Properties

Label 275.2.b.a.199.2
Level $275$
Weight $2$
Character 275.199
Analytic conductor $2.196$
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] = [275,2,Mod(199,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("275.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.2.b.a.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +2.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +2.00000i q^{7} +2.00000 q^{9} +1.00000 q^{11} +2.00000i q^{12} +4.00000i q^{13} -4.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} +4.00000i q^{18} +2.00000 q^{21} +2.00000i q^{22} -1.00000i q^{23} -8.00000 q^{26} -5.00000i q^{27} -4.00000i q^{28} +7.00000 q^{31} -8.00000i q^{32} -1.00000i q^{33} -4.00000 q^{34} -4.00000 q^{36} -3.00000i q^{37} +4.00000 q^{39} -8.00000 q^{41} +4.00000i q^{42} -6.00000i q^{43} -2.00000 q^{44} +2.00000 q^{46} -8.00000i q^{47} +4.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} -8.00000i q^{52} -6.00000i q^{53} +10.0000 q^{54} -5.00000 q^{59} +12.0000 q^{61} +14.0000i q^{62} +4.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} +7.00000i q^{67} -4.00000i q^{68} -1.00000 q^{69} -3.00000 q^{71} +4.00000i q^{73} +6.00000 q^{74} +2.00000i q^{77} +8.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -16.0000i q^{82} -6.00000i q^{83} -4.00000 q^{84} +12.0000 q^{86} -15.0000 q^{89} -8.00000 q^{91} +2.00000i q^{92} -7.00000i q^{93} +16.0000 q^{94} -8.00000 q^{96} +7.00000i q^{97} +6.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 4 q^{6} + 4 q^{9} + 2 q^{11} - 8 q^{14} - 8 q^{16} + 4 q^{21} - 16 q^{26} + 14 q^{31} - 8 q^{34} - 8 q^{36} + 8 q^{39} - 16 q^{41} - 4 q^{44} + 4 q^{46} + 6 q^{49} + 4 q^{51} + 20 q^{54} - 10 q^{59} + 24 q^{61} + 16 q^{64} + 4 q^{66} - 2 q^{69} - 6 q^{71} + 12 q^{74} + 20 q^{79} + 2 q^{81} - 8 q^{84} + 24 q^{86} - 30 q^{89} - 16 q^{91} + 32 q^{94} - 16 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000i 0.577350i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 4.00000i 0.942809i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000i 0.426401i
\(23\) − 1.00000i − 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.00000 −1.56893
\(27\) − 5.00000i − 0.962250i
\(28\) − 4.00000i − 0.755929i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) − 1.00000i − 0.174078i
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) −4.00000 −0.666667
\(37\) − 3.00000i − 0.493197i −0.969118 0.246598i \(-0.920687\pi\)
0.969118 0.246598i \(-0.0793129\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 4.00000i 0.617213i
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 4.00000i 0.577350i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) − 8.00000i − 1.10940i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 10.0000 1.36083
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 14.0000i 1.77800i
\(63\) 4.00000i 0.503953i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 8.00000i 0.905822i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 16.0000i − 1.76690i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 2.00000i 0.208514i
\(93\) − 7.00000i − 0.725866i
\(94\) 16.0000 1.65027
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 7.00000i 0.710742i 0.934725 + 0.355371i \(0.115646\pi\)
−0.934725 + 0.355371i \(0.884354\pi\)
\(98\) 6.00000i 0.606092i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 4.00000i 0.396059i
\(103\) − 16.0000i − 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) − 18.0000i − 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) 10.0000i 0.962250i
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) − 8.00000i − 0.755929i
\(113\) 9.00000i 0.846649i 0.905978 + 0.423324i \(0.139137\pi\)
−0.905978 + 0.423324i \(0.860863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.00000i 0.739600i
\(118\) − 10.0000i − 0.920575i
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 24.0000i 2.17286i
\(123\) 8.00000i 0.721336i
\(124\) −14.0000 −1.25724
\(125\) 0 0
\(126\) −8.00000 −0.712697
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 0 0
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) 7.00000i 0.598050i 0.954245 + 0.299025i \(0.0966615\pi\)
−0.954245 + 0.299025i \(0.903339\pi\)
\(138\) − 2.00000i − 0.170251i
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) − 6.00000i − 0.503509i
\(143\) 4.00000i 0.334497i
\(144\) −8.00000 −0.666667
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) − 3.00000i − 0.247436i
\(148\) 6.00000i 0.493197i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 4.00000i 0.323381i
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) −8.00000 −0.640513
\(157\) 7.00000i 0.558661i 0.960195 + 0.279330i \(0.0901125\pi\)
−0.960195 + 0.279330i \(0.909888\pi\)
\(158\) 20.0000i 1.59111i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 2.00000i 0.157135i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 16.0000 1.24939
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 12.0000i 0.914991i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 5.00000i 0.375823i
\(178\) − 30.0000i − 2.24860i
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) − 16.0000i − 1.18600i
\(183\) − 12.0000i − 0.887066i
\(184\) 0 0
\(185\) 0 0
\(186\) 14.0000 1.02653
\(187\) 2.00000i 0.146254i
\(188\) 16.0000i 1.16692i
\(189\) 10.0000 0.727393
\(190\) 0 0
\(191\) 17.0000 1.23008 0.615038 0.788497i \(-0.289140\pi\)
0.615038 + 0.788497i \(0.289140\pi\)
\(192\) − 8.00000i − 0.577350i
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 4.00000i 0.281439i
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 32.0000 2.22955
\(207\) − 2.00000i − 0.139010i
\(208\) − 16.0000i − 1.10940i
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 3.00000i 0.205557i
\(214\) 36.0000 2.46091
\(215\) 0 0
\(216\) 0 0
\(217\) 14.0000i 0.950382i
\(218\) − 20.0000i − 1.35457i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) − 6.00000i − 0.402694i
\(223\) 19.0000i 1.27233i 0.771551 + 0.636167i \(0.219481\pi\)
−0.771551 + 0.636167i \(0.780519\pi\)
\(224\) 16.0000 1.06904
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) −16.0000 −1.04595
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) − 10.0000i − 0.649570i
\(238\) − 8.00000i − 0.518563i
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 2.00000i 0.128565i
\(243\) − 16.0000i − 1.02640i
\(244\) −24.0000 −1.53644
\(245\) 0 0
\(246\) −16.0000 −1.02012
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −23.0000 −1.45175 −0.725874 0.687828i \(-0.758564\pi\)
−0.725874 + 0.687828i \(0.758564\pi\)
\(252\) − 8.00000i − 0.503953i
\(253\) − 1.00000i − 0.0628695i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) − 12.0000i − 0.747087i
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) − 36.0000i − 2.22409i
\(263\) 14.0000i 0.863277i 0.902047 + 0.431638i \(0.142064\pi\)
−0.902047 + 0.431638i \(0.857936\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.0000i 0.917985i
\(268\) − 14.0000i − 0.855186i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) − 8.00000i − 0.485071i
\(273\) 8.00000i 0.484182i
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 2.00000 0.120386
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 20.0000i − 1.19952i
\(279\) 14.0000 0.838158
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) − 16.0000i − 0.952786i
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) − 16.0000i − 0.944450i
\(288\) − 16.0000i − 0.942809i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 7.00000 0.410347
\(292\) − 8.00000i − 0.468165i
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 0 0
\(297\) − 5.00000i − 0.290129i
\(298\) 20.0000i 1.15857i
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 4.00000i 0.230174i
\(303\) − 2.00000i − 0.114897i
\(304\) 0 0
\(305\) 0 0
\(306\) −8.00000 −0.457330
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) − 1.00000i − 0.0565233i −0.999601 0.0282617i \(-0.991003\pi\)
0.999601 0.0282617i \(-0.00899717\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −20.0000 −1.12509
\(317\) − 13.0000i − 0.730153i −0.930978 0.365076i \(-0.881043\pi\)
0.930978 0.365076i \(-0.118957\pi\)
\(318\) − 12.0000i − 0.672927i
\(319\) 0 0
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 4.00000i 0.222911i
\(323\) 0 0
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 6.00000i − 0.328798i
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) −8.00000 −0.436436
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) − 6.00000i − 0.326357i
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) 7.00000 0.379071
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) − 28.0000i − 1.50312i −0.659665 0.751559i \(-0.729302\pi\)
0.659665 0.751559i \(-0.270698\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 20.0000 1.06752
\(352\) − 8.00000i − 0.426401i
\(353\) − 21.0000i − 1.11772i −0.829263 0.558859i \(-0.811239\pi\)
0.829263 0.558859i \(-0.188761\pi\)
\(354\) −10.0000 −0.531494
\(355\) 0 0
\(356\) 30.0000 1.59000
\(357\) 4.00000i 0.211702i
\(358\) 30.0000i 1.58555i
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 14.0000i 0.735824i
\(363\) − 1.00000i − 0.0524864i
\(364\) 16.0000 0.838628
\(365\) 0 0
\(366\) 24.0000 1.25450
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −16.0000 −0.832927
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 14.0000i 0.725866i
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 20.0000i 1.02869i
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 34.0000i 1.73959i
\(383\) − 1.00000i − 0.0510976i −0.999674 0.0255488i \(-0.991867\pi\)
0.999674 0.0255488i \(-0.00813332\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) − 12.0000i − 0.609994i
\(388\) − 14.0000i − 0.710742i
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) −4.00000 −0.201517
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 14.0000i 0.698257i
\(403\) 28.0000i 1.39478i
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.00000i − 0.148704i
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) 7.00000 0.345285
\(412\) 32.0000i 1.57653i
\(413\) − 10.0000i − 0.492068i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 32.0000 1.56893
\(417\) 10.0000i 0.489702i
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 24.0000i 1.16830i
\(423\) − 16.0000i − 0.777947i
\(424\) 0 0
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 24.0000i 1.16144i
\(428\) 36.0000i 1.74013i
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 20.0000i 0.962250i
\(433\) − 11.0000i − 0.528626i −0.964437 0.264313i \(-0.914855\pi\)
0.964437 0.264313i \(-0.0851452\pi\)
\(434\) −28.0000 −1.34404
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) 0 0
\(438\) 8.00000i 0.382255i
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) − 16.0000i − 0.761042i
\(443\) − 11.0000i − 0.522626i −0.965254 0.261313i \(-0.915845\pi\)
0.965254 0.261313i \(-0.0841554\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −38.0000 −1.79935
\(447\) − 10.0000i − 0.472984i
\(448\) 16.0000i 0.755929i
\(449\) −35.0000 −1.65175 −0.825876 0.563852i \(-0.809319\pi\)
−0.825876 + 0.563852i \(0.809319\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) − 18.0000i − 0.846649i
\(453\) − 2.00000i − 0.0939682i
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) 12.0000i 0.561336i 0.959805 + 0.280668i \(0.0905560\pi\)
−0.959805 + 0.280668i \(0.909444\pi\)
\(458\) − 30.0000i − 1.40181i
\(459\) 10.0000 0.466760
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 4.00000i 0.186097i
\(463\) − 11.0000i − 0.511213i −0.966781 0.255607i \(-0.917725\pi\)
0.966781 0.255607i \(-0.0822752\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −48.0000 −2.22356
\(467\) 27.0000i 1.24941i 0.780860 + 0.624705i \(0.214781\pi\)
−0.780860 + 0.624705i \(0.785219\pi\)
\(468\) − 16.0000i − 0.739600i
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) 7.00000 0.322543
\(472\) 0 0
\(473\) − 6.00000i − 0.275880i
\(474\) 20.0000 0.918630
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) − 12.0000i − 0.549442i
\(478\) 60.0000i 2.74434i
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) − 16.0000i − 0.728780i
\(483\) − 2.00000i − 0.0910032i
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 32.0000 1.45155
\(487\) − 23.0000i − 1.04223i −0.853487 0.521115i \(-0.825516\pi\)
0.853487 0.521115i \(-0.174484\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) − 16.0000i − 0.721336i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −28.0000 −1.25724
\(497\) − 6.00000i − 0.269137i
\(498\) − 12.0000i − 0.537733i
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) − 46.0000i − 2.05308i
\(503\) − 26.0000i − 1.15928i −0.814872 0.579641i \(-0.803193\pi\)
0.814872 0.579641i \(-0.196807\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 3.00000i 0.133235i
\(508\) 16.0000i 0.709885i
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 32.0000i 1.41421i
\(513\) 0 0
\(514\) −4.00000 −0.176432
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) − 8.00000i − 0.351840i
\(518\) 12.0000i 0.527250i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 36.0000 1.57267
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 14.0000i 0.609850i
\(528\) 4.00000i 0.174078i
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) − 32.0000i − 1.38607i
\(534\) −30.0000 −1.29823
\(535\) 0 0
\(536\) 0 0
\(537\) − 15.0000i − 0.647298i
\(538\) − 20.0000i − 0.862261i
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) − 56.0000i − 2.40541i
\(543\) − 7.00000i − 0.300399i
\(544\) 16.0000 0.685994
\(545\) 0 0
\(546\) −16.0000 −0.684737
\(547\) − 8.00000i − 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) − 14.0000i − 0.598050i
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 20.0000i 0.850487i
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 28.0000i 1.18533i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) − 36.0000i − 1.51857i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 16.0000 0.673722
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) − 17.0000i − 0.710185i
\(574\) 32.0000 1.33565
\(575\) 0 0
\(576\) 16.0000 0.666667
\(577\) − 33.0000i − 1.37381i −0.726748 0.686904i \(-0.758969\pi\)
0.726748 0.686904i \(-0.241031\pi\)
\(578\) 26.0000i 1.08146i
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 14.0000i 0.580319i
\(583\) − 6.00000i − 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 6.00000i 0.247436i
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 12.0000i 0.493197i
\(593\) 44.0000i 1.80686i 0.428732 + 0.903432i \(0.358960\pi\)
−0.428732 + 0.903432i \(0.641040\pi\)
\(594\) 10.0000 0.410305
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) 8.00000i 0.327144i
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 24.0000i 0.978167i
\(603\) 14.0000i 0.570124i
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 4.00000 0.162489
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) − 8.00000i − 0.323381i
\(613\) − 16.0000i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) − 32.0000i − 1.28723i
\(619\) 25.0000 1.00483 0.502417 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 24.0000i 0.962312i
\(623\) − 30.0000i − 1.20192i
\(624\) −16.0000 −0.640513
\(625\) 0 0
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) − 14.0000i − 0.558661i
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) − 12.0000i − 0.476957i
\(634\) 26.0000 1.03259
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 12.0000i 0.475457i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) − 36.0000i − 1.42081i
\(643\) 29.0000i 1.14365i 0.820376 + 0.571824i \(0.193764\pi\)
−0.820376 + 0.571824i \(0.806236\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 7.00000i 0.275198i 0.990488 + 0.137599i \(0.0439386\pi\)
−0.990488 + 0.137599i \(0.956061\pi\)
\(648\) 0 0
\(649\) −5.00000 −0.196267
\(650\) 0 0
\(651\) 14.0000 0.548703
\(652\) − 8.00000i − 0.313304i
\(653\) − 41.0000i − 1.60445i −0.597019 0.802227i \(-0.703648\pi\)
0.597019 0.802227i \(-0.296352\pi\)
\(654\) −20.0000 −0.782062
\(655\) 0 0
\(656\) 32.0000 1.24939
\(657\) 8.00000i 0.312110i
\(658\) 32.0000i 1.24749i
\(659\) −10.0000 −0.389545 −0.194772 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 0 0
\(661\) 37.0000 1.43913 0.719567 0.694423i \(-0.244340\pi\)
0.719567 + 0.694423i \(0.244340\pi\)
\(662\) 14.0000i 0.544125i
\(663\) 8.00000i 0.310694i
\(664\) 0 0
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) 0 0
\(668\) − 24.0000i − 0.928588i
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) − 16.0000i − 0.617213i
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) −44.0000 −1.69482
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 18.0000i 0.691286i
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 14.0000i 0.536088i
\(683\) − 16.0000i − 0.612223i −0.951996 0.306111i \(-0.900972\pi\)
0.951996 0.306111i \(-0.0990280\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −40.0000 −1.52721
\(687\) 15.0000i 0.572286i
\(688\) 24.0000i 0.914991i
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 12.0000i 0.456172i
\(693\) 4.00000i 0.151947i
\(694\) 56.0000 2.12573
\(695\) 0 0
\(696\) 0 0
\(697\) − 16.0000i − 0.606043i
\(698\) − 60.0000i − 2.27103i
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 40.0000i 1.50970i
\(703\) 0 0
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 42.0000 1.58069
\(707\) 4.00000i 0.150435i
\(708\) − 10.0000i − 0.375823i
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) − 7.00000i − 0.262152i
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −30.0000 −1.12115
\(717\) − 30.0000i − 1.12037i
\(718\) 40.0000i 1.49279i
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) − 38.0000i − 1.41421i
\(723\) 8.00000i 0.297523i
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) − 3.00000i − 0.111264i −0.998451 0.0556319i \(-0.982283\pi\)
0.998451 0.0556319i \(-0.0177173\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 24.0000i 0.887066i
\(733\) − 36.0000i − 1.32969i −0.746981 0.664845i \(-0.768498\pi\)
0.746981 0.664845i \(-0.231502\pi\)
\(734\) −34.0000 −1.25496
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 7.00000i 0.257848i
\(738\) − 32.0000i − 1.17794i
\(739\) −50.0000 −1.83928 −0.919640 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 24.0000i 0.881068i
\(743\) 4.00000i 0.146746i 0.997305 + 0.0733729i \(0.0233763\pi\)
−0.997305 + 0.0733729i \(0.976624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 52.0000 1.90386
\(747\) − 12.0000i − 0.439057i
\(748\) − 4.00000i − 0.146254i
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 32.0000i 1.16692i
\(753\) 23.0000i 0.838167i
\(754\) 0 0
\(755\) 0 0
\(756\) −20.0000 −0.727393
\(757\) 22.0000i 0.799604i 0.916602 + 0.399802i \(0.130921\pi\)
−0.916602 + 0.399802i \(0.869079\pi\)
\(758\) 10.0000i 0.363216i
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) − 16.0000i − 0.579619i
\(763\) − 20.0000i − 0.724049i
\(764\) −34.0000 −1.23008
\(765\) 0 0
\(766\) 2.00000 0.0722629
\(767\) − 20.0000i − 0.722158i
\(768\) − 16.0000i − 0.577350i
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) − 8.00000i − 0.287926i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 24.0000 0.862662
\(775\) 0 0
\(776\) 0 0
\(777\) − 6.00000i − 0.215249i
\(778\) 30.0000i 1.07555i
\(779\) 0 0
\(780\) 0 0
\(781\) −3.00000 −0.107348
\(782\) 4.00000i 0.143040i
\(783\) 0 0
\(784\) −12.0000 −0.428571
\(785\) 0 0
\(786\) −36.0000 −1.28408
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) − 4.00000i − 0.142494i
\(789\) 14.0000 0.498413
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 48.0000i 1.70453i
\(794\) −4.00000 −0.141955
\(795\) 0 0
\(796\) 0 0
\(797\) − 53.0000i − 1.87736i −0.344795 0.938678i \(-0.612051\pi\)
0.344795 0.938678i \(-0.387949\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 4.00000i 0.141245i
\(803\) 4.00000i 0.141157i
\(804\) −14.0000 −0.493742
\(805\) 0 0
\(806\) −56.0000 −1.97252
\(807\) 10.0000i 0.352017i
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) 28.0000i 0.982003i
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) 0 0
\(818\) 60.0000i 2.09785i
\(819\) −16.0000 −0.559085
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 14.0000i 0.488306i
\(823\) 39.0000i 1.35945i 0.733465 + 0.679727i \(0.237902\pi\)
−0.733465 + 0.679727i \(0.762098\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) 52.0000i 1.80822i 0.427303 + 0.904109i \(0.359464\pi\)
−0.427303 + 0.904109i \(0.640536\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 32.0000i 1.10940i
\(833\) 6.00000i 0.207888i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) − 35.0000i − 1.20978i
\(838\) − 40.0000i − 1.38178i
\(839\) 5.00000 0.172619 0.0863096 0.996268i \(-0.472493\pi\)
0.0863096 + 0.996268i \(0.472493\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 44.0000i 1.51634i
\(843\) 18.0000i 0.619953i
\(844\) −24.0000 −0.826114
\(845\) 0 0
\(846\) 32.0000 1.10018
\(847\) 2.00000i 0.0687208i
\(848\) 24.0000i 0.824163i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) − 6.00000i − 0.205557i
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) −48.0000 −1.64253
\(855\) 0 0
\(856\) 0 0
\(857\) − 8.00000i − 0.273275i −0.990621 0.136637i \(-0.956370\pi\)
0.990621 0.136637i \(-0.0436295\pi\)
\(858\) 8.00000i 0.273115i
\(859\) 15.0000 0.511793 0.255897 0.966704i \(-0.417629\pi\)
0.255897 + 0.966704i \(0.417629\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) − 36.0000i − 1.22616i
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) −40.0000 −1.36083
\(865\) 0 0
\(866\) 22.0000 0.747590
\(867\) − 13.0000i − 0.441503i
\(868\) − 28.0000i − 0.950382i
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) −28.0000 −0.948744
\(872\) 0 0
\(873\) 14.0000i 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) 12.0000i 0.405211i 0.979260 + 0.202606i \(0.0649409\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) − 80.0000i − 2.69987i
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −43.0000 −1.44871 −0.724353 0.689429i \(-0.757862\pi\)
−0.724353 + 0.689429i \(0.757862\pi\)
\(882\) 12.0000i 0.404061i
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) 22.0000 0.739104
\(887\) 22.0000i 0.738688i 0.929293 + 0.369344i \(0.120418\pi\)
−0.929293 + 0.369344i \(0.879582\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) − 38.0000i − 1.27233i
\(893\) 0 0
\(894\) 20.0000 0.668900
\(895\) 0 0
\(896\) 0 0
\(897\) − 4.00000i − 0.133556i
\(898\) − 70.0000i − 2.33593i
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 16.0000i − 0.532742i
\(903\) − 12.0000i − 0.399335i
\(904\) 0 0
\(905\) 0 0
\(906\) 4.00000 0.132891
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) 36.0000i 1.19470i
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) − 6.00000i − 0.198571i
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) 30.0000 0.991228
\(917\) − 36.0000i − 1.18882i
\(918\) 20.0000i 0.660098i
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 24.0000i 0.790398i
\(923\) − 12.0000i − 0.394985i
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) 22.0000 0.722965
\(927\) − 32.0000i − 1.05102i
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 48.0000i − 1.57229i
\(933\) − 12.0000i − 0.392862i
\(934\) −54.0000 −1.76693
\(935\) 0 0
\(936\) 0 0
\(937\) − 8.00000i − 0.261349i −0.991425 0.130674i \(-0.958286\pi\)
0.991425 0.130674i \(-0.0417142\pi\)
\(938\) − 28.0000i − 0.914232i
\(939\) −1.00000 −0.0326338
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 14.0000i 0.456145i
\(943\) 8.00000i 0.260516i
\(944\) 20.0000 0.650945
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 27.0000i 0.877382i 0.898638 + 0.438691i \(0.144558\pi\)
−0.898638 + 0.438691i \(0.855442\pi\)
\(948\) 20.0000i 0.649570i
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) −13.0000 −0.421554
\(952\) 0 0
\(953\) 34.0000i 1.10137i 0.834714 + 0.550684i \(0.185633\pi\)
−0.834714 + 0.550684i \(0.814367\pi\)
\(954\) 24.0000 0.777029
\(955\) 0 0
\(956\) −60.0000 −1.94054
\(957\) 0 0
\(958\) − 40.0000i − 1.29234i
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 24.0000i 0.773791i
\(963\) − 36.0000i − 1.16008i
\(964\) 16.0000 0.515325
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 47.0000 1.50830 0.754151 0.656701i \(-0.228049\pi\)
0.754151 + 0.656701i \(0.228049\pi\)
\(972\) 32.0000i 1.02640i
\(973\) − 20.0000i − 0.641171i
\(974\) 46.0000 1.47394
\(975\) 0 0
\(976\) −48.0000 −1.53644
\(977\) 27.0000i 0.863807i 0.901920 + 0.431903i \(0.142158\pi\)
−0.901920 + 0.431903i \(0.857842\pi\)
\(978\) 8.00000i 0.255812i
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) − 16.0000i − 0.510581i
\(983\) 39.0000i 1.24391i 0.783054 + 0.621953i \(0.213661\pi\)
−0.783054 + 0.621953i \(0.786339\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) − 56.0000i − 1.77800i
\(993\) − 7.00000i − 0.222138i
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) − 38.0000i − 1.20347i −0.798695 0.601736i \(-0.794476\pi\)
0.798695 0.601736i \(-0.205524\pi\)
\(998\) − 40.0000i − 1.26618i
\(999\) −15.0000 −0.474579
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 275.2.b.a.199.2 2
3.2 odd 2 2475.2.c.a.199.1 2
4.3 odd 2 4400.2.b.h.4049.2 2
5.2 odd 4 11.2.a.a.1.1 1
5.3 odd 4 275.2.a.b.1.1 1
5.4 even 2 inner 275.2.b.a.199.1 2
15.2 even 4 99.2.a.d.1.1 1
15.8 even 4 2475.2.a.a.1.1 1
15.14 odd 2 2475.2.c.a.199.2 2
20.3 even 4 4400.2.a.i.1.1 1
20.7 even 4 176.2.a.b.1.1 1
20.19 odd 2 4400.2.b.h.4049.1 2
35.2 odd 12 539.2.e.h.67.1 2
35.12 even 12 539.2.e.g.67.1 2
35.17 even 12 539.2.e.g.177.1 2
35.27 even 4 539.2.a.a.1.1 1
35.32 odd 12 539.2.e.h.177.1 2
40.27 even 4 704.2.a.c.1.1 1
40.37 odd 4 704.2.a.h.1.1 1
45.2 even 12 891.2.e.b.595.1 2
45.7 odd 12 891.2.e.k.595.1 2
45.22 odd 12 891.2.e.k.298.1 2
45.32 even 12 891.2.e.b.298.1 2
55.2 even 20 121.2.c.a.81.1 4
55.7 even 20 121.2.c.a.27.1 4
55.17 even 20 121.2.c.a.3.1 4
55.27 odd 20 121.2.c.e.3.1 4
55.32 even 4 121.2.a.d.1.1 1
55.37 odd 20 121.2.c.e.27.1 4
55.42 odd 20 121.2.c.e.81.1 4
55.43 even 4 3025.2.a.a.1.1 1
55.47 odd 20 121.2.c.e.9.1 4
55.52 even 20 121.2.c.a.9.1 4
60.47 odd 4 1584.2.a.g.1.1 1
65.12 odd 4 1859.2.a.b.1.1 1
80.27 even 4 2816.2.c.f.1409.1 2
80.37 odd 4 2816.2.c.j.1409.2 2
80.67 even 4 2816.2.c.f.1409.2 2
80.77 odd 4 2816.2.c.j.1409.1 2
85.67 odd 4 3179.2.a.a.1.1 1
95.37 even 4 3971.2.a.b.1.1 1
105.62 odd 4 4851.2.a.t.1.1 1
115.22 even 4 5819.2.a.a.1.1 1
120.77 even 4 6336.2.a.br.1.1 1
120.107 odd 4 6336.2.a.bu.1.1 1
140.27 odd 4 8624.2.a.j.1.1 1
145.57 odd 4 9251.2.a.d.1.1 1
165.32 odd 4 1089.2.a.b.1.1 1
220.87 odd 4 1936.2.a.i.1.1 1
385.307 odd 4 5929.2.a.h.1.1 1
440.197 even 4 7744.2.a.x.1.1 1
440.307 odd 4 7744.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.2.a.a.1.1 1 5.2 odd 4
99.2.a.d.1.1 1 15.2 even 4
121.2.a.d.1.1 1 55.32 even 4
121.2.c.a.3.1 4 55.17 even 20
121.2.c.a.9.1 4 55.52 even 20
121.2.c.a.27.1 4 55.7 even 20
121.2.c.a.81.1 4 55.2 even 20
121.2.c.e.3.1 4 55.27 odd 20
121.2.c.e.9.1 4 55.47 odd 20
121.2.c.e.27.1 4 55.37 odd 20
121.2.c.e.81.1 4 55.42 odd 20
176.2.a.b.1.1 1 20.7 even 4
275.2.a.b.1.1 1 5.3 odd 4
275.2.b.a.199.1 2 5.4 even 2 inner
275.2.b.a.199.2 2 1.1 even 1 trivial
539.2.a.a.1.1 1 35.27 even 4
539.2.e.g.67.1 2 35.12 even 12
539.2.e.g.177.1 2 35.17 even 12
539.2.e.h.67.1 2 35.2 odd 12
539.2.e.h.177.1 2 35.32 odd 12
704.2.a.c.1.1 1 40.27 even 4
704.2.a.h.1.1 1 40.37 odd 4
891.2.e.b.298.1 2 45.32 even 12
891.2.e.b.595.1 2 45.2 even 12
891.2.e.k.298.1 2 45.22 odd 12
891.2.e.k.595.1 2 45.7 odd 12
1089.2.a.b.1.1 1 165.32 odd 4
1584.2.a.g.1.1 1 60.47 odd 4
1859.2.a.b.1.1 1 65.12 odd 4
1936.2.a.i.1.1 1 220.87 odd 4
2475.2.a.a.1.1 1 15.8 even 4
2475.2.c.a.199.1 2 3.2 odd 2
2475.2.c.a.199.2 2 15.14 odd 2
2816.2.c.f.1409.1 2 80.27 even 4
2816.2.c.f.1409.2 2 80.67 even 4
2816.2.c.j.1409.1 2 80.77 odd 4
2816.2.c.j.1409.2 2 80.37 odd 4
3025.2.a.a.1.1 1 55.43 even 4
3179.2.a.a.1.1 1 85.67 odd 4
3971.2.a.b.1.1 1 95.37 even 4
4400.2.a.i.1.1 1 20.3 even 4
4400.2.b.h.4049.1 2 20.19 odd 2
4400.2.b.h.4049.2 2 4.3 odd 2
4851.2.a.t.1.1 1 105.62 odd 4
5819.2.a.a.1.1 1 115.22 even 4
5929.2.a.h.1.1 1 385.307 odd 4
6336.2.a.br.1.1 1 120.77 even 4
6336.2.a.bu.1.1 1 120.107 odd 4
7744.2.a.k.1.1 1 440.307 odd 4
7744.2.a.x.1.1 1 440.197 even 4
8624.2.a.j.1.1 1 140.27 odd 4
9251.2.a.d.1.1 1 145.57 odd 4