Properties

Label 1950.2.a.bf.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
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] = [1950,2,Mod(1,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, 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("1950.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.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: \(15.5708283941\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1950.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.47214 q^{11} -1.00000 q^{12} -1.00000 q^{13} -3.23607 q^{14} +1.00000 q^{16} -7.23607 q^{17} +1.00000 q^{18} -2.76393 q^{19} +3.23607 q^{21} +4.47214 q^{22} -2.76393 q^{23} -1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} -3.23607 q^{28} -3.70820 q^{29} -4.00000 q^{31} +1.00000 q^{32} -4.47214 q^{33} -7.23607 q^{34} +1.00000 q^{36} +10.9443 q^{37} -2.76393 q^{38} +1.00000 q^{39} +3.52786 q^{41} +3.23607 q^{42} +2.47214 q^{43} +4.47214 q^{44} -2.76393 q^{46} -12.9443 q^{47} -1.00000 q^{48} +3.47214 q^{49} +7.23607 q^{51} -1.00000 q^{52} +0.472136 q^{53} -1.00000 q^{54} -3.23607 q^{56} +2.76393 q^{57} -3.70820 q^{58} -8.47214 q^{59} -10.9443 q^{61} -4.00000 q^{62} -3.23607 q^{63} +1.00000 q^{64} -4.47214 q^{66} -7.23607 q^{68} +2.76393 q^{69} -2.47214 q^{71} +1.00000 q^{72} -13.2361 q^{73} +10.9443 q^{74} -2.76393 q^{76} -14.4721 q^{77} +1.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} +3.52786 q^{82} +4.94427 q^{83} +3.23607 q^{84} +2.47214 q^{86} +3.70820 q^{87} +4.47214 q^{88} +0.472136 q^{89} +3.23607 q^{91} -2.76393 q^{92} +4.00000 q^{93} -12.9443 q^{94} -1.00000 q^{96} +3.70820 q^{97} +3.47214 q^{98} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{12} - 2 q^{13} - 2 q^{14} + 2 q^{16} - 10 q^{17} + 2 q^{18} - 10 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{24} - 2 q^{26} - 2 q^{27} - 2 q^{28} + 6 q^{29} - 8 q^{31} + 2 q^{32} - 10 q^{34} + 2 q^{36} + 4 q^{37} - 10 q^{38} + 2 q^{39} + 16 q^{41} + 2 q^{42} - 4 q^{43} - 10 q^{46} - 8 q^{47} - 2 q^{48} - 2 q^{49} + 10 q^{51} - 2 q^{52} - 8 q^{53} - 2 q^{54} - 2 q^{56} + 10 q^{57} + 6 q^{58} - 8 q^{59} - 4 q^{61} - 8 q^{62} - 2 q^{63} + 2 q^{64} - 10 q^{68} + 10 q^{69} + 4 q^{71} + 2 q^{72} - 22 q^{73} + 4 q^{74} - 10 q^{76} - 20 q^{77} + 2 q^{78} - 8 q^{79} + 2 q^{81} + 16 q^{82} - 8 q^{83} + 2 q^{84} - 4 q^{86} - 6 q^{87} - 8 q^{89} + 2 q^{91} - 10 q^{92} + 8 q^{93} - 8 q^{94} - 2 q^{96} - 6 q^{97} - 2 q^{98}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −3.23607 −1.22312 −0.611559 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.47214 1.34840 0.674200 0.738549i \(-0.264489\pi\)
0.674200 + 0.738549i \(0.264489\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −3.23607 −0.864876
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.23607 −1.75500 −0.877502 0.479573i \(-0.840792\pi\)
−0.877502 + 0.479573i \(0.840792\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.76393 −0.634089 −0.317045 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0 0
\(21\) 3.23607 0.706168
\(22\) 4.47214 0.953463
\(23\) −2.76393 −0.576320 −0.288160 0.957582i \(-0.593043\pi\)
−0.288160 + 0.957582i \(0.593043\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −3.23607 −0.611559
\(29\) −3.70820 −0.688596 −0.344298 0.938860i \(-0.611883\pi\)
−0.344298 + 0.938860i \(0.611883\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.47214 −0.778499
\(34\) −7.23607 −1.24098
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.9443 1.79923 0.899614 0.436687i \(-0.143848\pi\)
0.899614 + 0.436687i \(0.143848\pi\)
\(38\) −2.76393 −0.448369
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.52786 0.550960 0.275480 0.961307i \(-0.411163\pi\)
0.275480 + 0.961307i \(0.411163\pi\)
\(42\) 3.23607 0.499336
\(43\) 2.47214 0.376997 0.188499 0.982073i \(-0.439638\pi\)
0.188499 + 0.982073i \(0.439638\pi\)
\(44\) 4.47214 0.674200
\(45\) 0 0
\(46\) −2.76393 −0.407520
\(47\) −12.9443 −1.88812 −0.944058 0.329779i \(-0.893026\pi\)
−0.944058 + 0.329779i \(0.893026\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) 7.23607 1.01325
\(52\) −1.00000 −0.138675
\(53\) 0.472136 0.0648529 0.0324264 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −3.23607 −0.432438
\(57\) 2.76393 0.366092
\(58\) −3.70820 −0.486911
\(59\) −8.47214 −1.10298 −0.551489 0.834182i \(-0.685940\pi\)
−0.551489 + 0.834182i \(0.685940\pi\)
\(60\) 0 0
\(61\) −10.9443 −1.40127 −0.700635 0.713520i \(-0.747100\pi\)
−0.700635 + 0.713520i \(0.747100\pi\)
\(62\) −4.00000 −0.508001
\(63\) −3.23607 −0.407706
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.47214 −0.550482
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −7.23607 −0.877502
\(69\) 2.76393 0.332738
\(70\) 0 0
\(71\) −2.47214 −0.293389 −0.146694 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) 1.00000 0.117851
\(73\) −13.2361 −1.54916 −0.774582 0.632473i \(-0.782040\pi\)
−0.774582 + 0.632473i \(0.782040\pi\)
\(74\) 10.9443 1.27225
\(75\) 0 0
\(76\) −2.76393 −0.317045
\(77\) −14.4721 −1.64925
\(78\) 1.00000 0.113228
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.52786 0.389587
\(83\) 4.94427 0.542704 0.271352 0.962480i \(-0.412529\pi\)
0.271352 + 0.962480i \(0.412529\pi\)
\(84\) 3.23607 0.353084
\(85\) 0 0
\(86\) 2.47214 0.266577
\(87\) 3.70820 0.397561
\(88\) 4.47214 0.476731
\(89\) 0.472136 0.0500463 0.0250232 0.999687i \(-0.492034\pi\)
0.0250232 + 0.999687i \(0.492034\pi\)
\(90\) 0 0
\(91\) 3.23607 0.339232
\(92\) −2.76393 −0.288160
\(93\) 4.00000 0.414781
\(94\) −12.9443 −1.33510
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 3.70820 0.376511 0.188256 0.982120i \(-0.439717\pi\)
0.188256 + 0.982120i \(0.439717\pi\)
\(98\) 3.47214 0.350739
\(99\) 4.47214 0.449467
\(100\) 0 0
\(101\) −9.23607 −0.919023 −0.459512 0.888172i \(-0.651976\pi\)
−0.459512 + 0.888172i \(0.651976\pi\)
\(102\) 7.23607 0.716477
\(103\) −8.47214 −0.834784 −0.417392 0.908726i \(-0.637056\pi\)
−0.417392 + 0.908726i \(0.637056\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 0.472136 0.0458579
\(107\) 1.52786 0.147704 0.0738521 0.997269i \(-0.476471\pi\)
0.0738521 + 0.997269i \(0.476471\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.29180 −0.219514 −0.109757 0.993958i \(-0.535007\pi\)
−0.109757 + 0.993958i \(0.535007\pi\)
\(110\) 0 0
\(111\) −10.9443 −1.03878
\(112\) −3.23607 −0.305780
\(113\) −18.6525 −1.75468 −0.877339 0.479872i \(-0.840683\pi\)
−0.877339 + 0.479872i \(0.840683\pi\)
\(114\) 2.76393 0.258866
\(115\) 0 0
\(116\) −3.70820 −0.344298
\(117\) −1.00000 −0.0924500
\(118\) −8.47214 −0.779923
\(119\) 23.4164 2.14658
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) −10.9443 −0.990848
\(123\) −3.52786 −0.318097
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −3.23607 −0.288292
\(127\) −21.4164 −1.90040 −0.950199 0.311642i \(-0.899121\pi\)
−0.950199 + 0.311642i \(0.899121\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.47214 −0.217659
\(130\) 0 0
\(131\) 9.70820 0.848210 0.424105 0.905613i \(-0.360589\pi\)
0.424105 + 0.905613i \(0.360589\pi\)
\(132\) −4.47214 −0.389249
\(133\) 8.94427 0.775567
\(134\) 0 0
\(135\) 0 0
\(136\) −7.23607 −0.620488
\(137\) 19.8885 1.69919 0.849596 0.527433i \(-0.176846\pi\)
0.849596 + 0.527433i \(0.176846\pi\)
\(138\) 2.76393 0.235282
\(139\) −6.47214 −0.548959 −0.274480 0.961593i \(-0.588506\pi\)
−0.274480 + 0.961593i \(0.588506\pi\)
\(140\) 0 0
\(141\) 12.9443 1.09010
\(142\) −2.47214 −0.207457
\(143\) −4.47214 −0.373979
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −13.2361 −1.09542
\(147\) −3.47214 −0.286377
\(148\) 10.9443 0.899614
\(149\) 13.5279 1.10825 0.554123 0.832435i \(-0.313054\pi\)
0.554123 + 0.832435i \(0.313054\pi\)
\(150\) 0 0
\(151\) −19.4164 −1.58008 −0.790042 0.613052i \(-0.789942\pi\)
−0.790042 + 0.613052i \(0.789942\pi\)
\(152\) −2.76393 −0.224184
\(153\) −7.23607 −0.585001
\(154\) −14.4721 −1.16620
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 18.9443 1.51192 0.755959 0.654619i \(-0.227171\pi\)
0.755959 + 0.654619i \(0.227171\pi\)
\(158\) −4.00000 −0.318223
\(159\) −0.472136 −0.0374428
\(160\) 0 0
\(161\) 8.94427 0.704907
\(162\) 1.00000 0.0785674
\(163\) −12.9443 −1.01387 −0.506937 0.861983i \(-0.669222\pi\)
−0.506937 + 0.861983i \(0.669222\pi\)
\(164\) 3.52786 0.275480
\(165\) 0 0
\(166\) 4.94427 0.383750
\(167\) 19.4164 1.50249 0.751243 0.660025i \(-0.229454\pi\)
0.751243 + 0.660025i \(0.229454\pi\)
\(168\) 3.23607 0.249668
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.76393 −0.211363
\(172\) 2.47214 0.188499
\(173\) 22.9443 1.74442 0.872210 0.489131i \(-0.162686\pi\)
0.872210 + 0.489131i \(0.162686\pi\)
\(174\) 3.70820 0.281118
\(175\) 0 0
\(176\) 4.47214 0.337100
\(177\) 8.47214 0.636805
\(178\) 0.472136 0.0353881
\(179\) 8.18034 0.611427 0.305714 0.952124i \(-0.401105\pi\)
0.305714 + 0.952124i \(0.401105\pi\)
\(180\) 0 0
\(181\) 17.4164 1.29455 0.647276 0.762256i \(-0.275908\pi\)
0.647276 + 0.762256i \(0.275908\pi\)
\(182\) 3.23607 0.239873
\(183\) 10.9443 0.809024
\(184\) −2.76393 −0.203760
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −32.3607 −2.36645
\(188\) −12.9443 −0.944058
\(189\) 3.23607 0.235389
\(190\) 0 0
\(191\) 22.4721 1.62603 0.813013 0.582245i \(-0.197826\pi\)
0.813013 + 0.582245i \(0.197826\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.29180 0.596857 0.298428 0.954432i \(-0.403538\pi\)
0.298428 + 0.954432i \(0.403538\pi\)
\(194\) 3.70820 0.266234
\(195\) 0 0
\(196\) 3.47214 0.248010
\(197\) 14.9443 1.06474 0.532368 0.846513i \(-0.321302\pi\)
0.532368 + 0.846513i \(0.321302\pi\)
\(198\) 4.47214 0.317821
\(199\) −0.944272 −0.0669377 −0.0334688 0.999440i \(-0.510655\pi\)
−0.0334688 + 0.999440i \(0.510655\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −9.23607 −0.649847
\(203\) 12.0000 0.842235
\(204\) 7.23607 0.506626
\(205\) 0 0
\(206\) −8.47214 −0.590282
\(207\) −2.76393 −0.192107
\(208\) −1.00000 −0.0693375
\(209\) −12.3607 −0.855006
\(210\) 0 0
\(211\) 13.8885 0.956127 0.478063 0.878325i \(-0.341339\pi\)
0.478063 + 0.878325i \(0.341339\pi\)
\(212\) 0.472136 0.0324264
\(213\) 2.47214 0.169388
\(214\) 1.52786 0.104443
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 12.9443 0.878714
\(218\) −2.29180 −0.155220
\(219\) 13.2361 0.894411
\(220\) 0 0
\(221\) 7.23607 0.486751
\(222\) −10.9443 −0.734531
\(223\) −16.7639 −1.12260 −0.561298 0.827614i \(-0.689698\pi\)
−0.561298 + 0.827614i \(0.689698\pi\)
\(224\) −3.23607 −0.216219
\(225\) 0 0
\(226\) −18.6525 −1.24074
\(227\) −7.05573 −0.468305 −0.234153 0.972200i \(-0.575232\pi\)
−0.234153 + 0.972200i \(0.575232\pi\)
\(228\) 2.76393 0.183046
\(229\) 9.70820 0.641536 0.320768 0.947158i \(-0.396059\pi\)
0.320768 + 0.947158i \(0.396059\pi\)
\(230\) 0 0
\(231\) 14.4721 0.952197
\(232\) −3.70820 −0.243456
\(233\) −20.1803 −1.32206 −0.661029 0.750360i \(-0.729880\pi\)
−0.661029 + 0.750360i \(0.729880\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −8.47214 −0.551489
\(237\) 4.00000 0.259828
\(238\) 23.4164 1.51786
\(239\) −21.8885 −1.41585 −0.707926 0.706287i \(-0.750369\pi\)
−0.707926 + 0.706287i \(0.750369\pi\)
\(240\) 0 0
\(241\) 3.52786 0.227250 0.113625 0.993524i \(-0.463754\pi\)
0.113625 + 0.993524i \(0.463754\pi\)
\(242\) 9.00000 0.578542
\(243\) −1.00000 −0.0641500
\(244\) −10.9443 −0.700635
\(245\) 0 0
\(246\) −3.52786 −0.224928
\(247\) 2.76393 0.175865
\(248\) −4.00000 −0.254000
\(249\) −4.94427 −0.313331
\(250\) 0 0
\(251\) 2.29180 0.144657 0.0723284 0.997381i \(-0.476957\pi\)
0.0723284 + 0.997381i \(0.476957\pi\)
\(252\) −3.23607 −0.203853
\(253\) −12.3607 −0.777109
\(254\) −21.4164 −1.34378
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −0.763932 −0.0476528 −0.0238264 0.999716i \(-0.507585\pi\)
−0.0238264 + 0.999716i \(0.507585\pi\)
\(258\) −2.47214 −0.153908
\(259\) −35.4164 −2.20067
\(260\) 0 0
\(261\) −3.70820 −0.229532
\(262\) 9.70820 0.599775
\(263\) −10.1803 −0.627747 −0.313873 0.949465i \(-0.601627\pi\)
−0.313873 + 0.949465i \(0.601627\pi\)
\(264\) −4.47214 −0.275241
\(265\) 0 0
\(266\) 8.94427 0.548408
\(267\) −0.472136 −0.0288943
\(268\) 0 0
\(269\) 6.76393 0.412404 0.206202 0.978509i \(-0.433890\pi\)
0.206202 + 0.978509i \(0.433890\pi\)
\(270\) 0 0
\(271\) −11.4164 −0.693497 −0.346749 0.937958i \(-0.612714\pi\)
−0.346749 + 0.937958i \(0.612714\pi\)
\(272\) −7.23607 −0.438751
\(273\) −3.23607 −0.195856
\(274\) 19.8885 1.20151
\(275\) 0 0
\(276\) 2.76393 0.166369
\(277\) −20.8328 −1.25172 −0.625861 0.779934i \(-0.715252\pi\)
−0.625861 + 0.779934i \(0.715252\pi\)
\(278\) −6.47214 −0.388173
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 24.4721 1.45989 0.729943 0.683508i \(-0.239547\pi\)
0.729943 + 0.683508i \(0.239547\pi\)
\(282\) 12.9443 0.770820
\(283\) 11.4164 0.678635 0.339318 0.940672i \(-0.389804\pi\)
0.339318 + 0.940672i \(0.389804\pi\)
\(284\) −2.47214 −0.146694
\(285\) 0 0
\(286\) −4.47214 −0.264443
\(287\) −11.4164 −0.673889
\(288\) 1.00000 0.0589256
\(289\) 35.3607 2.08004
\(290\) 0 0
\(291\) −3.70820 −0.217379
\(292\) −13.2361 −0.774582
\(293\) −2.94427 −0.172006 −0.0860031 0.996295i \(-0.527409\pi\)
−0.0860031 + 0.996295i \(0.527409\pi\)
\(294\) −3.47214 −0.202499
\(295\) 0 0
\(296\) 10.9443 0.636123
\(297\) −4.47214 −0.259500
\(298\) 13.5279 0.783648
\(299\) 2.76393 0.159842
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −19.4164 −1.11729
\(303\) 9.23607 0.530598
\(304\) −2.76393 −0.158522
\(305\) 0 0
\(306\) −7.23607 −0.413658
\(307\) 24.3607 1.39034 0.695169 0.718847i \(-0.255330\pi\)
0.695169 + 0.718847i \(0.255330\pi\)
\(308\) −14.4721 −0.824626
\(309\) 8.47214 0.481963
\(310\) 0 0
\(311\) −22.4721 −1.27428 −0.637139 0.770749i \(-0.719882\pi\)
−0.637139 + 0.770749i \(0.719882\pi\)
\(312\) 1.00000 0.0566139
\(313\) 19.4164 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(314\) 18.9443 1.06909
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −13.4164 −0.753541 −0.376770 0.926307i \(-0.622965\pi\)
−0.376770 + 0.926307i \(0.622965\pi\)
\(318\) −0.472136 −0.0264761
\(319\) −16.5836 −0.928503
\(320\) 0 0
\(321\) −1.52786 −0.0852771
\(322\) 8.94427 0.498445
\(323\) 20.0000 1.11283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.9443 −0.716917
\(327\) 2.29180 0.126737
\(328\) 3.52786 0.194794
\(329\) 41.8885 2.30939
\(330\) 0 0
\(331\) −2.76393 −0.151919 −0.0759597 0.997111i \(-0.524202\pi\)
−0.0759597 + 0.997111i \(0.524202\pi\)
\(332\) 4.94427 0.271352
\(333\) 10.9443 0.599742
\(334\) 19.4164 1.06242
\(335\) 0 0
\(336\) 3.23607 0.176542
\(337\) 6.47214 0.352560 0.176280 0.984340i \(-0.443594\pi\)
0.176280 + 0.984340i \(0.443594\pi\)
\(338\) 1.00000 0.0543928
\(339\) 18.6525 1.01306
\(340\) 0 0
\(341\) −17.8885 −0.968719
\(342\) −2.76393 −0.149456
\(343\) 11.4164 0.616428
\(344\) 2.47214 0.133289
\(345\) 0 0
\(346\) 22.9443 1.23349
\(347\) −1.52786 −0.0820200 −0.0410100 0.999159i \(-0.513058\pi\)
−0.0410100 + 0.999159i \(0.513058\pi\)
\(348\) 3.70820 0.198781
\(349\) −15.2361 −0.815568 −0.407784 0.913078i \(-0.633698\pi\)
−0.407784 + 0.913078i \(0.633698\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 4.47214 0.238366
\(353\) 26.9443 1.43410 0.717049 0.697022i \(-0.245492\pi\)
0.717049 + 0.697022i \(0.245492\pi\)
\(354\) 8.47214 0.450289
\(355\) 0 0
\(356\) 0.472136 0.0250232
\(357\) −23.4164 −1.23933
\(358\) 8.18034 0.432344
\(359\) 6.47214 0.341586 0.170793 0.985307i \(-0.445367\pi\)
0.170793 + 0.985307i \(0.445367\pi\)
\(360\) 0 0
\(361\) −11.3607 −0.597931
\(362\) 17.4164 0.915386
\(363\) −9.00000 −0.472377
\(364\) 3.23607 0.169616
\(365\) 0 0
\(366\) 10.9443 0.572066
\(367\) −0.472136 −0.0246453 −0.0123226 0.999924i \(-0.503923\pi\)
−0.0123226 + 0.999924i \(0.503923\pi\)
\(368\) −2.76393 −0.144080
\(369\) 3.52786 0.183653
\(370\) 0 0
\(371\) −1.52786 −0.0793227
\(372\) 4.00000 0.207390
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) −32.3607 −1.67333
\(375\) 0 0
\(376\) −12.9443 −0.667550
\(377\) 3.70820 0.190982
\(378\) 3.23607 0.166445
\(379\) −35.1246 −1.80423 −0.902115 0.431496i \(-0.857986\pi\)
−0.902115 + 0.431496i \(0.857986\pi\)
\(380\) 0 0
\(381\) 21.4164 1.09720
\(382\) 22.4721 1.14977
\(383\) −5.52786 −0.282461 −0.141230 0.989977i \(-0.545106\pi\)
−0.141230 + 0.989977i \(0.545106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 8.29180 0.422041
\(387\) 2.47214 0.125666
\(388\) 3.70820 0.188256
\(389\) 12.6525 0.641506 0.320753 0.947163i \(-0.396064\pi\)
0.320753 + 0.947163i \(0.396064\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) 3.47214 0.175369
\(393\) −9.70820 −0.489714
\(394\) 14.9443 0.752882
\(395\) 0 0
\(396\) 4.47214 0.224733
\(397\) −17.4164 −0.874104 −0.437052 0.899436i \(-0.643978\pi\)
−0.437052 + 0.899436i \(0.643978\pi\)
\(398\) −0.944272 −0.0473321
\(399\) −8.94427 −0.447774
\(400\) 0 0
\(401\) −24.4721 −1.22208 −0.611040 0.791600i \(-0.709249\pi\)
−0.611040 + 0.791600i \(0.709249\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −9.23607 −0.459512
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 48.9443 2.42608
\(408\) 7.23607 0.358239
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −19.8885 −0.981030
\(412\) −8.47214 −0.417392
\(413\) 27.4164 1.34907
\(414\) −2.76393 −0.135840
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 6.47214 0.316942
\(418\) −12.3607 −0.604581
\(419\) 27.5967 1.34819 0.674095 0.738645i \(-0.264534\pi\)
0.674095 + 0.738645i \(0.264534\pi\)
\(420\) 0 0
\(421\) −13.7082 −0.668097 −0.334048 0.942556i \(-0.608415\pi\)
−0.334048 + 0.942556i \(0.608415\pi\)
\(422\) 13.8885 0.676084
\(423\) −12.9443 −0.629372
\(424\) 0.472136 0.0229289
\(425\) 0 0
\(426\) 2.47214 0.119775
\(427\) 35.4164 1.71392
\(428\) 1.52786 0.0738521
\(429\) 4.47214 0.215917
\(430\) 0 0
\(431\) −14.8328 −0.714472 −0.357236 0.934014i \(-0.616281\pi\)
−0.357236 + 0.934014i \(0.616281\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 9.52786 0.457880 0.228940 0.973441i \(-0.426474\pi\)
0.228940 + 0.973441i \(0.426474\pi\)
\(434\) 12.9443 0.621345
\(435\) 0 0
\(436\) −2.29180 −0.109757
\(437\) 7.63932 0.365438
\(438\) 13.2361 0.632444
\(439\) −4.94427 −0.235977 −0.117989 0.993015i \(-0.537645\pi\)
−0.117989 + 0.993015i \(0.537645\pi\)
\(440\) 0 0
\(441\) 3.47214 0.165340
\(442\) 7.23607 0.344185
\(443\) 22.4721 1.06768 0.533842 0.845584i \(-0.320748\pi\)
0.533842 + 0.845584i \(0.320748\pi\)
\(444\) −10.9443 −0.519392
\(445\) 0 0
\(446\) −16.7639 −0.793795
\(447\) −13.5279 −0.639846
\(448\) −3.23607 −0.152890
\(449\) 0.472136 0.0222815 0.0111407 0.999938i \(-0.496454\pi\)
0.0111407 + 0.999938i \(0.496454\pi\)
\(450\) 0 0
\(451\) 15.7771 0.742914
\(452\) −18.6525 −0.877339
\(453\) 19.4164 0.912262
\(454\) −7.05573 −0.331142
\(455\) 0 0
\(456\) 2.76393 0.129433
\(457\) 3.70820 0.173462 0.0867312 0.996232i \(-0.472358\pi\)
0.0867312 + 0.996232i \(0.472358\pi\)
\(458\) 9.70820 0.453635
\(459\) 7.23607 0.337751
\(460\) 0 0
\(461\) 41.8885 1.95094 0.975472 0.220124i \(-0.0706461\pi\)
0.975472 + 0.220124i \(0.0706461\pi\)
\(462\) 14.4721 0.673305
\(463\) 22.2918 1.03599 0.517994 0.855384i \(-0.326679\pi\)
0.517994 + 0.855384i \(0.326679\pi\)
\(464\) −3.70820 −0.172149
\(465\) 0 0
\(466\) −20.1803 −0.934836
\(467\) −6.47214 −0.299495 −0.149747 0.988724i \(-0.547846\pi\)
−0.149747 + 0.988724i \(0.547846\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −18.9443 −0.872906
\(472\) −8.47214 −0.389962
\(473\) 11.0557 0.508343
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 23.4164 1.07329
\(477\) 0.472136 0.0216176
\(478\) −21.8885 −1.00116
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −10.9443 −0.499016
\(482\) 3.52786 0.160690
\(483\) −8.94427 −0.406978
\(484\) 9.00000 0.409091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 23.5967 1.06927 0.534635 0.845083i \(-0.320449\pi\)
0.534635 + 0.845083i \(0.320449\pi\)
\(488\) −10.9443 −0.495424
\(489\) 12.9443 0.585360
\(490\) 0 0
\(491\) −16.1803 −0.730209 −0.365104 0.930967i \(-0.618967\pi\)
−0.365104 + 0.930967i \(0.618967\pi\)
\(492\) −3.52786 −0.159048
\(493\) 26.8328 1.20849
\(494\) 2.76393 0.124355
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 8.00000 0.358849
\(498\) −4.94427 −0.221558
\(499\) −39.1246 −1.75146 −0.875729 0.482803i \(-0.839619\pi\)
−0.875729 + 0.482803i \(0.839619\pi\)
\(500\) 0 0
\(501\) −19.4164 −0.867461
\(502\) 2.29180 0.102288
\(503\) 30.7639 1.37170 0.685848 0.727745i \(-0.259431\pi\)
0.685848 + 0.727745i \(0.259431\pi\)
\(504\) −3.23607 −0.144146
\(505\) 0 0
\(506\) −12.3607 −0.549499
\(507\) −1.00000 −0.0444116
\(508\) −21.4164 −0.950199
\(509\) −15.0557 −0.667333 −0.333667 0.942691i \(-0.608286\pi\)
−0.333667 + 0.942691i \(0.608286\pi\)
\(510\) 0 0
\(511\) 42.8328 1.89481
\(512\) 1.00000 0.0441942
\(513\) 2.76393 0.122031
\(514\) −0.763932 −0.0336956
\(515\) 0 0
\(516\) −2.47214 −0.108830
\(517\) −57.8885 −2.54594
\(518\) −35.4164 −1.55611
\(519\) −22.9443 −1.00714
\(520\) 0 0
\(521\) 20.8328 0.912702 0.456351 0.889800i \(-0.349156\pi\)
0.456351 + 0.889800i \(0.349156\pi\)
\(522\) −3.70820 −0.162304
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 9.70820 0.424105
\(525\) 0 0
\(526\) −10.1803 −0.443884
\(527\) 28.9443 1.26083
\(528\) −4.47214 −0.194625
\(529\) −15.3607 −0.667856
\(530\) 0 0
\(531\) −8.47214 −0.367659
\(532\) 8.94427 0.387783
\(533\) −3.52786 −0.152809
\(534\) −0.472136 −0.0204313
\(535\) 0 0
\(536\) 0 0
\(537\) −8.18034 −0.353008
\(538\) 6.76393 0.291614
\(539\) 15.5279 0.668832
\(540\) 0 0
\(541\) −4.76393 −0.204817 −0.102409 0.994742i \(-0.532655\pi\)
−0.102409 + 0.994742i \(0.532655\pi\)
\(542\) −11.4164 −0.490377
\(543\) −17.4164 −0.747410
\(544\) −7.23607 −0.310244
\(545\) 0 0
\(546\) −3.23607 −0.138491
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 19.8885 0.849596
\(549\) −10.9443 −0.467090
\(550\) 0 0
\(551\) 10.2492 0.436632
\(552\) 2.76393 0.117641
\(553\) 12.9443 0.550446
\(554\) −20.8328 −0.885102
\(555\) 0 0
\(556\) −6.47214 −0.274480
\(557\) −19.3050 −0.817977 −0.408989 0.912540i \(-0.634118\pi\)
−0.408989 + 0.912540i \(0.634118\pi\)
\(558\) −4.00000 −0.169334
\(559\) −2.47214 −0.104560
\(560\) 0 0
\(561\) 32.3607 1.36627
\(562\) 24.4721 1.03229
\(563\) 19.4164 0.818304 0.409152 0.912466i \(-0.365825\pi\)
0.409152 + 0.912466i \(0.365825\pi\)
\(564\) 12.9443 0.545052
\(565\) 0 0
\(566\) 11.4164 0.479867
\(567\) −3.23607 −0.135902
\(568\) −2.47214 −0.103729
\(569\) 34.9443 1.46494 0.732470 0.680799i \(-0.238367\pi\)
0.732470 + 0.680799i \(0.238367\pi\)
\(570\) 0 0
\(571\) −8.58359 −0.359212 −0.179606 0.983739i \(-0.557482\pi\)
−0.179606 + 0.983739i \(0.557482\pi\)
\(572\) −4.47214 −0.186989
\(573\) −22.4721 −0.938787
\(574\) −11.4164 −0.476512
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 43.1246 1.79530 0.897651 0.440708i \(-0.145273\pi\)
0.897651 + 0.440708i \(0.145273\pi\)
\(578\) 35.3607 1.47081
\(579\) −8.29180 −0.344595
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) −3.70820 −0.153710
\(583\) 2.11146 0.0874476
\(584\) −13.2361 −0.547712
\(585\) 0 0
\(586\) −2.94427 −0.121627
\(587\) 2.11146 0.0871491 0.0435746 0.999050i \(-0.486125\pi\)
0.0435746 + 0.999050i \(0.486125\pi\)
\(588\) −3.47214 −0.143188
\(589\) 11.0557 0.455543
\(590\) 0 0
\(591\) −14.9443 −0.614725
\(592\) 10.9443 0.449807
\(593\) −3.52786 −0.144872 −0.0724360 0.997373i \(-0.523077\pi\)
−0.0724360 + 0.997373i \(0.523077\pi\)
\(594\) −4.47214 −0.183494
\(595\) 0 0
\(596\) 13.5279 0.554123
\(597\) 0.944272 0.0386465
\(598\) 2.76393 0.113026
\(599\) −6.11146 −0.249707 −0.124854 0.992175i \(-0.539846\pi\)
−0.124854 + 0.992175i \(0.539846\pi\)
\(600\) 0 0
\(601\) 13.4164 0.547267 0.273633 0.961834i \(-0.411775\pi\)
0.273633 + 0.961834i \(0.411775\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) −19.4164 −0.790042
\(605\) 0 0
\(606\) 9.23607 0.375190
\(607\) −1.41641 −0.0574902 −0.0287451 0.999587i \(-0.509151\pi\)
−0.0287451 + 0.999587i \(0.509151\pi\)
\(608\) −2.76393 −0.112092
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 12.9443 0.523669
\(612\) −7.23607 −0.292501
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 24.3607 0.983117
\(615\) 0 0
\(616\) −14.4721 −0.583099
\(617\) −35.8885 −1.44482 −0.722409 0.691466i \(-0.756965\pi\)
−0.722409 + 0.691466i \(0.756965\pi\)
\(618\) 8.47214 0.340799
\(619\) −25.2361 −1.01432 −0.507162 0.861851i \(-0.669305\pi\)
−0.507162 + 0.861851i \(0.669305\pi\)
\(620\) 0 0
\(621\) 2.76393 0.110913
\(622\) −22.4721 −0.901051
\(623\) −1.52786 −0.0612126
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 19.4164 0.776036
\(627\) 12.3607 0.493638
\(628\) 18.9443 0.755959
\(629\) −79.1935 −3.15765
\(630\) 0 0
\(631\) −12.5836 −0.500945 −0.250472 0.968124i \(-0.580586\pi\)
−0.250472 + 0.968124i \(0.580586\pi\)
\(632\) −4.00000 −0.159111
\(633\) −13.8885 −0.552020
\(634\) −13.4164 −0.532834
\(635\) 0 0
\(636\) −0.472136 −0.0187214
\(637\) −3.47214 −0.137571
\(638\) −16.5836 −0.656551
\(639\) −2.47214 −0.0977962
\(640\) 0 0
\(641\) −23.3050 −0.920490 −0.460245 0.887792i \(-0.652238\pi\)
−0.460245 + 0.887792i \(0.652238\pi\)
\(642\) −1.52786 −0.0603000
\(643\) −19.0557 −0.751485 −0.375742 0.926724i \(-0.622612\pi\)
−0.375742 + 0.926724i \(0.622612\pi\)
\(644\) 8.94427 0.352454
\(645\) 0 0
\(646\) 20.0000 0.786889
\(647\) 3.70820 0.145785 0.0728923 0.997340i \(-0.476777\pi\)
0.0728923 + 0.997340i \(0.476777\pi\)
\(648\) 1.00000 0.0392837
\(649\) −37.8885 −1.48726
\(650\) 0 0
\(651\) −12.9443 −0.507326
\(652\) −12.9443 −0.506937
\(653\) 30.3607 1.18811 0.594053 0.804426i \(-0.297527\pi\)
0.594053 + 0.804426i \(0.297527\pi\)
\(654\) 2.29180 0.0896163
\(655\) 0 0
\(656\) 3.52786 0.137740
\(657\) −13.2361 −0.516388
\(658\) 41.8885 1.63299
\(659\) −10.0689 −0.392228 −0.196114 0.980581i \(-0.562832\pi\)
−0.196114 + 0.980581i \(0.562832\pi\)
\(660\) 0 0
\(661\) −41.7082 −1.62226 −0.811131 0.584865i \(-0.801147\pi\)
−0.811131 + 0.584865i \(0.801147\pi\)
\(662\) −2.76393 −0.107423
\(663\) −7.23607 −0.281026
\(664\) 4.94427 0.191875
\(665\) 0 0
\(666\) 10.9443 0.424082
\(667\) 10.2492 0.396852
\(668\) 19.4164 0.751243
\(669\) 16.7639 0.648131
\(670\) 0 0
\(671\) −48.9443 −1.88947
\(672\) 3.23607 0.124834
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 6.47214 0.249297
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −27.5279 −1.05798 −0.528991 0.848628i \(-0.677429\pi\)
−0.528991 + 0.848628i \(0.677429\pi\)
\(678\) 18.6525 0.716344
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) 7.05573 0.270376
\(682\) −17.8885 −0.684988
\(683\) 41.8885 1.60282 0.801410 0.598115i \(-0.204083\pi\)
0.801410 + 0.598115i \(0.204083\pi\)
\(684\) −2.76393 −0.105682
\(685\) 0 0
\(686\) 11.4164 0.435880
\(687\) −9.70820 −0.370391
\(688\) 2.47214 0.0942493
\(689\) −0.472136 −0.0179869
\(690\) 0 0
\(691\) 0.291796 0.0111004 0.00555022 0.999985i \(-0.498233\pi\)
0.00555022 + 0.999985i \(0.498233\pi\)
\(692\) 22.9443 0.872210
\(693\) −14.4721 −0.549751
\(694\) −1.52786 −0.0579969
\(695\) 0 0
\(696\) 3.70820 0.140559
\(697\) −25.5279 −0.966937
\(698\) −15.2361 −0.576694
\(699\) 20.1803 0.763291
\(700\) 0 0
\(701\) −24.0689 −0.909069 −0.454535 0.890729i \(-0.650194\pi\)
−0.454535 + 0.890729i \(0.650194\pi\)
\(702\) 1.00000 0.0377426
\(703\) −30.2492 −1.14087
\(704\) 4.47214 0.168550
\(705\) 0 0
\(706\) 26.9443 1.01406
\(707\) 29.8885 1.12407
\(708\) 8.47214 0.318402
\(709\) 32.5410 1.22210 0.611052 0.791591i \(-0.290747\pi\)
0.611052 + 0.791591i \(0.290747\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0.472136 0.0176940
\(713\) 11.0557 0.414040
\(714\) −23.4164 −0.876337
\(715\) 0 0
\(716\) 8.18034 0.305714
\(717\) 21.8885 0.817443
\(718\) 6.47214 0.241538
\(719\) 20.9443 0.781090 0.390545 0.920584i \(-0.372287\pi\)
0.390545 + 0.920584i \(0.372287\pi\)
\(720\) 0 0
\(721\) 27.4164 1.02104
\(722\) −11.3607 −0.422801
\(723\) −3.52786 −0.131203
\(724\) 17.4164 0.647276
\(725\) 0 0
\(726\) −9.00000 −0.334021
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 3.23607 0.119937
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −17.8885 −0.661632
\(732\) 10.9443 0.404512
\(733\) −33.4164 −1.23426 −0.617132 0.786860i \(-0.711705\pi\)
−0.617132 + 0.786860i \(0.711705\pi\)
\(734\) −0.472136 −0.0174269
\(735\) 0 0
\(736\) −2.76393 −0.101880
\(737\) 0 0
\(738\) 3.52786 0.129862
\(739\) −0.291796 −0.0107339 −0.00536695 0.999986i \(-0.501708\pi\)
−0.00536695 + 0.999986i \(0.501708\pi\)
\(740\) 0 0
\(741\) −2.76393 −0.101536
\(742\) −1.52786 −0.0560897
\(743\) −28.3607 −1.04045 −0.520226 0.854029i \(-0.674152\pi\)
−0.520226 + 0.854029i \(0.674152\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) 4.94427 0.180901
\(748\) −32.3607 −1.18322
\(749\) −4.94427 −0.180660
\(750\) 0 0
\(751\) −46.8328 −1.70895 −0.854477 0.519489i \(-0.826122\pi\)
−0.854477 + 0.519489i \(0.826122\pi\)
\(752\) −12.9443 −0.472029
\(753\) −2.29180 −0.0835177
\(754\) 3.70820 0.135045
\(755\) 0 0
\(756\) 3.23607 0.117695
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −35.1246 −1.27578
\(759\) 12.3607 0.448664
\(760\) 0 0
\(761\) 7.52786 0.272885 0.136442 0.990648i \(-0.456433\pi\)
0.136442 + 0.990648i \(0.456433\pi\)
\(762\) 21.4164 0.775835
\(763\) 7.41641 0.268492
\(764\) 22.4721 0.813013
\(765\) 0 0
\(766\) −5.52786 −0.199730
\(767\) 8.47214 0.305911
\(768\) −1.00000 −0.0360844
\(769\) −4.83282 −0.174276 −0.0871379 0.996196i \(-0.527772\pi\)
−0.0871379 + 0.996196i \(0.527772\pi\)
\(770\) 0 0
\(771\) 0.763932 0.0275123
\(772\) 8.29180 0.298428
\(773\) 2.94427 0.105898 0.0529491 0.998597i \(-0.483138\pi\)
0.0529491 + 0.998597i \(0.483138\pi\)
\(774\) 2.47214 0.0888591
\(775\) 0 0
\(776\) 3.70820 0.133117
\(777\) 35.4164 1.27056
\(778\) 12.6525 0.453613
\(779\) −9.75078 −0.349358
\(780\) 0 0
\(781\) −11.0557 −0.395605
\(782\) 20.0000 0.715199
\(783\) 3.70820 0.132520
\(784\) 3.47214 0.124005
\(785\) 0 0
\(786\) −9.70820 −0.346280
\(787\) −28.9443 −1.03175 −0.515876 0.856663i \(-0.672533\pi\)
−0.515876 + 0.856663i \(0.672533\pi\)
\(788\) 14.9443 0.532368
\(789\) 10.1803 0.362430
\(790\) 0 0
\(791\) 60.3607 2.14618
\(792\) 4.47214 0.158910
\(793\) 10.9443 0.388642
\(794\) −17.4164 −0.618085
\(795\) 0 0
\(796\) −0.944272 −0.0334688
\(797\) −9.05573 −0.320770 −0.160385 0.987055i \(-0.551274\pi\)
−0.160385 + 0.987055i \(0.551274\pi\)
\(798\) −8.94427 −0.316624
\(799\) 93.6656 3.31365
\(800\) 0 0
\(801\) 0.472136 0.0166821
\(802\) −24.4721 −0.864141
\(803\) −59.1935 −2.08889
\(804\) 0 0
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −6.76393 −0.238102
\(808\) −9.23607 −0.324924
\(809\) 31.5279 1.10846 0.554230 0.832363i \(-0.313013\pi\)
0.554230 + 0.832363i \(0.313013\pi\)
\(810\) 0 0
\(811\) 4.29180 0.150705 0.0753527 0.997157i \(-0.475992\pi\)
0.0753527 + 0.997157i \(0.475992\pi\)
\(812\) 12.0000 0.421117
\(813\) 11.4164 0.400391
\(814\) 48.9443 1.71550
\(815\) 0 0
\(816\) 7.23607 0.253313
\(817\) −6.83282 −0.239050
\(818\) 6.00000 0.209785
\(819\) 3.23607 0.113077
\(820\) 0 0
\(821\) −30.4721 −1.06348 −0.531742 0.846906i \(-0.678463\pi\)
−0.531742 + 0.846906i \(0.678463\pi\)
\(822\) −19.8885 −0.693693
\(823\) −48.2492 −1.68186 −0.840931 0.541142i \(-0.817992\pi\)
−0.840931 + 0.541142i \(0.817992\pi\)
\(824\) −8.47214 −0.295141
\(825\) 0 0
\(826\) 27.4164 0.953939
\(827\) 41.8885 1.45661 0.728304 0.685254i \(-0.240309\pi\)
0.728304 + 0.685254i \(0.240309\pi\)
\(828\) −2.76393 −0.0960533
\(829\) 42.7214 1.48377 0.741887 0.670525i \(-0.233931\pi\)
0.741887 + 0.670525i \(0.233931\pi\)
\(830\) 0 0
\(831\) 20.8328 0.722682
\(832\) −1.00000 −0.0346688
\(833\) −25.1246 −0.870516
\(834\) 6.47214 0.224112
\(835\) 0 0
\(836\) −12.3607 −0.427503
\(837\) 4.00000 0.138260
\(838\) 27.5967 0.953314
\(839\) −51.7771 −1.78754 −0.893772 0.448522i \(-0.851951\pi\)
−0.893772 + 0.448522i \(0.851951\pi\)
\(840\) 0 0
\(841\) −15.2492 −0.525835
\(842\) −13.7082 −0.472416
\(843\) −24.4721 −0.842865
\(844\) 13.8885 0.478063
\(845\) 0 0
\(846\) −12.9443 −0.445033
\(847\) −29.1246 −1.00073
\(848\) 0.472136 0.0162132
\(849\) −11.4164 −0.391810
\(850\) 0 0
\(851\) −30.2492 −1.03693
\(852\) 2.47214 0.0846940
\(853\) −41.4164 −1.41807 −0.709035 0.705173i \(-0.750869\pi\)
−0.709035 + 0.705173i \(0.750869\pi\)
\(854\) 35.4164 1.21192
\(855\) 0 0
\(856\) 1.52786 0.0522213
\(857\) 0.180340 0.00616029 0.00308015 0.999995i \(-0.499020\pi\)
0.00308015 + 0.999995i \(0.499020\pi\)
\(858\) 4.47214 0.152676
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 11.4164 0.389070
\(862\) −14.8328 −0.505208
\(863\) −22.4721 −0.764960 −0.382480 0.923964i \(-0.624930\pi\)
−0.382480 + 0.923964i \(0.624930\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 9.52786 0.323770
\(867\) −35.3607 −1.20091
\(868\) 12.9443 0.439357
\(869\) −17.8885 −0.606827
\(870\) 0 0
\(871\) 0 0
\(872\) −2.29180 −0.0776100
\(873\) 3.70820 0.125504
\(874\) 7.63932 0.258404
\(875\) 0 0
\(876\) 13.2361 0.447205
\(877\) −24.4721 −0.826365 −0.413183 0.910648i \(-0.635583\pi\)
−0.413183 + 0.910648i \(0.635583\pi\)
\(878\) −4.94427 −0.166861
\(879\) 2.94427 0.0993078
\(880\) 0 0
\(881\) 34.3607 1.15764 0.578820 0.815455i \(-0.303513\pi\)
0.578820 + 0.815455i \(0.303513\pi\)
\(882\) 3.47214 0.116913
\(883\) −25.8885 −0.871219 −0.435609 0.900136i \(-0.643467\pi\)
−0.435609 + 0.900136i \(0.643467\pi\)
\(884\) 7.23607 0.243375
\(885\) 0 0
\(886\) 22.4721 0.754966
\(887\) 10.1803 0.341822 0.170911 0.985286i \(-0.445329\pi\)
0.170911 + 0.985286i \(0.445329\pi\)
\(888\) −10.9443 −0.367266
\(889\) 69.3050 2.32441
\(890\) 0 0
\(891\) 4.47214 0.149822
\(892\) −16.7639 −0.561298
\(893\) 35.7771 1.19723
\(894\) −13.5279 −0.452439
\(895\) 0 0
\(896\) −3.23607 −0.108109
\(897\) −2.76393 −0.0922850
\(898\) 0.472136 0.0157554
\(899\) 14.8328 0.494702
\(900\) 0 0
\(901\) −3.41641 −0.113817
\(902\) 15.7771 0.525320
\(903\) 8.00000 0.266223
\(904\) −18.6525 −0.620372
\(905\) 0 0
\(906\) 19.4164 0.645067
\(907\) −45.5279 −1.51173 −0.755864 0.654729i \(-0.772783\pi\)
−0.755864 + 0.654729i \(0.772783\pi\)
\(908\) −7.05573 −0.234153
\(909\) −9.23607 −0.306341
\(910\) 0 0
\(911\) −11.0557 −0.366293 −0.183146 0.983086i \(-0.558628\pi\)
−0.183146 + 0.983086i \(0.558628\pi\)
\(912\) 2.76393 0.0915229
\(913\) 22.1115 0.731782
\(914\) 3.70820 0.122656
\(915\) 0 0
\(916\) 9.70820 0.320768
\(917\) −31.4164 −1.03746
\(918\) 7.23607 0.238826
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −24.3607 −0.802712
\(922\) 41.8885 1.37953
\(923\) 2.47214 0.0813713
\(924\) 14.4721 0.476098
\(925\) 0 0
\(926\) 22.2918 0.732554
\(927\) −8.47214 −0.278261
\(928\) −3.70820 −0.121728
\(929\) 32.4721 1.06538 0.532688 0.846312i \(-0.321182\pi\)
0.532688 + 0.846312i \(0.321182\pi\)
\(930\) 0 0
\(931\) −9.59675 −0.314521
\(932\) −20.1803 −0.661029
\(933\) 22.4721 0.735705
\(934\) −6.47214 −0.211775
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) −19.7771 −0.646089 −0.323045 0.946384i \(-0.604706\pi\)
−0.323045 + 0.946384i \(0.604706\pi\)
\(938\) 0 0
\(939\) −19.4164 −0.633631
\(940\) 0 0
\(941\) −37.5279 −1.22337 −0.611687 0.791100i \(-0.709509\pi\)
−0.611687 + 0.791100i \(0.709509\pi\)
\(942\) −18.9443 −0.617238
\(943\) −9.75078 −0.317529
\(944\) −8.47214 −0.275745
\(945\) 0 0
\(946\) 11.0557 0.359453
\(947\) −4.94427 −0.160667 −0.0803336 0.996768i \(-0.525599\pi\)
−0.0803336 + 0.996768i \(0.525599\pi\)
\(948\) 4.00000 0.129914
\(949\) 13.2361 0.429661
\(950\) 0 0
\(951\) 13.4164 0.435057
\(952\) 23.4164 0.758930
\(953\) −29.1246 −0.943439 −0.471719 0.881749i \(-0.656366\pi\)
−0.471719 + 0.881749i \(0.656366\pi\)
\(954\) 0.472136 0.0152860
\(955\) 0 0
\(956\) −21.8885 −0.707926
\(957\) 16.5836 0.536071
\(958\) −36.0000 −1.16311
\(959\) −64.3607 −2.07831
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −10.9443 −0.352857
\(963\) 1.52786 0.0492347
\(964\) 3.52786 0.113625
\(965\) 0 0
\(966\) −8.94427 −0.287777
\(967\) 56.5410 1.81824 0.909118 0.416538i \(-0.136757\pi\)
0.909118 + 0.416538i \(0.136757\pi\)
\(968\) 9.00000 0.289271
\(969\) −20.0000 −0.642493
\(970\) 0 0
\(971\) 23.2361 0.745681 0.372840 0.927895i \(-0.378384\pi\)
0.372840 + 0.927895i \(0.378384\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 20.9443 0.671443
\(974\) 23.5967 0.756089
\(975\) 0 0
\(976\) −10.9443 −0.350318
\(977\) 10.3607 0.331468 0.165734 0.986171i \(-0.447001\pi\)
0.165734 + 0.986171i \(0.447001\pi\)
\(978\) 12.9443 0.413912
\(979\) 2.11146 0.0674824
\(980\) 0 0
\(981\) −2.29180 −0.0731714
\(982\) −16.1803 −0.516335
\(983\) 7.41641 0.236547 0.118273 0.992981i \(-0.462264\pi\)
0.118273 + 0.992981i \(0.462264\pi\)
\(984\) −3.52786 −0.112464
\(985\) 0 0
\(986\) 26.8328 0.854531
\(987\) −41.8885 −1.33333
\(988\) 2.76393 0.0879324
\(989\) −6.83282 −0.217271
\(990\) 0 0
\(991\) 24.9443 0.792381 0.396190 0.918168i \(-0.370332\pi\)
0.396190 + 0.918168i \(0.370332\pi\)
\(992\) −4.00000 −0.127000
\(993\) 2.76393 0.0877107
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) −4.94427 −0.156665
\(997\) −43.8885 −1.38996 −0.694982 0.719027i \(-0.744588\pi\)
−0.694982 + 0.719027i \(0.744588\pi\)
\(998\) −39.1246 −1.23847
\(999\) −10.9443 −0.346261
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 1950.2.a.bf.1.1 2
3.2 odd 2 5850.2.a.cf.1.1 2
5.2 odd 4 390.2.e.e.79.3 yes 4
5.3 odd 4 390.2.e.e.79.2 4
5.4 even 2 1950.2.a.be.1.2 2
15.2 even 4 1170.2.e.e.469.2 4
15.8 even 4 1170.2.e.e.469.3 4
15.14 odd 2 5850.2.a.cm.1.2 2
20.3 even 4 3120.2.l.k.1249.4 4
20.7 even 4 3120.2.l.k.1249.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.e.79.2 4 5.3 odd 4
390.2.e.e.79.3 yes 4 5.2 odd 4
1170.2.e.e.469.2 4 15.2 even 4
1170.2.e.e.469.3 4 15.8 even 4
1950.2.a.be.1.2 2 5.4 even 2
1950.2.a.bf.1.1 2 1.1 even 1 trivial
3120.2.l.k.1249.1 4 20.7 even 4
3120.2.l.k.1249.4 4 20.3 even 4
5850.2.a.cf.1.1 2 3.2 odd 2
5850.2.a.cm.1.2 2 15.14 odd 2