Properties

Label 3822.2.c.b.883.1
Level $3822$
Weight $2$
Character 3822.883
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3822,2,Mod(883,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3822.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 883.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3822.883
Dual form 3822.2.c.b.883.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000i q^{11} +1.00000 q^{12} +(-3.00000 - 2.00000i) q^{13} +1.00000i q^{15} +1.00000 q^{16} -7.00000 q^{17} -1.00000i q^{18} +3.00000i q^{19} +1.00000i q^{20} -3.00000 q^{22} +1.00000 q^{23} -1.00000i q^{24} +4.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -1.00000 q^{27} -1.00000 q^{29} +1.00000 q^{30} -8.00000i q^{31} -1.00000i q^{32} +3.00000i q^{33} +7.00000i q^{34} -1.00000 q^{36} +1.00000i q^{37} +3.00000 q^{38} +(3.00000 + 2.00000i) q^{39} +1.00000 q^{40} +4.00000i q^{41} -5.00000 q^{43} +3.00000i q^{44} -1.00000i q^{45} -1.00000i q^{46} -1.00000 q^{48} -4.00000i q^{50} +7.00000 q^{51} +(3.00000 + 2.00000i) q^{52} -6.00000 q^{53} +1.00000i q^{54} -3.00000 q^{55} -3.00000i q^{57} +1.00000i q^{58} +10.0000i q^{59} -1.00000i q^{60} +13.0000 q^{61} -8.00000 q^{62} -1.00000 q^{64} +(-2.00000 + 3.00000i) q^{65} +3.00000 q^{66} +8.00000i q^{67} +7.00000 q^{68} -1.00000 q^{69} -6.00000i q^{71} +1.00000i q^{72} -13.0000i q^{73} +1.00000 q^{74} -4.00000 q^{75} -3.00000i q^{76} +(2.00000 - 3.00000i) q^{78} -12.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +4.00000 q^{82} +2.00000i q^{83} +7.00000i q^{85} +5.00000i q^{86} +1.00000 q^{87} +3.00000 q^{88} +12.0000i q^{89} -1.00000 q^{90} -1.00000 q^{92} +8.00000i q^{93} +3.00000 q^{95} +1.00000i q^{96} +6.00000i q^{97} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{13} + 2 q^{16} - 14 q^{17} - 6 q^{22} + 2 q^{23} + 8 q^{25} - 4 q^{26} - 2 q^{27} - 2 q^{29} + 2 q^{30} - 2 q^{36} + 6 q^{38} + 6 q^{39} + 2 q^{40} - 10 q^{43} - 2 q^{48} + 14 q^{51} + 6 q^{52} - 12 q^{53} - 6 q^{55} + 26 q^{61} - 16 q^{62} - 2 q^{64} - 4 q^{65} + 6 q^{66} + 14 q^{68} - 2 q^{69} + 2 q^{74} - 8 q^{75} + 4 q^{78} - 24 q^{79} + 2 q^{81} + 8 q^{82} + 2 q^{87} + 6 q^{88} - 2 q^{90} - 2 q^{92} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times\).

\(n\) \(1471\) \(2549\) \(3433\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i −0.974679 0.223607i \(-0.928217\pi\)
0.974679 0.223607i \(-0.0717831\pi\)
\(6\) 1.00000i 0.408248i
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.00000i 0.904534i −0.891883 0.452267i \(-0.850615\pi\)
0.891883 0.452267i \(-0.149385\pi\)
\(12\) 1.00000 0.288675
\(13\) −3.00000 2.00000i −0.832050 0.554700i
\(14\) 0 0
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 3.00000i 0.688247i 0.938924 + 0.344124i \(0.111824\pi\)
−0.938924 + 0.344124i \(0.888176\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 4.00000 0.800000
\(26\) −2.00000 + 3.00000i −0.392232 + 0.588348i
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.00000i 1.43684i −0.695608 0.718421i \(-0.744865\pi\)
0.695608 0.718421i \(-0.255135\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 3.00000i 0.522233i
\(34\) 7.00000i 1.20049i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 1.00000i 0.164399i 0.996616 + 0.0821995i \(0.0261945\pi\)
−0.996616 + 0.0821995i \(0.973806\pi\)
\(38\) 3.00000 0.486664
\(39\) 3.00000 + 2.00000i 0.480384 + 0.320256i
\(40\) 1.00000 0.158114
\(41\) 4.00000i 0.624695i 0.949968 + 0.312348i \(0.101115\pi\)
−0.949968 + 0.312348i \(0.898885\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 3.00000i 0.452267i
\(45\) 1.00000i 0.149071i
\(46\) 1.00000i 0.147442i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000i 0.565685i
\(51\) 7.00000 0.980196
\(52\) 3.00000 + 2.00000i 0.416025 + 0.277350i
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 3.00000i 0.397360i
\(58\) 1.00000i 0.131306i
\(59\) 10.0000i 1.30189i 0.759125 + 0.650945i \(0.225627\pi\)
−0.759125 + 0.650945i \(0.774373\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −2.00000 + 3.00000i −0.248069 + 0.372104i
\(66\) 3.00000 0.369274
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 7.00000 0.848875
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 6.00000i 0.712069i −0.934473 0.356034i \(-0.884129\pi\)
0.934473 0.356034i \(-0.115871\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 13.0000i 1.52153i −0.649025 0.760767i \(-0.724823\pi\)
0.649025 0.760767i \(-0.275177\pi\)
\(74\) 1.00000 0.116248
\(75\) −4.00000 −0.461880
\(76\) 3.00000i 0.344124i
\(77\) 0 0
\(78\) 2.00000 3.00000i 0.226455 0.339683i
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) 7.00000i 0.759257i
\(86\) 5.00000i 0.539164i
\(87\) 1.00000 0.107211
\(88\) 3.00000 0.319801
\(89\) 12.0000i 1.27200i 0.771690 + 0.635999i \(0.219412\pi\)
−0.771690 + 0.635999i \(0.780588\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 1.00000i 0.102062i
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) 3.00000i 0.301511i
\(100\) −4.00000 −0.400000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 7.00000i 0.693103i
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 2.00000 3.00000i 0.196116 0.294174i
\(105\) 0 0
\(106\) 6.00000i 0.582772i
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) 7.00000i 0.670478i −0.942133 0.335239i \(-0.891183\pi\)
0.942133 0.335239i \(-0.108817\pi\)
\(110\) 3.00000i 0.286039i
\(111\) 1.00000i 0.0949158i
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) −3.00000 −0.280976
\(115\) 1.00000i 0.0932505i
\(116\) 1.00000 0.0928477
\(117\) −3.00000 2.00000i −0.277350 0.184900i
\(118\) 10.0000 0.920575
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 2.00000 0.181818
\(122\) 13.0000i 1.17696i
\(123\) 4.00000i 0.360668i
\(124\) 8.00000i 0.718421i
\(125\) 9.00000i 0.804984i
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 5.00000 0.440225
\(130\) 3.00000 + 2.00000i 0.263117 + 0.175412i
\(131\) 17.0000 1.48530 0.742648 0.669681i \(-0.233569\pi\)
0.742648 + 0.669681i \(0.233569\pi\)
\(132\) 3.00000i 0.261116i
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 1.00000i 0.0860663i
\(136\) 7.00000i 0.600245i
\(137\) 17.0000i 1.45241i 0.687479 + 0.726204i \(0.258717\pi\)
−0.687479 + 0.726204i \(0.741283\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −6.00000 + 9.00000i −0.501745 + 0.752618i
\(144\) 1.00000 0.0833333
\(145\) 1.00000i 0.0830455i
\(146\) −13.0000 −1.07589
\(147\) 0 0
\(148\) 1.00000i 0.0821995i
\(149\) 22.0000i 1.80231i 0.433497 + 0.901155i \(0.357280\pi\)
−0.433497 + 0.901155i \(0.642720\pi\)
\(150\) 4.00000i 0.326599i
\(151\) 11.0000i 0.895167i 0.894242 + 0.447584i \(0.147715\pi\)
−0.894242 + 0.447584i \(0.852285\pi\)
\(152\) −3.00000 −0.243332
\(153\) −7.00000 −0.565916
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −3.00000 2.00000i −0.240192 0.160128i
\(157\) −19.0000 −1.51637 −0.758183 0.652042i \(-0.773912\pi\)
−0.758183 + 0.652042i \(0.773912\pi\)
\(158\) 12.0000i 0.954669i
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 4.00000i 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 4.00000i 0.312348i
\(165\) 3.00000 0.233550
\(166\) 2.00000 0.155230
\(167\) 15.0000i 1.16073i 0.814355 + 0.580367i \(0.197091\pi\)
−0.814355 + 0.580367i \(0.802909\pi\)
\(168\) 0 0
\(169\) 5.00000 + 12.0000i 0.384615 + 0.923077i
\(170\) 7.00000 0.536875
\(171\) 3.00000i 0.229416i
\(172\) 5.00000 0.381246
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 1.00000i 0.0758098i
\(175\) 0 0
\(176\) 3.00000i 0.226134i
\(177\) 10.0000i 0.751646i
\(178\) 12.0000 0.899438
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) −13.0000 −0.960988
\(184\) 1.00000i 0.0737210i
\(185\) 1.00000 0.0735215
\(186\) 8.00000 0.586588
\(187\) 21.0000i 1.53567i
\(188\) 0 0
\(189\) 0 0
\(190\) 3.00000i 0.217643i
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 1.00000 0.0721688
\(193\) 8.00000i 0.575853i 0.957653 + 0.287926i \(0.0929658\pi\)
−0.957653 + 0.287926i \(0.907034\pi\)
\(194\) 6.00000 0.430775
\(195\) 2.00000 3.00000i 0.143223 0.214834i
\(196\) 0 0
\(197\) 14.0000i 0.997459i −0.866758 0.498729i \(-0.833800\pi\)
0.866758 0.498729i \(-0.166200\pi\)
\(198\) −3.00000 −0.213201
\(199\) −9.00000 −0.637993 −0.318997 0.947756i \(-0.603346\pi\)
−0.318997 + 0.947756i \(0.603346\pi\)
\(200\) 4.00000i 0.282843i
\(201\) 8.00000i 0.564276i
\(202\) 14.0000i 0.985037i
\(203\) 0 0
\(204\) −7.00000 −0.490098
\(205\) 4.00000 0.279372
\(206\) 1.00000i 0.0696733i
\(207\) 1.00000 0.0695048
\(208\) −3.00000 2.00000i −0.208013 0.138675i
\(209\) 9.00000 0.622543
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 6.00000 0.412082
\(213\) 6.00000i 0.411113i
\(214\) 2.00000i 0.136717i
\(215\) 5.00000i 0.340997i
\(216\) 1.00000i 0.0680414i
\(217\) 0 0
\(218\) −7.00000 −0.474100
\(219\) 13.0000i 0.878459i
\(220\) 3.00000 0.202260
\(221\) 21.0000 + 14.0000i 1.41261 + 0.941742i
\(222\) −1.00000 −0.0671156
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 10.0000i 0.665190i
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 3.00000i 0.198680i
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 1.00000i 0.0656532i
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −2.00000 + 3.00000i −0.130744 + 0.196116i
\(235\) 0 0
\(236\) 10.0000i 0.650945i
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) 8.00000i 0.517477i −0.965947 0.258738i \(-0.916693\pi\)
0.965947 0.258738i \(-0.0833068\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 26.0000i 1.67481i 0.546585 + 0.837404i \(0.315928\pi\)
−0.546585 + 0.837404i \(0.684072\pi\)
\(242\) 2.00000i 0.128565i
\(243\) −1.00000 −0.0641500
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) −4.00000 −0.255031
\(247\) 6.00000 9.00000i 0.381771 0.572656i
\(248\) 8.00000 0.508001
\(249\) 2.00000i 0.126745i
\(250\) −9.00000 −0.569210
\(251\) −17.0000 −1.07303 −0.536515 0.843891i \(-0.680260\pi\)
−0.536515 + 0.843891i \(0.680260\pi\)
\(252\) 0 0
\(253\) 3.00000i 0.188608i
\(254\) 4.00000i 0.250982i
\(255\) 7.00000i 0.438357i
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 5.00000i 0.311286i
\(259\) 0 0
\(260\) 2.00000 3.00000i 0.124035 0.186052i
\(261\) −1.00000 −0.0618984
\(262\) 17.0000i 1.05026i
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −3.00000 −0.184637
\(265\) 6.00000i 0.368577i
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 8.00000i 0.488678i
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 1.00000 0.0608581
\(271\) 20.0000i 1.21491i 0.794353 + 0.607457i \(0.207810\pi\)
−0.794353 + 0.607457i \(0.792190\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) 17.0000 1.02701
\(275\) 12.0000i 0.723627i
\(276\) 1.00000 0.0601929
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 8.00000i 0.479808i
\(279\) 8.00000i 0.478947i
\(280\) 0 0
\(281\) 10.0000i 0.596550i 0.954480 + 0.298275i \(0.0964112\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 6.00000i 0.356034i
\(285\) −3.00000 −0.177705
\(286\) 9.00000 + 6.00000i 0.532181 + 0.354787i
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 32.0000 1.88235
\(290\) 1.00000 0.0587220
\(291\) 6.00000i 0.351726i
\(292\) 13.0000i 0.760767i
\(293\) 14.0000i 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 0 0
\(295\) 10.0000 0.582223
\(296\) −1.00000 −0.0581238
\(297\) 3.00000i 0.174078i
\(298\) 22.0000 1.27443
\(299\) −3.00000 2.00000i −0.173494 0.115663i
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 11.0000 0.632979
\(303\) 14.0000 0.804279
\(304\) 3.00000i 0.172062i
\(305\) 13.0000i 0.744378i
\(306\) 7.00000i 0.400163i
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 8.00000i 0.454369i
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) −2.00000 + 3.00000i −0.113228 + 0.169842i
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 19.0000i 1.07223i
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 24.0000i 1.34797i −0.738743 0.673987i \(-0.764580\pi\)
0.738743 0.673987i \(-0.235420\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 3.00000i 0.167968i
\(320\) 1.00000i 0.0559017i
\(321\) −2.00000 −0.111629
\(322\) 0 0
\(323\) 21.0000i 1.16847i
\(324\) −1.00000 −0.0555556
\(325\) −12.0000 8.00000i −0.665640 0.443760i
\(326\) −4.00000 −0.221540
\(327\) 7.00000i 0.387101i
\(328\) −4.00000 −0.220863
\(329\) 0 0
\(330\) 3.00000i 0.165145i
\(331\) 8.00000i 0.439720i −0.975531 0.219860i \(-0.929440\pi\)
0.975531 0.219860i \(-0.0705600\pi\)
\(332\) 2.00000i 0.109764i
\(333\) 1.00000i 0.0547997i
\(334\) 15.0000 0.820763
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 12.0000 5.00000i 0.652714 0.271964i
\(339\) 10.0000 0.543125
\(340\) 7.00000i 0.379628i
\(341\) −24.0000 −1.29967
\(342\) 3.00000 0.162221
\(343\) 0 0
\(344\) 5.00000i 0.269582i
\(345\) 1.00000i 0.0538382i
\(346\) 6.00000i 0.322562i
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 16.0000i 0.856460i −0.903670 0.428230i \(-0.859137\pi\)
0.903670 0.428230i \(-0.140863\pi\)
\(350\) 0 0
\(351\) 3.00000 + 2.00000i 0.160128 + 0.106752i
\(352\) −3.00000 −0.159901
\(353\) 16.0000i 0.851594i −0.904819 0.425797i \(-0.859994\pi\)
0.904819 0.425797i \(-0.140006\pi\)
\(354\) −10.0000 −0.531494
\(355\) −6.00000 −0.318447
\(356\) 12.0000i 0.635999i
\(357\) 0 0
\(358\) 6.00000i 0.317110i
\(359\) 26.0000i 1.37223i 0.727494 + 0.686114i \(0.240685\pi\)
−0.727494 + 0.686114i \(0.759315\pi\)
\(360\) 1.00000 0.0527046
\(361\) 10.0000 0.526316
\(362\) 26.0000i 1.36653i
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −13.0000 −0.680451
\(366\) 13.0000i 0.679521i
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 1.00000 0.0521286
\(369\) 4.00000i 0.208232i
\(370\) 1.00000i 0.0519875i
\(371\) 0 0
\(372\) 8.00000i 0.414781i
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 21.0000 1.08588
\(375\) 9.00000i 0.464758i
\(376\) 0 0
\(377\) 3.00000 + 2.00000i 0.154508 + 0.103005i
\(378\) 0 0
\(379\) 4.00000i 0.205466i −0.994709 0.102733i \(-0.967241\pi\)
0.994709 0.102733i \(-0.0327588\pi\)
\(380\) −3.00000 −0.153897
\(381\) 4.00000 0.204926
\(382\) 9.00000i 0.460480i
\(383\) 17.0000i 0.868659i 0.900754 + 0.434330i \(0.143015\pi\)
−0.900754 + 0.434330i \(0.856985\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) −5.00000 −0.254164
\(388\) 6.00000i 0.304604i
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −3.00000 2.00000i −0.151911 0.101274i
\(391\) −7.00000 −0.354005
\(392\) 0 0
\(393\) −17.0000 −0.857537
\(394\) −14.0000 −0.705310
\(395\) 12.0000i 0.603786i
\(396\) 3.00000i 0.150756i
\(397\) 8.00000i 0.401508i 0.979642 + 0.200754i \(0.0643393\pi\)
−0.979642 + 0.200754i \(0.935661\pi\)
\(398\) 9.00000i 0.451129i
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 6.00000i 0.299626i 0.988714 + 0.149813i \(0.0478671\pi\)
−0.988714 + 0.149813i \(0.952133\pi\)
\(402\) −8.00000 −0.399004
\(403\) −16.0000 + 24.0000i −0.797017 + 1.19553i
\(404\) 14.0000 0.696526
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) 7.00000i 0.346552i
\(409\) 29.0000i 1.43396i −0.697095 0.716979i \(-0.745524\pi\)
0.697095 0.716979i \(-0.254476\pi\)
\(410\) 4.00000i 0.197546i
\(411\) 17.0000i 0.838548i
\(412\) −1.00000 −0.0492665
\(413\) 0 0
\(414\) 1.00000i 0.0491473i
\(415\) 2.00000 0.0981761
\(416\) −2.00000 + 3.00000i −0.0980581 + 0.147087i
\(417\) −8.00000 −0.391762
\(418\) 9.00000i 0.440204i
\(419\) 11.0000 0.537385 0.268693 0.963226i \(-0.413408\pi\)
0.268693 + 0.963226i \(0.413408\pi\)
\(420\) 0 0
\(421\) 34.0000i 1.65706i 0.559946 + 0.828529i \(0.310822\pi\)
−0.559946 + 0.828529i \(0.689178\pi\)
\(422\) 15.0000i 0.730189i
\(423\) 0 0
\(424\) 6.00000i 0.291386i
\(425\) −28.0000 −1.35820
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) −2.00000 −0.0966736
\(429\) 6.00000 9.00000i 0.289683 0.434524i
\(430\) 5.00000 0.241121
\(431\) 14.0000i 0.674356i 0.941441 + 0.337178i \(0.109472\pi\)
−0.941441 + 0.337178i \(0.890528\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) 1.00000i 0.0479463i
\(436\) 7.00000i 0.335239i
\(437\) 3.00000i 0.143509i
\(438\) 13.0000 0.621164
\(439\) −27.0000 −1.28864 −0.644320 0.764756i \(-0.722859\pi\)
−0.644320 + 0.764756i \(0.722859\pi\)
\(440\) 3.00000i 0.143019i
\(441\) 0 0
\(442\) 14.0000 21.0000i 0.665912 0.998868i
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 1.00000i 0.0474579i
\(445\) 12.0000 0.568855
\(446\) 4.00000 0.189405
\(447\) 22.0000i 1.04056i
\(448\) 0 0
\(449\) 9.00000i 0.424736i 0.977190 + 0.212368i \(0.0681176\pi\)
−0.977190 + 0.212368i \(0.931882\pi\)
\(450\) 4.00000i 0.188562i
\(451\) 12.0000 0.565058
\(452\) 10.0000 0.470360
\(453\) 11.0000i 0.516825i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 28.0000i 1.30978i −0.755722 0.654892i \(-0.772714\pi\)
0.755722 0.654892i \(-0.227286\pi\)
\(458\) 10.0000 0.467269
\(459\) 7.00000 0.326732
\(460\) 1.00000i 0.0466252i
\(461\) 27.0000i 1.25752i −0.777601 0.628758i \(-0.783564\pi\)
0.777601 0.628758i \(-0.216436\pi\)
\(462\) 0 0
\(463\) 15.0000i 0.697109i 0.937288 + 0.348555i \(0.113327\pi\)
−0.937288 + 0.348555i \(0.886673\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 8.00000 0.370991
\(466\) 18.0000i 0.833834i
\(467\) −9.00000 −0.416470 −0.208235 0.978079i \(-0.566772\pi\)
−0.208235 + 0.978079i \(0.566772\pi\)
\(468\) 3.00000 + 2.00000i 0.138675 + 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) 19.0000 0.875474
\(472\) −10.0000 −0.460287
\(473\) 15.0000i 0.689701i
\(474\) 12.0000i 0.551178i
\(475\) 12.0000i 0.550598i
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −8.00000 −0.365911
\(479\) 3.00000i 0.137073i −0.997649 0.0685367i \(-0.978167\pi\)
0.997649 0.0685367i \(-0.0218330\pi\)
\(480\) 1.00000 0.0456435
\(481\) 2.00000 3.00000i 0.0911922 0.136788i
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 6.00000 0.272446
\(486\) 1.00000i 0.0453609i
\(487\) 24.0000i 1.08754i 0.839233 + 0.543772i \(0.183004\pi\)
−0.839233 + 0.543772i \(0.816996\pi\)
\(488\) 13.0000i 0.588482i
\(489\) 4.00000i 0.180886i
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 4.00000i 0.180334i
\(493\) 7.00000 0.315264
\(494\) −9.00000 6.00000i −0.404929 0.269953i
\(495\) −3.00000 −0.134840
\(496\) 8.00000i 0.359211i
\(497\) 0 0
\(498\) −2.00000 −0.0896221
\(499\) 14.0000i 0.626726i −0.949633 0.313363i \(-0.898544\pi\)
0.949633 0.313363i \(-0.101456\pi\)
\(500\) 9.00000i 0.402492i
\(501\) 15.0000i 0.670151i
\(502\) 17.0000i 0.758747i
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 14.0000i 0.622992i
\(506\) −3.00000 −0.133366
\(507\) −5.00000 12.0000i −0.222058 0.532939i
\(508\) 4.00000 0.177471
\(509\) 27.0000i 1.19675i −0.801215 0.598377i \(-0.795813\pi\)
0.801215 0.598377i \(-0.204187\pi\)
\(510\) −7.00000 −0.309965
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 3.00000i 0.132453i
\(514\) 6.00000i 0.264649i
\(515\) 1.00000i 0.0440653i
\(516\) −5.00000 −0.220113
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) −3.00000 2.00000i −0.131559 0.0877058i
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 1.00000i 0.0437688i
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −17.0000 −0.742648
\(525\) 0 0
\(526\) 24.0000i 1.04645i
\(527\) 56.0000i 2.43940i
\(528\) 3.00000i 0.130558i
\(529\) −22.0000 −0.956522
\(530\) 6.00000 0.260623
\(531\) 10.0000i 0.433963i
\(532\) 0 0
\(533\) 8.00000 12.0000i 0.346518 0.519778i
\(534\) −12.0000 −0.519291
\(535\) 2.00000i 0.0864675i
\(536\) −8.00000 −0.345547
\(537\) 6.00000 0.258919
\(538\) 4.00000i 0.172452i
\(539\) 0 0
\(540\) 1.00000i 0.0430331i
\(541\) 25.0000i 1.07483i −0.843317 0.537417i \(-0.819400\pi\)
0.843317 0.537417i \(-0.180600\pi\)
\(542\) 20.0000 0.859074
\(543\) −26.0000 −1.11577
\(544\) 7.00000i 0.300123i
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 17.0000i 0.726204i
\(549\) 13.0000 0.554826
\(550\) −12.0000 −0.511682
\(551\) 3.00000i 0.127804i
\(552\) 1.00000i 0.0425628i
\(553\) 0 0
\(554\) 28.0000i 1.18961i
\(555\) −1.00000 −0.0424476
\(556\) −8.00000 −0.339276
\(557\) 30.0000i 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) −8.00000 −0.338667
\(559\) 15.0000 + 10.0000i 0.634432 + 0.422955i
\(560\) 0 0
\(561\) 21.0000i 0.886621i
\(562\) 10.0000 0.421825
\(563\) −41.0000 −1.72794 −0.863972 0.503540i \(-0.832031\pi\)
−0.863972 + 0.503540i \(0.832031\pi\)
\(564\) 0 0
\(565\) 10.0000i 0.420703i
\(566\) 14.0000i 0.588464i
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 3.00000i 0.125656i
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 6.00000 9.00000i 0.250873 0.376309i
\(573\) 9.00000 0.375980
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) −1.00000 −0.0416667
\(577\) 38.0000i 1.58196i 0.611842 + 0.790980i \(0.290429\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 32.0000i 1.33102i
\(579\) 8.00000i 0.332469i
\(580\) 1.00000i 0.0415227i
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) 18.0000i 0.745484i
\(584\) 13.0000 0.537944
\(585\) −2.00000 + 3.00000i −0.0826898 + 0.124035i
\(586\) −14.0000 −0.578335
\(587\) 16.0000i 0.660391i 0.943913 + 0.330195i \(0.107115\pi\)
−0.943913 + 0.330195i \(0.892885\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 10.0000i 0.411693i
\(591\) 14.0000i 0.575883i
\(592\) 1.00000i 0.0410997i
\(593\) 16.0000i 0.657041i −0.944497 0.328521i \(-0.893450\pi\)
0.944497 0.328521i \(-0.106550\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 22.0000i 0.901155i
\(597\) 9.00000 0.368345
\(598\) −2.00000 + 3.00000i −0.0817861 + 0.122679i
\(599\) −3.00000 −0.122577 −0.0612883 0.998120i \(-0.519521\pi\)
−0.0612883 + 0.998120i \(0.519521\pi\)
\(600\) 4.00000i 0.163299i
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 8.00000i 0.325785i
\(604\) 11.0000i 0.447584i
\(605\) 2.00000i 0.0813116i
\(606\) 14.0000i 0.568711i
\(607\) 11.0000 0.446476 0.223238 0.974764i \(-0.428337\pi\)
0.223238 + 0.974764i \(0.428337\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) −13.0000 −0.526355
\(611\) 0 0
\(612\) 7.00000 0.282958
\(613\) 31.0000i 1.25208i −0.779792 0.626039i \(-0.784675\pi\)
0.779792 0.626039i \(-0.215325\pi\)
\(614\) 16.0000 0.645707
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) 37.0000i 1.48956i 0.667308 + 0.744782i \(0.267447\pi\)
−0.667308 + 0.744782i \(0.732553\pi\)
\(618\) 1.00000i 0.0402259i
\(619\) 23.0000i 0.924448i −0.886763 0.462224i \(-0.847052\pi\)
0.886763 0.462224i \(-0.152948\pi\)
\(620\) 8.00000 0.321288
\(621\) −1.00000 −0.0401286
\(622\) 4.00000i 0.160385i
\(623\) 0 0
\(624\) 3.00000 + 2.00000i 0.120096 + 0.0800641i
\(625\) 11.0000 0.440000
\(626\) 26.0000i 1.03917i
\(627\) −9.00000 −0.359425
\(628\) 19.0000 0.758183
\(629\) 7.00000i 0.279108i
\(630\) 0 0
\(631\) 47.0000i 1.87104i 0.353273 + 0.935520i \(0.385069\pi\)
−0.353273 + 0.935520i \(0.614931\pi\)
\(632\) 12.0000i 0.477334i
\(633\) −15.0000 −0.596196
\(634\) −24.0000 −0.953162
\(635\) 4.00000i 0.158735i
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 3.00000 0.118771
\(639\) 6.00000i 0.237356i
\(640\) 1.00000 0.0395285
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 2.00000i 0.0789337i
\(643\) 39.0000i 1.53801i −0.639243 0.769005i \(-0.720752\pi\)
0.639243 0.769005i \(-0.279248\pi\)
\(644\) 0 0
\(645\) 5.00000i 0.196875i
\(646\) −21.0000 −0.826234
\(647\) −22.0000 −0.864909 −0.432455 0.901656i \(-0.642352\pi\)
−0.432455 + 0.901656i \(0.642352\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 30.0000 1.17760
\(650\) −8.00000 + 12.0000i −0.313786 + 0.470679i
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) −35.0000 −1.36966 −0.684828 0.728705i \(-0.740123\pi\)
−0.684828 + 0.728705i \(0.740123\pi\)
\(654\) 7.00000 0.273722
\(655\) 17.0000i 0.664245i
\(656\) 4.00000i 0.156174i
\(657\) 13.0000i 0.507178i
\(658\) 0 0
\(659\) 26.0000 1.01282 0.506408 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) −3.00000 −0.116775
\(661\) 10.0000i 0.388955i −0.980907 0.194477i \(-0.937699\pi\)
0.980907 0.194477i \(-0.0623011\pi\)
\(662\) −8.00000 −0.310929
\(663\) −21.0000 14.0000i −0.815572 0.543715i
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) −1.00000 −0.0387202
\(668\) 15.0000i 0.580367i
\(669\) 4.00000i 0.154649i
\(670\) 8.00000i 0.309067i
\(671\) 39.0000i 1.50558i
\(672\) 0 0
\(673\) −3.00000 −0.115642 −0.0578208 0.998327i \(-0.518415\pi\)
−0.0578208 + 0.998327i \(0.518415\pi\)
\(674\) 23.0000i 0.885927i
\(675\) −4.00000 −0.153960
\(676\) −5.00000 12.0000i −0.192308 0.461538i
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 10.0000i 0.384048i
\(679\) 0 0
\(680\) −7.00000 −0.268438
\(681\) 20.0000i 0.766402i
\(682\) 24.0000i 0.919007i
\(683\) 9.00000i 0.344375i −0.985064 0.172188i \(-0.944916\pi\)
0.985064 0.172188i \(-0.0550836\pi\)
\(684\) 3.00000i 0.114708i
\(685\) 17.0000 0.649537
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) −5.00000 −0.190623
\(689\) 18.0000 + 12.0000i 0.685745 + 0.457164i
\(690\) 1.00000 0.0380693
\(691\) 16.0000i 0.608669i −0.952565 0.304334i \(-0.901566\pi\)
0.952565 0.304334i \(-0.0984340\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 24.0000i 0.911028i
\(695\) 8.00000i 0.303457i
\(696\) 1.00000i 0.0379049i
\(697\) 28.0000i 1.06058i
\(698\) −16.0000 −0.605609
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 2.00000 3.00000i 0.0754851 0.113228i
\(703\) −3.00000 −0.113147
\(704\) 3.00000i 0.113067i
\(705\) 0 0
\(706\) −16.0000 −0.602168
\(707\) 0 0
\(708\) 10.0000i 0.375823i
\(709\) 30.0000i 1.12667i 0.826227 + 0.563337i \(0.190483\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 6.00000i 0.225176i
\(711\) −12.0000 −0.450035
\(712\) −12.0000 −0.449719
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) 9.00000 + 6.00000i 0.336581 + 0.224387i
\(716\) 6.00000 0.224231
\(717\) 8.00000i 0.298765i
\(718\) 26.0000 0.970311
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 0 0
\(722\) 10.0000i 0.372161i
\(723\) 26.0000i 0.966950i
\(724\) −26.0000 −0.966282
\(725\) −4.00000 −0.148556
\(726\) 2.00000i 0.0742270i
\(727\) −35.0000 −1.29808 −0.649039 0.760755i \(-0.724829\pi\)
−0.649039 + 0.760755i \(0.724829\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 13.0000i 0.481152i
\(731\) 35.0000 1.29452
\(732\) 13.0000 0.480494
\(733\) 8.00000i 0.295487i −0.989026 0.147743i \(-0.952799\pi\)
0.989026 0.147743i \(-0.0472010\pi\)
\(734\) 32.0000i 1.18114i
\(735\) 0 0
\(736\) 1.00000i 0.0368605i
\(737\) 24.0000 0.884051
\(738\) 4.00000 0.147242
\(739\) 16.0000i 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) −1.00000 −0.0367607
\(741\) −6.00000 + 9.00000i −0.220416 + 0.330623i
\(742\) 0 0
\(743\) 40.0000i 1.46746i −0.679442 0.733729i \(-0.737778\pi\)
0.679442 0.733729i \(-0.262222\pi\)
\(744\) −8.00000 −0.293294
\(745\) 22.0000 0.806018
\(746\) 6.00000i 0.219676i
\(747\) 2.00000i 0.0731762i
\(748\) 21.0000i 0.767836i
\(749\) 0 0
\(750\) 9.00000 0.328634
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 17.0000 0.619514
\(754\) 2.00000 3.00000i 0.0728357 0.109254i
\(755\) 11.0000 0.400331
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −4.00000 −0.145287
\(759\) 3.00000i 0.108893i
\(760\) 3.00000i 0.108821i
\(761\) 30.0000i 1.08750i −0.839248 0.543750i \(-0.817004\pi\)
0.839248 0.543750i \(-0.182996\pi\)
\(762\) 4.00000i 0.144905i
\(763\) 0 0
\(764\) 9.00000 0.325609
\(765\) 7.00000i 0.253086i
\(766\) 17.0000 0.614235
\(767\) 20.0000 30.0000i 0.722158 1.08324i
\(768\) −1.00000 −0.0360844
\(769\) 39.0000i 1.40638i 0.711004 + 0.703188i \(0.248241\pi\)
−0.711004 + 0.703188i \(0.751759\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 8.00000i 0.287926i
\(773\) 21.0000i 0.755318i 0.925945 + 0.377659i \(0.123271\pi\)
−0.925945 + 0.377659i \(0.876729\pi\)
\(774\) 5.00000i 0.179721i
\(775\) 32.0000i 1.14947i
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) 18.0000i 0.645331i
\(779\) −12.0000 −0.429945
\(780\) −2.00000 + 3.00000i −0.0716115 + 0.107417i
\(781\) −18.0000 −0.644091
\(782\) 7.00000i 0.250319i
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 19.0000i 0.678139i
\(786\) 17.0000i 0.606370i
\(787\) 17.0000i 0.605985i 0.952993 + 0.302992i \(0.0979856\pi\)
−0.952993 + 0.302992i \(0.902014\pi\)
\(788\) 14.0000i 0.498729i
\(789\) 24.0000 0.854423
\(790\) 12.0000 0.426941
\(791\) 0 0
\(792\) 3.00000 0.106600
\(793\) −39.0000 26.0000i −1.38493 0.923287i
\(794\) 8.00000 0.283909
\(795\) 6.00000i 0.212798i
\(796\) 9.00000 0.318997
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000i 0.141421i
\(801\) 12.0000i 0.423999i
\(802\) 6.00000 0.211867
\(803\) −39.0000 −1.37628
\(804\) 8.00000i 0.282138i
\(805\) 0 0
\(806\) 24.0000 + 16.0000i 0.845364 + 0.563576i
\(807\) −4.00000 −0.140807
\(808\) 14.0000i 0.492518i
\(809\) 52.0000 1.82822 0.914111 0.405463i \(-0.132890\pi\)
0.914111 + 0.405463i \(0.132890\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 7.00000i 0.245803i 0.992419 + 0.122902i \(0.0392200\pi\)
−0.992419 + 0.122902i \(0.960780\pi\)
\(812\) 0 0
\(813\) 20.0000i 0.701431i
\(814\) 3.00000i 0.105150i
\(815\) −4.00000 −0.140114
\(816\) 7.00000 0.245049
\(817\) 15.0000i 0.524784i
\(818\) −29.0000 −1.01396
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 36.0000i 1.25641i −0.778048 0.628204i \(-0.783790\pi\)
0.778048 0.628204i \(-0.216210\pi\)
\(822\) −17.0000 −0.592943
\(823\) 12.0000 0.418294 0.209147 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) 1.00000i 0.0348367i
\(825\) 12.0000i 0.417786i
\(826\) 0 0
\(827\) 9.00000i 0.312961i 0.987681 + 0.156480i \(0.0500148\pi\)
−0.987681 + 0.156480i \(0.949985\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 2.00000i 0.0694210i
\(831\) −28.0000 −0.971309
\(832\) 3.00000 + 2.00000i 0.104006 + 0.0693375i
\(833\) 0 0
\(834\) 8.00000i 0.277017i
\(835\) 15.0000 0.519096
\(836\) −9.00000 −0.311272
\(837\) 8.00000i 0.276520i
\(838\) 11.0000i 0.379989i
\(839\) 28.0000i 0.966667i −0.875436 0.483334i \(-0.839426\pi\)
0.875436 0.483334i \(-0.160574\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 34.0000 1.17172
\(843\) 10.0000i 0.344418i
\(844\) −15.0000 −0.516321
\(845\) 12.0000 5.00000i 0.412813 0.172005i
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −14.0000 −0.480479
\(850\) 28.0000i 0.960392i
\(851\) 1.00000i 0.0342796i
\(852\) 6.00000i 0.205557i
\(853\) 32.0000i 1.09566i 0.836590 + 0.547830i \(0.184546\pi\)
−0.836590 + 0.547830i \(0.815454\pi\)
\(854\) 0 0
\(855\) 3.00000 0.102598
\(856\) 2.00000i 0.0683586i
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) −9.00000 6.00000i −0.307255 0.204837i
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 5.00000i 0.170499i
\(861\) 0 0
\(862\) 14.0000 0.476842
\(863\) 28.0000i 0.953131i −0.879139 0.476566i \(-0.841881\pi\)
0.879139 0.476566i \(-0.158119\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 6.00000i 0.204006i
\(866\) 8.00000i 0.271851i
\(867\) −32.0000 −1.08678
\(868\) 0 0
\(869\) 36.0000i 1.22122i
\(870\) −1.00000 −0.0339032
\(871\) 16.0000 24.0000i 0.542139 0.813209i
\(872\) 7.00000 0.237050
\(873\) 6.00000i 0.203069i
\(874\) 3.00000 0.101477
\(875\) 0 0
\(876\) 13.0000i 0.439229i
\(877\) 2.00000i 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) 27.0000i 0.911206i
\(879\) 14.0000i 0.472208i
\(880\) −3.00000 −0.101130
\(881\) −51.0000 −1.71823 −0.859117 0.511780i \(-0.828986\pi\)
−0.859117 + 0.511780i \(0.828986\pi\)
\(882\) 0 0
\(883\) 3.00000 0.100958 0.0504790 0.998725i \(-0.483925\pi\)
0.0504790 + 0.998725i \(0.483925\pi\)
\(884\) −21.0000 14.0000i −0.706306 0.470871i
\(885\) −10.0000 −0.336146
\(886\) 34.0000i 1.14225i
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 1.00000 0.0335578
\(889\) 0 0
\(890\) 12.0000i 0.402241i
\(891\) 3.00000i 0.100504i
\(892\) 4.00000i 0.133930i
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) 6.00000i 0.200558i
\(896\) 0 0
\(897\) 3.00000 + 2.00000i 0.100167 + 0.0667781i
\(898\) 9.00000 0.300334
\(899\) 8.00000i 0.266815i
\(900\) −4.00000 −0.133333
\(901\) 42.0000 1.39922
\(902\) 12.0000i 0.399556i
\(903\) 0 0
\(904\) 10.0000i 0.332595i
\(905\) 26.0000i 0.864269i
\(906\) −11.0000 −0.365451
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 20.0000i 0.663723i
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 57.0000 1.88849 0.944247 0.329238i \(-0.106792\pi\)
0.944247 + 0.329238i \(0.106792\pi\)
\(912\) 3.00000i 0.0993399i
\(913\) 6.00000 0.198571
\(914\) −28.0000 −0.926158
\(915\) 13.0000i 0.429767i
\(916\) 10.0000i 0.330409i
\(917\) 0 0
\(918\) 7.00000i 0.231034i
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 1.00000 0.0329690
\(921\) 16.0000i 0.527218i
\(922\) −27.0000 −0.889198
\(923\) −12.0000 + 18.0000i −0.394985 + 0.592477i
\(924\) 0 0
\(925\) 4.00000i 0.131519i
\(926\) 15.0000 0.492931
\(927\) 1.00000 0.0328443
\(928\) 1.00000i 0.0328266i
\(929\) 42.0000i 1.37798i −0.724773 0.688988i \(-0.758055\pi\)
0.724773 0.688988i \(-0.241945\pi\)
\(930\) 8.00000i 0.262330i
\(931\) 0 0
\(932\) −18.0000 −0.589610
\(933\) −4.00000 −0.130954
\(934\) 9.00000i 0.294489i
\(935\) 21.0000 0.686773
\(936\) 2.00000 3.00000i 0.0653720 0.0980581i
\(937\) −4.00000 −0.130674 −0.0653372 0.997863i \(-0.520812\pi\)
−0.0653372 + 0.997863i \(0.520812\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 38.0000i 1.23876i 0.785090 + 0.619382i \(0.212617\pi\)
−0.785090 + 0.619382i \(0.787383\pi\)
\(942\) 19.0000i 0.619053i
\(943\) 4.00000i 0.130258i
\(944\) 10.0000i 0.325472i
\(945\) 0 0
\(946\) 15.0000 0.487692
\(947\) 25.0000i 0.812391i 0.913786 + 0.406195i \(0.133145\pi\)
−0.913786 + 0.406195i \(0.866855\pi\)
\(948\) −12.0000 −0.389742
\(949\) −26.0000 + 39.0000i −0.843996 + 1.26599i
\(950\) 12.0000 0.389331
\(951\) 24.0000i 0.778253i
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 6.00000i 0.194257i
\(955\) 9.00000i 0.291233i
\(956\) 8.00000i 0.258738i
\(957\) 3.00000i 0.0969762i
\(958\) −3.00000 −0.0969256
\(959\) 0 0
\(960\) 1.00000i 0.0322749i
\(961\) −33.0000 −1.06452
\(962\) −3.00000 2.00000i −0.0967239 0.0644826i
\(963\) 2.00000 0.0644491
\(964\) 26.0000i 0.837404i
\(965\) 8.00000 0.257529
\(966\) 0 0
\(967\) 43.0000i 1.38279i 0.722478 + 0.691393i \(0.243003\pi\)
−0.722478 + 0.691393i \(0.756997\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 21.0000i 0.674617i
\(970\) 6.00000i 0.192648i
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 24.0000 0.769010
\(975\) 12.0000 + 8.00000i 0.384308 + 0.256205i
\(976\) 13.0000 0.416120
\(977\) 15.0000i 0.479893i −0.970786 0.239946i \(-0.922870\pi\)
0.970786 0.239946i \(-0.0771298\pi\)
\(978\) 4.00000 0.127906
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 7.00000i 0.223493i
\(982\) 40.0000i 1.27645i
\(983\) 51.0000i 1.62665i −0.581811 0.813324i \(-0.697656\pi\)
0.581811 0.813324i \(-0.302344\pi\)
\(984\) 4.00000 0.127515
\(985\) −14.0000 −0.446077
\(986\) 7.00000i 0.222925i
\(987\) 0 0
\(988\) −6.00000 + 9.00000i −0.190885 + 0.286328i
\(989\) −5.00000 −0.158991
\(990\) 3.00000i 0.0953463i
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −8.00000 −0.254000
\(993\) 8.00000i 0.253872i
\(994\) 0 0
\(995\) 9.00000i 0.285319i
\(996\) 2.00000i 0.0633724i
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −14.0000 −0.443162
\(999\) 1.00000i 0.0316386i
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 3822.2.c.b.883.1 2
7.6 odd 2 546.2.c.c.337.1 2
13.12 even 2 inner 3822.2.c.b.883.2 2
21.20 even 2 1638.2.c.e.883.2 2
28.27 even 2 4368.2.h.h.337.2 2
91.34 even 4 7098.2.a.y.1.1 1
91.83 even 4 7098.2.a.o.1.1 1
91.90 odd 2 546.2.c.c.337.2 yes 2
273.272 even 2 1638.2.c.e.883.1 2
364.363 even 2 4368.2.h.h.337.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.c.337.1 2 7.6 odd 2
546.2.c.c.337.2 yes 2 91.90 odd 2
1638.2.c.e.883.1 2 273.272 even 2
1638.2.c.e.883.2 2 21.20 even 2
3822.2.c.b.883.1 2 1.1 even 1 trivial
3822.2.c.b.883.2 2 13.12 even 2 inner
4368.2.h.h.337.1 2 364.363 even 2
4368.2.h.h.337.2 2 28.27 even 2
7098.2.a.o.1.1 1 91.83 even 4
7098.2.a.y.1.1 1 91.34 even 4