Properties

Label 7440.2.a.bs.1.3
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
Analytic rank $1$
Dimension $3$
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] = [7440,2,Mod(1,7440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7440.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.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: \(59.4086991038\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 7440.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -0.485863 q^{7} +1.00000 q^{9} -5.02827 q^{11} +3.51414 q^{13} -1.00000 q^{15} -1.32088 q^{17} +6.64177 q^{19} -0.485863 q^{21} -0.292611 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.86330 q^{29} -1.00000 q^{31} -5.02827 q^{33} +0.485863 q^{35} +5.51414 q^{37} +3.51414 q^{39} -7.02827 q^{41} +1.02827 q^{43} -1.00000 q^{45} +6.93438 q^{47} -6.76394 q^{49} -1.32088 q^{51} -1.70739 q^{53} +5.02827 q^{55} +6.64177 q^{57} +2.19325 q^{59} -2.00000 q^{61} -0.485863 q^{63} -3.51414 q^{65} -9.12763 q^{67} -0.292611 q^{69} -13.4768 q^{71} +12.5424 q^{73} +1.00000 q^{75} +2.44305 q^{77} +0.349158 q^{79} +1.00000 q^{81} -10.9344 q^{83} +1.32088 q^{85} -9.86330 q^{87} +5.03374 q^{89} -1.70739 q^{91} -1.00000 q^{93} -6.64177 q^{95} +10.4431 q^{97} -5.02827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.485863 −0.183639 −0.0918195 0.995776i \(-0.529268\pi\)
−0.0918195 + 0.995776i \(0.529268\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.02827 −1.51608 −0.758041 0.652207i \(-0.773843\pi\)
−0.758041 + 0.652207i \(0.773843\pi\)
\(12\) 0 0
\(13\) 3.51414 0.974646 0.487323 0.873222i \(-0.337973\pi\)
0.487323 + 0.873222i \(0.337973\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.32088 −0.320362 −0.160181 0.987088i \(-0.551208\pi\)
−0.160181 + 0.987088i \(0.551208\pi\)
\(18\) 0 0
\(19\) 6.64177 1.52373 0.761863 0.647738i \(-0.224285\pi\)
0.761863 + 0.647738i \(0.224285\pi\)
\(20\) 0 0
\(21\) −0.485863 −0.106024
\(22\) 0 0
\(23\) −0.292611 −0.0610135 −0.0305068 0.999535i \(-0.509712\pi\)
−0.0305068 + 0.999535i \(0.509712\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.86330 −1.83157 −0.915784 0.401671i \(-0.868429\pi\)
−0.915784 + 0.401671i \(0.868429\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −5.02827 −0.875310
\(34\) 0 0
\(35\) 0.485863 0.0821258
\(36\) 0 0
\(37\) 5.51414 0.906519 0.453259 0.891379i \(-0.350261\pi\)
0.453259 + 0.891379i \(0.350261\pi\)
\(38\) 0 0
\(39\) 3.51414 0.562712
\(40\) 0 0
\(41\) −7.02827 −1.09763 −0.548816 0.835943i \(-0.684921\pi\)
−0.548816 + 0.835943i \(0.684921\pi\)
\(42\) 0 0
\(43\) 1.02827 0.156810 0.0784051 0.996922i \(-0.475017\pi\)
0.0784051 + 0.996922i \(0.475017\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.93438 1.01148 0.505742 0.862685i \(-0.331219\pi\)
0.505742 + 0.862685i \(0.331219\pi\)
\(48\) 0 0
\(49\) −6.76394 −0.966277
\(50\) 0 0
\(51\) −1.32088 −0.184961
\(52\) 0 0
\(53\) −1.70739 −0.234528 −0.117264 0.993101i \(-0.537412\pi\)
−0.117264 + 0.993101i \(0.537412\pi\)
\(54\) 0 0
\(55\) 5.02827 0.678012
\(56\) 0 0
\(57\) 6.64177 0.879724
\(58\) 0 0
\(59\) 2.19325 0.285537 0.142769 0.989756i \(-0.454400\pi\)
0.142769 + 0.989756i \(0.454400\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −0.485863 −0.0612130
\(64\) 0 0
\(65\) −3.51414 −0.435875
\(66\) 0 0
\(67\) −9.12763 −1.11512 −0.557559 0.830137i \(-0.688262\pi\)
−0.557559 + 0.830137i \(0.688262\pi\)
\(68\) 0 0
\(69\) −0.292611 −0.0352262
\(70\) 0 0
\(71\) −13.4768 −1.59940 −0.799700 0.600399i \(-0.795008\pi\)
−0.799700 + 0.600399i \(0.795008\pi\)
\(72\) 0 0
\(73\) 12.5424 1.46798 0.733989 0.679161i \(-0.237656\pi\)
0.733989 + 0.679161i \(0.237656\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 2.44305 0.278412
\(78\) 0 0
\(79\) 0.349158 0.0392834 0.0196417 0.999807i \(-0.493747\pi\)
0.0196417 + 0.999807i \(0.493747\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.9344 −1.20020 −0.600102 0.799923i \(-0.704873\pi\)
−0.600102 + 0.799923i \(0.704873\pi\)
\(84\) 0 0
\(85\) 1.32088 0.143270
\(86\) 0 0
\(87\) −9.86330 −1.05746
\(88\) 0 0
\(89\) 5.03374 0.533575 0.266788 0.963755i \(-0.414038\pi\)
0.266788 + 0.963755i \(0.414038\pi\)
\(90\) 0 0
\(91\) −1.70739 −0.178983
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −6.64177 −0.681431
\(96\) 0 0
\(97\) 10.4431 1.06033 0.530166 0.847894i \(-0.322130\pi\)
0.530166 + 0.847894i \(0.322130\pi\)
\(98\) 0 0
\(99\) −5.02827 −0.505361
\(100\) 0 0
\(101\) −11.6135 −1.15559 −0.577793 0.816183i \(-0.696086\pi\)
−0.577793 + 0.816183i \(0.696086\pi\)
\(102\) 0 0
\(103\) 3.76940 0.371410 0.185705 0.982606i \(-0.440543\pi\)
0.185705 + 0.982606i \(0.440543\pi\)
\(104\) 0 0
\(105\) 0.485863 0.0474154
\(106\) 0 0
\(107\) 6.73566 0.651161 0.325581 0.945514i \(-0.394440\pi\)
0.325581 + 0.945514i \(0.394440\pi\)
\(108\) 0 0
\(109\) −3.90611 −0.374137 −0.187069 0.982347i \(-0.559899\pi\)
−0.187069 + 0.982347i \(0.559899\pi\)
\(110\) 0 0
\(111\) 5.51414 0.523379
\(112\) 0 0
\(113\) −15.0848 −1.41906 −0.709530 0.704675i \(-0.751093\pi\)
−0.709530 + 0.704675i \(0.751093\pi\)
\(114\) 0 0
\(115\) 0.292611 0.0272861
\(116\) 0 0
\(117\) 3.51414 0.324882
\(118\) 0 0
\(119\) 0.641769 0.0588309
\(120\) 0 0
\(121\) 14.2835 1.29850
\(122\) 0 0
\(123\) −7.02827 −0.633718
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.25526 0.555065 0.277532 0.960716i \(-0.410483\pi\)
0.277532 + 0.960716i \(0.410483\pi\)
\(128\) 0 0
\(129\) 1.02827 0.0905345
\(130\) 0 0
\(131\) 22.1186 1.93251 0.966254 0.257592i \(-0.0829291\pi\)
0.966254 + 0.257592i \(0.0829291\pi\)
\(132\) 0 0
\(133\) −3.22699 −0.279816
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −14.6044 −1.24774 −0.623870 0.781528i \(-0.714441\pi\)
−0.623870 + 0.781528i \(0.714441\pi\)
\(138\) 0 0
\(139\) −12.8970 −1.09391 −0.546956 0.837161i \(-0.684214\pi\)
−0.546956 + 0.837161i \(0.684214\pi\)
\(140\) 0 0
\(141\) 6.93438 0.583980
\(142\) 0 0
\(143\) −17.6700 −1.47764
\(144\) 0 0
\(145\) 9.86330 0.819102
\(146\) 0 0
\(147\) −6.76394 −0.557880
\(148\) 0 0
\(149\) 16.3118 1.33632 0.668158 0.744020i \(-0.267083\pi\)
0.668158 + 0.744020i \(0.267083\pi\)
\(150\) 0 0
\(151\) −19.3774 −1.57691 −0.788457 0.615090i \(-0.789119\pi\)
−0.788457 + 0.615090i \(0.789119\pi\)
\(152\) 0 0
\(153\) −1.32088 −0.106787
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −3.80128 −0.303375 −0.151688 0.988428i \(-0.548471\pi\)
−0.151688 + 0.988428i \(0.548471\pi\)
\(158\) 0 0
\(159\) −1.70739 −0.135405
\(160\) 0 0
\(161\) 0.142169 0.0112045
\(162\) 0 0
\(163\) −12.5424 −0.982397 −0.491199 0.871048i \(-0.663441\pi\)
−0.491199 + 0.871048i \(0.663441\pi\)
\(164\) 0 0
\(165\) 5.02827 0.391451
\(166\) 0 0
\(167\) −22.2553 −1.72216 −0.861082 0.508466i \(-0.830213\pi\)
−0.861082 + 0.508466i \(0.830213\pi\)
\(168\) 0 0
\(169\) −0.650842 −0.0500647
\(170\) 0 0
\(171\) 6.64177 0.507909
\(172\) 0 0
\(173\) −10.9717 −0.834165 −0.417082 0.908869i \(-0.636947\pi\)
−0.417082 + 0.908869i \(0.636947\pi\)
\(174\) 0 0
\(175\) −0.485863 −0.0367278
\(176\) 0 0
\(177\) 2.19325 0.164855
\(178\) 0 0
\(179\) 15.9253 1.19031 0.595157 0.803610i \(-0.297090\pi\)
0.595157 + 0.803610i \(0.297090\pi\)
\(180\) 0 0
\(181\) −8.05655 −0.598838 −0.299419 0.954122i \(-0.596793\pi\)
−0.299419 + 0.954122i \(0.596793\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −5.51414 −0.405407
\(186\) 0 0
\(187\) 6.64177 0.485694
\(188\) 0 0
\(189\) −0.485863 −0.0353413
\(190\) 0 0
\(191\) −6.19325 −0.448128 −0.224064 0.974574i \(-0.571932\pi\)
−0.224064 + 0.974574i \(0.571932\pi\)
\(192\) 0 0
\(193\) −2.25526 −0.162337 −0.0811687 0.996700i \(-0.525865\pi\)
−0.0811687 + 0.996700i \(0.525865\pi\)
\(194\) 0 0
\(195\) −3.51414 −0.251653
\(196\) 0 0
\(197\) 9.32088 0.664086 0.332043 0.943264i \(-0.392262\pi\)
0.332043 + 0.943264i \(0.392262\pi\)
\(198\) 0 0
\(199\) −16.5479 −1.17305 −0.586524 0.809932i \(-0.699504\pi\)
−0.586524 + 0.809932i \(0.699504\pi\)
\(200\) 0 0
\(201\) −9.12763 −0.643814
\(202\) 0 0
\(203\) 4.79221 0.336347
\(204\) 0 0
\(205\) 7.02827 0.490876
\(206\) 0 0
\(207\) −0.292611 −0.0203378
\(208\) 0 0
\(209\) −33.3966 −2.31009
\(210\) 0 0
\(211\) −25.0101 −1.72177 −0.860884 0.508801i \(-0.830089\pi\)
−0.860884 + 0.508801i \(0.830089\pi\)
\(212\) 0 0
\(213\) −13.4768 −0.923414
\(214\) 0 0
\(215\) −1.02827 −0.0701277
\(216\) 0 0
\(217\) 0.485863 0.0329825
\(218\) 0 0
\(219\) 12.5424 0.847538
\(220\) 0 0
\(221\) −4.64177 −0.312239
\(222\) 0 0
\(223\) −19.2835 −1.29132 −0.645661 0.763625i \(-0.723418\pi\)
−0.645661 + 0.763625i \(0.723418\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −23.4340 −1.55537 −0.777684 0.628655i \(-0.783606\pi\)
−0.777684 + 0.628655i \(0.783606\pi\)
\(228\) 0 0
\(229\) −25.0283 −1.65391 −0.826957 0.562265i \(-0.809930\pi\)
−0.826957 + 0.562265i \(0.809930\pi\)
\(230\) 0 0
\(231\) 2.44305 0.160741
\(232\) 0 0
\(233\) 18.7175 1.22623 0.613113 0.789995i \(-0.289917\pi\)
0.613113 + 0.789995i \(0.289917\pi\)
\(234\) 0 0
\(235\) −6.93438 −0.452349
\(236\) 0 0
\(237\) 0.349158 0.0226803
\(238\) 0 0
\(239\) 3.86876 0.250249 0.125125 0.992141i \(-0.460067\pi\)
0.125125 + 0.992141i \(0.460067\pi\)
\(240\) 0 0
\(241\) −1.61350 −0.103934 −0.0519672 0.998649i \(-0.516549\pi\)
−0.0519672 + 0.998649i \(0.516549\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.76394 0.432132
\(246\) 0 0
\(247\) 23.3401 1.48509
\(248\) 0 0
\(249\) −10.9344 −0.692938
\(250\) 0 0
\(251\) 25.4713 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(252\) 0 0
\(253\) 1.47133 0.0925015
\(254\) 0 0
\(255\) 1.32088 0.0827170
\(256\) 0 0
\(257\) 18.0192 1.12401 0.562003 0.827135i \(-0.310031\pi\)
0.562003 + 0.827135i \(0.310031\pi\)
\(258\) 0 0
\(259\) −2.67912 −0.166472
\(260\) 0 0
\(261\) −9.86330 −0.610523
\(262\) 0 0
\(263\) −1.15951 −0.0714987 −0.0357494 0.999361i \(-0.511382\pi\)
−0.0357494 + 0.999361i \(0.511382\pi\)
\(264\) 0 0
\(265\) 1.70739 0.104884
\(266\) 0 0
\(267\) 5.03374 0.308060
\(268\) 0 0
\(269\) −7.80675 −0.475986 −0.237993 0.971267i \(-0.576489\pi\)
−0.237993 + 0.971267i \(0.576489\pi\)
\(270\) 0 0
\(271\) −17.7831 −1.08025 −0.540124 0.841585i \(-0.681623\pi\)
−0.540124 + 0.841585i \(0.681623\pi\)
\(272\) 0 0
\(273\) −1.70739 −0.103336
\(274\) 0 0
\(275\) −5.02827 −0.303216
\(276\) 0 0
\(277\) 25.9390 1.55853 0.779263 0.626697i \(-0.215594\pi\)
0.779263 + 0.626697i \(0.215594\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −13.0283 −0.777202 −0.388601 0.921406i \(-0.627041\pi\)
−0.388601 + 0.921406i \(0.627041\pi\)
\(282\) 0 0
\(283\) −15.9006 −0.945195 −0.472598 0.881278i \(-0.656684\pi\)
−0.472598 + 0.881278i \(0.656684\pi\)
\(284\) 0 0
\(285\) −6.64177 −0.393424
\(286\) 0 0
\(287\) 3.41478 0.201568
\(288\) 0 0
\(289\) −15.2553 −0.897368
\(290\) 0 0
\(291\) 10.4431 0.612183
\(292\) 0 0
\(293\) −29.4340 −1.71955 −0.859776 0.510672i \(-0.829397\pi\)
−0.859776 + 0.510672i \(0.829397\pi\)
\(294\) 0 0
\(295\) −2.19325 −0.127696
\(296\) 0 0
\(297\) −5.02827 −0.291770
\(298\) 0 0
\(299\) −1.02827 −0.0594666
\(300\) 0 0
\(301\) −0.499600 −0.0287965
\(302\) 0 0
\(303\) −11.6135 −0.667178
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −9.96812 −0.568911 −0.284455 0.958689i \(-0.591813\pi\)
−0.284455 + 0.958689i \(0.591813\pi\)
\(308\) 0 0
\(309\) 3.76940 0.214434
\(310\) 0 0
\(311\) −15.9945 −0.906967 −0.453483 0.891265i \(-0.649819\pi\)
−0.453483 + 0.891265i \(0.649819\pi\)
\(312\) 0 0
\(313\) −22.8542 −1.29180 −0.645899 0.763423i \(-0.723517\pi\)
−0.645899 + 0.763423i \(0.723517\pi\)
\(314\) 0 0
\(315\) 0.485863 0.0273753
\(316\) 0 0
\(317\) 14.6044 0.820266 0.410133 0.912026i \(-0.365482\pi\)
0.410133 + 0.912026i \(0.365482\pi\)
\(318\) 0 0
\(319\) 49.5953 2.77681
\(320\) 0 0
\(321\) 6.73566 0.375948
\(322\) 0 0
\(323\) −8.77301 −0.488143
\(324\) 0 0
\(325\) 3.51414 0.194929
\(326\) 0 0
\(327\) −3.90611 −0.216008
\(328\) 0 0
\(329\) −3.36916 −0.185748
\(330\) 0 0
\(331\) 8.16137 0.448589 0.224295 0.974521i \(-0.427992\pi\)
0.224295 + 0.974521i \(0.427992\pi\)
\(332\) 0 0
\(333\) 5.51414 0.302173
\(334\) 0 0
\(335\) 9.12763 0.498696
\(336\) 0 0
\(337\) −11.8825 −0.647281 −0.323640 0.946180i \(-0.604907\pi\)
−0.323640 + 0.946180i \(0.604907\pi\)
\(338\) 0 0
\(339\) −15.0848 −0.819295
\(340\) 0 0
\(341\) 5.02827 0.272296
\(342\) 0 0
\(343\) 6.68739 0.361085
\(344\) 0 0
\(345\) 0.292611 0.0157536
\(346\) 0 0
\(347\) 18.3684 0.986065 0.493033 0.870011i \(-0.335888\pi\)
0.493033 + 0.870011i \(0.335888\pi\)
\(348\) 0 0
\(349\) 11.2462 0.601995 0.300997 0.953625i \(-0.402680\pi\)
0.300997 + 0.953625i \(0.402680\pi\)
\(350\) 0 0
\(351\) 3.51414 0.187571
\(352\) 0 0
\(353\) 25.3593 1.34974 0.674869 0.737937i \(-0.264200\pi\)
0.674869 + 0.737937i \(0.264200\pi\)
\(354\) 0 0
\(355\) 13.4768 0.715274
\(356\) 0 0
\(357\) 0.641769 0.0339660
\(358\) 0 0
\(359\) −15.2890 −0.806923 −0.403461 0.914997i \(-0.632193\pi\)
−0.403461 + 0.914997i \(0.632193\pi\)
\(360\) 0 0
\(361\) 25.1131 1.32174
\(362\) 0 0
\(363\) 14.2835 0.749691
\(364\) 0 0
\(365\) −12.5424 −0.656500
\(366\) 0 0
\(367\) −6.05655 −0.316149 −0.158075 0.987427i \(-0.550529\pi\)
−0.158075 + 0.987427i \(0.550529\pi\)
\(368\) 0 0
\(369\) −7.02827 −0.365877
\(370\) 0 0
\(371\) 0.829557 0.0430685
\(372\) 0 0
\(373\) 8.06748 0.417718 0.208859 0.977946i \(-0.433025\pi\)
0.208859 + 0.977946i \(0.433025\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −34.6610 −1.78513
\(378\) 0 0
\(379\) 36.8114 1.89088 0.945438 0.325803i \(-0.105635\pi\)
0.945438 + 0.325803i \(0.105635\pi\)
\(380\) 0 0
\(381\) 6.25526 0.320467
\(382\) 0 0
\(383\) 3.63270 0.185622 0.0928111 0.995684i \(-0.470415\pi\)
0.0928111 + 0.995684i \(0.470415\pi\)
\(384\) 0 0
\(385\) −2.44305 −0.124509
\(386\) 0 0
\(387\) 1.02827 0.0522701
\(388\) 0 0
\(389\) 15.2781 0.774629 0.387315 0.921948i \(-0.373403\pi\)
0.387315 + 0.921948i \(0.373403\pi\)
\(390\) 0 0
\(391\) 0.386505 0.0195464
\(392\) 0 0
\(393\) 22.1186 1.11573
\(394\) 0 0
\(395\) −0.349158 −0.0175681
\(396\) 0 0
\(397\) −18.5852 −0.932766 −0.466383 0.884583i \(-0.654443\pi\)
−0.466383 + 0.884583i \(0.654443\pi\)
\(398\) 0 0
\(399\) −3.22699 −0.161552
\(400\) 0 0
\(401\) 6.19325 0.309276 0.154638 0.987971i \(-0.450579\pi\)
0.154638 + 0.987971i \(0.450579\pi\)
\(402\) 0 0
\(403\) −3.51414 −0.175052
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −27.7266 −1.37436
\(408\) 0 0
\(409\) −32.0565 −1.58509 −0.792547 0.609811i \(-0.791245\pi\)
−0.792547 + 0.609811i \(0.791245\pi\)
\(410\) 0 0
\(411\) −14.6044 −0.720383
\(412\) 0 0
\(413\) −1.06562 −0.0524357
\(414\) 0 0
\(415\) 10.9344 0.536748
\(416\) 0 0
\(417\) −12.8970 −0.631570
\(418\) 0 0
\(419\) −9.78860 −0.478205 −0.239102 0.970994i \(-0.576853\pi\)
−0.239102 + 0.970994i \(0.576853\pi\)
\(420\) 0 0
\(421\) −14.2745 −0.695695 −0.347847 0.937551i \(-0.613087\pi\)
−0.347847 + 0.937551i \(0.613087\pi\)
\(422\) 0 0
\(423\) 6.93438 0.337161
\(424\) 0 0
\(425\) −1.32088 −0.0640723
\(426\) 0 0
\(427\) 0.971726 0.0470251
\(428\) 0 0
\(429\) −17.6700 −0.853118
\(430\) 0 0
\(431\) 22.8350 1.09992 0.549962 0.835190i \(-0.314642\pi\)
0.549962 + 0.835190i \(0.314642\pi\)
\(432\) 0 0
\(433\) 13.8259 0.664433 0.332216 0.943203i \(-0.392204\pi\)
0.332216 + 0.943203i \(0.392204\pi\)
\(434\) 0 0
\(435\) 9.86330 0.472909
\(436\) 0 0
\(437\) −1.94345 −0.0929679
\(438\) 0 0
\(439\) −39.8506 −1.90197 −0.950983 0.309243i \(-0.899924\pi\)
−0.950983 + 0.309243i \(0.899924\pi\)
\(440\) 0 0
\(441\) −6.76394 −0.322092
\(442\) 0 0
\(443\) 10.1504 0.482262 0.241131 0.970493i \(-0.422482\pi\)
0.241131 + 0.970493i \(0.422482\pi\)
\(444\) 0 0
\(445\) −5.03374 −0.238622
\(446\) 0 0
\(447\) 16.3118 0.771522
\(448\) 0 0
\(449\) −3.75020 −0.176983 −0.0884914 0.996077i \(-0.528205\pi\)
−0.0884914 + 0.996077i \(0.528205\pi\)
\(450\) 0 0
\(451\) 35.3401 1.66410
\(452\) 0 0
\(453\) −19.3774 −0.910431
\(454\) 0 0
\(455\) 1.70739 0.0800436
\(456\) 0 0
\(457\) −1.70193 −0.0796127 −0.0398064 0.999207i \(-0.512674\pi\)
−0.0398064 + 0.999207i \(0.512674\pi\)
\(458\) 0 0
\(459\) −1.32088 −0.0616536
\(460\) 0 0
\(461\) 7.40931 0.345086 0.172543 0.985002i \(-0.444802\pi\)
0.172543 + 0.985002i \(0.444802\pi\)
\(462\) 0 0
\(463\) 30.3865 1.41218 0.706090 0.708122i \(-0.250457\pi\)
0.706090 + 0.708122i \(0.250457\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) −17.1040 −0.791480 −0.395740 0.918363i \(-0.629512\pi\)
−0.395740 + 0.918363i \(0.629512\pi\)
\(468\) 0 0
\(469\) 4.43478 0.204779
\(470\) 0 0
\(471\) −3.80128 −0.175154
\(472\) 0 0
\(473\) −5.17044 −0.237737
\(474\) 0 0
\(475\) 6.64177 0.304745
\(476\) 0 0
\(477\) −1.70739 −0.0781760
\(478\) 0 0
\(479\) −21.2890 −0.972719 −0.486360 0.873759i \(-0.661676\pi\)
−0.486360 + 0.873759i \(0.661676\pi\)
\(480\) 0 0
\(481\) 19.3774 0.883535
\(482\) 0 0
\(483\) 0.142169 0.00646890
\(484\) 0 0
\(485\) −10.4431 −0.474195
\(486\) 0 0
\(487\) −20.4996 −0.928926 −0.464463 0.885593i \(-0.653753\pi\)
−0.464463 + 0.885593i \(0.653753\pi\)
\(488\) 0 0
\(489\) −12.5424 −0.567187
\(490\) 0 0
\(491\) 31.0667 1.40202 0.701010 0.713152i \(-0.252733\pi\)
0.701010 + 0.713152i \(0.252733\pi\)
\(492\) 0 0
\(493\) 13.0283 0.586764
\(494\) 0 0
\(495\) 5.02827 0.226004
\(496\) 0 0
\(497\) 6.54787 0.293712
\(498\) 0 0
\(499\) −15.3401 −0.686717 −0.343358 0.939205i \(-0.611565\pi\)
−0.343358 + 0.939205i \(0.611565\pi\)
\(500\) 0 0
\(501\) −22.2553 −0.994292
\(502\) 0 0
\(503\) 14.9344 0.665891 0.332946 0.942946i \(-0.391957\pi\)
0.332946 + 0.942946i \(0.391957\pi\)
\(504\) 0 0
\(505\) 11.6135 0.516794
\(506\) 0 0
\(507\) −0.650842 −0.0289049
\(508\) 0 0
\(509\) −1.22153 −0.0541432 −0.0270716 0.999633i \(-0.508618\pi\)
−0.0270716 + 0.999633i \(0.508618\pi\)
\(510\) 0 0
\(511\) −6.09389 −0.269578
\(512\) 0 0
\(513\) 6.64177 0.293241
\(514\) 0 0
\(515\) −3.76940 −0.166100
\(516\) 0 0
\(517\) −34.8680 −1.53349
\(518\) 0 0
\(519\) −10.9717 −0.481605
\(520\) 0 0
\(521\) 4.57429 0.200403 0.100202 0.994967i \(-0.468051\pi\)
0.100202 + 0.994967i \(0.468051\pi\)
\(522\) 0 0
\(523\) 29.7831 1.30233 0.651163 0.758938i \(-0.274281\pi\)
0.651163 + 0.758938i \(0.274281\pi\)
\(524\) 0 0
\(525\) −0.485863 −0.0212048
\(526\) 0 0
\(527\) 1.32088 0.0575386
\(528\) 0 0
\(529\) −22.9144 −0.996277
\(530\) 0 0
\(531\) 2.19325 0.0951790
\(532\) 0 0
\(533\) −24.6983 −1.06980
\(534\) 0 0
\(535\) −6.73566 −0.291208
\(536\) 0 0
\(537\) 15.9253 0.687228
\(538\) 0 0
\(539\) 34.0109 1.46495
\(540\) 0 0
\(541\) −3.04748 −0.131021 −0.0655106 0.997852i \(-0.520868\pi\)
−0.0655106 + 0.997852i \(0.520868\pi\)
\(542\) 0 0
\(543\) −8.05655 −0.345740
\(544\) 0 0
\(545\) 3.90611 0.167319
\(546\) 0 0
\(547\) −9.82595 −0.420127 −0.210064 0.977688i \(-0.567367\pi\)
−0.210064 + 0.977688i \(0.567367\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −65.5097 −2.79081
\(552\) 0 0
\(553\) −0.169643 −0.00721396
\(554\) 0 0
\(555\) −5.51414 −0.234062
\(556\) 0 0
\(557\) −38.7922 −1.64368 −0.821839 0.569719i \(-0.807052\pi\)
−0.821839 + 0.569719i \(0.807052\pi\)
\(558\) 0 0
\(559\) 3.61350 0.152835
\(560\) 0 0
\(561\) 6.64177 0.280416
\(562\) 0 0
\(563\) −2.73566 −0.115294 −0.0576472 0.998337i \(-0.518360\pi\)
−0.0576472 + 0.998337i \(0.518360\pi\)
\(564\) 0 0
\(565\) 15.0848 0.634623
\(566\) 0 0
\(567\) −0.485863 −0.0204043
\(568\) 0 0
\(569\) 4.39197 0.184121 0.0920605 0.995753i \(-0.470655\pi\)
0.0920605 + 0.995753i \(0.470655\pi\)
\(570\) 0 0
\(571\) −9.67004 −0.404679 −0.202339 0.979315i \(-0.564854\pi\)
−0.202339 + 0.979315i \(0.564854\pi\)
\(572\) 0 0
\(573\) −6.19325 −0.258727
\(574\) 0 0
\(575\) −0.292611 −0.0122027
\(576\) 0 0
\(577\) 31.4148 1.30781 0.653907 0.756575i \(-0.273129\pi\)
0.653907 + 0.756575i \(0.273129\pi\)
\(578\) 0 0
\(579\) −2.25526 −0.0937256
\(580\) 0 0
\(581\) 5.31261 0.220404
\(582\) 0 0
\(583\) 8.58522 0.355564
\(584\) 0 0
\(585\) −3.51414 −0.145292
\(586\) 0 0
\(587\) 27.2161 1.12333 0.561664 0.827366i \(-0.310162\pi\)
0.561664 + 0.827366i \(0.310162\pi\)
\(588\) 0 0
\(589\) −6.64177 −0.273669
\(590\) 0 0
\(591\) 9.32088 0.383410
\(592\) 0 0
\(593\) 32.5561 1.33692 0.668460 0.743748i \(-0.266954\pi\)
0.668460 + 0.743748i \(0.266954\pi\)
\(594\) 0 0
\(595\) −0.641769 −0.0263100
\(596\) 0 0
\(597\) −16.5479 −0.677259
\(598\) 0 0
\(599\) −27.8067 −1.13615 −0.568076 0.822976i \(-0.692312\pi\)
−0.568076 + 0.822976i \(0.692312\pi\)
\(600\) 0 0
\(601\) 17.6519 0.720036 0.360018 0.932945i \(-0.382771\pi\)
0.360018 + 0.932945i \(0.382771\pi\)
\(602\) 0 0
\(603\) −9.12763 −0.371706
\(604\) 0 0
\(605\) −14.2835 −0.580708
\(606\) 0 0
\(607\) −44.9673 −1.82517 −0.912584 0.408890i \(-0.865916\pi\)
−0.912584 + 0.408890i \(0.865916\pi\)
\(608\) 0 0
\(609\) 4.79221 0.194190
\(610\) 0 0
\(611\) 24.3684 0.985838
\(612\) 0 0
\(613\) 33.7694 1.36393 0.681967 0.731383i \(-0.261125\pi\)
0.681967 + 0.731383i \(0.261125\pi\)
\(614\) 0 0
\(615\) 7.02827 0.283407
\(616\) 0 0
\(617\) −26.1131 −1.05127 −0.525637 0.850709i \(-0.676173\pi\)
−0.525637 + 0.850709i \(0.676173\pi\)
\(618\) 0 0
\(619\) 20.8597 0.838422 0.419211 0.907889i \(-0.362307\pi\)
0.419211 + 0.907889i \(0.362307\pi\)
\(620\) 0 0
\(621\) −0.292611 −0.0117421
\(622\) 0 0
\(623\) −2.44571 −0.0979852
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −33.3966 −1.33373
\(628\) 0 0
\(629\) −7.28354 −0.290414
\(630\) 0 0
\(631\) 20.5105 0.816511 0.408256 0.912868i \(-0.366137\pi\)
0.408256 + 0.912868i \(0.366137\pi\)
\(632\) 0 0
\(633\) −25.0101 −0.994063
\(634\) 0 0
\(635\) −6.25526 −0.248233
\(636\) 0 0
\(637\) −23.7694 −0.941778
\(638\) 0 0
\(639\) −13.4768 −0.533134
\(640\) 0 0
\(641\) 23.9945 0.947727 0.473864 0.880598i \(-0.342859\pi\)
0.473864 + 0.880598i \(0.342859\pi\)
\(642\) 0 0
\(643\) 13.6700 0.539094 0.269547 0.962987i \(-0.413126\pi\)
0.269547 + 0.962987i \(0.413126\pi\)
\(644\) 0 0
\(645\) −1.02827 −0.0404882
\(646\) 0 0
\(647\) 11.1523 0.438442 0.219221 0.975675i \(-0.429648\pi\)
0.219221 + 0.975675i \(0.429648\pi\)
\(648\) 0 0
\(649\) −11.0283 −0.432898
\(650\) 0 0
\(651\) 0.485863 0.0190425
\(652\) 0 0
\(653\) −39.1896 −1.53361 −0.766805 0.641881i \(-0.778154\pi\)
−0.766805 + 0.641881i \(0.778154\pi\)
\(654\) 0 0
\(655\) −22.1186 −0.864244
\(656\) 0 0
\(657\) 12.5424 0.489326
\(658\) 0 0
\(659\) 31.6464 1.23277 0.616385 0.787445i \(-0.288597\pi\)
0.616385 + 0.787445i \(0.288597\pi\)
\(660\) 0 0
\(661\) −3.59535 −0.139843 −0.0699215 0.997553i \(-0.522275\pi\)
−0.0699215 + 0.997553i \(0.522275\pi\)
\(662\) 0 0
\(663\) −4.64177 −0.180271
\(664\) 0 0
\(665\) 3.22699 0.125137
\(666\) 0 0
\(667\) 2.88611 0.111750
\(668\) 0 0
\(669\) −19.2835 −0.745545
\(670\) 0 0
\(671\) 10.0565 0.388229
\(672\) 0 0
\(673\) −7.24073 −0.279110 −0.139555 0.990214i \(-0.544567\pi\)
−0.139555 + 0.990214i \(0.544567\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 38.5188 1.48040 0.740199 0.672388i \(-0.234731\pi\)
0.740199 + 0.672388i \(0.234731\pi\)
\(678\) 0 0
\(679\) −5.07389 −0.194718
\(680\) 0 0
\(681\) −23.4340 −0.897992
\(682\) 0 0
\(683\) −40.0192 −1.53129 −0.765646 0.643262i \(-0.777581\pi\)
−0.765646 + 0.643262i \(0.777581\pi\)
\(684\) 0 0
\(685\) 14.6044 0.558006
\(686\) 0 0
\(687\) −25.0283 −0.954888
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 11.8013 0.448942 0.224471 0.974481i \(-0.427935\pi\)
0.224471 + 0.974481i \(0.427935\pi\)
\(692\) 0 0
\(693\) 2.44305 0.0928039
\(694\) 0 0
\(695\) 12.8970 0.489212
\(696\) 0 0
\(697\) 9.28354 0.351639
\(698\) 0 0
\(699\) 18.7175 0.707962
\(700\) 0 0
\(701\) 5.17044 0.195285 0.0976425 0.995222i \(-0.468870\pi\)
0.0976425 + 0.995222i \(0.468870\pi\)
\(702\) 0 0
\(703\) 36.6236 1.38129
\(704\) 0 0
\(705\) −6.93438 −0.261164
\(706\) 0 0
\(707\) 5.64257 0.212211
\(708\) 0 0
\(709\) 47.7722 1.79412 0.897062 0.441906i \(-0.145697\pi\)
0.897062 + 0.441906i \(0.145697\pi\)
\(710\) 0 0
\(711\) 0.349158 0.0130945
\(712\) 0 0
\(713\) 0.292611 0.0109584
\(714\) 0 0
\(715\) 17.6700 0.660822
\(716\) 0 0
\(717\) 3.86876 0.144481
\(718\) 0 0
\(719\) −23.0848 −0.860919 −0.430459 0.902610i \(-0.641648\pi\)
−0.430459 + 0.902610i \(0.641648\pi\)
\(720\) 0 0
\(721\) −1.83141 −0.0682054
\(722\) 0 0
\(723\) −1.61350 −0.0600065
\(724\) 0 0
\(725\) −9.86330 −0.366314
\(726\) 0 0
\(727\) 22.2125 0.823814 0.411907 0.911226i \(-0.364863\pi\)
0.411907 + 0.911226i \(0.364863\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.35823 −0.0502360
\(732\) 0 0
\(733\) −43.9072 −1.62175 −0.810874 0.585221i \(-0.801008\pi\)
−0.810874 + 0.585221i \(0.801008\pi\)
\(734\) 0 0
\(735\) 6.76394 0.249492
\(736\) 0 0
\(737\) 45.8962 1.69061
\(738\) 0 0
\(739\) −6.86690 −0.252603 −0.126302 0.991992i \(-0.540311\pi\)
−0.126302 + 0.991992i \(0.540311\pi\)
\(740\) 0 0
\(741\) 23.3401 0.857419
\(742\) 0 0
\(743\) −29.2726 −1.07391 −0.536954 0.843612i \(-0.680425\pi\)
−0.536954 + 0.843612i \(0.680425\pi\)
\(744\) 0 0
\(745\) −16.3118 −0.597619
\(746\) 0 0
\(747\) −10.9344 −0.400068
\(748\) 0 0
\(749\) −3.27261 −0.119579
\(750\) 0 0
\(751\) −9.67004 −0.352865 −0.176432 0.984313i \(-0.556456\pi\)
−0.176432 + 0.984313i \(0.556456\pi\)
\(752\) 0 0
\(753\) 25.4713 0.928227
\(754\) 0 0
\(755\) 19.3774 0.705217
\(756\) 0 0
\(757\) −49.4104 −1.79585 −0.897925 0.440148i \(-0.854926\pi\)
−0.897925 + 0.440148i \(0.854926\pi\)
\(758\) 0 0
\(759\) 1.47133 0.0534058
\(760\) 0 0
\(761\) 47.2599 1.71317 0.856586 0.516005i \(-0.172581\pi\)
0.856586 + 0.516005i \(0.172581\pi\)
\(762\) 0 0
\(763\) 1.89783 0.0687062
\(764\) 0 0
\(765\) 1.32088 0.0477567
\(766\) 0 0
\(767\) 7.70739 0.278298
\(768\) 0 0
\(769\) 9.49053 0.342237 0.171119 0.985250i \(-0.445262\pi\)
0.171119 + 0.985250i \(0.445262\pi\)
\(770\) 0 0
\(771\) 18.0192 0.648946
\(772\) 0 0
\(773\) 41.7567 1.50188 0.750942 0.660368i \(-0.229600\pi\)
0.750942 + 0.660368i \(0.229600\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) −2.67912 −0.0961127
\(778\) 0 0
\(779\) −46.6802 −1.67249
\(780\) 0 0
\(781\) 67.7650 2.42482
\(782\) 0 0
\(783\) −9.86330 −0.352485
\(784\) 0 0
\(785\) 3.80128 0.135674
\(786\) 0 0
\(787\) 40.0950 1.42923 0.714615 0.699518i \(-0.246602\pi\)
0.714615 + 0.699518i \(0.246602\pi\)
\(788\) 0 0
\(789\) −1.15951 −0.0412798
\(790\) 0 0
\(791\) 7.32916 0.260595
\(792\) 0 0
\(793\) −7.02827 −0.249581
\(794\) 0 0
\(795\) 1.70739 0.0605549
\(796\) 0 0
\(797\) −31.1150 −1.10215 −0.551074 0.834456i \(-0.685782\pi\)
−0.551074 + 0.834456i \(0.685782\pi\)
\(798\) 0 0
\(799\) −9.15951 −0.324040
\(800\) 0 0
\(801\) 5.03374 0.177858
\(802\) 0 0
\(803\) −63.0667 −2.22557
\(804\) 0 0
\(805\) −0.142169 −0.00501079
\(806\) 0 0
\(807\) −7.80675 −0.274811
\(808\) 0 0
\(809\) 46.6291 1.63939 0.819696 0.572799i \(-0.194143\pi\)
0.819696 + 0.572799i \(0.194143\pi\)
\(810\) 0 0
\(811\) 47.0283 1.65139 0.825693 0.564120i \(-0.190784\pi\)
0.825693 + 0.564120i \(0.190784\pi\)
\(812\) 0 0
\(813\) −17.7831 −0.623682
\(814\) 0 0
\(815\) 12.5424 0.439341
\(816\) 0 0
\(817\) 6.82956 0.238936
\(818\) 0 0
\(819\) −1.70739 −0.0596610
\(820\) 0 0
\(821\) 35.3912 1.23516 0.617580 0.786508i \(-0.288113\pi\)
0.617580 + 0.786508i \(0.288113\pi\)
\(822\) 0 0
\(823\) 17.2161 0.600114 0.300057 0.953921i \(-0.402994\pi\)
0.300057 + 0.953921i \(0.402994\pi\)
\(824\) 0 0
\(825\) −5.02827 −0.175062
\(826\) 0 0
\(827\) −16.3310 −0.567885 −0.283942 0.958841i \(-0.591642\pi\)
−0.283942 + 0.958841i \(0.591642\pi\)
\(828\) 0 0
\(829\) −29.0667 −1.00953 −0.504764 0.863258i \(-0.668420\pi\)
−0.504764 + 0.863258i \(0.668420\pi\)
\(830\) 0 0
\(831\) 25.9390 0.899815
\(832\) 0 0
\(833\) 8.93438 0.309558
\(834\) 0 0
\(835\) 22.2553 0.770175
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −27.6646 −0.955087 −0.477544 0.878608i \(-0.658473\pi\)
−0.477544 + 0.878608i \(0.658473\pi\)
\(840\) 0 0
\(841\) 68.2846 2.35464
\(842\) 0 0
\(843\) −13.0283 −0.448718
\(844\) 0 0
\(845\) 0.650842 0.0223896
\(846\) 0 0
\(847\) −6.93984 −0.238456
\(848\) 0 0
\(849\) −15.9006 −0.545709
\(850\) 0 0
\(851\) −1.61350 −0.0553099
\(852\) 0 0
\(853\) 32.5105 1.11314 0.556570 0.830801i \(-0.312117\pi\)
0.556570 + 0.830801i \(0.312117\pi\)
\(854\) 0 0
\(855\) −6.64177 −0.227144
\(856\) 0 0
\(857\) −45.9144 −1.56841 −0.784203 0.620505i \(-0.786928\pi\)
−0.784203 + 0.620505i \(0.786928\pi\)
\(858\) 0 0
\(859\) 19.6026 0.668831 0.334415 0.942426i \(-0.391461\pi\)
0.334415 + 0.942426i \(0.391461\pi\)
\(860\) 0 0
\(861\) 3.41478 0.116375
\(862\) 0 0
\(863\) −46.1696 −1.57163 −0.785816 0.618460i \(-0.787757\pi\)
−0.785816 + 0.618460i \(0.787757\pi\)
\(864\) 0 0
\(865\) 10.9717 0.373050
\(866\) 0 0
\(867\) −15.2553 −0.518096
\(868\) 0 0
\(869\) −1.75566 −0.0595568
\(870\) 0 0
\(871\) −32.0757 −1.08685
\(872\) 0 0
\(873\) 10.4431 0.353444
\(874\) 0 0
\(875\) 0.485863 0.0164252
\(876\) 0 0
\(877\) 11.3017 0.381631 0.190815 0.981626i \(-0.438887\pi\)
0.190815 + 0.981626i \(0.438887\pi\)
\(878\) 0 0
\(879\) −29.4340 −0.992784
\(880\) 0 0
\(881\) −9.60803 −0.323703 −0.161851 0.986815i \(-0.551747\pi\)
−0.161851 + 0.986815i \(0.551747\pi\)
\(882\) 0 0
\(883\) 33.5279 1.12830 0.564151 0.825671i \(-0.309203\pi\)
0.564151 + 0.825671i \(0.309203\pi\)
\(884\) 0 0
\(885\) −2.19325 −0.0737254
\(886\) 0 0
\(887\) 9.76394 0.327841 0.163920 0.986474i \(-0.447586\pi\)
0.163920 + 0.986474i \(0.447586\pi\)
\(888\) 0 0
\(889\) −3.03920 −0.101932
\(890\) 0 0
\(891\) −5.02827 −0.168454
\(892\) 0 0
\(893\) 46.0565 1.54122
\(894\) 0 0
\(895\) −15.9253 −0.532324
\(896\) 0 0
\(897\) −1.02827 −0.0343331
\(898\) 0 0
\(899\) 9.86330 0.328959
\(900\) 0 0
\(901\) 2.25526 0.0751337
\(902\) 0 0
\(903\) −0.499600 −0.0166257
\(904\) 0 0
\(905\) 8.05655 0.267809
\(906\) 0 0
\(907\) 9.18418 0.304956 0.152478 0.988307i \(-0.451275\pi\)
0.152478 + 0.988307i \(0.451275\pi\)
\(908\) 0 0
\(909\) −11.6135 −0.385195
\(910\) 0 0
\(911\) 12.2553 0.406035 0.203018 0.979175i \(-0.434925\pi\)
0.203018 + 0.979175i \(0.434925\pi\)
\(912\) 0 0
\(913\) 54.9811 1.81961
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) −10.7466 −0.354884
\(918\) 0 0
\(919\) 47.5663 1.56907 0.784533 0.620087i \(-0.212903\pi\)
0.784533 + 0.620087i \(0.212903\pi\)
\(920\) 0 0
\(921\) −9.96812 −0.328461
\(922\) 0 0
\(923\) −47.3593 −1.55885
\(924\) 0 0
\(925\) 5.51414 0.181304
\(926\) 0 0
\(927\) 3.76940 0.123803
\(928\) 0 0
\(929\) 23.5333 0.772104 0.386052 0.922477i \(-0.373839\pi\)
0.386052 + 0.922477i \(0.373839\pi\)
\(930\) 0 0
\(931\) −44.9245 −1.47234
\(932\) 0 0
\(933\) −15.9945 −0.523638
\(934\) 0 0
\(935\) −6.64177 −0.217209
\(936\) 0 0
\(937\) −56.7258 −1.85315 −0.926575 0.376109i \(-0.877262\pi\)
−0.926575 + 0.376109i \(0.877262\pi\)
\(938\) 0 0
\(939\) −22.8542 −0.745819
\(940\) 0 0
\(941\) −33.5333 −1.09316 −0.546578 0.837408i \(-0.684070\pi\)
−0.546578 + 0.837408i \(0.684070\pi\)
\(942\) 0 0
\(943\) 2.05655 0.0669704
\(944\) 0 0
\(945\) 0.485863 0.0158051
\(946\) 0 0
\(947\) 52.0685 1.69200 0.846000 0.533183i \(-0.179004\pi\)
0.846000 + 0.533183i \(0.179004\pi\)
\(948\) 0 0
\(949\) 44.0757 1.43076
\(950\) 0 0
\(951\) 14.6044 0.473581
\(952\) 0 0
\(953\) −11.1896 −0.362468 −0.181234 0.983440i \(-0.558009\pi\)
−0.181234 + 0.983440i \(0.558009\pi\)
\(954\) 0 0
\(955\) 6.19325 0.200409
\(956\) 0 0
\(957\) 49.5953 1.60319
\(958\) 0 0
\(959\) 7.09575 0.229134
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 6.73566 0.217054
\(964\) 0 0
\(965\) 2.25526 0.0725995
\(966\) 0 0
\(967\) −36.8296 −1.18436 −0.592179 0.805806i \(-0.701732\pi\)
−0.592179 + 0.805806i \(0.701732\pi\)
\(968\) 0 0
\(969\) −8.77301 −0.281830
\(970\) 0 0
\(971\) −18.7494 −0.601697 −0.300848 0.953672i \(-0.597270\pi\)
−0.300848 + 0.953672i \(0.597270\pi\)
\(972\) 0 0
\(973\) 6.26619 0.200885
\(974\) 0 0
\(975\) 3.51414 0.112542
\(976\) 0 0
\(977\) −50.2262 −1.60688 −0.803439 0.595387i \(-0.796999\pi\)
−0.803439 + 0.595387i \(0.796999\pi\)
\(978\) 0 0
\(979\) −25.3110 −0.808943
\(980\) 0 0
\(981\) −3.90611 −0.124712
\(982\) 0 0
\(983\) −15.5279 −0.495262 −0.247631 0.968854i \(-0.579652\pi\)
−0.247631 + 0.968854i \(0.579652\pi\)
\(984\) 0 0
\(985\) −9.32088 −0.296988
\(986\) 0 0
\(987\) −3.36916 −0.107242
\(988\) 0 0
\(989\) −0.300884 −0.00956755
\(990\) 0 0
\(991\) 20.1022 0.638566 0.319283 0.947659i \(-0.396558\pi\)
0.319283 + 0.947659i \(0.396558\pi\)
\(992\) 0 0
\(993\) 8.16137 0.258993
\(994\) 0 0
\(995\) 16.5479 0.524603
\(996\) 0 0
\(997\) −20.8186 −0.659333 −0.329666 0.944098i \(-0.606936\pi\)
−0.329666 + 0.944098i \(0.606936\pi\)
\(998\) 0 0
\(999\) 5.51414 0.174460
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 7440.2.a.bs.1.3 3
4.3 odd 2 465.2.a.e.1.3 3
12.11 even 2 1395.2.a.j.1.1 3
20.3 even 4 2325.2.c.k.1024.1 6
20.7 even 4 2325.2.c.k.1024.6 6
20.19 odd 2 2325.2.a.r.1.1 3
60.59 even 2 6975.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.e.1.3 3 4.3 odd 2
1395.2.a.j.1.1 3 12.11 even 2
2325.2.a.r.1.1 3 20.19 odd 2
2325.2.c.k.1024.1 6 20.3 even 4
2325.2.c.k.1024.6 6 20.7 even 4
6975.2.a.bf.1.3 3 60.59 even 2
7440.2.a.bs.1.3 3 1.1 even 1 trivial