Properties

Label 435.2.c.a.349.2
Level $435$
Weight $2$
Character 435.349
Analytic conductor $3.473$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(349,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.c (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: \(3.47349248793\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 435.349
Dual form 435.2.c.a.349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} -2.00000 q^{6} +2.00000i q^{7} -1.00000 q^{9} +(2.00000 - 4.00000i) q^{10} -3.00000 q^{11} -2.00000i q^{12} +4.00000i q^{13} -4.00000 q^{14} +(1.00000 - 2.00000i) q^{15} -4.00000 q^{16} -8.00000i q^{17} -2.00000i q^{18} +(4.00000 + 2.00000i) q^{20} -2.00000 q^{21} -6.00000i q^{22} -1.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -8.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -1.00000 q^{29} +(4.00000 + 2.00000i) q^{30} -8.00000 q^{31} -8.00000i q^{32} -3.00000i q^{33} +16.0000 q^{34} +(2.00000 - 4.00000i) q^{35} +2.00000 q^{36} +7.00000i q^{37} -4.00000 q^{39} +7.00000 q^{41} -4.00000i q^{42} +9.00000i q^{43} +6.00000 q^{44} +(2.00000 + 1.00000i) q^{45} +2.00000 q^{46} +12.0000i q^{47} -4.00000i q^{48} +3.00000 q^{49} +(-8.00000 + 6.00000i) q^{50} +8.00000 q^{51} -8.00000i q^{52} +9.00000i q^{53} +2.00000 q^{54} +(6.00000 + 3.00000i) q^{55} -2.00000i q^{58} -10.0000 q^{59} +(-2.00000 + 4.00000i) q^{60} +2.00000 q^{61} -16.0000i q^{62} -2.00000i q^{63} +8.00000 q^{64} +(4.00000 - 8.00000i) q^{65} +6.00000 q^{66} -8.00000i q^{67} +16.0000i q^{68} +1.00000 q^{69} +(8.00000 + 4.00000i) q^{70} -8.00000 q^{71} -1.00000i q^{73} -14.0000 q^{74} +(-4.00000 + 3.00000i) q^{75} -6.00000i q^{77} -8.00000i q^{78} +10.0000 q^{79} +(8.00000 + 4.00000i) q^{80} +1.00000 q^{81} +14.0000i q^{82} +9.00000i q^{83} +4.00000 q^{84} +(-8.00000 + 16.0000i) q^{85} -18.0000 q^{86} -1.00000i q^{87} -10.0000 q^{89} +(-2.00000 + 4.00000i) q^{90} -8.00000 q^{91} +2.00000i q^{92} -8.00000i q^{93} -24.0000 q^{94} +8.00000 q^{96} -13.0000i q^{97} +6.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 4 q^{5} - 4 q^{6} - 2 q^{9} + 4 q^{10} - 6 q^{11} - 8 q^{14} + 2 q^{15} - 8 q^{16} + 8 q^{20} - 4 q^{21} + 6 q^{25} - 16 q^{26} - 2 q^{29} + 8 q^{30} - 16 q^{31} + 32 q^{34} + 4 q^{35} + 4 q^{36}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/435\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(146\) \(262\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −2.00000 −1.00000
\(5\) −2.00000 1.00000i −0.894427 0.447214i
\(6\) −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
\(9\) −1.00000 −0.333333
\(10\) 2.00000 4.00000i 0.632456 1.26491i
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 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
\(15\) 1.00000 2.00000i 0.258199 0.516398i
\(16\) −4.00000 −1.00000
\(17\) 8.00000i 1.94029i −0.242536 0.970143i \(-0.577979\pi\)
0.242536 0.970143i \(-0.422021\pi\)
\(18\) 2.00000i 0.471405i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 4.00000 + 2.00000i 0.894427 + 0.447214i
\(21\) −2.00000 −0.436436
\(22\) 6.00000i 1.27920i
\(23\) 1.00000i 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) −8.00000 −1.56893
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) −1.00000 −0.185695
\(30\) 4.00000 + 2.00000i 0.730297 + 0.365148i
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 8.00000i 1.41421i
\(33\) 3.00000i 0.522233i
\(34\) 16.0000 2.74398
\(35\) 2.00000 4.00000i 0.338062 0.676123i
\(36\) 2.00000 0.333333
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 9.00000i 1.37249i 0.727372 + 0.686244i \(0.240742\pi\)
−0.727372 + 0.686244i \(0.759258\pi\)
\(44\) 6.00000 0.904534
\(45\) 2.00000 + 1.00000i 0.298142 + 0.149071i
\(46\) 2.00000 0.294884
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 4.00000i 0.577350i
\(49\) 3.00000 0.428571
\(50\) −8.00000 + 6.00000i −1.13137 + 0.848528i
\(51\) 8.00000 1.12022
\(52\) 8.00000i 1.10940i
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) 2.00000 0.272166
\(55\) 6.00000 + 3.00000i 0.809040 + 0.404520i
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000i 0.262613i
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −2.00000 + 4.00000i −0.258199 + 0.516398i
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 16.0000i 2.03200i
\(63\) 2.00000i 0.251976i
\(64\) 8.00000 1.00000
\(65\) 4.00000 8.00000i 0.496139 0.992278i
\(66\) 6.00000 0.738549
\(67\) 8.00000i 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 16.0000i 1.94029i
\(69\) 1.00000 0.120386
\(70\) 8.00000 + 4.00000i 0.956183 + 0.478091i
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 1.00000i 0.117041i −0.998286 0.0585206i \(-0.981362\pi\)
0.998286 0.0585206i \(-0.0186383\pi\)
\(74\) −14.0000 −1.62747
\(75\) −4.00000 + 3.00000i −0.461880 + 0.346410i
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 8.00000i 0.905822i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 8.00000 + 4.00000i 0.894427 + 0.447214i
\(81\) 1.00000 0.111111
\(82\) 14.0000i 1.54604i
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 4.00000 0.436436
\(85\) −8.00000 + 16.0000i −0.867722 + 1.73544i
\(86\) −18.0000 −1.94099
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 + 4.00000i −0.210819 + 0.421637i
\(91\) −8.00000 −0.838628
\(92\) 2.00000i 0.208514i
\(93\) 8.00000i 0.829561i
\(94\) −24.0000 −2.47541
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) 13.0000i 1.31995i −0.751288 0.659975i \(-0.770567\pi\)
0.751288 0.659975i \(-0.229433\pi\)
\(98\) 6.00000i 0.606092i
\(99\) 3.00000 0.301511
\(100\) −6.00000 8.00000i −0.600000 0.800000i
\(101\) 17.0000 1.69156 0.845782 0.533529i \(-0.179135\pi\)
0.845782 + 0.533529i \(0.179135\pi\)
\(102\) 16.0000i 1.58424i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 4.00000 + 2.00000i 0.390360 + 0.195180i
\(106\) −18.0000 −1.74831
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 2.00000i 0.192450i
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) −6.00000 + 12.0000i −0.572078 + 1.14416i
\(111\) −7.00000 −0.664411
\(112\) 8.00000i 0.755929i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −1.00000 + 2.00000i −0.0932505 + 0.186501i
\(116\) 2.00000 0.185695
\(117\) 4.00000i 0.369800i
\(118\) 20.0000i 1.84115i
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 4.00000i 0.362143i
\(123\) 7.00000i 0.631169i
\(124\) 16.0000 1.43684
\(125\) −2.00000 11.0000i −0.178885 0.983870i
\(126\) 4.00000 0.356348
\(127\) 3.00000i 0.266207i −0.991102 0.133103i \(-0.957506\pi\)
0.991102 0.133103i \(-0.0424943\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 16.0000 + 8.00000i 1.40329 + 0.701646i
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 6.00000i 0.522233i
\(133\) 0 0
\(134\) 16.0000 1.38219
\(135\) −1.00000 + 2.00000i −0.0860663 + 0.172133i
\(136\) 0 0
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 2.00000i 0.170251i
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) −4.00000 + 8.00000i −0.338062 + 0.676123i
\(141\) −12.0000 −1.01058
\(142\) 16.0000i 1.34269i
\(143\) 12.0000i 1.00349i
\(144\) 4.00000 0.333333
\(145\) 2.00000 + 1.00000i 0.166091 + 0.0830455i
\(146\) 2.00000 0.165521
\(147\) 3.00000i 0.247436i
\(148\) 14.0000i 1.15079i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −6.00000 8.00000i −0.489898 0.653197i
\(151\) −3.00000 −0.244137 −0.122068 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 0 0
\(153\) 8.00000i 0.646762i
\(154\) 12.0000 0.966988
\(155\) 16.0000 + 8.00000i 1.28515 + 0.642575i
\(156\) 8.00000 0.640513
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 20.0000i 1.59111i
\(159\) −9.00000 −0.713746
\(160\) −8.00000 + 16.0000i −0.632456 + 1.26491i
\(161\) 2.00000 0.157622
\(162\) 2.00000i 0.157135i
\(163\) 1.00000i 0.0783260i −0.999233 0.0391630i \(-0.987531\pi\)
0.999233 0.0391630i \(-0.0124692\pi\)
\(164\) −14.0000 −1.09322
\(165\) −3.00000 + 6.00000i −0.233550 + 0.467099i
\(166\) −18.0000 −1.39707
\(167\) 8.00000i 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) −32.0000 16.0000i −2.45429 1.22714i
\(171\) 0 0
\(172\) 18.0000i 1.37249i
\(173\) 9.00000i 0.684257i 0.939653 + 0.342129i \(0.111148\pi\)
−0.939653 + 0.342129i \(0.888852\pi\)
\(174\) 2.00000 0.151620
\(175\) −8.00000 + 6.00000i −0.604743 + 0.453557i
\(176\) 12.0000 0.904534
\(177\) 10.0000i 0.751646i
\(178\) 20.0000i 1.49906i
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) −4.00000 2.00000i −0.298142 0.149071i
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 16.0000i 1.18600i
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) 7.00000 14.0000i 0.514650 1.02930i
\(186\) 16.0000 1.17318
\(187\) 24.0000i 1.75505i
\(188\) 24.0000i 1.75038i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 8.00000i 0.577350i
\(193\) 6.00000i 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) 26.0000 1.86669
\(195\) 8.00000 + 4.00000i 0.572892 + 0.286446i
\(196\) −6.00000 −0.428571
\(197\) 13.0000i 0.926212i −0.886303 0.463106i \(-0.846735\pi\)
0.886303 0.463106i \(-0.153265\pi\)
\(198\) 6.00000i 0.426401i
\(199\) 25.0000 1.77220 0.886102 0.463491i \(-0.153403\pi\)
0.886102 + 0.463491i \(0.153403\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 34.0000i 2.39223i
\(203\) 2.00000i 0.140372i
\(204\) −16.0000 −1.12022
\(205\) −14.0000 7.00000i −0.977802 0.488901i
\(206\) −8.00000 −0.557386
\(207\) 1.00000i 0.0695048i
\(208\) 16.0000i 1.10940i
\(209\) 0 0
\(210\) −4.00000 + 8.00000i −0.276026 + 0.552052i
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 18.0000i 1.23625i
\(213\) 8.00000i 0.548151i
\(214\) −24.0000 −1.64061
\(215\) 9.00000 18.0000i 0.613795 1.22759i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 10.0000i 0.677285i
\(219\) 1.00000 0.0675737
\(220\) −12.0000 6.00000i −0.809040 0.404520i
\(221\) 32.0000 2.15255
\(222\) 14.0000i 0.939618i
\(223\) 6.00000i 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) 16.0000 1.06904
\(225\) −3.00000 4.00000i −0.200000 0.266667i
\(226\) −28.0000 −1.86253
\(227\) 3.00000i 0.199117i −0.995032 0.0995585i \(-0.968257\pi\)
0.995032 0.0995585i \(-0.0317430\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) −4.00000 2.00000i −0.263752 0.131876i
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) 1.00000i 0.0655122i −0.999463 0.0327561i \(-0.989572\pi\)
0.999463 0.0327561i \(-0.0104285\pi\)
\(234\) 8.00000 0.522976
\(235\) 12.0000 24.0000i 0.782794 1.56559i
\(236\) 20.0000 1.30189
\(237\) 10.0000i 0.649570i
\(238\) 32.0000i 2.07425i
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) −4.00000 + 8.00000i −0.258199 + 0.516398i
\(241\) −3.00000 −0.193247 −0.0966235 0.995321i \(-0.530804\pi\)
−0.0966235 + 0.995321i \(0.530804\pi\)
\(242\) 4.00000i 0.257130i
\(243\) 1.00000i 0.0641500i
\(244\) −4.00000 −0.256074
\(245\) −6.00000 3.00000i −0.383326 0.191663i
\(246\) −14.0000 −0.892607
\(247\) 0 0
\(248\) 0 0
\(249\) −9.00000 −0.570352
\(250\) 22.0000 4.00000i 1.39140 0.252982i
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 3.00000i 0.188608i
\(254\) 6.00000 0.376473
\(255\) −16.0000 8.00000i −1.00196 0.500979i
\(256\) 16.0000 1.00000
\(257\) 23.0000i 1.43470i −0.696713 0.717350i \(-0.745355\pi\)
0.696713 0.717350i \(-0.254645\pi\)
\(258\) 18.0000i 1.12063i
\(259\) −14.0000 −0.869918
\(260\) −8.00000 + 16.0000i −0.496139 + 0.992278i
\(261\) 1.00000 0.0618984
\(262\) 24.0000i 1.48272i
\(263\) 4.00000i 0.246651i 0.992366 + 0.123325i \(0.0393559\pi\)
−0.992366 + 0.123325i \(0.960644\pi\)
\(264\) 0 0
\(265\) 9.00000 18.0000i 0.552866 1.10573i
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) 16.0000i 0.977356i
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −4.00000 2.00000i −0.243432 0.121716i
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 32.0000i 1.94029i
\(273\) 8.00000i 0.484182i
\(274\) −4.00000 −0.241649
\(275\) −9.00000 12.0000i −0.542720 0.723627i
\(276\) −2.00000 −0.120386
\(277\) 28.0000i 1.68236i −0.540758 0.841178i \(-0.681862\pi\)
0.540758 0.841178i \(-0.318138\pi\)
\(278\) 10.0000i 0.599760i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 24.0000i 1.42918i
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 14.0000i 0.826394i
\(288\) 8.00000i 0.471405i
\(289\) −47.0000 −2.76471
\(290\) −2.00000 + 4.00000i −0.117444 + 0.234888i
\(291\) 13.0000 0.762073
\(292\) 2.00000i 0.117041i
\(293\) 6.00000i 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) −6.00000 −0.349927
\(295\) 20.0000 + 10.0000i 1.16445 + 0.582223i
\(296\) 0 0
\(297\) 3.00000i 0.174078i
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 8.00000 6.00000i 0.461880 0.346410i
\(301\) −18.0000 −1.03750
\(302\) 6.00000i 0.345261i
\(303\) 17.0000i 0.976624i
\(304\) 0 0
\(305\) −4.00000 2.00000i −0.229039 0.114520i
\(306\) −16.0000 −0.914659
\(307\) 17.0000i 0.970241i 0.874447 + 0.485121i \(0.161224\pi\)
−0.874447 + 0.485121i \(0.838776\pi\)
\(308\) 12.0000i 0.683763i
\(309\) −4.00000 −0.227552
\(310\) −16.0000 + 32.0000i −0.908739 + 1.81748i
\(311\) 17.0000 0.963982 0.481991 0.876176i \(-0.339914\pi\)
0.481991 + 0.876176i \(0.339914\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) −4.00000 −0.225733
\(315\) −2.00000 + 4.00000i −0.112687 + 0.225374i
\(316\) −20.0000 −1.12509
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 18.0000i 1.00939i
\(319\) 3.00000 0.167968
\(320\) −16.0000 8.00000i −0.894427 0.447214i
\(321\) −12.0000 −0.669775
\(322\) 4.00000i 0.222911i
\(323\) 0 0
\(324\) −2.00000 −0.111111
\(325\) −16.0000 + 12.0000i −0.887520 + 0.665640i
\(326\) 2.00000 0.110770
\(327\) 5.00000i 0.276501i
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) −12.0000 6.00000i −0.660578 0.330289i
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 18.0000i 0.987878i
\(333\) 7.00000i 0.383598i
\(334\) 16.0000 0.875481
\(335\) −8.00000 + 16.0000i −0.437087 + 0.874173i
\(336\) 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
\(339\) −14.0000 −0.760376
\(340\) 16.0000 32.0000i 0.867722 1.73544i
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) −2.00000 1.00000i −0.107676 0.0538382i
\(346\) −18.0000 −0.967686
\(347\) 3.00000i 0.161048i −0.996753 0.0805242i \(-0.974341\pi\)
0.996753 0.0805242i \(-0.0256594\pi\)
\(348\) 2.00000i 0.107211i
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) −12.0000 16.0000i −0.641427 0.855236i
\(351\) 4.00000 0.213504
\(352\) 24.0000i 1.27920i
\(353\) 26.0000i 1.38384i −0.721974 0.691920i \(-0.756765\pi\)
0.721974 0.691920i \(-0.243235\pi\)
\(354\) 20.0000 1.06299
\(355\) 16.0000 + 8.00000i 0.849192 + 0.424596i
\(356\) 20.0000 1.06000
\(357\) 16.0000i 0.846810i
\(358\) 20.0000i 1.05703i
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 34.0000i 1.78700i
\(363\) 2.00000i 0.104973i
\(364\) 16.0000 0.838628
\(365\) −1.00000 + 2.00000i −0.0523424 + 0.104685i
\(366\) −4.00000 −0.209083
\(367\) 3.00000i 0.156599i −0.996930 0.0782994i \(-0.975051\pi\)
0.996930 0.0782994i \(-0.0249490\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −7.00000 −0.364405
\(370\) 28.0000 + 14.0000i 1.45565 + 0.727825i
\(371\) −18.0000 −0.934513
\(372\) 16.0000i 0.829561i
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) −48.0000 −2.48202
\(375\) 11.0000 2.00000i 0.568038 0.103280i
\(376\) 0 0
\(377\) 4.00000i 0.206010i
\(378\) 4.00000i 0.205738i
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) 3.00000 0.153695
\(382\) 6.00000i 0.306987i
\(383\) 39.0000i 1.99281i 0.0847358 + 0.996403i \(0.472995\pi\)
−0.0847358 + 0.996403i \(0.527005\pi\)
\(384\) 0 0
\(385\) −6.00000 + 12.0000i −0.305788 + 0.611577i
\(386\) 12.0000 0.610784
\(387\) 9.00000i 0.457496i
\(388\) 26.0000i 1.31995i
\(389\) 25.0000 1.26755 0.633775 0.773517i \(-0.281504\pi\)
0.633775 + 0.773517i \(0.281504\pi\)
\(390\) −8.00000 + 16.0000i −0.405096 + 0.810191i
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 26.0000 1.30986
\(395\) −20.0000 10.0000i −1.00631 0.503155i
\(396\) −6.00000 −0.301511
\(397\) 8.00000i 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 50.0000i 2.50627i
\(399\) 0 0
\(400\) −12.0000 16.0000i −0.600000 0.800000i
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 16.0000i 0.798007i
\(403\) 32.0000i 1.59403i
\(404\) −34.0000 −1.69156
\(405\) −2.00000 1.00000i −0.0993808 0.0496904i
\(406\) 4.00000 0.198517
\(407\) 21.0000i 1.04093i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 14.0000 28.0000i 0.691411 1.38282i
\(411\) −2.00000 −0.0986527
\(412\) 8.00000i 0.394132i
\(413\) 20.0000i 0.984136i
\(414\) −2.00000 −0.0982946
\(415\) 9.00000 18.0000i 0.441793 0.883585i
\(416\) 32.0000 1.56893
\(417\) 5.00000i 0.244851i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −8.00000 4.00000i −0.390360 0.195180i
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 24.0000i 1.16830i
\(423\) 12.0000i 0.583460i
\(424\) 0 0
\(425\) 32.0000 24.0000i 1.55223 1.16417i
\(426\) 16.0000 0.775203
\(427\) 4.00000i 0.193574i
\(428\) 24.0000i 1.16008i
\(429\) 12.0000 0.579365
\(430\) 36.0000 + 18.0000i 1.73607 + 0.868037i
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 19.0000i 0.913082i 0.889702 + 0.456541i \(0.150912\pi\)
−0.889702 + 0.456541i \(0.849088\pi\)
\(434\) 32.0000 1.53605
\(435\) −1.00000 + 2.00000i −0.0479463 + 0.0958927i
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 2.00000i 0.0955637i
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 64.0000i 3.04417i
\(443\) 6.00000i 0.285069i −0.989790 0.142534i \(-0.954475\pi\)
0.989790 0.142534i \(-0.0455251\pi\)
\(444\) 14.0000 0.664411
\(445\) 20.0000 + 10.0000i 0.948091 + 0.474045i
\(446\) 12.0000 0.568216
\(447\) 0 0
\(448\) 16.0000i 0.755929i
\(449\) −5.00000 −0.235965 −0.117982 0.993016i \(-0.537643\pi\)
−0.117982 + 0.993016i \(0.537643\pi\)
\(450\) 8.00000 6.00000i 0.377124 0.282843i
\(451\) −21.0000 −0.988851
\(452\) 28.0000i 1.31701i
\(453\) 3.00000i 0.140952i
\(454\) 6.00000 0.281594
\(455\) 16.0000 + 8.00000i 0.750092 + 0.375046i
\(456\) 0 0
\(457\) 28.0000i 1.30978i −0.755722 0.654892i \(-0.772714\pi\)
0.755722 0.654892i \(-0.227286\pi\)
\(458\) 40.0000i 1.86908i
\(459\) −8.00000 −0.373408
\(460\) 2.00000 4.00000i 0.0932505 0.186501i
\(461\) −13.0000 −0.605470 −0.302735 0.953075i \(-0.597900\pi\)
−0.302735 + 0.953075i \(0.597900\pi\)
\(462\) 12.0000i 0.558291i
\(463\) 6.00000i 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(464\) 4.00000 0.185695
\(465\) −8.00000 + 16.0000i −0.370991 + 0.741982i
\(466\) 2.00000 0.0926482
\(467\) 42.0000i 1.94353i 0.235954 + 0.971764i \(0.424178\pi\)
−0.235954 + 0.971764i \(0.575822\pi\)
\(468\) 8.00000i 0.369800i
\(469\) 16.0000 0.738811
\(470\) 48.0000 + 24.0000i 2.21407 + 1.10704i
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 27.0000i 1.24146i
\(474\) −20.0000 −0.918630
\(475\) 0 0
\(476\) −32.0000 −1.46672
\(477\) 9.00000i 0.412082i
\(478\) 60.0000i 2.74434i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −16.0000 8.00000i −0.730297 0.365148i
\(481\) −28.0000 −1.27669
\(482\) 6.00000i 0.273293i
\(483\) 2.00000i 0.0910032i
\(484\) 4.00000 0.181818
\(485\) −13.0000 + 26.0000i −0.590300 + 1.18060i
\(486\) −2.00000 −0.0907218
\(487\) 22.0000i 0.996915i 0.866914 + 0.498458i \(0.166100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 6.00000 12.0000i 0.271052 0.542105i
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 14.0000i 0.631169i
\(493\) 8.00000i 0.360302i
\(494\) 0 0
\(495\) −6.00000 3.00000i −0.269680 0.134840i
\(496\) 32.0000 1.43684
\(497\) 16.0000i 0.717698i
\(498\) 18.0000i 0.806599i
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 4.00000 + 22.0000i 0.178885 + 0.983870i
\(501\) 8.00000 0.357414
\(502\) 24.0000i 1.07117i
\(503\) 4.00000i 0.178351i 0.996016 + 0.0891756i \(0.0284232\pi\)
−0.996016 + 0.0891756i \(0.971577\pi\)
\(504\) 0 0
\(505\) −34.0000 17.0000i −1.51298 0.756490i
\(506\) −6.00000 −0.266733
\(507\) 3.00000i 0.133235i
\(508\) 6.00000i 0.266207i
\(509\) 40.0000 1.77297 0.886484 0.462758i \(-0.153140\pi\)
0.886484 + 0.462758i \(0.153140\pi\)
\(510\) 16.0000 32.0000i 0.708492 1.41698i
\(511\) 2.00000 0.0884748
\(512\) 32.0000i 1.41421i
\(513\) 0 0
\(514\) 46.0000 2.02897
\(515\) 4.00000 8.00000i 0.176261 0.352522i
\(516\) 18.0000 0.792406
\(517\) 36.0000i 1.58328i
\(518\) 28.0000i 1.23025i
\(519\) −9.00000 −0.395056
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) −24.0000 −1.04844
\(525\) −6.00000 8.00000i −0.261861 0.349149i
\(526\) −8.00000 −0.348817
\(527\) 64.0000i 2.78788i
\(528\) 12.0000i 0.522233i
\(529\) 22.0000 0.956522
\(530\) 36.0000 + 18.0000i 1.56374 + 0.781870i
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) 28.0000i 1.21281i
\(534\) 20.0000 0.865485
\(535\) 12.0000 24.0000i 0.518805 1.03761i
\(536\) 0 0
\(537\) 10.0000i 0.431532i
\(538\) 20.0000i 0.862261i
\(539\) −9.00000 −0.387657
\(540\) 2.00000 4.00000i 0.0860663 0.172133i
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 24.0000i 1.03089i
\(543\) 17.0000i 0.729540i
\(544\) −64.0000 −2.74398
\(545\) −10.0000 5.00000i −0.428353 0.214176i
\(546\) 16.0000 0.684737
\(547\) 8.00000i 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 4.00000i 0.170872i
\(549\) −2.00000 −0.0853579
\(550\) 24.0000 18.0000i 1.02336 0.767523i
\(551\) 0 0
\(552\) 0 0
\(553\) 20.0000i 0.850487i
\(554\) 56.0000 2.37921
\(555\) 14.0000 + 7.00000i 0.594267 + 0.297133i
\(556\) 10.0000 0.424094
\(557\) 47.0000i 1.99145i 0.0923462 + 0.995727i \(0.470563\pi\)
−0.0923462 + 0.995727i \(0.529437\pi\)
\(558\) 16.0000i 0.677334i
\(559\) −36.0000 −1.52264
\(560\) −8.00000 + 16.0000i −0.338062 + 0.676123i
\(561\) −24.0000 −1.01328
\(562\) 36.0000i 1.51857i
\(563\) 46.0000i 1.93867i −0.245745 0.969334i \(-0.579033\pi\)
0.245745 0.969334i \(-0.420967\pi\)
\(564\) 24.0000 1.01058
\(565\) 14.0000 28.0000i 0.588984 1.17797i
\(566\) −28.0000 −1.17693
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −23.0000 −0.962520 −0.481260 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(572\) 24.0000i 1.00349i
\(573\) 3.00000i 0.125327i
\(574\) −28.0000 −1.16870
\(575\) 4.00000 3.00000i 0.166812 0.125109i
\(576\) −8.00000 −0.333333
\(577\) 18.0000i 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) 94.0000i 3.90988i
\(579\) 6.00000 0.249351
\(580\) −4.00000 2.00000i −0.166091 0.0830455i
\(581\) −18.0000 −0.746766
\(582\) 26.0000i 1.07773i
\(583\) 27.0000i 1.11823i
\(584\) 0 0
\(585\) −4.00000 + 8.00000i −0.165380 + 0.330759i
\(586\) 12.0000 0.495715
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 6.00000i 0.247436i
\(589\) 0 0
\(590\) −20.0000 + 40.0000i −0.823387 + 1.64677i
\(591\) 13.0000 0.534749
\(592\) 28.0000i 1.15079i
\(593\) 6.00000i 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) −6.00000 −0.246183
\(595\) −32.0000 16.0000i −1.31187 0.655936i
\(596\) 0 0
\(597\) 25.0000i 1.02318i
\(598\) 8.00000i 0.327144i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 36.0000i 1.46725i
\(603\) 8.00000i 0.325785i
\(604\) 6.00000 0.244137
\(605\) 4.00000 + 2.00000i 0.162623 + 0.0813116i
\(606\) −34.0000 −1.38116
\(607\) 8.00000i 0.324710i −0.986732 0.162355i \(-0.948091\pi\)
0.986732 0.162355i \(-0.0519090\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 4.00000 8.00000i 0.161955 0.323911i
\(611\) −48.0000 −1.94187
\(612\) 16.0000i 0.646762i
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) −34.0000 −1.37213
\(615\) 7.00000 14.0000i 0.282267 0.564534i
\(616\) 0 0
\(617\) 18.0000i 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 8.00000i 0.321807i
\(619\) 30.0000 1.20580 0.602901 0.797816i \(-0.294011\pi\)
0.602901 + 0.797816i \(0.294011\pi\)
\(620\) −32.0000 16.0000i −1.28515 0.642575i
\(621\) −1.00000 −0.0401286
\(622\) 34.0000i 1.36328i
\(623\) 20.0000i 0.801283i
\(624\) 16.0000 0.640513
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) 4.00000i 0.159617i
\(629\) 56.0000 2.23287
\(630\) −8.00000 4.00000i −0.318728 0.159364i
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) −24.0000 −0.953162
\(635\) −3.00000 + 6.00000i −0.119051 + 0.238103i
\(636\) 18.0000 0.713746
\(637\) 12.0000i 0.475457i
\(638\) 6.00000i 0.237542i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −43.0000 −1.69840 −0.849199 0.528073i \(-0.822915\pi\)
−0.849199 + 0.528073i \(0.822915\pi\)
\(642\) 24.0000i 0.947204i
\(643\) 16.0000i 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) −4.00000 −0.157622
\(645\) 18.0000 + 9.00000i 0.708749 + 0.354375i
\(646\) 0 0
\(647\) 33.0000i 1.29736i −0.761060 0.648682i \(-0.775321\pi\)
0.761060 0.648682i \(-0.224679\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) −24.0000 32.0000i −0.941357 1.25514i
\(651\) 16.0000 0.627089
\(652\) 2.00000i 0.0783260i
\(653\) 36.0000i 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) −10.0000 −0.391031
\(655\) −24.0000 12.0000i −0.937758 0.468879i
\(656\) −28.0000 −1.09322
\(657\) 1.00000i 0.0390137i
\(658\) 48.0000i 1.87123i
\(659\) −25.0000 −0.973862 −0.486931 0.873441i \(-0.661884\pi\)
−0.486931 + 0.873441i \(0.661884\pi\)
\(660\) 6.00000 12.0000i 0.233550 0.467099i
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 64.0000i 2.48743i
\(663\) 32.0000i 1.24278i
\(664\) 0 0
\(665\) 0 0
\(666\) 14.0000 0.542489
\(667\) 1.00000i 0.0387202i
\(668\) 16.0000i 0.619059i
\(669\) 6.00000 0.231973
\(670\) −32.0000 16.0000i −1.23627 0.618134i
\(671\) −6.00000 −0.231627
\(672\) 16.0000i 0.617213i
\(673\) 6.00000i 0.231283i −0.993291 0.115642i \(-0.963108\pi\)
0.993291 0.115642i \(-0.0368924\pi\)
\(674\) −44.0000 −1.69482
\(675\) 4.00000 3.00000i 0.153960 0.115470i
\(676\) 6.00000 0.230769
\(677\) 22.0000i 0.845529i 0.906240 + 0.422764i \(0.138940\pi\)
−0.906240 + 0.422764i \(0.861060\pi\)
\(678\) 28.0000i 1.07533i
\(679\) 26.0000 0.997788
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 48.0000i 1.83801i
\(683\) 39.0000i 1.49229i 0.665782 + 0.746147i \(0.268098\pi\)
−0.665782 + 0.746147i \(0.731902\pi\)
\(684\) 0 0
\(685\) 2.00000 4.00000i 0.0764161 0.152832i
\(686\) −40.0000 −1.52721
\(687\) 20.0000i 0.763048i
\(688\) 36.0000i 1.37249i
\(689\) −36.0000 −1.37149
\(690\) 2.00000 4.00000i 0.0761387 0.152277i
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 6.00000i 0.227921i
\(694\) 6.00000 0.227757
\(695\) 10.0000 + 5.00000i 0.379322 + 0.189661i
\(696\) 0 0
\(697\) 56.0000i 2.12115i
\(698\) 10.0000i 0.378506i
\(699\) 1.00000 0.0378235
\(700\) 16.0000 12.0000i 0.604743 0.453557i
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 8.00000i 0.301941i
\(703\) 0 0
\(704\) −24.0000 −0.904534
\(705\) 24.0000 + 12.0000i 0.903892 + 0.451946i
\(706\) 52.0000 1.95705
\(707\) 34.0000i 1.27870i
\(708\) 20.0000i 0.751646i
\(709\) 45.0000 1.69001 0.845005 0.534758i \(-0.179597\pi\)
0.845005 + 0.534758i \(0.179597\pi\)
\(710\) −16.0000 + 32.0000i −0.600469 + 1.20094i
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) 8.00000i 0.299602i
\(714\) −32.0000 −1.19757
\(715\) −12.0000 + 24.0000i −0.448775 + 0.897549i
\(716\) 20.0000 0.747435
\(717\) 30.0000i 1.12037i
\(718\) 30.0000i 1.11959i
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) −8.00000 4.00000i −0.298142 0.149071i
\(721\) −8.00000 −0.297936
\(722\) 38.0000i 1.41421i
\(723\) 3.00000i 0.111571i
\(724\) −34.0000 −1.26360
\(725\) −3.00000 4.00000i −0.111417 0.148556i
\(726\) 4.00000 0.148454
\(727\) 8.00000i 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) −4.00000 2.00000i −0.148047 0.0740233i
\(731\) 72.0000 2.66302
\(732\) 4.00000i 0.147844i
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 6.00000 0.221464
\(735\) 3.00000 6.00000i 0.110657 0.221313i
\(736\) −8.00000 −0.294884
\(737\) 24.0000i 0.884051i
\(738\) 14.0000i 0.515347i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −14.0000 + 28.0000i −0.514650 + 1.02930i
\(741\) 0 0
\(742\) 36.0000i 1.32160i
\(743\) 44.0000i 1.61420i 0.590412 + 0.807102i \(0.298965\pi\)
−0.590412 + 0.807102i \(0.701035\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −8.00000 −0.292901
\(747\) 9.00000i 0.329293i
\(748\) 48.0000i 1.75505i
\(749\) −24.0000 −0.876941
\(750\) 4.00000 + 22.0000i 0.146059 + 0.803326i
\(751\) 22.0000 0.802791 0.401396 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 48.0000i 1.75038i
\(753\) 12.0000i 0.437304i
\(754\) 8.00000 0.291343
\(755\) 6.00000 + 3.00000i 0.218362 + 0.109181i
\(756\) −4.00000 −0.145479
\(757\) 3.00000i 0.109037i −0.998513 0.0545184i \(-0.982638\pi\)
0.998513 0.0545184i \(-0.0173624\pi\)
\(758\) 20.0000i 0.726433i
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 6.00000i 0.217357i
\(763\) 10.0000i 0.362024i
\(764\) 6.00000 0.217072
\(765\) 8.00000 16.0000i 0.289241 0.578481i
\(766\) −78.0000 −2.81825
\(767\) 40.0000i 1.44432i
\(768\) 16.0000i 0.577350i
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) −24.0000 12.0000i −0.864900 0.432450i
\(771\) 23.0000 0.828325
\(772\) 12.0000i 0.431889i
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 18.0000 0.646997
\(775\) −24.0000 32.0000i −0.862105 1.14947i
\(776\) 0 0
\(777\) 14.0000i 0.502247i
\(778\) 50.0000i 1.79259i
\(779\) 0 0
\(780\) −16.0000 8.00000i −0.572892 0.286446i
\(781\) 24.0000 0.858788
\(782\) 16.0000i 0.572159i
\(783\) 1.00000i 0.0357371i
\(784\) −12.0000 −0.428571
\(785\) 2.00000 4.00000i 0.0713831 0.142766i
\(786\) −24.0000 −0.856052
\(787\) 38.0000i 1.35455i −0.735728 0.677277i \(-0.763160\pi\)
0.735728 0.677277i \(-0.236840\pi\)
\(788\) 26.0000i 0.926212i
\(789\) −4.00000 −0.142404
\(790\) 20.0000 40.0000i 0.711568 1.42314i
\(791\) −28.0000 −0.995565
\(792\) 0 0
\(793\) 8.00000i 0.284088i
\(794\) 16.0000 0.567819
\(795\) 18.0000 + 9.00000i 0.638394 + 0.319197i
\(796\) −50.0000 −1.77220
\(797\) 8.00000i 0.283375i −0.989911 0.141687i \(-0.954747\pi\)
0.989911 0.141687i \(-0.0452527\pi\)
\(798\) 0 0
\(799\) 96.0000 3.39624
\(800\) 32.0000 24.0000i 1.13137 0.848528i
\(801\) 10.0000 0.353333
\(802\) 56.0000i 1.97743i
\(803\) 3.00000i 0.105868i
\(804\) −16.0000 −0.564276
\(805\) −4.00000 2.00000i −0.140981 0.0704907i
\(806\) 64.0000 2.25430
\(807\) 10.0000i 0.352017i
\(808\) 0 0
\(809\) 45.0000 1.58212 0.791058 0.611741i \(-0.209531\pi\)
0.791058 + 0.611741i \(0.209531\pi\)
\(810\) 2.00000 4.00000i 0.0702728 0.140546i
\(811\) −13.0000 −0.456492 −0.228246 0.973604i \(-0.573299\pi\)
−0.228246 + 0.973604i \(0.573299\pi\)
\(812\) 4.00000i 0.140372i
\(813\) 12.0000i 0.420858i
\(814\) 42.0000 1.47210
\(815\) −1.00000 + 2.00000i −0.0350285 + 0.0700569i
\(816\) −32.0000 −1.12022
\(817\) 0 0
\(818\) 20.0000i 0.699284i
\(819\) 8.00000 0.279543
\(820\) 28.0000 + 14.0000i 0.977802 + 0.488901i
\(821\) −48.0000 −1.67521 −0.837606 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) 4.00000i 0.139516i
\(823\) 16.0000i 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) 0 0
\(825\) 12.0000 9.00000i 0.417786 0.313340i
\(826\) 40.0000 1.39178
\(827\) 22.0000i 0.765015i 0.923952 + 0.382507i \(0.124939\pi\)
−0.923952 + 0.382507i \(0.875061\pi\)
\(828\) 2.00000i 0.0695048i
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 36.0000 + 18.0000i 1.24958 + 0.624789i
\(831\) 28.0000 0.971309
\(832\) 32.0000i 1.10940i
\(833\) 24.0000i 0.831551i
\(834\) 10.0000 0.346272
\(835\) −8.00000 + 16.0000i −0.276851 + 0.553703i
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 56.0000i 1.92989i
\(843\) 18.0000i 0.619953i
\(844\) −24.0000 −0.826114
\(845\) 6.00000 + 3.00000i 0.206406 + 0.103203i
\(846\) 24.0000 0.825137
\(847\) 4.00000i 0.137442i
\(848\) 36.0000i 1.23625i
\(849\) −14.0000 −0.480479
\(850\) 48.0000 + 64.0000i 1.64639 + 2.19518i
\(851\) 7.00000 0.239957
\(852\) 16.0000i 0.548151i
\(853\) 19.0000i 0.650548i 0.945620 + 0.325274i \(0.105456\pi\)
−0.945620 + 0.325274i \(0.894544\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 0 0
\(857\) 7.00000i 0.239115i 0.992827 + 0.119558i \(0.0381477\pi\)
−0.992827 + 0.119558i \(0.961852\pi\)
\(858\) 24.0000i 0.819346i
\(859\) 30.0000 1.02359 0.511793 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) −18.0000 + 36.0000i −0.613795 + 1.22759i
\(861\) −14.0000 −0.477119
\(862\) 24.0000i 0.817443i
\(863\) 44.0000i 1.49778i 0.662696 + 0.748889i \(0.269412\pi\)
−0.662696 + 0.748889i \(0.730588\pi\)
\(864\) −8.00000 −0.272166
\(865\) 9.00000 18.0000i 0.306009 0.612018i
\(866\) −38.0000 −1.29129
\(867\) 47.0000i 1.59620i
\(868\) 32.0000i 1.08615i
\(869\) −30.0000 −1.01768
\(870\) −4.00000 2.00000i −0.135613 0.0678064i
\(871\) 32.0000 1.08428
\(872\) 0 0
\(873\) 13.0000i 0.439983i
\(874\) 0 0
\(875\) 22.0000 4.00000i 0.743736 0.135225i
\(876\) −2.00000 −0.0675737
\(877\) 52.0000i 1.75592i 0.478738 + 0.877958i \(0.341094\pi\)
−0.478738 + 0.877958i \(0.658906\pi\)
\(878\) 40.0000i 1.34993i
\(879\) 6.00000 0.202375
\(880\) −24.0000 12.0000i −0.809040 0.404520i
\(881\) 57.0000 1.92038 0.960189 0.279350i \(-0.0901189\pi\)
0.960189 + 0.279350i \(0.0901189\pi\)
\(882\) 6.00000i 0.202031i
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) −64.0000 −2.15255
\(885\) −10.0000 + 20.0000i −0.336146 + 0.672293i
\(886\) 12.0000 0.403148
\(887\) 22.0000i 0.738688i 0.929293 + 0.369344i \(0.120418\pi\)
−0.929293 + 0.369344i \(0.879582\pi\)
\(888\) 0 0
\(889\) 6.00000 0.201234
\(890\) −20.0000 + 40.0000i −0.670402 + 1.34080i
\(891\) −3.00000 −0.100504
\(892\) 12.0000i 0.401790i
\(893\) 0 0
\(894\) 0 0
\(895\) 20.0000 + 10.0000i 0.668526 + 0.334263i
\(896\) 0 0
\(897\) 4.00000i 0.133556i
\(898\) 10.0000i 0.333704i
\(899\) 8.00000 0.266815
\(900\) 6.00000 + 8.00000i 0.200000 + 0.266667i
\(901\) 72.0000 2.39867
\(902\) 42.0000i 1.39845i
\(903\) 18.0000i 0.599002i
\(904\) 0 0
\(905\) −34.0000 17.0000i −1.13020 0.565099i
\(906\) 6.00000 0.199337
\(907\) 37.0000i 1.22856i 0.789086 + 0.614282i \(0.210554\pi\)
−0.789086 + 0.614282i \(0.789446\pi\)
\(908\) 6.00000i 0.199117i
\(909\) −17.0000 −0.563854
\(910\) −16.0000 + 32.0000i −0.530395 + 1.06079i
\(911\) 27.0000 0.894550 0.447275 0.894397i \(-0.352395\pi\)
0.447275 + 0.894397i \(0.352395\pi\)
\(912\) 0 0
\(913\) 27.0000i 0.893570i
\(914\) 56.0000 1.85232
\(915\) 2.00000 4.00000i 0.0661180 0.132236i
\(916\) 40.0000 1.32164
\(917\) 24.0000i 0.792550i
\(918\) 16.0000i 0.528079i
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −17.0000 −0.560169
\(922\) 26.0000i 0.856264i
\(923\) 32.0000i 1.05329i
\(924\) −12.0000 −0.394771
\(925\) −28.0000 + 21.0000i −0.920634 + 0.690476i
\(926\) 12.0000 0.394344
\(927\) 4.00000i 0.131377i
\(928\) 8.00000i 0.262613i
\(929\) −20.0000 −0.656179 −0.328089 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) −32.0000 16.0000i −1.04932 0.524661i
\(931\) 0 0
\(932\) 2.00000i 0.0655122i
\(933\) 17.0000i 0.556555i
\(934\) −84.0000 −2.74856
\(935\) 24.0000 48.0000i 0.784884 1.56977i
\(936\) 0 0
\(937\) 22.0000i 0.718709i 0.933201 + 0.359354i \(0.117003\pi\)
−0.933201 + 0.359354i \(0.882997\pi\)
\(938\) 32.0000i 1.04484i
\(939\) −4.00000 −0.130535
\(940\) −24.0000 + 48.0000i −0.782794 + 1.56559i
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 4.00000i 0.130327i
\(943\) 7.00000i 0.227951i
\(944\) 40.0000 1.30189
\(945\) −4.00000 2.00000i −0.130120 0.0650600i
\(946\) 54.0000 1.75569
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 20.0000i 0.649570i
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 6.00000i 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 18.0000 0.582772
\(955\) 6.00000 + 3.00000i 0.194155 + 0.0970777i
\(956\) 60.0000 1.94054
\(957\) 3.00000i 0.0969762i
\(958\) 0 0
\(959\) −4.00000 −0.129167
\(960\) 8.00000 16.0000i 0.258199 0.516398i
\(961\) 33.0000 1.06452
\(962\) 56.0000i 1.80551i
\(963\) 12.0000i 0.386695i
\(964\) 6.00000 0.193247
\(965\) −6.00000 + 12.0000i −0.193147 + 0.386294i
\(966\) −4.00000 −0.128698
\(967\) 3.00000i 0.0964735i −0.998836 0.0482367i \(-0.984640\pi\)
0.998836 0.0482367i \(-0.0153602\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −52.0000 26.0000i −1.66962 0.834810i
\(971\) −3.00000 −0.0962746 −0.0481373 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) 2.00000i 0.0641500i
\(973\) 10.0000i 0.320585i
\(974\) −44.0000 −1.40985
\(975\) −12.0000 16.0000i −0.384308 0.512410i
\(976\) −8.00000 −0.256074
\(977\) 13.0000i 0.415907i −0.978139 0.207953i \(-0.933320\pi\)
0.978139 0.207953i \(-0.0666802\pi\)
\(978\) 2.00000i 0.0639529i
\(979\) 30.0000 0.958804
\(980\) 12.0000 + 6.00000i 0.383326 + 0.191663i
\(981\) −5.00000 −0.159638
\(982\) 56.0000i 1.78703i
\(983\) 36.0000i 1.14822i −0.818778 0.574111i \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) 0 0
\(985\) −13.0000 + 26.0000i −0.414214 + 0.828429i
\(986\) −16.0000 −0.509544
\(987\) 24.0000i 0.763928i
\(988\) 0 0
\(989\) 9.00000 0.286183
\(990\) 6.00000 12.0000i 0.190693 0.381385i
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) 64.0000i 2.03200i
\(993\) 32.0000i 1.01549i
\(994\) 32.0000 1.01498
\(995\) −50.0000 25.0000i −1.58511 0.792553i
\(996\) 18.0000 0.570352
\(997\) 7.00000i 0.221692i 0.993838 + 0.110846i \(0.0353561\pi\)
−0.993838 + 0.110846i \(0.964644\pi\)
\(998\) 40.0000i 1.26618i
\(999\) 7.00000 0.221470
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 435.2.c.a.349.2 yes 2
3.2 odd 2 1305.2.c.a.784.1 2
5.2 odd 4 2175.2.a.a.1.1 1
5.3 odd 4 2175.2.a.j.1.1 1
5.4 even 2 inner 435.2.c.a.349.1 2
15.2 even 4 6525.2.a.l.1.1 1
15.8 even 4 6525.2.a.b.1.1 1
15.14 odd 2 1305.2.c.a.784.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.a.349.1 2 5.4 even 2 inner
435.2.c.a.349.2 yes 2 1.1 even 1 trivial
1305.2.c.a.784.1 2 3.2 odd 2
1305.2.c.a.784.2 2 15.14 odd 2
2175.2.a.a.1.1 1 5.2 odd 4
2175.2.a.j.1.1 1 5.3 odd 4
6525.2.a.b.1.1 1 15.8 even 4
6525.2.a.l.1.1 1 15.2 even 4