Properties

Label 9450.2.a.ec.1.2
Level $9450$
Weight $2$
Character 9450.1
Self dual yes
Analytic conductor $75.459$
Analytic rank $1$
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] = [9450,2,Mod(1,9450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, 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("9450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9450.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: \(75.4586299101\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 9450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +5.29150 q^{11} +4.64575 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -6.29150 q^{19} -5.29150 q^{22} -1.64575 q^{23} -4.64575 q^{26} -1.00000 q^{28} -10.6458 q^{29} -7.29150 q^{31} -1.00000 q^{32} -1.00000 q^{34} +9.64575 q^{37} +6.29150 q^{38} +7.29150 q^{41} -0.354249 q^{43} +5.29150 q^{44} +1.64575 q^{46} -0.645751 q^{47} +1.00000 q^{49} +4.64575 q^{52} -2.64575 q^{53} +1.00000 q^{56} +10.6458 q^{58} -13.2915 q^{59} +2.64575 q^{61} +7.29150 q^{62} +1.00000 q^{64} -4.93725 q^{67} +1.00000 q^{68} -11.2915 q^{71} +5.29150 q^{73} -9.64575 q^{74} -6.29150 q^{76} -5.29150 q^{77} -15.9373 q^{79} -7.29150 q^{82} -3.64575 q^{83} +0.354249 q^{86} -5.29150 q^{88} -1.00000 q^{89} -4.64575 q^{91} -1.64575 q^{92} +0.645751 q^{94} +12.9373 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{23} - 4 q^{26} - 2 q^{28} - 16 q^{29} - 4 q^{31} - 2 q^{32} - 2 q^{34} + 14 q^{37} + 2 q^{38} + 4 q^{41} - 6 q^{43} - 2 q^{46} + 4 q^{47} + 2 q^{49} + 4 q^{52} + 2 q^{56} + 16 q^{58} - 16 q^{59} + 4 q^{62} + 2 q^{64} + 6 q^{67} + 2 q^{68} - 12 q^{71} - 14 q^{74} - 2 q^{76} - 16 q^{79} - 4 q^{82} - 2 q^{83} + 6 q^{86} - 2 q^{89} - 4 q^{91} + 2 q^{92} - 4 q^{94} + 10 q^{97} - 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 5.29150 1.59545 0.797724 0.603023i \(-0.206037\pi\)
0.797724 + 0.603023i \(0.206037\pi\)
\(12\) 0 0
\(13\) 4.64575 1.28850 0.644250 0.764815i \(-0.277170\pi\)
0.644250 + 0.764815i \(0.277170\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) −6.29150 −1.44337 −0.721685 0.692222i \(-0.756632\pi\)
−0.721685 + 0.692222i \(0.756632\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.29150 −1.12815
\(23\) −1.64575 −0.343163 −0.171581 0.985170i \(-0.554888\pi\)
−0.171581 + 0.985170i \(0.554888\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.64575 −0.911107
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −10.6458 −1.97687 −0.988433 0.151657i \(-0.951539\pi\)
−0.988433 + 0.151657i \(0.951539\pi\)
\(30\) 0 0
\(31\) −7.29150 −1.30959 −0.654796 0.755805i \(-0.727246\pi\)
−0.654796 + 0.755805i \(0.727246\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 0 0
\(37\) 9.64575 1.58575 0.792876 0.609383i \(-0.208583\pi\)
0.792876 + 0.609383i \(0.208583\pi\)
\(38\) 6.29150 1.02062
\(39\) 0 0
\(40\) 0 0
\(41\) 7.29150 1.13874 0.569371 0.822081i \(-0.307187\pi\)
0.569371 + 0.822081i \(0.307187\pi\)
\(42\) 0 0
\(43\) −0.354249 −0.0540224 −0.0270112 0.999635i \(-0.508599\pi\)
−0.0270112 + 0.999635i \(0.508599\pi\)
\(44\) 5.29150 0.797724
\(45\) 0 0
\(46\) 1.64575 0.242653
\(47\) −0.645751 −0.0941925 −0.0470963 0.998890i \(-0.514997\pi\)
−0.0470963 + 0.998890i \(0.514997\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 4.64575 0.644250
\(53\) −2.64575 −0.363422 −0.181711 0.983352i \(-0.558164\pi\)
−0.181711 + 0.983352i \(0.558164\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 10.6458 1.39786
\(59\) −13.2915 −1.73041 −0.865203 0.501422i \(-0.832811\pi\)
−0.865203 + 0.501422i \(0.832811\pi\)
\(60\) 0 0
\(61\) 2.64575 0.338754 0.169377 0.985551i \(-0.445824\pi\)
0.169377 + 0.985551i \(0.445824\pi\)
\(62\) 7.29150 0.926022
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.93725 −0.603182 −0.301591 0.953437i \(-0.597518\pi\)
−0.301591 + 0.953437i \(0.597518\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) −11.2915 −1.34005 −0.670027 0.742336i \(-0.733718\pi\)
−0.670027 + 0.742336i \(0.733718\pi\)
\(72\) 0 0
\(73\) 5.29150 0.619324 0.309662 0.950847i \(-0.399784\pi\)
0.309662 + 0.950847i \(0.399784\pi\)
\(74\) −9.64575 −1.12130
\(75\) 0 0
\(76\) −6.29150 −0.721685
\(77\) −5.29150 −0.603023
\(78\) 0 0
\(79\) −15.9373 −1.79308 −0.896541 0.442962i \(-0.853928\pi\)
−0.896541 + 0.442962i \(0.853928\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.29150 −0.805212
\(83\) −3.64575 −0.400173 −0.200087 0.979778i \(-0.564122\pi\)
−0.200087 + 0.979778i \(0.564122\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.354249 0.0381996
\(87\) 0 0
\(88\) −5.29150 −0.564076
\(89\) −1.00000 −0.106000 −0.0529999 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) −4.64575 −0.487007
\(92\) −1.64575 −0.171581
\(93\) 0 0
\(94\) 0.645751 0.0666042
\(95\) 0 0
\(96\) 0 0
\(97\) 12.9373 1.31358 0.656790 0.754074i \(-0.271914\pi\)
0.656790 + 0.754074i \(0.271914\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) −16.9373 −1.68532 −0.842660 0.538446i \(-0.819012\pi\)
−0.842660 + 0.538446i \(0.819012\pi\)
\(102\) 0 0
\(103\) −17.8745 −1.76123 −0.880614 0.473835i \(-0.842869\pi\)
−0.880614 + 0.473835i \(0.842869\pi\)
\(104\) −4.64575 −0.455553
\(105\) 0 0
\(106\) 2.64575 0.256978
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −1.64575 −0.157634 −0.0788172 0.996889i \(-0.525114\pi\)
−0.0788172 + 0.996889i \(0.525114\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −8.58301 −0.807421 −0.403711 0.914887i \(-0.632280\pi\)
−0.403711 + 0.914887i \(0.632280\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.6458 −0.988433
\(117\) 0 0
\(118\) 13.2915 1.22358
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 17.0000 1.54545
\(122\) −2.64575 −0.239535
\(123\) 0 0
\(124\) −7.29150 −0.654796
\(125\) 0 0
\(126\) 0 0
\(127\) 14.6458 1.29960 0.649800 0.760105i \(-0.274853\pi\)
0.649800 + 0.760105i \(0.274853\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −18.9373 −1.65456 −0.827278 0.561793i \(-0.810112\pi\)
−0.827278 + 0.561793i \(0.810112\pi\)
\(132\) 0 0
\(133\) 6.29150 0.545542
\(134\) 4.93725 0.426514
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −7.64575 −0.653221 −0.326610 0.945159i \(-0.605906\pi\)
−0.326610 + 0.945159i \(0.605906\pi\)
\(138\) 0 0
\(139\) 4.58301 0.388725 0.194363 0.980930i \(-0.437736\pi\)
0.194363 + 0.980930i \(0.437736\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 11.2915 0.947562
\(143\) 24.5830 2.05573
\(144\) 0 0
\(145\) 0 0
\(146\) −5.29150 −0.437928
\(147\) 0 0
\(148\) 9.64575 0.792876
\(149\) 14.5203 1.18955 0.594773 0.803894i \(-0.297242\pi\)
0.594773 + 0.803894i \(0.297242\pi\)
\(150\) 0 0
\(151\) 13.2288 1.07654 0.538270 0.842772i \(-0.319078\pi\)
0.538270 + 0.842772i \(0.319078\pi\)
\(152\) 6.29150 0.510308
\(153\) 0 0
\(154\) 5.29150 0.426401
\(155\) 0 0
\(156\) 0 0
\(157\) −6.64575 −0.530389 −0.265194 0.964195i \(-0.585436\pi\)
−0.265194 + 0.964195i \(0.585436\pi\)
\(158\) 15.9373 1.26790
\(159\) 0 0
\(160\) 0 0
\(161\) 1.64575 0.129703
\(162\) 0 0
\(163\) −19.8745 −1.55669 −0.778346 0.627836i \(-0.783941\pi\)
−0.778346 + 0.627836i \(0.783941\pi\)
\(164\) 7.29150 0.569371
\(165\) 0 0
\(166\) 3.64575 0.282965
\(167\) 25.1660 1.94740 0.973702 0.227825i \(-0.0731613\pi\)
0.973702 + 0.227825i \(0.0731613\pi\)
\(168\) 0 0
\(169\) 8.58301 0.660231
\(170\) 0 0
\(171\) 0 0
\(172\) −0.354249 −0.0270112
\(173\) −3.64575 −0.277181 −0.138591 0.990350i \(-0.544257\pi\)
−0.138591 + 0.990350i \(0.544257\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.29150 0.398862
\(177\) 0 0
\(178\) 1.00000 0.0749532
\(179\) −3.58301 −0.267806 −0.133903 0.990994i \(-0.542751\pi\)
−0.133903 + 0.990994i \(0.542751\pi\)
\(180\) 0 0
\(181\) 17.3542 1.28993 0.644966 0.764212i \(-0.276872\pi\)
0.644966 + 0.764212i \(0.276872\pi\)
\(182\) 4.64575 0.344366
\(183\) 0 0
\(184\) 1.64575 0.121326
\(185\) 0 0
\(186\) 0 0
\(187\) 5.29150 0.386953
\(188\) −0.645751 −0.0470963
\(189\) 0 0
\(190\) 0 0
\(191\) 6.58301 0.476330 0.238165 0.971225i \(-0.423454\pi\)
0.238165 + 0.971225i \(0.423454\pi\)
\(192\) 0 0
\(193\) −16.8745 −1.21465 −0.607327 0.794452i \(-0.707758\pi\)
−0.607327 + 0.794452i \(0.707758\pi\)
\(194\) −12.9373 −0.928841
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.6458 −1.32845 −0.664227 0.747531i \(-0.731239\pi\)
−0.664227 + 0.747531i \(0.731239\pi\)
\(198\) 0 0
\(199\) 19.5203 1.38375 0.691877 0.722015i \(-0.256784\pi\)
0.691877 + 0.722015i \(0.256784\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 16.9373 1.19170
\(203\) 10.6458 0.747185
\(204\) 0 0
\(205\) 0 0
\(206\) 17.8745 1.24538
\(207\) 0 0
\(208\) 4.64575 0.322125
\(209\) −33.2915 −2.30282
\(210\) 0 0
\(211\) 25.2915 1.74114 0.870569 0.492046i \(-0.163751\pi\)
0.870569 + 0.492046i \(0.163751\pi\)
\(212\) −2.64575 −0.181711
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 0 0
\(217\) 7.29150 0.494979
\(218\) 1.64575 0.111464
\(219\) 0 0
\(220\) 0 0
\(221\) 4.64575 0.312507
\(222\) 0 0
\(223\) 21.5203 1.44110 0.720552 0.693401i \(-0.243889\pi\)
0.720552 + 0.693401i \(0.243889\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 8.58301 0.570933
\(227\) −6.35425 −0.421746 −0.210873 0.977513i \(-0.567631\pi\)
−0.210873 + 0.977513i \(0.567631\pi\)
\(228\) 0 0
\(229\) 12.5203 0.827362 0.413681 0.910422i \(-0.364243\pi\)
0.413681 + 0.910422i \(0.364243\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.6458 0.698928
\(233\) 1.64575 0.107817 0.0539084 0.998546i \(-0.482832\pi\)
0.0539084 + 0.998546i \(0.482832\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −13.2915 −0.865203
\(237\) 0 0
\(238\) 1.00000 0.0648204
\(239\) −14.3542 −0.928499 −0.464250 0.885704i \(-0.653676\pi\)
−0.464250 + 0.885704i \(0.653676\pi\)
\(240\) 0 0
\(241\) −12.9373 −0.833362 −0.416681 0.909053i \(-0.636807\pi\)
−0.416681 + 0.909053i \(0.636807\pi\)
\(242\) −17.0000 −1.09280
\(243\) 0 0
\(244\) 2.64575 0.169377
\(245\) 0 0
\(246\) 0 0
\(247\) −29.2288 −1.85978
\(248\) 7.29150 0.463011
\(249\) 0 0
\(250\) 0 0
\(251\) 3.87451 0.244557 0.122278 0.992496i \(-0.460980\pi\)
0.122278 + 0.992496i \(0.460980\pi\)
\(252\) 0 0
\(253\) −8.70850 −0.547499
\(254\) −14.6458 −0.918956
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.29150 −0.142940 −0.0714700 0.997443i \(-0.522769\pi\)
−0.0714700 + 0.997443i \(0.522769\pi\)
\(258\) 0 0
\(259\) −9.64575 −0.599358
\(260\) 0 0
\(261\) 0 0
\(262\) 18.9373 1.16995
\(263\) 19.8745 1.22551 0.612757 0.790271i \(-0.290060\pi\)
0.612757 + 0.790271i \(0.290060\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.29150 −0.385757
\(267\) 0 0
\(268\) −4.93725 −0.301591
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −15.2915 −0.928893 −0.464446 0.885601i \(-0.653747\pi\)
−0.464446 + 0.885601i \(0.653747\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 7.64575 0.461897
\(275\) 0 0
\(276\) 0 0
\(277\) 14.7085 0.883748 0.441874 0.897077i \(-0.354314\pi\)
0.441874 + 0.897077i \(0.354314\pi\)
\(278\) −4.58301 −0.274870
\(279\) 0 0
\(280\) 0 0
\(281\) −13.5203 −0.806551 −0.403276 0.915079i \(-0.632128\pi\)
−0.403276 + 0.915079i \(0.632128\pi\)
\(282\) 0 0
\(283\) 12.1660 0.723194 0.361597 0.932334i \(-0.382232\pi\)
0.361597 + 0.932334i \(0.382232\pi\)
\(284\) −11.2915 −0.670027
\(285\) 0 0
\(286\) −24.5830 −1.45362
\(287\) −7.29150 −0.430404
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 5.29150 0.309662
\(293\) 31.5203 1.84143 0.920717 0.390232i \(-0.127605\pi\)
0.920717 + 0.390232i \(0.127605\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.64575 −0.560648
\(297\) 0 0
\(298\) −14.5203 −0.841136
\(299\) −7.64575 −0.442165
\(300\) 0 0
\(301\) 0.354249 0.0204186
\(302\) −13.2288 −0.761229
\(303\) 0 0
\(304\) −6.29150 −0.360842
\(305\) 0 0
\(306\) 0 0
\(307\) −17.2915 −0.986878 −0.493439 0.869780i \(-0.664260\pi\)
−0.493439 + 0.869780i \(0.664260\pi\)
\(308\) −5.29150 −0.301511
\(309\) 0 0
\(310\) 0 0
\(311\) 21.2915 1.20733 0.603665 0.797238i \(-0.293706\pi\)
0.603665 + 0.797238i \(0.293706\pi\)
\(312\) 0 0
\(313\) −12.2288 −0.691210 −0.345605 0.938380i \(-0.612326\pi\)
−0.345605 + 0.938380i \(0.612326\pi\)
\(314\) 6.64575 0.375041
\(315\) 0 0
\(316\) −15.9373 −0.896541
\(317\) −29.9373 −1.68144 −0.840722 0.541467i \(-0.817869\pi\)
−0.840722 + 0.541467i \(0.817869\pi\)
\(318\) 0 0
\(319\) −56.3320 −3.15399
\(320\) 0 0
\(321\) 0 0
\(322\) −1.64575 −0.0917141
\(323\) −6.29150 −0.350069
\(324\) 0 0
\(325\) 0 0
\(326\) 19.8745 1.10075
\(327\) 0 0
\(328\) −7.29150 −0.402606
\(329\) 0.645751 0.0356014
\(330\) 0 0
\(331\) −9.64575 −0.530178 −0.265089 0.964224i \(-0.585401\pi\)
−0.265089 + 0.964224i \(0.585401\pi\)
\(332\) −3.64575 −0.200087
\(333\) 0 0
\(334\) −25.1660 −1.37702
\(335\) 0 0
\(336\) 0 0
\(337\) 26.1660 1.42535 0.712677 0.701493i \(-0.247483\pi\)
0.712677 + 0.701493i \(0.247483\pi\)
\(338\) −8.58301 −0.466854
\(339\) 0 0
\(340\) 0 0
\(341\) −38.5830 −2.08939
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0.354249 0.0190998
\(345\) 0 0
\(346\) 3.64575 0.195997
\(347\) −9.70850 −0.521179 −0.260590 0.965450i \(-0.583917\pi\)
−0.260590 + 0.965450i \(0.583917\pi\)
\(348\) 0 0
\(349\) −9.35425 −0.500721 −0.250361 0.968153i \(-0.580549\pi\)
−0.250361 + 0.968153i \(0.580549\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.29150 −0.282038
\(353\) −16.4575 −0.875945 −0.437973 0.898988i \(-0.644303\pi\)
−0.437973 + 0.898988i \(0.644303\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.00000 −0.0529999
\(357\) 0 0
\(358\) 3.58301 0.189368
\(359\) −32.5830 −1.71967 −0.859833 0.510576i \(-0.829432\pi\)
−0.859833 + 0.510576i \(0.829432\pi\)
\(360\) 0 0
\(361\) 20.5830 1.08332
\(362\) −17.3542 −0.912119
\(363\) 0 0
\(364\) −4.64575 −0.243504
\(365\) 0 0
\(366\) 0 0
\(367\) −0.937254 −0.0489243 −0.0244621 0.999701i \(-0.507787\pi\)
−0.0244621 + 0.999701i \(0.507787\pi\)
\(368\) −1.64575 −0.0857907
\(369\) 0 0
\(370\) 0 0
\(371\) 2.64575 0.137361
\(372\) 0 0
\(373\) 23.8745 1.23618 0.618088 0.786109i \(-0.287908\pi\)
0.618088 + 0.786109i \(0.287908\pi\)
\(374\) −5.29150 −0.273617
\(375\) 0 0
\(376\) 0.645751 0.0333021
\(377\) −49.4575 −2.54719
\(378\) 0 0
\(379\) −2.22876 −0.114484 −0.0572418 0.998360i \(-0.518231\pi\)
−0.0572418 + 0.998360i \(0.518231\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6.58301 −0.336816
\(383\) 27.2915 1.39453 0.697265 0.716813i \(-0.254400\pi\)
0.697265 + 0.716813i \(0.254400\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.8745 0.858890
\(387\) 0 0
\(388\) 12.9373 0.656790
\(389\) 34.3948 1.74388 0.871942 0.489609i \(-0.162861\pi\)
0.871942 + 0.489609i \(0.162861\pi\)
\(390\) 0 0
\(391\) −1.64575 −0.0832292
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 18.6458 0.939359
\(395\) 0 0
\(396\) 0 0
\(397\) −4.52026 −0.226865 −0.113433 0.993546i \(-0.536185\pi\)
−0.113433 + 0.993546i \(0.536185\pi\)
\(398\) −19.5203 −0.978462
\(399\) 0 0
\(400\) 0 0
\(401\) 24.9373 1.24531 0.622654 0.782498i \(-0.286055\pi\)
0.622654 + 0.782498i \(0.286055\pi\)
\(402\) 0 0
\(403\) −33.8745 −1.68741
\(404\) −16.9373 −0.842660
\(405\) 0 0
\(406\) −10.6458 −0.528340
\(407\) 51.0405 2.52998
\(408\) 0 0
\(409\) −8.22876 −0.406886 −0.203443 0.979087i \(-0.565213\pi\)
−0.203443 + 0.979087i \(0.565213\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −17.8745 −0.880614
\(413\) 13.2915 0.654032
\(414\) 0 0
\(415\) 0 0
\(416\) −4.64575 −0.227777
\(417\) 0 0
\(418\) 33.2915 1.62834
\(419\) −22.3542 −1.09208 −0.546038 0.837760i \(-0.683865\pi\)
−0.546038 + 0.837760i \(0.683865\pi\)
\(420\) 0 0
\(421\) −4.58301 −0.223362 −0.111681 0.993744i \(-0.535623\pi\)
−0.111681 + 0.993744i \(0.535623\pi\)
\(422\) −25.2915 −1.23117
\(423\) 0 0
\(424\) 2.64575 0.128489
\(425\) 0 0
\(426\) 0 0
\(427\) −2.64575 −0.128037
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) 0 0
\(431\) −33.8745 −1.63168 −0.815839 0.578279i \(-0.803724\pi\)
−0.815839 + 0.578279i \(0.803724\pi\)
\(432\) 0 0
\(433\) −0.708497 −0.0340482 −0.0170241 0.999855i \(-0.505419\pi\)
−0.0170241 + 0.999855i \(0.505419\pi\)
\(434\) −7.29150 −0.350003
\(435\) 0 0
\(436\) −1.64575 −0.0788172
\(437\) 10.3542 0.495311
\(438\) 0 0
\(439\) −15.0627 −0.718906 −0.359453 0.933163i \(-0.617037\pi\)
−0.359453 + 0.933163i \(0.617037\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.64575 −0.220976
\(443\) −12.4170 −0.589949 −0.294975 0.955505i \(-0.595311\pi\)
−0.294975 + 0.955505i \(0.595311\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −21.5203 −1.01901
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −8.22876 −0.388339 −0.194170 0.980968i \(-0.562201\pi\)
−0.194170 + 0.980968i \(0.562201\pi\)
\(450\) 0 0
\(451\) 38.5830 1.81680
\(452\) −8.58301 −0.403711
\(453\) 0 0
\(454\) 6.35425 0.298220
\(455\) 0 0
\(456\) 0 0
\(457\) −38.8745 −1.81847 −0.909236 0.416280i \(-0.863334\pi\)
−0.909236 + 0.416280i \(0.863334\pi\)
\(458\) −12.5203 −0.585033
\(459\) 0 0
\(460\) 0 0
\(461\) −3.52026 −0.163955 −0.0819774 0.996634i \(-0.526124\pi\)
−0.0819774 + 0.996634i \(0.526124\pi\)
\(462\) 0 0
\(463\) 6.64575 0.308854 0.154427 0.988004i \(-0.450647\pi\)
0.154427 + 0.988004i \(0.450647\pi\)
\(464\) −10.6458 −0.494217
\(465\) 0 0
\(466\) −1.64575 −0.0762380
\(467\) −28.7085 −1.32847 −0.664235 0.747523i \(-0.731243\pi\)
−0.664235 + 0.747523i \(0.731243\pi\)
\(468\) 0 0
\(469\) 4.93725 0.227981
\(470\) 0 0
\(471\) 0 0
\(472\) 13.2915 0.611791
\(473\) −1.87451 −0.0861900
\(474\) 0 0
\(475\) 0 0
\(476\) −1.00000 −0.0458349
\(477\) 0 0
\(478\) 14.3542 0.656548
\(479\) −7.35425 −0.336024 −0.168012 0.985785i \(-0.553735\pi\)
−0.168012 + 0.985785i \(0.553735\pi\)
\(480\) 0 0
\(481\) 44.8118 2.04324
\(482\) 12.9373 0.589276
\(483\) 0 0
\(484\) 17.0000 0.772727
\(485\) 0 0
\(486\) 0 0
\(487\) −0.708497 −0.0321051 −0.0160525 0.999871i \(-0.505110\pi\)
−0.0160525 + 0.999871i \(0.505110\pi\)
\(488\) −2.64575 −0.119768
\(489\) 0 0
\(490\) 0 0
\(491\) −12.8745 −0.581018 −0.290509 0.956872i \(-0.593825\pi\)
−0.290509 + 0.956872i \(0.593825\pi\)
\(492\) 0 0
\(493\) −10.6458 −0.479461
\(494\) 29.2288 1.31506
\(495\) 0 0
\(496\) −7.29150 −0.327398
\(497\) 11.2915 0.506493
\(498\) 0 0
\(499\) 13.0627 0.584769 0.292384 0.956301i \(-0.405551\pi\)
0.292384 + 0.956301i \(0.405551\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.87451 −0.172928
\(503\) −2.70850 −0.120766 −0.0603830 0.998175i \(-0.519232\pi\)
−0.0603830 + 0.998175i \(0.519232\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.70850 0.387140
\(507\) 0 0
\(508\) 14.6458 0.649800
\(509\) 7.52026 0.333330 0.166665 0.986014i \(-0.446700\pi\)
0.166665 + 0.986014i \(0.446700\pi\)
\(510\) 0 0
\(511\) −5.29150 −0.234082
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 2.29150 0.101074
\(515\) 0 0
\(516\) 0 0
\(517\) −3.41699 −0.150279
\(518\) 9.64575 0.423810
\(519\) 0 0
\(520\) 0 0
\(521\) −29.5830 −1.29605 −0.648027 0.761617i \(-0.724406\pi\)
−0.648027 + 0.761617i \(0.724406\pi\)
\(522\) 0 0
\(523\) 26.7490 1.16965 0.584826 0.811158i \(-0.301163\pi\)
0.584826 + 0.811158i \(0.301163\pi\)
\(524\) −18.9373 −0.827278
\(525\) 0 0
\(526\) −19.8745 −0.866570
\(527\) −7.29150 −0.317623
\(528\) 0 0
\(529\) −20.2915 −0.882239
\(530\) 0 0
\(531\) 0 0
\(532\) 6.29150 0.272771
\(533\) 33.8745 1.46727
\(534\) 0 0
\(535\) 0 0
\(536\) 4.93725 0.213257
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 5.29150 0.227921
\(540\) 0 0
\(541\) −13.6458 −0.586677 −0.293338 0.956009i \(-0.594766\pi\)
−0.293338 + 0.956009i \(0.594766\pi\)
\(542\) 15.2915 0.656826
\(543\) 0 0
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −7.64575 −0.326610
\(549\) 0 0
\(550\) 0 0
\(551\) 66.9778 2.85335
\(552\) 0 0
\(553\) 15.9373 0.677721
\(554\) −14.7085 −0.624904
\(555\) 0 0
\(556\) 4.58301 0.194363
\(557\) 4.52026 0.191530 0.0957648 0.995404i \(-0.469470\pi\)
0.0957648 + 0.995404i \(0.469470\pi\)
\(558\) 0 0
\(559\) −1.64575 −0.0696079
\(560\) 0 0
\(561\) 0 0
\(562\) 13.5203 0.570318
\(563\) 23.0627 0.971979 0.485989 0.873965i \(-0.338459\pi\)
0.485989 + 0.873965i \(0.338459\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −12.1660 −0.511376
\(567\) 0 0
\(568\) 11.2915 0.473781
\(569\) −37.1660 −1.55808 −0.779040 0.626974i \(-0.784293\pi\)
−0.779040 + 0.626974i \(0.784293\pi\)
\(570\) 0 0
\(571\) 12.5830 0.526582 0.263291 0.964716i \(-0.415192\pi\)
0.263291 + 0.964716i \(0.415192\pi\)
\(572\) 24.5830 1.02787
\(573\) 0 0
\(574\) 7.29150 0.304341
\(575\) 0 0
\(576\) 0 0
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 16.0000 0.665512
\(579\) 0 0
\(580\) 0 0
\(581\) 3.64575 0.151251
\(582\) 0 0
\(583\) −14.0000 −0.579821
\(584\) −5.29150 −0.218964
\(585\) 0 0
\(586\) −31.5203 −1.30209
\(587\) −31.5203 −1.30098 −0.650490 0.759515i \(-0.725436\pi\)
−0.650490 + 0.759515i \(0.725436\pi\)
\(588\) 0 0
\(589\) 45.8745 1.89023
\(590\) 0 0
\(591\) 0 0
\(592\) 9.64575 0.396438
\(593\) 15.0000 0.615976 0.307988 0.951390i \(-0.400344\pi\)
0.307988 + 0.951390i \(0.400344\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.5203 0.594773
\(597\) 0 0
\(598\) 7.64575 0.312658
\(599\) −38.8118 −1.58581 −0.792903 0.609348i \(-0.791431\pi\)
−0.792903 + 0.609348i \(0.791431\pi\)
\(600\) 0 0
\(601\) −15.2915 −0.623753 −0.311877 0.950123i \(-0.600958\pi\)
−0.311877 + 0.950123i \(0.600958\pi\)
\(602\) −0.354249 −0.0144381
\(603\) 0 0
\(604\) 13.2288 0.538270
\(605\) 0 0
\(606\) 0 0
\(607\) 0.811762 0.0329484 0.0164742 0.999864i \(-0.494756\pi\)
0.0164742 + 0.999864i \(0.494756\pi\)
\(608\) 6.29150 0.255154
\(609\) 0 0
\(610\) 0 0
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) −22.9373 −0.926427 −0.463213 0.886247i \(-0.653304\pi\)
−0.463213 + 0.886247i \(0.653304\pi\)
\(614\) 17.2915 0.697828
\(615\) 0 0
\(616\) 5.29150 0.213201
\(617\) 5.06275 0.203818 0.101909 0.994794i \(-0.467505\pi\)
0.101909 + 0.994794i \(0.467505\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −21.2915 −0.853711
\(623\) 1.00000 0.0400642
\(624\) 0 0
\(625\) 0 0
\(626\) 12.2288 0.488759
\(627\) 0 0
\(628\) −6.64575 −0.265194
\(629\) 9.64575 0.384601
\(630\) 0 0
\(631\) −27.8118 −1.10717 −0.553584 0.832793i \(-0.686740\pi\)
−0.553584 + 0.832793i \(0.686740\pi\)
\(632\) 15.9373 0.633950
\(633\) 0 0
\(634\) 29.9373 1.18896
\(635\) 0 0
\(636\) 0 0
\(637\) 4.64575 0.184071
\(638\) 56.3320 2.23021
\(639\) 0 0
\(640\) 0 0
\(641\) −35.2915 −1.39393 −0.696965 0.717105i \(-0.745467\pi\)
−0.696965 + 0.717105i \(0.745467\pi\)
\(642\) 0 0
\(643\) −2.29150 −0.0903680 −0.0451840 0.998979i \(-0.514387\pi\)
−0.0451840 + 0.998979i \(0.514387\pi\)
\(644\) 1.64575 0.0648517
\(645\) 0 0
\(646\) 6.29150 0.247536
\(647\) −11.3542 −0.446382 −0.223191 0.974775i \(-0.571647\pi\)
−0.223191 + 0.974775i \(0.571647\pi\)
\(648\) 0 0
\(649\) −70.3320 −2.76077
\(650\) 0 0
\(651\) 0 0
\(652\) −19.8745 −0.778346
\(653\) 27.8118 1.08836 0.544179 0.838969i \(-0.316841\pi\)
0.544179 + 0.838969i \(0.316841\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 7.29150 0.284685
\(657\) 0 0
\(658\) −0.645751 −0.0251740
\(659\) 5.58301 0.217483 0.108742 0.994070i \(-0.465318\pi\)
0.108742 + 0.994070i \(0.465318\pi\)
\(660\) 0 0
\(661\) −3.16601 −0.123144 −0.0615718 0.998103i \(-0.519611\pi\)
−0.0615718 + 0.998103i \(0.519611\pi\)
\(662\) 9.64575 0.374893
\(663\) 0 0
\(664\) 3.64575 0.141483
\(665\) 0 0
\(666\) 0 0
\(667\) 17.5203 0.678387
\(668\) 25.1660 0.973702
\(669\) 0 0
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) −26.5830 −1.02470 −0.512350 0.858777i \(-0.671225\pi\)
−0.512350 + 0.858777i \(0.671225\pi\)
\(674\) −26.1660 −1.00788
\(675\) 0 0
\(676\) 8.58301 0.330116
\(677\) 9.29150 0.357101 0.178551 0.983931i \(-0.442859\pi\)
0.178551 + 0.983931i \(0.442859\pi\)
\(678\) 0 0
\(679\) −12.9373 −0.496486
\(680\) 0 0
\(681\) 0 0
\(682\) 38.5830 1.47742
\(683\) −2.58301 −0.0988359 −0.0494180 0.998778i \(-0.515737\pi\)
−0.0494180 + 0.998778i \(0.515737\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −0.354249 −0.0135056
\(689\) −12.2915 −0.468269
\(690\) 0 0
\(691\) −22.7085 −0.863872 −0.431936 0.901904i \(-0.642169\pi\)
−0.431936 + 0.901904i \(0.642169\pi\)
\(692\) −3.64575 −0.138591
\(693\) 0 0
\(694\) 9.70850 0.368530
\(695\) 0 0
\(696\) 0 0
\(697\) 7.29150 0.276185
\(698\) 9.35425 0.354064
\(699\) 0 0
\(700\) 0 0
\(701\) −13.2915 −0.502013 −0.251007 0.967985i \(-0.580762\pi\)
−0.251007 + 0.967985i \(0.580762\pi\)
\(702\) 0 0
\(703\) −60.6863 −2.28883
\(704\) 5.29150 0.199431
\(705\) 0 0
\(706\) 16.4575 0.619387
\(707\) 16.9373 0.636991
\(708\) 0 0
\(709\) 1.87451 0.0703986 0.0351993 0.999380i \(-0.488793\pi\)
0.0351993 + 0.999380i \(0.488793\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.00000 0.0374766
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) −3.58301 −0.133903
\(717\) 0 0
\(718\) 32.5830 1.21599
\(719\) 28.6458 1.06831 0.534153 0.845388i \(-0.320631\pi\)
0.534153 + 0.845388i \(0.320631\pi\)
\(720\) 0 0
\(721\) 17.8745 0.665681
\(722\) −20.5830 −0.766020
\(723\) 0 0
\(724\) 17.3542 0.644966
\(725\) 0 0
\(726\) 0 0
\(727\) −25.6458 −0.951148 −0.475574 0.879676i \(-0.657760\pi\)
−0.475574 + 0.879676i \(0.657760\pi\)
\(728\) 4.64575 0.172183
\(729\) 0 0
\(730\) 0 0
\(731\) −0.354249 −0.0131024
\(732\) 0 0
\(733\) −37.8118 −1.39661 −0.698305 0.715801i \(-0.746062\pi\)
−0.698305 + 0.715801i \(0.746062\pi\)
\(734\) 0.937254 0.0345947
\(735\) 0 0
\(736\) 1.64575 0.0606632
\(737\) −26.1255 −0.962345
\(738\) 0 0
\(739\) 48.3320 1.77792 0.888961 0.457983i \(-0.151428\pi\)
0.888961 + 0.457983i \(0.151428\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.64575 −0.0971286
\(743\) −49.7490 −1.82511 −0.912557 0.408949i \(-0.865895\pi\)
−0.912557 + 0.408949i \(0.865895\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −23.8745 −0.874108
\(747\) 0 0
\(748\) 5.29150 0.193476
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −22.5830 −0.824066 −0.412033 0.911169i \(-0.635181\pi\)
−0.412033 + 0.911169i \(0.635181\pi\)
\(752\) −0.645751 −0.0235481
\(753\) 0 0
\(754\) 49.4575 1.80114
\(755\) 0 0
\(756\) 0 0
\(757\) −28.8118 −1.04718 −0.523591 0.851970i \(-0.675408\pi\)
−0.523591 + 0.851970i \(0.675408\pi\)
\(758\) 2.22876 0.0809521
\(759\) 0 0
\(760\) 0 0
\(761\) −1.83399 −0.0664821 −0.0332410 0.999447i \(-0.510583\pi\)
−0.0332410 + 0.999447i \(0.510583\pi\)
\(762\) 0 0
\(763\) 1.64575 0.0595802
\(764\) 6.58301 0.238165
\(765\) 0 0
\(766\) −27.2915 −0.986082
\(767\) −61.7490 −2.22963
\(768\) 0 0
\(769\) −32.0000 −1.15395 −0.576975 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16.8745 −0.607327
\(773\) 25.1660 0.905158 0.452579 0.891724i \(-0.350504\pi\)
0.452579 + 0.891724i \(0.350504\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −12.9373 −0.464420
\(777\) 0 0
\(778\) −34.3948 −1.23311
\(779\) −45.8745 −1.64362
\(780\) 0 0
\(781\) −59.7490 −2.13799
\(782\) 1.64575 0.0588519
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −19.4170 −0.692141 −0.346071 0.938208i \(-0.612484\pi\)
−0.346071 + 0.938208i \(0.612484\pi\)
\(788\) −18.6458 −0.664227
\(789\) 0 0
\(790\) 0 0
\(791\) 8.58301 0.305177
\(792\) 0 0
\(793\) 12.2915 0.436484
\(794\) 4.52026 0.160418
\(795\) 0 0
\(796\) 19.5203 0.691877
\(797\) 3.18824 0.112933 0.0564666 0.998404i \(-0.482017\pi\)
0.0564666 + 0.998404i \(0.482017\pi\)
\(798\) 0 0
\(799\) −0.645751 −0.0228450
\(800\) 0 0
\(801\) 0 0
\(802\) −24.9373 −0.880565
\(803\) 28.0000 0.988099
\(804\) 0 0
\(805\) 0 0
\(806\) 33.8745 1.19318
\(807\) 0 0
\(808\) 16.9373 0.595851
\(809\) 24.5830 0.864292 0.432146 0.901804i \(-0.357756\pi\)
0.432146 + 0.901804i \(0.357756\pi\)
\(810\) 0 0
\(811\) 15.1660 0.532551 0.266275 0.963897i \(-0.414207\pi\)
0.266275 + 0.963897i \(0.414207\pi\)
\(812\) 10.6458 0.373593
\(813\) 0 0
\(814\) −51.0405 −1.78897
\(815\) 0 0
\(816\) 0 0
\(817\) 2.22876 0.0779743
\(818\) 8.22876 0.287712
\(819\) 0 0
\(820\) 0 0
\(821\) 47.2288 1.64829 0.824147 0.566375i \(-0.191655\pi\)
0.824147 + 0.566375i \(0.191655\pi\)
\(822\) 0 0
\(823\) 21.0405 0.733426 0.366713 0.930334i \(-0.380483\pi\)
0.366713 + 0.930334i \(0.380483\pi\)
\(824\) 17.8745 0.622688
\(825\) 0 0
\(826\) −13.2915 −0.462471
\(827\) −22.4170 −0.779515 −0.389758 0.920917i \(-0.627441\pi\)
−0.389758 + 0.920917i \(0.627441\pi\)
\(828\) 0 0
\(829\) 12.6458 0.439205 0.219603 0.975589i \(-0.429524\pi\)
0.219603 + 0.975589i \(0.429524\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4.64575 0.161062
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) 0 0
\(836\) −33.2915 −1.15141
\(837\) 0 0
\(838\) 22.3542 0.772215
\(839\) −2.18824 −0.0755464 −0.0377732 0.999286i \(-0.512026\pi\)
−0.0377732 + 0.999286i \(0.512026\pi\)
\(840\) 0 0
\(841\) 84.3320 2.90800
\(842\) 4.58301 0.157941
\(843\) 0 0
\(844\) 25.2915 0.870569
\(845\) 0 0
\(846\) 0 0
\(847\) −17.0000 −0.584127
\(848\) −2.64575 −0.0908555
\(849\) 0 0
\(850\) 0 0
\(851\) −15.8745 −0.544171
\(852\) 0 0
\(853\) −30.3320 −1.03855 −0.519274 0.854608i \(-0.673798\pi\)
−0.519274 + 0.854608i \(0.673798\pi\)
\(854\) 2.64575 0.0905357
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) 23.7085 0.809867 0.404933 0.914346i \(-0.367295\pi\)
0.404933 + 0.914346i \(0.367295\pi\)
\(858\) 0 0
\(859\) 50.3320 1.71731 0.858653 0.512557i \(-0.171302\pi\)
0.858653 + 0.512557i \(0.171302\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 33.8745 1.15377
\(863\) −35.6458 −1.21340 −0.606698 0.794933i \(-0.707506\pi\)
−0.606698 + 0.794933i \(0.707506\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0.708497 0.0240757
\(867\) 0 0
\(868\) 7.29150 0.247490
\(869\) −84.3320 −2.86077
\(870\) 0 0
\(871\) −22.9373 −0.777199
\(872\) 1.64575 0.0557322
\(873\) 0 0
\(874\) −10.3542 −0.350238
\(875\) 0 0
\(876\) 0 0
\(877\) −25.8745 −0.873720 −0.436860 0.899529i \(-0.643909\pi\)
−0.436860 + 0.899529i \(0.643909\pi\)
\(878\) 15.0627 0.508343
\(879\) 0 0
\(880\) 0 0
\(881\) 46.8745 1.57924 0.789621 0.613595i \(-0.210277\pi\)
0.789621 + 0.613595i \(0.210277\pi\)
\(882\) 0 0
\(883\) −20.7085 −0.696896 −0.348448 0.937328i \(-0.613291\pi\)
−0.348448 + 0.937328i \(0.613291\pi\)
\(884\) 4.64575 0.156254
\(885\) 0 0
\(886\) 12.4170 0.417157
\(887\) 18.1255 0.608594 0.304297 0.952577i \(-0.401578\pi\)
0.304297 + 0.952577i \(0.401578\pi\)
\(888\) 0 0
\(889\) −14.6458 −0.491203
\(890\) 0 0
\(891\) 0 0
\(892\) 21.5203 0.720552
\(893\) 4.06275 0.135955
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 8.22876 0.274597
\(899\) 77.6235 2.58889
\(900\) 0 0
\(901\) −2.64575 −0.0881428
\(902\) −38.5830 −1.28467
\(903\) 0 0
\(904\) 8.58301 0.285467
\(905\) 0 0
\(906\) 0 0
\(907\) −12.7085 −0.421979 −0.210989 0.977488i \(-0.567669\pi\)
−0.210989 + 0.977488i \(0.567669\pi\)
\(908\) −6.35425 −0.210873
\(909\) 0 0
\(910\) 0 0
\(911\) −13.1660 −0.436209 −0.218105 0.975925i \(-0.569987\pi\)
−0.218105 + 0.975925i \(0.569987\pi\)
\(912\) 0 0
\(913\) −19.2915 −0.638456
\(914\) 38.8745 1.28585
\(915\) 0 0
\(916\) 12.5203 0.413681
\(917\) 18.9373 0.625363
\(918\) 0 0
\(919\) 49.0405 1.61770 0.808849 0.588017i \(-0.200091\pi\)
0.808849 + 0.588017i \(0.200091\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 3.52026 0.115934
\(923\) −52.4575 −1.72666
\(924\) 0 0
\(925\) 0 0
\(926\) −6.64575 −0.218393
\(927\) 0 0
\(928\) 10.6458 0.349464
\(929\) −10.8745 −0.356781 −0.178391 0.983960i \(-0.557089\pi\)
−0.178391 + 0.983960i \(0.557089\pi\)
\(930\) 0 0
\(931\) −6.29150 −0.206196
\(932\) 1.64575 0.0539084
\(933\) 0 0
\(934\) 28.7085 0.939371
\(935\) 0 0
\(936\) 0 0
\(937\) 16.2288 0.530170 0.265085 0.964225i \(-0.414600\pi\)
0.265085 + 0.964225i \(0.414600\pi\)
\(938\) −4.93725 −0.161207
\(939\) 0 0
\(940\) 0 0
\(941\) −15.6458 −0.510037 −0.255018 0.966936i \(-0.582082\pi\)
−0.255018 + 0.966936i \(0.582082\pi\)
\(942\) 0 0
\(943\) −12.0000 −0.390774
\(944\) −13.2915 −0.432602
\(945\) 0 0
\(946\) 1.87451 0.0609455
\(947\) 18.7085 0.607944 0.303972 0.952681i \(-0.401687\pi\)
0.303972 + 0.952681i \(0.401687\pi\)
\(948\) 0 0
\(949\) 24.5830 0.797998
\(950\) 0 0
\(951\) 0 0
\(952\) 1.00000 0.0324102
\(953\) −18.1033 −0.586422 −0.293211 0.956048i \(-0.594724\pi\)
−0.293211 + 0.956048i \(0.594724\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −14.3542 −0.464250
\(957\) 0 0
\(958\) 7.35425 0.237605
\(959\) 7.64575 0.246894
\(960\) 0 0
\(961\) 22.1660 0.715033
\(962\) −44.8118 −1.44479
\(963\) 0 0
\(964\) −12.9373 −0.416681
\(965\) 0 0
\(966\) 0 0
\(967\) 45.8118 1.47321 0.736603 0.676325i \(-0.236428\pi\)
0.736603 + 0.676325i \(0.236428\pi\)
\(968\) −17.0000 −0.546401
\(969\) 0 0
\(970\) 0 0
\(971\) −3.77124 −0.121025 −0.0605125 0.998167i \(-0.519273\pi\)
−0.0605125 + 0.998167i \(0.519273\pi\)
\(972\) 0 0
\(973\) −4.58301 −0.146924
\(974\) 0.708497 0.0227017
\(975\) 0 0
\(976\) 2.64575 0.0846884
\(977\) 22.9373 0.733828 0.366914 0.930255i \(-0.380414\pi\)
0.366914 + 0.930255i \(0.380414\pi\)
\(978\) 0 0
\(979\) −5.29150 −0.169117
\(980\) 0 0
\(981\) 0 0
\(982\) 12.8745 0.410842
\(983\) 13.4797 0.429937 0.214968 0.976621i \(-0.431035\pi\)
0.214968 + 0.976621i \(0.431035\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 10.6458 0.339030
\(987\) 0 0
\(988\) −29.2288 −0.929891
\(989\) 0.583005 0.0185385
\(990\) 0 0
\(991\) −33.9373 −1.07805 −0.539026 0.842289i \(-0.681208\pi\)
−0.539026 + 0.842289i \(0.681208\pi\)
\(992\) 7.29150 0.231505
\(993\) 0 0
\(994\) −11.2915 −0.358145
\(995\) 0 0
\(996\) 0 0
\(997\) −50.4575 −1.59801 −0.799003 0.601327i \(-0.794639\pi\)
−0.799003 + 0.601327i \(0.794639\pi\)
\(998\) −13.0627 −0.413494
\(999\) 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 9450.2.a.ec.1.2 2
3.2 odd 2 9450.2.a.eq.1.1 yes 2
5.4 even 2 9450.2.a.et.1.2 yes 2
15.14 odd 2 9450.2.a.ej.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9450.2.a.ec.1.2 2 1.1 even 1 trivial
9450.2.a.ej.1.1 yes 2 15.14 odd 2
9450.2.a.eq.1.1 yes 2 3.2 odd 2
9450.2.a.et.1.2 yes 2 5.4 even 2