Properties

Label 7440.2.a.bs.1.1
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 1.1
Root \(-2.08613\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -5.08613 q^{7} +1.00000 q^{9} +4.17226 q^{11} -1.08613 q^{13} -1.00000 q^{15} +0.648061 q^{17} +2.70388 q^{19} -5.08613 q^{21} -7.52420 q^{23} +1.00000 q^{25} +1.00000 q^{27} +5.90645 q^{29} -1.00000 q^{31} +4.17226 q^{33} +5.08613 q^{35} +0.913870 q^{37} -1.08613 q^{39} +2.17226 q^{41} -8.17226 q^{43} -1.00000 q^{45} +10.2281 q^{47} +18.8687 q^{49} +0.648061 q^{51} +5.52420 q^{53} -4.17226 q^{55} +2.70388 q^{57} -0.438069 q^{59} -2.00000 q^{61} -5.08613 q^{63} +1.08613 q^{65} -9.79001 q^{67} -7.52420 q^{69} -2.96969 q^{71} -1.25839 q^{73} +1.00000 q^{75} -21.2207 q^{77} -10.8203 q^{79} +1.00000 q^{81} -14.2281 q^{83} -0.648061 q^{85} +5.90645 q^{87} +18.1903 q^{89} +5.52420 q^{91} -1.00000 q^{93} -2.70388 q^{95} -13.2207 q^{97} +4.17226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(7\) −5.08613 −1.92238 −0.961188 0.275893i \(-0.911026\pi\)
−0.961188 + 0.275893i \(0.911026\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.17226 1.25798 0.628992 0.777412i \(-0.283468\pi\)
0.628992 + 0.777412i \(0.283468\pi\)
\(12\) 0 0
\(13\) −1.08613 −0.301238 −0.150619 0.988592i \(-0.548127\pi\)
−0.150619 + 0.988592i \(0.548127\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0.648061 0.157178 0.0785889 0.996907i \(-0.474959\pi\)
0.0785889 + 0.996907i \(0.474959\pi\)
\(18\) 0 0
\(19\) 2.70388 0.620312 0.310156 0.950686i \(-0.399619\pi\)
0.310156 + 0.950686i \(0.399619\pi\)
\(20\) 0 0
\(21\) −5.08613 −1.10988
\(22\) 0 0
\(23\) −7.52420 −1.56890 −0.784452 0.620189i \(-0.787056\pi\)
−0.784452 + 0.620189i \(0.787056\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.90645 1.09680 0.548400 0.836216i \(-0.315237\pi\)
0.548400 + 0.836216i \(0.315237\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 4.17226 0.726297
\(34\) 0 0
\(35\) 5.08613 0.859713
\(36\) 0 0
\(37\) 0.913870 0.150239 0.0751196 0.997175i \(-0.476066\pi\)
0.0751196 + 0.997175i \(0.476066\pi\)
\(38\) 0 0
\(39\) −1.08613 −0.173920
\(40\) 0 0
\(41\) 2.17226 0.339250 0.169625 0.985509i \(-0.445744\pi\)
0.169625 + 0.985509i \(0.445744\pi\)
\(42\) 0 0
\(43\) −8.17226 −1.24626 −0.623129 0.782119i \(-0.714139\pi\)
−0.623129 + 0.782119i \(0.714139\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 10.2281 1.49192 0.745959 0.665992i \(-0.231991\pi\)
0.745959 + 0.665992i \(0.231991\pi\)
\(48\) 0 0
\(49\) 18.8687 2.69553
\(50\) 0 0
\(51\) 0.648061 0.0907467
\(52\) 0 0
\(53\) 5.52420 0.758807 0.379404 0.925231i \(-0.376129\pi\)
0.379404 + 0.925231i \(0.376129\pi\)
\(54\) 0 0
\(55\) −4.17226 −0.562587
\(56\) 0 0
\(57\) 2.70388 0.358137
\(58\) 0 0
\(59\) −0.438069 −0.0570318 −0.0285159 0.999593i \(-0.509078\pi\)
−0.0285159 + 0.999593i \(0.509078\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −5.08613 −0.640792
\(64\) 0 0
\(65\) 1.08613 0.134718
\(66\) 0 0
\(67\) −9.79001 −1.19604 −0.598020 0.801481i \(-0.704046\pi\)
−0.598020 + 0.801481i \(0.704046\pi\)
\(68\) 0 0
\(69\) −7.52420 −0.905807
\(70\) 0 0
\(71\) −2.96969 −0.352437 −0.176219 0.984351i \(-0.556387\pi\)
−0.176219 + 0.984351i \(0.556387\pi\)
\(72\) 0 0
\(73\) −1.25839 −0.147283 −0.0736417 0.997285i \(-0.523462\pi\)
−0.0736417 + 0.997285i \(0.523462\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −21.2207 −2.41832
\(78\) 0 0
\(79\) −10.8203 −1.21738 −0.608691 0.793408i \(-0.708305\pi\)
−0.608691 + 0.793408i \(0.708305\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.2281 −1.56173 −0.780867 0.624697i \(-0.785222\pi\)
−0.780867 + 0.624697i \(0.785222\pi\)
\(84\) 0 0
\(85\) −0.648061 −0.0702921
\(86\) 0 0
\(87\) 5.90645 0.633238
\(88\) 0 0
\(89\) 18.1903 1.92817 0.964086 0.265589i \(-0.0855663\pi\)
0.964086 + 0.265589i \(0.0855663\pi\)
\(90\) 0 0
\(91\) 5.52420 0.579093
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −2.70388 −0.277412
\(96\) 0 0
\(97\) −13.2207 −1.34235 −0.671177 0.741297i \(-0.734211\pi\)
−0.671177 + 0.741297i \(0.734211\pi\)
\(98\) 0 0
\(99\) 4.17226 0.419328
\(100\) 0 0
\(101\) −16.8761 −1.67924 −0.839619 0.543175i \(-0.817222\pi\)
−0.839619 + 0.543175i \(0.817222\pi\)
\(102\) 0 0
\(103\) 0.493887 0.0486641 0.0243321 0.999704i \(-0.492254\pi\)
0.0243321 + 0.999704i \(0.492254\pi\)
\(104\) 0 0
\(105\) 5.08613 0.496355
\(106\) 0 0
\(107\) −9.69646 −0.937392 −0.468696 0.883359i \(-0.655276\pi\)
−0.468696 + 0.883359i \(0.655276\pi\)
\(108\) 0 0
\(109\) −16.4003 −1.57087 −0.785434 0.618946i \(-0.787560\pi\)
−0.785434 + 0.618946i \(0.787560\pi\)
\(110\) 0 0
\(111\) 0.913870 0.0867407
\(112\) 0 0
\(113\) 12.5168 1.17748 0.588740 0.808323i \(-0.299624\pi\)
0.588740 + 0.808323i \(0.299624\pi\)
\(114\) 0 0
\(115\) 7.52420 0.701635
\(116\) 0 0
\(117\) −1.08613 −0.100413
\(118\) 0 0
\(119\) −3.29612 −0.302155
\(120\) 0 0
\(121\) 6.40776 0.582523
\(122\) 0 0
\(123\) 2.17226 0.195866
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 7.58002 0.672618 0.336309 0.941752i \(-0.390821\pi\)
0.336309 + 0.941752i \(0.390821\pi\)
\(128\) 0 0
\(129\) −8.17226 −0.719527
\(130\) 0 0
\(131\) 7.67357 0.670443 0.335221 0.942139i \(-0.391189\pi\)
0.335221 + 0.942139i \(0.391189\pi\)
\(132\) 0 0
\(133\) −13.7523 −1.19247
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −4.75970 −0.406648 −0.203324 0.979111i \(-0.565175\pi\)
−0.203324 + 0.979111i \(0.565175\pi\)
\(138\) 0 0
\(139\) −10.2839 −0.872269 −0.436134 0.899882i \(-0.643653\pi\)
−0.436134 + 0.899882i \(0.643653\pi\)
\(140\) 0 0
\(141\) 10.2281 0.861360
\(142\) 0 0
\(143\) −4.53162 −0.378953
\(144\) 0 0
\(145\) −5.90645 −0.490504
\(146\) 0 0
\(147\) 18.8687 1.55627
\(148\) 0 0
\(149\) −0.764504 −0.0626306 −0.0313153 0.999510i \(-0.509970\pi\)
−0.0313153 + 0.999510i \(0.509970\pi\)
\(150\) 0 0
\(151\) 0.992582 0.0807751 0.0403876 0.999184i \(-0.487141\pi\)
0.0403876 + 0.999184i \(0.487141\pi\)
\(152\) 0 0
\(153\) 0.648061 0.0523926
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 15.9245 1.27092 0.635458 0.772135i \(-0.280811\pi\)
0.635458 + 0.772135i \(0.280811\pi\)
\(158\) 0 0
\(159\) 5.52420 0.438098
\(160\) 0 0
\(161\) 38.2691 3.01602
\(162\) 0 0
\(163\) 1.25839 0.0985648 0.0492824 0.998785i \(-0.484307\pi\)
0.0492824 + 0.998785i \(0.484307\pi\)
\(164\) 0 0
\(165\) −4.17226 −0.324810
\(166\) 0 0
\(167\) −23.5800 −1.82468 −0.912338 0.409437i \(-0.865725\pi\)
−0.912338 + 0.409437i \(0.865725\pi\)
\(168\) 0 0
\(169\) −11.8203 −0.909255
\(170\) 0 0
\(171\) 2.70388 0.206771
\(172\) 0 0
\(173\) −20.1723 −1.53367 −0.766834 0.641845i \(-0.778169\pi\)
−0.766834 + 0.641845i \(0.778169\pi\)
\(174\) 0 0
\(175\) −5.08613 −0.384475
\(176\) 0 0
\(177\) −0.438069 −0.0329273
\(178\) 0 0
\(179\) 4.11164 0.307318 0.153659 0.988124i \(-0.450894\pi\)
0.153659 + 0.988124i \(0.450894\pi\)
\(180\) 0 0
\(181\) 10.3445 0.768902 0.384451 0.923145i \(-0.374391\pi\)
0.384451 + 0.923145i \(0.374391\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −0.913870 −0.0671890
\(186\) 0 0
\(187\) 2.70388 0.197727
\(188\) 0 0
\(189\) −5.08613 −0.369962
\(190\) 0 0
\(191\) −3.56193 −0.257732 −0.128866 0.991662i \(-0.541134\pi\)
−0.128866 + 0.991662i \(0.541134\pi\)
\(192\) 0 0
\(193\) −3.58002 −0.257695 −0.128848 0.991664i \(-0.541128\pi\)
−0.128848 + 0.991664i \(0.541128\pi\)
\(194\) 0 0
\(195\) 1.08613 0.0777794
\(196\) 0 0
\(197\) 7.35194 0.523804 0.261902 0.965094i \(-0.415650\pi\)
0.261902 + 0.965094i \(0.415650\pi\)
\(198\) 0 0
\(199\) −25.1042 −1.77959 −0.889795 0.456360i \(-0.849153\pi\)
−0.889795 + 0.456360i \(0.849153\pi\)
\(200\) 0 0
\(201\) −9.79001 −0.690534
\(202\) 0 0
\(203\) −30.0410 −2.10846
\(204\) 0 0
\(205\) −2.17226 −0.151717
\(206\) 0 0
\(207\) −7.52420 −0.522968
\(208\) 0 0
\(209\) 11.2813 0.780343
\(210\) 0 0
\(211\) 14.4051 0.991691 0.495846 0.868411i \(-0.334858\pi\)
0.495846 + 0.868411i \(0.334858\pi\)
\(212\) 0 0
\(213\) −2.96969 −0.203480
\(214\) 0 0
\(215\) 8.17226 0.557344
\(216\) 0 0
\(217\) 5.08613 0.345269
\(218\) 0 0
\(219\) −1.25839 −0.0850342
\(220\) 0 0
\(221\) −0.703878 −0.0473480
\(222\) 0 0
\(223\) −11.4078 −0.763920 −0.381960 0.924179i \(-0.624751\pi\)
−0.381960 + 0.924179i \(0.624751\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 15.3371 1.01796 0.508980 0.860779i \(-0.330023\pi\)
0.508980 + 0.860779i \(0.330023\pi\)
\(228\) 0 0
\(229\) −15.8277 −1.04593 −0.522963 0.852355i \(-0.675174\pi\)
−0.522963 + 0.852355i \(0.675174\pi\)
\(230\) 0 0
\(231\) −21.2207 −1.39622
\(232\) 0 0
\(233\) −27.9293 −1.82971 −0.914856 0.403780i \(-0.867696\pi\)
−0.914856 + 0.403780i \(0.867696\pi\)
\(234\) 0 0
\(235\) −10.2281 −0.667206
\(236\) 0 0
\(237\) −10.8203 −0.702855
\(238\) 0 0
\(239\) 10.4562 0.676353 0.338176 0.941083i \(-0.390190\pi\)
0.338176 + 0.941083i \(0.390190\pi\)
\(240\) 0 0
\(241\) −6.87614 −0.442931 −0.221466 0.975168i \(-0.571084\pi\)
−0.221466 + 0.975168i \(0.571084\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −18.8687 −1.20548
\(246\) 0 0
\(247\) −2.93676 −0.186862
\(248\) 0 0
\(249\) −14.2281 −0.901668
\(250\) 0 0
\(251\) −7.39292 −0.466637 −0.233318 0.972400i \(-0.574958\pi\)
−0.233318 + 0.972400i \(0.574958\pi\)
\(252\) 0 0
\(253\) −31.3929 −1.97366
\(254\) 0 0
\(255\) −0.648061 −0.0405831
\(256\) 0 0
\(257\) −6.28870 −0.392279 −0.196139 0.980576i \(-0.562840\pi\)
−0.196139 + 0.980576i \(0.562840\pi\)
\(258\) 0 0
\(259\) −4.64806 −0.288816
\(260\) 0 0
\(261\) 5.90645 0.365600
\(262\) 0 0
\(263\) 14.6284 0.902027 0.451013 0.892517i \(-0.351063\pi\)
0.451013 + 0.892517i \(0.351063\pi\)
\(264\) 0 0
\(265\) −5.52420 −0.339349
\(266\) 0 0
\(267\) 18.1903 1.11323
\(268\) 0 0
\(269\) −10.4381 −0.636420 −0.318210 0.948020i \(-0.603082\pi\)
−0.318210 + 0.948020i \(0.603082\pi\)
\(270\) 0 0
\(271\) 32.1574 1.95342 0.976712 0.214554i \(-0.0688297\pi\)
0.976712 + 0.214554i \(0.0688297\pi\)
\(272\) 0 0
\(273\) 5.52420 0.334340
\(274\) 0 0
\(275\) 4.17226 0.251597
\(276\) 0 0
\(277\) −32.5397 −1.95512 −0.977560 0.210658i \(-0.932439\pi\)
−0.977560 + 0.210658i \(0.932439\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −3.82774 −0.228344 −0.114172 0.993461i \(-0.536421\pi\)
−0.114172 + 0.993461i \(0.536421\pi\)
\(282\) 0 0
\(283\) −6.03773 −0.358906 −0.179453 0.983767i \(-0.557433\pi\)
−0.179453 + 0.983767i \(0.557433\pi\)
\(284\) 0 0
\(285\) −2.70388 −0.160164
\(286\) 0 0
\(287\) −11.0484 −0.652166
\(288\) 0 0
\(289\) −16.5800 −0.975295
\(290\) 0 0
\(291\) −13.2207 −0.775009
\(292\) 0 0
\(293\) 9.33710 0.545479 0.272740 0.962088i \(-0.412070\pi\)
0.272740 + 0.962088i \(0.412070\pi\)
\(294\) 0 0
\(295\) 0.438069 0.0255054
\(296\) 0 0
\(297\) 4.17226 0.242099
\(298\) 0 0
\(299\) 8.17226 0.472614
\(300\) 0 0
\(301\) 41.5652 2.39578
\(302\) 0 0
\(303\) −16.8761 −0.969509
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −26.4184 −1.50778 −0.753890 0.657001i \(-0.771825\pi\)
−0.753890 + 0.657001i \(0.771825\pi\)
\(308\) 0 0
\(309\) 0.493887 0.0280962
\(310\) 0 0
\(311\) 6.36261 0.360790 0.180395 0.983594i \(-0.442262\pi\)
0.180395 + 0.983594i \(0.442262\pi\)
\(312\) 0 0
\(313\) 8.02289 0.453481 0.226740 0.973955i \(-0.427193\pi\)
0.226740 + 0.973955i \(0.427193\pi\)
\(314\) 0 0
\(315\) 5.08613 0.286571
\(316\) 0 0
\(317\) 4.75970 0.267331 0.133666 0.991026i \(-0.457325\pi\)
0.133666 + 0.991026i \(0.457325\pi\)
\(318\) 0 0
\(319\) 24.6433 1.37976
\(320\) 0 0
\(321\) −9.69646 −0.541204
\(322\) 0 0
\(323\) 1.75228 0.0974993
\(324\) 0 0
\(325\) −1.08613 −0.0602477
\(326\) 0 0
\(327\) −16.4003 −0.906941
\(328\) 0 0
\(329\) −52.0213 −2.86803
\(330\) 0 0
\(331\) 21.9804 1.20815 0.604075 0.796928i \(-0.293543\pi\)
0.604075 + 0.796928i \(0.293543\pi\)
\(332\) 0 0
\(333\) 0.913870 0.0500798
\(334\) 0 0
\(335\) 9.79001 0.534885
\(336\) 0 0
\(337\) 28.1952 1.53589 0.767944 0.640517i \(-0.221280\pi\)
0.767944 + 0.640517i \(0.221280\pi\)
\(338\) 0 0
\(339\) 12.5168 0.679818
\(340\) 0 0
\(341\) −4.17226 −0.225941
\(342\) 0 0
\(343\) −60.3659 −3.25945
\(344\) 0 0
\(345\) 7.52420 0.405089
\(346\) 0 0
\(347\) −17.1090 −0.918461 −0.459230 0.888317i \(-0.651875\pi\)
−0.459230 + 0.888317i \(0.651875\pi\)
\(348\) 0 0
\(349\) −2.53643 −0.135772 −0.0678859 0.997693i \(-0.521625\pi\)
−0.0678859 + 0.997693i \(0.521625\pi\)
\(350\) 0 0
\(351\) −1.08613 −0.0579733
\(352\) 0 0
\(353\) −25.2255 −1.34262 −0.671308 0.741178i \(-0.734267\pi\)
−0.671308 + 0.741178i \(0.734267\pi\)
\(354\) 0 0
\(355\) 2.96969 0.157615
\(356\) 0 0
\(357\) −3.29612 −0.174449
\(358\) 0 0
\(359\) −29.7704 −1.57122 −0.785610 0.618722i \(-0.787651\pi\)
−0.785610 + 0.618722i \(0.787651\pi\)
\(360\) 0 0
\(361\) −11.6890 −0.615213
\(362\) 0 0
\(363\) 6.40776 0.336320
\(364\) 0 0
\(365\) 1.25839 0.0658672
\(366\) 0 0
\(367\) 12.3445 0.644379 0.322189 0.946675i \(-0.395581\pi\)
0.322189 + 0.946675i \(0.395581\pi\)
\(368\) 0 0
\(369\) 2.17226 0.113083
\(370\) 0 0
\(371\) −28.0968 −1.45871
\(372\) 0 0
\(373\) 34.3807 1.78016 0.890082 0.455800i \(-0.150647\pi\)
0.890082 + 0.455800i \(0.150647\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −6.41518 −0.330398
\(378\) 0 0
\(379\) −22.3297 −1.14700 −0.573499 0.819206i \(-0.694415\pi\)
−0.573499 + 0.819206i \(0.694415\pi\)
\(380\) 0 0
\(381\) 7.58002 0.388336
\(382\) 0 0
\(383\) −15.4126 −0.787545 −0.393773 0.919208i \(-0.628830\pi\)
−0.393773 + 0.919208i \(0.628830\pi\)
\(384\) 0 0
\(385\) 21.2207 1.08150
\(386\) 0 0
\(387\) −8.17226 −0.415419
\(388\) 0 0
\(389\) −14.9549 −0.758241 −0.379121 0.925347i \(-0.623773\pi\)
−0.379121 + 0.925347i \(0.623773\pi\)
\(390\) 0 0
\(391\) −4.87614 −0.246597
\(392\) 0 0
\(393\) 7.67357 0.387080
\(394\) 0 0
\(395\) 10.8203 0.544429
\(396\) 0 0
\(397\) −33.0484 −1.65865 −0.829326 0.558765i \(-0.811275\pi\)
−0.829326 + 0.558765i \(0.811275\pi\)
\(398\) 0 0
\(399\) −13.7523 −0.688475
\(400\) 0 0
\(401\) 3.56193 0.177874 0.0889372 0.996037i \(-0.471653\pi\)
0.0889372 + 0.996037i \(0.471653\pi\)
\(402\) 0 0
\(403\) 1.08613 0.0541040
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 3.81290 0.188999
\(408\) 0 0
\(409\) −13.6555 −0.675220 −0.337610 0.941286i \(-0.609618\pi\)
−0.337610 + 0.941286i \(0.609618\pi\)
\(410\) 0 0
\(411\) −4.75970 −0.234778
\(412\) 0 0
\(413\) 2.22808 0.109637
\(414\) 0 0
\(415\) 14.2281 0.698429
\(416\) 0 0
\(417\) −10.2839 −0.503605
\(418\) 0 0
\(419\) 17.7948 0.869334 0.434667 0.900591i \(-0.356866\pi\)
0.434667 + 0.900591i \(0.356866\pi\)
\(420\) 0 0
\(421\) 8.70869 0.424435 0.212218 0.977222i \(-0.431931\pi\)
0.212218 + 0.977222i \(0.431931\pi\)
\(422\) 0 0
\(423\) 10.2281 0.497306
\(424\) 0 0
\(425\) 0.648061 0.0314356
\(426\) 0 0
\(427\) 10.1723 0.492270
\(428\) 0 0
\(429\) −4.53162 −0.218789
\(430\) 0 0
\(431\) 16.2658 0.783496 0.391748 0.920072i \(-0.371870\pi\)
0.391748 + 0.920072i \(0.371870\pi\)
\(432\) 0 0
\(433\) −7.85063 −0.377277 −0.188639 0.982047i \(-0.560408\pi\)
−0.188639 + 0.982047i \(0.560408\pi\)
\(434\) 0 0
\(435\) −5.90645 −0.283193
\(436\) 0 0
\(437\) −20.3445 −0.973210
\(438\) 0 0
\(439\) −16.2233 −0.774294 −0.387147 0.922018i \(-0.626539\pi\)
−0.387147 + 0.922018i \(0.626539\pi\)
\(440\) 0 0
\(441\) 18.8687 0.898510
\(442\) 0 0
\(443\) −20.7449 −0.985618 −0.492809 0.870138i \(-0.664030\pi\)
−0.492809 + 0.870138i \(0.664030\pi\)
\(444\) 0 0
\(445\) −18.1903 −0.862305
\(446\) 0 0
\(447\) −0.764504 −0.0361598
\(448\) 0 0
\(449\) −24.7826 −1.16956 −0.584781 0.811191i \(-0.698820\pi\)
−0.584781 + 0.811191i \(0.698820\pi\)
\(450\) 0 0
\(451\) 9.06324 0.426771
\(452\) 0 0
\(453\) 0.992582 0.0466356
\(454\) 0 0
\(455\) −5.52420 −0.258978
\(456\) 0 0
\(457\) 27.8868 1.30449 0.652245 0.758008i \(-0.273827\pi\)
0.652245 + 0.758008i \(0.273827\pi\)
\(458\) 0 0
\(459\) 0.648061 0.0302489
\(460\) 0 0
\(461\) −29.4110 −1.36981 −0.684904 0.728634i \(-0.740156\pi\)
−0.684904 + 0.728634i \(0.740156\pi\)
\(462\) 0 0
\(463\) 25.1239 1.16760 0.583802 0.811896i \(-0.301564\pi\)
0.583802 + 0.811896i \(0.301564\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) 34.8055 1.61061 0.805303 0.592864i \(-0.202003\pi\)
0.805303 + 0.592864i \(0.202003\pi\)
\(468\) 0 0
\(469\) 49.7933 2.29924
\(470\) 0 0
\(471\) 15.9245 0.733764
\(472\) 0 0
\(473\) −34.0968 −1.56777
\(474\) 0 0
\(475\) 2.70388 0.124062
\(476\) 0 0
\(477\) 5.52420 0.252936
\(478\) 0 0
\(479\) −35.7704 −1.63439 −0.817195 0.576362i \(-0.804472\pi\)
−0.817195 + 0.576362i \(0.804472\pi\)
\(480\) 0 0
\(481\) −0.992582 −0.0452578
\(482\) 0 0
\(483\) 38.2691 1.74130
\(484\) 0 0
\(485\) 13.2207 0.600319
\(486\) 0 0
\(487\) 21.5652 0.977212 0.488606 0.872505i \(-0.337506\pi\)
0.488606 + 0.872505i \(0.337506\pi\)
\(488\) 0 0
\(489\) 1.25839 0.0569064
\(490\) 0 0
\(491\) −26.7497 −1.20720 −0.603598 0.797289i \(-0.706267\pi\)
−0.603598 + 0.797289i \(0.706267\pi\)
\(492\) 0 0
\(493\) 3.82774 0.172393
\(494\) 0 0
\(495\) −4.17226 −0.187529
\(496\) 0 0
\(497\) 15.1042 0.677517
\(498\) 0 0
\(499\) 10.9368 0.489597 0.244798 0.969574i \(-0.421278\pi\)
0.244798 + 0.969574i \(0.421278\pi\)
\(500\) 0 0
\(501\) −23.5800 −1.05348
\(502\) 0 0
\(503\) 18.2281 0.812750 0.406375 0.913706i \(-0.366793\pi\)
0.406375 + 0.913706i \(0.366793\pi\)
\(504\) 0 0
\(505\) 16.8761 0.750978
\(506\) 0 0
\(507\) −11.8203 −0.524959
\(508\) 0 0
\(509\) 10.6103 0.470295 0.235147 0.971960i \(-0.424443\pi\)
0.235147 + 0.971960i \(0.424443\pi\)
\(510\) 0 0
\(511\) 6.40034 0.283134
\(512\) 0 0
\(513\) 2.70388 0.119379
\(514\) 0 0
\(515\) −0.493887 −0.0217633
\(516\) 0 0
\(517\) 42.6742 1.87681
\(518\) 0 0
\(519\) −20.1723 −0.885464
\(520\) 0 0
\(521\) −25.6768 −1.12492 −0.562461 0.826824i \(-0.690145\pi\)
−0.562461 + 0.826824i \(0.690145\pi\)
\(522\) 0 0
\(523\) −20.1574 −0.881423 −0.440711 0.897649i \(-0.645274\pi\)
−0.440711 + 0.897649i \(0.645274\pi\)
\(524\) 0 0
\(525\) −5.08613 −0.221977
\(526\) 0 0
\(527\) −0.648061 −0.0282300
\(528\) 0 0
\(529\) 33.6136 1.46146
\(530\) 0 0
\(531\) −0.438069 −0.0190106
\(532\) 0 0
\(533\) −2.35936 −0.102195
\(534\) 0 0
\(535\) 9.69646 0.419215
\(536\) 0 0
\(537\) 4.11164 0.177430
\(538\) 0 0
\(539\) 78.7252 3.39094
\(540\) 0 0
\(541\) 30.4610 1.30962 0.654810 0.755794i \(-0.272749\pi\)
0.654810 + 0.755794i \(0.272749\pi\)
\(542\) 0 0
\(543\) 10.3445 0.443926
\(544\) 0 0
\(545\) 16.4003 0.702513
\(546\) 0 0
\(547\) 11.8506 0.506697 0.253348 0.967375i \(-0.418468\pi\)
0.253348 + 0.967375i \(0.418468\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 15.9703 0.680359
\(552\) 0 0
\(553\) 55.0336 2.34027
\(554\) 0 0
\(555\) −0.913870 −0.0387916
\(556\) 0 0
\(557\) −3.95902 −0.167749 −0.0838745 0.996476i \(-0.526729\pi\)
−0.0838745 + 0.996476i \(0.526729\pi\)
\(558\) 0 0
\(559\) 8.87614 0.375421
\(560\) 0 0
\(561\) 2.70388 0.114158
\(562\) 0 0
\(563\) 13.6965 0.577237 0.288618 0.957444i \(-0.406804\pi\)
0.288618 + 0.957444i \(0.406804\pi\)
\(564\) 0 0
\(565\) −12.5168 −0.526585
\(566\) 0 0
\(567\) −5.08613 −0.213597
\(568\) 0 0
\(569\) 21.4865 0.900760 0.450380 0.892837i \(-0.351289\pi\)
0.450380 + 0.892837i \(0.351289\pi\)
\(570\) 0 0
\(571\) 3.46838 0.145147 0.0725736 0.997363i \(-0.476879\pi\)
0.0725736 + 0.997363i \(0.476879\pi\)
\(572\) 0 0
\(573\) −3.56193 −0.148802
\(574\) 0 0
\(575\) −7.52420 −0.313781
\(576\) 0 0
\(577\) 16.9516 0.705704 0.352852 0.935679i \(-0.385212\pi\)
0.352852 + 0.935679i \(0.385212\pi\)
\(578\) 0 0
\(579\) −3.58002 −0.148780
\(580\) 0 0
\(581\) 72.3659 3.00224
\(582\) 0 0
\(583\) 23.0484 0.954567
\(584\) 0 0
\(585\) 1.08613 0.0449060
\(586\) 0 0
\(587\) −6.97294 −0.287804 −0.143902 0.989592i \(-0.545965\pi\)
−0.143902 + 0.989592i \(0.545965\pi\)
\(588\) 0 0
\(589\) −2.70388 −0.111411
\(590\) 0 0
\(591\) 7.35194 0.302418
\(592\) 0 0
\(593\) −27.9097 −1.14611 −0.573057 0.819515i \(-0.694243\pi\)
−0.573057 + 0.819515i \(0.694243\pi\)
\(594\) 0 0
\(595\) 3.29612 0.135128
\(596\) 0 0
\(597\) −25.1042 −1.02745
\(598\) 0 0
\(599\) −30.4381 −1.24367 −0.621833 0.783150i \(-0.713612\pi\)
−0.621833 + 0.783150i \(0.713612\pi\)
\(600\) 0 0
\(601\) −25.7013 −1.04838 −0.524188 0.851602i \(-0.675631\pi\)
−0.524188 + 0.851602i \(0.675631\pi\)
\(602\) 0 0
\(603\) −9.79001 −0.398680
\(604\) 0 0
\(605\) −6.40776 −0.260512
\(606\) 0 0
\(607\) 22.7119 0.921849 0.460925 0.887439i \(-0.347518\pi\)
0.460925 + 0.887439i \(0.347518\pi\)
\(608\) 0 0
\(609\) −30.0410 −1.21732
\(610\) 0 0
\(611\) −11.1090 −0.449423
\(612\) 0 0
\(613\) 30.4939 1.23164 0.615818 0.787888i \(-0.288826\pi\)
0.615818 + 0.787888i \(0.288826\pi\)
\(614\) 0 0
\(615\) −2.17226 −0.0875940
\(616\) 0 0
\(617\) 10.6890 0.430325 0.215162 0.976578i \(-0.430972\pi\)
0.215162 + 0.976578i \(0.430972\pi\)
\(618\) 0 0
\(619\) 12.3397 0.495975 0.247987 0.968763i \(-0.420231\pi\)
0.247987 + 0.968763i \(0.420231\pi\)
\(620\) 0 0
\(621\) −7.52420 −0.301936
\(622\) 0 0
\(623\) −92.5185 −3.70667
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.2813 0.450531
\(628\) 0 0
\(629\) 0.592243 0.0236143
\(630\) 0 0
\(631\) 23.1600 0.921986 0.460993 0.887404i \(-0.347493\pi\)
0.460993 + 0.887404i \(0.347493\pi\)
\(632\) 0 0
\(633\) 14.4051 0.572553
\(634\) 0 0
\(635\) −7.58002 −0.300804
\(636\) 0 0
\(637\) −20.4939 −0.811997
\(638\) 0 0
\(639\) −2.96969 −0.117479
\(640\) 0 0
\(641\) 1.63739 0.0646731 0.0323366 0.999477i \(-0.489705\pi\)
0.0323366 + 0.999477i \(0.489705\pi\)
\(642\) 0 0
\(643\) 0.531618 0.0209650 0.0104825 0.999945i \(-0.496663\pi\)
0.0104825 + 0.999945i \(0.496663\pi\)
\(644\) 0 0
\(645\) 8.17226 0.321782
\(646\) 0 0
\(647\) 9.86391 0.387790 0.193895 0.981022i \(-0.437888\pi\)
0.193895 + 0.981022i \(0.437888\pi\)
\(648\) 0 0
\(649\) −1.82774 −0.0717451
\(650\) 0 0
\(651\) 5.08613 0.199341
\(652\) 0 0
\(653\) −43.8081 −1.71434 −0.857172 0.515031i \(-0.827780\pi\)
−0.857172 + 0.515031i \(0.827780\pi\)
\(654\) 0 0
\(655\) −7.67357 −0.299831
\(656\) 0 0
\(657\) −1.25839 −0.0490945
\(658\) 0 0
\(659\) −34.0639 −1.32694 −0.663470 0.748203i \(-0.730917\pi\)
−0.663470 + 0.748203i \(0.730917\pi\)
\(660\) 0 0
\(661\) 21.3567 0.830681 0.415341 0.909666i \(-0.363662\pi\)
0.415341 + 0.909666i \(0.363662\pi\)
\(662\) 0 0
\(663\) −0.703878 −0.0273364
\(664\) 0 0
\(665\) 13.7523 0.533290
\(666\) 0 0
\(667\) −44.4413 −1.72077
\(668\) 0 0
\(669\) −11.4078 −0.441049
\(670\) 0 0
\(671\) −8.34452 −0.322137
\(672\) 0 0
\(673\) 28.8990 1.11398 0.556988 0.830521i \(-0.311957\pi\)
0.556988 + 0.830521i \(0.311957\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −27.8539 −1.07051 −0.535256 0.844690i \(-0.679785\pi\)
−0.535256 + 0.844690i \(0.679785\pi\)
\(678\) 0 0
\(679\) 67.2420 2.58051
\(680\) 0 0
\(681\) 15.3371 0.587719
\(682\) 0 0
\(683\) −15.7113 −0.601176 −0.300588 0.953754i \(-0.597183\pi\)
−0.300588 + 0.953754i \(0.597183\pi\)
\(684\) 0 0
\(685\) 4.75970 0.181859
\(686\) 0 0
\(687\) −15.8277 −0.603866
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −7.92454 −0.301464 −0.150732 0.988575i \(-0.548163\pi\)
−0.150732 + 0.988575i \(0.548163\pi\)
\(692\) 0 0
\(693\) −21.2207 −0.806106
\(694\) 0 0
\(695\) 10.2839 0.390090
\(696\) 0 0
\(697\) 1.40776 0.0533226
\(698\) 0 0
\(699\) −27.9293 −1.05638
\(700\) 0 0
\(701\) 34.0968 1.28782 0.643909 0.765102i \(-0.277311\pi\)
0.643909 + 0.765102i \(0.277311\pi\)
\(702\) 0 0
\(703\) 2.47099 0.0931953
\(704\) 0 0
\(705\) −10.2281 −0.385212
\(706\) 0 0
\(707\) 85.8342 3.22813
\(708\) 0 0
\(709\) −46.8826 −1.76071 −0.880357 0.474311i \(-0.842697\pi\)
−0.880357 + 0.474311i \(0.842697\pi\)
\(710\) 0 0
\(711\) −10.8203 −0.405794
\(712\) 0 0
\(713\) 7.52420 0.281783
\(714\) 0 0
\(715\) 4.53162 0.169473
\(716\) 0 0
\(717\) 10.4562 0.390492
\(718\) 0 0
\(719\) 4.51678 0.168448 0.0842238 0.996447i \(-0.473159\pi\)
0.0842238 + 0.996447i \(0.473159\pi\)
\(720\) 0 0
\(721\) −2.51197 −0.0935508
\(722\) 0 0
\(723\) −6.87614 −0.255726
\(724\) 0 0
\(725\) 5.90645 0.219360
\(726\) 0 0
\(727\) −4.72677 −0.175306 −0.0876531 0.996151i \(-0.527937\pi\)
−0.0876531 + 0.996151i \(0.527937\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.29612 −0.195884
\(732\) 0 0
\(733\) −1.87875 −0.0693932 −0.0346966 0.999398i \(-0.511046\pi\)
−0.0346966 + 0.999398i \(0.511046\pi\)
\(734\) 0 0
\(735\) −18.8687 −0.695983
\(736\) 0 0
\(737\) −40.8465 −1.50460
\(738\) 0 0
\(739\) 16.1526 0.594184 0.297092 0.954849i \(-0.403983\pi\)
0.297092 + 0.954849i \(0.403983\pi\)
\(740\) 0 0
\(741\) −2.93676 −0.107885
\(742\) 0 0
\(743\) 23.3175 0.855435 0.427717 0.903913i \(-0.359318\pi\)
0.427717 + 0.903913i \(0.359318\pi\)
\(744\) 0 0
\(745\) 0.764504 0.0280092
\(746\) 0 0
\(747\) −14.2281 −0.520578
\(748\) 0 0
\(749\) 49.3175 1.80202
\(750\) 0 0
\(751\) 3.46838 0.126563 0.0632815 0.997996i \(-0.479843\pi\)
0.0632815 + 0.997996i \(0.479843\pi\)
\(752\) 0 0
\(753\) −7.39292 −0.269413
\(754\) 0 0
\(755\) −0.992582 −0.0361237
\(756\) 0 0
\(757\) 41.9326 1.52407 0.762033 0.647538i \(-0.224201\pi\)
0.762033 + 0.647538i \(0.224201\pi\)
\(758\) 0 0
\(759\) −31.3929 −1.13949
\(760\) 0 0
\(761\) −13.1877 −0.478055 −0.239028 0.971013i \(-0.576829\pi\)
−0.239028 + 0.971013i \(0.576829\pi\)
\(762\) 0 0
\(763\) 83.4143 3.01980
\(764\) 0 0
\(765\) −0.648061 −0.0234307
\(766\) 0 0
\(767\) 0.475800 0.0171802
\(768\) 0 0
\(769\) −47.6816 −1.71944 −0.859722 0.510763i \(-0.829363\pi\)
−0.859722 + 0.510763i \(0.829363\pi\)
\(770\) 0 0
\(771\) −6.28870 −0.226482
\(772\) 0 0
\(773\) 30.6236 1.10145 0.550727 0.834685i \(-0.314350\pi\)
0.550727 + 0.834685i \(0.314350\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) −4.64806 −0.166748
\(778\) 0 0
\(779\) 5.87353 0.210441
\(780\) 0 0
\(781\) −12.3903 −0.443360
\(782\) 0 0
\(783\) 5.90645 0.211079
\(784\) 0 0
\(785\) −15.9245 −0.568371
\(786\) 0 0
\(787\) −26.9219 −0.959663 −0.479832 0.877361i \(-0.659302\pi\)
−0.479832 + 0.877361i \(0.659302\pi\)
\(788\) 0 0
\(789\) 14.6284 0.520785
\(790\) 0 0
\(791\) −63.6620 −2.26356
\(792\) 0 0
\(793\) 2.17226 0.0771392
\(794\) 0 0
\(795\) −5.52420 −0.195923
\(796\) 0 0
\(797\) −23.9197 −0.847280 −0.423640 0.905831i \(-0.639248\pi\)
−0.423640 + 0.905831i \(0.639248\pi\)
\(798\) 0 0
\(799\) 6.62842 0.234497
\(800\) 0 0
\(801\) 18.1903 0.642724
\(802\) 0 0
\(803\) −5.25033 −0.185280
\(804\) 0 0
\(805\) −38.2691 −1.34881
\(806\) 0 0
\(807\) −10.4381 −0.367437
\(808\) 0 0
\(809\) 34.8336 1.22468 0.612342 0.790593i \(-0.290228\pi\)
0.612342 + 0.790593i \(0.290228\pi\)
\(810\) 0 0
\(811\) 37.8277 1.32831 0.664156 0.747594i \(-0.268791\pi\)
0.664156 + 0.747594i \(0.268791\pi\)
\(812\) 0 0
\(813\) 32.1574 1.12781
\(814\) 0 0
\(815\) −1.25839 −0.0440795
\(816\) 0 0
\(817\) −22.0968 −0.773069
\(818\) 0 0
\(819\) 5.52420 0.193031
\(820\) 0 0
\(821\) −31.6439 −1.10438 −0.552190 0.833718i \(-0.686208\pi\)
−0.552190 + 0.833718i \(0.686208\pi\)
\(822\) 0 0
\(823\) −16.9729 −0.591639 −0.295820 0.955244i \(-0.595593\pi\)
−0.295820 + 0.955244i \(0.595593\pi\)
\(824\) 0 0
\(825\) 4.17226 0.145259
\(826\) 0 0
\(827\) 25.0532 0.871185 0.435593 0.900144i \(-0.356539\pi\)
0.435593 + 0.900144i \(0.356539\pi\)
\(828\) 0 0
\(829\) 28.7497 0.998517 0.499259 0.866453i \(-0.333606\pi\)
0.499259 + 0.866453i \(0.333606\pi\)
\(830\) 0 0
\(831\) −32.5397 −1.12879
\(832\) 0 0
\(833\) 12.2281 0.423678
\(834\) 0 0
\(835\) 23.5800 0.816020
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) 7.83099 0.270356 0.135178 0.990821i \(-0.456839\pi\)
0.135178 + 0.990821i \(0.456839\pi\)
\(840\) 0 0
\(841\) 5.88617 0.202971
\(842\) 0 0
\(843\) −3.82774 −0.131834
\(844\) 0 0
\(845\) 11.8203 0.406631
\(846\) 0 0
\(847\) −32.5907 −1.11983
\(848\) 0 0
\(849\) −6.03773 −0.207214
\(850\) 0 0
\(851\) −6.87614 −0.235711
\(852\) 0 0
\(853\) 35.1600 1.20386 0.601928 0.798550i \(-0.294399\pi\)
0.601928 + 0.798550i \(0.294399\pi\)
\(854\) 0 0
\(855\) −2.70388 −0.0924707
\(856\) 0 0
\(857\) 10.6136 0.362553 0.181276 0.983432i \(-0.441977\pi\)
0.181276 + 0.983432i \(0.441977\pi\)
\(858\) 0 0
\(859\) −19.8491 −0.677242 −0.338621 0.940923i \(-0.609960\pi\)
−0.338621 + 0.940923i \(0.609960\pi\)
\(860\) 0 0
\(861\) −11.0484 −0.376528
\(862\) 0 0
\(863\) 9.03356 0.307506 0.153753 0.988109i \(-0.450864\pi\)
0.153753 + 0.988109i \(0.450864\pi\)
\(864\) 0 0
\(865\) 20.1723 0.685877
\(866\) 0 0
\(867\) −16.5800 −0.563087
\(868\) 0 0
\(869\) −45.1452 −1.53145
\(870\) 0 0
\(871\) 10.6332 0.360293
\(872\) 0 0
\(873\) −13.2207 −0.447452
\(874\) 0 0
\(875\) 5.08613 0.171943
\(876\) 0 0
\(877\) 33.6406 1.13596 0.567982 0.823041i \(-0.307724\pi\)
0.567982 + 0.823041i \(0.307724\pi\)
\(878\) 0 0
\(879\) 9.33710 0.314933
\(880\) 0 0
\(881\) 7.48647 0.252226 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(882\) 0 0
\(883\) −17.7374 −0.596912 −0.298456 0.954423i \(-0.596472\pi\)
−0.298456 + 0.954423i \(0.596472\pi\)
\(884\) 0 0
\(885\) 0.438069 0.0147255
\(886\) 0 0
\(887\) −15.8687 −0.532819 −0.266410 0.963860i \(-0.585837\pi\)
−0.266410 + 0.963860i \(0.585837\pi\)
\(888\) 0 0
\(889\) −38.5530 −1.29302
\(890\) 0 0
\(891\) 4.17226 0.139776
\(892\) 0 0
\(893\) 27.6555 0.925455
\(894\) 0 0
\(895\) −4.11164 −0.137437
\(896\) 0 0
\(897\) 8.17226 0.272864
\(898\) 0 0
\(899\) −5.90645 −0.196991
\(900\) 0 0
\(901\) 3.58002 0.119268
\(902\) 0 0
\(903\) 41.5652 1.38320
\(904\) 0 0
\(905\) −10.3445 −0.343864
\(906\) 0 0
\(907\) −8.55451 −0.284048 −0.142024 0.989863i \(-0.545361\pi\)
−0.142024 + 0.989863i \(0.545361\pi\)
\(908\) 0 0
\(909\) −16.8761 −0.559746
\(910\) 0 0
\(911\) 13.5800 0.449926 0.224963 0.974367i \(-0.427774\pi\)
0.224963 + 0.974367i \(0.427774\pi\)
\(912\) 0 0
\(913\) −59.3632 −1.96464
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) −39.0288 −1.28884
\(918\) 0 0
\(919\) −52.3148 −1.72571 −0.862854 0.505454i \(-0.831325\pi\)
−0.862854 + 0.505454i \(0.831325\pi\)
\(920\) 0 0
\(921\) −26.4184 −0.870517
\(922\) 0 0
\(923\) 3.22547 0.106168
\(924\) 0 0
\(925\) 0.913870 0.0300479
\(926\) 0 0
\(927\) 0.493887 0.0162214
\(928\) 0 0
\(929\) −5.37483 −0.176343 −0.0881713 0.996105i \(-0.528102\pi\)
−0.0881713 + 0.996105i \(0.528102\pi\)
\(930\) 0 0
\(931\) 51.0187 1.67207
\(932\) 0 0
\(933\) 6.36261 0.208302
\(934\) 0 0
\(935\) −2.70388 −0.0884263
\(936\) 0 0
\(937\) 58.9433 1.92559 0.962796 0.270228i \(-0.0870992\pi\)
0.962796 + 0.270228i \(0.0870992\pi\)
\(938\) 0 0
\(939\) 8.02289 0.261817
\(940\) 0 0
\(941\) −4.62517 −0.150776 −0.0753881 0.997154i \(-0.524020\pi\)
−0.0753881 + 0.997154i \(0.524020\pi\)
\(942\) 0 0
\(943\) −16.3445 −0.532251
\(944\) 0 0
\(945\) 5.08613 0.165452
\(946\) 0 0
\(947\) 23.8591 0.775317 0.387658 0.921803i \(-0.373284\pi\)
0.387658 + 0.921803i \(0.373284\pi\)
\(948\) 0 0
\(949\) 1.36678 0.0443674
\(950\) 0 0
\(951\) 4.75970 0.154344
\(952\) 0 0
\(953\) −15.8081 −0.512074 −0.256037 0.966667i \(-0.582417\pi\)
−0.256037 + 0.966667i \(0.582417\pi\)
\(954\) 0 0
\(955\) 3.56193 0.115261
\(956\) 0 0
\(957\) 24.6433 0.796603
\(958\) 0 0
\(959\) 24.2084 0.781731
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −9.69646 −0.312464
\(964\) 0 0
\(965\) 3.58002 0.115245
\(966\) 0 0
\(967\) −7.90320 −0.254150 −0.127075 0.991893i \(-0.540559\pi\)
−0.127075 + 0.991893i \(0.540559\pi\)
\(968\) 0 0
\(969\) 1.75228 0.0562913
\(970\) 0 0
\(971\) 44.3478 1.42319 0.711594 0.702591i \(-0.247974\pi\)
0.711594 + 0.702591i \(0.247974\pi\)
\(972\) 0 0
\(973\) 52.3052 1.67683
\(974\) 0 0
\(975\) −1.08613 −0.0347840
\(976\) 0 0
\(977\) 23.3781 0.747931 0.373966 0.927443i \(-0.377998\pi\)
0.373966 + 0.927443i \(0.377998\pi\)
\(978\) 0 0
\(979\) 75.8949 2.42561
\(980\) 0 0
\(981\) −16.4003 −0.523623
\(982\) 0 0
\(983\) 35.7374 1.13985 0.569924 0.821698i \(-0.306973\pi\)
0.569924 + 0.821698i \(0.306973\pi\)
\(984\) 0 0
\(985\) −7.35194 −0.234252
\(986\) 0 0
\(987\) −52.0213 −1.65586
\(988\) 0 0
\(989\) 61.4897 1.95526
\(990\) 0 0
\(991\) −61.4143 −1.95089 −0.975444 0.220247i \(-0.929314\pi\)
−0.975444 + 0.220247i \(0.929314\pi\)
\(992\) 0 0
\(993\) 21.9804 0.697526
\(994\) 0 0
\(995\) 25.1042 0.795857
\(996\) 0 0
\(997\) 52.8220 1.67289 0.836445 0.548051i \(-0.184630\pi\)
0.836445 + 0.548051i \(0.184630\pi\)
\(998\) 0 0
\(999\) 0.913870 0.0289136
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.1.1 3
4.3 odd 2 465.2.a.e.1.1 3
12.11 even 2 1395.2.a.j.1.3 3
20.3 even 4 2325.2.c.k.1024.5 6
20.7 even 4 2325.2.c.k.1024.2 6
20.19 odd 2 2325.2.a.r.1.3 3
60.59 even 2 6975.2.a.bf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.e.1.1 3 4.3 odd 2
1395.2.a.j.1.3 3 12.11 even 2
2325.2.a.r.1.3 3 20.19 odd 2
2325.2.c.k.1024.2 6 20.7 even 4
2325.2.c.k.1024.5 6 20.3 even 4
6975.2.a.bf.1.1 3 60.59 even 2
7440.2.a.bs.1.1 3 1.1 even 1 trivial