Properties

Label 414.2.a.f.1.1
Level $414$
Weight $2$
Character 414.1
Self dual yes
Analytic conductor $3.306$
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] = [414,2,Mod(1,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("414.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.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: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 414.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -4.47214 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -4.47214 q^{7} -1.00000 q^{8} +1.23607 q^{10} +5.23607 q^{11} +4.47214 q^{13} +4.47214 q^{14} +1.00000 q^{16} +4.00000 q^{17} +5.70820 q^{19} -1.23607 q^{20} -5.23607 q^{22} -1.00000 q^{23} -3.47214 q^{25} -4.47214 q^{26} -4.47214 q^{28} +4.47214 q^{29} -2.47214 q^{31} -1.00000 q^{32} -4.00000 q^{34} +5.52786 q^{35} +11.2361 q^{37} -5.70820 q^{38} +1.23607 q^{40} +2.00000 q^{41} -4.76393 q^{43} +5.23607 q^{44} +1.00000 q^{46} -4.00000 q^{47} +13.0000 q^{49} +3.47214 q^{50} +4.47214 q^{52} -5.23607 q^{53} -6.47214 q^{55} +4.47214 q^{56} -4.47214 q^{58} +8.94427 q^{59} +0.763932 q^{61} +2.47214 q^{62} +1.00000 q^{64} -5.52786 q^{65} +9.70820 q^{67} +4.00000 q^{68} -5.52786 q^{70} -8.94427 q^{71} -4.47214 q^{73} -11.2361 q^{74} +5.70820 q^{76} -23.4164 q^{77} +4.47214 q^{79} -1.23607 q^{80} -2.00000 q^{82} -13.2361 q^{83} -4.94427 q^{85} +4.76393 q^{86} -5.23607 q^{88} +10.4721 q^{89} -20.0000 q^{91} -1.00000 q^{92} +4.00000 q^{94} -7.05573 q^{95} +0.472136 q^{97} -13.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} + 6 q^{11} + 2 q^{16} + 8 q^{17} - 2 q^{19} + 2 q^{20} - 6 q^{22} - 2 q^{23} + 2 q^{25} + 4 q^{31} - 2 q^{32} - 8 q^{34} + 20 q^{35} + 18 q^{37} + 2 q^{38} - 2 q^{40} + 4 q^{41} - 14 q^{43} + 6 q^{44} + 2 q^{46} - 8 q^{47} + 26 q^{49} - 2 q^{50} - 6 q^{53} - 4 q^{55} + 6 q^{61} - 4 q^{62} + 2 q^{64} - 20 q^{65} + 6 q^{67} + 8 q^{68} - 20 q^{70} - 18 q^{74} - 2 q^{76} - 20 q^{77} + 2 q^{80} - 4 q^{82} - 22 q^{83} + 8 q^{85} + 14 q^{86} - 6 q^{88} + 12 q^{89} - 40 q^{91} - 2 q^{92} + 8 q^{94} - 32 q^{95} - 8 q^{97} - 26 q^{98}+O(q^{100}) \) 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
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) −4.47214 −1.69031 −0.845154 0.534522i \(-0.820491\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.23607 0.390879
\(11\) 5.23607 1.57873 0.789367 0.613922i \(-0.210409\pi\)
0.789367 + 0.613922i \(0.210409\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 4.47214 1.19523
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 5.70820 1.30955 0.654776 0.755823i \(-0.272763\pi\)
0.654776 + 0.755823i \(0.272763\pi\)
\(20\) −1.23607 −0.276393
\(21\) 0 0
\(22\) −5.23607 −1.11633
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) −4.47214 −0.877058
\(27\) 0 0
\(28\) −4.47214 −0.845154
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 5.52786 0.934380
\(36\) 0 0
\(37\) 11.2361 1.84720 0.923599 0.383360i \(-0.125233\pi\)
0.923599 + 0.383360i \(0.125233\pi\)
\(38\) −5.70820 −0.925993
\(39\) 0 0
\(40\) 1.23607 0.195440
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.76393 −0.726493 −0.363246 0.931693i \(-0.618332\pi\)
−0.363246 + 0.931693i \(0.618332\pi\)
\(44\) 5.23607 0.789367
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 13.0000 1.85714
\(50\) 3.47214 0.491034
\(51\) 0 0
\(52\) 4.47214 0.620174
\(53\) −5.23607 −0.719229 −0.359615 0.933101i \(-0.617092\pi\)
−0.359615 + 0.933101i \(0.617092\pi\)
\(54\) 0 0
\(55\) −6.47214 −0.872703
\(56\) 4.47214 0.597614
\(57\) 0 0
\(58\) −4.47214 −0.587220
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) 0.763932 0.0978115 0.0489057 0.998803i \(-0.484427\pi\)
0.0489057 + 0.998803i \(0.484427\pi\)
\(62\) 2.47214 0.313962
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.52786 −0.685647
\(66\) 0 0
\(67\) 9.70820 1.18605 0.593023 0.805186i \(-0.297934\pi\)
0.593023 + 0.805186i \(0.297934\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −5.52786 −0.660706
\(71\) −8.94427 −1.06149 −0.530745 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 0 0
\(73\) −4.47214 −0.523424 −0.261712 0.965146i \(-0.584287\pi\)
−0.261712 + 0.965146i \(0.584287\pi\)
\(74\) −11.2361 −1.30617
\(75\) 0 0
\(76\) 5.70820 0.654776
\(77\) −23.4164 −2.66855
\(78\) 0 0
\(79\) 4.47214 0.503155 0.251577 0.967837i \(-0.419051\pi\)
0.251577 + 0.967837i \(0.419051\pi\)
\(80\) −1.23607 −0.138197
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −13.2361 −1.45285 −0.726424 0.687247i \(-0.758819\pi\)
−0.726424 + 0.687247i \(0.758819\pi\)
\(84\) 0 0
\(85\) −4.94427 −0.536282
\(86\) 4.76393 0.513708
\(87\) 0 0
\(88\) −5.23607 −0.558167
\(89\) 10.4721 1.11004 0.555022 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) −7.05573 −0.723902
\(96\) 0 0
\(97\) 0.472136 0.0479381 0.0239691 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) −13.0000 −1.31320
\(99\) 0 0
\(100\) −3.47214 −0.347214
\(101\) −4.47214 −0.444994 −0.222497 0.974933i \(-0.571421\pi\)
−0.222497 + 0.974933i \(0.571421\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −4.47214 −0.438529
\(105\) 0 0
\(106\) 5.23607 0.508572
\(107\) 12.6525 1.22316 0.611581 0.791182i \(-0.290534\pi\)
0.611581 + 0.791182i \(0.290534\pi\)
\(108\) 0 0
\(109\) 4.76393 0.456302 0.228151 0.973626i \(-0.426732\pi\)
0.228151 + 0.973626i \(0.426732\pi\)
\(110\) 6.47214 0.617094
\(111\) 0 0
\(112\) −4.47214 −0.422577
\(113\) 5.52786 0.520018 0.260009 0.965606i \(-0.416275\pi\)
0.260009 + 0.965606i \(0.416275\pi\)
\(114\) 0 0
\(115\) 1.23607 0.115264
\(116\) 4.47214 0.415227
\(117\) 0 0
\(118\) −8.94427 −0.823387
\(119\) −17.8885 −1.63984
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) −0.763932 −0.0691632
\(123\) 0 0
\(124\) −2.47214 −0.222004
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 5.52786 0.484826
\(131\) 9.52786 0.832453 0.416227 0.909261i \(-0.363352\pi\)
0.416227 + 0.909261i \(0.363352\pi\)
\(132\) 0 0
\(133\) −25.5279 −2.21355
\(134\) −9.70820 −0.838661
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −3.05573 −0.261068 −0.130534 0.991444i \(-0.541669\pi\)
−0.130534 + 0.991444i \(0.541669\pi\)
\(138\) 0 0
\(139\) −16.9443 −1.43719 −0.718597 0.695427i \(-0.755215\pi\)
−0.718597 + 0.695427i \(0.755215\pi\)
\(140\) 5.52786 0.467190
\(141\) 0 0
\(142\) 8.94427 0.750587
\(143\) 23.4164 1.95818
\(144\) 0 0
\(145\) −5.52786 −0.459064
\(146\) 4.47214 0.370117
\(147\) 0 0
\(148\) 11.2361 0.923599
\(149\) 11.7082 0.959173 0.479587 0.877494i \(-0.340787\pi\)
0.479587 + 0.877494i \(0.340787\pi\)
\(150\) 0 0
\(151\) −14.4721 −1.17773 −0.588863 0.808233i \(-0.700424\pi\)
−0.588863 + 0.808233i \(0.700424\pi\)
\(152\) −5.70820 −0.462996
\(153\) 0 0
\(154\) 23.4164 1.88695
\(155\) 3.05573 0.245442
\(156\) 0 0
\(157\) −6.65248 −0.530925 −0.265463 0.964121i \(-0.585525\pi\)
−0.265463 + 0.964121i \(0.585525\pi\)
\(158\) −4.47214 −0.355784
\(159\) 0 0
\(160\) 1.23607 0.0977198
\(161\) 4.47214 0.352454
\(162\) 0 0
\(163\) −2.47214 −0.193633 −0.0968163 0.995302i \(-0.530866\pi\)
−0.0968163 + 0.995302i \(0.530866\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 13.2361 1.02732
\(167\) −16.9443 −1.31119 −0.655594 0.755114i \(-0.727582\pi\)
−0.655594 + 0.755114i \(0.727582\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 4.94427 0.379208
\(171\) 0 0
\(172\) −4.76393 −0.363246
\(173\) 17.4164 1.32414 0.662072 0.749440i \(-0.269677\pi\)
0.662072 + 0.749440i \(0.269677\pi\)
\(174\) 0 0
\(175\) 15.5279 1.17380
\(176\) 5.23607 0.394683
\(177\) 0 0
\(178\) −10.4721 −0.784920
\(179\) −19.4164 −1.45125 −0.725625 0.688090i \(-0.758449\pi\)
−0.725625 + 0.688090i \(0.758449\pi\)
\(180\) 0 0
\(181\) −11.2361 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(182\) 20.0000 1.48250
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) −13.8885 −1.02111
\(186\) 0 0
\(187\) 20.9443 1.53160
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 7.05573 0.511876
\(191\) −6.47214 −0.468307 −0.234154 0.972200i \(-0.575232\pi\)
−0.234154 + 0.972200i \(0.575232\pi\)
\(192\) 0 0
\(193\) 23.8885 1.71954 0.859768 0.510686i \(-0.170608\pi\)
0.859768 + 0.510686i \(0.170608\pi\)
\(194\) −0.472136 −0.0338974
\(195\) 0 0
\(196\) 13.0000 0.928571
\(197\) −2.94427 −0.209771 −0.104885 0.994484i \(-0.533448\pi\)
−0.104885 + 0.994484i \(0.533448\pi\)
\(198\) 0 0
\(199\) 17.4164 1.23462 0.617308 0.786721i \(-0.288223\pi\)
0.617308 + 0.786721i \(0.288223\pi\)
\(200\) 3.47214 0.245517
\(201\) 0 0
\(202\) 4.47214 0.314658
\(203\) −20.0000 −1.40372
\(204\) 0 0
\(205\) −2.47214 −0.172661
\(206\) 6.00000 0.418040
\(207\) 0 0
\(208\) 4.47214 0.310087
\(209\) 29.8885 2.06743
\(210\) 0 0
\(211\) −23.4164 −1.61205 −0.806026 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(212\) −5.23607 −0.359615
\(213\) 0 0
\(214\) −12.6525 −0.864905
\(215\) 5.88854 0.401595
\(216\) 0 0
\(217\) 11.0557 0.750512
\(218\) −4.76393 −0.322654
\(219\) 0 0
\(220\) −6.47214 −0.436351
\(221\) 17.8885 1.20331
\(222\) 0 0
\(223\) 19.4164 1.30022 0.650109 0.759841i \(-0.274723\pi\)
0.650109 + 0.759841i \(0.274723\pi\)
\(224\) 4.47214 0.298807
\(225\) 0 0
\(226\) −5.52786 −0.367708
\(227\) 9.23607 0.613019 0.306510 0.951868i \(-0.400839\pi\)
0.306510 + 0.951868i \(0.400839\pi\)
\(228\) 0 0
\(229\) 17.7082 1.17019 0.585096 0.810964i \(-0.301057\pi\)
0.585096 + 0.810964i \(0.301057\pi\)
\(230\) −1.23607 −0.0815039
\(231\) 0 0
\(232\) −4.47214 −0.293610
\(233\) −19.8885 −1.30294 −0.651471 0.758674i \(-0.725848\pi\)
−0.651471 + 0.758674i \(0.725848\pi\)
\(234\) 0 0
\(235\) 4.94427 0.322529
\(236\) 8.94427 0.582223
\(237\) 0 0
\(238\) 17.8885 1.15954
\(239\) 4.94427 0.319818 0.159909 0.987132i \(-0.448880\pi\)
0.159909 + 0.987132i \(0.448880\pi\)
\(240\) 0 0
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) −16.4164 −1.05529
\(243\) 0 0
\(244\) 0.763932 0.0489057
\(245\) −16.0689 −1.02660
\(246\) 0 0
\(247\) 25.5279 1.62430
\(248\) 2.47214 0.156981
\(249\) 0 0
\(250\) −10.4721 −0.662316
\(251\) 19.7082 1.24397 0.621985 0.783029i \(-0.286326\pi\)
0.621985 + 0.783029i \(0.286326\pi\)
\(252\) 0 0
\(253\) −5.23607 −0.329189
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.8885 0.741587 0.370793 0.928715i \(-0.379086\pi\)
0.370793 + 0.928715i \(0.379086\pi\)
\(258\) 0 0
\(259\) −50.2492 −3.12233
\(260\) −5.52786 −0.342824
\(261\) 0 0
\(262\) −9.52786 −0.588633
\(263\) −24.9443 −1.53813 −0.769065 0.639171i \(-0.779278\pi\)
−0.769065 + 0.639171i \(0.779278\pi\)
\(264\) 0 0
\(265\) 6.47214 0.397580
\(266\) 25.5279 1.56521
\(267\) 0 0
\(268\) 9.70820 0.593023
\(269\) 13.0557 0.796022 0.398011 0.917381i \(-0.369701\pi\)
0.398011 + 0.917381i \(0.369701\pi\)
\(270\) 0 0
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 3.05573 0.184603
\(275\) −18.1803 −1.09632
\(276\) 0 0
\(277\) −20.4721 −1.23005 −0.615026 0.788507i \(-0.710854\pi\)
−0.615026 + 0.788507i \(0.710854\pi\)
\(278\) 16.9443 1.01625
\(279\) 0 0
\(280\) −5.52786 −0.330353
\(281\) 13.5279 0.807005 0.403502 0.914979i \(-0.367793\pi\)
0.403502 + 0.914979i \(0.367793\pi\)
\(282\) 0 0
\(283\) −3.81966 −0.227055 −0.113528 0.993535i \(-0.536215\pi\)
−0.113528 + 0.993535i \(0.536215\pi\)
\(284\) −8.94427 −0.530745
\(285\) 0 0
\(286\) −23.4164 −1.38464
\(287\) −8.94427 −0.527964
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 5.52786 0.324607
\(291\) 0 0
\(292\) −4.47214 −0.261712
\(293\) −0.291796 −0.0170469 −0.00852345 0.999964i \(-0.502713\pi\)
−0.00852345 + 0.999964i \(0.502713\pi\)
\(294\) 0 0
\(295\) −11.0557 −0.643689
\(296\) −11.2361 −0.653083
\(297\) 0 0
\(298\) −11.7082 −0.678238
\(299\) −4.47214 −0.258630
\(300\) 0 0
\(301\) 21.3050 1.22800
\(302\) 14.4721 0.832778
\(303\) 0 0
\(304\) 5.70820 0.327388
\(305\) −0.944272 −0.0540689
\(306\) 0 0
\(307\) 15.4164 0.879861 0.439930 0.898032i \(-0.355003\pi\)
0.439930 + 0.898032i \(0.355003\pi\)
\(308\) −23.4164 −1.33427
\(309\) 0 0
\(310\) −3.05573 −0.173554
\(311\) 20.9443 1.18764 0.593820 0.804598i \(-0.297619\pi\)
0.593820 + 0.804598i \(0.297619\pi\)
\(312\) 0 0
\(313\) 15.5279 0.877687 0.438843 0.898564i \(-0.355388\pi\)
0.438843 + 0.898564i \(0.355388\pi\)
\(314\) 6.65248 0.375421
\(315\) 0 0
\(316\) 4.47214 0.251577
\(317\) −19.5279 −1.09679 −0.548397 0.836218i \(-0.684762\pi\)
−0.548397 + 0.836218i \(0.684762\pi\)
\(318\) 0 0
\(319\) 23.4164 1.31107
\(320\) −1.23607 −0.0690983
\(321\) 0 0
\(322\) −4.47214 −0.249222
\(323\) 22.8328 1.27045
\(324\) 0 0
\(325\) −15.5279 −0.861331
\(326\) 2.47214 0.136919
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) 17.8885 0.986227
\(330\) 0 0
\(331\) −10.4721 −0.575601 −0.287800 0.957690i \(-0.592924\pi\)
−0.287800 + 0.957690i \(0.592924\pi\)
\(332\) −13.2361 −0.726424
\(333\) 0 0
\(334\) 16.9443 0.927149
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −19.8885 −1.08340 −0.541699 0.840573i \(-0.682219\pi\)
−0.541699 + 0.840573i \(0.682219\pi\)
\(338\) −7.00000 −0.380750
\(339\) 0 0
\(340\) −4.94427 −0.268141
\(341\) −12.9443 −0.700972
\(342\) 0 0
\(343\) −26.8328 −1.44884
\(344\) 4.76393 0.256854
\(345\) 0 0
\(346\) −17.4164 −0.936312
\(347\) −30.4721 −1.63583 −0.817915 0.575339i \(-0.804870\pi\)
−0.817915 + 0.575339i \(0.804870\pi\)
\(348\) 0 0
\(349\) 3.88854 0.208149 0.104074 0.994570i \(-0.466812\pi\)
0.104074 + 0.994570i \(0.466812\pi\)
\(350\) −15.5279 −0.829999
\(351\) 0 0
\(352\) −5.23607 −0.279083
\(353\) 3.88854 0.206966 0.103483 0.994631i \(-0.467001\pi\)
0.103483 + 0.994631i \(0.467001\pi\)
\(354\) 0 0
\(355\) 11.0557 0.586777
\(356\) 10.4721 0.555022
\(357\) 0 0
\(358\) 19.4164 1.02619
\(359\) −29.3050 −1.54666 −0.773328 0.634006i \(-0.781409\pi\)
−0.773328 + 0.634006i \(0.781409\pi\)
\(360\) 0 0
\(361\) 13.5836 0.714926
\(362\) 11.2361 0.590555
\(363\) 0 0
\(364\) −20.0000 −1.04828
\(365\) 5.52786 0.289342
\(366\) 0 0
\(367\) 9.41641 0.491532 0.245766 0.969329i \(-0.420960\pi\)
0.245766 + 0.969329i \(0.420960\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 13.8885 0.722031
\(371\) 23.4164 1.21572
\(372\) 0 0
\(373\) −35.5967 −1.84313 −0.921565 0.388224i \(-0.873089\pi\)
−0.921565 + 0.388224i \(0.873089\pi\)
\(374\) −20.9443 −1.08300
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 20.0000 1.03005
\(378\) 0 0
\(379\) 13.7082 0.704143 0.352072 0.935973i \(-0.385477\pi\)
0.352072 + 0.935973i \(0.385477\pi\)
\(380\) −7.05573 −0.361951
\(381\) 0 0
\(382\) 6.47214 0.331143
\(383\) −7.05573 −0.360531 −0.180265 0.983618i \(-0.557696\pi\)
−0.180265 + 0.983618i \(0.557696\pi\)
\(384\) 0 0
\(385\) 28.9443 1.47514
\(386\) −23.8885 −1.21589
\(387\) 0 0
\(388\) 0.472136 0.0239691
\(389\) −23.7082 −1.20205 −0.601027 0.799229i \(-0.705242\pi\)
−0.601027 + 0.799229i \(0.705242\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) −13.0000 −0.656599
\(393\) 0 0
\(394\) 2.94427 0.148330
\(395\) −5.52786 −0.278137
\(396\) 0 0
\(397\) 26.9443 1.35229 0.676147 0.736767i \(-0.263648\pi\)
0.676147 + 0.736767i \(0.263648\pi\)
\(398\) −17.4164 −0.873006
\(399\) 0 0
\(400\) −3.47214 −0.173607
\(401\) −8.94427 −0.446656 −0.223328 0.974743i \(-0.571692\pi\)
−0.223328 + 0.974743i \(0.571692\pi\)
\(402\) 0 0
\(403\) −11.0557 −0.550725
\(404\) −4.47214 −0.222497
\(405\) 0 0
\(406\) 20.0000 0.992583
\(407\) 58.8328 2.91623
\(408\) 0 0
\(409\) −35.8885 −1.77457 −0.887287 0.461217i \(-0.847413\pi\)
−0.887287 + 0.461217i \(0.847413\pi\)
\(410\) 2.47214 0.122090
\(411\) 0 0
\(412\) −6.00000 −0.295599
\(413\) −40.0000 −1.96827
\(414\) 0 0
\(415\) 16.3607 0.803114
\(416\) −4.47214 −0.219265
\(417\) 0 0
\(418\) −29.8885 −1.46190
\(419\) −28.0689 −1.37125 −0.685627 0.727953i \(-0.740472\pi\)
−0.685627 + 0.727953i \(0.740472\pi\)
\(420\) 0 0
\(421\) −0.763932 −0.0372318 −0.0186159 0.999827i \(-0.505926\pi\)
−0.0186159 + 0.999827i \(0.505926\pi\)
\(422\) 23.4164 1.13989
\(423\) 0 0
\(424\) 5.23607 0.254286
\(425\) −13.8885 −0.673693
\(426\) 0 0
\(427\) −3.41641 −0.165332
\(428\) 12.6525 0.611581
\(429\) 0 0
\(430\) −5.88854 −0.283971
\(431\) 23.4164 1.12793 0.563964 0.825799i \(-0.309276\pi\)
0.563964 + 0.825799i \(0.309276\pi\)
\(432\) 0 0
\(433\) −21.4164 −1.02921 −0.514603 0.857428i \(-0.672061\pi\)
−0.514603 + 0.857428i \(0.672061\pi\)
\(434\) −11.0557 −0.530692
\(435\) 0 0
\(436\) 4.76393 0.228151
\(437\) −5.70820 −0.273060
\(438\) 0 0
\(439\) 19.0557 0.909480 0.454740 0.890624i \(-0.349732\pi\)
0.454740 + 0.890624i \(0.349732\pi\)
\(440\) 6.47214 0.308547
\(441\) 0 0
\(442\) −17.8885 −0.850871
\(443\) 15.0557 0.715319 0.357660 0.933852i \(-0.383575\pi\)
0.357660 + 0.933852i \(0.383575\pi\)
\(444\) 0 0
\(445\) −12.9443 −0.613617
\(446\) −19.4164 −0.919394
\(447\) 0 0
\(448\) −4.47214 −0.211289
\(449\) −15.8885 −0.749827 −0.374913 0.927060i \(-0.622328\pi\)
−0.374913 + 0.927060i \(0.622328\pi\)
\(450\) 0 0
\(451\) 10.4721 0.493114
\(452\) 5.52786 0.260009
\(453\) 0 0
\(454\) −9.23607 −0.433470
\(455\) 24.7214 1.15896
\(456\) 0 0
\(457\) 27.5279 1.28770 0.643850 0.765152i \(-0.277336\pi\)
0.643850 + 0.765152i \(0.277336\pi\)
\(458\) −17.7082 −0.827450
\(459\) 0 0
\(460\) 1.23607 0.0576320
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 4.47214 0.207614
\(465\) 0 0
\(466\) 19.8885 0.921319
\(467\) 17.8197 0.824596 0.412298 0.911049i \(-0.364726\pi\)
0.412298 + 0.911049i \(0.364726\pi\)
\(468\) 0 0
\(469\) −43.4164 −2.00478
\(470\) −4.94427 −0.228062
\(471\) 0 0
\(472\) −8.94427 −0.411693
\(473\) −24.9443 −1.14694
\(474\) 0 0
\(475\) −19.8197 −0.909388
\(476\) −17.8885 −0.819920
\(477\) 0 0
\(478\) −4.94427 −0.226146
\(479\) 28.9443 1.32250 0.661249 0.750167i \(-0.270027\pi\)
0.661249 + 0.750167i \(0.270027\pi\)
\(480\) 0 0
\(481\) 50.2492 2.29117
\(482\) 12.4721 0.568090
\(483\) 0 0
\(484\) 16.4164 0.746200
\(485\) −0.583592 −0.0264996
\(486\) 0 0
\(487\) 9.88854 0.448093 0.224046 0.974578i \(-0.428073\pi\)
0.224046 + 0.974578i \(0.428073\pi\)
\(488\) −0.763932 −0.0345816
\(489\) 0 0
\(490\) 16.0689 0.725918
\(491\) −29.3050 −1.32251 −0.661257 0.750159i \(-0.729977\pi\)
−0.661257 + 0.750159i \(0.729977\pi\)
\(492\) 0 0
\(493\) 17.8885 0.805659
\(494\) −25.5279 −1.14855
\(495\) 0 0
\(496\) −2.47214 −0.111002
\(497\) 40.0000 1.79425
\(498\) 0 0
\(499\) 0.583592 0.0261252 0.0130626 0.999915i \(-0.495842\pi\)
0.0130626 + 0.999915i \(0.495842\pi\)
\(500\) 10.4721 0.468328
\(501\) 0 0
\(502\) −19.7082 −0.879620
\(503\) −10.4721 −0.466929 −0.233465 0.972365i \(-0.575006\pi\)
−0.233465 + 0.972365i \(0.575006\pi\)
\(504\) 0 0
\(505\) 5.52786 0.245987
\(506\) 5.23607 0.232772
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 33.4164 1.48116 0.740578 0.671970i \(-0.234552\pi\)
0.740578 + 0.671970i \(0.234552\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −11.8885 −0.524381
\(515\) 7.41641 0.326806
\(516\) 0 0
\(517\) −20.9443 −0.921128
\(518\) 50.2492 2.20782
\(519\) 0 0
\(520\) 5.52786 0.242413
\(521\) −16.0000 −0.700973 −0.350486 0.936568i \(-0.613984\pi\)
−0.350486 + 0.936568i \(0.613984\pi\)
\(522\) 0 0
\(523\) 25.7082 1.12414 0.562071 0.827089i \(-0.310005\pi\)
0.562071 + 0.827089i \(0.310005\pi\)
\(524\) 9.52786 0.416227
\(525\) 0 0
\(526\) 24.9443 1.08762
\(527\) −9.88854 −0.430752
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −6.47214 −0.281132
\(531\) 0 0
\(532\) −25.5279 −1.10677
\(533\) 8.94427 0.387419
\(534\) 0 0
\(535\) −15.6393 −0.676147
\(536\) −9.70820 −0.419331
\(537\) 0 0
\(538\) −13.0557 −0.562872
\(539\) 68.0689 2.93193
\(540\) 0 0
\(541\) 8.11146 0.348739 0.174369 0.984680i \(-0.444211\pi\)
0.174369 + 0.984680i \(0.444211\pi\)
\(542\) 16.9443 0.727819
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −5.88854 −0.252238
\(546\) 0 0
\(547\) 41.3050 1.76607 0.883036 0.469305i \(-0.155496\pi\)
0.883036 + 0.469305i \(0.155496\pi\)
\(548\) −3.05573 −0.130534
\(549\) 0 0
\(550\) 18.1803 0.775212
\(551\) 25.5279 1.08752
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 20.4721 0.869778
\(555\) 0 0
\(556\) −16.9443 −0.718597
\(557\) 15.1246 0.640850 0.320425 0.947274i \(-0.396174\pi\)
0.320425 + 0.947274i \(0.396174\pi\)
\(558\) 0 0
\(559\) −21.3050 −0.901103
\(560\) 5.52786 0.233595
\(561\) 0 0
\(562\) −13.5279 −0.570639
\(563\) −24.6525 −1.03898 −0.519489 0.854477i \(-0.673878\pi\)
−0.519489 + 0.854477i \(0.673878\pi\)
\(564\) 0 0
\(565\) −6.83282 −0.287459
\(566\) 3.81966 0.160552
\(567\) 0 0
\(568\) 8.94427 0.375293
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −14.2918 −0.598093 −0.299047 0.954239i \(-0.596669\pi\)
−0.299047 + 0.954239i \(0.596669\pi\)
\(572\) 23.4164 0.979089
\(573\) 0 0
\(574\) 8.94427 0.373327
\(575\) 3.47214 0.144798
\(576\) 0 0
\(577\) −34.3607 −1.43045 −0.715227 0.698892i \(-0.753677\pi\)
−0.715227 + 0.698892i \(0.753677\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) −5.52786 −0.229532
\(581\) 59.1935 2.45576
\(582\) 0 0
\(583\) −27.4164 −1.13547
\(584\) 4.47214 0.185058
\(585\) 0 0
\(586\) 0.291796 0.0120540
\(587\) −6.47214 −0.267134 −0.133567 0.991040i \(-0.542643\pi\)
−0.133567 + 0.991040i \(0.542643\pi\)
\(588\) 0 0
\(589\) −14.1115 −0.581452
\(590\) 11.0557 0.455157
\(591\) 0 0
\(592\) 11.2361 0.461800
\(593\) −33.7771 −1.38706 −0.693529 0.720428i \(-0.743945\pi\)
−0.693529 + 0.720428i \(0.743945\pi\)
\(594\) 0 0
\(595\) 22.1115 0.906481
\(596\) 11.7082 0.479587
\(597\) 0 0
\(598\) 4.47214 0.182879
\(599\) −33.8885 −1.38465 −0.692324 0.721587i \(-0.743413\pi\)
−0.692324 + 0.721587i \(0.743413\pi\)
\(600\) 0 0
\(601\) −10.3607 −0.422621 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(602\) −21.3050 −0.868325
\(603\) 0 0
\(604\) −14.4721 −0.588863
\(605\) −20.2918 −0.824979
\(606\) 0 0
\(607\) −17.5279 −0.711434 −0.355717 0.934594i \(-0.615763\pi\)
−0.355717 + 0.934594i \(0.615763\pi\)
\(608\) −5.70820 −0.231498
\(609\) 0 0
\(610\) 0.944272 0.0382325
\(611\) −17.8885 −0.723693
\(612\) 0 0
\(613\) −25.1246 −1.01477 −0.507387 0.861718i \(-0.669388\pi\)
−0.507387 + 0.861718i \(0.669388\pi\)
\(614\) −15.4164 −0.622156
\(615\) 0 0
\(616\) 23.4164 0.943474
\(617\) −20.3607 −0.819690 −0.409845 0.912155i \(-0.634417\pi\)
−0.409845 + 0.912155i \(0.634417\pi\)
\(618\) 0 0
\(619\) 18.2918 0.735209 0.367605 0.929982i \(-0.380178\pi\)
0.367605 + 0.929982i \(0.380178\pi\)
\(620\) 3.05573 0.122721
\(621\) 0 0
\(622\) −20.9443 −0.839789
\(623\) −46.8328 −1.87632
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) −15.5279 −0.620618
\(627\) 0 0
\(628\) −6.65248 −0.265463
\(629\) 44.9443 1.79205
\(630\) 0 0
\(631\) −6.00000 −0.238856 −0.119428 0.992843i \(-0.538106\pi\)
−0.119428 + 0.992843i \(0.538106\pi\)
\(632\) −4.47214 −0.177892
\(633\) 0 0
\(634\) 19.5279 0.775551
\(635\) 4.94427 0.196207
\(636\) 0 0
\(637\) 58.1378 2.30350
\(638\) −23.4164 −0.927064
\(639\) 0 0
\(640\) 1.23607 0.0488599
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 0 0
\(643\) −32.5410 −1.28329 −0.641646 0.767001i \(-0.721748\pi\)
−0.641646 + 0.767001i \(0.721748\pi\)
\(644\) 4.47214 0.176227
\(645\) 0 0
\(646\) −22.8328 −0.898345
\(647\) −12.9443 −0.508892 −0.254446 0.967087i \(-0.581893\pi\)
−0.254446 + 0.967087i \(0.581893\pi\)
\(648\) 0 0
\(649\) 46.8328 1.83835
\(650\) 15.5279 0.609053
\(651\) 0 0
\(652\) −2.47214 −0.0968163
\(653\) −9.41641 −0.368493 −0.184246 0.982880i \(-0.558984\pi\)
−0.184246 + 0.982880i \(0.558984\pi\)
\(654\) 0 0
\(655\) −11.7771 −0.460169
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) −17.8885 −0.697368
\(659\) 32.6525 1.27196 0.635980 0.771706i \(-0.280596\pi\)
0.635980 + 0.771706i \(0.280596\pi\)
\(660\) 0 0
\(661\) 3.81966 0.148568 0.0742838 0.997237i \(-0.476333\pi\)
0.0742838 + 0.997237i \(0.476333\pi\)
\(662\) 10.4721 0.407011
\(663\) 0 0
\(664\) 13.2361 0.513659
\(665\) 31.5542 1.22362
\(666\) 0 0
\(667\) −4.47214 −0.173162
\(668\) −16.9443 −0.655594
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 4.47214 0.172388 0.0861941 0.996278i \(-0.472529\pi\)
0.0861941 + 0.996278i \(0.472529\pi\)
\(674\) 19.8885 0.766078
\(675\) 0 0
\(676\) 7.00000 0.269231
\(677\) −19.1246 −0.735019 −0.367509 0.930020i \(-0.619789\pi\)
−0.367509 + 0.930020i \(0.619789\pi\)
\(678\) 0 0
\(679\) −2.11146 −0.0810303
\(680\) 4.94427 0.189604
\(681\) 0 0
\(682\) 12.9443 0.495662
\(683\) −14.4721 −0.553761 −0.276880 0.960904i \(-0.589301\pi\)
−0.276880 + 0.960904i \(0.589301\pi\)
\(684\) 0 0
\(685\) 3.77709 0.144315
\(686\) 26.8328 1.02448
\(687\) 0 0
\(688\) −4.76393 −0.181623
\(689\) −23.4164 −0.892094
\(690\) 0 0
\(691\) 16.5836 0.630870 0.315435 0.948947i \(-0.397850\pi\)
0.315435 + 0.948947i \(0.397850\pi\)
\(692\) 17.4164 0.662072
\(693\) 0 0
\(694\) 30.4721 1.15671
\(695\) 20.9443 0.794462
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) −3.88854 −0.147184
\(699\) 0 0
\(700\) 15.5279 0.586898
\(701\) 3.12461 0.118015 0.0590075 0.998258i \(-0.481206\pi\)
0.0590075 + 0.998258i \(0.481206\pi\)
\(702\) 0 0
\(703\) 64.1378 2.41900
\(704\) 5.23607 0.197342
\(705\) 0 0
\(706\) −3.88854 −0.146347
\(707\) 20.0000 0.752177
\(708\) 0 0
\(709\) 35.0132 1.31495 0.657473 0.753478i \(-0.271625\pi\)
0.657473 + 0.753478i \(0.271625\pi\)
\(710\) −11.0557 −0.414914
\(711\) 0 0
\(712\) −10.4721 −0.392460
\(713\) 2.47214 0.0925822
\(714\) 0 0
\(715\) −28.9443 −1.08245
\(716\) −19.4164 −0.725625
\(717\) 0 0
\(718\) 29.3050 1.09365
\(719\) 3.05573 0.113959 0.0569797 0.998375i \(-0.481853\pi\)
0.0569797 + 0.998375i \(0.481853\pi\)
\(720\) 0 0
\(721\) 26.8328 0.999306
\(722\) −13.5836 −0.505529
\(723\) 0 0
\(724\) −11.2361 −0.417585
\(725\) −15.5279 −0.576690
\(726\) 0 0
\(727\) −19.3050 −0.715981 −0.357991 0.933725i \(-0.616538\pi\)
−0.357991 + 0.933725i \(0.616538\pi\)
\(728\) 20.0000 0.741249
\(729\) 0 0
\(730\) −5.52786 −0.204595
\(731\) −19.0557 −0.704802
\(732\) 0 0
\(733\) −21.7082 −0.801811 −0.400905 0.916119i \(-0.631304\pi\)
−0.400905 + 0.916119i \(0.631304\pi\)
\(734\) −9.41641 −0.347566
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 50.8328 1.87245
\(738\) 0 0
\(739\) −32.9443 −1.21187 −0.605937 0.795512i \(-0.707202\pi\)
−0.605937 + 0.795512i \(0.707202\pi\)
\(740\) −13.8885 −0.510553
\(741\) 0 0
\(742\) −23.4164 −0.859643
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) −14.4721 −0.530218
\(746\) 35.5967 1.30329
\(747\) 0 0
\(748\) 20.9443 0.765798
\(749\) −56.5836 −2.06752
\(750\) 0 0
\(751\) −18.0000 −0.656829 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) −20.0000 −0.728357
\(755\) 17.8885 0.651031
\(756\) 0 0
\(757\) −1.34752 −0.0489766 −0.0244883 0.999700i \(-0.507796\pi\)
−0.0244883 + 0.999700i \(0.507796\pi\)
\(758\) −13.7082 −0.497904
\(759\) 0 0
\(760\) 7.05573 0.255938
\(761\) 5.05573 0.183270 0.0916350 0.995793i \(-0.470791\pi\)
0.0916350 + 0.995793i \(0.470791\pi\)
\(762\) 0 0
\(763\) −21.3050 −0.771291
\(764\) −6.47214 −0.234154
\(765\) 0 0
\(766\) 7.05573 0.254934
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) 12.4721 0.449757 0.224878 0.974387i \(-0.427802\pi\)
0.224878 + 0.974387i \(0.427802\pi\)
\(770\) −28.9443 −1.04308
\(771\) 0 0
\(772\) 23.8885 0.859768
\(773\) 38.1803 1.37325 0.686626 0.727011i \(-0.259091\pi\)
0.686626 + 0.727011i \(0.259091\pi\)
\(774\) 0 0
\(775\) 8.58359 0.308332
\(776\) −0.472136 −0.0169487
\(777\) 0 0
\(778\) 23.7082 0.849980
\(779\) 11.4164 0.409035
\(780\) 0 0
\(781\) −46.8328 −1.67581
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 13.0000 0.464286
\(785\) 8.22291 0.293488
\(786\) 0 0
\(787\) 35.2361 1.25603 0.628015 0.778201i \(-0.283868\pi\)
0.628015 + 0.778201i \(0.283868\pi\)
\(788\) −2.94427 −0.104885
\(789\) 0 0
\(790\) 5.52786 0.196673
\(791\) −24.7214 −0.878990
\(792\) 0 0
\(793\) 3.41641 0.121320
\(794\) −26.9443 −0.956216
\(795\) 0 0
\(796\) 17.4164 0.617308
\(797\) −41.5967 −1.47343 −0.736716 0.676202i \(-0.763625\pi\)
−0.736716 + 0.676202i \(0.763625\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 3.47214 0.122759
\(801\) 0 0
\(802\) 8.94427 0.315833
\(803\) −23.4164 −0.826347
\(804\) 0 0
\(805\) −5.52786 −0.194832
\(806\) 11.0557 0.389421
\(807\) 0 0
\(808\) 4.47214 0.157329
\(809\) −42.9443 −1.50984 −0.754920 0.655817i \(-0.772324\pi\)
−0.754920 + 0.655817i \(0.772324\pi\)
\(810\) 0 0
\(811\) 41.3050 1.45041 0.725207 0.688531i \(-0.241744\pi\)
0.725207 + 0.688531i \(0.241744\pi\)
\(812\) −20.0000 −0.701862
\(813\) 0 0
\(814\) −58.8328 −2.06209
\(815\) 3.05573 0.107037
\(816\) 0 0
\(817\) −27.1935 −0.951380
\(818\) 35.8885 1.25481
\(819\) 0 0
\(820\) −2.47214 −0.0863307
\(821\) −39.5279 −1.37953 −0.689766 0.724032i \(-0.742287\pi\)
−0.689766 + 0.724032i \(0.742287\pi\)
\(822\) 0 0
\(823\) −52.3607 −1.82518 −0.912589 0.408877i \(-0.865920\pi\)
−0.912589 + 0.408877i \(0.865920\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) 40.0000 1.39178
\(827\) 21.5967 0.750993 0.375496 0.926824i \(-0.377472\pi\)
0.375496 + 0.926824i \(0.377472\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) −16.3607 −0.567887
\(831\) 0 0
\(832\) 4.47214 0.155043
\(833\) 52.0000 1.80169
\(834\) 0 0
\(835\) 20.9443 0.724806
\(836\) 29.8885 1.03372
\(837\) 0 0
\(838\) 28.0689 0.969623
\(839\) 45.8885 1.58425 0.792124 0.610360i \(-0.208975\pi\)
0.792124 + 0.610360i \(0.208975\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0.763932 0.0263268
\(843\) 0 0
\(844\) −23.4164 −0.806026
\(845\) −8.65248 −0.297654
\(846\) 0 0
\(847\) −73.4164 −2.52262
\(848\) −5.23607 −0.179807
\(849\) 0 0
\(850\) 13.8885 0.476373
\(851\) −11.2361 −0.385167
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 3.41641 0.116907
\(855\) 0 0
\(856\) −12.6525 −0.432453
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 5.88854 0.200798
\(861\) 0 0
\(862\) −23.4164 −0.797566
\(863\) −14.8328 −0.504915 −0.252457 0.967608i \(-0.581239\pi\)
−0.252457 + 0.967608i \(0.581239\pi\)
\(864\) 0 0
\(865\) −21.5279 −0.731969
\(866\) 21.4164 0.727759
\(867\) 0 0
\(868\) 11.0557 0.375256
\(869\) 23.4164 0.794347
\(870\) 0 0
\(871\) 43.4164 1.47111
\(872\) −4.76393 −0.161327
\(873\) 0 0
\(874\) 5.70820 0.193083
\(875\) −46.8328 −1.58324
\(876\) 0 0
\(877\) −48.2492 −1.62926 −0.814630 0.579981i \(-0.803060\pi\)
−0.814630 + 0.579981i \(0.803060\pi\)
\(878\) −19.0557 −0.643100
\(879\) 0 0
\(880\) −6.47214 −0.218176
\(881\) −39.4164 −1.32797 −0.663986 0.747745i \(-0.731137\pi\)
−0.663986 + 0.747745i \(0.731137\pi\)
\(882\) 0 0
\(883\) 13.5279 0.455249 0.227624 0.973749i \(-0.426904\pi\)
0.227624 + 0.973749i \(0.426904\pi\)
\(884\) 17.8885 0.601657
\(885\) 0 0
\(886\) −15.0557 −0.505807
\(887\) 16.9443 0.568933 0.284466 0.958686i \(-0.408184\pi\)
0.284466 + 0.958686i \(0.408184\pi\)
\(888\) 0 0
\(889\) 17.8885 0.599963
\(890\) 12.9443 0.433893
\(891\) 0 0
\(892\) 19.4164 0.650109
\(893\) −22.8328 −0.764071
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 4.47214 0.149404
\(897\) 0 0
\(898\) 15.8885 0.530208
\(899\) −11.0557 −0.368729
\(900\) 0 0
\(901\) −20.9443 −0.697755
\(902\) −10.4721 −0.348684
\(903\) 0 0
\(904\) −5.52786 −0.183854
\(905\) 13.8885 0.461671
\(906\) 0 0
\(907\) 4.18034 0.138806 0.0694030 0.997589i \(-0.477891\pi\)
0.0694030 + 0.997589i \(0.477891\pi\)
\(908\) 9.23607 0.306510
\(909\) 0 0
\(910\) −24.7214 −0.819505
\(911\) −16.5836 −0.549439 −0.274719 0.961524i \(-0.588585\pi\)
−0.274719 + 0.961524i \(0.588585\pi\)
\(912\) 0 0
\(913\) −69.3050 −2.29366
\(914\) −27.5279 −0.910541
\(915\) 0 0
\(916\) 17.7082 0.585096
\(917\) −42.6099 −1.40710
\(918\) 0 0
\(919\) −16.4721 −0.543366 −0.271683 0.962387i \(-0.587580\pi\)
−0.271683 + 0.962387i \(0.587580\pi\)
\(920\) −1.23607 −0.0407520
\(921\) 0 0
\(922\) 22.0000 0.724531
\(923\) −40.0000 −1.31662
\(924\) 0 0
\(925\) −39.0132 −1.28274
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) −4.47214 −0.146805
\(929\) 24.8328 0.814738 0.407369 0.913264i \(-0.366446\pi\)
0.407369 + 0.913264i \(0.366446\pi\)
\(930\) 0 0
\(931\) 74.2067 2.43202
\(932\) −19.8885 −0.651471
\(933\) 0 0
\(934\) −17.8197 −0.583077
\(935\) −25.8885 −0.846646
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 43.4164 1.41760
\(939\) 0 0
\(940\) 4.94427 0.161264
\(941\) −38.1803 −1.24464 −0.622322 0.782762i \(-0.713810\pi\)
−0.622322 + 0.782762i \(0.713810\pi\)
\(942\) 0 0
\(943\) −2.00000 −0.0651290
\(944\) 8.94427 0.291111
\(945\) 0 0
\(946\) 24.9443 0.811008
\(947\) 47.1935 1.53358 0.766791 0.641897i \(-0.221852\pi\)
0.766791 + 0.641897i \(0.221852\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 19.8197 0.643035
\(951\) 0 0
\(952\) 17.8885 0.579771
\(953\) 47.7771 1.54765 0.773826 0.633398i \(-0.218341\pi\)
0.773826 + 0.633398i \(0.218341\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 4.94427 0.159909
\(957\) 0 0
\(958\) −28.9443 −0.935147
\(959\) 13.6656 0.441286
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) −50.2492 −1.62010
\(963\) 0 0
\(964\) −12.4721 −0.401700
\(965\) −29.5279 −0.950536
\(966\) 0 0
\(967\) 15.4164 0.495758 0.247879 0.968791i \(-0.420266\pi\)
0.247879 + 0.968791i \(0.420266\pi\)
\(968\) −16.4164 −0.527643
\(969\) 0 0
\(970\) 0.583592 0.0187380
\(971\) 19.1246 0.613738 0.306869 0.951752i \(-0.400719\pi\)
0.306869 + 0.951752i \(0.400719\pi\)
\(972\) 0 0
\(973\) 75.7771 2.42930
\(974\) −9.88854 −0.316849
\(975\) 0 0
\(976\) 0.763932 0.0244529
\(977\) −1.16718 −0.0373415 −0.0186708 0.999826i \(-0.505943\pi\)
−0.0186708 + 0.999826i \(0.505943\pi\)
\(978\) 0 0
\(979\) 54.8328 1.75246
\(980\) −16.0689 −0.513302
\(981\) 0 0
\(982\) 29.3050 0.935159
\(983\) −0.583592 −0.0186137 −0.00930685 0.999957i \(-0.502963\pi\)
−0.00930685 + 0.999957i \(0.502963\pi\)
\(984\) 0 0
\(985\) 3.63932 0.115958
\(986\) −17.8885 −0.569687
\(987\) 0 0
\(988\) 25.5279 0.812150
\(989\) 4.76393 0.151484
\(990\) 0 0
\(991\) −10.4721 −0.332658 −0.166329 0.986070i \(-0.553191\pi\)
−0.166329 + 0.986070i \(0.553191\pi\)
\(992\) 2.47214 0.0784904
\(993\) 0 0
\(994\) −40.0000 −1.26872
\(995\) −21.5279 −0.682479
\(996\) 0 0
\(997\) 5.05573 0.160117 0.0800583 0.996790i \(-0.474489\pi\)
0.0800583 + 0.996790i \(0.474489\pi\)
\(998\) −0.583592 −0.0184733
\(999\) 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 414.2.a.f.1.1 2
3.2 odd 2 138.2.a.d.1.2 2
4.3 odd 2 3312.2.a.bc.1.1 2
12.11 even 2 1104.2.a.j.1.2 2
15.2 even 4 3450.2.d.x.2899.3 4
15.8 even 4 3450.2.d.x.2899.2 4
15.14 odd 2 3450.2.a.be.1.2 2
21.20 even 2 6762.2.a.cb.1.1 2
23.22 odd 2 9522.2.a.q.1.2 2
24.5 odd 2 4416.2.a.bh.1.1 2
24.11 even 2 4416.2.a.bl.1.1 2
69.68 even 2 3174.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.d.1.2 2 3.2 odd 2
414.2.a.f.1.1 2 1.1 even 1 trivial
1104.2.a.j.1.2 2 12.11 even 2
3174.2.a.s.1.1 2 69.68 even 2
3312.2.a.bc.1.1 2 4.3 odd 2
3450.2.a.be.1.2 2 15.14 odd 2
3450.2.d.x.2899.2 4 15.8 even 4
3450.2.d.x.2899.3 4 15.2 even 4
4416.2.a.bh.1.1 2 24.5 odd 2
4416.2.a.bl.1.1 2 24.11 even 2
6762.2.a.cb.1.1 2 21.20 even 2
9522.2.a.q.1.2 2 23.22 odd 2