Properties

Label 5550.2.a.cb.1.2
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
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] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, 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("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.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: \(44.3169731218\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.41421 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.41421 q^{7} +1.00000 q^{8} +1.00000 q^{9} -6.41421 q^{11} +1.00000 q^{12} -5.24264 q^{13} +2.41421 q^{14} +1.00000 q^{16} -3.58579 q^{17} +1.00000 q^{18} -2.17157 q^{19} +2.41421 q^{21} -6.41421 q^{22} -3.00000 q^{23} +1.00000 q^{24} -5.24264 q^{26} +1.00000 q^{27} +2.41421 q^{28} +7.07107 q^{29} -3.65685 q^{31} +1.00000 q^{32} -6.41421 q^{33} -3.58579 q^{34} +1.00000 q^{36} +1.00000 q^{37} -2.17157 q^{38} -5.24264 q^{39} -6.58579 q^{41} +2.41421 q^{42} -10.4853 q^{43} -6.41421 q^{44} -3.00000 q^{46} +11.8995 q^{47} +1.00000 q^{48} -1.17157 q^{49} -3.58579 q^{51} -5.24264 q^{52} +3.82843 q^{53} +1.00000 q^{54} +2.41421 q^{56} -2.17157 q^{57} +7.07107 q^{58} -13.0711 q^{59} -6.48528 q^{61} -3.65685 q^{62} +2.41421 q^{63} +1.00000 q^{64} -6.41421 q^{66} +0.828427 q^{67} -3.58579 q^{68} -3.00000 q^{69} -9.41421 q^{71} +1.00000 q^{72} -2.17157 q^{73} +1.00000 q^{74} -2.17157 q^{76} -15.4853 q^{77} -5.24264 q^{78} +12.2426 q^{79} +1.00000 q^{81} -6.58579 q^{82} -6.89949 q^{83} +2.41421 q^{84} -10.4853 q^{86} +7.07107 q^{87} -6.41421 q^{88} +2.89949 q^{89} -12.6569 q^{91} -3.00000 q^{92} -3.65685 q^{93} +11.8995 q^{94} +1.00000 q^{96} +6.72792 q^{97} -1.17157 q^{98} -6.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 10 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 10 q^{17} + 2 q^{18} - 10 q^{19} + 2 q^{21} - 10 q^{22} - 6 q^{23} + 2 q^{24} - 2 q^{26} + 2 q^{27} + 2 q^{28} + 4 q^{31} + 2 q^{32} - 10 q^{33} - 10 q^{34} + 2 q^{36} + 2 q^{37} - 10 q^{38} - 2 q^{39} - 16 q^{41} + 2 q^{42} - 4 q^{43} - 10 q^{44} - 6 q^{46} + 4 q^{47} + 2 q^{48} - 8 q^{49} - 10 q^{51} - 2 q^{52} + 2 q^{53} + 2 q^{54} + 2 q^{56} - 10 q^{57} - 12 q^{59} + 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} - 10 q^{66} - 4 q^{67} - 10 q^{68} - 6 q^{69} - 16 q^{71} + 2 q^{72} - 10 q^{73} + 2 q^{74} - 10 q^{76} - 14 q^{77} - 2 q^{78} + 16 q^{79} + 2 q^{81} - 16 q^{82} + 6 q^{83} + 2 q^{84} - 4 q^{86} - 10 q^{88} - 14 q^{89} - 14 q^{91} - 6 q^{92} + 4 q^{93} + 4 q^{94} + 2 q^{96} - 12 q^{97} - 8 q^{98} - 10 q^{99}+O(q^{100}) \) 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
\(7\) 2.41421 0.912487 0.456243 0.889855i \(-0.349195\pi\)
0.456243 + 0.889855i \(0.349195\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.41421 −1.93396 −0.966979 0.254856i \(-0.917972\pi\)
−0.966979 + 0.254856i \(0.917972\pi\)
\(12\) 1.00000 0.288675
\(13\) −5.24264 −1.45405 −0.727023 0.686613i \(-0.759097\pi\)
−0.727023 + 0.686613i \(0.759097\pi\)
\(14\) 2.41421 0.645226
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.58579 −0.869681 −0.434840 0.900508i \(-0.643195\pi\)
−0.434840 + 0.900508i \(0.643195\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.17157 −0.498193 −0.249096 0.968479i \(-0.580134\pi\)
−0.249096 + 0.968479i \(0.580134\pi\)
\(20\) 0 0
\(21\) 2.41421 0.526825
\(22\) −6.41421 −1.36751
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −5.24264 −1.02817
\(27\) 1.00000 0.192450
\(28\) 2.41421 0.456243
\(29\) 7.07107 1.31306 0.656532 0.754298i \(-0.272023\pi\)
0.656532 + 0.754298i \(0.272023\pi\)
\(30\) 0 0
\(31\) −3.65685 −0.656790 −0.328395 0.944540i \(-0.606508\pi\)
−0.328395 + 0.944540i \(0.606508\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.41421 −1.11657
\(34\) −3.58579 −0.614957
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −2.17157 −0.352276
\(39\) −5.24264 −0.839494
\(40\) 0 0
\(41\) −6.58579 −1.02853 −0.514264 0.857632i \(-0.671935\pi\)
−0.514264 + 0.857632i \(0.671935\pi\)
\(42\) 2.41421 0.372521
\(43\) −10.4853 −1.59899 −0.799495 0.600672i \(-0.794900\pi\)
−0.799495 + 0.600672i \(0.794900\pi\)
\(44\) −6.41421 −0.966979
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 11.8995 1.73572 0.867860 0.496809i \(-0.165495\pi\)
0.867860 + 0.496809i \(0.165495\pi\)
\(48\) 1.00000 0.144338
\(49\) −1.17157 −0.167368
\(50\) 0 0
\(51\) −3.58579 −0.502111
\(52\) −5.24264 −0.727023
\(53\) 3.82843 0.525875 0.262937 0.964813i \(-0.415309\pi\)
0.262937 + 0.964813i \(0.415309\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.41421 0.322613
\(57\) −2.17157 −0.287632
\(58\) 7.07107 0.928477
\(59\) −13.0711 −1.70171 −0.850854 0.525402i \(-0.823915\pi\)
−0.850854 + 0.525402i \(0.823915\pi\)
\(60\) 0 0
\(61\) −6.48528 −0.830355 −0.415178 0.909740i \(-0.636281\pi\)
−0.415178 + 0.909740i \(0.636281\pi\)
\(62\) −3.65685 −0.464421
\(63\) 2.41421 0.304162
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.41421 −0.789535
\(67\) 0.828427 0.101208 0.0506042 0.998719i \(-0.483885\pi\)
0.0506042 + 0.998719i \(0.483885\pi\)
\(68\) −3.58579 −0.434840
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −9.41421 −1.11726 −0.558631 0.829416i \(-0.688673\pi\)
−0.558631 + 0.829416i \(0.688673\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.17157 −0.254163 −0.127082 0.991892i \(-0.540561\pi\)
−0.127082 + 0.991892i \(0.540561\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −2.17157 −0.249096
\(77\) −15.4853 −1.76471
\(78\) −5.24264 −0.593612
\(79\) 12.2426 1.37740 0.688702 0.725044i \(-0.258181\pi\)
0.688702 + 0.725044i \(0.258181\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.58579 −0.727278
\(83\) −6.89949 −0.757318 −0.378659 0.925536i \(-0.623615\pi\)
−0.378659 + 0.925536i \(0.623615\pi\)
\(84\) 2.41421 0.263412
\(85\) 0 0
\(86\) −10.4853 −1.13066
\(87\) 7.07107 0.758098
\(88\) −6.41421 −0.683757
\(89\) 2.89949 0.307346 0.153673 0.988122i \(-0.450890\pi\)
0.153673 + 0.988122i \(0.450890\pi\)
\(90\) 0 0
\(91\) −12.6569 −1.32680
\(92\) −3.00000 −0.312772
\(93\) −3.65685 −0.379198
\(94\) 11.8995 1.22734
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 6.72792 0.683117 0.341558 0.939861i \(-0.389045\pi\)
0.341558 + 0.939861i \(0.389045\pi\)
\(98\) −1.17157 −0.118347
\(99\) −6.41421 −0.644653
\(100\) 0 0
\(101\) −5.31371 −0.528734 −0.264367 0.964422i \(-0.585163\pi\)
−0.264367 + 0.964422i \(0.585163\pi\)
\(102\) −3.58579 −0.355046
\(103\) 15.0711 1.48500 0.742498 0.669848i \(-0.233641\pi\)
0.742498 + 0.669848i \(0.233641\pi\)
\(104\) −5.24264 −0.514083
\(105\) 0 0
\(106\) 3.82843 0.371850
\(107\) −16.0711 −1.55365 −0.776824 0.629717i \(-0.783171\pi\)
−0.776824 + 0.629717i \(0.783171\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.75736 0.455672 0.227836 0.973699i \(-0.426835\pi\)
0.227836 + 0.973699i \(0.426835\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 2.41421 0.228122
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) −2.17157 −0.203386
\(115\) 0 0
\(116\) 7.07107 0.656532
\(117\) −5.24264 −0.484682
\(118\) −13.0711 −1.20329
\(119\) −8.65685 −0.793573
\(120\) 0 0
\(121\) 30.1421 2.74019
\(122\) −6.48528 −0.587150
\(123\) −6.58579 −0.593820
\(124\) −3.65685 −0.328395
\(125\) 0 0
\(126\) 2.41421 0.215075
\(127\) −9.58579 −0.850601 −0.425300 0.905052i \(-0.639832\pi\)
−0.425300 + 0.905052i \(0.639832\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.4853 −0.923178
\(130\) 0 0
\(131\) 1.89949 0.165960 0.0829798 0.996551i \(-0.473556\pi\)
0.0829798 + 0.996551i \(0.473556\pi\)
\(132\) −6.41421 −0.558286
\(133\) −5.24264 −0.454595
\(134\) 0.828427 0.0715652
\(135\) 0 0
\(136\) −3.58579 −0.307479
\(137\) −17.5563 −1.49994 −0.749970 0.661472i \(-0.769932\pi\)
−0.749970 + 0.661472i \(0.769932\pi\)
\(138\) −3.00000 −0.255377
\(139\) −2.48528 −0.210799 −0.105399 0.994430i \(-0.533612\pi\)
−0.105399 + 0.994430i \(0.533612\pi\)
\(140\) 0 0
\(141\) 11.8995 1.00212
\(142\) −9.41421 −0.790023
\(143\) 33.6274 2.81207
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.17157 −0.179721
\(147\) −1.17157 −0.0966297
\(148\) 1.00000 0.0821995
\(149\) 3.51472 0.287937 0.143968 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(150\) 0 0
\(151\) 2.41421 0.196466 0.0982330 0.995163i \(-0.468681\pi\)
0.0982330 + 0.995163i \(0.468681\pi\)
\(152\) −2.17157 −0.176138
\(153\) −3.58579 −0.289894
\(154\) −15.4853 −1.24784
\(155\) 0 0
\(156\) −5.24264 −0.419747
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 12.2426 0.973972
\(159\) 3.82843 0.303614
\(160\) 0 0
\(161\) −7.24264 −0.570800
\(162\) 1.00000 0.0785674
\(163\) −15.1421 −1.18602 −0.593012 0.805194i \(-0.702061\pi\)
−0.593012 + 0.805194i \(0.702061\pi\)
\(164\) −6.58579 −0.514264
\(165\) 0 0
\(166\) −6.89949 −0.535505
\(167\) 18.6569 1.44371 0.721855 0.692044i \(-0.243290\pi\)
0.721855 + 0.692044i \(0.243290\pi\)
\(168\) 2.41421 0.186261
\(169\) 14.4853 1.11425
\(170\) 0 0
\(171\) −2.17157 −0.166064
\(172\) −10.4853 −0.799495
\(173\) −0.656854 −0.0499397 −0.0249699 0.999688i \(-0.507949\pi\)
−0.0249699 + 0.999688i \(0.507949\pi\)
\(174\) 7.07107 0.536056
\(175\) 0 0
\(176\) −6.41421 −0.483490
\(177\) −13.0711 −0.982482
\(178\) 2.89949 0.217326
\(179\) −18.8284 −1.40730 −0.703651 0.710545i \(-0.748448\pi\)
−0.703651 + 0.710545i \(0.748448\pi\)
\(180\) 0 0
\(181\) 20.5858 1.53013 0.765065 0.643953i \(-0.222707\pi\)
0.765065 + 0.643953i \(0.222707\pi\)
\(182\) −12.6569 −0.938188
\(183\) −6.48528 −0.479406
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) −3.65685 −0.268134
\(187\) 23.0000 1.68193
\(188\) 11.8995 0.867860
\(189\) 2.41421 0.175608
\(190\) 0 0
\(191\) 8.65685 0.626388 0.313194 0.949689i \(-0.398601\pi\)
0.313194 + 0.949689i \(0.398601\pi\)
\(192\) 1.00000 0.0721688
\(193\) 9.31371 0.670415 0.335208 0.942144i \(-0.391194\pi\)
0.335208 + 0.942144i \(0.391194\pi\)
\(194\) 6.72792 0.483037
\(195\) 0 0
\(196\) −1.17157 −0.0836838
\(197\) −15.1421 −1.07883 −0.539416 0.842039i \(-0.681355\pi\)
−0.539416 + 0.842039i \(0.681355\pi\)
\(198\) −6.41421 −0.455838
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0.828427 0.0584327
\(202\) −5.31371 −0.373871
\(203\) 17.0711 1.19815
\(204\) −3.58579 −0.251055
\(205\) 0 0
\(206\) 15.0711 1.05005
\(207\) −3.00000 −0.208514
\(208\) −5.24264 −0.363512
\(209\) 13.9289 0.963484
\(210\) 0 0
\(211\) 10.3848 0.714917 0.357459 0.933929i \(-0.383643\pi\)
0.357459 + 0.933929i \(0.383643\pi\)
\(212\) 3.82843 0.262937
\(213\) −9.41421 −0.645051
\(214\) −16.0711 −1.09860
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −8.82843 −0.599313
\(218\) 4.75736 0.322209
\(219\) −2.17157 −0.146741
\(220\) 0 0
\(221\) 18.7990 1.26456
\(222\) 1.00000 0.0671156
\(223\) −9.65685 −0.646671 −0.323335 0.946284i \(-0.604804\pi\)
−0.323335 + 0.946284i \(0.604804\pi\)
\(224\) 2.41421 0.161306
\(225\) 0 0
\(226\) 2.82843 0.188144
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −2.17157 −0.143816
\(229\) 18.7279 1.23758 0.618788 0.785558i \(-0.287624\pi\)
0.618788 + 0.785558i \(0.287624\pi\)
\(230\) 0 0
\(231\) −15.4853 −1.01886
\(232\) 7.07107 0.464238
\(233\) 2.48528 0.162816 0.0814081 0.996681i \(-0.474058\pi\)
0.0814081 + 0.996681i \(0.474058\pi\)
\(234\) −5.24264 −0.342722
\(235\) 0 0
\(236\) −13.0711 −0.850854
\(237\) 12.2426 0.795245
\(238\) −8.65685 −0.561141
\(239\) 20.6274 1.33428 0.667138 0.744934i \(-0.267519\pi\)
0.667138 + 0.744934i \(0.267519\pi\)
\(240\) 0 0
\(241\) 2.58579 0.166565 0.0832826 0.996526i \(-0.473460\pi\)
0.0832826 + 0.996526i \(0.473460\pi\)
\(242\) 30.1421 1.93761
\(243\) 1.00000 0.0641500
\(244\) −6.48528 −0.415178
\(245\) 0 0
\(246\) −6.58579 −0.419894
\(247\) 11.3848 0.724396
\(248\) −3.65685 −0.232210
\(249\) −6.89949 −0.437238
\(250\) 0 0
\(251\) 4.82843 0.304768 0.152384 0.988321i \(-0.451305\pi\)
0.152384 + 0.988321i \(0.451305\pi\)
\(252\) 2.41421 0.152081
\(253\) 19.2426 1.20977
\(254\) −9.58579 −0.601466
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.58579 0.597945 0.298972 0.954262i \(-0.403356\pi\)
0.298972 + 0.954262i \(0.403356\pi\)
\(258\) −10.4853 −0.652785
\(259\) 2.41421 0.150012
\(260\) 0 0
\(261\) 7.07107 0.437688
\(262\) 1.89949 0.117351
\(263\) −31.4142 −1.93708 −0.968542 0.248851i \(-0.919947\pi\)
−0.968542 + 0.248851i \(0.919947\pi\)
\(264\) −6.41421 −0.394768
\(265\) 0 0
\(266\) −5.24264 −0.321447
\(267\) 2.89949 0.177446
\(268\) 0.828427 0.0506042
\(269\) −25.6274 −1.56253 −0.781266 0.624199i \(-0.785426\pi\)
−0.781266 + 0.624199i \(0.785426\pi\)
\(270\) 0 0
\(271\) −6.82843 −0.414797 −0.207399 0.978256i \(-0.566500\pi\)
−0.207399 + 0.978256i \(0.566500\pi\)
\(272\) −3.58579 −0.217420
\(273\) −12.6569 −0.766028
\(274\) −17.5563 −1.06062
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) 2.89949 0.174214 0.0871069 0.996199i \(-0.472238\pi\)
0.0871069 + 0.996199i \(0.472238\pi\)
\(278\) −2.48528 −0.149057
\(279\) −3.65685 −0.218930
\(280\) 0 0
\(281\) −13.3848 −0.798469 −0.399234 0.916849i \(-0.630724\pi\)
−0.399234 + 0.916849i \(0.630724\pi\)
\(282\) 11.8995 0.708605
\(283\) 18.7990 1.11748 0.558742 0.829342i \(-0.311284\pi\)
0.558742 + 0.829342i \(0.311284\pi\)
\(284\) −9.41421 −0.558631
\(285\) 0 0
\(286\) 33.6274 1.98843
\(287\) −15.8995 −0.938518
\(288\) 1.00000 0.0589256
\(289\) −4.14214 −0.243655
\(290\) 0 0
\(291\) 6.72792 0.394398
\(292\) −2.17157 −0.127082
\(293\) 8.65685 0.505739 0.252869 0.967500i \(-0.418626\pi\)
0.252869 + 0.967500i \(0.418626\pi\)
\(294\) −1.17157 −0.0683275
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −6.41421 −0.372190
\(298\) 3.51472 0.203602
\(299\) 15.7279 0.909569
\(300\) 0 0
\(301\) −25.3137 −1.45906
\(302\) 2.41421 0.138922
\(303\) −5.31371 −0.305265
\(304\) −2.17157 −0.124548
\(305\) 0 0
\(306\) −3.58579 −0.204986
\(307\) 13.8995 0.793286 0.396643 0.917973i \(-0.370175\pi\)
0.396643 + 0.917973i \(0.370175\pi\)
\(308\) −15.4853 −0.882356
\(309\) 15.0711 0.857363
\(310\) 0 0
\(311\) −12.8284 −0.727433 −0.363717 0.931510i \(-0.618492\pi\)
−0.363717 + 0.931510i \(0.618492\pi\)
\(312\) −5.24264 −0.296806
\(313\) 2.72792 0.154191 0.0770956 0.997024i \(-0.475435\pi\)
0.0770956 + 0.997024i \(0.475435\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 12.2426 0.688702
\(317\) −21.1716 −1.18911 −0.594557 0.804053i \(-0.702673\pi\)
−0.594557 + 0.804053i \(0.702673\pi\)
\(318\) 3.82843 0.214688
\(319\) −45.3553 −2.53941
\(320\) 0 0
\(321\) −16.0711 −0.897000
\(322\) −7.24264 −0.403617
\(323\) 7.78680 0.433269
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −15.1421 −0.838645
\(327\) 4.75736 0.263083
\(328\) −6.58579 −0.363639
\(329\) 28.7279 1.58382
\(330\) 0 0
\(331\) 23.4558 1.28925 0.644625 0.764499i \(-0.277014\pi\)
0.644625 + 0.764499i \(0.277014\pi\)
\(332\) −6.89949 −0.378659
\(333\) 1.00000 0.0547997
\(334\) 18.6569 1.02086
\(335\) 0 0
\(336\) 2.41421 0.131706
\(337\) −27.9706 −1.52365 −0.761827 0.647781i \(-0.775697\pi\)
−0.761827 + 0.647781i \(0.775697\pi\)
\(338\) 14.4853 0.787895
\(339\) 2.82843 0.153619
\(340\) 0 0
\(341\) 23.4558 1.27021
\(342\) −2.17157 −0.117425
\(343\) −19.7279 −1.06521
\(344\) −10.4853 −0.565328
\(345\) 0 0
\(346\) −0.656854 −0.0353127
\(347\) 23.5563 1.26457 0.632286 0.774735i \(-0.282117\pi\)
0.632286 + 0.774735i \(0.282117\pi\)
\(348\) 7.07107 0.379049
\(349\) 28.0416 1.50103 0.750517 0.660851i \(-0.229805\pi\)
0.750517 + 0.660851i \(0.229805\pi\)
\(350\) 0 0
\(351\) −5.24264 −0.279831
\(352\) −6.41421 −0.341879
\(353\) −16.1421 −0.859159 −0.429580 0.903029i \(-0.641338\pi\)
−0.429580 + 0.903029i \(0.641338\pi\)
\(354\) −13.0711 −0.694719
\(355\) 0 0
\(356\) 2.89949 0.153673
\(357\) −8.65685 −0.458169
\(358\) −18.8284 −0.995113
\(359\) 29.0711 1.53431 0.767156 0.641460i \(-0.221671\pi\)
0.767156 + 0.641460i \(0.221671\pi\)
\(360\) 0 0
\(361\) −14.2843 −0.751804
\(362\) 20.5858 1.08196
\(363\) 30.1421 1.58205
\(364\) −12.6569 −0.663399
\(365\) 0 0
\(366\) −6.48528 −0.338991
\(367\) −15.2426 −0.795659 −0.397830 0.917459i \(-0.630237\pi\)
−0.397830 + 0.917459i \(0.630237\pi\)
\(368\) −3.00000 −0.156386
\(369\) −6.58579 −0.342842
\(370\) 0 0
\(371\) 9.24264 0.479854
\(372\) −3.65685 −0.189599
\(373\) −1.75736 −0.0909926 −0.0454963 0.998965i \(-0.514487\pi\)
−0.0454963 + 0.998965i \(0.514487\pi\)
\(374\) 23.0000 1.18930
\(375\) 0 0
\(376\) 11.8995 0.613670
\(377\) −37.0711 −1.90926
\(378\) 2.41421 0.124174
\(379\) −28.7279 −1.47565 −0.737827 0.674990i \(-0.764148\pi\)
−0.737827 + 0.674990i \(0.764148\pi\)
\(380\) 0 0
\(381\) −9.58579 −0.491095
\(382\) 8.65685 0.442923
\(383\) 14.6569 0.748930 0.374465 0.927241i \(-0.377826\pi\)
0.374465 + 0.927241i \(0.377826\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 9.31371 0.474055
\(387\) −10.4853 −0.532997
\(388\) 6.72792 0.341558
\(389\) −22.9706 −1.16465 −0.582327 0.812955i \(-0.697858\pi\)
−0.582327 + 0.812955i \(0.697858\pi\)
\(390\) 0 0
\(391\) 10.7574 0.544023
\(392\) −1.17157 −0.0591734
\(393\) 1.89949 0.0958168
\(394\) −15.1421 −0.762850
\(395\) 0 0
\(396\) −6.41421 −0.322326
\(397\) 8.48528 0.425864 0.212932 0.977067i \(-0.431699\pi\)
0.212932 + 0.977067i \(0.431699\pi\)
\(398\) 20.0000 1.00251
\(399\) −5.24264 −0.262460
\(400\) 0 0
\(401\) −15.0416 −0.751143 −0.375572 0.926793i \(-0.622554\pi\)
−0.375572 + 0.926793i \(0.622554\pi\)
\(402\) 0.828427 0.0413182
\(403\) 19.1716 0.955004
\(404\) −5.31371 −0.264367
\(405\) 0 0
\(406\) 17.0711 0.847223
\(407\) −6.41421 −0.317941
\(408\) −3.58579 −0.177523
\(409\) 32.3848 1.60132 0.800662 0.599116i \(-0.204481\pi\)
0.800662 + 0.599116i \(0.204481\pi\)
\(410\) 0 0
\(411\) −17.5563 −0.865991
\(412\) 15.0711 0.742498
\(413\) −31.5563 −1.55279
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) −5.24264 −0.257042
\(417\) −2.48528 −0.121705
\(418\) 13.9289 0.681286
\(419\) −27.7279 −1.35460 −0.677299 0.735708i \(-0.736850\pi\)
−0.677299 + 0.735708i \(0.736850\pi\)
\(420\) 0 0
\(421\) −13.4558 −0.655798 −0.327899 0.944713i \(-0.606341\pi\)
−0.327899 + 0.944713i \(0.606341\pi\)
\(422\) 10.3848 0.505523
\(423\) 11.8995 0.578573
\(424\) 3.82843 0.185925
\(425\) 0 0
\(426\) −9.41421 −0.456120
\(427\) −15.6569 −0.757688
\(428\) −16.0711 −0.776824
\(429\) 33.6274 1.62355
\(430\) 0 0
\(431\) 29.8284 1.43678 0.718392 0.695638i \(-0.244878\pi\)
0.718392 + 0.695638i \(0.244878\pi\)
\(432\) 1.00000 0.0481125
\(433\) −23.4853 −1.12863 −0.564315 0.825559i \(-0.690860\pi\)
−0.564315 + 0.825559i \(0.690860\pi\)
\(434\) −8.82843 −0.423778
\(435\) 0 0
\(436\) 4.75736 0.227836
\(437\) 6.51472 0.311641
\(438\) −2.17157 −0.103762
\(439\) 24.7279 1.18020 0.590100 0.807330i \(-0.299088\pi\)
0.590100 + 0.807330i \(0.299088\pi\)
\(440\) 0 0
\(441\) −1.17157 −0.0557892
\(442\) 18.7990 0.894177
\(443\) −23.6569 −1.12397 −0.561986 0.827147i \(-0.689962\pi\)
−0.561986 + 0.827147i \(0.689962\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) −9.65685 −0.457265
\(447\) 3.51472 0.166240
\(448\) 2.41421 0.114061
\(449\) −13.7990 −0.651215 −0.325607 0.945505i \(-0.605569\pi\)
−0.325607 + 0.945505i \(0.605569\pi\)
\(450\) 0 0
\(451\) 42.2426 1.98913
\(452\) 2.82843 0.133038
\(453\) 2.41421 0.113430
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −2.17157 −0.101693
\(457\) −0.970563 −0.0454010 −0.0227005 0.999742i \(-0.507226\pi\)
−0.0227005 + 0.999742i \(0.507226\pi\)
\(458\) 18.7279 0.875098
\(459\) −3.58579 −0.167370
\(460\) 0 0
\(461\) 20.8284 0.970077 0.485038 0.874493i \(-0.338806\pi\)
0.485038 + 0.874493i \(0.338806\pi\)
\(462\) −15.4853 −0.720440
\(463\) −29.8995 −1.38955 −0.694774 0.719228i \(-0.744495\pi\)
−0.694774 + 0.719228i \(0.744495\pi\)
\(464\) 7.07107 0.328266
\(465\) 0 0
\(466\) 2.48528 0.115128
\(467\) −40.2426 −1.86221 −0.931104 0.364755i \(-0.881153\pi\)
−0.931104 + 0.364755i \(0.881153\pi\)
\(468\) −5.24264 −0.242341
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 0 0
\(472\) −13.0711 −0.601645
\(473\) 67.2548 3.09238
\(474\) 12.2426 0.562323
\(475\) 0 0
\(476\) −8.65685 −0.396786
\(477\) 3.82843 0.175292
\(478\) 20.6274 0.943476
\(479\) 6.17157 0.281986 0.140993 0.990011i \(-0.454970\pi\)
0.140993 + 0.990011i \(0.454970\pi\)
\(480\) 0 0
\(481\) −5.24264 −0.239044
\(482\) 2.58579 0.117779
\(483\) −7.24264 −0.329552
\(484\) 30.1421 1.37010
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −30.0000 −1.35943 −0.679715 0.733476i \(-0.737896\pi\)
−0.679715 + 0.733476i \(0.737896\pi\)
\(488\) −6.48528 −0.293575
\(489\) −15.1421 −0.684751
\(490\) 0 0
\(491\) −16.4142 −0.740763 −0.370382 0.928880i \(-0.620773\pi\)
−0.370382 + 0.928880i \(0.620773\pi\)
\(492\) −6.58579 −0.296910
\(493\) −25.3553 −1.14195
\(494\) 11.3848 0.512225
\(495\) 0 0
\(496\) −3.65685 −0.164198
\(497\) −22.7279 −1.01949
\(498\) −6.89949 −0.309174
\(499\) 7.48528 0.335087 0.167544 0.985865i \(-0.446417\pi\)
0.167544 + 0.985865i \(0.446417\pi\)
\(500\) 0 0
\(501\) 18.6569 0.833527
\(502\) 4.82843 0.215503
\(503\) −23.6569 −1.05481 −0.527403 0.849615i \(-0.676834\pi\)
−0.527403 + 0.849615i \(0.676834\pi\)
\(504\) 2.41421 0.107538
\(505\) 0 0
\(506\) 19.2426 0.855440
\(507\) 14.4853 0.643314
\(508\) −9.58579 −0.425300
\(509\) −30.3137 −1.34363 −0.671816 0.740718i \(-0.734485\pi\)
−0.671816 + 0.740718i \(0.734485\pi\)
\(510\) 0 0
\(511\) −5.24264 −0.231921
\(512\) 1.00000 0.0441942
\(513\) −2.17157 −0.0958773
\(514\) 9.58579 0.422811
\(515\) 0 0
\(516\) −10.4853 −0.461589
\(517\) −76.3259 −3.35681
\(518\) 2.41421 0.106074
\(519\) −0.656854 −0.0288327
\(520\) 0 0
\(521\) −16.0416 −0.702797 −0.351398 0.936226i \(-0.614294\pi\)
−0.351398 + 0.936226i \(0.614294\pi\)
\(522\) 7.07107 0.309492
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 1.89949 0.0829798
\(525\) 0 0
\(526\) −31.4142 −1.36972
\(527\) 13.1127 0.571198
\(528\) −6.41421 −0.279143
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −13.0711 −0.567236
\(532\) −5.24264 −0.227297
\(533\) 34.5269 1.49553
\(534\) 2.89949 0.125473
\(535\) 0 0
\(536\) 0.828427 0.0357826
\(537\) −18.8284 −0.812507
\(538\) −25.6274 −1.10488
\(539\) 7.51472 0.323682
\(540\) 0 0
\(541\) 36.2132 1.55693 0.778464 0.627690i \(-0.215999\pi\)
0.778464 + 0.627690i \(0.215999\pi\)
\(542\) −6.82843 −0.293306
\(543\) 20.5858 0.883421
\(544\) −3.58579 −0.153739
\(545\) 0 0
\(546\) −12.6569 −0.541663
\(547\) 0.372583 0.0159305 0.00796525 0.999968i \(-0.497465\pi\)
0.00796525 + 0.999968i \(0.497465\pi\)
\(548\) −17.5563 −0.749970
\(549\) −6.48528 −0.276785
\(550\) 0 0
\(551\) −15.3553 −0.654159
\(552\) −3.00000 −0.127688
\(553\) 29.5563 1.25686
\(554\) 2.89949 0.123188
\(555\) 0 0
\(556\) −2.48528 −0.105399
\(557\) 37.5563 1.59131 0.795657 0.605748i \(-0.207126\pi\)
0.795657 + 0.605748i \(0.207126\pi\)
\(558\) −3.65685 −0.154807
\(559\) 54.9706 2.32501
\(560\) 0 0
\(561\) 23.0000 0.971061
\(562\) −13.3848 −0.564603
\(563\) −16.5858 −0.699008 −0.349504 0.936935i \(-0.613650\pi\)
−0.349504 + 0.936935i \(0.613650\pi\)
\(564\) 11.8995 0.501059
\(565\) 0 0
\(566\) 18.7990 0.790180
\(567\) 2.41421 0.101387
\(568\) −9.41421 −0.395012
\(569\) 37.0416 1.55287 0.776433 0.630200i \(-0.217027\pi\)
0.776433 + 0.630200i \(0.217027\pi\)
\(570\) 0 0
\(571\) −28.0416 −1.17351 −0.586753 0.809766i \(-0.699594\pi\)
−0.586753 + 0.809766i \(0.699594\pi\)
\(572\) 33.6274 1.40603
\(573\) 8.65685 0.361645
\(574\) −15.8995 −0.663632
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −9.51472 −0.396103 −0.198051 0.980192i \(-0.563461\pi\)
−0.198051 + 0.980192i \(0.563461\pi\)
\(578\) −4.14214 −0.172290
\(579\) 9.31371 0.387065
\(580\) 0 0
\(581\) −16.6569 −0.691043
\(582\) 6.72792 0.278881
\(583\) −24.5563 −1.01702
\(584\) −2.17157 −0.0898603
\(585\) 0 0
\(586\) 8.65685 0.357611
\(587\) 2.82843 0.116742 0.0583708 0.998295i \(-0.481409\pi\)
0.0583708 + 0.998295i \(0.481409\pi\)
\(588\) −1.17157 −0.0483149
\(589\) 7.94113 0.327208
\(590\) 0 0
\(591\) −15.1421 −0.622864
\(592\) 1.00000 0.0410997
\(593\) −4.97056 −0.204117 −0.102058 0.994778i \(-0.532543\pi\)
−0.102058 + 0.994778i \(0.532543\pi\)
\(594\) −6.41421 −0.263178
\(595\) 0 0
\(596\) 3.51472 0.143968
\(597\) 20.0000 0.818546
\(598\) 15.7279 0.643163
\(599\) 32.8701 1.34303 0.671517 0.740989i \(-0.265643\pi\)
0.671517 + 0.740989i \(0.265643\pi\)
\(600\) 0 0
\(601\) 9.97056 0.406708 0.203354 0.979105i \(-0.434816\pi\)
0.203354 + 0.979105i \(0.434816\pi\)
\(602\) −25.3137 −1.03171
\(603\) 0.828427 0.0337362
\(604\) 2.41421 0.0982330
\(605\) 0 0
\(606\) −5.31371 −0.215855
\(607\) 11.8579 0.481296 0.240648 0.970612i \(-0.422640\pi\)
0.240648 + 0.970612i \(0.422640\pi\)
\(608\) −2.17157 −0.0880689
\(609\) 17.0711 0.691755
\(610\) 0 0
\(611\) −62.3848 −2.52382
\(612\) −3.58579 −0.144947
\(613\) −1.31371 −0.0530602 −0.0265301 0.999648i \(-0.508446\pi\)
−0.0265301 + 0.999648i \(0.508446\pi\)
\(614\) 13.8995 0.560938
\(615\) 0 0
\(616\) −15.4853 −0.623920
\(617\) 35.6985 1.43717 0.718583 0.695441i \(-0.244791\pi\)
0.718583 + 0.695441i \(0.244791\pi\)
\(618\) 15.0711 0.606247
\(619\) −7.21320 −0.289923 −0.144962 0.989437i \(-0.546306\pi\)
−0.144962 + 0.989437i \(0.546306\pi\)
\(620\) 0 0
\(621\) −3.00000 −0.120386
\(622\) −12.8284 −0.514373
\(623\) 7.00000 0.280449
\(624\) −5.24264 −0.209874
\(625\) 0 0
\(626\) 2.72792 0.109030
\(627\) 13.9289 0.556268
\(628\) 0 0
\(629\) −3.58579 −0.142975
\(630\) 0 0
\(631\) 7.51472 0.299156 0.149578 0.988750i \(-0.452208\pi\)
0.149578 + 0.988750i \(0.452208\pi\)
\(632\) 12.2426 0.486986
\(633\) 10.3848 0.412758
\(634\) −21.1716 −0.840831
\(635\) 0 0
\(636\) 3.82843 0.151807
\(637\) 6.14214 0.243360
\(638\) −45.3553 −1.79564
\(639\) −9.41421 −0.372421
\(640\) 0 0
\(641\) −26.8284 −1.05966 −0.529830 0.848104i \(-0.677744\pi\)
−0.529830 + 0.848104i \(0.677744\pi\)
\(642\) −16.0711 −0.634274
\(643\) 18.4558 0.727827 0.363914 0.931433i \(-0.381440\pi\)
0.363914 + 0.931433i \(0.381440\pi\)
\(644\) −7.24264 −0.285400
\(645\) 0 0
\(646\) 7.78680 0.306367
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 1.00000 0.0392837
\(649\) 83.8406 3.29103
\(650\) 0 0
\(651\) −8.82843 −0.346013
\(652\) −15.1421 −0.593012
\(653\) −20.4853 −0.801651 −0.400826 0.916154i \(-0.631277\pi\)
−0.400826 + 0.916154i \(0.631277\pi\)
\(654\) 4.75736 0.186027
\(655\) 0 0
\(656\) −6.58579 −0.257132
\(657\) −2.17157 −0.0847211
\(658\) 28.7279 1.11993
\(659\) 51.1127 1.99107 0.995534 0.0944035i \(-0.0300944\pi\)
0.995534 + 0.0944035i \(0.0300944\pi\)
\(660\) 0 0
\(661\) 25.2426 0.981825 0.490912 0.871209i \(-0.336663\pi\)
0.490912 + 0.871209i \(0.336663\pi\)
\(662\) 23.4558 0.911637
\(663\) 18.7990 0.730092
\(664\) −6.89949 −0.267752
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) −21.2132 −0.821379
\(668\) 18.6569 0.721855
\(669\) −9.65685 −0.373356
\(670\) 0 0
\(671\) 41.5980 1.60587
\(672\) 2.41421 0.0931303
\(673\) −0.0294373 −0.00113472 −0.000567361 1.00000i \(-0.500181\pi\)
−0.000567361 1.00000i \(0.500181\pi\)
\(674\) −27.9706 −1.07739
\(675\) 0 0
\(676\) 14.4853 0.557126
\(677\) −1.20101 −0.0461586 −0.0230793 0.999734i \(-0.507347\pi\)
−0.0230793 + 0.999734i \(0.507347\pi\)
\(678\) 2.82843 0.108625
\(679\) 16.2426 0.623335
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 23.4558 0.898171
\(683\) 16.9289 0.647768 0.323884 0.946097i \(-0.395011\pi\)
0.323884 + 0.946097i \(0.395011\pi\)
\(684\) −2.17157 −0.0830322
\(685\) 0 0
\(686\) −19.7279 −0.753216
\(687\) 18.7279 0.714515
\(688\) −10.4853 −0.399748
\(689\) −20.0711 −0.764647
\(690\) 0 0
\(691\) 30.5858 1.16354 0.581769 0.813354i \(-0.302361\pi\)
0.581769 + 0.813354i \(0.302361\pi\)
\(692\) −0.656854 −0.0249699
\(693\) −15.4853 −0.588237
\(694\) 23.5563 0.894187
\(695\) 0 0
\(696\) 7.07107 0.268028
\(697\) 23.6152 0.894490
\(698\) 28.0416 1.06139
\(699\) 2.48528 0.0940020
\(700\) 0 0
\(701\) 14.6274 0.552470 0.276235 0.961090i \(-0.410913\pi\)
0.276235 + 0.961090i \(0.410913\pi\)
\(702\) −5.24264 −0.197871
\(703\) −2.17157 −0.0819024
\(704\) −6.41421 −0.241745
\(705\) 0 0
\(706\) −16.1421 −0.607517
\(707\) −12.8284 −0.482463
\(708\) −13.0711 −0.491241
\(709\) 44.0711 1.65512 0.827562 0.561375i \(-0.189727\pi\)
0.827562 + 0.561375i \(0.189727\pi\)
\(710\) 0 0
\(711\) 12.2426 0.459135
\(712\) 2.89949 0.108663
\(713\) 10.9706 0.410851
\(714\) −8.65685 −0.323975
\(715\) 0 0
\(716\) −18.8284 −0.703651
\(717\) 20.6274 0.770345
\(718\) 29.0711 1.08492
\(719\) 1.41421 0.0527413 0.0263706 0.999652i \(-0.491605\pi\)
0.0263706 + 0.999652i \(0.491605\pi\)
\(720\) 0 0
\(721\) 36.3848 1.35504
\(722\) −14.2843 −0.531606
\(723\) 2.58579 0.0961664
\(724\) 20.5858 0.765065
\(725\) 0 0
\(726\) 30.1421 1.11868
\(727\) −20.3848 −0.756030 −0.378015 0.925800i \(-0.623393\pi\)
−0.378015 + 0.925800i \(0.623393\pi\)
\(728\) −12.6569 −0.469094
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 37.5980 1.39061
\(732\) −6.48528 −0.239703
\(733\) 10.9706 0.405207 0.202603 0.979261i \(-0.435060\pi\)
0.202603 + 0.979261i \(0.435060\pi\)
\(734\) −15.2426 −0.562616
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) −5.31371 −0.195733
\(738\) −6.58579 −0.242426
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 11.3848 0.418230
\(742\) 9.24264 0.339308
\(743\) −51.3553 −1.88404 −0.942022 0.335550i \(-0.891078\pi\)
−0.942022 + 0.335550i \(0.891078\pi\)
\(744\) −3.65685 −0.134067
\(745\) 0 0
\(746\) −1.75736 −0.0643415
\(747\) −6.89949 −0.252439
\(748\) 23.0000 0.840963
\(749\) −38.7990 −1.41768
\(750\) 0 0
\(751\) −35.1127 −1.28128 −0.640640 0.767841i \(-0.721331\pi\)
−0.640640 + 0.767841i \(0.721331\pi\)
\(752\) 11.8995 0.433930
\(753\) 4.82843 0.175958
\(754\) −37.0711 −1.35005
\(755\) 0 0
\(756\) 2.41421 0.0878041
\(757\) 16.2132 0.589279 0.294639 0.955608i \(-0.404800\pi\)
0.294639 + 0.955608i \(0.404800\pi\)
\(758\) −28.7279 −1.04345
\(759\) 19.2426 0.698464
\(760\) 0 0
\(761\) 25.3137 0.917621 0.458811 0.888534i \(-0.348276\pi\)
0.458811 + 0.888534i \(0.348276\pi\)
\(762\) −9.58579 −0.347256
\(763\) 11.4853 0.415795
\(764\) 8.65685 0.313194
\(765\) 0 0
\(766\) 14.6569 0.529574
\(767\) 68.5269 2.47436
\(768\) 1.00000 0.0360844
\(769\) −19.5563 −0.705220 −0.352610 0.935770i \(-0.614706\pi\)
−0.352610 + 0.935770i \(0.614706\pi\)
\(770\) 0 0
\(771\) 9.58579 0.345224
\(772\) 9.31371 0.335208
\(773\) −14.6569 −0.527170 −0.263585 0.964636i \(-0.584905\pi\)
−0.263585 + 0.964636i \(0.584905\pi\)
\(774\) −10.4853 −0.376886
\(775\) 0 0
\(776\) 6.72792 0.241518
\(777\) 2.41421 0.0866094
\(778\) −22.9706 −0.823535
\(779\) 14.3015 0.512405
\(780\) 0 0
\(781\) 60.3848 2.16074
\(782\) 10.7574 0.384682
\(783\) 7.07107 0.252699
\(784\) −1.17157 −0.0418419
\(785\) 0 0
\(786\) 1.89949 0.0677527
\(787\) 38.4853 1.37185 0.685926 0.727671i \(-0.259397\pi\)
0.685926 + 0.727671i \(0.259397\pi\)
\(788\) −15.1421 −0.539416
\(789\) −31.4142 −1.11838
\(790\) 0 0
\(791\) 6.82843 0.242791
\(792\) −6.41421 −0.227919
\(793\) 34.0000 1.20738
\(794\) 8.48528 0.301131
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 50.8701 1.80191 0.900955 0.433913i \(-0.142867\pi\)
0.900955 + 0.433913i \(0.142867\pi\)
\(798\) −5.24264 −0.185587
\(799\) −42.6690 −1.50952
\(800\) 0 0
\(801\) 2.89949 0.102449
\(802\) −15.0416 −0.531138
\(803\) 13.9289 0.491541
\(804\) 0.828427 0.0292164
\(805\) 0 0
\(806\) 19.1716 0.675290
\(807\) −25.6274 −0.902128
\(808\) −5.31371 −0.186936
\(809\) 11.7868 0.414402 0.207201 0.978298i \(-0.433565\pi\)
0.207201 + 0.978298i \(0.433565\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 17.0711 0.599077
\(813\) −6.82843 −0.239483
\(814\) −6.41421 −0.224818
\(815\) 0 0
\(816\) −3.58579 −0.125528
\(817\) 22.7696 0.796606
\(818\) 32.3848 1.13231
\(819\) −12.6569 −0.442266
\(820\) 0 0
\(821\) −51.2843 −1.78983 −0.894917 0.446233i \(-0.852765\pi\)
−0.894917 + 0.446233i \(0.852765\pi\)
\(822\) −17.5563 −0.612348
\(823\) −7.72792 −0.269378 −0.134689 0.990888i \(-0.543004\pi\)
−0.134689 + 0.990888i \(0.543004\pi\)
\(824\) 15.0711 0.525026
\(825\) 0 0
\(826\) −31.5563 −1.09799
\(827\) 7.07107 0.245885 0.122943 0.992414i \(-0.460767\pi\)
0.122943 + 0.992414i \(0.460767\pi\)
\(828\) −3.00000 −0.104257
\(829\) −19.2426 −0.668325 −0.334162 0.942516i \(-0.608453\pi\)
−0.334162 + 0.942516i \(0.608453\pi\)
\(830\) 0 0
\(831\) 2.89949 0.100582
\(832\) −5.24264 −0.181756
\(833\) 4.20101 0.145556
\(834\) −2.48528 −0.0860583
\(835\) 0 0
\(836\) 13.9289 0.481742
\(837\) −3.65685 −0.126399
\(838\) −27.7279 −0.957845
\(839\) 21.2132 0.732361 0.366181 0.930544i \(-0.380665\pi\)
0.366181 + 0.930544i \(0.380665\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) −13.4558 −0.463719
\(843\) −13.3848 −0.460996
\(844\) 10.3848 0.357459
\(845\) 0 0
\(846\) 11.8995 0.409113
\(847\) 72.7696 2.50039
\(848\) 3.82843 0.131469
\(849\) 18.7990 0.645180
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) −9.41421 −0.322526
\(853\) −42.8406 −1.46683 −0.733417 0.679779i \(-0.762076\pi\)
−0.733417 + 0.679779i \(0.762076\pi\)
\(854\) −15.6569 −0.535767
\(855\) 0 0
\(856\) −16.0711 −0.549298
\(857\) −56.4975 −1.92992 −0.964958 0.262403i \(-0.915485\pi\)
−0.964958 + 0.262403i \(0.915485\pi\)
\(858\) 33.6274 1.14802
\(859\) −23.4853 −0.801307 −0.400654 0.916230i \(-0.631217\pi\)
−0.400654 + 0.916230i \(0.631217\pi\)
\(860\) 0 0
\(861\) −15.8995 −0.541853
\(862\) 29.8284 1.01596
\(863\) −25.5563 −0.869948 −0.434974 0.900443i \(-0.643242\pi\)
−0.434974 + 0.900443i \(0.643242\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −23.4853 −0.798062
\(867\) −4.14214 −0.140674
\(868\) −8.82843 −0.299656
\(869\) −78.5269 −2.66384
\(870\) 0 0
\(871\) −4.34315 −0.147162
\(872\) 4.75736 0.161105
\(873\) 6.72792 0.227706
\(874\) 6.51472 0.220364
\(875\) 0 0
\(876\) −2.17157 −0.0733706
\(877\) 40.2426 1.35890 0.679449 0.733723i \(-0.262219\pi\)
0.679449 + 0.733723i \(0.262219\pi\)
\(878\) 24.7279 0.834527
\(879\) 8.65685 0.291988
\(880\) 0 0
\(881\) 18.6274 0.627574 0.313787 0.949493i \(-0.398402\pi\)
0.313787 + 0.949493i \(0.398402\pi\)
\(882\) −1.17157 −0.0394489
\(883\) −22.0294 −0.741350 −0.370675 0.928763i \(-0.620874\pi\)
−0.370675 + 0.928763i \(0.620874\pi\)
\(884\) 18.7990 0.632278
\(885\) 0 0
\(886\) −23.6569 −0.794768
\(887\) 20.1421 0.676307 0.338153 0.941091i \(-0.390198\pi\)
0.338153 + 0.941091i \(0.390198\pi\)
\(888\) 1.00000 0.0335578
\(889\) −23.1421 −0.776162
\(890\) 0 0
\(891\) −6.41421 −0.214884
\(892\) −9.65685 −0.323335
\(893\) −25.8406 −0.864723
\(894\) 3.51472 0.117550
\(895\) 0 0
\(896\) 2.41421 0.0806532
\(897\) 15.7279 0.525140
\(898\) −13.7990 −0.460478
\(899\) −25.8579 −0.862408
\(900\) 0 0
\(901\) −13.7279 −0.457343
\(902\) 42.2426 1.40653
\(903\) −25.3137 −0.842387
\(904\) 2.82843 0.0940721
\(905\) 0 0
\(906\) 2.41421 0.0802069
\(907\) 2.79899 0.0929389 0.0464695 0.998920i \(-0.485203\pi\)
0.0464695 + 0.998920i \(0.485203\pi\)
\(908\) −12.0000 −0.398234
\(909\) −5.31371 −0.176245
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −2.17157 −0.0719080
\(913\) 44.2548 1.46462
\(914\) −0.970563 −0.0321034
\(915\) 0 0
\(916\) 18.7279 0.618788
\(917\) 4.58579 0.151436
\(918\) −3.58579 −0.118349
\(919\) 16.2426 0.535795 0.267898 0.963447i \(-0.413671\pi\)
0.267898 + 0.963447i \(0.413671\pi\)
\(920\) 0 0
\(921\) 13.8995 0.458004
\(922\) 20.8284 0.685948
\(923\) 49.3553 1.62455
\(924\) −15.4853 −0.509428
\(925\) 0 0
\(926\) −29.8995 −0.982558
\(927\) 15.0711 0.494999
\(928\) 7.07107 0.232119
\(929\) 2.48528 0.0815394 0.0407697 0.999169i \(-0.487019\pi\)
0.0407697 + 0.999169i \(0.487019\pi\)
\(930\) 0 0
\(931\) 2.54416 0.0833813
\(932\) 2.48528 0.0814081
\(933\) −12.8284 −0.419984
\(934\) −40.2426 −1.31678
\(935\) 0 0
\(936\) −5.24264 −0.171361
\(937\) 49.9411 1.63151 0.815753 0.578401i \(-0.196323\pi\)
0.815753 + 0.578401i \(0.196323\pi\)
\(938\) 2.00000 0.0653023
\(939\) 2.72792 0.0890224
\(940\) 0 0
\(941\) −37.6569 −1.22758 −0.613789 0.789470i \(-0.710356\pi\)
−0.613789 + 0.789470i \(0.710356\pi\)
\(942\) 0 0
\(943\) 19.7574 0.643388
\(944\) −13.0711 −0.425427
\(945\) 0 0
\(946\) 67.2548 2.18664
\(947\) 28.0416 0.911231 0.455615 0.890177i \(-0.349419\pi\)
0.455615 + 0.890177i \(0.349419\pi\)
\(948\) 12.2426 0.397622
\(949\) 11.3848 0.369565
\(950\) 0 0
\(951\) −21.1716 −0.686535
\(952\) −8.65685 −0.280570
\(953\) 0.585786 0.0189755 0.00948774 0.999955i \(-0.496980\pi\)
0.00948774 + 0.999955i \(0.496980\pi\)
\(954\) 3.82843 0.123950
\(955\) 0 0
\(956\) 20.6274 0.667138
\(957\) −45.3553 −1.46613
\(958\) 6.17157 0.199394
\(959\) −42.3848 −1.36868
\(960\) 0 0
\(961\) −17.6274 −0.568626
\(962\) −5.24264 −0.169030
\(963\) −16.0711 −0.517883
\(964\) 2.58579 0.0832826
\(965\) 0 0
\(966\) −7.24264 −0.233028
\(967\) −59.8995 −1.92624 −0.963119 0.269076i \(-0.913282\pi\)
−0.963119 + 0.269076i \(0.913282\pi\)
\(968\) 30.1421 0.968805
\(969\) 7.78680 0.250148
\(970\) 0 0
\(971\) 10.9706 0.352062 0.176031 0.984385i \(-0.443674\pi\)
0.176031 + 0.984385i \(0.443674\pi\)
\(972\) 1.00000 0.0320750
\(973\) −6.00000 −0.192351
\(974\) −30.0000 −0.961262
\(975\) 0 0
\(976\) −6.48528 −0.207589
\(977\) −38.6985 −1.23807 −0.619037 0.785362i \(-0.712477\pi\)
−0.619037 + 0.785362i \(0.712477\pi\)
\(978\) −15.1421 −0.484192
\(979\) −18.5980 −0.594394
\(980\) 0 0
\(981\) 4.75736 0.151891
\(982\) −16.4142 −0.523799
\(983\) −25.1127 −0.800971 −0.400485 0.916303i \(-0.631158\pi\)
−0.400485 + 0.916303i \(0.631158\pi\)
\(984\) −6.58579 −0.209947
\(985\) 0 0
\(986\) −25.3553 −0.807478
\(987\) 28.7279 0.914420
\(988\) 11.3848 0.362198
\(989\) 31.4558 1.00024
\(990\) 0 0
\(991\) 14.6274 0.464655 0.232328 0.972638i \(-0.425366\pi\)
0.232328 + 0.972638i \(0.425366\pi\)
\(992\) −3.65685 −0.116105
\(993\) 23.4558 0.744349
\(994\) −22.7279 −0.720886
\(995\) 0 0
\(996\) −6.89949 −0.218619
\(997\) −52.2132 −1.65361 −0.826804 0.562490i \(-0.809844\pi\)
−0.826804 + 0.562490i \(0.809844\pi\)
\(998\) 7.48528 0.236942
\(999\) 1.00000 0.0316386
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 5550.2.a.cb.1.2 2
5.2 odd 4 1110.2.d.g.889.4 yes 4
5.3 odd 4 1110.2.d.g.889.2 4
5.4 even 2 5550.2.a.bs.1.1 2
15.2 even 4 3330.2.d.k.1999.1 4
15.8 even 4 3330.2.d.k.1999.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.d.g.889.2 4 5.3 odd 4
1110.2.d.g.889.4 yes 4 5.2 odd 4
3330.2.d.k.1999.1 4 15.2 even 4
3330.2.d.k.1999.3 4 15.8 even 4
5550.2.a.bs.1.1 2 5.4 even 2
5550.2.a.cb.1.2 2 1.1 even 1 trivial