Properties

Label 3822.2.a.bx.1.4
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3822,2,Mod(1,3822)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3822.1");
 
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.a (trivial)

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: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.10304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.16053\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.16053 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.16053 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.16053 q^{10} +1.51929 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.16053 q^{15} +1.00000 q^{16} +5.84035 q^{17} -1.00000 q^{18} -1.45774 q^{19} +1.16053 q^{20} -1.51929 q^{22} +5.77880 q^{23} +1.00000 q^{24} -3.65317 q^{25} -1.00000 q^{26} -1.00000 q^{27} +3.59771 q^{29} +1.16053 q^{30} +7.29809 q^{31} -1.00000 q^{32} -1.51929 q^{33} -5.84035 q^{34} +1.00000 q^{36} +1.95648 q^{37} +1.45774 q^{38} -1.00000 q^{39} -1.16053 q^{40} -9.88388 q^{41} -1.77880 q^{43} +1.51929 q^{44} +1.16053 q^{45} -5.77880 q^{46} -11.5867 q^{47} -1.00000 q^{48} +3.65317 q^{50} -5.84035 q^{51} +1.00000 q^{52} +9.58667 q^{53} +1.00000 q^{54} +1.76319 q^{55} +1.45774 q^{57} -3.59771 q^{58} +0.0156151 q^{59} -1.16053 q^{60} +9.20988 q^{61} -7.29809 q^{62} +1.00000 q^{64} +1.16053 q^{65} +1.51929 q^{66} +4.07842 q^{67} +5.84035 q^{68} -5.77880 q^{69} -0.358761 q^{71} -1.00000 q^{72} +3.53491 q^{73} -1.95648 q^{74} +3.65317 q^{75} -1.45774 q^{76} +1.00000 q^{78} +10.2374 q^{79} +1.16053 q^{80} +1.00000 q^{81} +9.88388 q^{82} -7.33579 q^{83} +6.77792 q^{85} +1.77880 q^{86} -3.59771 q^{87} -1.51929 q^{88} -9.20406 q^{89} -1.16053 q^{90} +5.77880 q^{92} -7.29809 q^{93} +11.5867 q^{94} -1.69175 q^{95} +1.00000 q^{96} +5.19543 q^{97} +1.51929 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} + 6 q^{10} + 2 q^{11} - 4 q^{12} + 4 q^{13} + 6 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 2 q^{19} - 6 q^{20} - 2 q^{22} - 2 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{26} - 4 q^{27} + 6 q^{29} - 6 q^{30} - 4 q^{32} - 2 q^{33} + 2 q^{34} + 4 q^{36} + 6 q^{37} + 2 q^{38} - 4 q^{39} + 6 q^{40} - 16 q^{41} + 18 q^{43} + 2 q^{44} - 6 q^{45} + 2 q^{46} - 16 q^{47} - 4 q^{48} - 6 q^{50} + 2 q^{51} + 4 q^{52} + 8 q^{53} + 4 q^{54} - 2 q^{55} + 2 q^{57} - 6 q^{58} - 16 q^{59} + 6 q^{60} + 2 q^{61} + 4 q^{64} - 6 q^{65} + 2 q^{66} + 12 q^{67} - 2 q^{68} + 2 q^{69} - 8 q^{71} - 4 q^{72} - 6 q^{73} - 6 q^{74} - 6 q^{75} - 2 q^{76} + 4 q^{78} - 24 q^{79} - 6 q^{80} + 4 q^{81} + 16 q^{82} - 28 q^{83} + 38 q^{85} - 18 q^{86} - 6 q^{87} - 2 q^{88} - 28 q^{89} + 6 q^{90} - 2 q^{92} + 16 q^{94} + 22 q^{95} + 4 q^{96} + 4 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.16053 0.519005 0.259503 0.965742i \(-0.416441\pi\)
0.259503 + 0.965742i \(0.416441\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.16053 −0.366992
\(11\) 1.51929 0.458084 0.229042 0.973417i \(-0.426441\pi\)
0.229042 + 0.973417i \(0.426441\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.16053 −0.299648
\(16\) 1.00000 0.250000
\(17\) 5.84035 1.41649 0.708247 0.705965i \(-0.249486\pi\)
0.708247 + 0.705965i \(0.249486\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.45774 −0.334428 −0.167214 0.985921i \(-0.553477\pi\)
−0.167214 + 0.985921i \(0.553477\pi\)
\(20\) 1.16053 0.259503
\(21\) 0 0
\(22\) −1.51929 −0.323914
\(23\) 5.77880 1.20496 0.602482 0.798133i \(-0.294179\pi\)
0.602482 + 0.798133i \(0.294179\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.65317 −0.730633
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.59771 0.668079 0.334039 0.942559i \(-0.391588\pi\)
0.334039 + 0.942559i \(0.391588\pi\)
\(30\) 1.16053 0.211883
\(31\) 7.29809 1.31078 0.655388 0.755292i \(-0.272505\pi\)
0.655388 + 0.755292i \(0.272505\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.51929 −0.264475
\(34\) −5.84035 −1.00161
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.95648 0.321643 0.160821 0.986984i \(-0.448586\pi\)
0.160821 + 0.986984i \(0.448586\pi\)
\(38\) 1.45774 0.236476
\(39\) −1.00000 −0.160128
\(40\) −1.16053 −0.183496
\(41\) −9.88388 −1.54360 −0.771801 0.635864i \(-0.780644\pi\)
−0.771801 + 0.635864i \(0.780644\pi\)
\(42\) 0 0
\(43\) −1.77880 −0.271265 −0.135632 0.990759i \(-0.543307\pi\)
−0.135632 + 0.990759i \(0.543307\pi\)
\(44\) 1.51929 0.229042
\(45\) 1.16053 0.173002
\(46\) −5.77880 −0.852038
\(47\) −11.5867 −1.69009 −0.845045 0.534695i \(-0.820426\pi\)
−0.845045 + 0.534695i \(0.820426\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 3.65317 0.516636
\(51\) −5.84035 −0.817813
\(52\) 1.00000 0.138675
\(53\) 9.58667 1.31683 0.658415 0.752655i \(-0.271227\pi\)
0.658415 + 0.752655i \(0.271227\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.76319 0.237748
\(56\) 0 0
\(57\) 1.45774 0.193082
\(58\) −3.59771 −0.472403
\(59\) 0.0156151 0.00203291 0.00101645 0.999999i \(-0.499676\pi\)
0.00101645 + 0.999999i \(0.499676\pi\)
\(60\) −1.16053 −0.149824
\(61\) 9.20988 1.17920 0.589602 0.807694i \(-0.299284\pi\)
0.589602 + 0.807694i \(0.299284\pi\)
\(62\) −7.29809 −0.926859
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.16053 0.143946
\(66\) 1.51929 0.187012
\(67\) 4.07842 0.498259 0.249129 0.968470i \(-0.419856\pi\)
0.249129 + 0.968470i \(0.419856\pi\)
\(68\) 5.84035 0.708247
\(69\) −5.77880 −0.695686
\(70\) 0 0
\(71\) −0.358761 −0.0425771 −0.0212885 0.999773i \(-0.506777\pi\)
−0.0212885 + 0.999773i \(0.506777\pi\)
\(72\) −1.00000 −0.117851
\(73\) 3.53491 0.413730 0.206865 0.978370i \(-0.433674\pi\)
0.206865 + 0.978370i \(0.433674\pi\)
\(74\) −1.95648 −0.227436
\(75\) 3.65317 0.421831
\(76\) −1.45774 −0.167214
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 10.2374 1.15180 0.575900 0.817520i \(-0.304652\pi\)
0.575900 + 0.817520i \(0.304652\pi\)
\(80\) 1.16053 0.129751
\(81\) 1.00000 0.111111
\(82\) 9.88388 1.09149
\(83\) −7.33579 −0.805208 −0.402604 0.915374i \(-0.631895\pi\)
−0.402604 + 0.915374i \(0.631895\pi\)
\(84\) 0 0
\(85\) 6.77792 0.735168
\(86\) 1.77880 0.191813
\(87\) −3.59771 −0.385715
\(88\) −1.51929 −0.161957
\(89\) −9.20406 −0.975628 −0.487814 0.872948i \(-0.662206\pi\)
−0.487814 + 0.872948i \(0.662206\pi\)
\(90\) −1.16053 −0.122331
\(91\) 0 0
\(92\) 5.77880 0.602482
\(93\) −7.29809 −0.756777
\(94\) 11.5867 1.19507
\(95\) −1.69175 −0.173570
\(96\) 1.00000 0.102062
\(97\) 5.19543 0.527516 0.263758 0.964589i \(-0.415038\pi\)
0.263758 + 0.964589i \(0.415038\pi\)
\(98\) 0 0
\(99\) 1.51929 0.152695
\(100\) −3.65317 −0.365317
\(101\) 12.0273 1.19676 0.598379 0.801213i \(-0.295812\pi\)
0.598379 + 0.801213i \(0.295812\pi\)
\(102\) 5.84035 0.578281
\(103\) −7.81128 −0.769669 −0.384834 0.922986i \(-0.625741\pi\)
−0.384834 + 0.922986i \(0.625741\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −9.58667 −0.931140
\(107\) −5.10480 −0.493500 −0.246750 0.969079i \(-0.579363\pi\)
−0.246750 + 0.969079i \(0.579363\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.20037 0.593888 0.296944 0.954895i \(-0.404033\pi\)
0.296944 + 0.954895i \(0.404033\pi\)
\(110\) −1.76319 −0.168113
\(111\) −1.95648 −0.185700
\(112\) 0 0
\(113\) −17.0659 −1.60542 −0.802710 0.596369i \(-0.796609\pi\)
−0.802710 + 0.596369i \(0.796609\pi\)
\(114\) −1.45774 −0.136530
\(115\) 6.70648 0.625382
\(116\) 3.59771 0.334039
\(117\) 1.00000 0.0924500
\(118\) −0.0156151 −0.00143748
\(119\) 0 0
\(120\) 1.16053 0.105942
\(121\) −8.69175 −0.790159
\(122\) −9.20988 −0.833824
\(123\) 9.88388 0.891199
\(124\) 7.29809 0.655388
\(125\) −10.0423 −0.898208
\(126\) 0 0
\(127\) 17.1651 1.52316 0.761578 0.648073i \(-0.224425\pi\)
0.761578 + 0.648073i \(0.224425\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.77880 0.156615
\(130\) −1.16053 −0.101785
\(131\) −12.9127 −1.12819 −0.564093 0.825711i \(-0.690774\pi\)
−0.564093 + 0.825711i \(0.690774\pi\)
\(132\) −1.51929 −0.132237
\(133\) 0 0
\(134\) −4.07842 −0.352322
\(135\) −1.16053 −0.0998826
\(136\) −5.84035 −0.500806
\(137\) 10.4348 0.891503 0.445751 0.895157i \(-0.352937\pi\)
0.445751 + 0.895157i \(0.352937\pi\)
\(138\) 5.77880 0.491924
\(139\) −20.7071 −1.75635 −0.878176 0.478337i \(-0.841240\pi\)
−0.878176 + 0.478337i \(0.841240\pi\)
\(140\) 0 0
\(141\) 11.5867 0.975774
\(142\) 0.358761 0.0301065
\(143\) 1.51929 0.127050
\(144\) 1.00000 0.0833333
\(145\) 4.17526 0.346736
\(146\) −3.53491 −0.292551
\(147\) 0 0
\(148\) 1.95648 0.160821
\(149\) −2.50736 −0.205411 −0.102706 0.994712i \(-0.532750\pi\)
−0.102706 + 0.994712i \(0.532750\pi\)
\(150\) −3.65317 −0.298280
\(151\) 1.67614 0.136402 0.0682010 0.997672i \(-0.478274\pi\)
0.0682010 + 0.997672i \(0.478274\pi\)
\(152\) 1.45774 0.118238
\(153\) 5.84035 0.472165
\(154\) 0 0
\(155\) 8.46967 0.680300
\(156\) −1.00000 −0.0800641
\(157\) 10.8551 0.866330 0.433165 0.901315i \(-0.357397\pi\)
0.433165 + 0.901315i \(0.357397\pi\)
\(158\) −10.2374 −0.814446
\(159\) −9.58667 −0.760273
\(160\) −1.16053 −0.0917481
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 8.00699 0.627156 0.313578 0.949563i \(-0.398472\pi\)
0.313578 + 0.949563i \(0.398472\pi\)
\(164\) −9.88388 −0.771801
\(165\) −1.76319 −0.137264
\(166\) 7.33579 0.569368
\(167\) −0.339091 −0.0262397 −0.0131198 0.999914i \(-0.504176\pi\)
−0.0131198 + 0.999914i \(0.504176\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −6.77792 −0.519842
\(171\) −1.45774 −0.111476
\(172\) −1.77880 −0.135632
\(173\) 2.22577 0.169222 0.0846112 0.996414i \(-0.473035\pi\)
0.0846112 + 0.996414i \(0.473035\pi\)
\(174\) 3.59771 0.272742
\(175\) 0 0
\(176\) 1.51929 0.114521
\(177\) −0.0156151 −0.00117370
\(178\) 9.20406 0.689873
\(179\) −6.45280 −0.482305 −0.241152 0.970487i \(-0.577525\pi\)
−0.241152 + 0.970487i \(0.577525\pi\)
\(180\) 1.16053 0.0865009
\(181\) −0.0238555 −0.00177317 −0.000886584 1.00000i \(-0.500282\pi\)
−0.000886584 1.00000i \(0.500282\pi\)
\(182\) 0 0
\(183\) −9.20988 −0.680814
\(184\) −5.77880 −0.426019
\(185\) 2.27055 0.166934
\(186\) 7.29809 0.535122
\(187\) 8.87321 0.648873
\(188\) −11.5867 −0.845045
\(189\) 0 0
\(190\) 1.69175 0.122733
\(191\) 9.20164 0.665808 0.332904 0.942961i \(-0.391972\pi\)
0.332904 + 0.942961i \(0.391972\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 23.2522 1.67373 0.836863 0.547412i \(-0.184387\pi\)
0.836863 + 0.547412i \(0.184387\pi\)
\(194\) −5.19543 −0.373010
\(195\) −1.16053 −0.0831074
\(196\) 0 0
\(197\) 21.7521 1.54978 0.774888 0.632098i \(-0.217806\pi\)
0.774888 + 0.632098i \(0.217806\pi\)
\(198\) −1.51929 −0.107971
\(199\) 17.4902 1.23985 0.619924 0.784661i \(-0.287163\pi\)
0.619924 + 0.784661i \(0.287163\pi\)
\(200\) 3.65317 0.258318
\(201\) −4.07842 −0.287670
\(202\) −12.0273 −0.846236
\(203\) 0 0
\(204\) −5.84035 −0.408907
\(205\) −11.4706 −0.801138
\(206\) 7.81128 0.544238
\(207\) 5.77880 0.401654
\(208\) 1.00000 0.0693375
\(209\) −2.21473 −0.153196
\(210\) 0 0
\(211\) 4.87070 0.335313 0.167656 0.985845i \(-0.446380\pi\)
0.167656 + 0.985845i \(0.446380\pi\)
\(212\) 9.58667 0.658415
\(213\) 0.358761 0.0245819
\(214\) 5.10480 0.348957
\(215\) −2.06435 −0.140788
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −6.20037 −0.419942
\(219\) −3.53491 −0.238867
\(220\) 1.76319 0.118874
\(221\) 5.84035 0.392865
\(222\) 1.95648 0.131310
\(223\) −8.01165 −0.536500 −0.268250 0.963349i \(-0.586445\pi\)
−0.268250 + 0.963349i \(0.586445\pi\)
\(224\) 0 0
\(225\) −3.65317 −0.243544
\(226\) 17.0659 1.13520
\(227\) −23.8532 −1.58319 −0.791595 0.611046i \(-0.790749\pi\)
−0.791595 + 0.611046i \(0.790749\pi\)
\(228\) 1.45774 0.0965411
\(229\) 14.4935 0.957759 0.478879 0.877881i \(-0.341043\pi\)
0.478879 + 0.877881i \(0.341043\pi\)
\(230\) −6.70648 −0.442212
\(231\) 0 0
\(232\) −3.59771 −0.236202
\(233\) 17.0723 1.11844 0.559222 0.829018i \(-0.311100\pi\)
0.559222 + 0.829018i \(0.311100\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −13.4467 −0.877166
\(236\) 0.0156151 0.00101645
\(237\) −10.2374 −0.664992
\(238\) 0 0
\(239\) −19.8943 −1.28685 −0.643427 0.765507i \(-0.722488\pi\)
−0.643427 + 0.765507i \(0.722488\pi\)
\(240\) −1.16053 −0.0749120
\(241\) −6.03374 −0.388667 −0.194334 0.980936i \(-0.562254\pi\)
−0.194334 + 0.980936i \(0.562254\pi\)
\(242\) 8.69175 0.558727
\(243\) −1.00000 −0.0641500
\(244\) 9.20988 0.589602
\(245\) 0 0
\(246\) −9.88388 −0.630173
\(247\) −1.45774 −0.0927537
\(248\) −7.29809 −0.463429
\(249\) 7.33579 0.464887
\(250\) 10.0423 0.635129
\(251\) 10.5817 0.667913 0.333957 0.942588i \(-0.391616\pi\)
0.333957 + 0.942588i \(0.391616\pi\)
\(252\) 0 0
\(253\) 8.77969 0.551974
\(254\) −17.1651 −1.07703
\(255\) −6.77792 −0.424450
\(256\) 1.00000 0.0625000
\(257\) −2.27512 −0.141918 −0.0709592 0.997479i \(-0.522606\pi\)
−0.0709592 + 0.997479i \(0.522606\pi\)
\(258\) −1.77880 −0.110743
\(259\) 0 0
\(260\) 1.16053 0.0719731
\(261\) 3.59771 0.222693
\(262\) 12.9127 0.797748
\(263\) 14.9853 0.924031 0.462016 0.886872i \(-0.347126\pi\)
0.462016 + 0.886872i \(0.347126\pi\)
\(264\) 1.51929 0.0935060
\(265\) 11.1256 0.683442
\(266\) 0 0
\(267\) 9.20406 0.563279
\(268\) 4.07842 0.249129
\(269\) −4.34656 −0.265014 −0.132507 0.991182i \(-0.542303\pi\)
−0.132507 + 0.991182i \(0.542303\pi\)
\(270\) 1.16053 0.0706277
\(271\) −5.32195 −0.323285 −0.161643 0.986849i \(-0.551679\pi\)
−0.161643 + 0.986849i \(0.551679\pi\)
\(272\) 5.84035 0.354124
\(273\) 0 0
\(274\) −10.4348 −0.630387
\(275\) −5.55023 −0.334691
\(276\) −5.77880 −0.347843
\(277\) −22.0052 −1.32216 −0.661082 0.750314i \(-0.729902\pi\)
−0.661082 + 0.750314i \(0.729902\pi\)
\(278\) 20.7071 1.24193
\(279\) 7.29809 0.436925
\(280\) 0 0
\(281\) 21.6348 1.29062 0.645311 0.763920i \(-0.276728\pi\)
0.645311 + 0.763920i \(0.276728\pi\)
\(282\) −11.5867 −0.689976
\(283\) −16.2530 −0.966143 −0.483072 0.875581i \(-0.660479\pi\)
−0.483072 + 0.875581i \(0.660479\pi\)
\(284\) −0.358761 −0.0212885
\(285\) 1.69175 0.100211
\(286\) −1.51929 −0.0898376
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 17.1097 1.00646
\(290\) −4.17526 −0.245180
\(291\) −5.19543 −0.304561
\(292\) 3.53491 0.206865
\(293\) −4.43020 −0.258815 −0.129407 0.991592i \(-0.541307\pi\)
−0.129407 + 0.991592i \(0.541307\pi\)
\(294\) 0 0
\(295\) 0.0181218 0.00105509
\(296\) −1.95648 −0.113718
\(297\) −1.51929 −0.0881583
\(298\) 2.50736 0.145248
\(299\) 5.77880 0.334197
\(300\) 3.65317 0.210916
\(301\) 0 0
\(302\) −1.67614 −0.0964508
\(303\) −12.0273 −0.690949
\(304\) −1.45774 −0.0836070
\(305\) 10.6884 0.612014
\(306\) −5.84035 −0.333871
\(307\) −3.75611 −0.214372 −0.107186 0.994239i \(-0.534184\pi\)
−0.107186 + 0.994239i \(0.534184\pi\)
\(308\) 0 0
\(309\) 7.81128 0.444368
\(310\) −8.46967 −0.481045
\(311\) 28.1348 1.59538 0.797688 0.603070i \(-0.206056\pi\)
0.797688 + 0.603070i \(0.206056\pi\)
\(312\) 1.00000 0.0566139
\(313\) 33.5490 1.89630 0.948150 0.317823i \(-0.102952\pi\)
0.948150 + 0.317823i \(0.102952\pi\)
\(314\) −10.8551 −0.612588
\(315\) 0 0
\(316\) 10.2374 0.575900
\(317\) 16.4714 0.925128 0.462564 0.886586i \(-0.346930\pi\)
0.462564 + 0.886586i \(0.346930\pi\)
\(318\) 9.58667 0.537594
\(319\) 5.46598 0.306036
\(320\) 1.16053 0.0648757
\(321\) 5.10480 0.284922
\(322\) 0 0
\(323\) −8.51371 −0.473716
\(324\) 1.00000 0.0555556
\(325\) −3.65317 −0.202641
\(326\) −8.00699 −0.443466
\(327\) −6.20037 −0.342881
\(328\) 9.88388 0.545746
\(329\) 0 0
\(330\) 1.76319 0.0970602
\(331\) −1.98477 −0.109093 −0.0545465 0.998511i \(-0.517371\pi\)
−0.0545465 + 0.998511i \(0.517371\pi\)
\(332\) −7.33579 −0.402604
\(333\) 1.95648 0.107214
\(334\) 0.339091 0.0185543
\(335\) 4.73314 0.258599
\(336\) 0 0
\(337\) 19.3707 1.05519 0.527594 0.849496i \(-0.323094\pi\)
0.527594 + 0.849496i \(0.323094\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 17.0659 0.926890
\(340\) 6.77792 0.367584
\(341\) 11.0879 0.600445
\(342\) 1.45774 0.0788255
\(343\) 0 0
\(344\) 1.77880 0.0959065
\(345\) −6.70648 −0.361065
\(346\) −2.22577 −0.119658
\(347\) 15.3921 0.826293 0.413146 0.910665i \(-0.364430\pi\)
0.413146 + 0.910665i \(0.364430\pi\)
\(348\) −3.59771 −0.192858
\(349\) 29.0585 1.55546 0.777732 0.628596i \(-0.216370\pi\)
0.777732 + 0.628596i \(0.216370\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −1.51929 −0.0809785
\(353\) 17.8955 0.952483 0.476242 0.879314i \(-0.341999\pi\)
0.476242 + 0.879314i \(0.341999\pi\)
\(354\) 0.0156151 0.000829931 0
\(355\) −0.416353 −0.0220977
\(356\) −9.20406 −0.487814
\(357\) 0 0
\(358\) 6.45280 0.341041
\(359\) −9.71445 −0.512709 −0.256354 0.966583i \(-0.582521\pi\)
−0.256354 + 0.966583i \(0.582521\pi\)
\(360\) −1.16053 −0.0611654
\(361\) −16.8750 −0.888158
\(362\) 0.0238555 0.00125382
\(363\) 8.69175 0.456199
\(364\) 0 0
\(365\) 4.10237 0.214728
\(366\) 9.20988 0.481408
\(367\) 4.84530 0.252922 0.126461 0.991972i \(-0.459638\pi\)
0.126461 + 0.991972i \(0.459638\pi\)
\(368\) 5.77880 0.301241
\(369\) −9.88388 −0.514534
\(370\) −2.27055 −0.118040
\(371\) 0 0
\(372\) −7.29809 −0.378389
\(373\) 15.1001 0.781856 0.390928 0.920421i \(-0.372154\pi\)
0.390928 + 0.920421i \(0.372154\pi\)
\(374\) −8.87321 −0.458823
\(375\) 10.0423 0.518581
\(376\) 11.5867 0.597537
\(377\) 3.59771 0.185292
\(378\) 0 0
\(379\) 7.30508 0.375237 0.187618 0.982242i \(-0.439923\pi\)
0.187618 + 0.982242i \(0.439923\pi\)
\(380\) −1.69175 −0.0867850
\(381\) −17.1651 −0.879395
\(382\) −9.20164 −0.470797
\(383\) 8.36800 0.427585 0.213792 0.976879i \(-0.431418\pi\)
0.213792 + 0.976879i \(0.431418\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −23.2522 −1.18350
\(387\) −1.77880 −0.0904216
\(388\) 5.19543 0.263758
\(389\) 11.8042 0.598497 0.299248 0.954175i \(-0.403264\pi\)
0.299248 + 0.954175i \(0.403264\pi\)
\(390\) 1.16053 0.0587658
\(391\) 33.7503 1.70682
\(392\) 0 0
\(393\) 12.9127 0.651358
\(394\) −21.7521 −1.09586
\(395\) 11.8809 0.597791
\(396\) 1.51929 0.0763473
\(397\) −6.64950 −0.333729 −0.166864 0.985980i \(-0.553364\pi\)
−0.166864 + 0.985980i \(0.553364\pi\)
\(398\) −17.4902 −0.876706
\(399\) 0 0
\(400\) −3.65317 −0.182658
\(401\) −2.59619 −0.129647 −0.0648237 0.997897i \(-0.520649\pi\)
−0.0648237 + 0.997897i \(0.520649\pi\)
\(402\) 4.07842 0.203413
\(403\) 7.29809 0.363544
\(404\) 12.0273 0.598379
\(405\) 1.16053 0.0576673
\(406\) 0 0
\(407\) 2.97246 0.147339
\(408\) 5.84035 0.289141
\(409\) 9.20824 0.455318 0.227659 0.973741i \(-0.426893\pi\)
0.227659 + 0.973741i \(0.426893\pi\)
\(410\) 11.4706 0.566490
\(411\) −10.4348 −0.514709
\(412\) −7.81128 −0.384834
\(413\) 0 0
\(414\) −5.77880 −0.284013
\(415\) −8.51342 −0.417907
\(416\) −1.00000 −0.0490290
\(417\) 20.7071 1.01403
\(418\) 2.21473 0.108326
\(419\) −10.5383 −0.514830 −0.257415 0.966301i \(-0.582871\pi\)
−0.257415 + 0.966301i \(0.582871\pi\)
\(420\) 0 0
\(421\) 17.6400 0.859720 0.429860 0.902895i \(-0.358563\pi\)
0.429860 + 0.902895i \(0.358563\pi\)
\(422\) −4.87070 −0.237102
\(423\) −11.5867 −0.563363
\(424\) −9.58667 −0.465570
\(425\) −21.3358 −1.03494
\(426\) −0.358761 −0.0173820
\(427\) 0 0
\(428\) −5.10480 −0.246750
\(429\) −1.51929 −0.0733521
\(430\) 2.06435 0.0995520
\(431\) −9.42944 −0.454200 −0.227100 0.973871i \(-0.572924\pi\)
−0.227100 + 0.973871i \(0.572924\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −15.4316 −0.741595 −0.370798 0.928714i \(-0.620916\pi\)
−0.370798 + 0.928714i \(0.620916\pi\)
\(434\) 0 0
\(435\) −4.17526 −0.200188
\(436\) 6.20037 0.296944
\(437\) −8.42398 −0.402974
\(438\) 3.53491 0.168904
\(439\) 27.0938 1.29311 0.646557 0.762865i \(-0.276208\pi\)
0.646557 + 0.762865i \(0.276208\pi\)
\(440\) −1.76319 −0.0840566
\(441\) 0 0
\(442\) −5.84035 −0.277797
\(443\) 34.8250 1.65458 0.827292 0.561771i \(-0.189880\pi\)
0.827292 + 0.561771i \(0.189880\pi\)
\(444\) −1.95648 −0.0928502
\(445\) −10.6816 −0.506356
\(446\) 8.01165 0.379363
\(447\) 2.50736 0.118594
\(448\) 0 0
\(449\) −20.9556 −0.988954 −0.494477 0.869191i \(-0.664640\pi\)
−0.494477 + 0.869191i \(0.664640\pi\)
\(450\) 3.65317 0.172212
\(451\) −15.0165 −0.707099
\(452\) −17.0659 −0.802710
\(453\) −1.67614 −0.0787518
\(454\) 23.8532 1.11948
\(455\) 0 0
\(456\) −1.45774 −0.0682649
\(457\) 37.8171 1.76901 0.884505 0.466531i \(-0.154496\pi\)
0.884505 + 0.466531i \(0.154496\pi\)
\(458\) −14.4935 −0.677238
\(459\) −5.84035 −0.272604
\(460\) 6.70648 0.312691
\(461\) −24.0146 −1.11847 −0.559235 0.829009i \(-0.688905\pi\)
−0.559235 + 0.829009i \(0.688905\pi\)
\(462\) 0 0
\(463\) 17.8594 0.829996 0.414998 0.909822i \(-0.363782\pi\)
0.414998 + 0.909822i \(0.363782\pi\)
\(464\) 3.59771 0.167020
\(465\) −8.46967 −0.392771
\(466\) −17.0723 −0.790860
\(467\) −36.2355 −1.67678 −0.838390 0.545070i \(-0.816503\pi\)
−0.838390 + 0.545070i \(0.816503\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 13.4467 0.620250
\(471\) −10.8551 −0.500176
\(472\) −0.0156151 −0.000718741 0
\(473\) −2.70252 −0.124262
\(474\) 10.2374 0.470221
\(475\) 5.32536 0.244344
\(476\) 0 0
\(477\) 9.58667 0.438944
\(478\) 19.8943 0.909943
\(479\) −23.8606 −1.09022 −0.545110 0.838364i \(-0.683512\pi\)
−0.545110 + 0.838364i \(0.683512\pi\)
\(480\) 1.16053 0.0529708
\(481\) 1.95648 0.0892076
\(482\) 6.03374 0.274829
\(483\) 0 0
\(484\) −8.69175 −0.395080
\(485\) 6.02946 0.273784
\(486\) 1.00000 0.0453609
\(487\) −34.0551 −1.54318 −0.771592 0.636118i \(-0.780539\pi\)
−0.771592 + 0.636118i \(0.780539\pi\)
\(488\) −9.20988 −0.416912
\(489\) −8.00699 −0.362088
\(490\) 0 0
\(491\) −5.02422 −0.226740 −0.113370 0.993553i \(-0.536165\pi\)
−0.113370 + 0.993553i \(0.536165\pi\)
\(492\) 9.88388 0.445600
\(493\) 21.0119 0.946330
\(494\) 1.45774 0.0655868
\(495\) 1.76319 0.0792493
\(496\) 7.29809 0.327694
\(497\) 0 0
\(498\) −7.33579 −0.328725
\(499\) 24.1863 1.08273 0.541363 0.840789i \(-0.317908\pi\)
0.541363 + 0.840789i \(0.317908\pi\)
\(500\) −10.0423 −0.449104
\(501\) 0.339091 0.0151495
\(502\) −10.5817 −0.472286
\(503\) 31.6915 1.41305 0.706527 0.707686i \(-0.250261\pi\)
0.706527 + 0.707686i \(0.250261\pi\)
\(504\) 0 0
\(505\) 13.9580 0.621124
\(506\) −8.77969 −0.390305
\(507\) −1.00000 −0.0444116
\(508\) 17.1651 0.761578
\(509\) −36.1410 −1.60192 −0.800960 0.598717i \(-0.795677\pi\)
−0.800960 + 0.598717i \(0.795677\pi\)
\(510\) 6.77792 0.300131
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.45774 0.0643607
\(514\) 2.27512 0.100351
\(515\) −9.06524 −0.399462
\(516\) 1.77880 0.0783074
\(517\) −17.6035 −0.774203
\(518\) 0 0
\(519\) −2.22577 −0.0977006
\(520\) −1.16053 −0.0508927
\(521\) −11.5089 −0.504212 −0.252106 0.967700i \(-0.581123\pi\)
−0.252106 + 0.967700i \(0.581123\pi\)
\(522\) −3.59771 −0.157468
\(523\) 12.6798 0.554450 0.277225 0.960805i \(-0.410585\pi\)
0.277225 + 0.960805i \(0.410585\pi\)
\(524\) −12.9127 −0.564093
\(525\) 0 0
\(526\) −14.9853 −0.653389
\(527\) 42.6235 1.85671
\(528\) −1.51929 −0.0661187
\(529\) 10.3945 0.451937
\(530\) −11.1256 −0.483267
\(531\) 0.0156151 0.000677636 0
\(532\) 0 0
\(533\) −9.88388 −0.428118
\(534\) −9.20406 −0.398298
\(535\) −5.92429 −0.256129
\(536\) −4.07842 −0.176161
\(537\) 6.45280 0.278459
\(538\) 4.34656 0.187393
\(539\) 0 0
\(540\) −1.16053 −0.0499413
\(541\) −4.65014 −0.199925 −0.0999626 0.994991i \(-0.531872\pi\)
−0.0999626 + 0.994991i \(0.531872\pi\)
\(542\) 5.32195 0.228597
\(543\) 0.0238555 0.00102374
\(544\) −5.84035 −0.250403
\(545\) 7.19572 0.308231
\(546\) 0 0
\(547\) 17.7916 0.760715 0.380357 0.924840i \(-0.375801\pi\)
0.380357 + 0.924840i \(0.375801\pi\)
\(548\) 10.4348 0.445751
\(549\) 9.20988 0.393068
\(550\) 5.55023 0.236663
\(551\) −5.24453 −0.223424
\(552\) 5.77880 0.245962
\(553\) 0 0
\(554\) 22.0052 0.934911
\(555\) −2.27055 −0.0963795
\(556\) −20.7071 −0.878176
\(557\) 30.6373 1.29814 0.649072 0.760727i \(-0.275157\pi\)
0.649072 + 0.760727i \(0.275157\pi\)
\(558\) −7.29809 −0.308953
\(559\) −1.77880 −0.0752353
\(560\) 0 0
\(561\) −8.87321 −0.374627
\(562\) −21.6348 −0.912608
\(563\) 18.3741 0.774375 0.387188 0.922001i \(-0.373447\pi\)
0.387188 + 0.922001i \(0.373447\pi\)
\(564\) 11.5867 0.487887
\(565\) −19.8055 −0.833222
\(566\) 16.2530 0.683166
\(567\) 0 0
\(568\) 0.358761 0.0150533
\(569\) −17.9394 −0.752057 −0.376028 0.926608i \(-0.622710\pi\)
−0.376028 + 0.926608i \(0.622710\pi\)
\(570\) −1.69175 −0.0708597
\(571\) −18.6495 −0.780457 −0.390229 0.920718i \(-0.627604\pi\)
−0.390229 + 0.920718i \(0.627604\pi\)
\(572\) 1.51929 0.0635248
\(573\) −9.20164 −0.384404
\(574\) 0 0
\(575\) −21.1109 −0.880386
\(576\) 1.00000 0.0416667
\(577\) 36.3523 1.51337 0.756683 0.653782i \(-0.226819\pi\)
0.756683 + 0.653782i \(0.226819\pi\)
\(578\) −17.1097 −0.711672
\(579\) −23.2522 −0.966327
\(580\) 4.17526 0.173368
\(581\) 0 0
\(582\) 5.19543 0.215357
\(583\) 14.5650 0.603219
\(584\) −3.53491 −0.146276
\(585\) 1.16053 0.0479821
\(586\) 4.43020 0.183010
\(587\) 18.7622 0.774398 0.387199 0.921996i \(-0.373443\pi\)
0.387199 + 0.921996i \(0.373443\pi\)
\(588\) 0 0
\(589\) −10.6387 −0.438360
\(590\) −0.0181218 −0.000746061 0
\(591\) −21.7521 −0.894764
\(592\) 1.95648 0.0804106
\(593\) 12.0975 0.496783 0.248391 0.968660i \(-0.420098\pi\)
0.248391 + 0.968660i \(0.420098\pi\)
\(594\) 1.51929 0.0623373
\(595\) 0 0
\(596\) −2.50736 −0.102706
\(597\) −17.4902 −0.715827
\(598\) −5.77880 −0.236313
\(599\) −19.3009 −0.788613 −0.394307 0.918979i \(-0.629015\pi\)
−0.394307 + 0.918979i \(0.629015\pi\)
\(600\) −3.65317 −0.149140
\(601\) −17.6854 −0.721402 −0.360701 0.932682i \(-0.617462\pi\)
−0.360701 + 0.932682i \(0.617462\pi\)
\(602\) 0 0
\(603\) 4.07842 0.166086
\(604\) 1.67614 0.0682010
\(605\) −10.0870 −0.410097
\(606\) 12.0273 0.488574
\(607\) −33.1514 −1.34557 −0.672786 0.739837i \(-0.734903\pi\)
−0.672786 + 0.739837i \(0.734903\pi\)
\(608\) 1.45774 0.0591191
\(609\) 0 0
\(610\) −10.6884 −0.432759
\(611\) −11.5867 −0.468747
\(612\) 5.84035 0.236082
\(613\) −28.6489 −1.15712 −0.578558 0.815641i \(-0.696384\pi\)
−0.578558 + 0.815641i \(0.696384\pi\)
\(614\) 3.75611 0.151584
\(615\) 11.4706 0.462537
\(616\) 0 0
\(617\) −22.2380 −0.895270 −0.447635 0.894216i \(-0.647734\pi\)
−0.447635 + 0.894216i \(0.647734\pi\)
\(618\) −7.81128 −0.314216
\(619\) −38.5778 −1.55057 −0.775286 0.631611i \(-0.782394\pi\)
−0.775286 + 0.631611i \(0.782394\pi\)
\(620\) 8.46967 0.340150
\(621\) −5.77880 −0.231895
\(622\) −28.1348 −1.12810
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 6.61146 0.264459
\(626\) −33.5490 −1.34089
\(627\) 2.21473 0.0884478
\(628\) 10.8551 0.433165
\(629\) 11.4265 0.455605
\(630\) 0 0
\(631\) 43.5419 1.73337 0.866687 0.498852i \(-0.166245\pi\)
0.866687 + 0.498852i \(0.166245\pi\)
\(632\) −10.2374 −0.407223
\(633\) −4.87070 −0.193593
\(634\) −16.4714 −0.654164
\(635\) 19.9206 0.790527
\(636\) −9.58667 −0.380136
\(637\) 0 0
\(638\) −5.46598 −0.216400
\(639\) −0.358761 −0.0141924
\(640\) −1.16053 −0.0458740
\(641\) 12.2184 0.482598 0.241299 0.970451i \(-0.422427\pi\)
0.241299 + 0.970451i \(0.422427\pi\)
\(642\) −5.10480 −0.201471
\(643\) 26.4417 1.04276 0.521380 0.853325i \(-0.325417\pi\)
0.521380 + 0.853325i \(0.325417\pi\)
\(644\) 0 0
\(645\) 2.06435 0.0812839
\(646\) 8.51371 0.334967
\(647\) −8.39591 −0.330077 −0.165039 0.986287i \(-0.552775\pi\)
−0.165039 + 0.986287i \(0.552775\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0.0237238 0.000931242 0
\(650\) 3.65317 0.143289
\(651\) 0 0
\(652\) 8.00699 0.313578
\(653\) −23.3820 −0.915007 −0.457503 0.889208i \(-0.651256\pi\)
−0.457503 + 0.889208i \(0.651256\pi\)
\(654\) 6.20037 0.242454
\(655\) −14.9856 −0.585534
\(656\) −9.88388 −0.385901
\(657\) 3.53491 0.137910
\(658\) 0 0
\(659\) −45.7625 −1.78266 −0.891328 0.453359i \(-0.850225\pi\)
−0.891328 + 0.453359i \(0.850225\pi\)
\(660\) −1.76319 −0.0686319
\(661\) −0.169046 −0.00657512 −0.00328756 0.999995i \(-0.501046\pi\)
−0.00328756 + 0.999995i \(0.501046\pi\)
\(662\) 1.98477 0.0771404
\(663\) −5.84035 −0.226821
\(664\) 7.33579 0.284684
\(665\) 0 0
\(666\) −1.95648 −0.0758119
\(667\) 20.7905 0.805010
\(668\) −0.339091 −0.0131198
\(669\) 8.01165 0.309748
\(670\) −4.73314 −0.182857
\(671\) 13.9925 0.540175
\(672\) 0 0
\(673\) 47.1905 1.81906 0.909530 0.415639i \(-0.136442\pi\)
0.909530 + 0.415639i \(0.136442\pi\)
\(674\) −19.3707 −0.746131
\(675\) 3.65317 0.140610
\(676\) 1.00000 0.0384615
\(677\) 32.7187 1.25748 0.628742 0.777614i \(-0.283570\pi\)
0.628742 + 0.777614i \(0.283570\pi\)
\(678\) −17.0659 −0.655410
\(679\) 0 0
\(680\) −6.77792 −0.259921
\(681\) 23.8532 0.914055
\(682\) −11.0879 −0.424579
\(683\) −17.9035 −0.685060 −0.342530 0.939507i \(-0.611284\pi\)
−0.342530 + 0.939507i \(0.611284\pi\)
\(684\) −1.45774 −0.0557380
\(685\) 12.1099 0.462695
\(686\) 0 0
\(687\) −14.4935 −0.552962
\(688\) −1.77880 −0.0678162
\(689\) 9.58667 0.365223
\(690\) 6.70648 0.255311
\(691\) 7.03858 0.267760 0.133880 0.990998i \(-0.457256\pi\)
0.133880 + 0.990998i \(0.457256\pi\)
\(692\) 2.22577 0.0846112
\(693\) 0 0
\(694\) −15.3921 −0.584277
\(695\) −24.0312 −0.911557
\(696\) 3.59771 0.136371
\(697\) −57.7254 −2.18650
\(698\) −29.0585 −1.09988
\(699\) −17.0723 −0.645734
\(700\) 0 0
\(701\) −36.7808 −1.38919 −0.694596 0.719400i \(-0.744417\pi\)
−0.694596 + 0.719400i \(0.744417\pi\)
\(702\) 1.00000 0.0377426
\(703\) −2.85203 −0.107566
\(704\) 1.51929 0.0572605
\(705\) 13.4467 0.506432
\(706\) −17.8955 −0.673507
\(707\) 0 0
\(708\) −0.0156151 −0.000586850 0
\(709\) −23.0021 −0.863863 −0.431932 0.901906i \(-0.642168\pi\)
−0.431932 + 0.901906i \(0.642168\pi\)
\(710\) 0.416353 0.0156255
\(711\) 10.2374 0.383933
\(712\) 9.20406 0.344937
\(713\) 42.1742 1.57944
\(714\) 0 0
\(715\) 1.76319 0.0659394
\(716\) −6.45280 −0.241152
\(717\) 19.8943 0.742966
\(718\) 9.71445 0.362540
\(719\) 4.66817 0.174093 0.0870467 0.996204i \(-0.472257\pi\)
0.0870467 + 0.996204i \(0.472257\pi\)
\(720\) 1.16053 0.0432505
\(721\) 0 0
\(722\) 16.8750 0.628022
\(723\) 6.03374 0.224397
\(724\) −0.0238555 −0.000886584 0
\(725\) −13.1431 −0.488121
\(726\) −8.69175 −0.322581
\(727\) 25.6085 0.949767 0.474883 0.880049i \(-0.342490\pi\)
0.474883 + 0.880049i \(0.342490\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.10237 −0.151836
\(731\) −10.3888 −0.384245
\(732\) −9.20988 −0.340407
\(733\) 1.84962 0.0683174 0.0341587 0.999416i \(-0.489125\pi\)
0.0341587 + 0.999416i \(0.489125\pi\)
\(734\) −4.84530 −0.178843
\(735\) 0 0
\(736\) −5.77880 −0.213009
\(737\) 6.19631 0.228244
\(738\) 9.88388 0.363831
\(739\) −23.9488 −0.880973 −0.440486 0.897759i \(-0.645194\pi\)
−0.440486 + 0.897759i \(0.645194\pi\)
\(740\) 2.27055 0.0834671
\(741\) 1.45774 0.0535514
\(742\) 0 0
\(743\) −10.4754 −0.384305 −0.192153 0.981365i \(-0.561547\pi\)
−0.192153 + 0.981365i \(0.561547\pi\)
\(744\) 7.29809 0.267561
\(745\) −2.90988 −0.106610
\(746\) −15.1001 −0.552855
\(747\) −7.33579 −0.268403
\(748\) 8.87321 0.324437
\(749\) 0 0
\(750\) −10.0423 −0.366692
\(751\) −23.5849 −0.860624 −0.430312 0.902680i \(-0.641596\pi\)
−0.430312 + 0.902680i \(0.641596\pi\)
\(752\) −11.5867 −0.422522
\(753\) −10.5817 −0.385620
\(754\) −3.59771 −0.131021
\(755\) 1.94521 0.0707934
\(756\) 0 0
\(757\) −18.6602 −0.678218 −0.339109 0.940747i \(-0.610126\pi\)
−0.339109 + 0.940747i \(0.610126\pi\)
\(758\) −7.30508 −0.265332
\(759\) −8.77969 −0.318682
\(760\) 1.69175 0.0613663
\(761\) −27.1786 −0.985222 −0.492611 0.870250i \(-0.663957\pi\)
−0.492611 + 0.870250i \(0.663957\pi\)
\(762\) 17.1651 0.621826
\(763\) 0 0
\(764\) 9.20164 0.332904
\(765\) 6.77792 0.245056
\(766\) −8.36800 −0.302348
\(767\) 0.0156151 0.000563827 0
\(768\) −1.00000 −0.0360844
\(769\) −30.8339 −1.11190 −0.555949 0.831217i \(-0.687645\pi\)
−0.555949 + 0.831217i \(0.687645\pi\)
\(770\) 0 0
\(771\) 2.27512 0.0819366
\(772\) 23.2522 0.836863
\(773\) −14.0245 −0.504425 −0.252213 0.967672i \(-0.581158\pi\)
−0.252213 + 0.967672i \(0.581158\pi\)
\(774\) 1.77880 0.0639377
\(775\) −26.6612 −0.957697
\(776\) −5.19543 −0.186505
\(777\) 0 0
\(778\) −11.8042 −0.423201
\(779\) 14.4081 0.516224
\(780\) −1.16053 −0.0415537
\(781\) −0.545062 −0.0195039
\(782\) −33.7503 −1.20691
\(783\) −3.59771 −0.128572
\(784\) 0 0
\(785\) 12.5977 0.449630
\(786\) −12.9127 −0.460580
\(787\) −24.7524 −0.882328 −0.441164 0.897426i \(-0.645434\pi\)
−0.441164 + 0.897426i \(0.645434\pi\)
\(788\) 21.7521 0.774888
\(789\) −14.9853 −0.533490
\(790\) −11.8809 −0.422702
\(791\) 0 0
\(792\) −1.51929 −0.0539857
\(793\) 9.20988 0.327053
\(794\) 6.64950 0.235982
\(795\) −11.1256 −0.394586
\(796\) 17.4902 0.619924
\(797\) 17.5536 0.621782 0.310891 0.950446i \(-0.399373\pi\)
0.310891 + 0.950446i \(0.399373\pi\)
\(798\) 0 0
\(799\) −67.6703 −2.39400
\(800\) 3.65317 0.129159
\(801\) −9.20406 −0.325209
\(802\) 2.59619 0.0916745
\(803\) 5.37056 0.189523
\(804\) −4.07842 −0.143835
\(805\) 0 0
\(806\) −7.29809 −0.257064
\(807\) 4.34656 0.153006
\(808\) −12.0273 −0.423118
\(809\) −0.153431 −0.00539434 −0.00269717 0.999996i \(-0.500859\pi\)
−0.00269717 + 0.999996i \(0.500859\pi\)
\(810\) −1.16053 −0.0407769
\(811\) 41.7585 1.46634 0.733169 0.680046i \(-0.238040\pi\)
0.733169 + 0.680046i \(0.238040\pi\)
\(812\) 0 0
\(813\) 5.32195 0.186649
\(814\) −2.97246 −0.104185
\(815\) 9.29236 0.325497
\(816\) −5.84035 −0.204453
\(817\) 2.59303 0.0907185
\(818\) −9.20824 −0.321959
\(819\) 0 0
\(820\) −11.4706 −0.400569
\(821\) −50.4865 −1.76199 −0.880996 0.473124i \(-0.843126\pi\)
−0.880996 + 0.473124i \(0.843126\pi\)
\(822\) 10.4348 0.363954
\(823\) −52.6062 −1.83374 −0.916868 0.399190i \(-0.869292\pi\)
−0.916868 + 0.399190i \(0.869292\pi\)
\(824\) 7.81128 0.272119
\(825\) 5.55023 0.193234
\(826\) 0 0
\(827\) 2.35509 0.0818946 0.0409473 0.999161i \(-0.486962\pi\)
0.0409473 + 0.999161i \(0.486962\pi\)
\(828\) 5.77880 0.200827
\(829\) 1.43169 0.0497248 0.0248624 0.999691i \(-0.492085\pi\)
0.0248624 + 0.999691i \(0.492085\pi\)
\(830\) 8.51342 0.295505
\(831\) 22.0052 0.763351
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −20.7071 −0.717028
\(835\) −0.393526 −0.0136185
\(836\) −2.21473 −0.0765981
\(837\) −7.29809 −0.252259
\(838\) 10.5383 0.364040
\(839\) −22.5382 −0.778105 −0.389053 0.921216i \(-0.627198\pi\)
−0.389053 + 0.921216i \(0.627198\pi\)
\(840\) 0 0
\(841\) −16.0565 −0.553671
\(842\) −17.6400 −0.607914
\(843\) −21.6348 −0.745141
\(844\) 4.87070 0.167656
\(845\) 1.16053 0.0399235
\(846\) 11.5867 0.398358
\(847\) 0 0
\(848\) 9.58667 0.329208
\(849\) 16.2530 0.557803
\(850\) 21.3358 0.731812
\(851\) 11.3061 0.387567
\(852\) 0.358761 0.0122909
\(853\) 4.09925 0.140356 0.0701779 0.997534i \(-0.477643\pi\)
0.0701779 + 0.997534i \(0.477643\pi\)
\(854\) 0 0
\(855\) −1.69175 −0.0578567
\(856\) 5.10480 0.174479
\(857\) 11.8622 0.405204 0.202602 0.979261i \(-0.435060\pi\)
0.202602 + 0.979261i \(0.435060\pi\)
\(858\) 1.51929 0.0518678
\(859\) −44.5980 −1.52166 −0.760831 0.648950i \(-0.775209\pi\)
−0.760831 + 0.648950i \(0.775209\pi\)
\(860\) −2.06435 −0.0703939
\(861\) 0 0
\(862\) 9.42944 0.321168
\(863\) 9.83022 0.334625 0.167312 0.985904i \(-0.446491\pi\)
0.167312 + 0.985904i \(0.446491\pi\)
\(864\) 1.00000 0.0340207
\(865\) 2.58308 0.0878273
\(866\) 15.4316 0.524387
\(867\) −17.1097 −0.581077
\(868\) 0 0
\(869\) 15.5536 0.527621
\(870\) 4.17526 0.141555
\(871\) 4.07842 0.138192
\(872\) −6.20037 −0.209971
\(873\) 5.19543 0.175839
\(874\) 8.42398 0.284945
\(875\) 0 0
\(876\) −3.53491 −0.119433
\(877\) −38.3415 −1.29470 −0.647350 0.762193i \(-0.724123\pi\)
−0.647350 + 0.762193i \(0.724123\pi\)
\(878\) −27.0938 −0.914370
\(879\) 4.43020 0.149427
\(880\) 1.76319 0.0594370
\(881\) 14.2467 0.479983 0.239992 0.970775i \(-0.422855\pi\)
0.239992 + 0.970775i \(0.422855\pi\)
\(882\) 0 0
\(883\) −54.1329 −1.82172 −0.910858 0.412720i \(-0.864579\pi\)
−0.910858 + 0.412720i \(0.864579\pi\)
\(884\) 5.84035 0.196432
\(885\) −0.0181218 −0.000609156 0
\(886\) −34.8250 −1.16997
\(887\) −8.85788 −0.297419 −0.148709 0.988881i \(-0.547512\pi\)
−0.148709 + 0.988881i \(0.547512\pi\)
\(888\) 1.95648 0.0656550
\(889\) 0 0
\(890\) 10.6816 0.358048
\(891\) 1.51929 0.0508982
\(892\) −8.01165 −0.268250
\(893\) 16.8903 0.565214
\(894\) −2.50736 −0.0838588
\(895\) −7.48867 −0.250319
\(896\) 0 0
\(897\) −5.77880 −0.192949
\(898\) 20.9556 0.699296
\(899\) 26.2565 0.875702
\(900\) −3.65317 −0.121772
\(901\) 55.9896 1.86528
\(902\) 15.0165 0.499995
\(903\) 0 0
\(904\) 17.0659 0.567602
\(905\) −0.0276851 −0.000920284 0
\(906\) 1.67614 0.0556859
\(907\) −13.1980 −0.438232 −0.219116 0.975699i \(-0.570317\pi\)
−0.219116 + 0.975699i \(0.570317\pi\)
\(908\) −23.8532 −0.791595
\(909\) 12.0273 0.398919
\(910\) 0 0
\(911\) −38.6427 −1.28029 −0.640146 0.768254i \(-0.721126\pi\)
−0.640146 + 0.768254i \(0.721126\pi\)
\(912\) 1.45774 0.0482705
\(913\) −11.1452 −0.368853
\(914\) −37.8171 −1.25088
\(915\) −10.6884 −0.353346
\(916\) 14.4935 0.478879
\(917\) 0 0
\(918\) 5.84035 0.192760
\(919\) −10.5294 −0.347334 −0.173667 0.984804i \(-0.555562\pi\)
−0.173667 + 0.984804i \(0.555562\pi\)
\(920\) −6.70648 −0.221106
\(921\) 3.75611 0.123768
\(922\) 24.0146 0.790878
\(923\) −0.358761 −0.0118088
\(924\) 0 0
\(925\) −7.14733 −0.235003
\(926\) −17.8594 −0.586896
\(927\) −7.81128 −0.256556
\(928\) −3.59771 −0.118101
\(929\) −33.7331 −1.10675 −0.553373 0.832933i \(-0.686660\pi\)
−0.553373 + 0.832933i \(0.686660\pi\)
\(930\) 8.46967 0.277731
\(931\) 0 0
\(932\) 17.0723 0.559222
\(933\) −28.1348 −0.921091
\(934\) 36.2355 1.18566
\(935\) 10.2976 0.336769
\(936\) −1.00000 −0.0326860
\(937\) −6.05885 −0.197934 −0.0989669 0.995091i \(-0.531554\pi\)
−0.0989669 + 0.995091i \(0.531554\pi\)
\(938\) 0 0
\(939\) −33.5490 −1.09483
\(940\) −13.4467 −0.438583
\(941\) 48.1109 1.56837 0.784185 0.620527i \(-0.213081\pi\)
0.784185 + 0.620527i \(0.213081\pi\)
\(942\) 10.8551 0.353678
\(943\) −57.1170 −1.85998
\(944\) 0.0156151 0.000508227 0
\(945\) 0 0
\(946\) 2.70252 0.0878665
\(947\) −46.8777 −1.52332 −0.761660 0.647977i \(-0.775615\pi\)
−0.761660 + 0.647977i \(0.775615\pi\)
\(948\) −10.2374 −0.332496
\(949\) 3.53491 0.114748
\(950\) −5.32536 −0.172778
\(951\) −16.4714 −0.534123
\(952\) 0 0
\(953\) −34.3401 −1.11238 −0.556192 0.831054i \(-0.687738\pi\)
−0.556192 + 0.831054i \(0.687738\pi\)
\(954\) −9.58667 −0.310380
\(955\) 10.6788 0.345558
\(956\) −19.8943 −0.643427
\(957\) −5.46598 −0.176690
\(958\) 23.8606 0.770902
\(959\) 0 0
\(960\) −1.16053 −0.0374560
\(961\) 22.2622 0.718134
\(962\) −1.95648 −0.0630793
\(963\) −5.10480 −0.164500
\(964\) −6.03374 −0.194334
\(965\) 26.9849 0.868673
\(966\) 0 0
\(967\) −11.9586 −0.384564 −0.192282 0.981340i \(-0.561589\pi\)
−0.192282 + 0.981340i \(0.561589\pi\)
\(968\) 8.69175 0.279363
\(969\) 8.51371 0.273500
\(970\) −6.02946 −0.193594
\(971\) 5.64697 0.181220 0.0906100 0.995886i \(-0.471118\pi\)
0.0906100 + 0.995886i \(0.471118\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 34.0551 1.09120
\(975\) 3.65317 0.116995
\(976\) 9.20988 0.294801
\(977\) 54.2348 1.73512 0.867562 0.497329i \(-0.165686\pi\)
0.867562 + 0.497329i \(0.165686\pi\)
\(978\) 8.00699 0.256035
\(979\) −13.9837 −0.446919
\(980\) 0 0
\(981\) 6.20037 0.197963
\(982\) 5.02422 0.160329
\(983\) −35.0175 −1.11688 −0.558442 0.829543i \(-0.688601\pi\)
−0.558442 + 0.829543i \(0.688601\pi\)
\(984\) −9.88388 −0.315087
\(985\) 25.2440 0.804342
\(986\) −21.0119 −0.669156
\(987\) 0 0
\(988\) −1.45774 −0.0463768
\(989\) −10.2793 −0.326864
\(990\) −1.76319 −0.0560377
\(991\) −12.4835 −0.396552 −0.198276 0.980146i \(-0.563534\pi\)
−0.198276 + 0.980146i \(0.563534\pi\)
\(992\) −7.29809 −0.231715
\(993\) 1.98477 0.0629848
\(994\) 0 0
\(995\) 20.2980 0.643488
\(996\) 7.33579 0.232443
\(997\) 59.4282 1.88211 0.941054 0.338255i \(-0.109837\pi\)
0.941054 + 0.338255i \(0.109837\pi\)
\(998\) −24.1863 −0.765603
\(999\) −1.95648 −0.0619001
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.a.bx.1.4 4
7.6 odd 2 3822.2.a.by.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3822.2.a.bx.1.4 4 1.1 even 1 trivial
3822.2.a.by.1.1 yes 4 7.6 odd 2