Properties

Label 3344.2.a.s.1.3
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.13676.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 7x + 1 \) 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 1672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.42957\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.42957 q^{3} -3.22628 q^{5} -0.429567 q^{7} -0.956338 q^{9} +O(q^{10})\) \(q+1.42957 q^{3} -3.22628 q^{5} -0.429567 q^{7} -0.956338 q^{9} -1.00000 q^{11} +4.88213 q^{13} -4.61219 q^{15} +3.70939 q^{17} -1.00000 q^{19} -0.614095 q^{21} +3.32349 q^{23} +5.40890 q^{25} -5.65585 q^{27} -2.75305 q^{29} -0.699512 q^{31} -1.42957 q^{33} +1.38591 q^{35} -1.61409 q^{37} +6.97934 q^{39} -8.99809 q^{41} -7.40890 q^{43} +3.08542 q^{45} +10.9051 q^{47} -6.81547 q^{49} +5.30283 q^{51} +0.452567 q^{53} +3.22628 q^{55} -1.42957 q^{57} -2.38591 q^{59} -7.62207 q^{61} +0.410811 q^{63} -15.7511 q^{65} -7.47132 q^{67} +4.75115 q^{69} +1.32248 q^{71} -13.7643 q^{73} +7.73239 q^{75} +0.429567 q^{77} +2.11596 q^{79} -5.21640 q^{81} +8.64597 q^{83} -11.9676 q^{85} -3.93568 q^{87} +3.89201 q^{89} -2.09720 q^{91} -1.00000 q^{93} +3.22628 q^{95} -13.9793 q^{97} +0.956338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41} - 5 q^{43} - 7 q^{45} + 4 q^{47} - 13 q^{49} - 4 q^{51} - 18 q^{53} + 3 q^{55} - q^{57} - 6 q^{61} + 6 q^{63} - 24 q^{65} - q^{67} - 20 q^{69} + q^{71} - 6 q^{73} + q^{75} - 3 q^{77} + 4 q^{79} - 16 q^{81} + 25 q^{83} - 4 q^{85} + 9 q^{87} - 14 q^{89} - 13 q^{91} - 4 q^{93} + 3 q^{95} - 36 q^{97} - q^{99}+O(q^{100}) \) 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
\(3\) 1.42957 0.825361 0.412680 0.910876i \(-0.364593\pi\)
0.412680 + 0.910876i \(0.364593\pi\)
\(4\) 0 0
\(5\) −3.22628 −1.44284 −0.721419 0.692499i \(-0.756510\pi\)
−0.721419 + 0.692499i \(0.756510\pi\)
\(6\) 0 0
\(7\) −0.429567 −0.162361 −0.0811805 0.996699i \(-0.525869\pi\)
−0.0811805 + 0.996699i \(0.525869\pi\)
\(8\) 0 0
\(9\) −0.956338 −0.318779
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.88213 1.35406 0.677030 0.735955i \(-0.263267\pi\)
0.677030 + 0.735955i \(0.263267\pi\)
\(14\) 0 0
\(15\) −4.61219 −1.19086
\(16\) 0 0
\(17\) 3.70939 0.899660 0.449830 0.893114i \(-0.351485\pi\)
0.449830 + 0.893114i \(0.351485\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.614095 −0.134006
\(22\) 0 0
\(23\) 3.32349 0.692995 0.346498 0.938051i \(-0.387371\pi\)
0.346498 + 0.938051i \(0.387371\pi\)
\(24\) 0 0
\(25\) 5.40890 1.08178
\(26\) 0 0
\(27\) −5.65585 −1.08847
\(28\) 0 0
\(29\) −2.75305 −0.511229 −0.255615 0.966779i \(-0.582278\pi\)
−0.255615 + 0.966779i \(0.582278\pi\)
\(30\) 0 0
\(31\) −0.699512 −0.125636 −0.0628181 0.998025i \(-0.520009\pi\)
−0.0628181 + 0.998025i \(0.520009\pi\)
\(32\) 0 0
\(33\) −1.42957 −0.248856
\(34\) 0 0
\(35\) 1.38591 0.234261
\(36\) 0 0
\(37\) −1.61409 −0.265356 −0.132678 0.991159i \(-0.542358\pi\)
−0.132678 + 0.991159i \(0.542358\pi\)
\(38\) 0 0
\(39\) 6.97934 1.11759
\(40\) 0 0
\(41\) −8.99809 −1.40527 −0.702633 0.711552i \(-0.747992\pi\)
−0.702633 + 0.711552i \(0.747992\pi\)
\(42\) 0 0
\(43\) −7.40890 −1.12985 −0.564924 0.825143i \(-0.691094\pi\)
−0.564924 + 0.825143i \(0.691094\pi\)
\(44\) 0 0
\(45\) 3.08542 0.459947
\(46\) 0 0
\(47\) 10.9051 1.59068 0.795339 0.606165i \(-0.207293\pi\)
0.795339 + 0.606165i \(0.207293\pi\)
\(48\) 0 0
\(49\) −6.81547 −0.973639
\(50\) 0 0
\(51\) 5.30283 0.742544
\(52\) 0 0
\(53\) 0.452567 0.0621648 0.0310824 0.999517i \(-0.490105\pi\)
0.0310824 + 0.999517i \(0.490105\pi\)
\(54\) 0 0
\(55\) 3.22628 0.435032
\(56\) 0 0
\(57\) −1.42957 −0.189351
\(58\) 0 0
\(59\) −2.38591 −0.310618 −0.155309 0.987866i \(-0.549637\pi\)
−0.155309 + 0.987866i \(0.549637\pi\)
\(60\) 0 0
\(61\) −7.62207 −0.975906 −0.487953 0.872870i \(-0.662256\pi\)
−0.487953 + 0.872870i \(0.662256\pi\)
\(62\) 0 0
\(63\) 0.410811 0.0517574
\(64\) 0 0
\(65\) −15.7511 −1.95369
\(66\) 0 0
\(67\) −7.47132 −0.912767 −0.456384 0.889783i \(-0.650856\pi\)
−0.456384 + 0.889783i \(0.650856\pi\)
\(68\) 0 0
\(69\) 4.75115 0.571971
\(70\) 0 0
\(71\) 1.32248 0.156950 0.0784749 0.996916i \(-0.474995\pi\)
0.0784749 + 0.996916i \(0.474995\pi\)
\(72\) 0 0
\(73\) −13.7643 −1.61099 −0.805493 0.592605i \(-0.798099\pi\)
−0.805493 + 0.592605i \(0.798099\pi\)
\(74\) 0 0
\(75\) 7.73239 0.892860
\(76\) 0 0
\(77\) 0.429567 0.0489537
\(78\) 0 0
\(79\) 2.11596 0.238064 0.119032 0.992890i \(-0.462021\pi\)
0.119032 + 0.992890i \(0.462021\pi\)
\(80\) 0 0
\(81\) −5.21640 −0.579600
\(82\) 0 0
\(83\) 8.64597 0.949019 0.474509 0.880251i \(-0.342626\pi\)
0.474509 + 0.880251i \(0.342626\pi\)
\(84\) 0 0
\(85\) −11.9676 −1.29806
\(86\) 0 0
\(87\) −3.93568 −0.421949
\(88\) 0 0
\(89\) 3.89201 0.412553 0.206276 0.978494i \(-0.433865\pi\)
0.206276 + 0.978494i \(0.433865\pi\)
\(90\) 0 0
\(91\) −2.09720 −0.219847
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 3.22628 0.331010
\(96\) 0 0
\(97\) −13.9793 −1.41939 −0.709693 0.704511i \(-0.751167\pi\)
−0.709693 + 0.704511i \(0.751167\pi\)
\(98\) 0 0
\(99\) 0.956338 0.0961156
\(100\) 0 0
\(101\) 6.65585 0.662282 0.331141 0.943581i \(-0.392566\pi\)
0.331141 + 0.943581i \(0.392566\pi\)
\(102\) 0 0
\(103\) −16.4014 −1.61607 −0.808037 0.589132i \(-0.799470\pi\)
−0.808037 + 0.589132i \(0.799470\pi\)
\(104\) 0 0
\(105\) 1.98124 0.193350
\(106\) 0 0
\(107\) −9.01222 −0.871244 −0.435622 0.900130i \(-0.643472\pi\)
−0.435622 + 0.900130i \(0.643472\pi\)
\(108\) 0 0
\(109\) −7.71827 −0.739276 −0.369638 0.929176i \(-0.620518\pi\)
−0.369638 + 0.929176i \(0.620518\pi\)
\(110\) 0 0
\(111\) −2.30746 −0.219014
\(112\) 0 0
\(113\) −18.7511 −1.76396 −0.881980 0.471287i \(-0.843790\pi\)
−0.881980 + 0.471287i \(0.843790\pi\)
\(114\) 0 0
\(115\) −10.7225 −0.999879
\(116\) 0 0
\(117\) −4.66897 −0.431646
\(118\) 0 0
\(119\) −1.59343 −0.146070
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −12.8634 −1.15985
\(124\) 0 0
\(125\) −1.31924 −0.117997
\(126\) 0 0
\(127\) −0.762934 −0.0676995 −0.0338497 0.999427i \(-0.510777\pi\)
−0.0338497 + 0.999427i \(0.510777\pi\)
\(128\) 0 0
\(129\) −10.5915 −0.932532
\(130\) 0 0
\(131\) 16.9568 1.48152 0.740760 0.671770i \(-0.234466\pi\)
0.740760 + 0.671770i \(0.234466\pi\)
\(132\) 0 0
\(133\) 0.429567 0.0372482
\(134\) 0 0
\(135\) 18.2474 1.57048
\(136\) 0 0
\(137\) 16.9075 1.44450 0.722251 0.691631i \(-0.243107\pi\)
0.722251 + 0.691631i \(0.243107\pi\)
\(138\) 0 0
\(139\) −17.9577 −1.52315 −0.761575 0.648077i \(-0.775574\pi\)
−0.761575 + 0.648077i \(0.775574\pi\)
\(140\) 0 0
\(141\) 15.5896 1.31288
\(142\) 0 0
\(143\) −4.88213 −0.408265
\(144\) 0 0
\(145\) 8.88213 0.737621
\(146\) 0 0
\(147\) −9.74317 −0.803603
\(148\) 0 0
\(149\) −19.1633 −1.56992 −0.784959 0.619548i \(-0.787316\pi\)
−0.784959 + 0.619548i \(0.787316\pi\)
\(150\) 0 0
\(151\) 9.02997 0.734848 0.367424 0.930053i \(-0.380240\pi\)
0.367424 + 0.930053i \(0.380240\pi\)
\(152\) 0 0
\(153\) −3.54743 −0.286793
\(154\) 0 0
\(155\) 2.25683 0.181273
\(156\) 0 0
\(157\) −6.47423 −0.516700 −0.258350 0.966051i \(-0.583179\pi\)
−0.258350 + 0.966051i \(0.583179\pi\)
\(158\) 0 0
\(159\) 0.646975 0.0513084
\(160\) 0 0
\(161\) −1.42766 −0.112515
\(162\) 0 0
\(163\) 15.9962 1.25292 0.626459 0.779454i \(-0.284504\pi\)
0.626459 + 0.779454i \(0.284504\pi\)
\(164\) 0 0
\(165\) 4.61219 0.359058
\(166\) 0 0
\(167\) −2.85026 −0.220560 −0.110280 0.993901i \(-0.535175\pi\)
−0.110280 + 0.993901i \(0.535175\pi\)
\(168\) 0 0
\(169\) 10.8352 0.833479
\(170\) 0 0
\(171\) 0.956338 0.0731330
\(172\) 0 0
\(173\) −20.7991 −1.58132 −0.790661 0.612254i \(-0.790263\pi\)
−0.790661 + 0.612254i \(0.790263\pi\)
\(174\) 0 0
\(175\) −2.32349 −0.175639
\(176\) 0 0
\(177\) −3.41081 −0.256372
\(178\) 0 0
\(179\) −1.84038 −0.137556 −0.0687782 0.997632i \(-0.521910\pi\)
−0.0687782 + 0.997632i \(0.521910\pi\)
\(180\) 0 0
\(181\) −2.94556 −0.218941 −0.109471 0.993990i \(-0.534916\pi\)
−0.109471 + 0.993990i \(0.534916\pi\)
\(182\) 0 0
\(183\) −10.8963 −0.805475
\(184\) 0 0
\(185\) 5.20753 0.382865
\(186\) 0 0
\(187\) −3.70939 −0.271258
\(188\) 0 0
\(189\) 2.42957 0.176725
\(190\) 0 0
\(191\) 14.3732 1.04001 0.520005 0.854163i \(-0.325930\pi\)
0.520005 + 0.854163i \(0.325930\pi\)
\(192\) 0 0
\(193\) −9.62106 −0.692539 −0.346270 0.938135i \(-0.612552\pi\)
−0.346270 + 0.938135i \(0.612552\pi\)
\(194\) 0 0
\(195\) −22.5173 −1.61250
\(196\) 0 0
\(197\) −21.0671 −1.50097 −0.750484 0.660888i \(-0.770180\pi\)
−0.750484 + 0.660888i \(0.770180\pi\)
\(198\) 0 0
\(199\) −15.0911 −1.06978 −0.534888 0.844923i \(-0.679646\pi\)
−0.534888 + 0.844923i \(0.679646\pi\)
\(200\) 0 0
\(201\) −10.6808 −0.753362
\(202\) 0 0
\(203\) 1.18262 0.0830038
\(204\) 0 0
\(205\) 29.0304 2.02757
\(206\) 0 0
\(207\) −3.17838 −0.220912
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −0.272856 −0.0187842 −0.00939209 0.999956i \(-0.502990\pi\)
−0.00939209 + 0.999956i \(0.502990\pi\)
\(212\) 0 0
\(213\) 1.89058 0.129540
\(214\) 0 0
\(215\) 23.9032 1.63019
\(216\) 0 0
\(217\) 0.300488 0.0203984
\(218\) 0 0
\(219\) −19.6769 −1.32964
\(220\) 0 0
\(221\) 18.1098 1.21819
\(222\) 0 0
\(223\) 12.8760 0.862240 0.431120 0.902295i \(-0.358119\pi\)
0.431120 + 0.902295i \(0.358119\pi\)
\(224\) 0 0
\(225\) −5.17274 −0.344849
\(226\) 0 0
\(227\) −23.1046 −1.53351 −0.766753 0.641942i \(-0.778129\pi\)
−0.766753 + 0.641942i \(0.778129\pi\)
\(228\) 0 0
\(229\) 18.9187 1.25019 0.625093 0.780550i \(-0.285061\pi\)
0.625093 + 0.780550i \(0.285061\pi\)
\(230\) 0 0
\(231\) 0.614095 0.0404045
\(232\) 0 0
\(233\) 15.8802 1.04035 0.520174 0.854060i \(-0.325867\pi\)
0.520174 + 0.854060i \(0.325867\pi\)
\(234\) 0 0
\(235\) −35.1831 −2.29509
\(236\) 0 0
\(237\) 3.02491 0.196489
\(238\) 0 0
\(239\) 26.6793 1.72574 0.862870 0.505427i \(-0.168665\pi\)
0.862870 + 0.505427i \(0.168665\pi\)
\(240\) 0 0
\(241\) −2.78550 −0.179430 −0.0897150 0.995967i \(-0.528596\pi\)
−0.0897150 + 0.995967i \(0.528596\pi\)
\(242\) 0 0
\(243\) 9.51035 0.610089
\(244\) 0 0
\(245\) 21.9886 1.40480
\(246\) 0 0
\(247\) −4.88213 −0.310643
\(248\) 0 0
\(249\) 12.3600 0.783283
\(250\) 0 0
\(251\) −11.6014 −0.732274 −0.366137 0.930561i \(-0.619320\pi\)
−0.366137 + 0.930561i \(0.619320\pi\)
\(252\) 0 0
\(253\) −3.32349 −0.208946
\(254\) 0 0
\(255\) −17.1084 −1.07137
\(256\) 0 0
\(257\) −2.04600 −0.127626 −0.0638130 0.997962i \(-0.520326\pi\)
−0.0638130 + 0.997962i \(0.520326\pi\)
\(258\) 0 0
\(259\) 0.693362 0.0430834
\(260\) 0 0
\(261\) 2.63285 0.162969
\(262\) 0 0
\(263\) 26.1807 1.61437 0.807186 0.590297i \(-0.200989\pi\)
0.807186 + 0.590297i \(0.200989\pi\)
\(264\) 0 0
\(265\) −1.46011 −0.0896938
\(266\) 0 0
\(267\) 5.56389 0.340505
\(268\) 0 0
\(269\) 14.2864 0.871058 0.435529 0.900175i \(-0.356561\pi\)
0.435529 + 0.900175i \(0.356561\pi\)
\(270\) 0 0
\(271\) −11.4920 −0.698088 −0.349044 0.937106i \(-0.613494\pi\)
−0.349044 + 0.937106i \(0.613494\pi\)
\(272\) 0 0
\(273\) −2.99809 −0.181453
\(274\) 0 0
\(275\) −5.40890 −0.326169
\(276\) 0 0
\(277\) −20.6057 −1.23807 −0.619037 0.785362i \(-0.712477\pi\)
−0.619037 + 0.785362i \(0.712477\pi\)
\(278\) 0 0
\(279\) 0.668970 0.0400502
\(280\) 0 0
\(281\) 17.4051 1.03830 0.519150 0.854683i \(-0.326248\pi\)
0.519150 + 0.854683i \(0.326248\pi\)
\(282\) 0 0
\(283\) 2.92813 0.174059 0.0870297 0.996206i \(-0.472262\pi\)
0.0870297 + 0.996206i \(0.472262\pi\)
\(284\) 0 0
\(285\) 4.61219 0.273202
\(286\) 0 0
\(287\) 3.86528 0.228161
\(288\) 0 0
\(289\) −3.24041 −0.190612
\(290\) 0 0
\(291\) −19.9844 −1.17151
\(292\) 0 0
\(293\) 9.97319 0.582640 0.291320 0.956626i \(-0.405906\pi\)
0.291320 + 0.956626i \(0.405906\pi\)
\(294\) 0 0
\(295\) 7.69761 0.448172
\(296\) 0 0
\(297\) 5.65585 0.328186
\(298\) 0 0
\(299\) 16.2257 0.938357
\(300\) 0 0
\(301\) 3.18262 0.183443
\(302\) 0 0
\(303\) 9.51498 0.546622
\(304\) 0 0
\(305\) 24.5910 1.40807
\(306\) 0 0
\(307\) 10.9140 0.622895 0.311448 0.950263i \(-0.399186\pi\)
0.311448 + 0.950263i \(0.399186\pi\)
\(308\) 0 0
\(309\) −23.4468 −1.33384
\(310\) 0 0
\(311\) 9.14087 0.518331 0.259165 0.965833i \(-0.416552\pi\)
0.259165 + 0.965833i \(0.416552\pi\)
\(312\) 0 0
\(313\) 13.9193 0.786763 0.393381 0.919375i \(-0.371305\pi\)
0.393381 + 0.919375i \(0.371305\pi\)
\(314\) 0 0
\(315\) −1.32539 −0.0746775
\(316\) 0 0
\(317\) −28.2159 −1.58476 −0.792382 0.610025i \(-0.791159\pi\)
−0.792382 + 0.610025i \(0.791159\pi\)
\(318\) 0 0
\(319\) 2.75305 0.154141
\(320\) 0 0
\(321\) −12.8836 −0.719091
\(322\) 0 0
\(323\) −3.70939 −0.206396
\(324\) 0 0
\(325\) 26.4070 1.46480
\(326\) 0 0
\(327\) −11.0338 −0.610170
\(328\) 0 0
\(329\) −4.68449 −0.258264
\(330\) 0 0
\(331\) 6.29338 0.345915 0.172958 0.984929i \(-0.444668\pi\)
0.172958 + 0.984929i \(0.444668\pi\)
\(332\) 0 0
\(333\) 1.54362 0.0845899
\(334\) 0 0
\(335\) 24.1046 1.31698
\(336\) 0 0
\(337\) 9.57043 0.521335 0.260667 0.965429i \(-0.416057\pi\)
0.260667 + 0.965429i \(0.416057\pi\)
\(338\) 0 0
\(339\) −26.8060 −1.45590
\(340\) 0 0
\(341\) 0.699512 0.0378807
\(342\) 0 0
\(343\) 5.93467 0.320442
\(344\) 0 0
\(345\) −15.3286 −0.825261
\(346\) 0 0
\(347\) −4.46144 −0.239503 −0.119751 0.992804i \(-0.538210\pi\)
−0.119751 + 0.992804i \(0.538210\pi\)
\(348\) 0 0
\(349\) 13.1320 0.702939 0.351470 0.936199i \(-0.385682\pi\)
0.351470 + 0.936199i \(0.385682\pi\)
\(350\) 0 0
\(351\) −27.6126 −1.47385
\(352\) 0 0
\(353\) −21.1140 −1.12378 −0.561892 0.827210i \(-0.689926\pi\)
−0.561892 + 0.827210i \(0.689926\pi\)
\(354\) 0 0
\(355\) −4.26671 −0.226453
\(356\) 0 0
\(357\) −2.27792 −0.120560
\(358\) 0 0
\(359\) 8.78593 0.463704 0.231852 0.972751i \(-0.425522\pi\)
0.231852 + 0.972751i \(0.425522\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.42957 0.0750328
\(364\) 0 0
\(365\) 44.4074 2.32439
\(366\) 0 0
\(367\) −7.68672 −0.401244 −0.200622 0.979669i \(-0.564296\pi\)
−0.200622 + 0.979669i \(0.564296\pi\)
\(368\) 0 0
\(369\) 8.60522 0.447970
\(370\) 0 0
\(371\) −0.194408 −0.0100932
\(372\) 0 0
\(373\) 1.71040 0.0885610 0.0442805 0.999019i \(-0.485900\pi\)
0.0442805 + 0.999019i \(0.485900\pi\)
\(374\) 0 0
\(375\) −1.88595 −0.0973899
\(376\) 0 0
\(377\) −13.4408 −0.692235
\(378\) 0 0
\(379\) 12.7417 0.654497 0.327249 0.944938i \(-0.393879\pi\)
0.327249 + 0.944938i \(0.393879\pi\)
\(380\) 0 0
\(381\) −1.09067 −0.0558765
\(382\) 0 0
\(383\) 25.9652 1.32676 0.663380 0.748282i \(-0.269121\pi\)
0.663380 + 0.748282i \(0.269121\pi\)
\(384\) 0 0
\(385\) −1.38591 −0.0706323
\(386\) 0 0
\(387\) 7.08542 0.360172
\(388\) 0 0
\(389\) −2.83747 −0.143865 −0.0719327 0.997409i \(-0.522917\pi\)
−0.0719327 + 0.997409i \(0.522917\pi\)
\(390\) 0 0
\(391\) 12.3281 0.623460
\(392\) 0 0
\(393\) 24.2408 1.22279
\(394\) 0 0
\(395\) −6.82669 −0.343488
\(396\) 0 0
\(397\) 2.44226 0.122573 0.0612866 0.998120i \(-0.480480\pi\)
0.0612866 + 0.998120i \(0.480480\pi\)
\(398\) 0 0
\(399\) 0.614095 0.0307432
\(400\) 0 0
\(401\) 23.2666 1.16188 0.580938 0.813948i \(-0.302686\pi\)
0.580938 + 0.813948i \(0.302686\pi\)
\(402\) 0 0
\(403\) −3.41511 −0.170119
\(404\) 0 0
\(405\) 16.8296 0.836269
\(406\) 0 0
\(407\) 1.61409 0.0800077
\(408\) 0 0
\(409\) 11.6910 0.578084 0.289042 0.957316i \(-0.406663\pi\)
0.289042 + 0.957316i \(0.406663\pi\)
\(410\) 0 0
\(411\) 24.1704 1.19224
\(412\) 0 0
\(413\) 1.02491 0.0504323
\(414\) 0 0
\(415\) −27.8944 −1.36928
\(416\) 0 0
\(417\) −25.6717 −1.25715
\(418\) 0 0
\(419\) −33.5285 −1.63798 −0.818988 0.573811i \(-0.805465\pi\)
−0.818988 + 0.573811i \(0.805465\pi\)
\(420\) 0 0
\(421\) −6.60098 −0.321712 −0.160856 0.986978i \(-0.551425\pi\)
−0.160856 + 0.986978i \(0.551425\pi\)
\(422\) 0 0
\(423\) −10.4290 −0.507075
\(424\) 0 0
\(425\) 20.0638 0.973235
\(426\) 0 0
\(427\) 3.27419 0.158449
\(428\) 0 0
\(429\) −6.97934 −0.336966
\(430\) 0 0
\(431\) 6.24041 0.300590 0.150295 0.988641i \(-0.451978\pi\)
0.150295 + 0.988641i \(0.451978\pi\)
\(432\) 0 0
\(433\) −25.2610 −1.21397 −0.606983 0.794715i \(-0.707620\pi\)
−0.606983 + 0.794715i \(0.707620\pi\)
\(434\) 0 0
\(435\) 12.6976 0.608804
\(436\) 0 0
\(437\) −3.32349 −0.158984
\(438\) 0 0
\(439\) 0.948854 0.0452863 0.0226432 0.999744i \(-0.492792\pi\)
0.0226432 + 0.999744i \(0.492792\pi\)
\(440\) 0 0
\(441\) 6.51790 0.310376
\(442\) 0 0
\(443\) −15.1713 −0.720809 −0.360404 0.932796i \(-0.617361\pi\)
−0.360404 + 0.932796i \(0.617361\pi\)
\(444\) 0 0
\(445\) −12.5567 −0.595247
\(446\) 0 0
\(447\) −27.3952 −1.29575
\(448\) 0 0
\(449\) −35.6787 −1.68378 −0.841891 0.539647i \(-0.818558\pi\)
−0.841891 + 0.539647i \(0.818558\pi\)
\(450\) 0 0
\(451\) 8.99809 0.423704
\(452\) 0 0
\(453\) 12.9089 0.606515
\(454\) 0 0
\(455\) 6.76617 0.317203
\(456\) 0 0
\(457\) 22.8904 1.07077 0.535385 0.844608i \(-0.320167\pi\)
0.535385 + 0.844608i \(0.320167\pi\)
\(458\) 0 0
\(459\) −20.9798 −0.979252
\(460\) 0 0
\(461\) −26.7980 −1.24811 −0.624055 0.781381i \(-0.714516\pi\)
−0.624055 + 0.781381i \(0.714516\pi\)
\(462\) 0 0
\(463\) −14.6314 −0.679978 −0.339989 0.940429i \(-0.610423\pi\)
−0.339989 + 0.940429i \(0.610423\pi\)
\(464\) 0 0
\(465\) 3.22628 0.149615
\(466\) 0 0
\(467\) −15.7460 −0.728638 −0.364319 0.931274i \(-0.618698\pi\)
−0.364319 + 0.931274i \(0.618698\pi\)
\(468\) 0 0
\(469\) 3.20943 0.148198
\(470\) 0 0
\(471\) −9.25535 −0.426464
\(472\) 0 0
\(473\) 7.40890 0.340662
\(474\) 0 0
\(475\) −5.40890 −0.248178
\(476\) 0 0
\(477\) −0.432807 −0.0198169
\(478\) 0 0
\(479\) 8.90571 0.406912 0.203456 0.979084i \(-0.434783\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(480\) 0 0
\(481\) −7.88023 −0.359307
\(482\) 0 0
\(483\) −2.04094 −0.0928658
\(484\) 0 0
\(485\) 45.1013 2.04794
\(486\) 0 0
\(487\) −3.35073 −0.151836 −0.0759181 0.997114i \(-0.524189\pi\)
−0.0759181 + 0.997114i \(0.524189\pi\)
\(488\) 0 0
\(489\) 22.8676 1.03411
\(490\) 0 0
\(491\) 19.1601 0.864681 0.432341 0.901710i \(-0.357688\pi\)
0.432341 + 0.901710i \(0.357688\pi\)
\(492\) 0 0
\(493\) −10.2122 −0.459933
\(494\) 0 0
\(495\) −3.08542 −0.138679
\(496\) 0 0
\(497\) −0.568095 −0.0254826
\(498\) 0 0
\(499\) −28.4665 −1.27434 −0.637168 0.770725i \(-0.719894\pi\)
−0.637168 + 0.770725i \(0.719894\pi\)
\(500\) 0 0
\(501\) −4.07464 −0.182041
\(502\) 0 0
\(503\) −23.2407 −1.03625 −0.518127 0.855304i \(-0.673370\pi\)
−0.518127 + 0.855304i \(0.673370\pi\)
\(504\) 0 0
\(505\) −21.4737 −0.955565
\(506\) 0 0
\(507\) 15.4897 0.687921
\(508\) 0 0
\(509\) 20.3295 0.901089 0.450545 0.892754i \(-0.351230\pi\)
0.450545 + 0.892754i \(0.351230\pi\)
\(510\) 0 0
\(511\) 5.91268 0.261561
\(512\) 0 0
\(513\) 5.65585 0.249712
\(514\) 0 0
\(515\) 52.9154 2.33173
\(516\) 0 0
\(517\) −10.9051 −0.479607
\(518\) 0 0
\(519\) −29.7336 −1.30516
\(520\) 0 0
\(521\) 6.68539 0.292892 0.146446 0.989219i \(-0.453217\pi\)
0.146446 + 0.989219i \(0.453217\pi\)
\(522\) 0 0
\(523\) 26.3463 1.15205 0.576023 0.817434i \(-0.304604\pi\)
0.576023 + 0.817434i \(0.304604\pi\)
\(524\) 0 0
\(525\) −3.32158 −0.144966
\(526\) 0 0
\(527\) −2.59477 −0.113030
\(528\) 0 0
\(529\) −11.9544 −0.519758
\(530\) 0 0
\(531\) 2.28173 0.0990187
\(532\) 0 0
\(533\) −43.9299 −1.90282
\(534\) 0 0
\(535\) 29.0760 1.25706
\(536\) 0 0
\(537\) −2.63094 −0.113534
\(538\) 0 0
\(539\) 6.81547 0.293563
\(540\) 0 0
\(541\) 3.65958 0.157338 0.0786688 0.996901i \(-0.474933\pi\)
0.0786688 + 0.996901i \(0.474933\pi\)
\(542\) 0 0
\(543\) −4.21087 −0.180706
\(544\) 0 0
\(545\) 24.9013 1.06666
\(546\) 0 0
\(547\) 33.7064 1.44118 0.720592 0.693360i \(-0.243870\pi\)
0.720592 + 0.693360i \(0.243870\pi\)
\(548\) 0 0
\(549\) 7.28927 0.311099
\(550\) 0 0
\(551\) 2.75305 0.117284
\(552\) 0 0
\(553\) −0.908947 −0.0386523
\(554\) 0 0
\(555\) 7.44451 0.316002
\(556\) 0 0
\(557\) 26.3897 1.11817 0.559083 0.829112i \(-0.311153\pi\)
0.559083 + 0.829112i \(0.311153\pi\)
\(558\) 0 0
\(559\) −36.1713 −1.52988
\(560\) 0 0
\(561\) −5.30283 −0.223885
\(562\) 0 0
\(563\) −0.209105 −0.00881272 −0.00440636 0.999990i \(-0.501403\pi\)
−0.00440636 + 0.999990i \(0.501403\pi\)
\(564\) 0 0
\(565\) 60.4965 2.54511
\(566\) 0 0
\(567\) 2.24080 0.0941045
\(568\) 0 0
\(569\) −42.3233 −1.77429 −0.887143 0.461494i \(-0.847314\pi\)
−0.887143 + 0.461494i \(0.847314\pi\)
\(570\) 0 0
\(571\) −33.5628 −1.40456 −0.702280 0.711901i \(-0.747834\pi\)
−0.702280 + 0.711901i \(0.747834\pi\)
\(572\) 0 0
\(573\) 20.5475 0.858383
\(574\) 0 0
\(575\) 17.9764 0.749669
\(576\) 0 0
\(577\) 10.4287 0.434151 0.217076 0.976155i \(-0.430348\pi\)
0.217076 + 0.976155i \(0.430348\pi\)
\(578\) 0 0
\(579\) −13.7540 −0.571595
\(580\) 0 0
\(581\) −3.71402 −0.154084
\(582\) 0 0
\(583\) −0.452567 −0.0187434
\(584\) 0 0
\(585\) 15.0634 0.622796
\(586\) 0 0
\(587\) −37.5023 −1.54789 −0.773943 0.633256i \(-0.781718\pi\)
−0.773943 + 0.633256i \(0.781718\pi\)
\(588\) 0 0
\(589\) 0.699512 0.0288229
\(590\) 0 0
\(591\) −30.1168 −1.23884
\(592\) 0 0
\(593\) 23.8262 0.978425 0.489213 0.872165i \(-0.337284\pi\)
0.489213 + 0.872165i \(0.337284\pi\)
\(594\) 0 0
\(595\) 5.14087 0.210755
\(596\) 0 0
\(597\) −21.5737 −0.882952
\(598\) 0 0
\(599\) 2.89144 0.118141 0.0590705 0.998254i \(-0.481186\pi\)
0.0590705 + 0.998254i \(0.481186\pi\)
\(600\) 0 0
\(601\) 30.3469 1.23788 0.618938 0.785440i \(-0.287563\pi\)
0.618938 + 0.785440i \(0.287563\pi\)
\(602\) 0 0
\(603\) 7.14511 0.290971
\(604\) 0 0
\(605\) −3.22628 −0.131167
\(606\) 0 0
\(607\) 21.2851 0.863934 0.431967 0.901889i \(-0.357820\pi\)
0.431967 + 0.901889i \(0.357820\pi\)
\(608\) 0 0
\(609\) 1.69064 0.0685081
\(610\) 0 0
\(611\) 53.2403 2.15387
\(612\) 0 0
\(613\) −32.1384 −1.29806 −0.649029 0.760764i \(-0.724824\pi\)
−0.649029 + 0.760764i \(0.724824\pi\)
\(614\) 0 0
\(615\) 41.5009 1.67348
\(616\) 0 0
\(617\) −13.4507 −0.541503 −0.270752 0.962649i \(-0.587272\pi\)
−0.270752 + 0.962649i \(0.587272\pi\)
\(618\) 0 0
\(619\) −15.6849 −0.630430 −0.315215 0.949020i \(-0.602077\pi\)
−0.315215 + 0.949020i \(0.602077\pi\)
\(620\) 0 0
\(621\) −18.7971 −0.754304
\(622\) 0 0
\(623\) −1.67188 −0.0669825
\(624\) 0 0
\(625\) −22.7883 −0.911531
\(626\) 0 0
\(627\) 1.42957 0.0570914
\(628\) 0 0
\(629\) −5.98731 −0.238730
\(630\) 0 0
\(631\) 0.0557776 0.00222047 0.00111024 0.999999i \(-0.499647\pi\)
0.00111024 + 0.999999i \(0.499647\pi\)
\(632\) 0 0
\(633\) −0.390066 −0.0155037
\(634\) 0 0
\(635\) 2.46144 0.0976794
\(636\) 0 0
\(637\) −33.2740 −1.31837
\(638\) 0 0
\(639\) −1.26474 −0.0500324
\(640\) 0 0
\(641\) −18.3071 −0.723086 −0.361543 0.932355i \(-0.617750\pi\)
−0.361543 + 0.932355i \(0.617750\pi\)
\(642\) 0 0
\(643\) 48.8246 1.92546 0.962728 0.270472i \(-0.0871797\pi\)
0.962728 + 0.270472i \(0.0871797\pi\)
\(644\) 0 0
\(645\) 34.1713 1.34549
\(646\) 0 0
\(647\) −17.3919 −0.683747 −0.341873 0.939746i \(-0.611061\pi\)
−0.341873 + 0.939746i \(0.611061\pi\)
\(648\) 0 0
\(649\) 2.38591 0.0936550
\(650\) 0 0
\(651\) 0.429567 0.0168361
\(652\) 0 0
\(653\) −27.1216 −1.06135 −0.530676 0.847575i \(-0.678062\pi\)
−0.530676 + 0.847575i \(0.678062\pi\)
\(654\) 0 0
\(655\) −54.7073 −2.13759
\(656\) 0 0
\(657\) 13.1633 0.513549
\(658\) 0 0
\(659\) 16.0696 0.625982 0.312991 0.949756i \(-0.398669\pi\)
0.312991 + 0.949756i \(0.398669\pi\)
\(660\) 0 0
\(661\) 2.36233 0.0918841 0.0459420 0.998944i \(-0.485371\pi\)
0.0459420 + 0.998944i \(0.485371\pi\)
\(662\) 0 0
\(663\) 25.8891 1.00545
\(664\) 0 0
\(665\) −1.38591 −0.0537431
\(666\) 0 0
\(667\) −9.14974 −0.354279
\(668\) 0 0
\(669\) 18.4071 0.711659
\(670\) 0 0
\(671\) 7.62207 0.294247
\(672\) 0 0
\(673\) −10.1474 −0.391155 −0.195578 0.980688i \(-0.562658\pi\)
−0.195578 + 0.980688i \(0.562658\pi\)
\(674\) 0 0
\(675\) −30.5920 −1.17749
\(676\) 0 0
\(677\) −35.8623 −1.37830 −0.689151 0.724618i \(-0.742016\pi\)
−0.689151 + 0.724618i \(0.742016\pi\)
\(678\) 0 0
\(679\) 6.00506 0.230453
\(680\) 0 0
\(681\) −33.0296 −1.26570
\(682\) 0 0
\(683\) −11.5108 −0.440448 −0.220224 0.975449i \(-0.570679\pi\)
−0.220224 + 0.975449i \(0.570679\pi\)
\(684\) 0 0
\(685\) −54.5483 −2.08418
\(686\) 0 0
\(687\) 27.0456 1.03185
\(688\) 0 0
\(689\) 2.20949 0.0841750
\(690\) 0 0
\(691\) 7.57517 0.288173 0.144086 0.989565i \(-0.453976\pi\)
0.144086 + 0.989565i \(0.453976\pi\)
\(692\) 0 0
\(693\) −0.410811 −0.0156054
\(694\) 0 0
\(695\) 57.9365 2.19766
\(696\) 0 0
\(697\) −33.3775 −1.26426
\(698\) 0 0
\(699\) 22.7019 0.858663
\(700\) 0 0
\(701\) 32.9211 1.24341 0.621706 0.783251i \(-0.286440\pi\)
0.621706 + 0.783251i \(0.286440\pi\)
\(702\) 0 0
\(703\) 1.61409 0.0608767
\(704\) 0 0
\(705\) −50.2965 −1.89428
\(706\) 0 0
\(707\) −2.85913 −0.107529
\(708\) 0 0
\(709\) −38.2867 −1.43789 −0.718944 0.695068i \(-0.755374\pi\)
−0.718944 + 0.695068i \(0.755374\pi\)
\(710\) 0 0
\(711\) −2.02357 −0.0758899
\(712\) 0 0
\(713\) −2.32482 −0.0870652
\(714\) 0 0
\(715\) 15.7511 0.589060
\(716\) 0 0
\(717\) 38.1398 1.42436
\(718\) 0 0
\(719\) −27.2146 −1.01493 −0.507467 0.861671i \(-0.669418\pi\)
−0.507467 + 0.861671i \(0.669418\pi\)
\(720\) 0 0
\(721\) 7.04549 0.262388
\(722\) 0 0
\(723\) −3.98206 −0.148095
\(724\) 0 0
\(725\) −14.8910 −0.553038
\(726\) 0 0
\(727\) −2.44078 −0.0905235 −0.0452618 0.998975i \(-0.514412\pi\)
−0.0452618 + 0.998975i \(0.514412\pi\)
\(728\) 0 0
\(729\) 29.2449 1.08314
\(730\) 0 0
\(731\) −27.4825 −1.01648
\(732\) 0 0
\(733\) −31.3825 −1.15914 −0.579570 0.814922i \(-0.696779\pi\)
−0.579570 + 0.814922i \(0.696779\pi\)
\(734\) 0 0
\(735\) 31.4342 1.15947
\(736\) 0 0
\(737\) 7.47132 0.275210
\(738\) 0 0
\(739\) −14.9277 −0.549124 −0.274562 0.961569i \(-0.588533\pi\)
−0.274562 + 0.961569i \(0.588533\pi\)
\(740\) 0 0
\(741\) −6.97934 −0.256392
\(742\) 0 0
\(743\) 46.2868 1.69810 0.849049 0.528314i \(-0.177176\pi\)
0.849049 + 0.528314i \(0.177176\pi\)
\(744\) 0 0
\(745\) 61.8262 2.26514
\(746\) 0 0
\(747\) −8.26847 −0.302528
\(748\) 0 0
\(749\) 3.87135 0.141456
\(750\) 0 0
\(751\) −15.2102 −0.555028 −0.277514 0.960722i \(-0.589510\pi\)
−0.277514 + 0.960722i \(0.589510\pi\)
\(752\) 0 0
\(753\) −16.5850 −0.604391
\(754\) 0 0
\(755\) −29.1332 −1.06027
\(756\) 0 0
\(757\) 35.6145 1.29443 0.647215 0.762308i \(-0.275934\pi\)
0.647215 + 0.762308i \(0.275934\pi\)
\(758\) 0 0
\(759\) −4.75115 −0.172456
\(760\) 0 0
\(761\) 37.0701 1.34379 0.671896 0.740646i \(-0.265480\pi\)
0.671896 + 0.740646i \(0.265480\pi\)
\(762\) 0 0
\(763\) 3.31551 0.120030
\(764\) 0 0
\(765\) 11.4450 0.413796
\(766\) 0 0
\(767\) −11.6483 −0.420596
\(768\) 0 0
\(769\) 7.21311 0.260111 0.130056 0.991507i \(-0.458484\pi\)
0.130056 + 0.991507i \(0.458484\pi\)
\(770\) 0 0
\(771\) −2.92489 −0.105337
\(772\) 0 0
\(773\) −52.8299 −1.90016 −0.950081 0.312005i \(-0.898999\pi\)
−0.950081 + 0.312005i \(0.898999\pi\)
\(774\) 0 0
\(775\) −3.78360 −0.135911
\(776\) 0 0
\(777\) 0.991207 0.0355594
\(778\) 0 0
\(779\) 8.99809 0.322390
\(780\) 0 0
\(781\) −1.32248 −0.0473222
\(782\) 0 0
\(783\) 15.5709 0.556457
\(784\) 0 0
\(785\) 20.8877 0.745514
\(786\) 0 0
\(787\) −43.5012 −1.55065 −0.775324 0.631563i \(-0.782414\pi\)
−0.775324 + 0.631563i \(0.782414\pi\)
\(788\) 0 0
\(789\) 37.4271 1.33244
\(790\) 0 0
\(791\) 8.05488 0.286398
\(792\) 0 0
\(793\) −37.2120 −1.32144
\(794\) 0 0
\(795\) −2.08732 −0.0740298
\(796\) 0 0
\(797\) −20.5718 −0.728691 −0.364346 0.931264i \(-0.618707\pi\)
−0.364346 + 0.931264i \(0.618707\pi\)
\(798\) 0 0
\(799\) 40.4514 1.43107
\(800\) 0 0
\(801\) −3.72208 −0.131513
\(802\) 0 0
\(803\) 13.7643 0.485730
\(804\) 0 0
\(805\) 4.60604 0.162342
\(806\) 0 0
\(807\) 20.4234 0.718937
\(808\) 0 0
\(809\) 47.2365 1.66075 0.830374 0.557207i \(-0.188127\pi\)
0.830374 + 0.557207i \(0.188127\pi\)
\(810\) 0 0
\(811\) 2.86621 0.100646 0.0503230 0.998733i \(-0.483975\pi\)
0.0503230 + 0.998733i \(0.483975\pi\)
\(812\) 0 0
\(813\) −16.4286 −0.576175
\(814\) 0 0
\(815\) −51.6082 −1.80776
\(816\) 0 0
\(817\) 7.40890 0.259205
\(818\) 0 0
\(819\) 2.00564 0.0700826
\(820\) 0 0
\(821\) −10.3986 −0.362912 −0.181456 0.983399i \(-0.558081\pi\)
−0.181456 + 0.983399i \(0.558081\pi\)
\(822\) 0 0
\(823\) 55.6014 1.93814 0.969072 0.246780i \(-0.0793724\pi\)
0.969072 + 0.246780i \(0.0793724\pi\)
\(824\) 0 0
\(825\) −7.73239 −0.269207
\(826\) 0 0
\(827\) 25.2728 0.878820 0.439410 0.898287i \(-0.355187\pi\)
0.439410 + 0.898287i \(0.355187\pi\)
\(828\) 0 0
\(829\) −16.8845 −0.586422 −0.293211 0.956048i \(-0.594724\pi\)
−0.293211 + 0.956048i \(0.594724\pi\)
\(830\) 0 0
\(831\) −29.4572 −1.02186
\(832\) 0 0
\(833\) −25.2813 −0.875944
\(834\) 0 0
\(835\) 9.19574 0.318232
\(836\) 0 0
\(837\) 3.95634 0.136751
\(838\) 0 0
\(839\) 35.6042 1.22919 0.614597 0.788841i \(-0.289319\pi\)
0.614597 + 0.788841i \(0.289319\pi\)
\(840\) 0 0
\(841\) −21.4207 −0.738645
\(842\) 0 0
\(843\) 24.8817 0.856973
\(844\) 0 0
\(845\) −34.9575 −1.20258
\(846\) 0 0
\(847\) −0.429567 −0.0147601
\(848\) 0 0
\(849\) 4.18596 0.143662
\(850\) 0 0
\(851\) −5.36442 −0.183890
\(852\) 0 0
\(853\) 37.6592 1.28943 0.644713 0.764425i \(-0.276977\pi\)
0.644713 + 0.764425i \(0.276977\pi\)
\(854\) 0 0
\(855\) −3.08542 −0.105519
\(856\) 0 0
\(857\) −4.25693 −0.145414 −0.0727069 0.997353i \(-0.523164\pi\)
−0.0727069 + 0.997353i \(0.523164\pi\)
\(858\) 0 0
\(859\) 46.4679 1.58546 0.792732 0.609571i \(-0.208658\pi\)
0.792732 + 0.609571i \(0.208658\pi\)
\(860\) 0 0
\(861\) 5.52568 0.188315
\(862\) 0 0
\(863\) 53.8291 1.83236 0.916181 0.400764i \(-0.131255\pi\)
0.916181 + 0.400764i \(0.131255\pi\)
\(864\) 0 0
\(865\) 67.1036 2.28159
\(866\) 0 0
\(867\) −4.63238 −0.157324
\(868\) 0 0
\(869\) −2.11596 −0.0717790
\(870\) 0 0
\(871\) −36.4760 −1.23594
\(872\) 0 0
\(873\) 13.3690 0.452471
\(874\) 0 0
\(875\) 0.566703 0.0191581
\(876\) 0 0
\(877\) 22.8239 0.770708 0.385354 0.922769i \(-0.374079\pi\)
0.385354 + 0.922769i \(0.374079\pi\)
\(878\) 0 0
\(879\) 14.2573 0.480888
\(880\) 0 0
\(881\) 44.0631 1.48452 0.742262 0.670109i \(-0.233753\pi\)
0.742262 + 0.670109i \(0.233753\pi\)
\(882\) 0 0
\(883\) 11.8215 0.397826 0.198913 0.980017i \(-0.436259\pi\)
0.198913 + 0.980017i \(0.436259\pi\)
\(884\) 0 0
\(885\) 11.0042 0.369904
\(886\) 0 0
\(887\) −24.1079 −0.809466 −0.404733 0.914435i \(-0.632635\pi\)
−0.404733 + 0.914435i \(0.632635\pi\)
\(888\) 0 0
\(889\) 0.327731 0.0109918
\(890\) 0 0
\(891\) 5.21640 0.174756
\(892\) 0 0
\(893\) −10.9051 −0.364926
\(894\) 0 0
\(895\) 5.93758 0.198471
\(896\) 0 0
\(897\) 23.1957 0.774483
\(898\) 0 0
\(899\) 1.92580 0.0642289
\(900\) 0 0
\(901\) 1.67875 0.0559272
\(902\) 0 0
\(903\) 4.54977 0.151407
\(904\) 0 0
\(905\) 9.50320 0.315897
\(906\) 0 0
\(907\) −17.1371 −0.569027 −0.284513 0.958672i \(-0.591832\pi\)
−0.284513 + 0.958672i \(0.591832\pi\)
\(908\) 0 0
\(909\) −6.36524 −0.211122
\(910\) 0 0
\(911\) 2.22446 0.0736997 0.0368498 0.999321i \(-0.488268\pi\)
0.0368498 + 0.999321i \(0.488268\pi\)
\(912\) 0 0
\(913\) −8.64597 −0.286140
\(914\) 0 0
\(915\) 35.1544 1.16217
\(916\) 0 0
\(917\) −7.28407 −0.240541
\(918\) 0 0
\(919\) 8.86662 0.292483 0.146241 0.989249i \(-0.453282\pi\)
0.146241 + 0.989249i \(0.453282\pi\)
\(920\) 0 0
\(921\) 15.6023 0.514114
\(922\) 0 0
\(923\) 6.45654 0.212520
\(924\) 0 0
\(925\) −8.73049 −0.287057
\(926\) 0 0
\(927\) 15.6852 0.515171
\(928\) 0 0
\(929\) 27.8252 0.912914 0.456457 0.889745i \(-0.349118\pi\)
0.456457 + 0.889745i \(0.349118\pi\)
\(930\) 0 0
\(931\) 6.81547 0.223368
\(932\) 0 0
\(933\) 13.0675 0.427810
\(934\) 0 0
\(935\) 11.9676 0.391381
\(936\) 0 0
\(937\) −5.45009 −0.178047 −0.0890233 0.996030i \(-0.528375\pi\)
−0.0890233 + 0.996030i \(0.528375\pi\)
\(938\) 0 0
\(939\) 19.8985 0.649363
\(940\) 0 0
\(941\) −0.858747 −0.0279943 −0.0139972 0.999902i \(-0.504456\pi\)
−0.0139972 + 0.999902i \(0.504456\pi\)
\(942\) 0 0
\(943\) −29.9051 −0.973842
\(944\) 0 0
\(945\) −7.83847 −0.254986
\(946\) 0 0
\(947\) −35.6323 −1.15789 −0.578947 0.815365i \(-0.696536\pi\)
−0.578947 + 0.815365i \(0.696536\pi\)
\(948\) 0 0
\(949\) −67.1990 −2.18137
\(950\) 0 0
\(951\) −40.3366 −1.30800
\(952\) 0 0
\(953\) −11.2728 −0.365161 −0.182580 0.983191i \(-0.558445\pi\)
−0.182580 + 0.983191i \(0.558445\pi\)
\(954\) 0 0
\(955\) −46.3721 −1.50056
\(956\) 0 0
\(957\) 3.93568 0.127222
\(958\) 0 0
\(959\) −7.26289 −0.234531
\(960\) 0 0
\(961\) −30.5107 −0.984216
\(962\) 0 0
\(963\) 8.61873 0.277735
\(964\) 0 0
\(965\) 31.0403 0.999222
\(966\) 0 0
\(967\) −9.59850 −0.308667 −0.154333 0.988019i \(-0.549323\pi\)
−0.154333 + 0.988019i \(0.549323\pi\)
\(968\) 0 0
\(969\) −5.30283 −0.170351
\(970\) 0 0
\(971\) −25.8632 −0.829991 −0.414995 0.909824i \(-0.636217\pi\)
−0.414995 + 0.909824i \(0.636217\pi\)
\(972\) 0 0
\(973\) 7.71402 0.247300
\(974\) 0 0
\(975\) 37.7506 1.20899
\(976\) 0 0
\(977\) 13.3510 0.427136 0.213568 0.976928i \(-0.431492\pi\)
0.213568 + 0.976928i \(0.431492\pi\)
\(978\) 0 0
\(979\) −3.89201 −0.124389
\(980\) 0 0
\(981\) 7.38127 0.235666
\(982\) 0 0
\(983\) 47.6233 1.51895 0.759473 0.650539i \(-0.225457\pi\)
0.759473 + 0.650539i \(0.225457\pi\)
\(984\) 0 0
\(985\) 67.9684 2.16565
\(986\) 0 0
\(987\) −6.69679 −0.213161
\(988\) 0 0
\(989\) −24.6234 −0.782979
\(990\) 0 0
\(991\) 47.5586 1.51075 0.755374 0.655293i \(-0.227455\pi\)
0.755374 + 0.655293i \(0.227455\pi\)
\(992\) 0 0
\(993\) 8.99680 0.285505
\(994\) 0 0
\(995\) 48.6880 1.54351
\(996\) 0 0
\(997\) 17.7152 0.561046 0.280523 0.959847i \(-0.409492\pi\)
0.280523 + 0.959847i \(0.409492\pi\)
\(998\) 0 0
\(999\) 9.12908 0.288831
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 3344.2.a.s.1.3 4
4.3 odd 2 1672.2.a.f.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.f.1.2 4 4.3 odd 2
3344.2.a.s.1.3 4 1.1 even 1 trivial