Properties

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -2.00000i q^{7} +2.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -4.00000i q^{13} -4.00000 q^{14} -4.00000 q^{16} -2.00000i q^{17} -4.00000i q^{18} +2.00000 q^{21} -2.00000i q^{22} +1.00000i q^{23} -8.00000 q^{26} +5.00000i q^{27} +4.00000i q^{28} +7.00000 q^{31} +8.00000i q^{32} +1.00000i q^{33} -4.00000 q^{34} -4.00000 q^{36} +3.00000i q^{37} +4.00000 q^{39} -8.00000 q^{41} -4.00000i q^{42} +6.00000i q^{43} -2.00000 q^{44} +2.00000 q^{46} +8.00000i q^{47} -4.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} +8.00000i q^{52} +6.00000i q^{53} +10.0000 q^{54} -5.00000 q^{59} +12.0000 q^{61} -14.0000i q^{62} -4.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} -7.00000i q^{67} +4.00000i q^{68} -1.00000 q^{69} -3.00000 q^{71} -4.00000i q^{73} +6.00000 q^{74} -2.00000i q^{77} -8.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +16.0000i q^{82} +6.00000i q^{83} -4.00000 q^{84} +12.0000 q^{86} -15.0000 q^{89} -8.00000 q^{91} -2.00000i q^{92} +7.00000i q^{93} +16.0000 q^{94} -8.00000 q^{96} -7.00000i q^{97} -6.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 4 q^{6} + 4 q^{9}+O(q^{10}) \) 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 \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 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 \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) − 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 \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) 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 \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 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 \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) − 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 \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 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 \(-0.859362\pi\)
0.903971 0.427593i \(-0.140638\pi\)
\(68\) 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 \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 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 \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) −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 \(-0.884354\pi\)
0.934725 0.355371i \(-0.115646\pi\)
\(98\) − 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 \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) − 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 \(-0.860863\pi\)
0.905978 0.423324i \(-0.139137\pi\)
\(114\) 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 \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 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 \(-0.903339\pi\)
0.954245 0.299025i \(-0.0966615\pi\)
\(138\) 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 \(-0.909888\pi\)
0.960195 0.279330i \(-0.0901125\pi\)
\(158\) − 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 \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 16.0000 1.24939
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 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 \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 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 \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) − 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 \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) 16.0000 1.06904
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 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 \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) −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 \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 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 \(-0.857936\pi\)
0.902047 0.431638i \(-0.142064\pi\)
\(264\) 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 \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 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 \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 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 \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) 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 \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) 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 \(0.00899717\pi\)
−0.999601 + 0.0282617i \(0.991003\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −20.0000 −1.12509
\(317\) 13.0000i 0.730153i 0.930978 + 0.365076i \(0.118957\pi\)
−0.930978 + 0.365076i \(0.881043\pi\)
\(318\) 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 \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 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 \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 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 \(0.188761\pi\)
−0.829263 + 0.558859i \(0.811239\pi\)
\(354\) −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 \(-0.853667\pi\)
0.896177 0.443696i \(-0.146333\pi\)
\(368\) − 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 \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) −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 \(0.00813332\pi\)
−0.999674 + 0.0255488i \(0.991867\pi\)
\(384\) 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 \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 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 \(0.0851452\pi\)
−0.964437 + 0.264313i \(0.914855\pi\)
\(434\) −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 \(0.0841554\pi\)
−0.965254 + 0.261313i \(0.915845\pi\)
\(444\) 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 \(-0.909444\pi\)
0.959805 0.280668i \(-0.0905560\pi\)
\(458\) 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 \(0.0822752\pi\)
−0.966781 + 0.255607i \(0.917725\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −48.0000 −2.22356
\(467\) − 27.0000i − 1.24941i −0.780860 0.624705i \(-0.785219\pi\)
0.780860 0.624705i \(-0.214781\pi\)
\(468\) 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 \(0.174484\pi\)
−0.853487 + 0.521115i \(0.825516\pi\)
\(488\) 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 \(0.196807\pi\)
−0.814872 + 0.579641i \(0.803193\pi\)
\(504\) 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 \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 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 \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 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 \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) − 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 \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 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 \(0.241031\pi\)
−0.726748 + 0.686904i \(0.758969\pi\)
\(578\) − 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 \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) − 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 \(-0.641040\pi\)
0.428732 0.903432i \(-0.358960\pi\)
\(594\) 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 \(-0.852679\pi\)
0.894795 0.446476i \(-0.147321\pi\)
\(608\) 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 \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 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 \(-0.806236\pi\)
0.820376 0.571824i \(-0.193764\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) − 7.00000i − 0.275198i −0.990488 0.137599i \(-0.956061\pi\)
0.990488 0.137599i \(-0.0439386\pi\)
\(648\) 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 \(0.296352\pi\)
−0.597019 + 0.802227i \(0.703648\pi\)
\(654\) −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 \(-0.913032\pi\)
0.962908 0.269830i \(-0.0869676\pi\)
\(674\) −44.0000 −1.69482
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) − 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 \(0.0990280\pi\)
−0.951996 + 0.306111i \(0.900972\pi\)
\(684\) 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 \(0.0177173\pi\)
−0.998451 + 0.0556319i \(0.982283\pi\)
\(728\) 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 \(0.231502\pi\)
−0.746981 + 0.664845i \(0.768498\pi\)
\(734\) −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 \(-0.976624\pi\)
0.997305 0.0733729i \(-0.0233763\pi\)
\(744\) 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 \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) − 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 \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 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 \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 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 \(0.387949\pi\)
−0.344795 + 0.938678i \(0.612051\pi\)
\(798\) 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 \(-0.762098\pi\)
0.733465 0.679727i \(-0.237902\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) − 52.0000i − 1.80822i −0.427303 0.904109i \(-0.640536\pi\)
0.427303 0.904109i \(-0.359464\pi\)
\(828\) − 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 \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) −48.0000 −1.64253
\(855\) 0 0
\(856\) 0 0
\(857\) 8.00000i 0.273275i 0.990621 + 0.136637i \(0.0436295\pi\)
−0.990621 + 0.136637i \(0.956370\pi\)
\(858\) − 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 \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) −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 \(-0.935059\pi\)
0.979260 0.202606i \(-0.0649409\pi\)
\(878\) 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 \(-0.978560\pi\)
0.997732 0.0673054i \(-0.0214402\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) 22.0000 0.739104
\(887\) − 22.0000i − 0.738688i −0.929293 0.369344i \(-0.879582\pi\)
0.929293 0.369344i \(-0.120418\pi\)
\(888\) 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 \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) − 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 \(0.0417142\pi\)
−0.991425 + 0.130674i \(0.958286\pi\)
\(938\) 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 \(-0.855442\pi\)
0.898638 0.438691i \(-0.144558\pi\)
\(948\) − 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 \(-0.814367\pi\)
0.834714 0.550684i \(-0.185633\pi\)
\(954\) 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 \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 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 \(-0.857842\pi\)
0.901920 0.431903i \(-0.142158\pi\)
\(978\) − 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 \(-0.786339\pi\)
0.783054 0.621953i \(-0.213661\pi\)
\(984\) 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 \(0.205524\pi\)
−0.798695 + 0.601736i \(0.794476\pi\)
\(998\) 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.199.1 2
3.2 odd 2 2475.2.c.a.199.2 2
4.3 odd 2 4400.2.b.h.4049.1 2
5.2 odd 4 275.2.a.b.1.1 1
5.3 odd 4 11.2.a.a.1.1 1
5.4 even 2 inner 275.2.b.a.199.2 2
15.2 even 4 2475.2.a.a.1.1 1
15.8 even 4 99.2.a.d.1.1 1
15.14 odd 2 2475.2.c.a.199.1 2
20.3 even 4 176.2.a.b.1.1 1
20.7 even 4 4400.2.a.i.1.1 1
20.19 odd 2 4400.2.b.h.4049.2 2
35.3 even 12 539.2.e.g.177.1 2
35.13 even 4 539.2.a.a.1.1 1
35.18 odd 12 539.2.e.h.177.1 2
35.23 odd 12 539.2.e.h.67.1 2
35.33 even 12 539.2.e.g.67.1 2
40.3 even 4 704.2.a.c.1.1 1
40.13 odd 4 704.2.a.h.1.1 1
45.13 odd 12 891.2.e.k.298.1 2
45.23 even 12 891.2.e.b.298.1 2
45.38 even 12 891.2.e.b.595.1 2
45.43 odd 12 891.2.e.k.595.1 2
55.3 odd 20 121.2.c.e.9.1 4
55.8 even 20 121.2.c.a.9.1 4
55.13 even 20 121.2.c.a.81.1 4
55.18 even 20 121.2.c.a.27.1 4
55.28 even 20 121.2.c.a.3.1 4
55.32 even 4 3025.2.a.a.1.1 1
55.38 odd 20 121.2.c.e.3.1 4
55.43 even 4 121.2.a.d.1.1 1
55.48 odd 20 121.2.c.e.27.1 4
55.53 odd 20 121.2.c.e.81.1 4
60.23 odd 4 1584.2.a.g.1.1 1
65.38 odd 4 1859.2.a.b.1.1 1
80.3 even 4 2816.2.c.f.1409.2 2
80.13 odd 4 2816.2.c.j.1409.1 2
80.43 even 4 2816.2.c.f.1409.1 2
80.53 odd 4 2816.2.c.j.1409.2 2
85.33 odd 4 3179.2.a.a.1.1 1
95.18 even 4 3971.2.a.b.1.1 1
105.83 odd 4 4851.2.a.t.1.1 1
115.68 even 4 5819.2.a.a.1.1 1
120.53 even 4 6336.2.a.br.1.1 1
120.83 odd 4 6336.2.a.bu.1.1 1
140.83 odd 4 8624.2.a.j.1.1 1
145.28 odd 4 9251.2.a.d.1.1 1
165.98 odd 4 1089.2.a.b.1.1 1
220.43 odd 4 1936.2.a.i.1.1 1
385.153 odd 4 5929.2.a.h.1.1 1
440.43 odd 4 7744.2.a.k.1.1 1
440.373 even 4 7744.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.2.a.a.1.1 1 5.3 odd 4
99.2.a.d.1.1 1 15.8 even 4
121.2.a.d.1.1 1 55.43 even 4
121.2.c.a.3.1 4 55.28 even 20
121.2.c.a.9.1 4 55.8 even 20
121.2.c.a.27.1 4 55.18 even 20
121.2.c.a.81.1 4 55.13 even 20
121.2.c.e.3.1 4 55.38 odd 20
121.2.c.e.9.1 4 55.3 odd 20
121.2.c.e.27.1 4 55.48 odd 20
121.2.c.e.81.1 4 55.53 odd 20
176.2.a.b.1.1 1 20.3 even 4
275.2.a.b.1.1 1 5.2 odd 4
275.2.b.a.199.1 2 1.1 even 1 trivial
275.2.b.a.199.2 2 5.4 even 2 inner
539.2.a.a.1.1 1 35.13 even 4
539.2.e.g.67.1 2 35.33 even 12
539.2.e.g.177.1 2 35.3 even 12
539.2.e.h.67.1 2 35.23 odd 12
539.2.e.h.177.1 2 35.18 odd 12
704.2.a.c.1.1 1 40.3 even 4
704.2.a.h.1.1 1 40.13 odd 4
891.2.e.b.298.1 2 45.23 even 12
891.2.e.b.595.1 2 45.38 even 12
891.2.e.k.298.1 2 45.13 odd 12
891.2.e.k.595.1 2 45.43 odd 12
1089.2.a.b.1.1 1 165.98 odd 4
1584.2.a.g.1.1 1 60.23 odd 4
1859.2.a.b.1.1 1 65.38 odd 4
1936.2.a.i.1.1 1 220.43 odd 4
2475.2.a.a.1.1 1 15.2 even 4
2475.2.c.a.199.1 2 15.14 odd 2
2475.2.c.a.199.2 2 3.2 odd 2
2816.2.c.f.1409.1 2 80.43 even 4
2816.2.c.f.1409.2 2 80.3 even 4
2816.2.c.j.1409.1 2 80.13 odd 4
2816.2.c.j.1409.2 2 80.53 odd 4
3025.2.a.a.1.1 1 55.32 even 4
3179.2.a.a.1.1 1 85.33 odd 4
3971.2.a.b.1.1 1 95.18 even 4
4400.2.a.i.1.1 1 20.7 even 4
4400.2.b.h.4049.1 2 4.3 odd 2
4400.2.b.h.4049.2 2 20.19 odd 2
4851.2.a.t.1.1 1 105.83 odd 4
5819.2.a.a.1.1 1 115.68 even 4
5929.2.a.h.1.1 1 385.153 odd 4
6336.2.a.br.1.1 1 120.53 even 4
6336.2.a.bu.1.1 1 120.83 odd 4
7744.2.a.k.1.1 1 440.43 odd 4
7744.2.a.x.1.1 1 440.373 even 4
8624.2.a.j.1.1 1 140.83 odd 4
9251.2.a.d.1.1 1 145.28 odd 4