Properties

Label 7497.2.a.v.1.2
Level $7497$
Weight $2$
Character 7497.1
Self dual yes
Analytic conductor $59.864$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7497,2,Mod(1,7497)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7497, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7497.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7497.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: \(59.8638463954\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 7497.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.56155 q^{2} +4.56155 q^{4} +3.56155 q^{5} +6.56155 q^{8} +O(q^{10})\) \(q+2.56155 q^{2} +4.56155 q^{4} +3.56155 q^{5} +6.56155 q^{8} +9.12311 q^{10} -1.56155 q^{11} -0.438447 q^{13} +7.68466 q^{16} +1.00000 q^{17} +4.68466 q^{19} +16.2462 q^{20} -4.00000 q^{22} +2.43845 q^{23} +7.68466 q^{25} -1.12311 q^{26} +8.24621 q^{29} -3.12311 q^{31} +6.56155 q^{32} +2.56155 q^{34} -5.12311 q^{37} +12.0000 q^{38} +23.3693 q^{40} -3.56155 q^{41} +4.68466 q^{43} -7.12311 q^{44} +6.24621 q^{46} -11.1231 q^{47} +19.6847 q^{50} -2.00000 q^{52} -12.2462 q^{53} -5.56155 q^{55} +21.1231 q^{58} +7.12311 q^{59} -9.12311 q^{61} -8.00000 q^{62} +1.43845 q^{64} -1.56155 q^{65} +4.00000 q^{67} +4.56155 q^{68} +6.24621 q^{71} +12.2462 q^{73} -13.1231 q^{74} +21.3693 q^{76} -9.36932 q^{79} +27.3693 q^{80} -9.12311 q^{82} -0.876894 q^{83} +3.56155 q^{85} +12.0000 q^{86} -10.2462 q^{88} -1.12311 q^{89} +11.1231 q^{92} -28.4924 q^{94} +16.6847 q^{95} +2.87689 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} + 3 q^{5} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} + 3 q^{5} + 9 q^{8} + 10 q^{10} + q^{11} - 5 q^{13} + 3 q^{16} + 2 q^{17} - 3 q^{19} + 16 q^{20} - 8 q^{22} + 9 q^{23} + 3 q^{25} + 6 q^{26} + 2 q^{31} + 9 q^{32} + q^{34} - 2 q^{37} + 24 q^{38} + 22 q^{40} - 3 q^{41} - 3 q^{43} - 6 q^{44} - 4 q^{46} - 14 q^{47} + 27 q^{50} - 4 q^{52} - 8 q^{53} - 7 q^{55} + 34 q^{58} + 6 q^{59} - 10 q^{61} - 16 q^{62} + 7 q^{64} + q^{65} + 8 q^{67} + 5 q^{68} - 4 q^{71} + 8 q^{73} - 18 q^{74} + 18 q^{76} + 6 q^{79} + 30 q^{80} - 10 q^{82} - 10 q^{83} + 3 q^{85} + 24 q^{86} - 4 q^{88} + 6 q^{89} + 14 q^{92} - 24 q^{94} + 21 q^{95} + 14 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 1.81129 0.905646 0.424035i \(-0.139387\pi\)
0.905646 + 0.424035i \(0.139387\pi\)
\(3\) 0 0
\(4\) 4.56155 2.28078
\(5\) 3.56155 1.59277 0.796387 0.604787i \(-0.206742\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 6.56155 2.31986
\(9\) 0 0
\(10\) 9.12311 2.88498
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 0 0
\(13\) −0.438447 −0.121603 −0.0608017 0.998150i \(-0.519366\pi\)
−0.0608017 + 0.998150i \(0.519366\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 4.68466 1.07473 0.537367 0.843348i \(-0.319419\pi\)
0.537367 + 0.843348i \(0.319419\pi\)
\(20\) 16.2462 3.63276
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 2.43845 0.508451 0.254226 0.967145i \(-0.418179\pi\)
0.254226 + 0.967145i \(0.418179\pi\)
\(24\) 0 0
\(25\) 7.68466 1.53693
\(26\) −1.12311 −0.220259
\(27\) 0 0
\(28\) 0 0
\(29\) 8.24621 1.53128 0.765641 0.643268i \(-0.222422\pi\)
0.765641 + 0.643268i \(0.222422\pi\)
\(30\) 0 0
\(31\) −3.12311 −0.560926 −0.280463 0.959865i \(-0.590488\pi\)
−0.280463 + 0.959865i \(0.590488\pi\)
\(32\) 6.56155 1.15993
\(33\) 0 0
\(34\) 2.56155 0.439303
\(35\) 0 0
\(36\) 0 0
\(37\) −5.12311 −0.842233 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(38\) 12.0000 1.94666
\(39\) 0 0
\(40\) 23.3693 3.69501
\(41\) −3.56155 −0.556221 −0.278111 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(42\) 0 0
\(43\) 4.68466 0.714404 0.357202 0.934027i \(-0.383731\pi\)
0.357202 + 0.934027i \(0.383731\pi\)
\(44\) −7.12311 −1.07385
\(45\) 0 0
\(46\) 6.24621 0.920954
\(47\) −11.1231 −1.62247 −0.811236 0.584719i \(-0.801205\pi\)
−0.811236 + 0.584719i \(0.801205\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 19.6847 2.78383
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −12.2462 −1.68215 −0.841073 0.540921i \(-0.818076\pi\)
−0.841073 + 0.540921i \(0.818076\pi\)
\(54\) 0 0
\(55\) −5.56155 −0.749920
\(56\) 0 0
\(57\) 0 0
\(58\) 21.1231 2.77360
\(59\) 7.12311 0.927349 0.463675 0.886006i \(-0.346531\pi\)
0.463675 + 0.886006i \(0.346531\pi\)
\(60\) 0 0
\(61\) −9.12311 −1.16809 −0.584047 0.811720i \(-0.698532\pi\)
−0.584047 + 0.811720i \(0.698532\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.43845 0.179806
\(65\) −1.56155 −0.193687
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.56155 0.553170
\(69\) 0 0
\(70\) 0 0
\(71\) 6.24621 0.741289 0.370644 0.928775i \(-0.379137\pi\)
0.370644 + 0.928775i \(0.379137\pi\)
\(72\) 0 0
\(73\) 12.2462 1.43331 0.716655 0.697428i \(-0.245672\pi\)
0.716655 + 0.697428i \(0.245672\pi\)
\(74\) −13.1231 −1.52553
\(75\) 0 0
\(76\) 21.3693 2.45123
\(77\) 0 0
\(78\) 0 0
\(79\) −9.36932 −1.05413 −0.527065 0.849825i \(-0.676708\pi\)
−0.527065 + 0.849825i \(0.676708\pi\)
\(80\) 27.3693 3.05998
\(81\) 0 0
\(82\) −9.12311 −1.00748
\(83\) −0.876894 −0.0962517 −0.0481258 0.998841i \(-0.515325\pi\)
−0.0481258 + 0.998841i \(0.515325\pi\)
\(84\) 0 0
\(85\) 3.56155 0.386305
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) −10.2462 −1.09225
\(89\) −1.12311 −0.119049 −0.0595245 0.998227i \(-0.518958\pi\)
−0.0595245 + 0.998227i \(0.518958\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 11.1231 1.15966
\(93\) 0 0
\(94\) −28.4924 −2.93877
\(95\) 16.6847 1.71181
\(96\) 0 0
\(97\) 2.87689 0.292104 0.146052 0.989277i \(-0.453343\pi\)
0.146052 + 0.989277i \(0.453343\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 35.0540 3.50540
\(101\) 10.8769 1.08229 0.541146 0.840929i \(-0.317991\pi\)
0.541146 + 0.840929i \(0.317991\pi\)
\(102\) 0 0
\(103\) −16.6847 −1.64399 −0.821994 0.569496i \(-0.807138\pi\)
−0.821994 + 0.569496i \(0.807138\pi\)
\(104\) −2.87689 −0.282103
\(105\) 0 0
\(106\) −31.3693 −3.04686
\(107\) 4.68466 0.452883 0.226442 0.974025i \(-0.427291\pi\)
0.226442 + 0.974025i \(0.427291\pi\)
\(108\) 0 0
\(109\) −6.87689 −0.658687 −0.329344 0.944210i \(-0.606827\pi\)
−0.329344 + 0.944210i \(0.606827\pi\)
\(110\) −14.2462 −1.35832
\(111\) 0 0
\(112\) 0 0
\(113\) 0.438447 0.0412456 0.0206228 0.999787i \(-0.493435\pi\)
0.0206228 + 0.999787i \(0.493435\pi\)
\(114\) 0 0
\(115\) 8.68466 0.809849
\(116\) 37.6155 3.49251
\(117\) 0 0
\(118\) 18.2462 1.67970
\(119\) 0 0
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) −23.3693 −2.11576
\(123\) 0 0
\(124\) −14.2462 −1.27935
\(125\) 9.56155 0.855211
\(126\) 0 0
\(127\) 19.8078 1.75765 0.878827 0.477140i \(-0.158326\pi\)
0.878827 + 0.477140i \(0.158326\pi\)
\(128\) −9.43845 −0.834249
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 14.4384 1.26149 0.630746 0.775989i \(-0.282749\pi\)
0.630746 + 0.775989i \(0.282749\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 10.2462 0.885138
\(135\) 0 0
\(136\) 6.56155 0.562649
\(137\) −0.246211 −0.0210352 −0.0105176 0.999945i \(-0.503348\pi\)
−0.0105176 + 0.999945i \(0.503348\pi\)
\(138\) 0 0
\(139\) 0.876894 0.0743772 0.0371886 0.999308i \(-0.488160\pi\)
0.0371886 + 0.999308i \(0.488160\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 0.684658 0.0572540
\(144\) 0 0
\(145\) 29.3693 2.43899
\(146\) 31.3693 2.59614
\(147\) 0 0
\(148\) −23.3693 −1.92095
\(149\) −12.2462 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 30.7386 2.49323
\(153\) 0 0
\(154\) 0 0
\(155\) −11.1231 −0.893429
\(156\) 0 0
\(157\) −6.68466 −0.533494 −0.266747 0.963767i \(-0.585949\pi\)
−0.266747 + 0.963767i \(0.585949\pi\)
\(158\) −24.0000 −1.90934
\(159\) 0 0
\(160\) 23.3693 1.84751
\(161\) 0 0
\(162\) 0 0
\(163\) −15.1231 −1.18453 −0.592267 0.805742i \(-0.701767\pi\)
−0.592267 + 0.805742i \(0.701767\pi\)
\(164\) −16.2462 −1.26862
\(165\) 0 0
\(166\) −2.24621 −0.174340
\(167\) 19.8078 1.53277 0.766385 0.642381i \(-0.222053\pi\)
0.766385 + 0.642381i \(0.222053\pi\)
\(168\) 0 0
\(169\) −12.8078 −0.985213
\(170\) 9.12311 0.699710
\(171\) 0 0
\(172\) 21.3693 1.62940
\(173\) 1.80776 0.137442 0.0687209 0.997636i \(-0.478108\pi\)
0.0687209 + 0.997636i \(0.478108\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) −2.87689 −0.215632
\(179\) 0.876894 0.0655422 0.0327711 0.999463i \(-0.489567\pi\)
0.0327711 + 0.999463i \(0.489567\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 16.0000 1.17954
\(185\) −18.2462 −1.34149
\(186\) 0 0
\(187\) −1.56155 −0.114192
\(188\) −50.7386 −3.70050
\(189\) 0 0
\(190\) 42.7386 3.10059
\(191\) 4.87689 0.352880 0.176440 0.984311i \(-0.443542\pi\)
0.176440 + 0.984311i \(0.443542\pi\)
\(192\) 0 0
\(193\) −7.75379 −0.558130 −0.279065 0.960272i \(-0.590024\pi\)
−0.279065 + 0.960272i \(0.590024\pi\)
\(194\) 7.36932 0.529086
\(195\) 0 0
\(196\) 0 0
\(197\) 8.93087 0.636298 0.318149 0.948041i \(-0.396939\pi\)
0.318149 + 0.948041i \(0.396939\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 50.4233 3.56547
\(201\) 0 0
\(202\) 27.8617 1.96035
\(203\) 0 0
\(204\) 0 0
\(205\) −12.6847 −0.885935
\(206\) −42.7386 −2.97774
\(207\) 0 0
\(208\) −3.36932 −0.233620
\(209\) −7.31534 −0.506013
\(210\) 0 0
\(211\) 13.3693 0.920382 0.460191 0.887820i \(-0.347781\pi\)
0.460191 + 0.887820i \(0.347781\pi\)
\(212\) −55.8617 −3.83660
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 16.6847 1.13788
\(216\) 0 0
\(217\) 0 0
\(218\) −17.6155 −1.19307
\(219\) 0 0
\(220\) −25.3693 −1.71040
\(221\) −0.438447 −0.0294931
\(222\) 0 0
\(223\) 14.9309 0.999845 0.499922 0.866070i \(-0.333362\pi\)
0.499922 + 0.866070i \(0.333362\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.12311 0.0747079
\(227\) −14.0540 −0.932795 −0.466398 0.884575i \(-0.654448\pi\)
−0.466398 + 0.884575i \(0.654448\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 22.2462 1.46687
\(231\) 0 0
\(232\) 54.1080 3.55236
\(233\) 3.56155 0.233325 0.116663 0.993172i \(-0.462780\pi\)
0.116663 + 0.993172i \(0.462780\pi\)
\(234\) 0 0
\(235\) −39.6155 −2.58423
\(236\) 32.4924 2.11508
\(237\) 0 0
\(238\) 0 0
\(239\) 6.24621 0.404034 0.202017 0.979382i \(-0.435250\pi\)
0.202017 + 0.979382i \(0.435250\pi\)
\(240\) 0 0
\(241\) −3.36932 −0.217037 −0.108518 0.994094i \(-0.534611\pi\)
−0.108518 + 0.994094i \(0.534611\pi\)
\(242\) −21.9309 −1.40977
\(243\) 0 0
\(244\) −41.6155 −2.66416
\(245\) 0 0
\(246\) 0 0
\(247\) −2.05398 −0.130691
\(248\) −20.4924 −1.30127
\(249\) 0 0
\(250\) 24.4924 1.54904
\(251\) 8.49242 0.536037 0.268018 0.963414i \(-0.413631\pi\)
0.268018 + 0.963414i \(0.413631\pi\)
\(252\) 0 0
\(253\) −3.80776 −0.239392
\(254\) 50.7386 3.18363
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) −15.3693 −0.958712 −0.479356 0.877621i \(-0.659130\pi\)
−0.479356 + 0.877621i \(0.659130\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −7.12311 −0.441756
\(261\) 0 0
\(262\) 36.9848 2.28493
\(263\) 20.4924 1.26362 0.631808 0.775125i \(-0.282313\pi\)
0.631808 + 0.775125i \(0.282313\pi\)
\(264\) 0 0
\(265\) −43.6155 −2.67928
\(266\) 0 0
\(267\) 0 0
\(268\) 18.2462 1.11456
\(269\) 16.4384 1.00227 0.501135 0.865369i \(-0.332916\pi\)
0.501135 + 0.865369i \(0.332916\pi\)
\(270\) 0 0
\(271\) −19.8078 −1.20324 −0.601618 0.798784i \(-0.705477\pi\)
−0.601618 + 0.798784i \(0.705477\pi\)
\(272\) 7.68466 0.465951
\(273\) 0 0
\(274\) −0.630683 −0.0381010
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 2.24621 0.134719
\(279\) 0 0
\(280\) 0 0
\(281\) 10.8769 0.648861 0.324431 0.945910i \(-0.394827\pi\)
0.324431 + 0.945910i \(0.394827\pi\)
\(282\) 0 0
\(283\) −21.3693 −1.27027 −0.635137 0.772399i \(-0.719056\pi\)
−0.635137 + 0.772399i \(0.719056\pi\)
\(284\) 28.4924 1.69071
\(285\) 0 0
\(286\) 1.75379 0.103704
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 75.2311 4.41772
\(291\) 0 0
\(292\) 55.8617 3.26906
\(293\) 1.12311 0.0656125 0.0328063 0.999462i \(-0.489556\pi\)
0.0328063 + 0.999462i \(0.489556\pi\)
\(294\) 0 0
\(295\) 25.3693 1.47706
\(296\) −33.6155 −1.95386
\(297\) 0 0
\(298\) −31.3693 −1.81718
\(299\) −1.06913 −0.0618294
\(300\) 0 0
\(301\) 0 0
\(302\) 20.4924 1.17921
\(303\) 0 0
\(304\) 36.0000 2.06474
\(305\) −32.4924 −1.86051
\(306\) 0 0
\(307\) −32.4924 −1.85444 −0.927220 0.374516i \(-0.877809\pi\)
−0.927220 + 0.374516i \(0.877809\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −28.4924 −1.61826
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −33.6155 −1.90006 −0.950031 0.312156i \(-0.898949\pi\)
−0.950031 + 0.312156i \(0.898949\pi\)
\(314\) −17.1231 −0.966313
\(315\) 0 0
\(316\) −42.7386 −2.40424
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −12.8769 −0.720968
\(320\) 5.12311 0.286390
\(321\) 0 0
\(322\) 0 0
\(323\) 4.68466 0.260661
\(324\) 0 0
\(325\) −3.36932 −0.186896
\(326\) −38.7386 −2.14553
\(327\) 0 0
\(328\) −23.3693 −1.29035
\(329\) 0 0
\(330\) 0 0
\(331\) −34.9309 −1.91997 −0.959987 0.280044i \(-0.909651\pi\)
−0.959987 + 0.280044i \(0.909651\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 50.7386 2.77629
\(335\) 14.2462 0.778354
\(336\) 0 0
\(337\) −16.7386 −0.911811 −0.455906 0.890028i \(-0.650685\pi\)
−0.455906 + 0.890028i \(0.650685\pi\)
\(338\) −32.8078 −1.78451
\(339\) 0 0
\(340\) 16.2462 0.881075
\(341\) 4.87689 0.264099
\(342\) 0 0
\(343\) 0 0
\(344\) 30.7386 1.65732
\(345\) 0 0
\(346\) 4.63068 0.248947
\(347\) 8.49242 0.455897 0.227949 0.973673i \(-0.426798\pi\)
0.227949 + 0.973673i \(0.426798\pi\)
\(348\) 0 0
\(349\) −11.5616 −0.618876 −0.309438 0.950920i \(-0.600141\pi\)
−0.309438 + 0.950920i \(0.600141\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −10.2462 −0.546125
\(353\) −10.4924 −0.558455 −0.279228 0.960225i \(-0.590078\pi\)
−0.279228 + 0.960225i \(0.590078\pi\)
\(354\) 0 0
\(355\) 22.2462 1.18071
\(356\) −5.12311 −0.271524
\(357\) 0 0
\(358\) 2.24621 0.118716
\(359\) −14.2462 −0.751886 −0.375943 0.926643i \(-0.622681\pi\)
−0.375943 + 0.926643i \(0.622681\pi\)
\(360\) 0 0
\(361\) 2.94602 0.155054
\(362\) −15.3693 −0.807793
\(363\) 0 0
\(364\) 0 0
\(365\) 43.6155 2.28294
\(366\) 0 0
\(367\) 1.75379 0.0915470 0.0457735 0.998952i \(-0.485425\pi\)
0.0457735 + 0.998952i \(0.485425\pi\)
\(368\) 18.7386 0.976819
\(369\) 0 0
\(370\) −46.7386 −2.42983
\(371\) 0 0
\(372\) 0 0
\(373\) −0.246211 −0.0127483 −0.00637417 0.999980i \(-0.502029\pi\)
−0.00637417 + 0.999980i \(0.502029\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −72.9848 −3.76391
\(377\) −3.61553 −0.186209
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 76.1080 3.90426
\(381\) 0 0
\(382\) 12.4924 0.639168
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −19.8617 −1.01094
\(387\) 0 0
\(388\) 13.1231 0.666225
\(389\) −35.8617 −1.81826 −0.909131 0.416510i \(-0.863253\pi\)
−0.909131 + 0.416510i \(0.863253\pi\)
\(390\) 0 0
\(391\) 2.43845 0.123318
\(392\) 0 0
\(393\) 0 0
\(394\) 22.8769 1.15252
\(395\) −33.3693 −1.67899
\(396\) 0 0
\(397\) 19.3693 0.972118 0.486059 0.873926i \(-0.338434\pi\)
0.486059 + 0.873926i \(0.338434\pi\)
\(398\) −40.9848 −2.05438
\(399\) 0 0
\(400\) 59.0540 2.95270
\(401\) −39.1771 −1.95641 −0.978205 0.207641i \(-0.933421\pi\)
−0.978205 + 0.207641i \(0.933421\pi\)
\(402\) 0 0
\(403\) 1.36932 0.0682105
\(404\) 49.6155 2.46846
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 14.6847 0.726110 0.363055 0.931768i \(-0.381734\pi\)
0.363055 + 0.931768i \(0.381734\pi\)
\(410\) −32.4924 −1.60469
\(411\) 0 0
\(412\) −76.1080 −3.74957
\(413\) 0 0
\(414\) 0 0
\(415\) −3.12311 −0.153307
\(416\) −2.87689 −0.141051
\(417\) 0 0
\(418\) −18.7386 −0.916537
\(419\) −0.492423 −0.0240564 −0.0120282 0.999928i \(-0.503829\pi\)
−0.0120282 + 0.999928i \(0.503829\pi\)
\(420\) 0 0
\(421\) 24.4384 1.19106 0.595529 0.803334i \(-0.296943\pi\)
0.595529 + 0.803334i \(0.296943\pi\)
\(422\) 34.2462 1.66708
\(423\) 0 0
\(424\) −80.3542 −3.90234
\(425\) 7.68466 0.372761
\(426\) 0 0
\(427\) 0 0
\(428\) 21.3693 1.03292
\(429\) 0 0
\(430\) 42.7386 2.06104
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −26.6847 −1.28238 −0.641191 0.767381i \(-0.721560\pi\)
−0.641191 + 0.767381i \(0.721560\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −31.3693 −1.50232
\(437\) 11.4233 0.546450
\(438\) 0 0
\(439\) 22.2462 1.06175 0.530877 0.847449i \(-0.321863\pi\)
0.530877 + 0.847449i \(0.321863\pi\)
\(440\) −36.4924 −1.73971
\(441\) 0 0
\(442\) −1.12311 −0.0534207
\(443\) 31.1231 1.47870 0.739352 0.673319i \(-0.235132\pi\)
0.739352 + 0.673319i \(0.235132\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) 38.2462 1.81101
\(447\) 0 0
\(448\) 0 0
\(449\) −36.7386 −1.73380 −0.866902 0.498479i \(-0.833892\pi\)
−0.866902 + 0.498479i \(0.833892\pi\)
\(450\) 0 0
\(451\) 5.56155 0.261883
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) −36.0000 −1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) 13.8078 0.645900 0.322950 0.946416i \(-0.395325\pi\)
0.322950 + 0.946416i \(0.395325\pi\)
\(458\) −15.3693 −0.718161
\(459\) 0 0
\(460\) 39.6155 1.84708
\(461\) −8.24621 −0.384064 −0.192032 0.981389i \(-0.561508\pi\)
−0.192032 + 0.981389i \(0.561508\pi\)
\(462\) 0 0
\(463\) −40.9848 −1.90473 −0.952364 0.304965i \(-0.901355\pi\)
−0.952364 + 0.304965i \(0.901355\pi\)
\(464\) 63.3693 2.94185
\(465\) 0 0
\(466\) 9.12311 0.422620
\(467\) 21.3693 0.988854 0.494427 0.869219i \(-0.335378\pi\)
0.494427 + 0.869219i \(0.335378\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −101.477 −4.68080
\(471\) 0 0
\(472\) 46.7386 2.15132
\(473\) −7.31534 −0.336360
\(474\) 0 0
\(475\) 36.0000 1.65179
\(476\) 0 0
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) 24.3002 1.11030 0.555152 0.831749i \(-0.312660\pi\)
0.555152 + 0.831749i \(0.312660\pi\)
\(480\) 0 0
\(481\) 2.24621 0.102418
\(482\) −8.63068 −0.393117
\(483\) 0 0
\(484\) −39.0540 −1.77518
\(485\) 10.2462 0.465256
\(486\) 0 0
\(487\) 17.3693 0.787079 0.393539 0.919308i \(-0.371250\pi\)
0.393539 + 0.919308i \(0.371250\pi\)
\(488\) −59.8617 −2.70981
\(489\) 0 0
\(490\) 0 0
\(491\) 21.3693 0.964384 0.482192 0.876066i \(-0.339841\pi\)
0.482192 + 0.876066i \(0.339841\pi\)
\(492\) 0 0
\(493\) 8.24621 0.371391
\(494\) −5.26137 −0.236720
\(495\) 0 0
\(496\) −24.0000 −1.07763
\(497\) 0 0
\(498\) 0 0
\(499\) 13.3693 0.598493 0.299246 0.954176i \(-0.403265\pi\)
0.299246 + 0.954176i \(0.403265\pi\)
\(500\) 43.6155 1.95055
\(501\) 0 0
\(502\) 21.7538 0.970919
\(503\) −29.5616 −1.31808 −0.659042 0.752106i \(-0.729038\pi\)
−0.659042 + 0.752106i \(0.729038\pi\)
\(504\) 0 0
\(505\) 38.7386 1.72385
\(506\) −9.75379 −0.433609
\(507\) 0 0
\(508\) 90.3542 4.00882
\(509\) 25.1231 1.11356 0.556781 0.830659i \(-0.312036\pi\)
0.556781 + 0.830659i \(0.312036\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −50.4233 −2.22842
\(513\) 0 0
\(514\) −39.3693 −1.73651
\(515\) −59.4233 −2.61850
\(516\) 0 0
\(517\) 17.3693 0.763902
\(518\) 0 0
\(519\) 0 0
\(520\) −10.2462 −0.449326
\(521\) −35.5616 −1.55798 −0.778990 0.627036i \(-0.784268\pi\)
−0.778990 + 0.627036i \(0.784268\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 65.8617 2.87718
\(525\) 0 0
\(526\) 52.4924 2.28878
\(527\) −3.12311 −0.136045
\(528\) 0 0
\(529\) −17.0540 −0.741477
\(530\) −111.723 −4.85296
\(531\) 0 0
\(532\) 0 0
\(533\) 1.56155 0.0676384
\(534\) 0 0
\(535\) 16.6847 0.721341
\(536\) 26.2462 1.13366
\(537\) 0 0
\(538\) 42.1080 1.81540
\(539\) 0 0
\(540\) 0 0
\(541\) 34.1080 1.46642 0.733208 0.680005i \(-0.238022\pi\)
0.733208 + 0.680005i \(0.238022\pi\)
\(542\) −50.7386 −2.17941
\(543\) 0 0
\(544\) 6.56155 0.281324
\(545\) −24.4924 −1.04914
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −1.12311 −0.0479767
\(549\) 0 0
\(550\) −30.7386 −1.31070
\(551\) 38.6307 1.64572
\(552\) 0 0
\(553\) 0 0
\(554\) 15.3693 0.652980
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −26.4924 −1.12252 −0.561260 0.827640i \(-0.689683\pi\)
−0.561260 + 0.827640i \(0.689683\pi\)
\(558\) 0 0
\(559\) −2.05398 −0.0868739
\(560\) 0 0
\(561\) 0 0
\(562\) 27.8617 1.17528
\(563\) 31.1231 1.31168 0.655841 0.754899i \(-0.272314\pi\)
0.655841 + 0.754899i \(0.272314\pi\)
\(564\) 0 0
\(565\) 1.56155 0.0656950
\(566\) −54.7386 −2.30084
\(567\) 0 0
\(568\) 40.9848 1.71969
\(569\) −21.1231 −0.885527 −0.442763 0.896639i \(-0.646002\pi\)
−0.442763 + 0.896639i \(0.646002\pi\)
\(570\) 0 0
\(571\) −30.7386 −1.28637 −0.643186 0.765710i \(-0.722388\pi\)
−0.643186 + 0.765710i \(0.722388\pi\)
\(572\) 3.12311 0.130584
\(573\) 0 0
\(574\) 0 0
\(575\) 18.7386 0.781455
\(576\) 0 0
\(577\) 3.94602 0.164275 0.0821376 0.996621i \(-0.473825\pi\)
0.0821376 + 0.996621i \(0.473825\pi\)
\(578\) 2.56155 0.106547
\(579\) 0 0
\(580\) 133.970 5.56279
\(581\) 0 0
\(582\) 0 0
\(583\) 19.1231 0.791998
\(584\) 80.3542 3.32508
\(585\) 0 0
\(586\) 2.87689 0.118843
\(587\) −28.9848 −1.19633 −0.598166 0.801372i \(-0.704104\pi\)
−0.598166 + 0.801372i \(0.704104\pi\)
\(588\) 0 0
\(589\) −14.6307 −0.602847
\(590\) 64.9848 2.67538
\(591\) 0 0
\(592\) −39.3693 −1.61807
\(593\) 27.7538 1.13971 0.569856 0.821745i \(-0.306999\pi\)
0.569856 + 0.821745i \(0.306999\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −55.8617 −2.28819
\(597\) 0 0
\(598\) −2.73863 −0.111991
\(599\) 0.384472 0.0157091 0.00785455 0.999969i \(-0.497500\pi\)
0.00785455 + 0.999969i \(0.497500\pi\)
\(600\) 0 0
\(601\) 30.9848 1.26390 0.631949 0.775010i \(-0.282255\pi\)
0.631949 + 0.775010i \(0.282255\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 36.4924 1.48486
\(605\) −30.4924 −1.23969
\(606\) 0 0
\(607\) 9.36932 0.380289 0.190144 0.981756i \(-0.439104\pi\)
0.190144 + 0.981756i \(0.439104\pi\)
\(608\) 30.7386 1.24662
\(609\) 0 0
\(610\) −83.2311 −3.36993
\(611\) 4.87689 0.197298
\(612\) 0 0
\(613\) 14.6847 0.593108 0.296554 0.955016i \(-0.404163\pi\)
0.296554 + 0.955016i \(0.404163\pi\)
\(614\) −83.2311 −3.35893
\(615\) 0 0
\(616\) 0 0
\(617\) 44.2462 1.78129 0.890643 0.454704i \(-0.150255\pi\)
0.890643 + 0.454704i \(0.150255\pi\)
\(618\) 0 0
\(619\) −5.36932 −0.215811 −0.107906 0.994161i \(-0.534414\pi\)
−0.107906 + 0.994161i \(0.534414\pi\)
\(620\) −50.7386 −2.03771
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) −86.1080 −3.44157
\(627\) 0 0
\(628\) −30.4924 −1.21678
\(629\) −5.12311 −0.204272
\(630\) 0 0
\(631\) −0.684658 −0.0272558 −0.0136279 0.999907i \(-0.504338\pi\)
−0.0136279 + 0.999907i \(0.504338\pi\)
\(632\) −61.4773 −2.44543
\(633\) 0 0
\(634\) 46.1080 1.83118
\(635\) 70.5464 2.79955
\(636\) 0 0
\(637\) 0 0
\(638\) −32.9848 −1.30588
\(639\) 0 0
\(640\) −33.6155 −1.32877
\(641\) 28.9309 1.14270 0.571350 0.820706i \(-0.306420\pi\)
0.571350 + 0.820706i \(0.306420\pi\)
\(642\) 0 0
\(643\) 13.7538 0.542396 0.271198 0.962524i \(-0.412580\pi\)
0.271198 + 0.962524i \(0.412580\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −9.36932 −0.368346 −0.184173 0.982894i \(-0.558961\pi\)
−0.184173 + 0.982894i \(0.558961\pi\)
\(648\) 0 0
\(649\) −11.1231 −0.436620
\(650\) −8.63068 −0.338523
\(651\) 0 0
\(652\) −68.9848 −2.70166
\(653\) 32.9309 1.28868 0.644342 0.764737i \(-0.277131\pi\)
0.644342 + 0.764737i \(0.277131\pi\)
\(654\) 0 0
\(655\) 51.4233 2.00927
\(656\) −27.3693 −1.06859
\(657\) 0 0
\(658\) 0 0
\(659\) 9.86174 0.384159 0.192079 0.981379i \(-0.438477\pi\)
0.192079 + 0.981379i \(0.438477\pi\)
\(660\) 0 0
\(661\) −13.3153 −0.517907 −0.258953 0.965890i \(-0.583378\pi\)
−0.258953 + 0.965890i \(0.583378\pi\)
\(662\) −89.4773 −3.47763
\(663\) 0 0
\(664\) −5.75379 −0.223290
\(665\) 0 0
\(666\) 0 0
\(667\) 20.1080 0.778583
\(668\) 90.3542 3.49591
\(669\) 0 0
\(670\) 36.4924 1.40983
\(671\) 14.2462 0.549969
\(672\) 0 0
\(673\) −0.738634 −0.0284722 −0.0142361 0.999899i \(-0.504532\pi\)
−0.0142361 + 0.999899i \(0.504532\pi\)
\(674\) −42.8769 −1.65156
\(675\) 0 0
\(676\) −58.4233 −2.24705
\(677\) −1.31534 −0.0505527 −0.0252763 0.999681i \(-0.508047\pi\)
−0.0252763 + 0.999681i \(0.508047\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 23.3693 0.896172
\(681\) 0 0
\(682\) 12.4924 0.478360
\(683\) 9.56155 0.365863 0.182931 0.983126i \(-0.441441\pi\)
0.182931 + 0.983126i \(0.441441\pi\)
\(684\) 0 0
\(685\) −0.876894 −0.0335044
\(686\) 0 0
\(687\) 0 0
\(688\) 36.0000 1.37249
\(689\) 5.36932 0.204555
\(690\) 0 0
\(691\) 28.9848 1.10264 0.551318 0.834295i \(-0.314125\pi\)
0.551318 + 0.834295i \(0.314125\pi\)
\(692\) 8.24621 0.313474
\(693\) 0 0
\(694\) 21.7538 0.825763
\(695\) 3.12311 0.118466
\(696\) 0 0
\(697\) −3.56155 −0.134903
\(698\) −29.6155 −1.12096
\(699\) 0 0
\(700\) 0 0
\(701\) −15.3693 −0.580491 −0.290246 0.956952i \(-0.593737\pi\)
−0.290246 + 0.956952i \(0.593737\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −2.24621 −0.0846573
\(705\) 0 0
\(706\) −26.8769 −1.01153
\(707\) 0 0
\(708\) 0 0
\(709\) −44.7386 −1.68019 −0.840097 0.542436i \(-0.817502\pi\)
−0.840097 + 0.542436i \(0.817502\pi\)
\(710\) 56.9848 2.13860
\(711\) 0 0
\(712\) −7.36932 −0.276177
\(713\) −7.61553 −0.285204
\(714\) 0 0
\(715\) 2.43845 0.0911928
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −36.4924 −1.36189
\(719\) −11.8078 −0.440355 −0.220178 0.975460i \(-0.570664\pi\)
−0.220178 + 0.975460i \(0.570664\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 7.54640 0.280848
\(723\) 0 0
\(724\) −27.3693 −1.01717
\(725\) 63.3693 2.35348
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 111.723 4.13507
\(731\) 4.68466 0.173268
\(732\) 0 0
\(733\) 11.7538 0.434136 0.217068 0.976156i \(-0.430351\pi\)
0.217068 + 0.976156i \(0.430351\pi\)
\(734\) 4.49242 0.165818
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) −6.24621 −0.230082
\(738\) 0 0
\(739\) −20.6847 −0.760897 −0.380449 0.924802i \(-0.624230\pi\)
−0.380449 + 0.924802i \(0.624230\pi\)
\(740\) −83.2311 −3.05963
\(741\) 0 0
\(742\) 0 0
\(743\) −28.4924 −1.04529 −0.522643 0.852552i \(-0.675054\pi\)
−0.522643 + 0.852552i \(0.675054\pi\)
\(744\) 0 0
\(745\) −43.6155 −1.59795
\(746\) −0.630683 −0.0230909
\(747\) 0 0
\(748\) −7.12311 −0.260447
\(749\) 0 0
\(750\) 0 0
\(751\) −25.3693 −0.925740 −0.462870 0.886426i \(-0.653180\pi\)
−0.462870 + 0.886426i \(0.653180\pi\)
\(752\) −85.4773 −3.11704
\(753\) 0 0
\(754\) −9.26137 −0.337279
\(755\) 28.4924 1.03695
\(756\) 0 0
\(757\) 16.0540 0.583492 0.291746 0.956496i \(-0.405764\pi\)
0.291746 + 0.956496i \(0.405764\pi\)
\(758\) −30.7386 −1.11648
\(759\) 0 0
\(760\) 109.477 3.97116
\(761\) −15.7538 −0.571074 −0.285537 0.958368i \(-0.592172\pi\)
−0.285537 + 0.958368i \(0.592172\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 22.2462 0.804840
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) −3.12311 −0.112769
\(768\) 0 0
\(769\) −40.5464 −1.46214 −0.731070 0.682302i \(-0.760979\pi\)
−0.731070 + 0.682302i \(0.760979\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −35.3693 −1.27297
\(773\) −8.63068 −0.310424 −0.155212 0.987881i \(-0.549606\pi\)
−0.155212 + 0.987881i \(0.549606\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 18.8769 0.677641
\(777\) 0 0
\(778\) −91.8617 −3.29340
\(779\) −16.6847 −0.597790
\(780\) 0 0
\(781\) −9.75379 −0.349018
\(782\) 6.24621 0.223364
\(783\) 0 0
\(784\) 0 0
\(785\) −23.8078 −0.849736
\(786\) 0 0
\(787\) 10.2462 0.365238 0.182619 0.983184i \(-0.441543\pi\)
0.182619 + 0.983184i \(0.441543\pi\)
\(788\) 40.7386 1.45125
\(789\) 0 0
\(790\) −85.4773 −3.04114
\(791\) 0 0
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 49.6155 1.76079
\(795\) 0 0
\(796\) −72.9848 −2.58688
\(797\) −9.61553 −0.340599 −0.170300 0.985392i \(-0.554474\pi\)
−0.170300 + 0.985392i \(0.554474\pi\)
\(798\) 0 0
\(799\) −11.1231 −0.393507
\(800\) 50.4233 1.78273
\(801\) 0 0
\(802\) −100.354 −3.54363
\(803\) −19.1231 −0.674840
\(804\) 0 0
\(805\) 0 0
\(806\) 3.50758 0.123549
\(807\) 0 0
\(808\) 71.3693 2.51076
\(809\) −15.9460 −0.560632 −0.280316 0.959908i \(-0.590439\pi\)
−0.280316 + 0.959908i \(0.590439\pi\)
\(810\) 0 0
\(811\) 45.3693 1.59313 0.796566 0.604551i \(-0.206648\pi\)
0.796566 + 0.604551i \(0.206648\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 20.4924 0.718259
\(815\) −53.8617 −1.88669
\(816\) 0 0
\(817\) 21.9460 0.767794
\(818\) 37.6155 1.31520
\(819\) 0 0
\(820\) −57.8617 −2.02062
\(821\) 12.4384 0.434105 0.217052 0.976160i \(-0.430356\pi\)
0.217052 + 0.976160i \(0.430356\pi\)
\(822\) 0 0
\(823\) 3.50758 0.122266 0.0611332 0.998130i \(-0.480529\pi\)
0.0611332 + 0.998130i \(0.480529\pi\)
\(824\) −109.477 −3.81382
\(825\) 0 0
\(826\) 0 0
\(827\) −47.4233 −1.64907 −0.824535 0.565811i \(-0.808563\pi\)
−0.824535 + 0.565811i \(0.808563\pi\)
\(828\) 0 0
\(829\) −17.5076 −0.608063 −0.304032 0.952662i \(-0.598333\pi\)
−0.304032 + 0.952662i \(0.598333\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) −0.630683 −0.0218650
\(833\) 0 0
\(834\) 0 0
\(835\) 70.5464 2.44136
\(836\) −33.3693 −1.15410
\(837\) 0 0
\(838\) −1.26137 −0.0435732
\(839\) −26.0540 −0.899483 −0.449742 0.893159i \(-0.648484\pi\)
−0.449742 + 0.893159i \(0.648484\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 62.6004 2.15735
\(843\) 0 0
\(844\) 60.9848 2.09918
\(845\) −45.6155 −1.56922
\(846\) 0 0
\(847\) 0 0
\(848\) −94.1080 −3.23168
\(849\) 0 0
\(850\) 19.6847 0.675178
\(851\) −12.4924 −0.428235
\(852\) 0 0
\(853\) 28.7386 0.983992 0.491996 0.870597i \(-0.336267\pi\)
0.491996 + 0.870597i \(0.336267\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 30.7386 1.05062
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 76.1080 2.59526
\(861\) 0 0
\(862\) 61.4773 2.09392
\(863\) 9.75379 0.332023 0.166011 0.986124i \(-0.446911\pi\)
0.166011 + 0.986124i \(0.446911\pi\)
\(864\) 0 0
\(865\) 6.43845 0.218914
\(866\) −68.3542 −2.32277
\(867\) 0 0
\(868\) 0 0
\(869\) 14.6307 0.496312
\(870\) 0 0
\(871\) −1.75379 −0.0594249
\(872\) −45.1231 −1.52806
\(873\) 0 0
\(874\) 29.2614 0.989780
\(875\) 0 0
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 56.9848 1.92315
\(879\) 0 0
\(880\) −42.7386 −1.44072
\(881\) 40.2462 1.35593 0.677965 0.735095i \(-0.262862\pi\)
0.677965 + 0.735095i \(0.262862\pi\)
\(882\) 0 0
\(883\) −23.4233 −0.788257 −0.394128 0.919055i \(-0.628953\pi\)
−0.394128 + 0.919055i \(0.628953\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 79.7235 2.67836
\(887\) 18.4384 0.619102 0.309551 0.950883i \(-0.399821\pi\)
0.309551 + 0.950883i \(0.399821\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −10.2462 −0.343454
\(891\) 0 0
\(892\) 68.1080 2.28042
\(893\) −52.1080 −1.74373
\(894\) 0 0
\(895\) 3.12311 0.104394
\(896\) 0 0
\(897\) 0 0
\(898\) −94.1080 −3.14042
\(899\) −25.7538 −0.858937
\(900\) 0 0
\(901\) −12.2462 −0.407980
\(902\) 14.2462 0.474347
\(903\) 0 0
\(904\) 2.87689 0.0956841
\(905\) −21.3693 −0.710340
\(906\) 0 0
\(907\) −9.86174 −0.327454 −0.163727 0.986506i \(-0.552352\pi\)
−0.163727 + 0.986506i \(0.552352\pi\)
\(908\) −64.1080 −2.12750
\(909\) 0 0
\(910\) 0 0
\(911\) −24.3002 −0.805101 −0.402551 0.915398i \(-0.631876\pi\)
−0.402551 + 0.915398i \(0.631876\pi\)
\(912\) 0 0
\(913\) 1.36932 0.0453178
\(914\) 35.3693 1.16991
\(915\) 0 0
\(916\) −27.3693 −0.904308
\(917\) 0 0
\(918\) 0 0
\(919\) −16.6847 −0.550376 −0.275188 0.961390i \(-0.588740\pi\)
−0.275188 + 0.961390i \(0.588740\pi\)
\(920\) 56.9848 1.87873
\(921\) 0 0
\(922\) −21.1231 −0.695652
\(923\) −2.73863 −0.0901432
\(924\) 0 0
\(925\) −39.3693 −1.29446
\(926\) −104.985 −3.45002
\(927\) 0 0
\(928\) 54.1080 1.77618
\(929\) 3.06913 0.100695 0.0503474 0.998732i \(-0.483967\pi\)
0.0503474 + 0.998732i \(0.483967\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 16.2462 0.532162
\(933\) 0 0
\(934\) 54.7386 1.79110
\(935\) −5.56155 −0.181882
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −180.708 −5.89406
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) −8.68466 −0.282811
\(944\) 54.7386 1.78159
\(945\) 0 0
\(946\) −18.7386 −0.609246
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −5.36932 −0.174295
\(950\) 92.2159 2.99188
\(951\) 0 0
\(952\) 0 0
\(953\) −36.3542 −1.17763 −0.588813 0.808269i \(-0.700405\pi\)
−0.588813 + 0.808269i \(0.700405\pi\)
\(954\) 0 0
\(955\) 17.3693 0.562058
\(956\) 28.4924 0.921511
\(957\) 0 0
\(958\) 62.2462 2.01108
\(959\) 0 0
\(960\) 0 0
\(961\) −21.2462 −0.685362
\(962\) 5.75379 0.185510
\(963\) 0 0
\(964\) −15.3693 −0.495012
\(965\) −27.6155 −0.888975
\(966\) 0 0
\(967\) 42.4384 1.36473 0.682364 0.731012i \(-0.260952\pi\)
0.682364 + 0.731012i \(0.260952\pi\)
\(968\) −56.1771 −1.80560
\(969\) 0 0
\(970\) 26.2462 0.842715
\(971\) −43.6155 −1.39969 −0.699844 0.714295i \(-0.746747\pi\)
−0.699844 + 0.714295i \(0.746747\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 44.4924 1.42563
\(975\) 0 0
\(976\) −70.1080 −2.24410
\(977\) −8.24621 −0.263820 −0.131910 0.991262i \(-0.542111\pi\)
−0.131910 + 0.991262i \(0.542111\pi\)
\(978\) 0 0
\(979\) 1.75379 0.0560513
\(980\) 0 0
\(981\) 0 0
\(982\) 54.7386 1.74678
\(983\) 30.9309 0.986542 0.493271 0.869876i \(-0.335801\pi\)
0.493271 + 0.869876i \(0.335801\pi\)
\(984\) 0 0
\(985\) 31.8078 1.01348
\(986\) 21.1231 0.672697
\(987\) 0 0
\(988\) −9.36932 −0.298078
\(989\) 11.4233 0.363240
\(990\) 0 0
\(991\) 42.7386 1.35764 0.678819 0.734306i \(-0.262492\pi\)
0.678819 + 0.734306i \(0.262492\pi\)
\(992\) −20.4924 −0.650635
\(993\) 0 0
\(994\) 0 0
\(995\) −56.9848 −1.80654
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 34.2462 1.08404
\(999\) 0 0
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 7497.2.a.v.1.2 2
3.2 odd 2 2499.2.a.o.1.1 2
7.6 odd 2 153.2.a.e.1.2 2
21.20 even 2 51.2.a.b.1.1 2
28.27 even 2 2448.2.a.v.1.1 2
35.34 odd 2 3825.2.a.s.1.1 2
56.13 odd 2 9792.2.a.cy.1.2 2
56.27 even 2 9792.2.a.cz.1.2 2
84.83 odd 2 816.2.a.m.1.2 2
105.62 odd 4 1275.2.b.d.1174.1 4
105.83 odd 4 1275.2.b.d.1174.4 4
105.104 even 2 1275.2.a.n.1.2 2
119.118 odd 2 2601.2.a.t.1.2 2
168.83 odd 2 3264.2.a.bg.1.1 2
168.125 even 2 3264.2.a.bl.1.1 2
231.230 odd 2 6171.2.a.p.1.2 2
273.272 even 2 8619.2.a.q.1.2 2
357.20 odd 16 867.2.h.j.757.1 16
357.41 odd 16 867.2.h.j.712.4 16
357.62 odd 16 867.2.h.j.733.2 16
357.83 even 8 867.2.e.f.616.2 8
357.104 even 8 867.2.e.f.616.1 8
357.125 odd 16 867.2.h.j.733.1 16
357.146 odd 16 867.2.h.j.712.3 16
357.167 odd 16 867.2.h.j.757.2 16
357.209 odd 16 867.2.h.j.688.4 16
357.230 even 8 867.2.e.f.829.3 8
357.251 even 4 867.2.d.c.577.3 4
357.293 even 4 867.2.d.c.577.4 4
357.314 even 8 867.2.e.f.829.4 8
357.335 odd 16 867.2.h.j.688.3 16
357.356 even 2 867.2.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.2.a.b.1.1 2 21.20 even 2
153.2.a.e.1.2 2 7.6 odd 2
816.2.a.m.1.2 2 84.83 odd 2
867.2.a.f.1.1 2 357.356 even 2
867.2.d.c.577.3 4 357.251 even 4
867.2.d.c.577.4 4 357.293 even 4
867.2.e.f.616.1 8 357.104 even 8
867.2.e.f.616.2 8 357.83 even 8
867.2.e.f.829.3 8 357.230 even 8
867.2.e.f.829.4 8 357.314 even 8
867.2.h.j.688.3 16 357.335 odd 16
867.2.h.j.688.4 16 357.209 odd 16
867.2.h.j.712.3 16 357.146 odd 16
867.2.h.j.712.4 16 357.41 odd 16
867.2.h.j.733.1 16 357.125 odd 16
867.2.h.j.733.2 16 357.62 odd 16
867.2.h.j.757.1 16 357.20 odd 16
867.2.h.j.757.2 16 357.167 odd 16
1275.2.a.n.1.2 2 105.104 even 2
1275.2.b.d.1174.1 4 105.62 odd 4
1275.2.b.d.1174.4 4 105.83 odd 4
2448.2.a.v.1.1 2 28.27 even 2
2499.2.a.o.1.1 2 3.2 odd 2
2601.2.a.t.1.2 2 119.118 odd 2
3264.2.a.bg.1.1 2 168.83 odd 2
3264.2.a.bl.1.1 2 168.125 even 2
3825.2.a.s.1.1 2 35.34 odd 2
6171.2.a.p.1.2 2 231.230 odd 2
7497.2.a.v.1.2 2 1.1 even 1 trivial
8619.2.a.q.1.2 2 273.272 even 2
9792.2.a.cy.1.2 2 56.13 odd 2
9792.2.a.cz.1.2 2 56.27 even 2