Properties

Label 56.2.b.a.29.1
Level $56$
Weight $2$
Character 56.29
Analytic conductor $0.447$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,2,Mod(29,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 56.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 29.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 56.29
Dual form 56.2.b.a.29.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{4} +1.41421i q^{5} -2.00000 q^{6} +1.00000 q^{7} +2.82843i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{4} +1.41421i q^{5} -2.00000 q^{6} +1.00000 q^{7} +2.82843i q^{8} +1.00000 q^{9} +2.00000 q^{10} -2.82843i q^{11} +2.82843i q^{12} +4.24264i q^{13} -1.41421i q^{14} +2.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} -1.41421i q^{18} +4.24264i q^{19} -2.82843i q^{20} -1.41421i q^{21} -4.00000 q^{22} -6.00000 q^{23} +4.00000 q^{24} +3.00000 q^{25} +6.00000 q^{26} -5.65685i q^{27} -2.00000 q^{28} -2.82843i q^{29} -2.82843i q^{30} -4.00000 q^{31} -5.65685i q^{32} -4.00000 q^{33} +8.48528i q^{34} +1.41421i q^{35} -2.00000 q^{36} -8.48528i q^{37} +6.00000 q^{38} +6.00000 q^{39} -4.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} +8.48528i q^{43} +5.65685i q^{44} +1.41421i q^{45} +8.48528i q^{46} -5.65685i q^{48} +1.00000 q^{49} -4.24264i q^{50} +8.48528i q^{51} -8.48528i q^{52} +5.65685i q^{53} -8.00000 q^{54} +4.00000 q^{55} +2.82843i q^{56} +6.00000 q^{57} -4.00000 q^{58} +1.41421i q^{59} -4.00000 q^{60} -12.7279i q^{61} +5.65685i q^{62} +1.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} +5.65685i q^{66} +12.0000 q^{68} +8.48528i q^{69} +2.00000 q^{70} +2.82843i q^{72} +2.00000 q^{73} -12.0000 q^{74} -4.24264i q^{75} -8.48528i q^{76} -2.82843i q^{77} -8.48528i q^{78} +8.00000 q^{79} +5.65685i q^{80} -5.00000 q^{81} -8.48528i q^{82} -15.5563i q^{83} +2.82843i q^{84} -8.48528i q^{85} +12.0000 q^{86} -4.00000 q^{87} +8.00000 q^{88} +6.00000 q^{89} +2.00000 q^{90} +4.24264i q^{91} +12.0000 q^{92} +5.65685i q^{93} -6.00000 q^{95} -8.00000 q^{96} -10.0000 q^{97} -1.41421i q^{98} -2.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 4 q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 4 q^{6} + 2 q^{7} + 2 q^{9} + 4 q^{10} + 4 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{22} - 12 q^{23} + 8 q^{24} + 6 q^{25} + 12 q^{26} - 4 q^{28} - 8 q^{31} - 8 q^{33} - 4 q^{36} + 12 q^{38} + 12 q^{39} - 8 q^{40} + 12 q^{41} - 4 q^{42} + 2 q^{49} - 16 q^{54} + 8 q^{55} + 12 q^{57} - 8 q^{58} - 8 q^{60} + 2 q^{63} - 16 q^{64} - 12 q^{65} + 24 q^{68} + 4 q^{70} + 4 q^{73} - 24 q^{74} + 16 q^{79} - 10 q^{81} + 24 q^{86} - 8 q^{87} + 16 q^{88} + 12 q^{89} + 4 q^{90} + 24 q^{92} - 12 q^{95} - 16 q^{96} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i
\(3\) − 1.41421i − 0.816497i −0.912871 0.408248i \(-0.866140\pi\)
0.912871 0.408248i \(-0.133860\pi\)
\(4\) −2.00000 −1.00000
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 2.82843i 1.00000i
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) − 2.82843i − 0.852803i −0.904534 0.426401i \(-0.859781\pi\)
0.904534 0.426401i \(-0.140219\pi\)
\(12\) 2.82843i 0.816497i
\(13\) 4.24264i 1.17670i 0.808608 + 0.588348i \(0.200222\pi\)
−0.808608 + 0.588348i \(0.799778\pi\)
\(14\) − 1.41421i − 0.377964i
\(15\) 2.00000 0.516398
\(16\) 4.00000 1.00000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) − 1.41421i − 0.333333i
\(19\) 4.24264i 0.973329i 0.873589 + 0.486664i \(0.161786\pi\)
−0.873589 + 0.486664i \(0.838214\pi\)
\(20\) − 2.82843i − 0.632456i
\(21\) − 1.41421i − 0.308607i
\(22\) −4.00000 −0.852803
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 4.00000 0.816497
\(25\) 3.00000 0.600000
\(26\) 6.00000 1.17670
\(27\) − 5.65685i − 1.08866i
\(28\) −2.00000 −0.377964
\(29\) − 2.82843i − 0.525226i −0.964901 0.262613i \(-0.915416\pi\)
0.964901 0.262613i \(-0.0845842\pi\)
\(30\) − 2.82843i − 0.516398i
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 5.65685i − 1.00000i
\(33\) −4.00000 −0.696311
\(34\) 8.48528i 1.45521i
\(35\) 1.41421i 0.239046i
\(36\) −2.00000 −0.333333
\(37\) − 8.48528i − 1.39497i −0.716599 0.697486i \(-0.754302\pi\)
0.716599 0.697486i \(-0.245698\pi\)
\(38\) 6.00000 0.973329
\(39\) 6.00000 0.960769
\(40\) −4.00000 −0.632456
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 5.65685i 0.852803i
\(45\) 1.41421i 0.210819i
\(46\) 8.48528i 1.25109i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 5.65685i − 0.816497i
\(49\) 1.00000 0.142857
\(50\) − 4.24264i − 0.600000i
\(51\) 8.48528i 1.18818i
\(52\) − 8.48528i − 1.17670i
\(53\) 5.65685i 0.777029i 0.921443 + 0.388514i \(0.127012\pi\)
−0.921443 + 0.388514i \(0.872988\pi\)
\(54\) −8.00000 −1.08866
\(55\) 4.00000 0.539360
\(56\) 2.82843i 0.377964i
\(57\) 6.00000 0.794719
\(58\) −4.00000 −0.525226
\(59\) 1.41421i 0.184115i 0.995754 + 0.0920575i \(0.0293443\pi\)
−0.995754 + 0.0920575i \(0.970656\pi\)
\(60\) −4.00000 −0.516398
\(61\) − 12.7279i − 1.62964i −0.579712 0.814822i \(-0.696835\pi\)
0.579712 0.814822i \(-0.303165\pi\)
\(62\) 5.65685i 0.718421i
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) −6.00000 −0.744208
\(66\) 5.65685i 0.696311i
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 12.0000 1.45521
\(69\) 8.48528i 1.02151i
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 2.82843i 0.333333i
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −12.0000 −1.39497
\(75\) − 4.24264i − 0.489898i
\(76\) − 8.48528i − 0.973329i
\(77\) − 2.82843i − 0.322329i
\(78\) − 8.48528i − 0.960769i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 5.65685i 0.632456i
\(81\) −5.00000 −0.555556
\(82\) − 8.48528i − 0.937043i
\(83\) − 15.5563i − 1.70753i −0.520658 0.853766i \(-0.674313\pi\)
0.520658 0.853766i \(-0.325687\pi\)
\(84\) 2.82843i 0.308607i
\(85\) − 8.48528i − 0.920358i
\(86\) 12.0000 1.29399
\(87\) −4.00000 −0.428845
\(88\) 8.00000 0.852803
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 2.00000 0.210819
\(91\) 4.24264i 0.444750i
\(92\) 12.0000 1.25109
\(93\) 5.65685i 0.586588i
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) −8.00000 −0.816497
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) − 1.41421i − 0.142857i
\(99\) − 2.82843i − 0.284268i
\(100\) −6.00000 −0.600000
\(101\) 9.89949i 0.985037i 0.870302 + 0.492518i \(0.163924\pi\)
−0.870302 + 0.492518i \(0.836076\pi\)
\(102\) 12.0000 1.18818
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −12.0000 −1.17670
\(105\) 2.00000 0.195180
\(106\) 8.00000 0.777029
\(107\) 5.65685i 0.546869i 0.961891 + 0.273434i \(0.0881596\pi\)
−0.961891 + 0.273434i \(0.911840\pi\)
\(108\) 11.3137i 1.08866i
\(109\) 8.48528i 0.812743i 0.913708 + 0.406371i \(0.133206\pi\)
−0.913708 + 0.406371i \(0.866794\pi\)
\(110\) − 5.65685i − 0.539360i
\(111\) −12.0000 −1.13899
\(112\) 4.00000 0.377964
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) − 8.48528i − 0.794719i
\(115\) − 8.48528i − 0.791257i
\(116\) 5.65685i 0.525226i
\(117\) 4.24264i 0.392232i
\(118\) 2.00000 0.184115
\(119\) −6.00000 −0.550019
\(120\) 5.65685i 0.516398i
\(121\) 3.00000 0.272727
\(122\) −18.0000 −1.62964
\(123\) − 8.48528i − 0.765092i
\(124\) 8.00000 0.718421
\(125\) 11.3137i 1.01193i
\(126\) − 1.41421i − 0.125988i
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 12.0000 1.05654
\(130\) 8.48528i 0.744208i
\(131\) 1.41421i 0.123560i 0.998090 + 0.0617802i \(0.0196778\pi\)
−0.998090 + 0.0617802i \(0.980322\pi\)
\(132\) 8.00000 0.696311
\(133\) 4.24264i 0.367884i
\(134\) 0 0
\(135\) 8.00000 0.688530
\(136\) − 16.9706i − 1.45521i
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 12.0000 1.02151
\(139\) 4.24264i 0.359856i 0.983680 + 0.179928i \(0.0575865\pi\)
−0.983680 + 0.179928i \(0.942414\pi\)
\(140\) − 2.82843i − 0.239046i
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 4.00000 0.333333
\(145\) 4.00000 0.332182
\(146\) − 2.82843i − 0.234082i
\(147\) − 1.41421i − 0.116642i
\(148\) 16.9706i 1.39497i
\(149\) − 11.3137i − 0.926855i −0.886135 0.463428i \(-0.846619\pi\)
0.886135 0.463428i \(-0.153381\pi\)
\(150\) −6.00000 −0.489898
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −12.0000 −0.973329
\(153\) −6.00000 −0.485071
\(154\) −4.00000 −0.322329
\(155\) − 5.65685i − 0.454369i
\(156\) −12.0000 −0.960769
\(157\) − 12.7279i − 1.01580i −0.861416 0.507899i \(-0.830422\pi\)
0.861416 0.507899i \(-0.169578\pi\)
\(158\) − 11.3137i − 0.900070i
\(159\) 8.00000 0.634441
\(160\) 8.00000 0.632456
\(161\) −6.00000 −0.472866
\(162\) 7.07107i 0.555556i
\(163\) − 8.48528i − 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) −12.0000 −0.937043
\(165\) − 5.65685i − 0.440386i
\(166\) −22.0000 −1.70753
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 4.00000 0.308607
\(169\) −5.00000 −0.384615
\(170\) −12.0000 −0.920358
\(171\) 4.24264i 0.324443i
\(172\) − 16.9706i − 1.29399i
\(173\) 9.89949i 0.752645i 0.926489 + 0.376322i \(0.122811\pi\)
−0.926489 + 0.376322i \(0.877189\pi\)
\(174\) 5.65685i 0.428845i
\(175\) 3.00000 0.226779
\(176\) − 11.3137i − 0.852803i
\(177\) 2.00000 0.150329
\(178\) − 8.48528i − 0.635999i
\(179\) 5.65685i 0.422813i 0.977398 + 0.211407i \(0.0678044\pi\)
−0.977398 + 0.211407i \(0.932196\pi\)
\(180\) − 2.82843i − 0.210819i
\(181\) − 12.7279i − 0.946059i −0.881047 0.473029i \(-0.843160\pi\)
0.881047 0.473029i \(-0.156840\pi\)
\(182\) 6.00000 0.444750
\(183\) −18.0000 −1.33060
\(184\) − 16.9706i − 1.25109i
\(185\) 12.0000 0.882258
\(186\) 8.00000 0.586588
\(187\) 16.9706i 1.24101i
\(188\) 0 0
\(189\) − 5.65685i − 0.411476i
\(190\) 8.48528i 0.615587i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 11.3137i 0.816497i
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 14.1421i 1.01535i
\(195\) 8.48528i 0.607644i
\(196\) −2.00000 −0.142857
\(197\) 5.65685i 0.403034i 0.979485 + 0.201517i \(0.0645872\pi\)
−0.979485 + 0.201517i \(0.935413\pi\)
\(198\) −4.00000 −0.284268
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 8.48528i 0.600000i
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) − 2.82843i − 0.198517i
\(204\) − 16.9706i − 1.18818i
\(205\) 8.48528i 0.592638i
\(206\) 5.65685i 0.394132i
\(207\) −6.00000 −0.417029
\(208\) 16.9706i 1.17670i
\(209\) 12.0000 0.830057
\(210\) − 2.82843i − 0.195180i
\(211\) 16.9706i 1.16830i 0.811645 + 0.584151i \(0.198572\pi\)
−0.811645 + 0.584151i \(0.801428\pi\)
\(212\) − 11.3137i − 0.777029i
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −12.0000 −0.818393
\(216\) 16.0000 1.08866
\(217\) −4.00000 −0.271538
\(218\) 12.0000 0.812743
\(219\) − 2.82843i − 0.191127i
\(220\) −8.00000 −0.539360
\(221\) − 25.4558i − 1.71235i
\(222\) 16.9706i 1.13899i
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) − 5.65685i − 0.377964i
\(225\) 3.00000 0.200000
\(226\) 16.9706i 1.12887i
\(227\) 9.89949i 0.657053i 0.944495 + 0.328526i \(0.106552\pi\)
−0.944495 + 0.328526i \(0.893448\pi\)
\(228\) −12.0000 −0.794719
\(229\) − 4.24264i − 0.280362i −0.990126 0.140181i \(-0.955232\pi\)
0.990126 0.140181i \(-0.0447684\pi\)
\(230\) −12.0000 −0.791257
\(231\) −4.00000 −0.263181
\(232\) 8.00000 0.525226
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) − 2.82843i − 0.184115i
\(237\) − 11.3137i − 0.734904i
\(238\) 8.48528i 0.550019i
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 8.00000 0.516398
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) − 4.24264i − 0.272727i
\(243\) − 9.89949i − 0.635053i
\(244\) 25.4558i 1.62964i
\(245\) 1.41421i 0.0903508i
\(246\) −12.0000 −0.765092
\(247\) −18.0000 −1.14531
\(248\) − 11.3137i − 0.718421i
\(249\) −22.0000 −1.39419
\(250\) 16.0000 1.01193
\(251\) 18.3848i 1.16044i 0.814461 + 0.580218i \(0.197033\pi\)
−0.814461 + 0.580218i \(0.802967\pi\)
\(252\) −2.00000 −0.125988
\(253\) 16.9706i 1.06693i
\(254\) − 2.82843i − 0.177471i
\(255\) −12.0000 −0.751469
\(256\) 16.0000 1.00000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) − 16.9706i − 1.05654i
\(259\) − 8.48528i − 0.527250i
\(260\) 12.0000 0.744208
\(261\) − 2.82843i − 0.175075i
\(262\) 2.00000 0.123560
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) − 11.3137i − 0.696311i
\(265\) −8.00000 −0.491436
\(266\) 6.00000 0.367884
\(267\) − 8.48528i − 0.519291i
\(268\) 0 0
\(269\) − 7.07107i − 0.431131i −0.976489 0.215565i \(-0.930841\pi\)
0.976489 0.215565i \(-0.0691594\pi\)
\(270\) − 11.3137i − 0.688530i
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −24.0000 −1.45521
\(273\) 6.00000 0.363137
\(274\) − 8.48528i − 0.512615i
\(275\) − 8.48528i − 0.511682i
\(276\) − 16.9706i − 1.02151i
\(277\) 16.9706i 1.01966i 0.860274 + 0.509831i \(0.170292\pi\)
−0.860274 + 0.509831i \(0.829708\pi\)
\(278\) 6.00000 0.359856
\(279\) −4.00000 −0.239474
\(280\) −4.00000 −0.239046
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) − 12.7279i − 0.756596i −0.925684 0.378298i \(-0.876509\pi\)
0.925684 0.378298i \(-0.123491\pi\)
\(284\) 0 0
\(285\) 8.48528i 0.502625i
\(286\) − 16.9706i − 1.00349i
\(287\) 6.00000 0.354169
\(288\) − 5.65685i − 0.333333i
\(289\) 19.0000 1.11765
\(290\) − 5.65685i − 0.332182i
\(291\) 14.1421i 0.829027i
\(292\) −4.00000 −0.234082
\(293\) − 24.0416i − 1.40453i −0.711917 0.702264i \(-0.752173\pi\)
0.711917 0.702264i \(-0.247827\pi\)
\(294\) −2.00000 −0.116642
\(295\) −2.00000 −0.116445
\(296\) 24.0000 1.39497
\(297\) −16.0000 −0.928414
\(298\) −16.0000 −0.926855
\(299\) − 25.4558i − 1.47215i
\(300\) 8.48528i 0.489898i
\(301\) 8.48528i 0.489083i
\(302\) 14.1421i 0.813788i
\(303\) 14.0000 0.804279
\(304\) 16.9706i 0.973329i
\(305\) 18.0000 1.03068
\(306\) 8.48528i 0.485071i
\(307\) 12.7279i 0.726421i 0.931707 + 0.363210i \(0.118319\pi\)
−0.931707 + 0.363210i \(0.881681\pi\)
\(308\) 5.65685i 0.322329i
\(309\) 5.65685i 0.321807i
\(310\) −8.00000 −0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 16.9706i 0.960769i
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −18.0000 −1.01580
\(315\) 1.41421i 0.0796819i
\(316\) −16.0000 −0.900070
\(317\) 22.6274i 1.27088i 0.772149 + 0.635441i \(0.219182\pi\)
−0.772149 + 0.635441i \(0.780818\pi\)
\(318\) − 11.3137i − 0.634441i
\(319\) −8.00000 −0.447914
\(320\) − 11.3137i − 0.632456i
\(321\) 8.00000 0.446516
\(322\) 8.48528i 0.472866i
\(323\) − 25.4558i − 1.41640i
\(324\) 10.0000 0.555556
\(325\) 12.7279i 0.706018i
\(326\) −12.0000 −0.664619
\(327\) 12.0000 0.663602
\(328\) 16.9706i 0.937043i
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) − 25.4558i − 1.39918i −0.714545 0.699590i \(-0.753366\pi\)
0.714545 0.699590i \(-0.246634\pi\)
\(332\) 31.1127i 1.70753i
\(333\) − 8.48528i − 0.464991i
\(334\) − 33.9411i − 1.85718i
\(335\) 0 0
\(336\) − 5.65685i − 0.308607i
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 7.07107i 0.384615i
\(339\) 16.9706i 0.921714i
\(340\) 16.9706i 0.920358i
\(341\) 11.3137i 0.612672i
\(342\) 6.00000 0.324443
\(343\) 1.00000 0.0539949
\(344\) −24.0000 −1.29399
\(345\) −12.0000 −0.646058
\(346\) 14.0000 0.752645
\(347\) 14.1421i 0.759190i 0.925153 + 0.379595i \(0.123937\pi\)
−0.925153 + 0.379595i \(0.876063\pi\)
\(348\) 8.00000 0.428845
\(349\) − 4.24264i − 0.227103i −0.993532 0.113552i \(-0.963777\pi\)
0.993532 0.113552i \(-0.0362227\pi\)
\(350\) − 4.24264i − 0.226779i
\(351\) 24.0000 1.28103
\(352\) −16.0000 −0.852803
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) − 2.82843i − 0.150329i
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) 8.48528i 0.449089i
\(358\) 8.00000 0.422813
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) −4.00000 −0.210819
\(361\) 1.00000 0.0526316
\(362\) −18.0000 −0.946059
\(363\) − 4.24264i − 0.222681i
\(364\) − 8.48528i − 0.444750i
\(365\) 2.82843i 0.148047i
\(366\) 25.4558i 1.33060i
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −24.0000 −1.25109
\(369\) 6.00000 0.312348
\(370\) − 16.9706i − 0.882258i
\(371\) 5.65685i 0.293689i
\(372\) − 11.3137i − 0.586588i
\(373\) 33.9411i 1.75740i 0.477370 + 0.878702i \(0.341590\pi\)
−0.477370 + 0.878702i \(0.658410\pi\)
\(374\) 24.0000 1.24101
\(375\) 16.0000 0.826236
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) −8.00000 −0.411476
\(379\) − 25.4558i − 1.30758i −0.756677 0.653789i \(-0.773178\pi\)
0.756677 0.653789i \(-0.226822\pi\)
\(380\) 12.0000 0.615587
\(381\) − 2.82843i − 0.144905i
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 16.0000 0.816497
\(385\) 4.00000 0.203859
\(386\) 5.65685i 0.287926i
\(387\) 8.48528i 0.431331i
\(388\) 20.0000 1.01535
\(389\) − 2.82843i − 0.143407i −0.997426 0.0717035i \(-0.977156\pi\)
0.997426 0.0717035i \(-0.0228435\pi\)
\(390\) 12.0000 0.607644
\(391\) 36.0000 1.82060
\(392\) 2.82843i 0.142857i
\(393\) 2.00000 0.100887
\(394\) 8.00000 0.403034
\(395\) 11.3137i 0.569254i
\(396\) 5.65685i 0.284268i
\(397\) − 21.2132i − 1.06466i −0.846537 0.532330i \(-0.821317\pi\)
0.846537 0.532330i \(-0.178683\pi\)
\(398\) − 28.2843i − 1.41776i
\(399\) 6.00000 0.300376
\(400\) 12.0000 0.600000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) − 16.9706i − 0.845364i
\(404\) − 19.7990i − 0.985037i
\(405\) − 7.07107i − 0.351364i
\(406\) −4.00000 −0.198517
\(407\) −24.0000 −1.18964
\(408\) −24.0000 −1.18818
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 12.0000 0.592638
\(411\) − 8.48528i − 0.418548i
\(412\) 8.00000 0.394132
\(413\) 1.41421i 0.0695889i
\(414\) 8.48528i 0.417029i
\(415\) 22.0000 1.07994
\(416\) 24.0000 1.17670
\(417\) 6.00000 0.293821
\(418\) − 16.9706i − 0.830057i
\(419\) 26.8701i 1.31269i 0.754462 + 0.656344i \(0.227898\pi\)
−0.754462 + 0.656344i \(0.772102\pi\)
\(420\) −4.00000 −0.195180
\(421\) − 16.9706i − 0.827095i −0.910483 0.413547i \(-0.864290\pi\)
0.910483 0.413547i \(-0.135710\pi\)
\(422\) 24.0000 1.16830
\(423\) 0 0
\(424\) −16.0000 −0.777029
\(425\) −18.0000 −0.873128
\(426\) 0 0
\(427\) − 12.7279i − 0.615947i
\(428\) − 11.3137i − 0.546869i
\(429\) − 16.9706i − 0.819346i
\(430\) 16.9706i 0.818393i
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) − 22.6274i − 1.08866i
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 5.65685i 0.271538i
\(435\) − 5.65685i − 0.271225i
\(436\) − 16.9706i − 0.812743i
\(437\) − 25.4558i − 1.21772i
\(438\) −4.00000 −0.191127
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 11.3137i 0.539360i
\(441\) 1.00000 0.0476190
\(442\) −36.0000 −1.71235
\(443\) 22.6274i 1.07506i 0.843244 + 0.537531i \(0.180643\pi\)
−0.843244 + 0.537531i \(0.819357\pi\)
\(444\) 24.0000 1.13899
\(445\) 8.48528i 0.402241i
\(446\) 39.5980i 1.87502i
\(447\) −16.0000 −0.756774
\(448\) −8.00000 −0.377964
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) − 4.24264i − 0.200000i
\(451\) − 16.9706i − 0.799113i
\(452\) 24.0000 1.12887
\(453\) 14.1421i 0.664455i
\(454\) 14.0000 0.657053
\(455\) −6.00000 −0.281284
\(456\) 16.9706i 0.794719i
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −6.00000 −0.280362
\(459\) 33.9411i 1.58424i
\(460\) 16.9706i 0.791257i
\(461\) − 15.5563i − 0.724531i −0.932075 0.362266i \(-0.882003\pi\)
0.932075 0.362266i \(-0.117997\pi\)
\(462\) 5.65685i 0.263181i
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) − 11.3137i − 0.525226i
\(465\) −8.00000 −0.370991
\(466\) − 8.48528i − 0.393073i
\(467\) − 7.07107i − 0.327210i −0.986526 0.163605i \(-0.947688\pi\)
0.986526 0.163605i \(-0.0523123\pi\)
\(468\) − 8.48528i − 0.392232i
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) −4.00000 −0.184115
\(473\) 24.0000 1.10352
\(474\) −16.0000 −0.734904
\(475\) 12.7279i 0.583997i
\(476\) 12.0000 0.550019
\(477\) 5.65685i 0.259010i
\(478\) − 8.48528i − 0.388108i
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) − 11.3137i − 0.516398i
\(481\) 36.0000 1.64146
\(482\) 14.1421i 0.644157i
\(483\) 8.48528i 0.386094i
\(484\) −6.00000 −0.272727
\(485\) − 14.1421i − 0.642161i
\(486\) −14.0000 −0.635053
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 36.0000 1.62964
\(489\) −12.0000 −0.542659
\(490\) 2.00000 0.0903508
\(491\) 39.5980i 1.78703i 0.449032 + 0.893516i \(0.351769\pi\)
−0.449032 + 0.893516i \(0.648231\pi\)
\(492\) 16.9706i 0.765092i
\(493\) 16.9706i 0.764316i
\(494\) 25.4558i 1.14531i
\(495\) 4.00000 0.179787
\(496\) −16.0000 −0.718421
\(497\) 0 0
\(498\) 31.1127i 1.39419i
\(499\) − 16.9706i − 0.759707i −0.925047 0.379853i \(-0.875974\pi\)
0.925047 0.379853i \(-0.124026\pi\)
\(500\) − 22.6274i − 1.01193i
\(501\) − 33.9411i − 1.51638i
\(502\) 26.0000 1.16044
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 2.82843i 0.125988i
\(505\) −14.0000 −0.622992
\(506\) 24.0000 1.06693
\(507\) 7.07107i 0.314037i
\(508\) −4.00000 −0.177471
\(509\) 1.41421i 0.0626839i 0.999509 + 0.0313420i \(0.00997809\pi\)
−0.999509 + 0.0313420i \(0.990022\pi\)
\(510\) 16.9706i 0.751469i
\(511\) 2.00000 0.0884748
\(512\) − 22.6274i − 1.00000i
\(513\) 24.0000 1.05963
\(514\) 8.48528i 0.374270i
\(515\) − 5.65685i − 0.249271i
\(516\) −24.0000 −1.05654
\(517\) 0 0
\(518\) −12.0000 −0.527250
\(519\) 14.0000 0.614532
\(520\) − 16.9706i − 0.744208i
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −4.00000 −0.175075
\(523\) 29.6985i 1.29862i 0.760522 + 0.649312i \(0.224943\pi\)
−0.760522 + 0.649312i \(0.775057\pi\)
\(524\) − 2.82843i − 0.123560i
\(525\) − 4.24264i − 0.185164i
\(526\) − 33.9411i − 1.47990i
\(527\) 24.0000 1.04546
\(528\) −16.0000 −0.696311
\(529\) 13.0000 0.565217
\(530\) 11.3137i 0.491436i
\(531\) 1.41421i 0.0613716i
\(532\) − 8.48528i − 0.367884i
\(533\) 25.4558i 1.10262i
\(534\) −12.0000 −0.519291
\(535\) −8.00000 −0.345870
\(536\) 0 0
\(537\) 8.00000 0.345225
\(538\) −10.0000 −0.431131
\(539\) − 2.82843i − 0.121829i
\(540\) −16.0000 −0.688530
\(541\) 16.9706i 0.729621i 0.931082 + 0.364811i \(0.118866\pi\)
−0.931082 + 0.364811i \(0.881134\pi\)
\(542\) − 28.2843i − 1.21491i
\(543\) −18.0000 −0.772454
\(544\) 33.9411i 1.45521i
\(545\) −12.0000 −0.514024
\(546\) − 8.48528i − 0.363137i
\(547\) 8.48528i 0.362804i 0.983409 + 0.181402i \(0.0580636\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) −12.0000 −0.512615
\(549\) − 12.7279i − 0.543214i
\(550\) −12.0000 −0.511682
\(551\) 12.0000 0.511217
\(552\) −24.0000 −1.02151
\(553\) 8.00000 0.340195
\(554\) 24.0000 1.01966
\(555\) − 16.9706i − 0.720360i
\(556\) − 8.48528i − 0.359856i
\(557\) 5.65685i 0.239689i 0.992793 + 0.119844i \(0.0382395\pi\)
−0.992793 + 0.119844i \(0.961760\pi\)
\(558\) 5.65685i 0.239474i
\(559\) −36.0000 −1.52264
\(560\) 5.65685i 0.239046i
\(561\) 24.0000 1.01328
\(562\) 8.48528i 0.357930i
\(563\) 1.41421i 0.0596020i 0.999556 + 0.0298010i \(0.00948736\pi\)
−0.999556 + 0.0298010i \(0.990513\pi\)
\(564\) 0 0
\(565\) − 16.9706i − 0.713957i
\(566\) −18.0000 −0.756596
\(567\) −5.00000 −0.209980
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 12.0000 0.502625
\(571\) − 25.4558i − 1.06529i −0.846338 0.532647i \(-0.821197\pi\)
0.846338 0.532647i \(-0.178803\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) − 8.48528i − 0.354169i
\(575\) −18.0000 −0.750652
\(576\) −8.00000 −0.333333
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) − 26.8701i − 1.11765i
\(579\) 5.65685i 0.235091i
\(580\) −8.00000 −0.332182
\(581\) − 15.5563i − 0.645386i
\(582\) 20.0000 0.829027
\(583\) 16.0000 0.662652
\(584\) 5.65685i 0.234082i
\(585\) −6.00000 −0.248069
\(586\) −34.0000 −1.40453
\(587\) − 41.0122i − 1.69275i −0.532584 0.846377i \(-0.678779\pi\)
0.532584 0.846377i \(-0.321221\pi\)
\(588\) 2.82843i 0.116642i
\(589\) − 16.9706i − 0.699260i
\(590\) 2.82843i 0.116445i
\(591\) 8.00000 0.329076
\(592\) − 33.9411i − 1.39497i
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 22.6274i 0.928414i
\(595\) − 8.48528i − 0.347863i
\(596\) 22.6274i 0.926855i
\(597\) − 28.2843i − 1.15760i
\(598\) −36.0000 −1.47215
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 12.0000 0.489898
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 4.24264i 0.172488i
\(606\) − 19.7990i − 0.804279i
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 24.0000 0.973329
\(609\) −4.00000 −0.162088
\(610\) − 25.4558i − 1.03068i
\(611\) 0 0
\(612\) 12.0000 0.485071
\(613\) 8.48528i 0.342717i 0.985209 + 0.171359i \(0.0548157\pi\)
−0.985209 + 0.171359i \(0.945184\pi\)
\(614\) 18.0000 0.726421
\(615\) 12.0000 0.483887
\(616\) 8.00000 0.322329
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 8.00000 0.321807
\(619\) 4.24264i 0.170526i 0.996358 + 0.0852631i \(0.0271731\pi\)
−0.996358 + 0.0852631i \(0.972827\pi\)
\(620\) 11.3137i 0.454369i
\(621\) 33.9411i 1.36201i
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 24.0000 0.960769
\(625\) −1.00000 −0.0400000
\(626\) 14.1421i 0.565233i
\(627\) − 16.9706i − 0.677739i
\(628\) 25.4558i 1.01580i
\(629\) 50.9117i 2.02998i
\(630\) 2.00000 0.0796819
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 22.6274i 0.900070i
\(633\) 24.0000 0.953914
\(634\) 32.0000 1.27088
\(635\) 2.82843i 0.112243i
\(636\) −16.0000 −0.634441
\(637\) 4.24264i 0.168100i
\(638\) 11.3137i 0.447914i
\(639\) 0 0
\(640\) −16.0000 −0.632456
\(641\) −48.0000 −1.89589 −0.947943 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) − 11.3137i − 0.446516i
\(643\) − 21.2132i − 0.836567i −0.908317 0.418284i \(-0.862632\pi\)
0.908317 0.418284i \(-0.137368\pi\)
\(644\) 12.0000 0.472866
\(645\) 16.9706i 0.668215i
\(646\) −36.0000 −1.41640
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) − 14.1421i − 0.555556i
\(649\) 4.00000 0.157014
\(650\) 18.0000 0.706018
\(651\) 5.65685i 0.221710i
\(652\) 16.9706i 0.664619i
\(653\) − 36.7696i − 1.43890i −0.694542 0.719452i \(-0.744393\pi\)
0.694542 0.719452i \(-0.255607\pi\)
\(654\) − 16.9706i − 0.663602i
\(655\) −2.00000 −0.0781465
\(656\) 24.0000 0.937043
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) − 2.82843i − 0.110180i −0.998481 0.0550899i \(-0.982455\pi\)
0.998481 0.0550899i \(-0.0175446\pi\)
\(660\) 11.3137i 0.440386i
\(661\) 38.1838i 1.48518i 0.669748 + 0.742588i \(0.266402\pi\)
−0.669748 + 0.742588i \(0.733598\pi\)
\(662\) −36.0000 −1.39918
\(663\) −36.0000 −1.39812
\(664\) 44.0000 1.70753
\(665\) −6.00000 −0.232670
\(666\) −12.0000 −0.464991
\(667\) 16.9706i 0.657103i
\(668\) −48.0000 −1.85718
\(669\) 39.5980i 1.53095i
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) −8.00000 −0.308607
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) − 45.2548i − 1.74315i
\(675\) − 16.9706i − 0.653197i
\(676\) 10.0000 0.384615
\(677\) 9.89949i 0.380468i 0.981739 + 0.190234i \(0.0609248\pi\)
−0.981739 + 0.190234i \(0.939075\pi\)
\(678\) 24.0000 0.921714
\(679\) −10.0000 −0.383765
\(680\) 24.0000 0.920358
\(681\) 14.0000 0.536481
\(682\) 16.0000 0.612672
\(683\) 5.65685i 0.216454i 0.994126 + 0.108227i \(0.0345173\pi\)
−0.994126 + 0.108227i \(0.965483\pi\)
\(684\) − 8.48528i − 0.324443i
\(685\) 8.48528i 0.324206i
\(686\) − 1.41421i − 0.0539949i
\(687\) −6.00000 −0.228914
\(688\) 33.9411i 1.29399i
\(689\) −24.0000 −0.914327
\(690\) 16.9706i 0.646058i
\(691\) 12.7279i 0.484193i 0.970252 + 0.242096i \(0.0778351\pi\)
−0.970252 + 0.242096i \(0.922165\pi\)
\(692\) − 19.7990i − 0.752645i
\(693\) − 2.82843i − 0.107443i
\(694\) 20.0000 0.759190
\(695\) −6.00000 −0.227593
\(696\) − 11.3137i − 0.428845i
\(697\) −36.0000 −1.36360
\(698\) −6.00000 −0.227103
\(699\) − 8.48528i − 0.320943i
\(700\) −6.00000 −0.226779
\(701\) − 19.7990i − 0.747798i −0.927470 0.373899i \(-0.878021\pi\)
0.927470 0.373899i \(-0.121979\pi\)
\(702\) − 33.9411i − 1.28103i
\(703\) 36.0000 1.35777
\(704\) 22.6274i 0.852803i
\(705\) 0 0
\(706\) − 8.48528i − 0.319348i
\(707\) 9.89949i 0.372309i
\(708\) −4.00000 −0.150329
\(709\) 25.4558i 0.956014i 0.878356 + 0.478007i \(0.158641\pi\)
−0.878356 + 0.478007i \(0.841359\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 16.9706i 0.635999i
\(713\) 24.0000 0.898807
\(714\) 12.0000 0.449089
\(715\) 16.9706i 0.634663i
\(716\) − 11.3137i − 0.422813i
\(717\) − 8.48528i − 0.316889i
\(718\) 42.4264i 1.58334i
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 5.65685i 0.210819i
\(721\) −4.00000 −0.148968
\(722\) − 1.41421i − 0.0526316i
\(723\) 14.1421i 0.525952i
\(724\) 25.4558i 0.946059i
\(725\) − 8.48528i − 0.315135i
\(726\) −6.00000 −0.222681
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −12.0000 −0.444750
\(729\) −29.0000 −1.07407
\(730\) 4.00000 0.148047
\(731\) − 50.9117i − 1.88304i
\(732\) 36.0000 1.33060
\(733\) − 29.6985i − 1.09694i −0.836171 0.548469i \(-0.815211\pi\)
0.836171 0.548469i \(-0.184789\pi\)
\(734\) 39.5980i 1.46159i
\(735\) 2.00000 0.0737711
\(736\) 33.9411i 1.25109i
\(737\) 0 0
\(738\) − 8.48528i − 0.312348i
\(739\) − 42.4264i − 1.56068i −0.625355 0.780340i \(-0.715046\pi\)
0.625355 0.780340i \(-0.284954\pi\)
\(740\) −24.0000 −0.882258
\(741\) 25.4558i 0.935144i
\(742\) 8.00000 0.293689
\(743\) −30.0000 −1.10059 −0.550297 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) −16.0000 −0.586588
\(745\) 16.0000 0.586195
\(746\) 48.0000 1.75740
\(747\) − 15.5563i − 0.569177i
\(748\) − 33.9411i − 1.24101i
\(749\) 5.65685i 0.206697i
\(750\) − 22.6274i − 0.826236i
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 0 0
\(753\) 26.0000 0.947493
\(754\) − 16.9706i − 0.618031i
\(755\) − 14.1421i − 0.514685i
\(756\) 11.3137i 0.411476i
\(757\) − 25.4558i − 0.925208i −0.886565 0.462604i \(-0.846915\pi\)
0.886565 0.462604i \(-0.153085\pi\)
\(758\) −36.0000 −1.30758
\(759\) 24.0000 0.871145
\(760\) − 16.9706i − 0.615587i
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −4.00000 −0.144905
\(763\) 8.48528i 0.307188i
\(764\) 0 0
\(765\) − 8.48528i − 0.306786i
\(766\) 33.9411i 1.22634i
\(767\) −6.00000 −0.216647
\(768\) − 22.6274i − 0.816497i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) − 5.65685i − 0.203859i
\(771\) 8.48528i 0.305590i
\(772\) 8.00000 0.287926
\(773\) − 32.5269i − 1.16991i −0.811065 0.584956i \(-0.801112\pi\)
0.811065 0.584956i \(-0.198888\pi\)
\(774\) 12.0000 0.431331
\(775\) −12.0000 −0.431053
\(776\) − 28.2843i − 1.01535i
\(777\) −12.0000 −0.430498
\(778\) −4.00000 −0.143407
\(779\) 25.4558i 0.912050i
\(780\) − 16.9706i − 0.607644i
\(781\) 0 0
\(782\) − 50.9117i − 1.82060i
\(783\) −16.0000 −0.571793
\(784\) 4.00000 0.142857
\(785\) 18.0000 0.642448
\(786\) − 2.82843i − 0.100887i
\(787\) 38.1838i 1.36110i 0.732700 + 0.680552i \(0.238260\pi\)
−0.732700 + 0.680552i \(0.761740\pi\)
\(788\) − 11.3137i − 0.403034i
\(789\) − 33.9411i − 1.20834i
\(790\) 16.0000 0.569254
\(791\) −12.0000 −0.426671
\(792\) 8.00000 0.284268
\(793\) 54.0000 1.91760
\(794\) −30.0000 −1.06466
\(795\) 11.3137i 0.401256i
\(796\) −40.0000 −1.41776
\(797\) 26.8701i 0.951786i 0.879503 + 0.475893i \(0.157875\pi\)
−0.879503 + 0.475893i \(0.842125\pi\)
\(798\) − 8.48528i − 0.300376i
\(799\) 0 0
\(800\) − 16.9706i − 0.600000i
\(801\) 6.00000 0.212000
\(802\) 33.9411i 1.19850i
\(803\) − 5.65685i − 0.199626i
\(804\) 0 0
\(805\) − 8.48528i − 0.299067i
\(806\) −24.0000 −0.845364
\(807\) −10.0000 −0.352017
\(808\) −28.0000 −0.985037
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) −10.0000 −0.351364
\(811\) − 12.7279i − 0.446938i −0.974711 0.223469i \(-0.928262\pi\)
0.974711 0.223469i \(-0.0717381\pi\)
\(812\) 5.65685i 0.198517i
\(813\) − 28.2843i − 0.991973i
\(814\) 33.9411i 1.18964i
\(815\) 12.0000 0.420342
\(816\) 33.9411i 1.18818i
\(817\) −36.0000 −1.25948
\(818\) 31.1127i 1.08783i
\(819\) 4.24264i 0.148250i
\(820\) − 16.9706i − 0.592638i
\(821\) 22.6274i 0.789702i 0.918745 + 0.394851i \(0.129204\pi\)
−0.918745 + 0.394851i \(0.870796\pi\)
\(822\) −12.0000 −0.418548
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) − 11.3137i − 0.394132i
\(825\) −12.0000 −0.417786
\(826\) 2.00000 0.0695889
\(827\) − 45.2548i − 1.57366i −0.617167 0.786832i \(-0.711720\pi\)
0.617167 0.786832i \(-0.288280\pi\)
\(828\) 12.0000 0.417029
\(829\) 21.2132i 0.736765i 0.929674 + 0.368383i \(0.120088\pi\)
−0.929674 + 0.368383i \(0.879912\pi\)
\(830\) − 31.1127i − 1.07994i
\(831\) 24.0000 0.832551
\(832\) − 33.9411i − 1.17670i
\(833\) −6.00000 −0.207888
\(834\) − 8.48528i − 0.293821i
\(835\) 33.9411i 1.17458i
\(836\) −24.0000 −0.830057
\(837\) 22.6274i 0.782118i
\(838\) 38.0000 1.31269
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 5.65685i 0.195180i
\(841\) 21.0000 0.724138
\(842\) −24.0000 −0.827095
\(843\) 8.48528i 0.292249i
\(844\) − 33.9411i − 1.16830i
\(845\) − 7.07107i − 0.243252i
\(846\) 0 0
\(847\) 3.00000 0.103081
\(848\) 22.6274i 0.777029i
\(849\) −18.0000 −0.617758
\(850\) 25.4558i 0.873128i
\(851\) 50.9117i 1.74523i
\(852\) 0 0
\(853\) − 4.24264i − 0.145265i −0.997359 0.0726326i \(-0.976860\pi\)
0.997359 0.0726326i \(-0.0231401\pi\)
\(854\) −18.0000 −0.615947
\(855\) −6.00000 −0.205196
\(856\) −16.0000 −0.546869
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −24.0000 −0.819346
\(859\) 46.6690i 1.59233i 0.605081 + 0.796164i \(0.293141\pi\)
−0.605081 + 0.796164i \(0.706859\pi\)
\(860\) 24.0000 0.818393
\(861\) − 8.48528i − 0.289178i
\(862\) 8.48528i 0.289010i
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −32.0000 −1.08866
\(865\) −14.0000 −0.476014
\(866\) − 2.82843i − 0.0961139i
\(867\) − 26.8701i − 0.912555i
\(868\) 8.00000 0.271538
\(869\) − 22.6274i − 0.767583i
\(870\) −8.00000 −0.271225
\(871\) 0 0
\(872\) −24.0000 −0.812743
\(873\) −10.0000 −0.338449
\(874\) −36.0000 −1.21772
\(875\) 11.3137i 0.382473i
\(876\) 5.65685i 0.191127i
\(877\) − 42.4264i − 1.43264i −0.697773 0.716319i \(-0.745826\pi\)
0.697773 0.716319i \(-0.254174\pi\)
\(878\) 39.5980i 1.33637i
\(879\) −34.0000 −1.14679
\(880\) 16.0000 0.539360
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) − 1.41421i − 0.0476190i
\(883\) 50.9117i 1.71331i 0.515886 + 0.856657i \(0.327463\pi\)
−0.515886 + 0.856657i \(0.672537\pi\)
\(884\) 50.9117i 1.71235i
\(885\) 2.82843i 0.0950765i
\(886\) 32.0000 1.07506
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) − 33.9411i − 1.13899i
\(889\) 2.00000 0.0670778
\(890\) 12.0000 0.402241
\(891\) 14.1421i 0.473779i
\(892\) 56.0000 1.87502
\(893\) 0 0
\(894\) 22.6274i 0.756774i
\(895\) −8.00000 −0.267411
\(896\) 11.3137i 0.377964i
\(897\) −36.0000 −1.20201
\(898\) 25.4558i 0.849473i
\(899\) 11.3137i 0.377333i
\(900\) −6.00000 −0.200000
\(901\) − 33.9411i − 1.13074i
\(902\) −24.0000 −0.799113
\(903\) 12.0000 0.399335
\(904\) − 33.9411i − 1.12887i
\(905\) 18.0000 0.598340
\(906\) 20.0000 0.664455
\(907\) − 33.9411i − 1.12700i −0.826117 0.563498i \(-0.809455\pi\)
0.826117 0.563498i \(-0.190545\pi\)
\(908\) − 19.7990i − 0.657053i
\(909\) 9.89949i 0.328346i
\(910\) 8.48528i 0.281284i
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 24.0000 0.794719
\(913\) −44.0000 −1.45619
\(914\) 39.5980i 1.30978i
\(915\) − 25.4558i − 0.841544i
\(916\) 8.48528i 0.280362i
\(917\) 1.41421i 0.0467014i
\(918\) 48.0000 1.58424
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 24.0000 0.791257
\(921\) 18.0000 0.593120
\(922\) −22.0000 −0.724531
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) − 25.4558i − 0.836983i
\(926\) − 45.2548i − 1.48717i
\(927\) −4.00000 −0.131377
\(928\) −16.0000 −0.525226
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 11.3137i 0.370991i
\(931\) 4.24264i 0.139047i
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) −10.0000 −0.327210
\(935\) −24.0000 −0.784884
\(936\) −12.0000 −0.392232
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 14.1421i 0.461511i
\(940\) 0 0
\(941\) 52.3259i 1.70578i 0.522094 + 0.852888i \(0.325151\pi\)
−0.522094 + 0.852888i \(0.674849\pi\)
\(942\) 25.4558i 0.829396i
\(943\) −36.0000 −1.17232
\(944\) 5.65685i 0.184115i
\(945\) 8.00000 0.260240
\(946\) − 33.9411i − 1.10352i
\(947\) − 2.82843i − 0.0919115i −0.998943 0.0459558i \(-0.985367\pi\)
0.998943 0.0459558i \(-0.0146333\pi\)
\(948\) 22.6274i 0.734904i
\(949\) 8.48528i 0.275444i
\(950\) 18.0000 0.583997
\(951\) 32.0000 1.03767
\(952\) − 16.9706i − 0.550019i
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 8.00000 0.259010
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 11.3137i 0.365720i
\(958\) 16.9706i 0.548294i
\(959\) 6.00000 0.193750
\(960\) −16.0000 −0.516398
\(961\) −15.0000 −0.483871
\(962\) − 50.9117i − 1.64146i
\(963\) 5.65685i 0.182290i
\(964\) 20.0000 0.644157
\(965\) − 5.65685i − 0.182101i
\(966\) 12.0000 0.386094
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 8.48528i 0.272727i
\(969\) −36.0000 −1.15649
\(970\) −20.0000 −0.642161
\(971\) − 32.5269i − 1.04384i −0.852995 0.521919i \(-0.825216\pi\)
0.852995 0.521919i \(-0.174784\pi\)
\(972\) 19.7990i 0.635053i
\(973\) 4.24264i 0.136013i
\(974\) − 2.82843i − 0.0906287i
\(975\) 18.0000 0.576461
\(976\) − 50.9117i − 1.62964i
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 16.9706i 0.542659i
\(979\) − 16.9706i − 0.542382i
\(980\) − 2.82843i − 0.0903508i
\(981\) 8.48528i 0.270914i
\(982\) 56.0000 1.78703
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 24.0000 0.765092
\(985\) −8.00000 −0.254901
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) 36.0000 1.14531
\(989\) − 50.9117i − 1.61890i
\(990\) − 5.65685i − 0.179787i
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 22.6274i 0.718421i
\(993\) −36.0000 −1.14243
\(994\) 0 0
\(995\) 28.2843i 0.896672i
\(996\) 44.0000 1.39419
\(997\) − 21.2132i − 0.671829i −0.941893 0.335914i \(-0.890955\pi\)
0.941893 0.335914i \(-0.109045\pi\)
\(998\) −24.0000 −0.759707
\(999\) −48.0000 −1.51865
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 56.2.b.a.29.1 2
3.2 odd 2 504.2.c.a.253.2 2
4.3 odd 2 224.2.b.a.113.2 2
7.2 even 3 392.2.p.a.165.1 4
7.3 odd 6 392.2.p.b.373.2 4
7.4 even 3 392.2.p.a.373.2 4
7.5 odd 6 392.2.p.b.165.1 4
7.6 odd 2 392.2.b.b.197.1 2
8.3 odd 2 224.2.b.a.113.1 2
8.5 even 2 inner 56.2.b.a.29.2 yes 2
12.11 even 2 2016.2.c.a.1009.1 2
16.3 odd 4 1792.2.a.p.1.1 2
16.5 even 4 1792.2.a.n.1.1 2
16.11 odd 4 1792.2.a.p.1.2 2
16.13 even 4 1792.2.a.n.1.2 2
24.5 odd 2 504.2.c.a.253.1 2
24.11 even 2 2016.2.c.a.1009.2 2
28.3 even 6 1568.2.t.b.177.1 4
28.11 odd 6 1568.2.t.c.177.2 4
28.19 even 6 1568.2.t.b.753.2 4
28.23 odd 6 1568.2.t.c.753.1 4
28.27 even 2 1568.2.b.a.785.1 2
56.3 even 6 1568.2.t.b.177.2 4
56.5 odd 6 392.2.p.b.165.2 4
56.11 odd 6 1568.2.t.c.177.1 4
56.13 odd 2 392.2.b.b.197.2 2
56.19 even 6 1568.2.t.b.753.1 4
56.27 even 2 1568.2.b.a.785.2 2
56.37 even 6 392.2.p.a.165.2 4
56.45 odd 6 392.2.p.b.373.1 4
56.51 odd 6 1568.2.t.c.753.2 4
56.53 even 6 392.2.p.a.373.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.b.a.29.1 2 1.1 even 1 trivial
56.2.b.a.29.2 yes 2 8.5 even 2 inner
224.2.b.a.113.1 2 8.3 odd 2
224.2.b.a.113.2 2 4.3 odd 2
392.2.b.b.197.1 2 7.6 odd 2
392.2.b.b.197.2 2 56.13 odd 2
392.2.p.a.165.1 4 7.2 even 3
392.2.p.a.165.2 4 56.37 even 6
392.2.p.a.373.1 4 56.53 even 6
392.2.p.a.373.2 4 7.4 even 3
392.2.p.b.165.1 4 7.5 odd 6
392.2.p.b.165.2 4 56.5 odd 6
392.2.p.b.373.1 4 56.45 odd 6
392.2.p.b.373.2 4 7.3 odd 6
504.2.c.a.253.1 2 24.5 odd 2
504.2.c.a.253.2 2 3.2 odd 2
1568.2.b.a.785.1 2 28.27 even 2
1568.2.b.a.785.2 2 56.27 even 2
1568.2.t.b.177.1 4 28.3 even 6
1568.2.t.b.177.2 4 56.3 even 6
1568.2.t.b.753.1 4 56.19 even 6
1568.2.t.b.753.2 4 28.19 even 6
1568.2.t.c.177.1 4 56.11 odd 6
1568.2.t.c.177.2 4 28.11 odd 6
1568.2.t.c.753.1 4 28.23 odd 6
1568.2.t.c.753.2 4 56.51 odd 6
1792.2.a.n.1.1 2 16.5 even 4
1792.2.a.n.1.2 2 16.13 even 4
1792.2.a.p.1.1 2 16.3 odd 4
1792.2.a.p.1.2 2 16.11 odd 4
2016.2.c.a.1009.1 2 12.11 even 2
2016.2.c.a.1009.2 2 24.11 even 2