Properties

Label 4840.2.a.r.1.3
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.62620 q^{3} +1.00000 q^{5} -2.62620 q^{7} +3.89692 q^{9} +O(q^{10})\) \(q+2.62620 q^{3} +1.00000 q^{5} -2.62620 q^{7} +3.89692 q^{9} -1.72928 q^{13} +2.62620 q^{15} +0.626198 q^{17} -8.14931 q^{19} -6.89692 q^{21} -5.52311 q^{23} +1.00000 q^{25} +2.35548 q^{27} -2.89692 q^{29} -6.89692 q^{31} -2.62620 q^{35} -0.896916 q^{37} -4.54144 q^{39} -5.25240 q^{41} -9.52311 q^{43} +3.89692 q^{45} -0.270718 q^{47} -0.103084 q^{49} +1.64452 q^{51} -0.896916 q^{53} -21.4017 q^{57} +7.04623 q^{59} +9.40171 q^{61} -10.2341 q^{63} -1.72928 q^{65} -12.9817 q^{67} -14.5048 q^{69} +13.4017 q^{71} +11.5231 q^{73} +2.62620 q^{75} -12.0000 q^{79} -5.50479 q^{81} +12.5693 q^{83} +0.626198 q^{85} -7.60788 q^{87} +9.60788 q^{89} +4.54144 q^{91} -18.1127 q^{93} -8.14931 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 3 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + 3 q^{5} + q^{7} + 8 q^{9} - q^{15} - 7 q^{17} - 3 q^{19} - 17 q^{21} - 4 q^{23} + 3 q^{25} - 7 q^{27} - 5 q^{29} - 17 q^{31} + q^{35} + q^{37} - 24 q^{39} + 2 q^{41} - 16 q^{43} + 8 q^{45} - 6 q^{47} - 4 q^{49} + 19 q^{51} + q^{53} - 25 q^{57} - 4 q^{59} - 11 q^{61} + 10 q^{63} - 16 q^{67} - 8 q^{69} + q^{71} + 22 q^{73} - q^{75} - 36 q^{79} + 19 q^{81} - 7 q^{85} + 9 q^{87} - 3 q^{89} + 24 q^{91} + 13 q^{93} - 3 q^{95} + 18 q^{97}+O(q^{100}) \) 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
\(3\) 2.62620 1.51624 0.758118 0.652117i \(-0.226119\pi\)
0.758118 + 0.652117i \(0.226119\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.62620 −0.992610 −0.496305 0.868148i \(-0.665310\pi\)
−0.496305 + 0.868148i \(0.665310\pi\)
\(8\) 0 0
\(9\) 3.89692 1.29897
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.72928 −0.479616 −0.239808 0.970820i \(-0.577085\pi\)
−0.239808 + 0.970820i \(0.577085\pi\)
\(14\) 0 0
\(15\) 2.62620 0.678081
\(16\) 0 0
\(17\) 0.626198 0.151875 0.0759377 0.997113i \(-0.475805\pi\)
0.0759377 + 0.997113i \(0.475805\pi\)
\(18\) 0 0
\(19\) −8.14931 −1.86958 −0.934790 0.355200i \(-0.884413\pi\)
−0.934790 + 0.355200i \(0.884413\pi\)
\(20\) 0 0
\(21\) −6.89692 −1.50503
\(22\) 0 0
\(23\) −5.52311 −1.15165 −0.575824 0.817573i \(-0.695319\pi\)
−0.575824 + 0.817573i \(0.695319\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.35548 0.453312
\(28\) 0 0
\(29\) −2.89692 −0.537944 −0.268972 0.963148i \(-0.586684\pi\)
−0.268972 + 0.963148i \(0.586684\pi\)
\(30\) 0 0
\(31\) −6.89692 −1.23872 −0.619361 0.785106i \(-0.712608\pi\)
−0.619361 + 0.785106i \(0.712608\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.62620 −0.443908
\(36\) 0 0
\(37\) −0.896916 −0.147452 −0.0737261 0.997279i \(-0.523489\pi\)
−0.0737261 + 0.997279i \(0.523489\pi\)
\(38\) 0 0
\(39\) −4.54144 −0.727212
\(40\) 0 0
\(41\) −5.25240 −0.820286 −0.410143 0.912021i \(-0.634521\pi\)
−0.410143 + 0.912021i \(0.634521\pi\)
\(42\) 0 0
\(43\) −9.52311 −1.45226 −0.726131 0.687557i \(-0.758683\pi\)
−0.726131 + 0.687557i \(0.758683\pi\)
\(44\) 0 0
\(45\) 3.89692 0.580918
\(46\) 0 0
\(47\) −0.270718 −0.0394883 −0.0197442 0.999805i \(-0.506285\pi\)
−0.0197442 + 0.999805i \(0.506285\pi\)
\(48\) 0 0
\(49\) −0.103084 −0.0147262
\(50\) 0 0
\(51\) 1.64452 0.230279
\(52\) 0 0
\(53\) −0.896916 −0.123201 −0.0616005 0.998101i \(-0.519620\pi\)
−0.0616005 + 0.998101i \(0.519620\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −21.4017 −2.83473
\(58\) 0 0
\(59\) 7.04623 0.917341 0.458670 0.888606i \(-0.348326\pi\)
0.458670 + 0.888606i \(0.348326\pi\)
\(60\) 0 0
\(61\) 9.40171 1.20377 0.601883 0.798584i \(-0.294417\pi\)
0.601883 + 0.798584i \(0.294417\pi\)
\(62\) 0 0
\(63\) −10.2341 −1.28937
\(64\) 0 0
\(65\) −1.72928 −0.214491
\(66\) 0 0
\(67\) −12.9817 −1.58596 −0.792982 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(68\) 0 0
\(69\) −14.5048 −1.74617
\(70\) 0 0
\(71\) 13.4017 1.59049 0.795245 0.606288i \(-0.207342\pi\)
0.795245 + 0.606288i \(0.207342\pi\)
\(72\) 0 0
\(73\) 11.5231 1.34868 0.674339 0.738422i \(-0.264429\pi\)
0.674339 + 0.738422i \(0.264429\pi\)
\(74\) 0 0
\(75\) 2.62620 0.303247
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) −5.50479 −0.611644
\(82\) 0 0
\(83\) 12.5693 1.37966 0.689832 0.723969i \(-0.257684\pi\)
0.689832 + 0.723969i \(0.257684\pi\)
\(84\) 0 0
\(85\) 0.626198 0.0679207
\(86\) 0 0
\(87\) −7.60788 −0.815650
\(88\) 0 0
\(89\) 9.60788 1.01843 0.509216 0.860639i \(-0.329935\pi\)
0.509216 + 0.860639i \(0.329935\pi\)
\(90\) 0 0
\(91\) 4.54144 0.476072
\(92\) 0 0
\(93\) −18.1127 −1.87820
\(94\) 0 0
\(95\) −8.14931 −0.836102
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.25240 −0.920648 −0.460324 0.887751i \(-0.652267\pi\)
−0.460324 + 0.887751i \(0.652267\pi\)
\(102\) 0 0
\(103\) −8.27072 −0.814938 −0.407469 0.913219i \(-0.633589\pi\)
−0.407469 + 0.913219i \(0.633589\pi\)
\(104\) 0 0
\(105\) −6.89692 −0.673070
\(106\) 0 0
\(107\) 14.7755 1.42840 0.714201 0.699940i \(-0.246790\pi\)
0.714201 + 0.699940i \(0.246790\pi\)
\(108\) 0 0
\(109\) 16.8401 1.61299 0.806493 0.591244i \(-0.201363\pi\)
0.806493 + 0.591244i \(0.201363\pi\)
\(110\) 0 0
\(111\) −2.35548 −0.223572
\(112\) 0 0
\(113\) 14.2986 1.34510 0.672551 0.740051i \(-0.265199\pi\)
0.672551 + 0.740051i \(0.265199\pi\)
\(114\) 0 0
\(115\) −5.52311 −0.515033
\(116\) 0 0
\(117\) −6.73887 −0.623008
\(118\) 0 0
\(119\) −1.64452 −0.150753
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −13.7938 −1.24375
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 5.52311 0.490097 0.245049 0.969511i \(-0.421196\pi\)
0.245049 + 0.969511i \(0.421196\pi\)
\(128\) 0 0
\(129\) −25.0096 −2.20197
\(130\) 0 0
\(131\) −9.10308 −0.795340 −0.397670 0.917528i \(-0.630181\pi\)
−0.397670 + 0.917528i \(0.630181\pi\)
\(132\) 0 0
\(133\) 21.4017 1.85576
\(134\) 0 0
\(135\) 2.35548 0.202727
\(136\) 0 0
\(137\) 0.206167 0.0176141 0.00880704 0.999961i \(-0.497197\pi\)
0.00880704 + 0.999961i \(0.497197\pi\)
\(138\) 0 0
\(139\) 21.9634 1.86291 0.931454 0.363860i \(-0.118541\pi\)
0.931454 + 0.363860i \(0.118541\pi\)
\(140\) 0 0
\(141\) −0.710960 −0.0598736
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.89692 −0.240576
\(146\) 0 0
\(147\) −0.270718 −0.0223285
\(148\) 0 0
\(149\) −5.64452 −0.462417 −0.231209 0.972904i \(-0.574268\pi\)
−0.231209 + 0.972904i \(0.574268\pi\)
\(150\) 0 0
\(151\) −15.4586 −1.25800 −0.629000 0.777405i \(-0.716535\pi\)
−0.629000 + 0.777405i \(0.716535\pi\)
\(152\) 0 0
\(153\) 2.44024 0.197282
\(154\) 0 0
\(155\) −6.89692 −0.553974
\(156\) 0 0
\(157\) −15.4017 −1.22919 −0.614595 0.788843i \(-0.710681\pi\)
−0.614595 + 0.788843i \(0.710681\pi\)
\(158\) 0 0
\(159\) −2.35548 −0.186802
\(160\) 0 0
\(161\) 14.5048 1.14314
\(162\) 0 0
\(163\) 14.2139 1.11332 0.556658 0.830742i \(-0.312083\pi\)
0.556658 + 0.830742i \(0.312083\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.0848 −1.39944 −0.699720 0.714417i \(-0.746692\pi\)
−0.699720 + 0.714417i \(0.746692\pi\)
\(168\) 0 0
\(169\) −10.0096 −0.769968
\(170\) 0 0
\(171\) −31.7572 −2.42853
\(172\) 0 0
\(173\) 7.52311 0.571972 0.285986 0.958234i \(-0.407679\pi\)
0.285986 + 0.958234i \(0.407679\pi\)
\(174\) 0 0
\(175\) −2.62620 −0.198522
\(176\) 0 0
\(177\) 18.5048 1.39091
\(178\) 0 0
\(179\) 23.7572 1.77570 0.887848 0.460137i \(-0.152200\pi\)
0.887848 + 0.460137i \(0.152200\pi\)
\(180\) 0 0
\(181\) −18.2986 −1.36013 −0.680063 0.733154i \(-0.738048\pi\)
−0.680063 + 0.733154i \(0.738048\pi\)
\(182\) 0 0
\(183\) 24.6907 1.82519
\(184\) 0 0
\(185\) −0.896916 −0.0659426
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −6.18596 −0.449962
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 0.626198 0.0450747 0.0225374 0.999746i \(-0.492826\pi\)
0.0225374 + 0.999746i \(0.492826\pi\)
\(194\) 0 0
\(195\) −4.54144 −0.325219
\(196\) 0 0
\(197\) −25.0741 −1.78646 −0.893229 0.449602i \(-0.851566\pi\)
−0.893229 + 0.449602i \(0.851566\pi\)
\(198\) 0 0
\(199\) −8.14931 −0.577689 −0.288845 0.957376i \(-0.593271\pi\)
−0.288845 + 0.957376i \(0.593271\pi\)
\(200\) 0 0
\(201\) −34.0925 −2.40470
\(202\) 0 0
\(203\) 7.60788 0.533968
\(204\) 0 0
\(205\) −5.25240 −0.366843
\(206\) 0 0
\(207\) −21.5231 −1.49596
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 6.89692 0.474803 0.237402 0.971412i \(-0.423704\pi\)
0.237402 + 0.971412i \(0.423704\pi\)
\(212\) 0 0
\(213\) 35.1955 2.41156
\(214\) 0 0
\(215\) −9.52311 −0.649471
\(216\) 0 0
\(217\) 18.1127 1.22957
\(218\) 0 0
\(219\) 30.2620 2.04492
\(220\) 0 0
\(221\) −1.08287 −0.0728419
\(222\) 0 0
\(223\) −3.31695 −0.222119 −0.111060 0.993814i \(-0.535424\pi\)
−0.111060 + 0.993814i \(0.535424\pi\)
\(224\) 0 0
\(225\) 3.89692 0.259794
\(226\) 0 0
\(227\) 1.22449 0.0812722 0.0406361 0.999174i \(-0.487062\pi\)
0.0406361 + 0.999174i \(0.487062\pi\)
\(228\) 0 0
\(229\) −28.8034 −1.90338 −0.951692 0.307055i \(-0.900656\pi\)
−0.951692 + 0.307055i \(0.900656\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.1772 0.666732 0.333366 0.942798i \(-0.391816\pi\)
0.333366 + 0.942798i \(0.391816\pi\)
\(234\) 0 0
\(235\) −0.270718 −0.0176597
\(236\) 0 0
\(237\) −31.5144 −2.04708
\(238\) 0 0
\(239\) −1.79383 −0.116033 −0.0580167 0.998316i \(-0.518478\pi\)
−0.0580167 + 0.998316i \(0.518478\pi\)
\(240\) 0 0
\(241\) −21.2524 −1.36899 −0.684494 0.729019i \(-0.739977\pi\)
−0.684494 + 0.729019i \(0.739977\pi\)
\(242\) 0 0
\(243\) −21.5231 −1.38071
\(244\) 0 0
\(245\) −0.103084 −0.00658578
\(246\) 0 0
\(247\) 14.0925 0.896682
\(248\) 0 0
\(249\) 33.0096 2.09190
\(250\) 0 0
\(251\) −6.50479 −0.410579 −0.205289 0.978701i \(-0.565814\pi\)
−0.205289 + 0.978701i \(0.565814\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 1.64452 0.102984
\(256\) 0 0
\(257\) −2.71096 −0.169105 −0.0845525 0.996419i \(-0.526946\pi\)
−0.0845525 + 0.996419i \(0.526946\pi\)
\(258\) 0 0
\(259\) 2.35548 0.146362
\(260\) 0 0
\(261\) −11.2890 −0.698774
\(262\) 0 0
\(263\) 5.67243 0.349777 0.174888 0.984588i \(-0.444044\pi\)
0.174888 + 0.984588i \(0.444044\pi\)
\(264\) 0 0
\(265\) −0.896916 −0.0550971
\(266\) 0 0
\(267\) 25.2322 1.54418
\(268\) 0 0
\(269\) 6.71096 0.409174 0.204587 0.978848i \(-0.434415\pi\)
0.204587 + 0.978848i \(0.434415\pi\)
\(270\) 0 0
\(271\) 7.45856 0.453075 0.226538 0.974002i \(-0.427259\pi\)
0.226538 + 0.974002i \(0.427259\pi\)
\(272\) 0 0
\(273\) 11.9267 0.721837
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.9354 −0.717132 −0.358566 0.933504i \(-0.616734\pi\)
−0.358566 + 0.933504i \(0.616734\pi\)
\(278\) 0 0
\(279\) −26.8767 −1.60907
\(280\) 0 0
\(281\) −10.7476 −0.641148 −0.320574 0.947223i \(-0.603876\pi\)
−0.320574 + 0.947223i \(0.603876\pi\)
\(282\) 0 0
\(283\) −7.01832 −0.417196 −0.208598 0.978001i \(-0.566890\pi\)
−0.208598 + 0.978001i \(0.566890\pi\)
\(284\) 0 0
\(285\) −21.4017 −1.26773
\(286\) 0 0
\(287\) 13.7938 0.814224
\(288\) 0 0
\(289\) −16.6079 −0.976934
\(290\) 0 0
\(291\) 15.7572 0.923703
\(292\) 0 0
\(293\) 21.4865 1.25525 0.627626 0.778515i \(-0.284027\pi\)
0.627626 + 0.778515i \(0.284027\pi\)
\(294\) 0 0
\(295\) 7.04623 0.410247
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.55102 0.552350
\(300\) 0 0
\(301\) 25.0096 1.44153
\(302\) 0 0
\(303\) −24.2986 −1.39592
\(304\) 0 0
\(305\) 9.40171 0.538340
\(306\) 0 0
\(307\) −20.3265 −1.16010 −0.580048 0.814582i \(-0.696966\pi\)
−0.580048 + 0.814582i \(0.696966\pi\)
\(308\) 0 0
\(309\) −21.7205 −1.23564
\(310\) 0 0
\(311\) −5.40171 −0.306303 −0.153151 0.988203i \(-0.548942\pi\)
−0.153151 + 0.988203i \(0.548942\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 0 0
\(315\) −10.2341 −0.576625
\(316\) 0 0
\(317\) 6.39212 0.359017 0.179509 0.983756i \(-0.442549\pi\)
0.179509 + 0.983756i \(0.442549\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 38.8034 2.16580
\(322\) 0 0
\(323\) −5.10308 −0.283943
\(324\) 0 0
\(325\) −1.72928 −0.0959233
\(326\) 0 0
\(327\) 44.2253 2.44567
\(328\) 0 0
\(329\) 0.710960 0.0391965
\(330\) 0 0
\(331\) −14.9171 −0.819919 −0.409960 0.912104i \(-0.634457\pi\)
−0.409960 + 0.912104i \(0.634457\pi\)
\(332\) 0 0
\(333\) −3.49521 −0.191536
\(334\) 0 0
\(335\) −12.9817 −0.709265
\(336\) 0 0
\(337\) −18.1772 −0.990176 −0.495088 0.868843i \(-0.664864\pi\)
−0.495088 + 0.868843i \(0.664864\pi\)
\(338\) 0 0
\(339\) 37.5510 2.03949
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 18.6541 1.00723
\(344\) 0 0
\(345\) −14.5048 −0.780912
\(346\) 0 0
\(347\) −15.3169 −0.822257 −0.411128 0.911577i \(-0.634865\pi\)
−0.411128 + 0.911577i \(0.634865\pi\)
\(348\) 0 0
\(349\) 5.79383 0.310137 0.155068 0.987904i \(-0.450440\pi\)
0.155068 + 0.987904i \(0.450440\pi\)
\(350\) 0 0
\(351\) −4.07329 −0.217416
\(352\) 0 0
\(353\) 29.8863 1.59069 0.795343 0.606159i \(-0.207291\pi\)
0.795343 + 0.606159i \(0.207291\pi\)
\(354\) 0 0
\(355\) 13.4017 0.711289
\(356\) 0 0
\(357\) −4.31884 −0.228577
\(358\) 0 0
\(359\) 33.3082 1.75794 0.878970 0.476877i \(-0.158231\pi\)
0.878970 + 0.476877i \(0.158231\pi\)
\(360\) 0 0
\(361\) 47.4113 2.49533
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.5231 0.603147
\(366\) 0 0
\(367\) 9.11078 0.475579 0.237789 0.971317i \(-0.423577\pi\)
0.237789 + 0.971317i \(0.423577\pi\)
\(368\) 0 0
\(369\) −20.4681 −1.06553
\(370\) 0 0
\(371\) 2.35548 0.122290
\(372\) 0 0
\(373\) 7.52311 0.389532 0.194766 0.980850i \(-0.437605\pi\)
0.194766 + 0.980850i \(0.437605\pi\)
\(374\) 0 0
\(375\) 2.62620 0.135616
\(376\) 0 0
\(377\) 5.00958 0.258007
\(378\) 0 0
\(379\) −9.55102 −0.490603 −0.245301 0.969447i \(-0.578887\pi\)
−0.245301 + 0.969447i \(0.578887\pi\)
\(380\) 0 0
\(381\) 14.5048 0.743103
\(382\) 0 0
\(383\) −10.7755 −0.550603 −0.275301 0.961358i \(-0.588778\pi\)
−0.275301 + 0.961358i \(0.588778\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −37.1108 −1.88645
\(388\) 0 0
\(389\) 26.7110 1.35430 0.677150 0.735845i \(-0.263215\pi\)
0.677150 + 0.735845i \(0.263215\pi\)
\(390\) 0 0
\(391\) −3.45856 −0.174907
\(392\) 0 0
\(393\) −23.9065 −1.20592
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −18.5972 −0.933369 −0.466685 0.884424i \(-0.654552\pi\)
−0.466685 + 0.884424i \(0.654552\pi\)
\(398\) 0 0
\(399\) 56.2051 2.81378
\(400\) 0 0
\(401\) 19.4017 0.968875 0.484438 0.874826i \(-0.339024\pi\)
0.484438 + 0.874826i \(0.339024\pi\)
\(402\) 0 0
\(403\) 11.9267 0.594112
\(404\) 0 0
\(405\) −5.50479 −0.273535
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.80342 0.336407 0.168204 0.985752i \(-0.446203\pi\)
0.168204 + 0.985752i \(0.446203\pi\)
\(410\) 0 0
\(411\) 0.541436 0.0267071
\(412\) 0 0
\(413\) −18.5048 −0.910561
\(414\) 0 0
\(415\) 12.5693 0.617005
\(416\) 0 0
\(417\) 57.6801 2.82461
\(418\) 0 0
\(419\) 0.840061 0.0410397 0.0205198 0.999789i \(-0.493468\pi\)
0.0205198 + 0.999789i \(0.493468\pi\)
\(420\) 0 0
\(421\) 22.5972 1.10132 0.550661 0.834729i \(-0.314376\pi\)
0.550661 + 0.834729i \(0.314376\pi\)
\(422\) 0 0
\(423\) −1.05497 −0.0512942
\(424\) 0 0
\(425\) 0.626198 0.0303751
\(426\) 0 0
\(427\) −24.6907 −1.19487
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.129102 −0.00621861 −0.00310930 0.999995i \(-0.500990\pi\)
−0.00310930 + 0.999995i \(0.500990\pi\)
\(432\) 0 0
\(433\) 2.41233 0.115929 0.0579647 0.998319i \(-0.481539\pi\)
0.0579647 + 0.998319i \(0.481539\pi\)
\(434\) 0 0
\(435\) −7.60788 −0.364770
\(436\) 0 0
\(437\) 45.0096 2.15310
\(438\) 0 0
\(439\) 2.20617 0.105295 0.0526473 0.998613i \(-0.483234\pi\)
0.0526473 + 0.998613i \(0.483234\pi\)
\(440\) 0 0
\(441\) −0.401709 −0.0191290
\(442\) 0 0
\(443\) −29.5789 −1.40534 −0.702669 0.711517i \(-0.748008\pi\)
−0.702669 + 0.711517i \(0.748008\pi\)
\(444\) 0 0
\(445\) 9.60788 0.455457
\(446\) 0 0
\(447\) −14.8236 −0.701134
\(448\) 0 0
\(449\) −20.5048 −0.967681 −0.483840 0.875156i \(-0.660759\pi\)
−0.483840 + 0.875156i \(0.660759\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −40.5972 −1.90743
\(454\) 0 0
\(455\) 4.54144 0.212906
\(456\) 0 0
\(457\) 7.37380 0.344932 0.172466 0.985015i \(-0.444827\pi\)
0.172466 + 0.985015i \(0.444827\pi\)
\(458\) 0 0
\(459\) 1.47500 0.0688470
\(460\) 0 0
\(461\) −35.0664 −1.63321 −0.816603 0.577199i \(-0.804146\pi\)
−0.816603 + 0.577199i \(0.804146\pi\)
\(462\) 0 0
\(463\) −9.11078 −0.423414 −0.211707 0.977333i \(-0.567902\pi\)
−0.211707 + 0.977333i \(0.567902\pi\)
\(464\) 0 0
\(465\) −18.1127 −0.839955
\(466\) 0 0
\(467\) −34.3834 −1.59107 −0.795537 0.605905i \(-0.792811\pi\)
−0.795537 + 0.605905i \(0.792811\pi\)
\(468\) 0 0
\(469\) 34.0925 1.57424
\(470\) 0 0
\(471\) −40.4479 −1.86374
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −8.14931 −0.373916
\(476\) 0 0
\(477\) −3.49521 −0.160035
\(478\) 0 0
\(479\) 31.8863 1.45692 0.728461 0.685087i \(-0.240236\pi\)
0.728461 + 0.685087i \(0.240236\pi\)
\(480\) 0 0
\(481\) 1.55102 0.0707205
\(482\) 0 0
\(483\) 38.0925 1.73327
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) −35.0741 −1.58936 −0.794680 0.607028i \(-0.792362\pi\)
−0.794680 + 0.607028i \(0.792362\pi\)
\(488\) 0 0
\(489\) 37.3284 1.68805
\(490\) 0 0
\(491\) −20.0202 −0.903499 −0.451750 0.892145i \(-0.649200\pi\)
−0.451750 + 0.892145i \(0.649200\pi\)
\(492\) 0 0
\(493\) −1.81404 −0.0817004
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −35.1955 −1.57874
\(498\) 0 0
\(499\) 12.8401 0.574800 0.287400 0.957811i \(-0.407209\pi\)
0.287400 + 0.957811i \(0.407209\pi\)
\(500\) 0 0
\(501\) −47.4942 −2.12188
\(502\) 0 0
\(503\) 40.0279 1.78476 0.892378 0.451289i \(-0.149035\pi\)
0.892378 + 0.451289i \(0.149035\pi\)
\(504\) 0 0
\(505\) −9.25240 −0.411726
\(506\) 0 0
\(507\) −26.2872 −1.16745
\(508\) 0 0
\(509\) −42.5972 −1.88809 −0.944045 0.329817i \(-0.893013\pi\)
−0.944045 + 0.329817i \(0.893013\pi\)
\(510\) 0 0
\(511\) −30.2620 −1.33871
\(512\) 0 0
\(513\) −19.1955 −0.847504
\(514\) 0 0
\(515\) −8.27072 −0.364451
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 19.7572 0.867244
\(520\) 0 0
\(521\) 2.41233 0.105686 0.0528432 0.998603i \(-0.483172\pi\)
0.0528432 + 0.998603i \(0.483172\pi\)
\(522\) 0 0
\(523\) −10.3632 −0.453150 −0.226575 0.973994i \(-0.572753\pi\)
−0.226575 + 0.973994i \(0.572753\pi\)
\(524\) 0 0
\(525\) −6.89692 −0.301006
\(526\) 0 0
\(527\) −4.31884 −0.188131
\(528\) 0 0
\(529\) 7.50479 0.326295
\(530\) 0 0
\(531\) 27.4586 1.19160
\(532\) 0 0
\(533\) 9.08287 0.393423
\(534\) 0 0
\(535\) 14.7755 0.638801
\(536\) 0 0
\(537\) 62.3911 2.69237
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 24.5616 1.05599 0.527994 0.849248i \(-0.322944\pi\)
0.527994 + 0.849248i \(0.322944\pi\)
\(542\) 0 0
\(543\) −48.0558 −2.06227
\(544\) 0 0
\(545\) 16.8401 0.721349
\(546\) 0 0
\(547\) 25.8217 1.10406 0.552029 0.833825i \(-0.313854\pi\)
0.552029 + 0.833825i \(0.313854\pi\)
\(548\) 0 0
\(549\) 36.6377 1.56366
\(550\) 0 0
\(551\) 23.6079 1.00573
\(552\) 0 0
\(553\) 31.5144 1.34013
\(554\) 0 0
\(555\) −2.35548 −0.0999846
\(556\) 0 0
\(557\) −32.7755 −1.38874 −0.694371 0.719617i \(-0.744318\pi\)
−0.694371 + 0.719617i \(0.744318\pi\)
\(558\) 0 0
\(559\) 16.4681 0.696528
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.82174 0.413937 0.206968 0.978348i \(-0.433640\pi\)
0.206968 + 0.978348i \(0.433640\pi\)
\(564\) 0 0
\(565\) 14.2986 0.601548
\(566\) 0 0
\(567\) 14.4567 0.607123
\(568\) 0 0
\(569\) 1.19658 0.0501634 0.0250817 0.999685i \(-0.492015\pi\)
0.0250817 + 0.999685i \(0.492015\pi\)
\(570\) 0 0
\(571\) −6.22638 −0.260566 −0.130283 0.991477i \(-0.541589\pi\)
−0.130283 + 0.991477i \(0.541589\pi\)
\(572\) 0 0
\(573\) −31.5144 −1.31653
\(574\) 0 0
\(575\) −5.52311 −0.230330
\(576\) 0 0
\(577\) 43.3082 1.80294 0.901472 0.432837i \(-0.142487\pi\)
0.901472 + 0.432837i \(0.142487\pi\)
\(578\) 0 0
\(579\) 1.64452 0.0683439
\(580\) 0 0
\(581\) −33.0096 −1.36947
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.73887 −0.278618
\(586\) 0 0
\(587\) −19.7649 −0.815784 −0.407892 0.913030i \(-0.633736\pi\)
−0.407892 + 0.913030i \(0.633736\pi\)
\(588\) 0 0
\(589\) 56.2051 2.31589
\(590\) 0 0
\(591\) −65.8496 −2.70869
\(592\) 0 0
\(593\) 4.60599 0.189145 0.0945726 0.995518i \(-0.469852\pi\)
0.0945726 + 0.995518i \(0.469852\pi\)
\(594\) 0 0
\(595\) −1.64452 −0.0674188
\(596\) 0 0
\(597\) −21.4017 −0.875914
\(598\) 0 0
\(599\) −46.9527 −1.91844 −0.959218 0.282666i \(-0.908781\pi\)
−0.959218 + 0.282666i \(0.908781\pi\)
\(600\) 0 0
\(601\) −21.2524 −0.866903 −0.433452 0.901177i \(-0.642705\pi\)
−0.433452 + 0.901177i \(0.642705\pi\)
\(602\) 0 0
\(603\) −50.5885 −2.06012
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −34.3834 −1.39558 −0.697789 0.716303i \(-0.745833\pi\)
−0.697789 + 0.716303i \(0.745833\pi\)
\(608\) 0 0
\(609\) 19.9798 0.809622
\(610\) 0 0
\(611\) 0.468148 0.0189392
\(612\) 0 0
\(613\) −28.3632 −1.14558 −0.572789 0.819703i \(-0.694139\pi\)
−0.572789 + 0.819703i \(0.694139\pi\)
\(614\) 0 0
\(615\) −13.7938 −0.556221
\(616\) 0 0
\(617\) 45.5144 1.83234 0.916170 0.400790i \(-0.131264\pi\)
0.916170 + 0.400790i \(0.131264\pi\)
\(618\) 0 0
\(619\) −20.4277 −0.821060 −0.410530 0.911847i \(-0.634656\pi\)
−0.410530 + 0.911847i \(0.634656\pi\)
\(620\) 0 0
\(621\) −13.0096 −0.522057
\(622\) 0 0
\(623\) −25.2322 −1.01091
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.561647 −0.0223943
\(630\) 0 0
\(631\) −42.1127 −1.67648 −0.838239 0.545302i \(-0.816415\pi\)
−0.838239 + 0.545302i \(0.816415\pi\)
\(632\) 0 0
\(633\) 18.1127 0.719914
\(634\) 0 0
\(635\) 5.52311 0.219178
\(636\) 0 0
\(637\) 0.178261 0.00706295
\(638\) 0 0
\(639\) 52.2253 2.06600
\(640\) 0 0
\(641\) 21.9065 0.865255 0.432627 0.901573i \(-0.357587\pi\)
0.432627 + 0.901573i \(0.357587\pi\)
\(642\) 0 0
\(643\) 42.6262 1.68101 0.840507 0.541801i \(-0.182257\pi\)
0.840507 + 0.541801i \(0.182257\pi\)
\(644\) 0 0
\(645\) −25.0096 −0.984751
\(646\) 0 0
\(647\) 1.65222 0.0649553 0.0324777 0.999472i \(-0.489660\pi\)
0.0324777 + 0.999472i \(0.489660\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 47.5675 1.86432
\(652\) 0 0
\(653\) 14.3188 0.560339 0.280170 0.959950i \(-0.409609\pi\)
0.280170 + 0.959950i \(0.409609\pi\)
\(654\) 0 0
\(655\) −9.10308 −0.355687
\(656\) 0 0
\(657\) 44.9046 1.75190
\(658\) 0 0
\(659\) −38.9527 −1.51738 −0.758691 0.651450i \(-0.774161\pi\)
−0.758691 + 0.651450i \(0.774161\pi\)
\(660\) 0 0
\(661\) 39.6801 1.54338 0.771689 0.636000i \(-0.219412\pi\)
0.771689 + 0.636000i \(0.219412\pi\)
\(662\) 0 0
\(663\) −2.84384 −0.110446
\(664\) 0 0
\(665\) 21.4017 0.829923
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 0 0
\(669\) −8.71096 −0.336785
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 23.5433 0.907528 0.453764 0.891122i \(-0.350081\pi\)
0.453764 + 0.891122i \(0.350081\pi\)
\(674\) 0 0
\(675\) 2.35548 0.0906625
\(676\) 0 0
\(677\) 37.4865 1.44072 0.720361 0.693599i \(-0.243976\pi\)
0.720361 + 0.693599i \(0.243976\pi\)
\(678\) 0 0
\(679\) −15.7572 −0.604705
\(680\) 0 0
\(681\) 3.21575 0.123228
\(682\) 0 0
\(683\) −17.5433 −0.671277 −0.335638 0.941991i \(-0.608952\pi\)
−0.335638 + 0.941991i \(0.608952\pi\)
\(684\) 0 0
\(685\) 0.206167 0.00787725
\(686\) 0 0
\(687\) −75.6435 −2.88598
\(688\) 0 0
\(689\) 1.55102 0.0590892
\(690\) 0 0
\(691\) 19.2158 0.731002 0.365501 0.930811i \(-0.380898\pi\)
0.365501 + 0.930811i \(0.380898\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.9634 0.833118
\(696\) 0 0
\(697\) −3.28904 −0.124581
\(698\) 0 0
\(699\) 26.7274 1.01092
\(700\) 0 0
\(701\) −8.02021 −0.302919 −0.151460 0.988463i \(-0.548397\pi\)
−0.151460 + 0.988463i \(0.548397\pi\)
\(702\) 0 0
\(703\) 7.30925 0.275674
\(704\) 0 0
\(705\) −0.710960 −0.0267763
\(706\) 0 0
\(707\) 24.2986 0.913844
\(708\) 0 0
\(709\) −3.90754 −0.146751 −0.0733754 0.997304i \(-0.523377\pi\)
−0.0733754 + 0.997304i \(0.523377\pi\)
\(710\) 0 0
\(711\) −46.7630 −1.75375
\(712\) 0 0
\(713\) 38.0925 1.42657
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.71096 −0.175934
\(718\) 0 0
\(719\) −17.3459 −0.646893 −0.323446 0.946247i \(-0.604842\pi\)
−0.323446 + 0.946247i \(0.604842\pi\)
\(720\) 0 0
\(721\) 21.7205 0.808915
\(722\) 0 0
\(723\) −55.8130 −2.07571
\(724\) 0 0
\(725\) −2.89692 −0.107589
\(726\) 0 0
\(727\) −8.85258 −0.328324 −0.164162 0.986433i \(-0.552492\pi\)
−0.164162 + 0.986433i \(0.552492\pi\)
\(728\) 0 0
\(729\) −40.0096 −1.48184
\(730\) 0 0
\(731\) −5.96336 −0.220563
\(732\) 0 0
\(733\) 5.31695 0.196386 0.0981930 0.995167i \(-0.468694\pi\)
0.0981930 + 0.995167i \(0.468694\pi\)
\(734\) 0 0
\(735\) −0.270718 −0.00998559
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −9.55102 −0.351340 −0.175670 0.984449i \(-0.556209\pi\)
−0.175670 + 0.984449i \(0.556209\pi\)
\(740\) 0 0
\(741\) 37.0096 1.35958
\(742\) 0 0
\(743\) 6.79572 0.249311 0.124655 0.992200i \(-0.460217\pi\)
0.124655 + 0.992200i \(0.460217\pi\)
\(744\) 0 0
\(745\) −5.64452 −0.206799
\(746\) 0 0
\(747\) 48.9817 1.79215
\(748\) 0 0
\(749\) −38.8034 −1.41785
\(750\) 0 0
\(751\) −3.72159 −0.135803 −0.0679013 0.997692i \(-0.521630\pi\)
−0.0679013 + 0.997692i \(0.521630\pi\)
\(752\) 0 0
\(753\) −17.0829 −0.622534
\(754\) 0 0
\(755\) −15.4586 −0.562595
\(756\) 0 0
\(757\) 9.32946 0.339085 0.169543 0.985523i \(-0.445771\pi\)
0.169543 + 0.985523i \(0.445771\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.80342 −0.101624 −0.0508119 0.998708i \(-0.516181\pi\)
−0.0508119 + 0.998708i \(0.516181\pi\)
\(762\) 0 0
\(763\) −44.2253 −1.60106
\(764\) 0 0
\(765\) 2.44024 0.0882271
\(766\) 0 0
\(767\) −12.1849 −0.439972
\(768\) 0 0
\(769\) −29.4373 −1.06154 −0.530768 0.847517i \(-0.678097\pi\)
−0.530768 + 0.847517i \(0.678097\pi\)
\(770\) 0 0
\(771\) −7.11952 −0.256403
\(772\) 0 0
\(773\) −5.30925 −0.190960 −0.0954802 0.995431i \(-0.530439\pi\)
−0.0954802 + 0.995431i \(0.530439\pi\)
\(774\) 0 0
\(775\) −6.89692 −0.247745
\(776\) 0 0
\(777\) 6.18596 0.221920
\(778\) 0 0
\(779\) 42.8034 1.53359
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.82363 −0.243857
\(784\) 0 0
\(785\) −15.4017 −0.549711
\(786\) 0 0
\(787\) 48.1974 1.71805 0.859026 0.511931i \(-0.171070\pi\)
0.859026 + 0.511931i \(0.171070\pi\)
\(788\) 0 0
\(789\) 14.8969 0.530344
\(790\) 0 0
\(791\) −37.5510 −1.33516
\(792\) 0 0
\(793\) −16.2582 −0.577346
\(794\) 0 0
\(795\) −2.35548 −0.0835403
\(796\) 0 0
\(797\) −2.59725 −0.0919993 −0.0459997 0.998941i \(-0.514647\pi\)
−0.0459997 + 0.998941i \(0.514647\pi\)
\(798\) 0 0
\(799\) −0.169523 −0.00599730
\(800\) 0 0
\(801\) 37.4411 1.32292
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 14.5048 0.511227
\(806\) 0 0
\(807\) 17.6243 0.620405
\(808\) 0 0
\(809\) 18.3757 0.646055 0.323027 0.946390i \(-0.395299\pi\)
0.323027 + 0.946390i \(0.395299\pi\)
\(810\) 0 0
\(811\) −51.4942 −1.80820 −0.904102 0.427316i \(-0.859459\pi\)
−0.904102 + 0.427316i \(0.859459\pi\)
\(812\) 0 0
\(813\) 19.5877 0.686969
\(814\) 0 0
\(815\) 14.2139 0.497890
\(816\) 0 0
\(817\) 77.6068 2.71512
\(818\) 0 0
\(819\) 17.6976 0.618404
\(820\) 0 0
\(821\) −23.2890 −0.812793 −0.406397 0.913697i \(-0.633215\pi\)
−0.406397 + 0.913697i \(0.633215\pi\)
\(822\) 0 0
\(823\) −42.2899 −1.47413 −0.737066 0.675820i \(-0.763790\pi\)
−0.737066 + 0.675820i \(0.763790\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.64641 −0.0920247 −0.0460123 0.998941i \(-0.514651\pi\)
−0.0460123 + 0.998941i \(0.514651\pi\)
\(828\) 0 0
\(829\) 8.20617 0.285012 0.142506 0.989794i \(-0.454484\pi\)
0.142506 + 0.989794i \(0.454484\pi\)
\(830\) 0 0
\(831\) −31.3449 −1.08734
\(832\) 0 0
\(833\) −0.0645508 −0.00223655
\(834\) 0 0
\(835\) −18.0848 −0.625849
\(836\) 0 0
\(837\) −16.2455 −0.561528
\(838\) 0 0
\(839\) 22.6339 0.781409 0.390704 0.920516i \(-0.372231\pi\)
0.390704 + 0.920516i \(0.372231\pi\)
\(840\) 0 0
\(841\) −20.6079 −0.710616
\(842\) 0 0
\(843\) −28.2253 −0.972132
\(844\) 0 0
\(845\) −10.0096 −0.344340
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −18.4315 −0.632568
\(850\) 0 0
\(851\) 4.95377 0.169813
\(852\) 0 0
\(853\) 28.2341 0.966716 0.483358 0.875423i \(-0.339417\pi\)
0.483358 + 0.875423i \(0.339417\pi\)
\(854\) 0 0
\(855\) −31.7572 −1.08607
\(856\) 0 0
\(857\) −25.0539 −0.855826 −0.427913 0.903820i \(-0.640751\pi\)
−0.427913 + 0.903820i \(0.640751\pi\)
\(858\) 0 0
\(859\) 21.5510 0.735311 0.367656 0.929962i \(-0.380161\pi\)
0.367656 + 0.929962i \(0.380161\pi\)
\(860\) 0 0
\(861\) 36.2253 1.23456
\(862\) 0 0
\(863\) −20.6831 −0.704059 −0.352030 0.935989i \(-0.614508\pi\)
−0.352030 + 0.935989i \(0.614508\pi\)
\(864\) 0 0
\(865\) 7.52311 0.255794
\(866\) 0 0
\(867\) −43.6156 −1.48126
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 22.4490 0.760655
\(872\) 0 0
\(873\) 23.3815 0.791344
\(874\) 0 0
\(875\) −2.62620 −0.0887817
\(876\) 0 0
\(877\) −57.6714 −1.94742 −0.973712 0.227782i \(-0.926853\pi\)
−0.973712 + 0.227782i \(0.926853\pi\)
\(878\) 0 0
\(879\) 56.4277 1.90326
\(880\) 0 0
\(881\) −45.5144 −1.53342 −0.766709 0.641995i \(-0.778107\pi\)
−0.766709 + 0.641995i \(0.778107\pi\)
\(882\) 0 0
\(883\) −42.6820 −1.43636 −0.718182 0.695855i \(-0.755025\pi\)
−0.718182 + 0.695855i \(0.755025\pi\)
\(884\) 0 0
\(885\) 18.5048 0.622032
\(886\) 0 0
\(887\) −40.5693 −1.36219 −0.681093 0.732197i \(-0.738495\pi\)
−0.681093 + 0.732197i \(0.738495\pi\)
\(888\) 0 0
\(889\) −14.5048 −0.486475
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.20617 0.0738266
\(894\) 0 0
\(895\) 23.7572 0.794115
\(896\) 0 0
\(897\) 25.0829 0.837493
\(898\) 0 0
\(899\) 19.9798 0.666363
\(900\) 0 0
\(901\) −0.561647 −0.0187112
\(902\) 0 0
\(903\) 65.6801 2.18570
\(904\) 0 0
\(905\) −18.2986 −0.608267
\(906\) 0 0
\(907\) 50.1406 1.66489 0.832445 0.554107i \(-0.186940\pi\)
0.832445 + 0.554107i \(0.186940\pi\)
\(908\) 0 0
\(909\) −36.0558 −1.19590
\(910\) 0 0
\(911\) 32.5770 1.07933 0.539663 0.841881i \(-0.318552\pi\)
0.539663 + 0.841881i \(0.318552\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 24.6907 0.816251
\(916\) 0 0
\(917\) 23.9065 0.789462
\(918\) 0 0
\(919\) 10.5048 0.346521 0.173261 0.984876i \(-0.444570\pi\)
0.173261 + 0.984876i \(0.444570\pi\)
\(920\) 0 0
\(921\) −53.3815 −1.75898
\(922\) 0 0
\(923\) −23.1753 −0.762825
\(924\) 0 0
\(925\) −0.896916 −0.0294904
\(926\) 0 0
\(927\) −32.2303 −1.05858
\(928\) 0 0
\(929\) 34.5038 1.13203 0.566016 0.824394i \(-0.308484\pi\)
0.566016 + 0.824394i \(0.308484\pi\)
\(930\) 0 0
\(931\) 0.840061 0.0275319
\(932\) 0 0
\(933\) −14.1860 −0.464427
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 8.06455 0.263457 0.131729 0.991286i \(-0.457947\pi\)
0.131729 + 0.991286i \(0.457947\pi\)
\(938\) 0 0
\(939\) 5.25240 0.171405
\(940\) 0 0
\(941\) 8.81985 0.287519 0.143759 0.989613i \(-0.454081\pi\)
0.143759 + 0.989613i \(0.454081\pi\)
\(942\) 0 0
\(943\) 29.0096 0.944682
\(944\) 0 0
\(945\) −6.18596 −0.201229
\(946\) 0 0
\(947\) −48.1772 −1.56555 −0.782775 0.622305i \(-0.786196\pi\)
−0.782775 + 0.622305i \(0.786196\pi\)
\(948\) 0 0
\(949\) −19.9267 −0.646848
\(950\) 0 0
\(951\) 16.7870 0.544355
\(952\) 0 0
\(953\) −53.6358 −1.73743 −0.868717 0.495309i \(-0.835055\pi\)
−0.868717 + 0.495309i \(0.835055\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.541436 −0.0174839
\(960\) 0 0
\(961\) 16.5675 0.534434
\(962\) 0 0
\(963\) 57.5789 1.85546
\(964\) 0 0
\(965\) 0.626198 0.0201580
\(966\) 0 0
\(967\) 32.1214 1.03295 0.516477 0.856301i \(-0.327243\pi\)
0.516477 + 0.856301i \(0.327243\pi\)
\(968\) 0 0
\(969\) −13.4017 −0.430525
\(970\) 0 0
\(971\) −40.8959 −1.31241 −0.656206 0.754582i \(-0.727840\pi\)
−0.656206 + 0.754582i \(0.727840\pi\)
\(972\) 0 0
\(973\) −57.6801 −1.84914
\(974\) 0 0
\(975\) −4.54144 −0.145442
\(976\) 0 0
\(977\) −35.6801 −1.14151 −0.570754 0.821121i \(-0.693349\pi\)
−0.570754 + 0.821121i \(0.693349\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 65.6243 2.09522
\(982\) 0 0
\(983\) −28.9817 −0.924372 −0.462186 0.886783i \(-0.652935\pi\)
−0.462186 + 0.886783i \(0.652935\pi\)
\(984\) 0 0
\(985\) −25.0741 −0.798928
\(986\) 0 0
\(987\) 1.86712 0.0594311
\(988\) 0 0
\(989\) 52.5972 1.67250
\(990\) 0 0
\(991\) −15.5877 −0.495159 −0.247579 0.968868i \(-0.579635\pi\)
−0.247579 + 0.968868i \(0.579635\pi\)
\(992\) 0 0
\(993\) −39.1753 −1.24319
\(994\) 0 0
\(995\) −8.14931 −0.258351
\(996\) 0 0
\(997\) 19.1512 0.606525 0.303262 0.952907i \(-0.401924\pi\)
0.303262 + 0.952907i \(0.401924\pi\)
\(998\) 0 0
\(999\) −2.11267 −0.0668419
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 4840.2.a.r.1.3 yes 3
4.3 odd 2 9680.2.a.cg.1.1 3
11.10 odd 2 4840.2.a.q.1.3 3
44.43 even 2 9680.2.a.ch.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.q.1.3 3 11.10 odd 2
4840.2.a.r.1.3 yes 3 1.1 even 1 trivial
9680.2.a.cg.1.1 3 4.3 odd 2
9680.2.a.ch.1.1 3 44.43 even 2