 0.804709i \(0.297679\pi\)
\(350\) 0 0
\(351\) −435.877 −0.0662831
\(352\) 0 0
\(353\) 2480.46 0.373999 0.186999 0.982360i \(-0.440124\pi\)
0.186999 + 0.982360i \(0.440124\pi\)
\(354\) 0 0
\(355\) 4853.60 0.725640
\(356\) 0 0
\(357\) 844.717 0.125230
\(358\) 0 0
\(359\) −4310.17 −0.633655 −0.316827 0.948483i \(-0.602618\pi\)
−0.316827 + 0.948483i \(0.602618\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 589.923 0.0852973
\(364\) 0 0
\(365\) −933.670 −0.133892
\(366\) 0 0
\(367\) 8484.71 1.20681 0.603404 0.797436i \(-0.293811\pi\)
0.603404 + 0.797436i \(0.293811\pi\)
\(368\) 0 0
\(369\) 7238.15 1.02115
\(370\) 0 0
\(371\) 2096.39 0.293367
\(372\) 0 0
\(373\) 8039.16 1.11596 0.557978 0.829855i \(-0.311577\pi\)
0.557978 + 0.829855i \(0.311577\pi\)
\(374\) 0 0
\(375\) −4328.32 −0.596036
\(376\) 0 0
\(377\) 106.187 0.0145063
\(378\) 0 0
\(379\) 2976.22 0.403372 0.201686 0.979450i \(-0.435358\pi\)
0.201686 + 0.979450i \(0.435358\pi\)
\(380\) 0 0
\(381\) −3830.71 −0.515100
\(382\) 0 0
\(383\) 5118.15 0.682834 0.341417 0.939912i \(-0.389093\pi\)
0.341417 + 0.939912i \(0.389093\pi\)
\(384\) 0 0
\(385\) −3852.93 −0.510035
\(386\) 0 0
\(387\) 1958.25 0.257218
\(388\) 0 0
\(389\) −6758.74 −0.880929 −0.440465 0.897770i \(-0.645186\pi\)
−0.440465 + 0.897770i \(0.645186\pi\)
\(390\) 0 0
\(391\) 1625.62 0.210259
\(392\) 0 0
\(393\) −3446.23 −0.442339
\(394\) 0 0
\(395\) −3497.27 −0.445485
\(396\) 0 0
\(397\) 14529.0 1.83675 0.918373 0.395716i \(-0.129504\pi\)
0.918373 + 0.395716i \(0.129504\pi\)
\(398\) 0 0
\(399\) −1126.22 −0.141308
\(400\) 0 0
\(401\) 12416.4 1.54625 0.773124 0.634255i \(-0.218693\pi\)
0.773124 + 0.634255i \(0.218693\pi\)
\(402\) 0 0
\(403\) 331.233 0.0409426
\(404\) 0 0
\(405\) 155.017 0.0190195
\(406\) 0 0
\(407\) −6163.99 −0.750707
\(408\) 0 0
\(409\) 11615.8 1.40432 0.702159 0.712020i \(-0.252220\pi\)
0.702159 + 0.712020i \(0.252220\pi\)
\(410\) 0 0
\(411\) 6932.73 0.832035
\(412\) 0 0
\(413\) −2926.36 −0.348660
\(414\) 0 0
\(415\) 4821.25 0.570279
\(416\) 0 0
\(417\) −3654.47 −0.429161
\(418\) 0 0
\(419\) −16739.9 −1.95179 −0.975893 0.218248i \(-0.929966\pi\)
−0.975893 + 0.218248i \(0.929966\pi\)
\(420\) 0 0
\(421\) 6261.41 0.724852 0.362426 0.932013i \(-0.381949\pi\)
0.362426 + 0.932013i \(0.381949\pi\)
\(422\) 0 0
\(423\) 4986.17 0.573134
\(424\) 0 0
\(425\) 1220.23 0.139271
\(426\) 0 0
\(427\) 4969.64 0.563226
\(428\) 0 0
\(429\) 341.696 0.0384551
\(430\) 0 0
\(431\) 8893.01 0.993878 0.496939 0.867786i \(-0.334457\pi\)
0.496939 + 0.867786i \(0.334457\pi\)
\(432\) 0 0
\(433\) 6326.61 0.702166 0.351083 0.936344i \(-0.385814\pi\)
0.351083 + 0.936344i \(0.385814\pi\)
\(434\) 0 0
\(435\) −709.498 −0.0782019
\(436\) 0 0
\(437\) −2167.37 −0.237253
\(438\) 0 0
\(439\) −15823.4 −1.72030 −0.860149 0.510043i \(-0.829630\pi\)
−0.860149 + 0.510043i \(0.829630\pi\)
\(440\) 0 0
\(441\) 250.666 0.0270668
\(442\) 0 0
\(443\) 14056.3 1.50753 0.753763 0.657147i \(-0.228237\pi\)
0.753763 + 0.657147i \(0.228237\pi\)
\(444\) 0 0
\(445\) −2873.18 −0.306072
\(446\) 0 0
\(447\) −1548.03 −0.163802
\(448\) 0 0
\(449\) 15412.9 1.62000 0.809999 0.586431i \(-0.199468\pi\)
0.809999 + 0.586431i \(0.199468\pi\)
\(450\) 0 0
\(451\) −15087.6 −1.57527
\(452\) 0 0
\(453\) −5355.81 −0.555492
\(454\) 0 0
\(455\) 349.306 0.0359906
\(456\) 0 0
\(457\) 13336.9 1.36515 0.682573 0.730817i \(-0.260861\pi\)
0.682573 + 0.730817i \(0.260861\pi\)
\(458\) 0 0
\(459\) −2019.65 −0.205380
\(460\) 0 0
\(461\) −10226.9 −1.03322 −0.516612 0.856220i \(-0.672807\pi\)
−0.516612 + 0.856220i \(0.672807\pi\)
\(462\) 0 0
\(463\) 17797.4 1.78642 0.893212 0.449636i \(-0.148446\pi\)
0.893212 + 0.449636i \(0.148446\pi\)
\(464\) 0 0
\(465\) −2213.17 −0.220717
\(466\) 0 0
\(467\) 3100.03 0.307179 0.153589 0.988135i \(-0.450917\pi\)
0.153589 + 0.988135i \(0.450917\pi\)
\(468\) 0 0
\(469\) −1477.39 −0.145458
\(470\) 0 0
\(471\) 2573.77 0.251790
\(472\) 0 0
\(473\) −4081.89 −0.396798
\(474\) 0 0
\(475\) −1626.88 −0.157151
\(476\) 0 0
\(477\) −1885.04 −0.180943
\(478\) 0 0
\(479\) −6280.67 −0.599105 −0.299552 0.954080i \(-0.596837\pi\)
−0.299552 + 0.954080i \(0.596837\pi\)
\(480\) 0 0
\(481\) 558.826 0.0529735
\(482\) 0 0
\(483\) 6761.62 0.636987
\(484\) 0 0
\(485\) 9417.52 0.881706
\(486\) 0 0
\(487\) −7940.07 −0.738807 −0.369403 0.929269i \(-0.620438\pi\)
−0.369403 + 0.929269i \(0.620438\pi\)
\(488\) 0 0
\(489\) 4011.61 0.370984
\(490\) 0 0
\(491\) −17276.3 −1.58792 −0.793959 0.607971i \(-0.791984\pi\)
−0.793959 + 0.607971i \(0.791984\pi\)
\(492\) 0 0
\(493\) 492.020 0.0449482
\(494\) 0 0
\(495\) 3464.49 0.314580
\(496\) 0 0
\(497\) −13999.9 −1.26355
\(498\) 0 0
\(499\) −14939.6 −1.34026 −0.670128 0.742246i \(-0.733761\pi\)
−0.670128 + 0.742246i \(0.733761\pi\)
\(500\) 0 0
\(501\) 3508.28 0.312851
\(502\) 0 0
\(503\) 5559.39 0.492805 0.246402 0.969168i \(-0.420752\pi\)
0.246402 + 0.969168i \(0.420752\pi\)
\(504\) 0 0
\(505\) 10862.3 0.957157
\(506\) 0 0
\(507\) 7164.02 0.627545
\(508\) 0 0
\(509\) 11029.4 0.960452 0.480226 0.877145i \(-0.340555\pi\)
0.480226 + 0.877145i \(0.340555\pi\)
\(510\) 0 0
\(511\) 2693.12 0.233144
\(512\) 0 0
\(513\) 2692.71 0.231747
\(514\) 0 0
\(515\) −11845.6 −1.01355
\(516\) 0 0
\(517\) −10393.5 −0.884147
\(518\) 0 0
\(519\) −2704.49 −0.228736
\(520\) 0 0
\(521\) −7802.02 −0.656070 −0.328035 0.944665i \(-0.606386\pi\)
−0.328035 + 0.944665i \(0.606386\pi\)
\(522\) 0 0
\(523\) 23625.2 1.97525 0.987627 0.156820i \(-0.0501241\pi\)
0.987627 + 0.156820i \(0.0501241\pi\)
\(524\) 0 0
\(525\) 5075.44 0.421924
\(526\) 0 0
\(527\) 1534.78 0.126862
\(528\) 0 0
\(529\) 845.482 0.0694898
\(530\) 0 0
\(531\) 2631.33 0.215047
\(532\) 0 0
\(533\) 1367.84 0.111159
\(534\) 0 0
\(535\) −2581.93 −0.208647
\(536\) 0 0
\(537\) −1525.46 −0.122586
\(538\) 0 0
\(539\) −522.503 −0.0417548
\(540\) 0 0
\(541\) −24753.3 −1.96715 −0.983573 0.180509i \(-0.942226\pi\)
−0.983573 + 0.180509i \(0.942226\pi\)
\(542\) 0 0
\(543\) 3978.19 0.314402
\(544\) 0 0
\(545\) −7447.43 −0.585344
\(546\) 0 0
\(547\) −5655.38 −0.442060 −0.221030 0.975267i \(-0.570942\pi\)
−0.221030 + 0.975267i \(0.570942\pi\)
\(548\) 0 0
\(549\) −4468.61 −0.347388
\(550\) 0 0
\(551\) −655.989 −0.0507188
\(552\) 0 0
\(553\) 10087.7 0.775717
\(554\) 0 0
\(555\) −3733.86 −0.285574
\(556\) 0 0
\(557\) −14716.3 −1.11948 −0.559741 0.828668i \(-0.689099\pi\)
−0.559741 + 0.828668i \(0.689099\pi\)
\(558\) 0 0
\(559\) 370.063 0.0280000
\(560\) 0 0
\(561\) 1583.26 0.119154
\(562\) 0 0
\(563\) 4400.91 0.329443 0.164721 0.986340i \(-0.447328\pi\)
0.164721 + 0.986340i \(0.447328\pi\)
\(564\) 0 0
\(565\) −3196.94 −0.238046
\(566\) 0 0
\(567\) −447.140 −0.0331183
\(568\) 0 0
\(569\) 10176.3 0.749758 0.374879 0.927074i \(-0.377684\pi\)
0.374879 + 0.927074i \(0.377684\pi\)
\(570\) 0 0
\(571\) −6775.74 −0.496595 −0.248298 0.968684i \(-0.579871\pi\)
−0.248298 + 0.968684i \(0.579871\pi\)
\(572\) 0 0
\(573\) 507.427 0.0369949
\(574\) 0 0
\(575\) 9767.49 0.708404
\(576\) 0 0
\(577\) −3714.70 −0.268016 −0.134008 0.990980i \(-0.542785\pi\)
−0.134008 + 0.990980i \(0.542785\pi\)
\(578\) 0 0
\(579\) 9093.26 0.652683
\(580\) 0 0
\(581\) −13906.6 −0.993021
\(582\) 0 0
\(583\) 3929.29 0.279133
\(584\) 0 0
\(585\) −314.090 −0.0221983
\(586\) 0 0
\(587\) 9362.24 0.658298 0.329149 0.944278i \(-0.393238\pi\)
0.329149 + 0.944278i \(0.393238\pi\)
\(588\) 0 0
\(589\) −2046.26 −0.143149
\(590\) 0 0
\(591\) −15932.8 −1.10894
\(592\) 0 0
\(593\) −14437.5 −0.999793 −0.499897 0.866085i \(-0.666629\pi\)
−0.499897 + 0.866085i \(0.666629\pi\)
\(594\) 0 0
\(595\) 1618.52 0.111518
\(596\) 0 0
\(597\) −9872.02 −0.676775
\(598\) 0 0
\(599\) 19992.7 1.36374 0.681871 0.731473i \(-0.261167\pi\)
0.681871 + 0.731473i \(0.261167\pi\)
\(600\) 0 0
\(601\) 17371.4 1.17902 0.589511 0.807760i \(-0.299320\pi\)
0.589511 + 0.807760i \(0.299320\pi\)
\(602\) 0 0
\(603\) 1328.45 0.0897157
\(604\) 0 0
\(605\) 1130.32 0.0759574
\(606\) 0 0
\(607\) 14626.3 0.978025 0.489013 0.872277i \(-0.337357\pi\)
0.489013 + 0.872277i \(0.337357\pi\)
\(608\) 0 0
\(609\) 2046.51 0.136172
\(610\) 0 0
\(611\) 942.269 0.0623897
\(612\) 0 0
\(613\) −21052.2 −1.38710 −0.693549 0.720409i \(-0.743954\pi\)
−0.693549 + 0.720409i \(0.743954\pi\)
\(614\) 0 0
\(615\) −9139.39 −0.599245
\(616\) 0 0
\(617\) 1605.72 0.104771 0.0523855 0.998627i \(-0.483318\pi\)
0.0523855 + 0.998627i \(0.483318\pi\)
\(618\) 0 0
\(619\) −21457.6 −1.39330 −0.696652 0.717410i \(-0.745328\pi\)
−0.696652 + 0.717410i \(0.745328\pi\)
\(620\) 0 0
\(621\) −16166.5 −1.04467
\(622\) 0 0
\(623\) 8287.54 0.532959
\(624\) 0 0
\(625\) 2409.89 0.154233
\(626\) 0 0
\(627\) −2110.89 −0.134451
\(628\) 0 0
\(629\) 2589.34 0.164140
\(630\) 0 0
\(631\) −9732.44 −0.614013 −0.307007 0.951707i \(-0.599327\pi\)
−0.307007 + 0.951707i \(0.599327\pi\)
\(632\) 0 0
\(633\) 11247.6 0.706242
\(634\) 0 0
\(635\) −7339.84 −0.458697
\(636\) 0 0
\(637\) 47.3700 0.00294642
\(638\) 0 0
\(639\) 12588.5 0.779333
\(640\) 0 0
\(641\) −25594.0 −1.57707 −0.788536 0.614989i \(-0.789161\pi\)
−0.788536 + 0.614989i \(0.789161\pi\)
\(642\) 0 0
\(643\) 11938.5 0.732207 0.366104 0.930574i \(-0.380692\pi\)
0.366104 + 0.930574i \(0.380692\pi\)
\(644\) 0 0
\(645\) −2472.62 −0.150945
\(646\) 0 0
\(647\) −15685.4 −0.953105 −0.476552 0.879146i \(-0.658114\pi\)
−0.476552 + 0.879146i \(0.658114\pi\)
\(648\) 0 0
\(649\) −5484.91 −0.331743
\(650\) 0 0
\(651\) 6383.77 0.384331
\(652\) 0 0
\(653\) 11414.3 0.684037 0.342019 0.939693i \(-0.388889\pi\)
0.342019 + 0.939693i \(0.388889\pi\)
\(654\) 0 0
\(655\) −6603.16 −0.393903
\(656\) 0 0
\(657\) −2421.61 −0.143799
\(658\) 0 0
\(659\) −24500.2 −1.44824 −0.724122 0.689672i \(-0.757755\pi\)
−0.724122 + 0.689672i \(0.757755\pi\)
\(660\) 0 0
\(661\) 24821.0 1.46055 0.730276 0.683152i \(-0.239391\pi\)
0.730276 + 0.683152i \(0.239391\pi\)
\(662\) 0 0
\(663\) −143.538 −0.00840808
\(664\) 0 0
\(665\) −2157.90 −0.125835
\(666\) 0 0
\(667\) 3938.43 0.228631
\(668\) 0 0
\(669\) −7361.14 −0.425408
\(670\) 0 0
\(671\) 9314.65 0.535899
\(672\) 0 0
\(673\) −14862.9 −0.851297 −0.425649 0.904888i \(-0.639954\pi\)
−0.425649 + 0.904888i \(0.639954\pi\)
\(674\) 0 0
\(675\) −12135.0 −0.691964
\(676\) 0 0
\(677\) −1218.79 −0.0691902 −0.0345951 0.999401i \(-0.511014\pi\)
−0.0345951 + 0.999401i \(0.511014\pi\)
\(678\) 0 0
\(679\) −27164.3 −1.53530
\(680\) 0 0
\(681\) 6169.33 0.347150
\(682\) 0 0
\(683\) 10180.0 0.570319 0.285159 0.958480i \(-0.407953\pi\)
0.285159 + 0.958480i \(0.407953\pi\)
\(684\) 0 0
\(685\) 13283.5 0.740929
\(686\) 0 0
\(687\) −11771.7 −0.653736
\(688\) 0 0
\(689\) −356.228 −0.0196970
\(690\) 0 0
\(691\) −13712.9 −0.754941 −0.377470 0.926022i \(-0.623206\pi\)
−0.377470 + 0.926022i \(0.623206\pi\)
\(692\) 0 0
\(693\) −9993.14 −0.547775
\(694\) 0 0
\(695\) −7002.16 −0.382168
\(696\) 0 0
\(697\) 6337.95 0.344429
\(698\) 0 0
\(699\) 18110.3 0.979964
\(700\) 0 0
\(701\) −29853.2 −1.60848 −0.804238 0.594307i \(-0.797426\pi\)
−0.804238 + 0.594307i \(0.797426\pi\)
\(702\) 0 0
\(703\) −3452.26 −0.185212
\(704\) 0 0
\(705\) −6295.88 −0.336335
\(706\) 0 0
\(707\) −31331.6 −1.66669
\(708\) 0 0
\(709\) −11488.4 −0.608541 −0.304271 0.952586i \(-0.598413\pi\)
−0.304271 + 0.952586i \(0.598413\pi\)
\(710\) 0 0
\(711\) −9070.67 −0.478448
\(712\) 0 0
\(713\) 12285.3 0.645286
\(714\) 0 0
\(715\) 654.708 0.0342443
\(716\) 0 0
\(717\) 8474.91 0.441424
\(718\) 0 0
\(719\) 17536.5 0.909596 0.454798 0.890595i \(-0.349711\pi\)
0.454798 + 0.890595i \(0.349711\pi\)
\(720\) 0 0
\(721\) 34168.1 1.76489
\(722\) 0 0
\(723\) 11712.6 0.602484
\(724\) 0 0
\(725\) 2956.28 0.151439
\(726\) 0 0
\(727\) −27607.0 −1.40837 −0.704185 0.710016i \(-0.748688\pi\)
−0.704185 + 0.710016i \(0.748688\pi\)
\(728\) 0 0
\(729\) 12933.5 0.657089
\(730\) 0 0
\(731\) 1714.70 0.0867587
\(732\) 0 0
\(733\) −3519.63 −0.177354 −0.0886770 0.996060i \(-0.528264\pi\)
−0.0886770 + 0.996060i \(0.528264\pi\)
\(734\) 0 0
\(735\) −316.508 −0.0158838
\(736\) 0 0
\(737\) −2769.09 −0.138400
\(738\) 0 0
\(739\) −4557.41 −0.226857 −0.113428 0.993546i \(-0.536183\pi\)
−0.113428 + 0.993546i \(0.536183\pi\)
\(740\) 0 0
\(741\) 191.373 0.00948754
\(742\) 0 0
\(743\) −5330.53 −0.263201 −0.131600 0.991303i \(-0.542012\pi\)
−0.131600 + 0.991303i \(0.542012\pi\)
\(744\) 0 0
\(745\) −2966.11 −0.145866
\(746\) 0 0
\(747\) 12504.6 0.612477
\(748\) 0 0
\(749\) 7447.43 0.363315
\(750\) 0 0
\(751\) −22386.8 −1.08776 −0.543879 0.839163i \(-0.683045\pi\)
−0.543879 + 0.839163i \(0.683045\pi\)
\(752\) 0 0
\(753\) 19541.0 0.945703
\(754\) 0 0
\(755\) −10262.0 −0.494666
\(756\) 0 0
\(757\) 25315.6 1.21547 0.607736 0.794139i \(-0.292078\pi\)
0.607736 + 0.794139i \(0.292078\pi\)
\(758\) 0 0
\(759\) 12673.4 0.606080
\(760\) 0 0
\(761\) −18715.0 −0.891485 −0.445742 0.895161i \(-0.647060\pi\)
−0.445742 + 0.895161i \(0.647060\pi\)
\(762\) 0 0
\(763\) 21481.7 1.01925
\(764\) 0 0
\(765\) −1455.35 −0.0687820
\(766\) 0 0
\(767\) 497.260 0.0234094
\(768\) 0 0
\(769\) 20985.3 0.984070 0.492035 0.870575i \(-0.336253\pi\)
0.492035 + 0.870575i \(0.336253\pi\)
\(770\) 0 0
\(771\) −12579.0 −0.587576
\(772\) 0 0
\(773\) 6302.27 0.293243 0.146622 0.989193i \(-0.453160\pi\)
0.146622 + 0.989193i \(0.453160\pi\)
\(774\) 0 0
\(775\) 9221.66 0.427422
\(776\) 0 0
\(777\) 10770.1 0.497266
\(778\) 0 0
\(779\) −8450.11 −0.388648
\(780\) 0 0
\(781\) −26240.3 −1.20224
\(782\) 0 0
\(783\) −4893.05 −0.223325
\(784\) 0 0
\(785\) 4931.48 0.224219
\(786\) 0 0
\(787\) −4124.73 −0.186825 −0.0934123 0.995628i \(-0.529777\pi\)
−0.0934123 + 0.995628i \(0.529777\pi\)
\(788\) 0 0
\(789\) 12956.9 0.584636
\(790\) 0 0
\(791\) 9221.39 0.414507
\(792\) 0 0
\(793\) −844.464 −0.0378156
\(794\) 0 0
\(795\) 2380.18 0.106184
\(796\) 0 0
\(797\) 8090.48 0.359573 0.179786 0.983706i \(-0.442459\pi\)
0.179786 + 0.983706i \(0.442459\pi\)
\(798\) 0 0
\(799\) 4366.04 0.193316
\(800\) 0 0
\(801\) −7452.02 −0.328719
\(802\) 0 0
\(803\) 5047.75 0.221832
\(804\) 0 0
\(805\) 12955.6 0.567237
\(806\) 0 0
\(807\) −7069.44 −0.308372
\(808\) 0 0
\(809\) −16735.9 −0.727323 −0.363662 0.931531i \(-0.618474\pi\)
−0.363662 + 0.931531i \(0.618474\pi\)
\(810\) 0 0
\(811\) −8476.59 −0.367020 −0.183510 0.983018i \(-0.558746\pi\)
−0.183510 + 0.983018i \(0.558746\pi\)
\(812\) 0 0
\(813\) −7603.62 −0.328008
\(814\) 0 0
\(815\) 7686.46 0.330362
\(816\) 0 0
\(817\) −2286.14 −0.0978970
\(818\) 0 0
\(819\) 905.975 0.0386537
\(820\) 0 0
\(821\) 902.399 0.0383605 0.0191802 0.999816i \(-0.493894\pi\)
0.0191802 + 0.999816i \(0.493894\pi\)
\(822\) 0 0
\(823\) 23266.2 0.985431 0.492716 0.870190i \(-0.336004\pi\)
0.492716 + 0.870190i \(0.336004\pi\)
\(824\) 0 0
\(825\) 9512.96 0.401453
\(826\) 0 0
\(827\) 17103.5 0.719160 0.359580 0.933114i \(-0.382920\pi\)
0.359580 + 0.933114i \(0.382920\pi\)
\(828\) 0 0
\(829\) −6125.30 −0.256623 −0.128312 0.991734i \(-0.540956\pi\)
−0.128312 + 0.991734i \(0.540956\pi\)
\(830\) 0 0
\(831\) 20252.9 0.845445
\(832\) 0 0
\(833\) 219.491 0.00912955
\(834\) 0 0
\(835\) 6722.05 0.278594
\(836\) 0 0
\(837\) −15263.1 −0.630311
\(838\) 0 0
\(839\) −19918.5 −0.819620 −0.409810 0.912171i \(-0.634405\pi\)
−0.409810 + 0.912171i \(0.634405\pi\)
\(840\) 0 0
\(841\) −23197.0 −0.951124
\(842\) 0 0
\(843\) 26137.8 1.06789
\(844\) 0 0
\(845\) 13726.6 0.558829
\(846\) 0 0
\(847\) −3260.36 −0.132264
\(848\) 0 0
\(849\) −15889.0 −0.642294
\(850\) 0 0
\(851\) 20726.7 0.834901
\(852\) 0 0
\(853\) 18785.3 0.754042 0.377021 0.926205i \(-0.376948\pi\)
0.377021 + 0.926205i \(0.376948\pi\)
\(854\) 0 0
\(855\) 1940.35 0.0776124
\(856\) 0 0
\(857\) −33483.4 −1.33462 −0.667311 0.744779i \(-0.732555\pi\)
−0.667311 + 0.744779i \(0.732555\pi\)
\(858\) 0 0
\(859\) −7322.74 −0.290860 −0.145430 0.989369i \(-0.546457\pi\)
−0.145430 + 0.989369i \(0.546457\pi\)
\(860\) 0 0
\(861\) 26362.1 1.04346
\(862\) 0 0
\(863\) −905.563 −0.0357193 −0.0178596 0.999841i \(-0.505685\pi\)
−0.0178596 + 0.999841i \(0.505685\pi\)
\(864\) 0 0
\(865\) −5181.96 −0.203690
\(866\) 0 0
\(867\) 15424.6 0.604205
\(868\) 0 0
\(869\) 18907.5 0.738080
\(870\) 0 0
\(871\) 251.045 0.00976619
\(872\) 0 0
\(873\) 24425.7 0.946947
\(874\) 0 0
\(875\) 23921.6 0.924224
\(876\) 0 0
\(877\) −27155.7 −1.04559 −0.522795 0.852459i \(-0.675111\pi\)
−0.522795 + 0.852459i \(0.675111\pi\)
\(878\) 0 0
\(879\) 4158.32 0.159564
\(880\) 0 0
\(881\) 26094.1 0.997881 0.498941 0.866636i \(-0.333723\pi\)
0.498941 + 0.866636i \(0.333723\pi\)
\(882\) 0 0
\(883\) −29674.6 −1.13095 −0.565476 0.824765i \(-0.691307\pi\)
−0.565476 + 0.824765i \(0.691307\pi\)
\(884\) 0 0
\(885\) −3322.50 −0.126197
\(886\) 0 0
\(887\) 1734.72 0.0656665 0.0328333 0.999461i \(-0.489547\pi\)
0.0328333 + 0.999461i \(0.489547\pi\)
\(888\) 0 0
\(889\) 21171.4 0.798724
\(890\) 0 0
\(891\) −838.079 −0.0315115
\(892\) 0 0
\(893\) −5821.05 −0.218135
\(894\) 0 0
\(895\) −2922.87 −0.109163
\(896\) 0 0
\(897\) −1148.97 −0.0427680
\(898\) 0 0
\(899\) 3718.34 0.137946
\(900\) 0 0
\(901\) −1650.60 −0.0610315
\(902\) 0 0
\(903\) 7132.14 0.262838
\(904\) 0 0
\(905\) 7622.42 0.279975
\(906\) 0 0
\(907\) −6755.01 −0.247295 −0.123647 0.992326i \(-0.539459\pi\)
−0.123647 + 0.992326i \(0.539459\pi\)
\(908\) 0 0
\(909\) 28172.9 1.02798
\(910\) 0 0
\(911\) 35150.7 1.27837 0.639184 0.769054i \(-0.279272\pi\)
0.639184 + 0.769054i \(0.279272\pi\)
\(912\) 0 0
\(913\) −26065.4 −0.944840
\(914\) 0 0
\(915\) 5642.38 0.203859
\(916\) 0 0
\(917\) 19046.5 0.685899
\(918\) 0 0
\(919\) 28684.2 1.02960 0.514802 0.857309i \(-0.327866\pi\)
0.514802 + 0.857309i \(0.327866\pi\)
\(920\) 0 0
\(921\) −31764.6 −1.13646
\(922\) 0 0
\(923\) 2378.94 0.0848360
\(924\) 0 0
\(925\) 15557.9 0.553018
\(926\) 0 0
\(927\) −30723.4 −1.08855
\(928\) 0 0
\(929\) −3762.58 −0.132881 −0.0664405 0.997790i \(-0.521164\pi\)
−0.0664405 + 0.997790i \(0.521164\pi\)
\(930\) 0 0
\(931\) −292.638 −0.0103016
\(932\) 0 0
\(933\) −1058.73 −0.0371503
\(934\) 0 0
\(935\) 3033.62 0.106107
\(936\) 0 0
\(937\) 25465.7 0.887863 0.443932 0.896061i \(-0.353583\pi\)
0.443932 + 0.896061i \(0.353583\pi\)
\(938\) 0 0
\(939\) 10647.7 0.370048
\(940\) 0 0
\(941\) 40865.6 1.41571 0.707854 0.706359i \(-0.249663\pi\)
0.707854 + 0.706359i \(0.249663\pi\)
\(942\) 0 0
\(943\) 50732.8 1.75195
\(944\) 0 0
\(945\) −16095.9 −0.554074
\(946\) 0 0
\(947\) 8047.62 0.276148 0.138074 0.990422i \(-0.455909\pi\)
0.138074 + 0.990422i \(0.455909\pi\)
\(948\) 0 0
\(949\) −457.628 −0.0156536
\(950\) 0 0
\(951\) 24602.0 0.838880
\(952\) 0 0
\(953\) −21562.2 −0.732915 −0.366458 0.930435i \(-0.619430\pi\)
−0.366458 + 0.930435i \(0.619430\pi\)
\(954\) 0 0
\(955\) 972.257 0.0329440
\(956\) 0 0
\(957\) 3835.80 0.129565
\(958\) 0 0
\(959\) −38315.5 −1.29017
\(960\) 0 0
\(961\) −18192.2 −0.610661
\(962\) 0 0
\(963\) −6696.60 −0.224086
\(964\) 0 0
\(965\) 17423.2 0.581215
\(966\) 0 0
\(967\) 14667.4 0.487767 0.243883 0.969805i \(-0.421579\pi\)
0.243883 + 0.969805i \(0.421579\pi\)
\(968\) 0 0
\(969\) 886.735 0.0293974
\(970\) 0 0
\(971\) 60154.7 1.98811 0.994056 0.108867i \(-0.0347221\pi\)
0.994056 + 0.108867i \(0.0347221\pi\)
\(972\) 0 0
\(973\) 20197.4 0.665465
\(974\) 0 0
\(975\) −862.442 −0.0283285
\(976\) 0 0
\(977\) −29474.6 −0.965176 −0.482588 0.875848i \(-0.660303\pi\)
−0.482588 + 0.875848i \(0.660303\pi\)
\(978\) 0 0
\(979\) 15533.4 0.507100
\(980\) 0 0
\(981\) −19316.0 −0.628657
\(982\) 0 0
\(983\) −6813.42 −0.221073 −0.110536 0.993872i \(-0.535257\pi\)
−0.110536 + 0.993872i \(0.535257\pi\)
\(984\) 0 0
\(985\) −30528.0 −0.987517
\(986\) 0 0
\(987\) 18160.1 0.585657
\(988\) 0 0
\(989\) 13725.5 0.441301
\(990\) 0 0
\(991\) 11660.9 0.373786 0.186893 0.982380i \(-0.440158\pi\)
0.186893 + 0.982380i \(0.440158\pi\)
\(992\) 0 0
\(993\) 9356.91 0.299026
\(994\) 0 0
\(995\) −18915.3 −0.602669
\(996\) 0 0
\(997\) −37537.3 −1.19240 −0.596198 0.802838i \(-0.703323\pi\)
−0.596198 + 0.802838i \(0.703323\pi\)
\(998\) 0 0
\(999\) −25750.5 −0.815526
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 304.4.a.e.1.1 2
4.3 odd 2 152.4.a.a.1.2 2
8.3 odd 2 1216.4.a.m.1.1 2
8.5 even 2 1216.4.a.k.1.2 2
12.11 even 2 1368.4.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.a.1.2 2 4.3 odd 2
304.4.a.e.1.1 2 1.1 even 1 trivial
1216.4.a.k.1.2 2 8.5 even 2
1216.4.a.m.1.1 2 8.3 odd 2
1368.4.a.a.1.2 2 12.11 even 2