Properties

Label 3675.2.a.bt.1.2
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3675,2,Mod(1,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, 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("3675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.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: \(29.3450227428\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 735)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.14896\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.32813 q^{2} -1.00000 q^{3} -0.236068 q^{4} +1.32813 q^{6} +2.96979 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.32813 q^{2} -1.00000 q^{3} -0.236068 q^{4} +1.32813 q^{6} +2.96979 q^{8} +1.00000 q^{9} +4.47214 q^{11} +0.236068 q^{12} -0.763932 q^{13} -3.47214 q^{16} -5.23607 q^{17} -1.32813 q^{18} -1.64166 q^{19} -5.93958 q^{22} +4.29792 q^{23} -2.96979 q^{24} +1.01460 q^{26} -1.00000 q^{27} +2.00000 q^{29} -6.95418 q^{31} -1.32813 q^{32} -4.47214 q^{33} +6.95418 q^{34} -0.236068 q^{36} -8.59584 q^{37} +2.18034 q^{38} +0.763932 q^{39} +9.61045 q^{41} +5.31252 q^{43} -1.05573 q^{44} -5.70820 q^{46} -12.9443 q^{47} +3.47214 q^{48} +5.23607 q^{51} +0.180340 q^{52} -6.95418 q^{53} +1.32813 q^{54} +1.64166 q^{57} -2.65626 q^{58} -5.31252 q^{59} +13.9084 q^{61} +9.23607 q^{62} +8.70820 q^{64} +5.93958 q^{66} +8.59584 q^{67} +1.23607 q^{68} -4.29792 q^{69} -8.47214 q^{71} +2.96979 q^{72} -7.23607 q^{73} +11.4164 q^{74} +0.387543 q^{76} -1.01460 q^{78} -4.94427 q^{79} +1.00000 q^{81} -12.7639 q^{82} +2.47214 q^{83} -7.05573 q^{86} -2.00000 q^{87} +13.2813 q^{88} +12.8938 q^{89} -1.01460 q^{92} +6.95418 q^{93} +17.1917 q^{94} +1.32813 q^{96} +12.1803 q^{97} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{9} - 8 q^{12} - 12 q^{13} + 4 q^{16} - 12 q^{17} - 4 q^{27} + 8 q^{29} + 8 q^{36} - 36 q^{38} + 12 q^{39} - 40 q^{44} + 4 q^{46} - 16 q^{47} - 4 q^{48} + 12 q^{51} - 44 q^{52} + 28 q^{62} + 8 q^{64} - 4 q^{68} - 16 q^{71} - 20 q^{73} - 8 q^{74} + 16 q^{79} + 4 q^{81} - 60 q^{82} - 8 q^{83} - 64 q^{86} - 8 q^{87} + 4 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32813 −0.939130 −0.469565 0.882898i \(-0.655589\pi\)
−0.469565 + 0.882898i \(0.655589\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.236068 −0.118034
\(5\) 0 0
\(6\) 1.32813 0.542207
\(7\) 0 0
\(8\) 2.96979 1.04998
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.47214 1.34840 0.674200 0.738549i \(-0.264489\pi\)
0.674200 + 0.738549i \(0.264489\pi\)
\(12\) 0.236068 0.0681470
\(13\) −0.763932 −0.211877 −0.105938 0.994373i \(-0.533785\pi\)
−0.105938 + 0.994373i \(0.533785\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.47214 −0.868034
\(17\) −5.23607 −1.26993 −0.634967 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) −1.32813 −0.313043
\(19\) −1.64166 −0.376623 −0.188311 0.982109i \(-0.560301\pi\)
−0.188311 + 0.982109i \(0.560301\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.93958 −1.26632
\(23\) 4.29792 0.896179 0.448089 0.893989i \(-0.352105\pi\)
0.448089 + 0.893989i \(0.352105\pi\)
\(24\) −2.96979 −0.606206
\(25\) 0 0
\(26\) 1.01460 0.198980
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −6.95418 −1.24901 −0.624504 0.781021i \(-0.714699\pi\)
−0.624504 + 0.781021i \(0.714699\pi\)
\(32\) −1.32813 −0.234783
\(33\) −4.47214 −0.778499
\(34\) 6.95418 1.19263
\(35\) 0 0
\(36\) −0.236068 −0.0393447
\(37\) −8.59584 −1.41315 −0.706574 0.707639i \(-0.749760\pi\)
−0.706574 + 0.707639i \(0.749760\pi\)
\(38\) 2.18034 0.353698
\(39\) 0.763932 0.122327
\(40\) 0 0
\(41\) 9.61045 1.50090 0.750450 0.660927i \(-0.229837\pi\)
0.750450 + 0.660927i \(0.229837\pi\)
\(42\) 0 0
\(43\) 5.31252 0.810152 0.405076 0.914283i \(-0.367245\pi\)
0.405076 + 0.914283i \(0.367245\pi\)
\(44\) −1.05573 −0.159157
\(45\) 0 0
\(46\) −5.70820 −0.841629
\(47\) −12.9443 −1.88812 −0.944058 0.329779i \(-0.893026\pi\)
−0.944058 + 0.329779i \(0.893026\pi\)
\(48\) 3.47214 0.501160
\(49\) 0 0
\(50\) 0 0
\(51\) 5.23607 0.733196
\(52\) 0.180340 0.0250086
\(53\) −6.95418 −0.955231 −0.477615 0.878569i \(-0.658499\pi\)
−0.477615 + 0.878569i \(0.658499\pi\)
\(54\) 1.32813 0.180736
\(55\) 0 0
\(56\) 0 0
\(57\) 1.64166 0.217443
\(58\) −2.65626 −0.348784
\(59\) −5.31252 −0.691632 −0.345816 0.938302i \(-0.612398\pi\)
−0.345816 + 0.938302i \(0.612398\pi\)
\(60\) 0 0
\(61\) 13.9084 1.78078 0.890392 0.455194i \(-0.150430\pi\)
0.890392 + 0.455194i \(0.150430\pi\)
\(62\) 9.23607 1.17298
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) 0 0
\(66\) 5.93958 0.731112
\(67\) 8.59584 1.05015 0.525075 0.851056i \(-0.324037\pi\)
0.525075 + 0.851056i \(0.324037\pi\)
\(68\) 1.23607 0.149895
\(69\) −4.29792 −0.517409
\(70\) 0 0
\(71\) −8.47214 −1.00546 −0.502729 0.864444i \(-0.667671\pi\)
−0.502729 + 0.864444i \(0.667671\pi\)
\(72\) 2.96979 0.349993
\(73\) −7.23607 −0.846918 −0.423459 0.905915i \(-0.639184\pi\)
−0.423459 + 0.905915i \(0.639184\pi\)
\(74\) 11.4164 1.32713
\(75\) 0 0
\(76\) 0.387543 0.0444543
\(77\) 0 0
\(78\) −1.01460 −0.114881
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −12.7639 −1.40954
\(83\) 2.47214 0.271352 0.135676 0.990753i \(-0.456679\pi\)
0.135676 + 0.990753i \(0.456679\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.05573 −0.760839
\(87\) −2.00000 −0.214423
\(88\) 13.2813 1.41579
\(89\) 12.8938 1.36674 0.683368 0.730074i \(-0.260514\pi\)
0.683368 + 0.730074i \(0.260514\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.01460 −0.105780
\(93\) 6.95418 0.721115
\(94\) 17.1917 1.77319
\(95\) 0 0
\(96\) 1.32813 0.135552
\(97\) 12.1803 1.23673 0.618363 0.785893i \(-0.287796\pi\)
0.618363 + 0.785893i \(0.287796\pi\)
\(98\) 0 0
\(99\) 4.47214 0.449467
\(100\) 0 0
\(101\) −1.01460 −0.100957 −0.0504783 0.998725i \(-0.516075\pi\)
−0.0504783 + 0.998725i \(0.516075\pi\)
\(102\) −6.95418 −0.688567
\(103\) −10.4721 −1.03185 −0.515925 0.856634i \(-0.672552\pi\)
−0.515925 + 0.856634i \(0.672552\pi\)
\(104\) −2.26872 −0.222466
\(105\) 0 0
\(106\) 9.23607 0.897086
\(107\) 9.61045 0.929077 0.464538 0.885553i \(-0.346220\pi\)
0.464538 + 0.885553i \(0.346220\pi\)
\(108\) 0.236068 0.0227157
\(109\) 3.52786 0.337908 0.168954 0.985624i \(-0.445961\pi\)
0.168954 + 0.985624i \(0.445961\pi\)
\(110\) 0 0
\(111\) 8.59584 0.815881
\(112\) 0 0
\(113\) −15.5500 −1.46282 −0.731412 0.681936i \(-0.761138\pi\)
−0.731412 + 0.681936i \(0.761138\pi\)
\(114\) −2.18034 −0.204208
\(115\) 0 0
\(116\) −0.472136 −0.0438367
\(117\) −0.763932 −0.0706255
\(118\) 7.05573 0.649532
\(119\) 0 0
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) −18.4721 −1.67239
\(123\) −9.61045 −0.866545
\(124\) 1.64166 0.147425
\(125\) 0 0
\(126\) 0 0
\(127\) 13.9084 1.23417 0.617084 0.786897i \(-0.288314\pi\)
0.617084 + 0.786897i \(0.288314\pi\)
\(128\) −8.90937 −0.787485
\(129\) −5.31252 −0.467742
\(130\) 0 0
\(131\) 5.31252 0.464157 0.232079 0.972697i \(-0.425447\pi\)
0.232079 + 0.972697i \(0.425447\pi\)
\(132\) 1.05573 0.0918893
\(133\) 0 0
\(134\) −11.4164 −0.986227
\(135\) 0 0
\(136\) −15.5500 −1.33340
\(137\) −1.64166 −0.140256 −0.0701282 0.997538i \(-0.522341\pi\)
−0.0701282 + 0.997538i \(0.522341\pi\)
\(138\) 5.70820 0.485915
\(139\) 10.2375 0.868334 0.434167 0.900832i \(-0.357043\pi\)
0.434167 + 0.900832i \(0.357043\pi\)
\(140\) 0 0
\(141\) 12.9443 1.09010
\(142\) 11.2521 0.944256
\(143\) −3.41641 −0.285694
\(144\) −3.47214 −0.289345
\(145\) 0 0
\(146\) 9.61045 0.795366
\(147\) 0 0
\(148\) 2.02920 0.166800
\(149\) −1.05573 −0.0864886 −0.0432443 0.999065i \(-0.513769\pi\)
−0.0432443 + 0.999065i \(0.513769\pi\)
\(150\) 0 0
\(151\) 8.94427 0.727875 0.363937 0.931423i \(-0.381432\pi\)
0.363937 + 0.931423i \(0.381432\pi\)
\(152\) −4.87539 −0.395446
\(153\) −5.23607 −0.423311
\(154\) 0 0
\(155\) 0 0
\(156\) −0.180340 −0.0144387
\(157\) −22.6525 −1.80786 −0.903932 0.427676i \(-0.859332\pi\)
−0.903932 + 0.427676i \(0.859332\pi\)
\(158\) 6.56664 0.522414
\(159\) 6.95418 0.551503
\(160\) 0 0
\(161\) 0 0
\(162\) −1.32813 −0.104348
\(163\) −13.9084 −1.08939 −0.544694 0.838635i \(-0.683354\pi\)
−0.544694 + 0.838635i \(0.683354\pi\)
\(164\) −2.26872 −0.177157
\(165\) 0 0
\(166\) −3.28332 −0.254835
\(167\) −11.4164 −0.883428 −0.441714 0.897156i \(-0.645629\pi\)
−0.441714 + 0.897156i \(0.645629\pi\)
\(168\) 0 0
\(169\) −12.4164 −0.955108
\(170\) 0 0
\(171\) −1.64166 −0.125541
\(172\) −1.25412 −0.0956255
\(173\) −14.7639 −1.12248 −0.561240 0.827653i \(-0.689676\pi\)
−0.561240 + 0.827653i \(0.689676\pi\)
\(174\) 2.65626 0.201371
\(175\) 0 0
\(176\) −15.5279 −1.17046
\(177\) 5.31252 0.399314
\(178\) −17.1246 −1.28354
\(179\) −16.4721 −1.23119 −0.615593 0.788065i \(-0.711083\pi\)
−0.615593 + 0.788065i \(0.711083\pi\)
\(180\) 0 0
\(181\) −5.31252 −0.394877 −0.197438 0.980315i \(-0.563262\pi\)
−0.197438 + 0.980315i \(0.563262\pi\)
\(182\) 0 0
\(183\) −13.9084 −1.02814
\(184\) 12.7639 0.940970
\(185\) 0 0
\(186\) −9.23607 −0.677221
\(187\) −23.4164 −1.71238
\(188\) 3.05573 0.222862
\(189\) 0 0
\(190\) 0 0
\(191\) −12.4721 −0.902452 −0.451226 0.892410i \(-0.649013\pi\)
−0.451226 + 0.892410i \(0.649013\pi\)
\(192\) −8.70820 −0.628460
\(193\) 19.2209 1.38355 0.691775 0.722113i \(-0.256829\pi\)
0.691775 + 0.722113i \(0.256829\pi\)
\(194\) −16.1771 −1.16145
\(195\) 0 0
\(196\) 0 0
\(197\) 6.95418 0.495465 0.247733 0.968828i \(-0.420315\pi\)
0.247733 + 0.968828i \(0.420315\pi\)
\(198\) −5.93958 −0.422108
\(199\) −12.2667 −0.869564 −0.434782 0.900536i \(-0.643175\pi\)
−0.434782 + 0.900536i \(0.643175\pi\)
\(200\) 0 0
\(201\) −8.59584 −0.606304
\(202\) 1.34752 0.0948115
\(203\) 0 0
\(204\) −1.23607 −0.0865421
\(205\) 0 0
\(206\) 13.9084 0.969042
\(207\) 4.29792 0.298726
\(208\) 2.65248 0.183916
\(209\) −7.34173 −0.507838
\(210\) 0 0
\(211\) 4.94427 0.340378 0.170189 0.985411i \(-0.445562\pi\)
0.170189 + 0.985411i \(0.445562\pi\)
\(212\) 1.64166 0.112750
\(213\) 8.47214 0.580501
\(214\) −12.7639 −0.872524
\(215\) 0 0
\(216\) −2.96979 −0.202069
\(217\) 0 0
\(218\) −4.68547 −0.317340
\(219\) 7.23607 0.488968
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −11.4164 −0.766219
\(223\) −17.8885 −1.19791 −0.598953 0.800784i \(-0.704416\pi\)
−0.598953 + 0.800784i \(0.704416\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 20.6525 1.37378
\(227\) 10.4721 0.695060 0.347530 0.937669i \(-0.387020\pi\)
0.347530 + 0.937669i \(0.387020\pi\)
\(228\) −0.387543 −0.0256657
\(229\) −11.8792 −0.784997 −0.392499 0.919753i \(-0.628389\pi\)
−0.392499 + 0.919753i \(0.628389\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.93958 0.389953
\(233\) 12.2667 0.803619 0.401809 0.915723i \(-0.368381\pi\)
0.401809 + 0.915723i \(0.368381\pi\)
\(234\) 1.01460 0.0663266
\(235\) 0 0
\(236\) 1.25412 0.0816361
\(237\) 4.94427 0.321165
\(238\) 0 0
\(239\) −16.4721 −1.06549 −0.532747 0.846275i \(-0.678840\pi\)
−0.532747 + 0.846275i \(0.678840\pi\)
\(240\) 0 0
\(241\) 8.59584 0.553707 0.276854 0.960912i \(-0.410708\pi\)
0.276854 + 0.960912i \(0.410708\pi\)
\(242\) −11.9532 −0.768379
\(243\) −1.00000 −0.0641500
\(244\) −3.28332 −0.210193
\(245\) 0 0
\(246\) 12.7639 0.813799
\(247\) 1.25412 0.0797975
\(248\) −20.6525 −1.31143
\(249\) −2.47214 −0.156665
\(250\) 0 0
\(251\) −22.5042 −1.42045 −0.710227 0.703973i \(-0.751408\pi\)
−0.710227 + 0.703973i \(0.751408\pi\)
\(252\) 0 0
\(253\) 19.2209 1.20841
\(254\) −18.4721 −1.15904
\(255\) 0 0
\(256\) −5.58359 −0.348975
\(257\) −8.29180 −0.517228 −0.258614 0.965981i \(-0.583266\pi\)
−0.258614 + 0.965981i \(0.583266\pi\)
\(258\) 7.05573 0.439270
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −7.05573 −0.435904
\(263\) −21.4896 −1.32511 −0.662553 0.749015i \(-0.730527\pi\)
−0.662553 + 0.749015i \(0.730527\pi\)
\(264\) −13.2813 −0.817408
\(265\) 0 0
\(266\) 0 0
\(267\) −12.8938 −0.789086
\(268\) −2.02920 −0.123953
\(269\) −9.61045 −0.585959 −0.292980 0.956119i \(-0.594647\pi\)
−0.292980 + 0.956119i \(0.594647\pi\)
\(270\) 0 0
\(271\) −3.67086 −0.222989 −0.111495 0.993765i \(-0.535564\pi\)
−0.111495 + 0.993765i \(0.535564\pi\)
\(272\) 18.1803 1.10235
\(273\) 0 0
\(274\) 2.18034 0.131719
\(275\) 0 0
\(276\) 1.01460 0.0610719
\(277\) −5.31252 −0.319199 −0.159599 0.987182i \(-0.551020\pi\)
−0.159599 + 0.987182i \(0.551020\pi\)
\(278\) −13.5967 −0.815479
\(279\) −6.95418 −0.416336
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −17.1917 −1.02375
\(283\) 32.3607 1.92364 0.961821 0.273678i \(-0.0882403\pi\)
0.961821 + 0.273678i \(0.0882403\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 4.53744 0.268304
\(287\) 0 0
\(288\) −1.32813 −0.0782609
\(289\) 10.4164 0.612730
\(290\) 0 0
\(291\) −12.1803 −0.714024
\(292\) 1.70820 0.0999651
\(293\) 2.18034 0.127377 0.0636884 0.997970i \(-0.479714\pi\)
0.0636884 + 0.997970i \(0.479714\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −25.5279 −1.48378
\(297\) −4.47214 −0.259500
\(298\) 1.40215 0.0812241
\(299\) −3.28332 −0.189879
\(300\) 0 0
\(301\) 0 0
\(302\) −11.8792 −0.683569
\(303\) 1.01460 0.0582874
\(304\) 5.70007 0.326921
\(305\) 0 0
\(306\) 6.95418 0.397544
\(307\) 6.47214 0.369384 0.184692 0.982796i \(-0.440871\pi\)
0.184692 + 0.982796i \(0.440871\pi\)
\(308\) 0 0
\(309\) 10.4721 0.595739
\(310\) 0 0
\(311\) −13.9084 −0.788671 −0.394336 0.918966i \(-0.629025\pi\)
−0.394336 + 0.918966i \(0.629025\pi\)
\(312\) 2.26872 0.128441
\(313\) 6.65248 0.376020 0.188010 0.982167i \(-0.439796\pi\)
0.188010 + 0.982167i \(0.439796\pi\)
\(314\) 30.0855 1.69782
\(315\) 0 0
\(316\) 1.16718 0.0656592
\(317\) −24.1459 −1.35617 −0.678084 0.734985i \(-0.737189\pi\)
−0.678084 + 0.734985i \(0.737189\pi\)
\(318\) −9.23607 −0.517933
\(319\) 8.94427 0.500783
\(320\) 0 0
\(321\) −9.61045 −0.536403
\(322\) 0 0
\(323\) 8.59584 0.478286
\(324\) −0.236068 −0.0131149
\(325\) 0 0
\(326\) 18.4721 1.02308
\(327\) −3.52786 −0.195091
\(328\) 28.5410 1.57591
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −0.583592 −0.0320288
\(333\) −8.59584 −0.471049
\(334\) 15.1625 0.829654
\(335\) 0 0
\(336\) 0 0
\(337\) 13.9084 0.757637 0.378819 0.925471i \(-0.376330\pi\)
0.378819 + 0.925471i \(0.376330\pi\)
\(338\) 16.4906 0.896971
\(339\) 15.5500 0.844562
\(340\) 0 0
\(341\) −31.1001 −1.68416
\(342\) 2.18034 0.117899
\(343\) 0 0
\(344\) 15.7771 0.850644
\(345\) 0 0
\(346\) 19.6084 1.05416
\(347\) 16.1771 0.868432 0.434216 0.900809i \(-0.357026\pi\)
0.434216 + 0.900809i \(0.357026\pi\)
\(348\) 0.472136 0.0253091
\(349\) −31.1001 −1.66475 −0.832374 0.554214i \(-0.813019\pi\)
−0.832374 + 0.554214i \(0.813019\pi\)
\(350\) 0 0
\(351\) 0.763932 0.0407757
\(352\) −5.93958 −0.316581
\(353\) −1.23607 −0.0657893 −0.0328946 0.999459i \(-0.510473\pi\)
−0.0328946 + 0.999459i \(0.510473\pi\)
\(354\) −7.05573 −0.375008
\(355\) 0 0
\(356\) −3.04381 −0.161321
\(357\) 0 0
\(358\) 21.8772 1.15624
\(359\) 11.5279 0.608417 0.304209 0.952605i \(-0.401608\pi\)
0.304209 + 0.952605i \(0.401608\pi\)
\(360\) 0 0
\(361\) −16.3050 −0.858155
\(362\) 7.05573 0.370841
\(363\) −9.00000 −0.472377
\(364\) 0 0
\(365\) 0 0
\(366\) 18.4721 0.965554
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −14.9230 −0.777914
\(369\) 9.61045 0.500300
\(370\) 0 0
\(371\) 0 0
\(372\) −1.64166 −0.0851161
\(373\) 6.56664 0.340008 0.170004 0.985443i \(-0.445622\pi\)
0.170004 + 0.985443i \(0.445622\pi\)
\(374\) 31.1001 1.60815
\(375\) 0 0
\(376\) −38.4418 −1.98248
\(377\) −1.52786 −0.0786890
\(378\) 0 0
\(379\) 20.9443 1.07583 0.537917 0.842997i \(-0.319211\pi\)
0.537917 + 0.842997i \(0.319211\pi\)
\(380\) 0 0
\(381\) −13.9084 −0.712547
\(382\) 16.5646 0.847520
\(383\) −1.52786 −0.0780702 −0.0390351 0.999238i \(-0.512428\pi\)
−0.0390351 + 0.999238i \(0.512428\pi\)
\(384\) 8.90937 0.454655
\(385\) 0 0
\(386\) −25.5279 −1.29933
\(387\) 5.31252 0.270051
\(388\) −2.87539 −0.145976
\(389\) 23.8885 1.21120 0.605599 0.795770i \(-0.292934\pi\)
0.605599 + 0.795770i \(0.292934\pi\)
\(390\) 0 0
\(391\) −22.5042 −1.13809
\(392\) 0 0
\(393\) −5.31252 −0.267981
\(394\) −9.23607 −0.465306
\(395\) 0 0
\(396\) −1.05573 −0.0530523
\(397\) 4.76393 0.239095 0.119547 0.992828i \(-0.461856\pi\)
0.119547 + 0.992828i \(0.461856\pi\)
\(398\) 16.2918 0.816634
\(399\) 0 0
\(400\) 0 0
\(401\) −16.8328 −0.840591 −0.420295 0.907387i \(-0.638074\pi\)
−0.420295 + 0.907387i \(0.638074\pi\)
\(402\) 11.4164 0.569399
\(403\) 5.31252 0.264636
\(404\) 0.239515 0.0119163
\(405\) 0 0
\(406\) 0 0
\(407\) −38.4418 −1.90549
\(408\) 15.5500 0.769841
\(409\) −29.8459 −1.47579 −0.737893 0.674917i \(-0.764179\pi\)
−0.737893 + 0.674917i \(0.764179\pi\)
\(410\) 0 0
\(411\) 1.64166 0.0809771
\(412\) 2.47214 0.121793
\(413\) 0 0
\(414\) −5.70820 −0.280543
\(415\) 0 0
\(416\) 1.01460 0.0497449
\(417\) −10.2375 −0.501333
\(418\) 9.75078 0.476926
\(419\) −36.4126 −1.77887 −0.889436 0.457061i \(-0.848902\pi\)
−0.889436 + 0.457061i \(0.848902\pi\)
\(420\) 0 0
\(421\) 7.52786 0.366886 0.183443 0.983030i \(-0.441276\pi\)
0.183443 + 0.983030i \(0.441276\pi\)
\(422\) −6.56664 −0.319659
\(423\) −12.9443 −0.629372
\(424\) −20.6525 −1.00297
\(425\) 0 0
\(426\) −11.2521 −0.545166
\(427\) 0 0
\(428\) −2.26872 −0.109663
\(429\) 3.41641 0.164946
\(430\) 0 0
\(431\) 4.47214 0.215415 0.107708 0.994183i \(-0.465649\pi\)
0.107708 + 0.994183i \(0.465649\pi\)
\(432\) 3.47214 0.167053
\(433\) −22.2918 −1.07128 −0.535638 0.844448i \(-0.679929\pi\)
−0.535638 + 0.844448i \(0.679929\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.832816 −0.0398846
\(437\) −7.05573 −0.337521
\(438\) −9.61045 −0.459205
\(439\) 4.92498 0.235057 0.117528 0.993070i \(-0.462503\pi\)
0.117528 + 0.993070i \(0.462503\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5.31252 −0.252691
\(443\) −24.7729 −1.17700 −0.588499 0.808498i \(-0.700281\pi\)
−0.588499 + 0.808498i \(0.700281\pi\)
\(444\) −2.02920 −0.0963017
\(445\) 0 0
\(446\) 23.7583 1.12499
\(447\) 1.05573 0.0499342
\(448\) 0 0
\(449\) −34.9443 −1.64912 −0.824561 0.565773i \(-0.808578\pi\)
−0.824561 + 0.565773i \(0.808578\pi\)
\(450\) 0 0
\(451\) 42.9792 2.02381
\(452\) 3.67086 0.172663
\(453\) −8.94427 −0.420239
\(454\) −13.9084 −0.652752
\(455\) 0 0
\(456\) 4.87539 0.228311
\(457\) −24.5334 −1.14762 −0.573812 0.818987i \(-0.694536\pi\)
−0.573812 + 0.818987i \(0.694536\pi\)
\(458\) 15.7771 0.737215
\(459\) 5.23607 0.244399
\(460\) 0 0
\(461\) −25.5480 −1.18989 −0.594945 0.803766i \(-0.702826\pi\)
−0.594945 + 0.803766i \(0.702826\pi\)
\(462\) 0 0
\(463\) 20.4750 0.951554 0.475777 0.879566i \(-0.342167\pi\)
0.475777 + 0.879566i \(0.342167\pi\)
\(464\) −6.94427 −0.322380
\(465\) 0 0
\(466\) −16.2918 −0.754703
\(467\) 0.583592 0.0270054 0.0135027 0.999909i \(-0.495702\pi\)
0.0135027 + 0.999909i \(0.495702\pi\)
\(468\) 0.180340 0.00833621
\(469\) 0 0
\(470\) 0 0
\(471\) 22.6525 1.04377
\(472\) −15.7771 −0.726199
\(473\) 23.7583 1.09241
\(474\) −6.56664 −0.301616
\(475\) 0 0
\(476\) 0 0
\(477\) −6.95418 −0.318410
\(478\) 21.8772 1.00064
\(479\) 20.4750 0.935527 0.467764 0.883854i \(-0.345060\pi\)
0.467764 + 0.883854i \(0.345060\pi\)
\(480\) 0 0
\(481\) 6.56664 0.299413
\(482\) −11.4164 −0.520003
\(483\) 0 0
\(484\) −2.12461 −0.0965733
\(485\) 0 0
\(486\) 1.32813 0.0602452
\(487\) −5.31252 −0.240733 −0.120367 0.992729i \(-0.538407\pi\)
−0.120367 + 0.992729i \(0.538407\pi\)
\(488\) 41.3050 1.86979
\(489\) 13.9084 0.628958
\(490\) 0 0
\(491\) −26.3607 −1.18964 −0.594820 0.803859i \(-0.702777\pi\)
−0.594820 + 0.803859i \(0.702777\pi\)
\(492\) 2.26872 0.102282
\(493\) −10.4721 −0.471641
\(494\) −1.66563 −0.0749403
\(495\) 0 0
\(496\) 24.1459 1.08418
\(497\) 0 0
\(498\) 3.28332 0.147129
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) 11.4164 0.510047
\(502\) 29.8885 1.33399
\(503\) −38.4721 −1.71539 −0.857694 0.514161i \(-0.828103\pi\)
−0.857694 + 0.514161i \(0.828103\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −25.5279 −1.13485
\(507\) 12.4164 0.551432
\(508\) −3.28332 −0.145674
\(509\) −1.01460 −0.0449714 −0.0224857 0.999747i \(-0.507158\pi\)
−0.0224857 + 0.999747i \(0.507158\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 25.2345 1.11522
\(513\) 1.64166 0.0724811
\(514\) 11.0126 0.485745
\(515\) 0 0
\(516\) 1.25412 0.0552094
\(517\) −57.8885 −2.54594
\(518\) 0 0
\(519\) 14.7639 0.648065
\(520\) 0 0
\(521\) 3.04381 0.133352 0.0666758 0.997775i \(-0.478761\pi\)
0.0666758 + 0.997775i \(0.478761\pi\)
\(522\) −2.65626 −0.116261
\(523\) −32.9443 −1.44055 −0.720276 0.693687i \(-0.755985\pi\)
−0.720276 + 0.693687i \(0.755985\pi\)
\(524\) −1.25412 −0.0547863
\(525\) 0 0
\(526\) 28.5410 1.24445
\(527\) 36.4126 1.58616
\(528\) 15.5279 0.675764
\(529\) −4.52786 −0.196864
\(530\) 0 0
\(531\) −5.31252 −0.230544
\(532\) 0 0
\(533\) −7.34173 −0.318006
\(534\) 17.1246 0.741054
\(535\) 0 0
\(536\) 25.5279 1.10264
\(537\) 16.4721 0.710825
\(538\) 12.7639 0.550292
\(539\) 0 0
\(540\) 0 0
\(541\) −15.5279 −0.667595 −0.333798 0.942645i \(-0.608330\pi\)
−0.333798 + 0.942645i \(0.608330\pi\)
\(542\) 4.87539 0.209416
\(543\) 5.31252 0.227982
\(544\) 6.95418 0.298158
\(545\) 0 0
\(546\) 0 0
\(547\) −15.1625 −0.648301 −0.324151 0.946006i \(-0.605078\pi\)
−0.324151 + 0.946006i \(0.605078\pi\)
\(548\) 0.387543 0.0165550
\(549\) 13.9084 0.593595
\(550\) 0 0
\(551\) −3.28332 −0.139874
\(552\) −12.7639 −0.543269
\(553\) 0 0
\(554\) 7.05573 0.299769
\(555\) 0 0
\(556\) −2.41675 −0.102493
\(557\) −13.5208 −0.572896 −0.286448 0.958096i \(-0.592475\pi\)
−0.286448 + 0.958096i \(0.592475\pi\)
\(558\) 9.23607 0.390994
\(559\) −4.05841 −0.171652
\(560\) 0 0
\(561\) 23.4164 0.988642
\(562\) 29.2189 1.23252
\(563\) 21.8885 0.922492 0.461246 0.887272i \(-0.347403\pi\)
0.461246 + 0.887272i \(0.347403\pi\)
\(564\) −3.05573 −0.128669
\(565\) 0 0
\(566\) −42.9792 −1.80655
\(567\) 0 0
\(568\) −25.1605 −1.05571
\(569\) −16.8328 −0.705668 −0.352834 0.935686i \(-0.614782\pi\)
−0.352834 + 0.935686i \(0.614782\pi\)
\(570\) 0 0
\(571\) 10.8328 0.453339 0.226670 0.973972i \(-0.427216\pi\)
0.226670 + 0.973972i \(0.427216\pi\)
\(572\) 0.806504 0.0337216
\(573\) 12.4721 0.521031
\(574\) 0 0
\(575\) 0 0
\(576\) 8.70820 0.362842
\(577\) 14.6525 0.609991 0.304995 0.952354i \(-0.401345\pi\)
0.304995 + 0.952354i \(0.401345\pi\)
\(578\) −13.8344 −0.575433
\(579\) −19.2209 −0.798793
\(580\) 0 0
\(581\) 0 0
\(582\) 16.1771 0.670562
\(583\) −31.1001 −1.28803
\(584\) −21.4896 −0.889246
\(585\) 0 0
\(586\) −2.89578 −0.119623
\(587\) 8.94427 0.369170 0.184585 0.982817i \(-0.440906\pi\)
0.184585 + 0.982817i \(0.440906\pi\)
\(588\) 0 0
\(589\) 11.4164 0.470405
\(590\) 0 0
\(591\) −6.95418 −0.286057
\(592\) 29.8459 1.22666
\(593\) 4.29180 0.176243 0.0881215 0.996110i \(-0.471914\pi\)
0.0881215 + 0.996110i \(0.471914\pi\)
\(594\) 5.93958 0.243704
\(595\) 0 0
\(596\) 0.249224 0.0102086
\(597\) 12.2667 0.502043
\(598\) 4.36068 0.178321
\(599\) −17.4164 −0.711615 −0.355808 0.934559i \(-0.615794\pi\)
−0.355808 + 0.934559i \(0.615794\pi\)
\(600\) 0 0
\(601\) 17.1917 0.701264 0.350632 0.936513i \(-0.385967\pi\)
0.350632 + 0.936513i \(0.385967\pi\)
\(602\) 0 0
\(603\) 8.59584 0.350050
\(604\) −2.11146 −0.0859139
\(605\) 0 0
\(606\) −1.34752 −0.0547394
\(607\) 14.8328 0.602045 0.301023 0.953617i \(-0.402672\pi\)
0.301023 + 0.953617i \(0.402672\pi\)
\(608\) 2.18034 0.0884245
\(609\) 0 0
\(610\) 0 0
\(611\) 9.88854 0.400048
\(612\) 1.23607 0.0499651
\(613\) 23.7583 0.959590 0.479795 0.877381i \(-0.340711\pi\)
0.479795 + 0.877381i \(0.340711\pi\)
\(614\) −8.59584 −0.346900
\(615\) 0 0
\(616\) 0 0
\(617\) 18.8333 0.758202 0.379101 0.925355i \(-0.376233\pi\)
0.379101 + 0.925355i \(0.376233\pi\)
\(618\) −13.9084 −0.559477
\(619\) −41.3376 −1.66150 −0.830748 0.556648i \(-0.812087\pi\)
−0.830748 + 0.556648i \(0.812087\pi\)
\(620\) 0 0
\(621\) −4.29792 −0.172470
\(622\) 18.4721 0.740665
\(623\) 0 0
\(624\) −2.65248 −0.106184
\(625\) 0 0
\(626\) −8.83536 −0.353132
\(627\) 7.34173 0.293200
\(628\) 5.34752 0.213389
\(629\) 45.0084 1.79460
\(630\) 0 0
\(631\) 44.9443 1.78920 0.894602 0.446865i \(-0.147459\pi\)
0.894602 + 0.446865i \(0.147459\pi\)
\(632\) −14.6835 −0.584076
\(633\) −4.94427 −0.196517
\(634\) 32.0689 1.27362
\(635\) 0 0
\(636\) −1.64166 −0.0650961
\(637\) 0 0
\(638\) −11.8792 −0.470301
\(639\) −8.47214 −0.335153
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.7639 0.503752
\(643\) 16.3607 0.645202 0.322601 0.946535i \(-0.395443\pi\)
0.322601 + 0.946535i \(0.395443\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −11.4164 −0.449173
\(647\) 20.5836 0.809225 0.404612 0.914488i \(-0.367406\pi\)
0.404612 + 0.914488i \(0.367406\pi\)
\(648\) 2.96979 0.116664
\(649\) −23.7583 −0.932596
\(650\) 0 0
\(651\) 0 0
\(652\) 3.28332 0.128585
\(653\) 30.7125 1.20187 0.600937 0.799297i \(-0.294794\pi\)
0.600937 + 0.799297i \(0.294794\pi\)
\(654\) 4.68547 0.183216
\(655\) 0 0
\(656\) −33.3688 −1.30283
\(657\) −7.23607 −0.282306
\(658\) 0 0
\(659\) 0.472136 0.0183918 0.00919590 0.999958i \(-0.497073\pi\)
0.00919590 + 0.999958i \(0.497073\pi\)
\(660\) 0 0
\(661\) −13.9084 −0.540973 −0.270486 0.962724i \(-0.587185\pi\)
−0.270486 + 0.962724i \(0.587185\pi\)
\(662\) −10.6250 −0.412954
\(663\) −4.00000 −0.155347
\(664\) 7.34173 0.284914
\(665\) 0 0
\(666\) 11.4164 0.442377
\(667\) 8.59584 0.332832
\(668\) 2.69505 0.104275
\(669\) 17.8885 0.691611
\(670\) 0 0
\(671\) 62.2001 2.40121
\(672\) 0 0
\(673\) −42.9792 −1.65673 −0.828364 0.560191i \(-0.810728\pi\)
−0.828364 + 0.560191i \(0.810728\pi\)
\(674\) −18.4721 −0.711520
\(675\) 0 0
\(676\) 2.93112 0.112735
\(677\) 17.5967 0.676298 0.338149 0.941093i \(-0.390199\pi\)
0.338149 + 0.941093i \(0.390199\pi\)
\(678\) −20.6525 −0.793154
\(679\) 0 0
\(680\) 0 0
\(681\) −10.4721 −0.401293
\(682\) 41.3050 1.58165
\(683\) 20.2355 0.774290 0.387145 0.922019i \(-0.373461\pi\)
0.387145 + 0.922019i \(0.373461\pi\)
\(684\) 0.387543 0.0148181
\(685\) 0 0
\(686\) 0 0
\(687\) 11.8792 0.453218
\(688\) −18.4458 −0.703240
\(689\) 5.31252 0.202391
\(690\) 0 0
\(691\) 0.387543 0.0147428 0.00737142 0.999973i \(-0.497654\pi\)
0.00737142 + 0.999973i \(0.497654\pi\)
\(692\) 3.48529 0.132491
\(693\) 0 0
\(694\) −21.4853 −0.815571
\(695\) 0 0
\(696\) −5.93958 −0.225139
\(697\) −50.3210 −1.90604
\(698\) 41.3050 1.56342
\(699\) −12.2667 −0.463970
\(700\) 0 0
\(701\) −44.8328 −1.69331 −0.846656 0.532141i \(-0.821388\pi\)
−0.846656 + 0.532141i \(0.821388\pi\)
\(702\) −1.01460 −0.0382937
\(703\) 14.1115 0.532224
\(704\) 38.9443 1.46777
\(705\) 0 0
\(706\) 1.64166 0.0617847
\(707\) 0 0
\(708\) −1.25412 −0.0471326
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −4.94427 −0.185425
\(712\) 38.2918 1.43505
\(713\) −29.8885 −1.11933
\(714\) 0 0
\(715\) 0 0
\(716\) 3.88854 0.145322
\(717\) 16.4721 0.615163
\(718\) −15.3105 −0.571383
\(719\) 24.5334 0.914942 0.457471 0.889225i \(-0.348755\pi\)
0.457471 + 0.889225i \(0.348755\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 21.6551 0.805920
\(723\) −8.59584 −0.319683
\(724\) 1.25412 0.0466089
\(725\) 0 0
\(726\) 11.9532 0.443624
\(727\) 10.4721 0.388390 0.194195 0.980963i \(-0.437791\pi\)
0.194195 + 0.980963i \(0.437791\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −27.8167 −1.02884
\(732\) 3.28332 0.121355
\(733\) −11.8197 −0.436569 −0.218285 0.975885i \(-0.570046\pi\)
−0.218285 + 0.975885i \(0.570046\pi\)
\(734\) −21.2501 −0.784355
\(735\) 0 0
\(736\) −5.70820 −0.210407
\(737\) 38.4418 1.41602
\(738\) −12.7639 −0.469847
\(739\) 7.05573 0.259549 0.129775 0.991544i \(-0.458575\pi\)
0.129775 + 0.991544i \(0.458575\pi\)
\(740\) 0 0
\(741\) −1.25412 −0.0460711
\(742\) 0 0
\(743\) 4.29792 0.157675 0.0788377 0.996887i \(-0.474879\pi\)
0.0788377 + 0.996887i \(0.474879\pi\)
\(744\) 20.6525 0.757157
\(745\) 0 0
\(746\) −8.72136 −0.319312
\(747\) 2.47214 0.0904507
\(748\) 5.52786 0.202119
\(749\) 0 0
\(750\) 0 0
\(751\) 42.8328 1.56299 0.781496 0.623910i \(-0.214457\pi\)
0.781496 + 0.623910i \(0.214457\pi\)
\(752\) 44.9443 1.63895
\(753\) 22.5042 0.820099
\(754\) 2.02920 0.0738992
\(755\) 0 0
\(756\) 0 0
\(757\) −20.4750 −0.744177 −0.372088 0.928197i \(-0.621358\pi\)
−0.372088 + 0.928197i \(0.621358\pi\)
\(758\) −27.8167 −1.01035
\(759\) −19.2209 −0.697674
\(760\) 0 0
\(761\) 32.1147 1.16416 0.582078 0.813133i \(-0.302240\pi\)
0.582078 + 0.813133i \(0.302240\pi\)
\(762\) 18.4721 0.669175
\(763\) 0 0
\(764\) 2.94427 0.106520
\(765\) 0 0
\(766\) 2.02920 0.0733181
\(767\) 4.05841 0.146541
\(768\) 5.58359 0.201481
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 8.29180 0.298622
\(772\) −4.53744 −0.163306
\(773\) 16.6525 0.598948 0.299474 0.954104i \(-0.403189\pi\)
0.299474 + 0.954104i \(0.403189\pi\)
\(774\) −7.05573 −0.253613
\(775\) 0 0
\(776\) 36.1731 1.29854
\(777\) 0 0
\(778\) −31.7271 −1.13747
\(779\) −15.7771 −0.565273
\(780\) 0 0
\(781\) −37.8885 −1.35576
\(782\) 29.8885 1.06881
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) 7.05573 0.251669
\(787\) 23.0557 0.821848 0.410924 0.911670i \(-0.365206\pi\)
0.410924 + 0.911670i \(0.365206\pi\)
\(788\) −1.64166 −0.0584817
\(789\) 21.4896 0.765050
\(790\) 0 0
\(791\) 0 0
\(792\) 13.2813 0.471931
\(793\) −10.6250 −0.377307
\(794\) −6.32713 −0.224541
\(795\) 0 0
\(796\) 2.89578 0.102638
\(797\) −22.7639 −0.806340 −0.403170 0.915125i \(-0.632092\pi\)
−0.403170 + 0.915125i \(0.632092\pi\)
\(798\) 0 0
\(799\) 67.7771 2.39778
\(800\) 0 0
\(801\) 12.8938 0.455579
\(802\) 22.3562 0.789424
\(803\) −32.3607 −1.14198
\(804\) 2.02920 0.0715645
\(805\) 0 0
\(806\) −7.05573 −0.248527
\(807\) 9.61045 0.338304
\(808\) −3.01316 −0.106002
\(809\) −13.0557 −0.459015 −0.229507 0.973307i \(-0.573712\pi\)
−0.229507 + 0.973307i \(0.573712\pi\)
\(810\) 0 0
\(811\) −13.5208 −0.474780 −0.237390 0.971414i \(-0.576292\pi\)
−0.237390 + 0.971414i \(0.576292\pi\)
\(812\) 0 0
\(813\) 3.67086 0.128743
\(814\) 51.0557 1.78950
\(815\) 0 0
\(816\) −18.1803 −0.636439
\(817\) −8.72136 −0.305122
\(818\) 39.6393 1.38596
\(819\) 0 0
\(820\) 0 0
\(821\) −27.8885 −0.973317 −0.486658 0.873592i \(-0.661784\pi\)
−0.486658 + 0.873592i \(0.661784\pi\)
\(822\) −2.18034 −0.0760481
\(823\) −41.7251 −1.45445 −0.727223 0.686401i \(-0.759189\pi\)
−0.727223 + 0.686401i \(0.759189\pi\)
\(824\) −31.1001 −1.08342
\(825\) 0 0
\(826\) 0 0
\(827\) 30.8605 1.07313 0.536563 0.843860i \(-0.319722\pi\)
0.536563 + 0.843860i \(0.319722\pi\)
\(828\) −1.01460 −0.0352599
\(829\) 33.1293 1.15063 0.575313 0.817933i \(-0.304880\pi\)
0.575313 + 0.817933i \(0.304880\pi\)
\(830\) 0 0
\(831\) 5.31252 0.184289
\(832\) −6.65248 −0.230633
\(833\) 0 0
\(834\) 13.5967 0.470817
\(835\) 0 0
\(836\) 1.73315 0.0599421
\(837\) 6.95418 0.240372
\(838\) 48.3607 1.67059
\(839\) 4.05841 0.140112 0.0700559 0.997543i \(-0.477682\pi\)
0.0700559 + 0.997543i \(0.477682\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −9.99799 −0.344553
\(843\) 22.0000 0.757720
\(844\) −1.16718 −0.0401761
\(845\) 0 0
\(846\) 17.1917 0.591062
\(847\) 0 0
\(848\) 24.1459 0.829173
\(849\) −32.3607 −1.11062
\(850\) 0 0
\(851\) −36.9443 −1.26643
\(852\) −2.00000 −0.0685189
\(853\) −45.1246 −1.54504 −0.772519 0.634992i \(-0.781003\pi\)
−0.772519 + 0.634992i \(0.781003\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 28.5410 0.975512
\(857\) −4.87539 −0.166540 −0.0832700 0.996527i \(-0.526536\pi\)
−0.0832700 + 0.996527i \(0.526536\pi\)
\(858\) −4.53744 −0.154906
\(859\) 29.4584 1.00511 0.502554 0.864546i \(-0.332394\pi\)
0.502554 + 0.864546i \(0.332394\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −5.93958 −0.202303
\(863\) −23.5188 −0.800590 −0.400295 0.916386i \(-0.631092\pi\)
−0.400295 + 0.916386i \(0.631092\pi\)
\(864\) 1.32813 0.0451839
\(865\) 0 0
\(866\) 29.6064 1.00607
\(867\) −10.4164 −0.353760
\(868\) 0 0
\(869\) −22.1115 −0.750080
\(870\) 0 0
\(871\) −6.56664 −0.222502
\(872\) 10.4770 0.354797
\(873\) 12.1803 0.412242
\(874\) 9.37093 0.316976
\(875\) 0 0
\(876\) −1.70820 −0.0577149
\(877\) −55.6335 −1.87861 −0.939304 0.343085i \(-0.888528\pi\)
−0.939304 + 0.343085i \(0.888528\pi\)
\(878\) −6.54102 −0.220749
\(879\) −2.18034 −0.0735410
\(880\) 0 0
\(881\) −19.4604 −0.655638 −0.327819 0.944741i \(-0.606314\pi\)
−0.327819 + 0.944741i \(0.606314\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −0.944272 −0.0317593
\(885\) 0 0
\(886\) 32.9017 1.10535
\(887\) 35.7771 1.20128 0.600639 0.799521i \(-0.294913\pi\)
0.600639 + 0.799521i \(0.294913\pi\)
\(888\) 25.5279 0.856659
\(889\) 0 0
\(890\) 0 0
\(891\) 4.47214 0.149822
\(892\) 4.22291 0.141394
\(893\) 21.2501 0.711107
\(894\) −1.40215 −0.0468948
\(895\) 0 0
\(896\) 0 0
\(897\) 3.28332 0.109627
\(898\) 46.4106 1.54874
\(899\) −13.9084 −0.463870
\(900\) 0 0
\(901\) 36.4126 1.21308
\(902\) −57.0820 −1.90062
\(903\) 0 0
\(904\) −46.1803 −1.53594
\(905\) 0 0
\(906\) 11.8792 0.394659
\(907\) −21.2501 −0.705598 −0.352799 0.935699i \(-0.614770\pi\)
−0.352799 + 0.935699i \(0.614770\pi\)
\(908\) −2.47214 −0.0820407
\(909\) −1.01460 −0.0336522
\(910\) 0 0
\(911\) −1.41641 −0.0469277 −0.0234638 0.999725i \(-0.507469\pi\)
−0.0234638 + 0.999725i \(0.507469\pi\)
\(912\) −5.70007 −0.188748
\(913\) 11.0557 0.365891
\(914\) 32.5836 1.07777
\(915\) 0 0
\(916\) 2.80429 0.0926564
\(917\) 0 0
\(918\) −6.95418 −0.229522
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) −6.47214 −0.213264
\(922\) 33.9311 1.11746
\(923\) 6.47214 0.213033
\(924\) 0 0
\(925\) 0 0
\(926\) −27.1935 −0.893634
\(927\) −10.4721 −0.343950
\(928\) −2.65626 −0.0871961
\(929\) 4.29792 0.141010 0.0705051 0.997511i \(-0.477539\pi\)
0.0705051 + 0.997511i \(0.477539\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.89578 −0.0948543
\(933\) 13.9084 0.455340
\(934\) −0.775087 −0.0253616
\(935\) 0 0
\(936\) −2.26872 −0.0741554
\(937\) −40.5410 −1.32442 −0.662209 0.749319i \(-0.730381\pi\)
−0.662209 + 0.749319i \(0.730381\pi\)
\(938\) 0 0
\(939\) −6.65248 −0.217095
\(940\) 0 0
\(941\) 10.8646 0.354175 0.177087 0.984195i \(-0.443332\pi\)
0.177087 + 0.984195i \(0.443332\pi\)
\(942\) −30.0855 −0.980237
\(943\) 41.3050 1.34507
\(944\) 18.4458 0.600360
\(945\) 0 0
\(946\) −31.5542 −1.02591
\(947\) −37.4272 −1.21622 −0.608110 0.793853i \(-0.708072\pi\)
−0.608110 + 0.793853i \(0.708072\pi\)
\(948\) −1.16718 −0.0379084
\(949\) 5.52786 0.179442
\(950\) 0 0
\(951\) 24.1459 0.782984
\(952\) 0 0
\(953\) 33.5168 1.08572 0.542858 0.839825i \(-0.317342\pi\)
0.542858 + 0.839825i \(0.317342\pi\)
\(954\) 9.23607 0.299029
\(955\) 0 0
\(956\) 3.88854 0.125764
\(957\) −8.94427 −0.289127
\(958\) −27.1935 −0.878582
\(959\) 0 0
\(960\) 0 0
\(961\) 17.3607 0.560022
\(962\) −8.72136 −0.281188
\(963\) 9.61045 0.309692
\(964\) −2.02920 −0.0653562
\(965\) 0 0
\(966\) 0 0
\(967\) −2.02920 −0.0652548 −0.0326274 0.999468i \(-0.510387\pi\)
−0.0326274 + 0.999468i \(0.510387\pi\)
\(968\) 26.7281 0.859074
\(969\) −8.59584 −0.276138
\(970\) 0 0
\(971\) 4.53744 0.145613 0.0728066 0.997346i \(-0.476804\pi\)
0.0728066 + 0.997346i \(0.476804\pi\)
\(972\) 0.236068 0.00757188
\(973\) 0 0
\(974\) 7.05573 0.226080
\(975\) 0 0
\(976\) −48.2917 −1.54578
\(977\) −4.92498 −0.157564 −0.0787820 0.996892i \(-0.525103\pi\)
−0.0787820 + 0.996892i \(0.525103\pi\)
\(978\) −18.4721 −0.590674
\(979\) 57.6627 1.84291
\(980\) 0 0
\(981\) 3.52786 0.112636
\(982\) 35.0104 1.11723
\(983\) −14.8328 −0.473093 −0.236547 0.971620i \(-0.576016\pi\)
−0.236547 + 0.971620i \(0.576016\pi\)
\(984\) −28.5410 −0.909854
\(985\) 0 0
\(986\) 13.9084 0.442933
\(987\) 0 0
\(988\) −0.296057 −0.00941882
\(989\) 22.8328 0.726041
\(990\) 0 0
\(991\) 36.0000 1.14358 0.571789 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) 9.23607 0.293245
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) 0.583592 0.0184918
\(997\) −14.2918 −0.452626 −0.226313 0.974055i \(-0.572667\pi\)
−0.226313 + 0.974055i \(0.572667\pi\)
\(998\) −10.6250 −0.336330
\(999\) 8.59584 0.271960
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 3675.2.a.bt.1.2 4
5.2 odd 4 735.2.d.c.589.4 yes 8
5.3 odd 4 735.2.d.c.589.5 yes 8
5.4 even 2 3675.2.a.bv.1.3 4
7.6 odd 2 3675.2.a.bv.1.2 4
15.2 even 4 2205.2.d.m.1324.6 8
15.8 even 4 2205.2.d.m.1324.4 8
35.2 odd 12 735.2.q.h.214.3 16
35.3 even 12 735.2.q.h.79.4 16
35.12 even 12 735.2.q.h.214.4 16
35.13 even 4 735.2.d.c.589.6 yes 8
35.17 even 12 735.2.q.h.79.5 16
35.18 odd 12 735.2.q.h.79.3 16
35.23 odd 12 735.2.q.h.214.6 16
35.27 even 4 735.2.d.c.589.3 8
35.32 odd 12 735.2.q.h.79.6 16
35.33 even 12 735.2.q.h.214.5 16
35.34 odd 2 inner 3675.2.a.bt.1.3 4
105.62 odd 4 2205.2.d.m.1324.5 8
105.83 odd 4 2205.2.d.m.1324.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
735.2.d.c.589.3 8 35.27 even 4
735.2.d.c.589.4 yes 8 5.2 odd 4
735.2.d.c.589.5 yes 8 5.3 odd 4
735.2.d.c.589.6 yes 8 35.13 even 4
735.2.q.h.79.3 16 35.18 odd 12
735.2.q.h.79.4 16 35.3 even 12
735.2.q.h.79.5 16 35.17 even 12
735.2.q.h.79.6 16 35.32 odd 12
735.2.q.h.214.3 16 35.2 odd 12
735.2.q.h.214.4 16 35.12 even 12
735.2.q.h.214.5 16 35.33 even 12
735.2.q.h.214.6 16 35.23 odd 12
2205.2.d.m.1324.3 8 105.83 odd 4
2205.2.d.m.1324.4 8 15.8 even 4
2205.2.d.m.1324.5 8 105.62 odd 4
2205.2.d.m.1324.6 8 15.2 even 4
3675.2.a.bt.1.2 4 1.1 even 1 trivial
3675.2.a.bt.1.3 4 35.34 odd 2 inner
3675.2.a.bv.1.2 4 7.6 odd 2
3675.2.a.bv.1.3 4 5.4 even 2