Properties

Label 2320.2.a.s.1.3
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
Analytic rank $1$
Dimension $3$
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] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.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: \(18.5252932689\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.90321 q^{3} -1.00000 q^{5} -1.52543 q^{7} +5.42864 q^{9} +O(q^{10})\) \(q+2.90321 q^{3} -1.00000 q^{5} -1.52543 q^{7} +5.42864 q^{9} -4.90321 q^{11} -6.42864 q^{13} -2.90321 q^{15} +2.14764 q^{17} -2.28100 q^{19} -4.42864 q^{21} -6.90321 q^{23} +1.00000 q^{25} +7.05086 q^{27} +1.00000 q^{29} -1.71900 q^{31} -14.2351 q^{33} +1.52543 q^{35} +7.95407 q^{37} -18.6637 q^{39} -3.37778 q^{41} +1.09679 q^{43} -5.42864 q^{45} -12.7096 q^{47} -4.67307 q^{49} +6.23506 q^{51} +3.37778 q^{53} +4.90321 q^{55} -6.62222 q^{57} +3.18421 q^{59} -2.42864 q^{61} -8.28100 q^{63} +6.42864 q^{65} +1.09679 q^{67} -20.0415 q^{69} -3.57136 q^{71} +14.1891 q^{73} +2.90321 q^{75} +7.47949 q^{77} -0.341219 q^{79} +4.18421 q^{81} +7.33185 q^{83} -2.14764 q^{85} +2.90321 q^{87} +2.94914 q^{89} +9.80642 q^{91} -4.99063 q^{93} +2.28100 q^{95} -18.5763 q^{97} -26.6178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} - 6 q^{13} - 2 q^{15} - 14 q^{23} + 3 q^{25} + 8 q^{27} + 3 q^{29} - 12 q^{31} - 16 q^{33} - 2 q^{35} + 4 q^{37} - 16 q^{39} - 10 q^{41} + 10 q^{43} - 3 q^{45} - 18 q^{47} - q^{49} - 8 q^{51} + 10 q^{53} + 8 q^{55} - 20 q^{57} - 4 q^{59} + 6 q^{61} - 18 q^{63} + 6 q^{65} + 10 q^{67} - 20 q^{69} - 24 q^{71} - 4 q^{73} + 2 q^{75} - 4 q^{77} - 8 q^{79} - q^{81} + 2 q^{83} + 2 q^{87} + 22 q^{89} + 16 q^{91} + 12 q^{93} - 36 q^{97} - 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.90321 1.67617 0.838085 0.545540i \(-0.183675\pi\)
0.838085 + 0.545540i \(0.183675\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.52543 −0.576557 −0.288279 0.957547i \(-0.593083\pi\)
−0.288279 + 0.957547i \(0.593083\pi\)
\(8\) 0 0
\(9\) 5.42864 1.80955
\(10\) 0 0
\(11\) −4.90321 −1.47837 −0.739187 0.673500i \(-0.764790\pi\)
−0.739187 + 0.673500i \(0.764790\pi\)
\(12\) 0 0
\(13\) −6.42864 −1.78298 −0.891492 0.453037i \(-0.850341\pi\)
−0.891492 + 0.453037i \(0.850341\pi\)
\(14\) 0 0
\(15\) −2.90321 −0.749606
\(16\) 0 0
\(17\) 2.14764 0.520880 0.260440 0.965490i \(-0.416132\pi\)
0.260440 + 0.965490i \(0.416132\pi\)
\(18\) 0 0
\(19\) −2.28100 −0.523296 −0.261648 0.965163i \(-0.584266\pi\)
−0.261648 + 0.965163i \(0.584266\pi\)
\(20\) 0 0
\(21\) −4.42864 −0.966408
\(22\) 0 0
\(23\) −6.90321 −1.43942 −0.719710 0.694275i \(-0.755725\pi\)
−0.719710 + 0.694275i \(0.755725\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 7.05086 1.35694
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −1.71900 −0.308742 −0.154371 0.988013i \(-0.549335\pi\)
−0.154371 + 0.988013i \(0.549335\pi\)
\(32\) 0 0
\(33\) −14.2351 −2.47801
\(34\) 0 0
\(35\) 1.52543 0.257844
\(36\) 0 0
\(37\) 7.95407 1.30764 0.653820 0.756650i \(-0.273165\pi\)
0.653820 + 0.756650i \(0.273165\pi\)
\(38\) 0 0
\(39\) −18.6637 −2.98858
\(40\) 0 0
\(41\) −3.37778 −0.527521 −0.263761 0.964588i \(-0.584963\pi\)
−0.263761 + 0.964588i \(0.584963\pi\)
\(42\) 0 0
\(43\) 1.09679 0.167259 0.0836293 0.996497i \(-0.473349\pi\)
0.0836293 + 0.996497i \(0.473349\pi\)
\(44\) 0 0
\(45\) −5.42864 −0.809254
\(46\) 0 0
\(47\) −12.7096 −1.85389 −0.926945 0.375196i \(-0.877575\pi\)
−0.926945 + 0.375196i \(0.877575\pi\)
\(48\) 0 0
\(49\) −4.67307 −0.667582
\(50\) 0 0
\(51\) 6.23506 0.873084
\(52\) 0 0
\(53\) 3.37778 0.463974 0.231987 0.972719i \(-0.425477\pi\)
0.231987 + 0.972719i \(0.425477\pi\)
\(54\) 0 0
\(55\) 4.90321 0.661149
\(56\) 0 0
\(57\) −6.62222 −0.877134
\(58\) 0 0
\(59\) 3.18421 0.414549 0.207274 0.978283i \(-0.433541\pi\)
0.207274 + 0.978283i \(0.433541\pi\)
\(60\) 0 0
\(61\) −2.42864 −0.310955 −0.155478 0.987839i \(-0.549692\pi\)
−0.155478 + 0.987839i \(0.549692\pi\)
\(62\) 0 0
\(63\) −8.28100 −1.04331
\(64\) 0 0
\(65\) 6.42864 0.797375
\(66\) 0 0
\(67\) 1.09679 0.133994 0.0669970 0.997753i \(-0.478658\pi\)
0.0669970 + 0.997753i \(0.478658\pi\)
\(68\) 0 0
\(69\) −20.0415 −2.41271
\(70\) 0 0
\(71\) −3.57136 −0.423843 −0.211921 0.977287i \(-0.567972\pi\)
−0.211921 + 0.977287i \(0.567972\pi\)
\(72\) 0 0
\(73\) 14.1891 1.66071 0.830356 0.557233i \(-0.188137\pi\)
0.830356 + 0.557233i \(0.188137\pi\)
\(74\) 0 0
\(75\) 2.90321 0.335234
\(76\) 0 0
\(77\) 7.47949 0.852368
\(78\) 0 0
\(79\) −0.341219 −0.0383902 −0.0191951 0.999816i \(-0.506110\pi\)
−0.0191951 + 0.999816i \(0.506110\pi\)
\(80\) 0 0
\(81\) 4.18421 0.464912
\(82\) 0 0
\(83\) 7.33185 0.804775 0.402388 0.915469i \(-0.368180\pi\)
0.402388 + 0.915469i \(0.368180\pi\)
\(84\) 0 0
\(85\) −2.14764 −0.232945
\(86\) 0 0
\(87\) 2.90321 0.311257
\(88\) 0 0
\(89\) 2.94914 0.312609 0.156304 0.987709i \(-0.450042\pi\)
0.156304 + 0.987709i \(0.450042\pi\)
\(90\) 0 0
\(91\) 9.80642 1.02799
\(92\) 0 0
\(93\) −4.99063 −0.517504
\(94\) 0 0
\(95\) 2.28100 0.234025
\(96\) 0 0
\(97\) −18.5763 −1.88614 −0.943068 0.332600i \(-0.892074\pi\)
−0.943068 + 0.332600i \(0.892074\pi\)
\(98\) 0 0
\(99\) −26.6178 −2.67519
\(100\) 0 0
\(101\) 15.4193 1.53427 0.767137 0.641483i \(-0.221680\pi\)
0.767137 + 0.641483i \(0.221680\pi\)
\(102\) 0 0
\(103\) −7.76049 −0.764664 −0.382332 0.924025i \(-0.624879\pi\)
−0.382332 + 0.924025i \(0.624879\pi\)
\(104\) 0 0
\(105\) 4.42864 0.432191
\(106\) 0 0
\(107\) 3.03657 0.293556 0.146778 0.989169i \(-0.453110\pi\)
0.146778 + 0.989169i \(0.453110\pi\)
\(108\) 0 0
\(109\) −7.93978 −0.760493 −0.380246 0.924885i \(-0.624161\pi\)
−0.380246 + 0.924885i \(0.624161\pi\)
\(110\) 0 0
\(111\) 23.0923 2.19183
\(112\) 0 0
\(113\) 7.82071 0.735711 0.367855 0.929883i \(-0.380092\pi\)
0.367855 + 0.929883i \(0.380092\pi\)
\(114\) 0 0
\(115\) 6.90321 0.643728
\(116\) 0 0
\(117\) −34.8988 −3.22639
\(118\) 0 0
\(119\) −3.27607 −0.300317
\(120\) 0 0
\(121\) 13.0415 1.18559
\(122\) 0 0
\(123\) −9.80642 −0.884215
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 12.4429 1.10413 0.552066 0.833801i \(-0.313840\pi\)
0.552066 + 0.833801i \(0.313840\pi\)
\(128\) 0 0
\(129\) 3.18421 0.280354
\(130\) 0 0
\(131\) −4.08742 −0.357120 −0.178560 0.983929i \(-0.557144\pi\)
−0.178560 + 0.983929i \(0.557144\pi\)
\(132\) 0 0
\(133\) 3.47949 0.301710
\(134\) 0 0
\(135\) −7.05086 −0.606841
\(136\) 0 0
\(137\) −19.9541 −1.70479 −0.852395 0.522898i \(-0.824851\pi\)
−0.852395 + 0.522898i \(0.824851\pi\)
\(138\) 0 0
\(139\) −7.90813 −0.670759 −0.335380 0.942083i \(-0.608865\pi\)
−0.335380 + 0.942083i \(0.608865\pi\)
\(140\) 0 0
\(141\) −36.8988 −3.10744
\(142\) 0 0
\(143\) 31.5210 2.63592
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) −13.5669 −1.11898
\(148\) 0 0
\(149\) −16.1017 −1.31910 −0.659552 0.751659i \(-0.729254\pi\)
−0.659552 + 0.751659i \(0.729254\pi\)
\(150\) 0 0
\(151\) −5.67307 −0.461668 −0.230834 0.972993i \(-0.574145\pi\)
−0.230834 + 0.972993i \(0.574145\pi\)
\(152\) 0 0
\(153\) 11.6588 0.942557
\(154\) 0 0
\(155\) 1.71900 0.138074
\(156\) 0 0
\(157\) 1.89384 0.151145 0.0755726 0.997140i \(-0.475922\pi\)
0.0755726 + 0.997140i \(0.475922\pi\)
\(158\) 0 0
\(159\) 9.80642 0.777700
\(160\) 0 0
\(161\) 10.5303 0.829908
\(162\) 0 0
\(163\) 5.95407 0.466359 0.233179 0.972434i \(-0.425087\pi\)
0.233179 + 0.972434i \(0.425087\pi\)
\(164\) 0 0
\(165\) 14.2351 1.10820
\(166\) 0 0
\(167\) −4.23951 −0.328063 −0.164032 0.986455i \(-0.552450\pi\)
−0.164032 + 0.986455i \(0.552450\pi\)
\(168\) 0 0
\(169\) 28.3274 2.17903
\(170\) 0 0
\(171\) −12.3827 −0.946929
\(172\) 0 0
\(173\) 24.4099 1.85585 0.927925 0.372766i \(-0.121591\pi\)
0.927925 + 0.372766i \(0.121591\pi\)
\(174\) 0 0
\(175\) −1.52543 −0.115311
\(176\) 0 0
\(177\) 9.24443 0.694854
\(178\) 0 0
\(179\) 3.61285 0.270037 0.135018 0.990843i \(-0.456891\pi\)
0.135018 + 0.990843i \(0.456891\pi\)
\(180\) 0 0
\(181\) 18.0415 1.34101 0.670507 0.741904i \(-0.266077\pi\)
0.670507 + 0.741904i \(0.266077\pi\)
\(182\) 0 0
\(183\) −7.05086 −0.521214
\(184\) 0 0
\(185\) −7.95407 −0.584795
\(186\) 0 0
\(187\) −10.5303 −0.770055
\(188\) 0 0
\(189\) −10.7556 −0.782353
\(190\) 0 0
\(191\) −9.85236 −0.712892 −0.356446 0.934316i \(-0.616012\pi\)
−0.356446 + 0.934316i \(0.616012\pi\)
\(192\) 0 0
\(193\) 2.23951 0.161203 0.0806017 0.996746i \(-0.474316\pi\)
0.0806017 + 0.996746i \(0.474316\pi\)
\(194\) 0 0
\(195\) 18.6637 1.33654
\(196\) 0 0
\(197\) −10.5620 −0.752511 −0.376255 0.926516i \(-0.622788\pi\)
−0.376255 + 0.926516i \(0.622788\pi\)
\(198\) 0 0
\(199\) −3.18421 −0.225723 −0.112861 0.993611i \(-0.536002\pi\)
−0.112861 + 0.993611i \(0.536002\pi\)
\(200\) 0 0
\(201\) 3.18421 0.224597
\(202\) 0 0
\(203\) −1.52543 −0.107064
\(204\) 0 0
\(205\) 3.37778 0.235915
\(206\) 0 0
\(207\) −37.4750 −2.60470
\(208\) 0 0
\(209\) 11.1842 0.773628
\(210\) 0 0
\(211\) 5.76049 0.396569 0.198284 0.980145i \(-0.436463\pi\)
0.198284 + 0.980145i \(0.436463\pi\)
\(212\) 0 0
\(213\) −10.3684 −0.710432
\(214\) 0 0
\(215\) −1.09679 −0.0748003
\(216\) 0 0
\(217\) 2.62222 0.178008
\(218\) 0 0
\(219\) 41.1941 2.78364
\(220\) 0 0
\(221\) −13.8064 −0.928721
\(222\) 0 0
\(223\) −8.14764 −0.545607 −0.272803 0.962070i \(-0.587951\pi\)
−0.272803 + 0.962070i \(0.587951\pi\)
\(224\) 0 0
\(225\) 5.42864 0.361909
\(226\) 0 0
\(227\) −10.5161 −0.697975 −0.348988 0.937127i \(-0.613475\pi\)
−0.348988 + 0.937127i \(0.613475\pi\)
\(228\) 0 0
\(229\) 8.48886 0.560960 0.280480 0.959860i \(-0.409506\pi\)
0.280480 + 0.959860i \(0.409506\pi\)
\(230\) 0 0
\(231\) 21.7146 1.42871
\(232\) 0 0
\(233\) −20.5718 −1.34771 −0.673853 0.738866i \(-0.735362\pi\)
−0.673853 + 0.738866i \(0.735362\pi\)
\(234\) 0 0
\(235\) 12.7096 0.829085
\(236\) 0 0
\(237\) −0.990632 −0.0643485
\(238\) 0 0
\(239\) −0.815792 −0.0527692 −0.0263846 0.999652i \(-0.508399\pi\)
−0.0263846 + 0.999652i \(0.508399\pi\)
\(240\) 0 0
\(241\) −7.24443 −0.466655 −0.233327 0.972398i \(-0.574961\pi\)
−0.233327 + 0.972398i \(0.574961\pi\)
\(242\) 0 0
\(243\) −9.00492 −0.577666
\(244\) 0 0
\(245\) 4.67307 0.298552
\(246\) 0 0
\(247\) 14.6637 0.933029
\(248\) 0 0
\(249\) 21.2859 1.34894
\(250\) 0 0
\(251\) 20.4242 1.28916 0.644582 0.764535i \(-0.277031\pi\)
0.644582 + 0.764535i \(0.277031\pi\)
\(252\) 0 0
\(253\) 33.8479 2.12800
\(254\) 0 0
\(255\) −6.23506 −0.390455
\(256\) 0 0
\(257\) 3.08250 0.192281 0.0961405 0.995368i \(-0.469350\pi\)
0.0961405 + 0.995368i \(0.469350\pi\)
\(258\) 0 0
\(259\) −12.1334 −0.753930
\(260\) 0 0
\(261\) 5.42864 0.336024
\(262\) 0 0
\(263\) −20.0558 −1.23669 −0.618346 0.785906i \(-0.712197\pi\)
−0.618346 + 0.785906i \(0.712197\pi\)
\(264\) 0 0
\(265\) −3.37778 −0.207496
\(266\) 0 0
\(267\) 8.56199 0.523985
\(268\) 0 0
\(269\) −23.4608 −1.43043 −0.715214 0.698906i \(-0.753671\pi\)
−0.715214 + 0.698906i \(0.753671\pi\)
\(270\) 0 0
\(271\) 21.9353 1.33248 0.666238 0.745739i \(-0.267903\pi\)
0.666238 + 0.745739i \(0.267903\pi\)
\(272\) 0 0
\(273\) 28.4701 1.72309
\(274\) 0 0
\(275\) −4.90321 −0.295675
\(276\) 0 0
\(277\) 18.3368 1.10175 0.550875 0.834588i \(-0.314294\pi\)
0.550875 + 0.834588i \(0.314294\pi\)
\(278\) 0 0
\(279\) −9.33185 −0.558683
\(280\) 0 0
\(281\) 6.89877 0.411546 0.205773 0.978600i \(-0.434029\pi\)
0.205773 + 0.978600i \(0.434029\pi\)
\(282\) 0 0
\(283\) 29.0049 1.72416 0.862082 0.506769i \(-0.169160\pi\)
0.862082 + 0.506769i \(0.169160\pi\)
\(284\) 0 0
\(285\) 6.62222 0.392266
\(286\) 0 0
\(287\) 5.15257 0.304146
\(288\) 0 0
\(289\) −12.3876 −0.728684
\(290\) 0 0
\(291\) −53.9309 −3.16148
\(292\) 0 0
\(293\) 7.79213 0.455221 0.227611 0.973752i \(-0.426909\pi\)
0.227611 + 0.973752i \(0.426909\pi\)
\(294\) 0 0
\(295\) −3.18421 −0.185392
\(296\) 0 0
\(297\) −34.5718 −2.00606
\(298\) 0 0
\(299\) 44.3783 2.56646
\(300\) 0 0
\(301\) −1.67307 −0.0964342
\(302\) 0 0
\(303\) 44.7654 2.57171
\(304\) 0 0
\(305\) 2.42864 0.139063
\(306\) 0 0
\(307\) −5.92549 −0.338185 −0.169093 0.985600i \(-0.554084\pi\)
−0.169093 + 0.985600i \(0.554084\pi\)
\(308\) 0 0
\(309\) −22.5303 −1.28171
\(310\) 0 0
\(311\) −8.94470 −0.507207 −0.253604 0.967308i \(-0.581616\pi\)
−0.253604 + 0.967308i \(0.581616\pi\)
\(312\) 0 0
\(313\) −30.5116 −1.72462 −0.862309 0.506382i \(-0.830983\pi\)
−0.862309 + 0.506382i \(0.830983\pi\)
\(314\) 0 0
\(315\) 8.28100 0.466581
\(316\) 0 0
\(317\) −22.2306 −1.24860 −0.624298 0.781186i \(-0.714615\pi\)
−0.624298 + 0.781186i \(0.714615\pi\)
\(318\) 0 0
\(319\) −4.90321 −0.274527
\(320\) 0 0
\(321\) 8.81579 0.492050
\(322\) 0 0
\(323\) −4.89877 −0.272575
\(324\) 0 0
\(325\) −6.42864 −0.356597
\(326\) 0 0
\(327\) −23.0509 −1.27472
\(328\) 0 0
\(329\) 19.3876 1.06887
\(330\) 0 0
\(331\) 6.54770 0.359894 0.179947 0.983676i \(-0.442407\pi\)
0.179947 + 0.983676i \(0.442407\pi\)
\(332\) 0 0
\(333\) 43.1798 2.36624
\(334\) 0 0
\(335\) −1.09679 −0.0599239
\(336\) 0 0
\(337\) 2.66815 0.145343 0.0726717 0.997356i \(-0.476847\pi\)
0.0726717 + 0.997356i \(0.476847\pi\)
\(338\) 0 0
\(339\) 22.7052 1.23318
\(340\) 0 0
\(341\) 8.42864 0.456436
\(342\) 0 0
\(343\) 17.8064 0.961457
\(344\) 0 0
\(345\) 20.0415 1.07900
\(346\) 0 0
\(347\) −14.6780 −0.787956 −0.393978 0.919120i \(-0.628901\pi\)
−0.393978 + 0.919120i \(0.628901\pi\)
\(348\) 0 0
\(349\) −11.1240 −0.595453 −0.297727 0.954651i \(-0.596228\pi\)
−0.297727 + 0.954651i \(0.596228\pi\)
\(350\) 0 0
\(351\) −45.3274 −2.41940
\(352\) 0 0
\(353\) −13.4795 −0.717441 −0.358721 0.933445i \(-0.616787\pi\)
−0.358721 + 0.933445i \(0.616787\pi\)
\(354\) 0 0
\(355\) 3.57136 0.189548
\(356\) 0 0
\(357\) −9.51114 −0.503383
\(358\) 0 0
\(359\) −26.1891 −1.38221 −0.691105 0.722755i \(-0.742876\pi\)
−0.691105 + 0.722755i \(0.742876\pi\)
\(360\) 0 0
\(361\) −13.7971 −0.726161
\(362\) 0 0
\(363\) 37.8622 1.98725
\(364\) 0 0
\(365\) −14.1891 −0.742693
\(366\) 0 0
\(367\) 22.9862 1.19987 0.599935 0.800049i \(-0.295193\pi\)
0.599935 + 0.800049i \(0.295193\pi\)
\(368\) 0 0
\(369\) −18.3368 −0.954574
\(370\) 0 0
\(371\) −5.15257 −0.267508
\(372\) 0 0
\(373\) −24.2766 −1.25699 −0.628496 0.777813i \(-0.716329\pi\)
−0.628496 + 0.777813i \(0.716329\pi\)
\(374\) 0 0
\(375\) −2.90321 −0.149921
\(376\) 0 0
\(377\) −6.42864 −0.331092
\(378\) 0 0
\(379\) −29.7605 −1.52869 −0.764347 0.644805i \(-0.776938\pi\)
−0.764347 + 0.644805i \(0.776938\pi\)
\(380\) 0 0
\(381\) 36.1245 1.85071
\(382\) 0 0
\(383\) −23.8020 −1.21622 −0.608112 0.793851i \(-0.708073\pi\)
−0.608112 + 0.793851i \(0.708073\pi\)
\(384\) 0 0
\(385\) −7.47949 −0.381190
\(386\) 0 0
\(387\) 5.95407 0.302662
\(388\) 0 0
\(389\) −6.52051 −0.330603 −0.165301 0.986243i \(-0.552860\pi\)
−0.165301 + 0.986243i \(0.552860\pi\)
\(390\) 0 0
\(391\) −14.8256 −0.749765
\(392\) 0 0
\(393\) −11.8666 −0.598593
\(394\) 0 0
\(395\) 0.341219 0.0171686
\(396\) 0 0
\(397\) −14.7654 −0.741055 −0.370527 0.928822i \(-0.620823\pi\)
−0.370527 + 0.928822i \(0.620823\pi\)
\(398\) 0 0
\(399\) 10.1017 0.505718
\(400\) 0 0
\(401\) 6.81579 0.340364 0.170182 0.985413i \(-0.445564\pi\)
0.170182 + 0.985413i \(0.445564\pi\)
\(402\) 0 0
\(403\) 11.0509 0.550482
\(404\) 0 0
\(405\) −4.18421 −0.207915
\(406\) 0 0
\(407\) −39.0005 −1.93318
\(408\) 0 0
\(409\) −11.0825 −0.547994 −0.273997 0.961731i \(-0.588346\pi\)
−0.273997 + 0.961731i \(0.588346\pi\)
\(410\) 0 0
\(411\) −57.9309 −2.85752
\(412\) 0 0
\(413\) −4.85728 −0.239011
\(414\) 0 0
\(415\) −7.33185 −0.359906
\(416\) 0 0
\(417\) −22.9590 −1.12431
\(418\) 0 0
\(419\) −30.9719 −1.51308 −0.756538 0.653950i \(-0.773111\pi\)
−0.756538 + 0.653950i \(0.773111\pi\)
\(420\) 0 0
\(421\) 22.8988 1.11602 0.558009 0.829835i \(-0.311566\pi\)
0.558009 + 0.829835i \(0.311566\pi\)
\(422\) 0 0
\(423\) −68.9960 −3.35470
\(424\) 0 0
\(425\) 2.14764 0.104176
\(426\) 0 0
\(427\) 3.70471 0.179284
\(428\) 0 0
\(429\) 91.5121 4.41825
\(430\) 0 0
\(431\) 28.0830 1.35271 0.676355 0.736576i \(-0.263559\pi\)
0.676355 + 0.736576i \(0.263559\pi\)
\(432\) 0 0
\(433\) 23.0049 1.10555 0.552773 0.833332i \(-0.313570\pi\)
0.552773 + 0.833332i \(0.313570\pi\)
\(434\) 0 0
\(435\) −2.90321 −0.139198
\(436\) 0 0
\(437\) 15.7462 0.753243
\(438\) 0 0
\(439\) −11.7462 −0.560616 −0.280308 0.959910i \(-0.590437\pi\)
−0.280308 + 0.959910i \(0.590437\pi\)
\(440\) 0 0
\(441\) −25.3684 −1.20802
\(442\) 0 0
\(443\) −17.6874 −0.840352 −0.420176 0.907443i \(-0.638032\pi\)
−0.420176 + 0.907443i \(0.638032\pi\)
\(444\) 0 0
\(445\) −2.94914 −0.139803
\(446\) 0 0
\(447\) −46.7467 −2.21104
\(448\) 0 0
\(449\) 1.57136 0.0741571 0.0370785 0.999312i \(-0.488195\pi\)
0.0370785 + 0.999312i \(0.488195\pi\)
\(450\) 0 0
\(451\) 16.5620 0.779874
\(452\) 0 0
\(453\) −16.4701 −0.773834
\(454\) 0 0
\(455\) −9.80642 −0.459732
\(456\) 0 0
\(457\) 1.47949 0.0692078 0.0346039 0.999401i \(-0.488983\pi\)
0.0346039 + 0.999401i \(0.488983\pi\)
\(458\) 0 0
\(459\) 15.1427 0.706802
\(460\) 0 0
\(461\) −41.2543 −1.92140 −0.960702 0.277583i \(-0.910467\pi\)
−0.960702 + 0.277583i \(0.910467\pi\)
\(462\) 0 0
\(463\) 34.4242 1.59983 0.799914 0.600115i \(-0.204878\pi\)
0.799914 + 0.600115i \(0.204878\pi\)
\(464\) 0 0
\(465\) 4.99063 0.231435
\(466\) 0 0
\(467\) −15.1699 −0.701980 −0.350990 0.936379i \(-0.614155\pi\)
−0.350990 + 0.936379i \(0.614155\pi\)
\(468\) 0 0
\(469\) −1.67307 −0.0772552
\(470\) 0 0
\(471\) 5.49823 0.253345
\(472\) 0 0
\(473\) −5.37778 −0.247271
\(474\) 0 0
\(475\) −2.28100 −0.104659
\(476\) 0 0
\(477\) 18.3368 0.839583
\(478\) 0 0
\(479\) 18.9763 0.867051 0.433526 0.901141i \(-0.357269\pi\)
0.433526 + 0.901141i \(0.357269\pi\)
\(480\) 0 0
\(481\) −51.1338 −2.33150
\(482\) 0 0
\(483\) 30.5718 1.39107
\(484\) 0 0
\(485\) 18.5763 0.843506
\(486\) 0 0
\(487\) 32.3926 1.46785 0.733923 0.679232i \(-0.237687\pi\)
0.733923 + 0.679232i \(0.237687\pi\)
\(488\) 0 0
\(489\) 17.2859 0.781696
\(490\) 0 0
\(491\) −2.69673 −0.121702 −0.0608508 0.998147i \(-0.519381\pi\)
−0.0608508 + 0.998147i \(0.519381\pi\)
\(492\) 0 0
\(493\) 2.14764 0.0967250
\(494\) 0 0
\(495\) 26.6178 1.19638
\(496\) 0 0
\(497\) 5.44785 0.244370
\(498\) 0 0
\(499\) −14.5718 −0.652325 −0.326163 0.945314i \(-0.605756\pi\)
−0.326163 + 0.945314i \(0.605756\pi\)
\(500\) 0 0
\(501\) −12.3082 −0.549890
\(502\) 0 0
\(503\) 22.2494 0.992050 0.496025 0.868308i \(-0.334792\pi\)
0.496025 + 0.868308i \(0.334792\pi\)
\(504\) 0 0
\(505\) −15.4193 −0.686149
\(506\) 0 0
\(507\) 82.2405 3.65243
\(508\) 0 0
\(509\) −9.18421 −0.407083 −0.203541 0.979066i \(-0.565245\pi\)
−0.203541 + 0.979066i \(0.565245\pi\)
\(510\) 0 0
\(511\) −21.6445 −0.957496
\(512\) 0 0
\(513\) −16.0830 −0.710081
\(514\) 0 0
\(515\) 7.76049 0.341968
\(516\) 0 0
\(517\) 62.3180 2.74074
\(518\) 0 0
\(519\) 70.8671 3.11072
\(520\) 0 0
\(521\) 7.01921 0.307517 0.153759 0.988108i \(-0.450862\pi\)
0.153759 + 0.988108i \(0.450862\pi\)
\(522\) 0 0
\(523\) 7.29036 0.318785 0.159393 0.987215i \(-0.449046\pi\)
0.159393 + 0.987215i \(0.449046\pi\)
\(524\) 0 0
\(525\) −4.42864 −0.193282
\(526\) 0 0
\(527\) −3.69181 −0.160818
\(528\) 0 0
\(529\) 24.6543 1.07193
\(530\) 0 0
\(531\) 17.2859 0.750145
\(532\) 0 0
\(533\) 21.7146 0.940562
\(534\) 0 0
\(535\) −3.03657 −0.131282
\(536\) 0 0
\(537\) 10.4889 0.452628
\(538\) 0 0
\(539\) 22.9131 0.986935
\(540\) 0 0
\(541\) 30.9491 1.33061 0.665304 0.746573i \(-0.268302\pi\)
0.665304 + 0.746573i \(0.268302\pi\)
\(542\) 0 0
\(543\) 52.3783 2.24777
\(544\) 0 0
\(545\) 7.93978 0.340103
\(546\) 0 0
\(547\) 19.4237 0.830498 0.415249 0.909708i \(-0.363694\pi\)
0.415249 + 0.909708i \(0.363694\pi\)
\(548\) 0 0
\(549\) −13.1842 −0.562688
\(550\) 0 0
\(551\) −2.28100 −0.0971737
\(552\) 0 0
\(553\) 0.520505 0.0221341
\(554\) 0 0
\(555\) −23.0923 −0.980215
\(556\) 0 0
\(557\) 30.3497 1.28596 0.642979 0.765884i \(-0.277698\pi\)
0.642979 + 0.765884i \(0.277698\pi\)
\(558\) 0 0
\(559\) −7.05086 −0.298219
\(560\) 0 0
\(561\) −30.5718 −1.29074
\(562\) 0 0
\(563\) 33.1798 1.39836 0.699180 0.714946i \(-0.253549\pi\)
0.699180 + 0.714946i \(0.253549\pi\)
\(564\) 0 0
\(565\) −7.82071 −0.329020
\(566\) 0 0
\(567\) −6.38271 −0.268048
\(568\) 0 0
\(569\) −4.06022 −0.170213 −0.0851067 0.996372i \(-0.527123\pi\)
−0.0851067 + 0.996372i \(0.527123\pi\)
\(570\) 0 0
\(571\) 31.5496 1.32031 0.660154 0.751130i \(-0.270491\pi\)
0.660154 + 0.751130i \(0.270491\pi\)
\(572\) 0 0
\(573\) −28.6035 −1.19493
\(574\) 0 0
\(575\) −6.90321 −0.287884
\(576\) 0 0
\(577\) 33.7891 1.40666 0.703329 0.710865i \(-0.251696\pi\)
0.703329 + 0.710865i \(0.251696\pi\)
\(578\) 0 0
\(579\) 6.50177 0.270204
\(580\) 0 0
\(581\) −11.1842 −0.463999
\(582\) 0 0
\(583\) −16.5620 −0.685928
\(584\) 0 0
\(585\) 34.8988 1.44289
\(586\) 0 0
\(587\) 25.5669 1.05526 0.527630 0.849474i \(-0.323081\pi\)
0.527630 + 0.849474i \(0.323081\pi\)
\(588\) 0 0
\(589\) 3.92104 0.161564
\(590\) 0 0
\(591\) −30.6637 −1.26134
\(592\) 0 0
\(593\) 7.96836 0.327221 0.163611 0.986525i \(-0.447686\pi\)
0.163611 + 0.986525i \(0.447686\pi\)
\(594\) 0 0
\(595\) 3.27607 0.134306
\(596\) 0 0
\(597\) −9.24443 −0.378349
\(598\) 0 0
\(599\) −37.2815 −1.52328 −0.761640 0.648001i \(-0.775605\pi\)
−0.761640 + 0.648001i \(0.775605\pi\)
\(600\) 0 0
\(601\) −29.9496 −1.22167 −0.610835 0.791758i \(-0.709166\pi\)
−0.610835 + 0.791758i \(0.709166\pi\)
\(602\) 0 0
\(603\) 5.95407 0.242468
\(604\) 0 0
\(605\) −13.0415 −0.530212
\(606\) 0 0
\(607\) 31.8435 1.29249 0.646243 0.763132i \(-0.276339\pi\)
0.646243 + 0.763132i \(0.276339\pi\)
\(608\) 0 0
\(609\) −4.42864 −0.179458
\(610\) 0 0
\(611\) 81.7057 3.30546
\(612\) 0 0
\(613\) −2.65386 −0.107188 −0.0535942 0.998563i \(-0.517068\pi\)
−0.0535942 + 0.998563i \(0.517068\pi\)
\(614\) 0 0
\(615\) 9.80642 0.395433
\(616\) 0 0
\(617\) −18.3096 −0.737116 −0.368558 0.929605i \(-0.620148\pi\)
−0.368558 + 0.929605i \(0.620148\pi\)
\(618\) 0 0
\(619\) −12.8113 −0.514931 −0.257466 0.966287i \(-0.582887\pi\)
−0.257466 + 0.966287i \(0.582887\pi\)
\(620\) 0 0
\(621\) −48.6735 −1.95320
\(622\) 0 0
\(623\) −4.49871 −0.180237
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 32.4701 1.29673
\(628\) 0 0
\(629\) 17.0825 0.681124
\(630\) 0 0
\(631\) −30.2766 −1.20529 −0.602645 0.798009i \(-0.705887\pi\)
−0.602645 + 0.798009i \(0.705887\pi\)
\(632\) 0 0
\(633\) 16.7239 0.664716
\(634\) 0 0
\(635\) −12.4429 −0.493783
\(636\) 0 0
\(637\) 30.0415 1.19029
\(638\) 0 0
\(639\) −19.3876 −0.766963
\(640\) 0 0
\(641\) −4.50177 −0.177809 −0.0889046 0.996040i \(-0.528337\pi\)
−0.0889046 + 0.996040i \(0.528337\pi\)
\(642\) 0 0
\(643\) 40.0272 1.57852 0.789259 0.614060i \(-0.210465\pi\)
0.789259 + 0.614060i \(0.210465\pi\)
\(644\) 0 0
\(645\) −3.18421 −0.125378
\(646\) 0 0
\(647\) −27.3604 −1.07565 −0.537825 0.843057i \(-0.680754\pi\)
−0.537825 + 0.843057i \(0.680754\pi\)
\(648\) 0 0
\(649\) −15.6128 −0.612858
\(650\) 0 0
\(651\) 7.61285 0.298371
\(652\) 0 0
\(653\) 22.8430 0.893915 0.446958 0.894555i \(-0.352507\pi\)
0.446958 + 0.894555i \(0.352507\pi\)
\(654\) 0 0
\(655\) 4.08742 0.159709
\(656\) 0 0
\(657\) 77.0277 3.00514
\(658\) 0 0
\(659\) −15.0178 −0.585012 −0.292506 0.956264i \(-0.594489\pi\)
−0.292506 + 0.956264i \(0.594489\pi\)
\(660\) 0 0
\(661\) −4.65080 −0.180895 −0.0904475 0.995901i \(-0.528830\pi\)
−0.0904475 + 0.995901i \(0.528830\pi\)
\(662\) 0 0
\(663\) −40.0830 −1.55669
\(664\) 0 0
\(665\) −3.47949 −0.134929
\(666\) 0 0
\(667\) −6.90321 −0.267293
\(668\) 0 0
\(669\) −23.6543 −0.914529
\(670\) 0 0
\(671\) 11.9081 0.459708
\(672\) 0 0
\(673\) −44.8671 −1.72950 −0.864750 0.502202i \(-0.832523\pi\)
−0.864750 + 0.502202i \(0.832523\pi\)
\(674\) 0 0
\(675\) 7.05086 0.271388
\(676\) 0 0
\(677\) 27.2212 1.04620 0.523099 0.852272i \(-0.324776\pi\)
0.523099 + 0.852272i \(0.324776\pi\)
\(678\) 0 0
\(679\) 28.3368 1.08747
\(680\) 0 0
\(681\) −30.5303 −1.16993
\(682\) 0 0
\(683\) −12.0558 −0.461301 −0.230651 0.973037i \(-0.574085\pi\)
−0.230651 + 0.973037i \(0.574085\pi\)
\(684\) 0 0
\(685\) 19.9541 0.762406
\(686\) 0 0
\(687\) 24.6450 0.940264
\(688\) 0 0
\(689\) −21.7146 −0.827259
\(690\) 0 0
\(691\) −37.5812 −1.42966 −0.714828 0.699300i \(-0.753495\pi\)
−0.714828 + 0.699300i \(0.753495\pi\)
\(692\) 0 0
\(693\) 40.6035 1.54240
\(694\) 0 0
\(695\) 7.90813 0.299973
\(696\) 0 0
\(697\) −7.25428 −0.274775
\(698\) 0 0
\(699\) −59.7244 −2.25898
\(700\) 0 0
\(701\) 2.04149 0.0771059 0.0385530 0.999257i \(-0.487725\pi\)
0.0385530 + 0.999257i \(0.487725\pi\)
\(702\) 0 0
\(703\) −18.1432 −0.684284
\(704\) 0 0
\(705\) 36.8988 1.38969
\(706\) 0 0
\(707\) −23.5210 −0.884598
\(708\) 0 0
\(709\) −32.1432 −1.20716 −0.603582 0.797301i \(-0.706260\pi\)
−0.603582 + 0.797301i \(0.706260\pi\)
\(710\) 0 0
\(711\) −1.85236 −0.0694688
\(712\) 0 0
\(713\) 11.8666 0.444409
\(714\) 0 0
\(715\) −31.5210 −1.17882
\(716\) 0 0
\(717\) −2.36842 −0.0884501
\(718\) 0 0
\(719\) −1.01921 −0.0380102 −0.0190051 0.999819i \(-0.506050\pi\)
−0.0190051 + 0.999819i \(0.506050\pi\)
\(720\) 0 0
\(721\) 11.8381 0.440873
\(722\) 0 0
\(723\) −21.0321 −0.782193
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 24.1476 0.895587 0.447793 0.894137i \(-0.352210\pi\)
0.447793 + 0.894137i \(0.352210\pi\)
\(728\) 0 0
\(729\) −38.6958 −1.43318
\(730\) 0 0
\(731\) 2.35551 0.0871217
\(732\) 0 0
\(733\) 18.8845 0.697514 0.348757 0.937213i \(-0.386604\pi\)
0.348757 + 0.937213i \(0.386604\pi\)
\(734\) 0 0
\(735\) 13.5669 0.500423
\(736\) 0 0
\(737\) −5.37778 −0.198093
\(738\) 0 0
\(739\) 3.31312 0.121875 0.0609375 0.998142i \(-0.480591\pi\)
0.0609375 + 0.998142i \(0.480591\pi\)
\(740\) 0 0
\(741\) 42.5718 1.56392
\(742\) 0 0
\(743\) −2.99508 −0.109879 −0.0549394 0.998490i \(-0.517497\pi\)
−0.0549394 + 0.998490i \(0.517497\pi\)
\(744\) 0 0
\(745\) 16.1017 0.589921
\(746\) 0 0
\(747\) 39.8020 1.45628
\(748\) 0 0
\(749\) −4.63206 −0.169252
\(750\) 0 0
\(751\) −11.9956 −0.437724 −0.218862 0.975756i \(-0.570234\pi\)
−0.218862 + 0.975756i \(0.570234\pi\)
\(752\) 0 0
\(753\) 59.2958 2.16086
\(754\) 0 0
\(755\) 5.67307 0.206464
\(756\) 0 0
\(757\) 18.5763 0.675166 0.337583 0.941296i \(-0.390391\pi\)
0.337583 + 0.941296i \(0.390391\pi\)
\(758\) 0 0
\(759\) 98.2677 3.56689
\(760\) 0 0
\(761\) 2.59057 0.0939082 0.0469541 0.998897i \(-0.485049\pi\)
0.0469541 + 0.998897i \(0.485049\pi\)
\(762\) 0 0
\(763\) 12.1116 0.438468
\(764\) 0 0
\(765\) −11.6588 −0.421524
\(766\) 0 0
\(767\) −20.4701 −0.739133
\(768\) 0 0
\(769\) 32.7467 1.18088 0.590438 0.807083i \(-0.298955\pi\)
0.590438 + 0.807083i \(0.298955\pi\)
\(770\) 0 0
\(771\) 8.94914 0.322296
\(772\) 0 0
\(773\) 28.2208 1.01503 0.507515 0.861643i \(-0.330564\pi\)
0.507515 + 0.861643i \(0.330564\pi\)
\(774\) 0 0
\(775\) −1.71900 −0.0617484
\(776\) 0 0
\(777\) −35.2257 −1.26371
\(778\) 0 0
\(779\) 7.70471 0.276050
\(780\) 0 0
\(781\) 17.5111 0.626598
\(782\) 0 0
\(783\) 7.05086 0.251977
\(784\) 0 0
\(785\) −1.89384 −0.0675942
\(786\) 0 0
\(787\) −11.9857 −0.427244 −0.213622 0.976916i \(-0.568526\pi\)
−0.213622 + 0.976916i \(0.568526\pi\)
\(788\) 0 0
\(789\) −58.2262 −2.07291
\(790\) 0 0
\(791\) −11.9299 −0.424180
\(792\) 0 0
\(793\) 15.6128 0.554428
\(794\) 0 0
\(795\) −9.80642 −0.347798
\(796\) 0 0
\(797\) −33.0366 −1.17022 −0.585108 0.810956i \(-0.698948\pi\)
−0.585108 + 0.810956i \(0.698948\pi\)
\(798\) 0 0
\(799\) −27.2958 −0.965655
\(800\) 0 0
\(801\) 16.0098 0.565680
\(802\) 0 0
\(803\) −69.5723 −2.45515
\(804\) 0 0
\(805\) −10.5303 −0.371146
\(806\) 0 0
\(807\) −68.1116 −2.39764
\(808\) 0 0
\(809\) −32.1303 −1.12964 −0.564820 0.825214i \(-0.691055\pi\)
−0.564820 + 0.825214i \(0.691055\pi\)
\(810\) 0 0
\(811\) −15.3176 −0.537872 −0.268936 0.963158i \(-0.586672\pi\)
−0.268936 + 0.963158i \(0.586672\pi\)
\(812\) 0 0
\(813\) 63.6829 2.23346
\(814\) 0 0
\(815\) −5.95407 −0.208562
\(816\) 0 0
\(817\) −2.50177 −0.0875258
\(818\) 0 0
\(819\) 53.2355 1.86020
\(820\) 0 0
\(821\) −17.1427 −0.598285 −0.299143 0.954208i \(-0.596701\pi\)
−0.299143 + 0.954208i \(0.596701\pi\)
\(822\) 0 0
\(823\) 35.6400 1.24233 0.621167 0.783678i \(-0.286659\pi\)
0.621167 + 0.783678i \(0.286659\pi\)
\(824\) 0 0
\(825\) −14.2351 −0.495601
\(826\) 0 0
\(827\) −4.70964 −0.163770 −0.0818850 0.996642i \(-0.526094\pi\)
−0.0818850 + 0.996642i \(0.526094\pi\)
\(828\) 0 0
\(829\) 2.25380 0.0782777 0.0391388 0.999234i \(-0.487539\pi\)
0.0391388 + 0.999234i \(0.487539\pi\)
\(830\) 0 0
\(831\) 53.2355 1.84672
\(832\) 0 0
\(833\) −10.0361 −0.347730
\(834\) 0 0
\(835\) 4.23951 0.146714
\(836\) 0 0
\(837\) −12.1204 −0.418944
\(838\) 0 0
\(839\) 8.42419 0.290835 0.145418 0.989370i \(-0.453547\pi\)
0.145418 + 0.989370i \(0.453547\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 20.0286 0.689821
\(844\) 0 0
\(845\) −28.3274 −0.974492
\(846\) 0 0
\(847\) −19.8938 −0.683561
\(848\) 0 0
\(849\) 84.2074 2.88999
\(850\) 0 0
\(851\) −54.9086 −1.88224
\(852\) 0 0
\(853\) 51.8247 1.77444 0.887222 0.461342i \(-0.152632\pi\)
0.887222 + 0.461342i \(0.152632\pi\)
\(854\) 0 0
\(855\) 12.3827 0.423480
\(856\) 0 0
\(857\) −30.0415 −1.02620 −0.513099 0.858330i \(-0.671503\pi\)
−0.513099 + 0.858330i \(0.671503\pi\)
\(858\) 0 0
\(859\) −19.6874 −0.671724 −0.335862 0.941911i \(-0.609028\pi\)
−0.335862 + 0.941911i \(0.609028\pi\)
\(860\) 0 0
\(861\) 14.9590 0.509801
\(862\) 0 0
\(863\) −19.5986 −0.667143 −0.333571 0.942725i \(-0.608254\pi\)
−0.333571 + 0.942725i \(0.608254\pi\)
\(864\) 0 0
\(865\) −24.4099 −0.829962
\(866\) 0 0
\(867\) −35.9639 −1.22140
\(868\) 0 0
\(869\) 1.67307 0.0567550
\(870\) 0 0
\(871\) −7.05086 −0.238909
\(872\) 0 0
\(873\) −100.844 −3.41305
\(874\) 0 0
\(875\) 1.52543 0.0515689
\(876\) 0 0
\(877\) −16.2351 −0.548219 −0.274110 0.961698i \(-0.588383\pi\)
−0.274110 + 0.961698i \(0.588383\pi\)
\(878\) 0 0
\(879\) 22.6222 0.763028
\(880\) 0 0
\(881\) −20.7052 −0.697576 −0.348788 0.937202i \(-0.613407\pi\)
−0.348788 + 0.937202i \(0.613407\pi\)
\(882\) 0 0
\(883\) −8.75112 −0.294499 −0.147249 0.989099i \(-0.547042\pi\)
−0.147249 + 0.989099i \(0.547042\pi\)
\(884\) 0 0
\(885\) −9.24443 −0.310748
\(886\) 0 0
\(887\) −0.414349 −0.0139125 −0.00695625 0.999976i \(-0.502214\pi\)
−0.00695625 + 0.999976i \(0.502214\pi\)
\(888\) 0 0
\(889\) −18.9808 −0.636595
\(890\) 0 0
\(891\) −20.5161 −0.687314
\(892\) 0 0
\(893\) 28.9906 0.970135
\(894\) 0 0
\(895\) −3.61285 −0.120764
\(896\) 0 0
\(897\) 128.839 4.30183
\(898\) 0 0
\(899\) −1.71900 −0.0573320
\(900\) 0 0
\(901\) 7.25428 0.241675
\(902\) 0 0
\(903\) −4.85728 −0.161640
\(904\) 0 0
\(905\) −18.0415 −0.599719
\(906\) 0 0
\(907\) 46.9862 1.56015 0.780075 0.625686i \(-0.215181\pi\)
0.780075 + 0.625686i \(0.215181\pi\)
\(908\) 0 0
\(909\) 83.7057 2.77634
\(910\) 0 0
\(911\) 12.1704 0.403223 0.201612 0.979466i \(-0.435382\pi\)
0.201612 + 0.979466i \(0.435382\pi\)
\(912\) 0 0
\(913\) −35.9496 −1.18976
\(914\) 0 0
\(915\) 7.05086 0.233094
\(916\) 0 0
\(917\) 6.23506 0.205900
\(918\) 0 0
\(919\) 23.2672 0.767514 0.383757 0.923434i \(-0.374630\pi\)
0.383757 + 0.923434i \(0.374630\pi\)
\(920\) 0 0
\(921\) −17.2029 −0.566856
\(922\) 0 0
\(923\) 22.9590 0.755704
\(924\) 0 0
\(925\) 7.95407 0.261528
\(926\) 0 0
\(927\) −42.1289 −1.38369
\(928\) 0 0
\(929\) 44.7556 1.46838 0.734191 0.678943i \(-0.237562\pi\)
0.734191 + 0.678943i \(0.237562\pi\)
\(930\) 0 0
\(931\) 10.6593 0.349343
\(932\) 0 0
\(933\) −25.9684 −0.850166
\(934\) 0 0
\(935\) 10.5303 0.344379
\(936\) 0 0
\(937\) −22.7239 −0.742358 −0.371179 0.928561i \(-0.621046\pi\)
−0.371179 + 0.928561i \(0.621046\pi\)
\(938\) 0 0
\(939\) −88.5817 −2.89075
\(940\) 0 0
\(941\) 4.10171 0.133712 0.0668560 0.997763i \(-0.478703\pi\)
0.0668560 + 0.997763i \(0.478703\pi\)
\(942\) 0 0
\(943\) 23.3176 0.759324
\(944\) 0 0
\(945\) 10.7556 0.349879
\(946\) 0 0
\(947\) −16.6178 −0.540005 −0.270002 0.962860i \(-0.587025\pi\)
−0.270002 + 0.962860i \(0.587025\pi\)
\(948\) 0 0
\(949\) −91.2168 −2.96102
\(950\) 0 0
\(951\) −64.5402 −2.09286
\(952\) 0 0
\(953\) 2.85728 0.0925563 0.0462782 0.998929i \(-0.485264\pi\)
0.0462782 + 0.998929i \(0.485264\pi\)
\(954\) 0 0
\(955\) 9.85236 0.318815
\(956\) 0 0
\(957\) −14.2351 −0.460154
\(958\) 0 0
\(959\) 30.4385 0.982910
\(960\) 0 0
\(961\) −28.0450 −0.904678
\(962\) 0 0
\(963\) 16.4844 0.531203
\(964\) 0 0
\(965\) −2.23951 −0.0720923
\(966\) 0 0
\(967\) −16.8015 −0.540300 −0.270150 0.962818i \(-0.587073\pi\)
−0.270150 + 0.962818i \(0.587073\pi\)
\(968\) 0 0
\(969\) −14.2222 −0.456881
\(970\) 0 0
\(971\) −38.3640 −1.23116 −0.615579 0.788075i \(-0.711078\pi\)
−0.615579 + 0.788075i \(0.711078\pi\)
\(972\) 0 0
\(973\) 12.0633 0.386731
\(974\) 0 0
\(975\) −18.6637 −0.597717
\(976\) 0 0
\(977\) 31.6356 1.01211 0.506056 0.862500i \(-0.331103\pi\)
0.506056 + 0.862500i \(0.331103\pi\)
\(978\) 0 0
\(979\) −14.4603 −0.462153
\(980\) 0 0
\(981\) −43.1022 −1.37615
\(982\) 0 0
\(983\) 35.3733 1.12823 0.564117 0.825695i \(-0.309217\pi\)
0.564117 + 0.825695i \(0.309217\pi\)
\(984\) 0 0
\(985\) 10.5620 0.336533
\(986\) 0 0
\(987\) 56.2864 1.79162
\(988\) 0 0
\(989\) −7.57136 −0.240755
\(990\) 0 0
\(991\) −24.6953 −0.784474 −0.392237 0.919864i \(-0.628299\pi\)
−0.392237 + 0.919864i \(0.628299\pi\)
\(992\) 0 0
\(993\) 19.0094 0.603244
\(994\) 0 0
\(995\) 3.18421 0.100946
\(996\) 0 0
\(997\) −11.4050 −0.361199 −0.180600 0.983557i \(-0.557804\pi\)
−0.180600 + 0.983557i \(0.557804\pi\)
\(998\) 0 0
\(999\) 56.0830 1.77439
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 2320.2.a.s.1.3 3
4.3 odd 2 145.2.a.d.1.1 3
8.3 odd 2 9280.2.a.bu.1.3 3
8.5 even 2 9280.2.a.bm.1.1 3
12.11 even 2 1305.2.a.o.1.3 3
20.3 even 4 725.2.b.d.349.4 6
20.7 even 4 725.2.b.d.349.3 6
20.19 odd 2 725.2.a.d.1.3 3
28.27 even 2 7105.2.a.p.1.1 3
60.59 even 2 6525.2.a.bh.1.1 3
116.115 odd 2 4205.2.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.1 3 4.3 odd 2
725.2.a.d.1.3 3 20.19 odd 2
725.2.b.d.349.3 6 20.7 even 4
725.2.b.d.349.4 6 20.3 even 4
1305.2.a.o.1.3 3 12.11 even 2
2320.2.a.s.1.3 3 1.1 even 1 trivial
4205.2.a.e.1.3 3 116.115 odd 2
6525.2.a.bh.1.1 3 60.59 even 2
7105.2.a.p.1.1 3 28.27 even 2
9280.2.a.bm.1.1 3 8.5 even 2
9280.2.a.bu.1.3 3 8.3 odd 2