Properties

Label 2320.2.a.m.1.3
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) 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 580)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.571993\) 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)
 
\(f(q)\) \(=\) \(q+2.67282 q^{3} -1.00000 q^{5} +4.67282 q^{7} +4.14399 q^{9} +O(q^{10})\) \(q+2.67282 q^{3} -1.00000 q^{5} +4.67282 q^{7} +4.14399 q^{9} +0.672824 q^{11} -1.14399 q^{13} -2.67282 q^{15} +3.52884 q^{17} -5.52884 q^{19} +12.4896 q^{21} +3.81681 q^{23} +1.00000 q^{25} +3.05767 q^{27} -1.00000 q^{29} +1.52884 q^{31} +1.79834 q^{33} -4.67282 q^{35} +7.16246 q^{37} -3.05767 q^{39} +2.85601 q^{41} -8.96080 q^{43} -4.14399 q^{45} +6.67282 q^{47} +14.8353 q^{49} +9.43196 q^{51} +10.4896 q^{53} -0.672824 q^{55} -14.7776 q^{57} -10.7776 q^{59} -14.4896 q^{61} +19.3641 q^{63} +1.14399 q^{65} +7.81681 q^{67} +10.2017 q^{69} -4.48963 q^{71} -4.96080 q^{73} +2.67282 q^{75} +3.14399 q^{77} -2.38485 q^{79} -4.25934 q^{81} -14.0185 q^{83} -3.52884 q^{85} -2.67282 q^{87} -1.63362 q^{89} -5.34565 q^{91} +4.08631 q^{93} +5.52884 q^{95} -9.32718 q^{97} +2.78817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 11 q^{9} - 8 q^{11} - 2 q^{13} + 2 q^{15} + 2 q^{17} - 8 q^{19} + 16 q^{21} + 3 q^{25} - 8 q^{27} - 3 q^{29} - 4 q^{31} + 24 q^{33} - 4 q^{35} - 10 q^{37} + 8 q^{39} + 10 q^{41} - 14 q^{43} - 11 q^{45} + 10 q^{47} + 3 q^{49} + 24 q^{51} + 10 q^{53} + 8 q^{55} - 20 q^{57} - 8 q^{59} - 22 q^{61} + 8 q^{63} + 2 q^{65} + 12 q^{67} + 12 q^{69} + 8 q^{71} - 2 q^{73} - 2 q^{75} + 8 q^{77} + 23 q^{81} - 12 q^{83} - 2 q^{85} + 2 q^{87} + 18 q^{89} + 4 q^{91} + 28 q^{93} + 8 q^{95} - 38 q^{97} - 36 q^{99}+O(q^{100}) \) 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
\(3\) 2.67282 1.54316 0.771578 0.636135i \(-0.219468\pi\)
0.771578 + 0.636135i \(0.219468\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.67282 1.76616 0.883081 0.469221i \(-0.155465\pi\)
0.883081 + 0.469221i \(0.155465\pi\)
\(8\) 0 0
\(9\) 4.14399 1.38133
\(10\) 0 0
\(11\) 0.672824 0.202864 0.101432 0.994842i \(-0.467658\pi\)
0.101432 + 0.994842i \(0.467658\pi\)
\(12\) 0 0
\(13\) −1.14399 −0.317285 −0.158642 0.987336i \(-0.550712\pi\)
−0.158642 + 0.987336i \(0.550712\pi\)
\(14\) 0 0
\(15\) −2.67282 −0.690120
\(16\) 0 0
\(17\) 3.52884 0.855869 0.427934 0.903810i \(-0.359241\pi\)
0.427934 + 0.903810i \(0.359241\pi\)
\(18\) 0 0
\(19\) −5.52884 −1.26840 −0.634201 0.773168i \(-0.718671\pi\)
−0.634201 + 0.773168i \(0.718671\pi\)
\(20\) 0 0
\(21\) 12.4896 2.72546
\(22\) 0 0
\(23\) 3.81681 0.795860 0.397930 0.917416i \(-0.369729\pi\)
0.397930 + 0.917416i \(0.369729\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.05767 0.588450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 1.52884 0.274587 0.137294 0.990530i \(-0.456160\pi\)
0.137294 + 0.990530i \(0.456160\pi\)
\(32\) 0 0
\(33\) 1.79834 0.313051
\(34\) 0 0
\(35\) −4.67282 −0.789851
\(36\) 0 0
\(37\) 7.16246 1.17750 0.588750 0.808315i \(-0.299620\pi\)
0.588750 + 0.808315i \(0.299620\pi\)
\(38\) 0 0
\(39\) −3.05767 −0.489620
\(40\) 0 0
\(41\) 2.85601 0.446034 0.223017 0.974815i \(-0.428409\pi\)
0.223017 + 0.974815i \(0.428409\pi\)
\(42\) 0 0
\(43\) −8.96080 −1.36651 −0.683254 0.730180i \(-0.739436\pi\)
−0.683254 + 0.730180i \(0.739436\pi\)
\(44\) 0 0
\(45\) −4.14399 −0.617749
\(46\) 0 0
\(47\) 6.67282 0.973331 0.486666 0.873588i \(-0.338213\pi\)
0.486666 + 0.873588i \(0.338213\pi\)
\(48\) 0 0
\(49\) 14.8353 2.11933
\(50\) 0 0
\(51\) 9.43196 1.32074
\(52\) 0 0
\(53\) 10.4896 1.44086 0.720431 0.693527i \(-0.243944\pi\)
0.720431 + 0.693527i \(0.243944\pi\)
\(54\) 0 0
\(55\) −0.672824 −0.0907235
\(56\) 0 0
\(57\) −14.7776 −1.95734
\(58\) 0 0
\(59\) −10.7776 −1.40312 −0.701562 0.712608i \(-0.747514\pi\)
−0.701562 + 0.712608i \(0.747514\pi\)
\(60\) 0 0
\(61\) −14.4896 −1.85521 −0.927604 0.373566i \(-0.878135\pi\)
−0.927604 + 0.373566i \(0.878135\pi\)
\(62\) 0 0
\(63\) 19.3641 2.43965
\(64\) 0 0
\(65\) 1.14399 0.141894
\(66\) 0 0
\(67\) 7.81681 0.954975 0.477488 0.878638i \(-0.341548\pi\)
0.477488 + 0.878638i \(0.341548\pi\)
\(68\) 0 0
\(69\) 10.2017 1.22814
\(70\) 0 0
\(71\) −4.48963 −0.532822 −0.266411 0.963860i \(-0.585838\pi\)
−0.266411 + 0.963860i \(0.585838\pi\)
\(72\) 0 0
\(73\) −4.96080 −0.580617 −0.290309 0.956933i \(-0.593758\pi\)
−0.290309 + 0.956933i \(0.593758\pi\)
\(74\) 0 0
\(75\) 2.67282 0.308631
\(76\) 0 0
\(77\) 3.14399 0.358291
\(78\) 0 0
\(79\) −2.38485 −0.268317 −0.134158 0.990960i \(-0.542833\pi\)
−0.134158 + 0.990960i \(0.542833\pi\)
\(80\) 0 0
\(81\) −4.25934 −0.473259
\(82\) 0 0
\(83\) −14.0185 −1.53873 −0.769364 0.638811i \(-0.779427\pi\)
−0.769364 + 0.638811i \(0.779427\pi\)
\(84\) 0 0
\(85\) −3.52884 −0.382756
\(86\) 0 0
\(87\) −2.67282 −0.286557
\(88\) 0 0
\(89\) −1.63362 −0.173163 −0.0865817 0.996245i \(-0.527594\pi\)
−0.0865817 + 0.996245i \(0.527594\pi\)
\(90\) 0 0
\(91\) −5.34565 −0.560376
\(92\) 0 0
\(93\) 4.08631 0.423731
\(94\) 0 0
\(95\) 5.52884 0.567247
\(96\) 0 0
\(97\) −9.32718 −0.947031 −0.473516 0.880785i \(-0.657015\pi\)
−0.473516 + 0.880785i \(0.657015\pi\)
\(98\) 0 0
\(99\) 2.78817 0.280222
\(100\) 0 0
\(101\) −16.3249 −1.62439 −0.812195 0.583386i \(-0.801727\pi\)
−0.812195 + 0.583386i \(0.801727\pi\)
\(102\) 0 0
\(103\) 13.5288 1.33304 0.666518 0.745489i \(-0.267784\pi\)
0.666518 + 0.745489i \(0.267784\pi\)
\(104\) 0 0
\(105\) −12.4896 −1.21886
\(106\) 0 0
\(107\) −0.672824 −0.0650443 −0.0325222 0.999471i \(-0.510354\pi\)
−0.0325222 + 0.999471i \(0.510354\pi\)
\(108\) 0 0
\(109\) 13.5473 1.29760 0.648798 0.760960i \(-0.275272\pi\)
0.648798 + 0.760960i \(0.275272\pi\)
\(110\) 0 0
\(111\) 19.1440 1.81707
\(112\) 0 0
\(113\) 20.0185 1.88318 0.941590 0.336762i \(-0.109332\pi\)
0.941590 + 0.336762i \(0.109332\pi\)
\(114\) 0 0
\(115\) −3.81681 −0.355919
\(116\) 0 0
\(117\) −4.74066 −0.438275
\(118\) 0 0
\(119\) 16.4896 1.51160
\(120\) 0 0
\(121\) −10.5473 −0.958846
\(122\) 0 0
\(123\) 7.63362 0.688300
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.03920 0.269686 0.134843 0.990867i \(-0.456947\pi\)
0.134843 + 0.990867i \(0.456947\pi\)
\(128\) 0 0
\(129\) −23.9506 −2.10874
\(130\) 0 0
\(131\) 3.81681 0.333476 0.166738 0.986001i \(-0.446677\pi\)
0.166738 + 0.986001i \(0.446677\pi\)
\(132\) 0 0
\(133\) −25.8353 −2.24020
\(134\) 0 0
\(135\) −3.05767 −0.263163
\(136\) 0 0
\(137\) 16.1048 1.37592 0.687962 0.725746i \(-0.258505\pi\)
0.687962 + 0.725746i \(0.258505\pi\)
\(138\) 0 0
\(139\) −3.05767 −0.259349 −0.129674 0.991557i \(-0.541393\pi\)
−0.129674 + 0.991557i \(0.541393\pi\)
\(140\) 0 0
\(141\) 17.8353 1.50200
\(142\) 0 0
\(143\) −0.769701 −0.0643657
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 39.6521 3.27045
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −6.77761 −0.551554 −0.275777 0.961222i \(-0.588935\pi\)
−0.275777 + 0.961222i \(0.588935\pi\)
\(152\) 0 0
\(153\) 14.6235 1.18224
\(154\) 0 0
\(155\) −1.52884 −0.122799
\(156\) 0 0
\(157\) 8.96080 0.715149 0.357575 0.933885i \(-0.383604\pi\)
0.357575 + 0.933885i \(0.383604\pi\)
\(158\) 0 0
\(159\) 28.0369 2.22347
\(160\) 0 0
\(161\) 17.8353 1.40562
\(162\) 0 0
\(163\) 6.30644 0.493959 0.246979 0.969021i \(-0.420562\pi\)
0.246979 + 0.969021i \(0.420562\pi\)
\(164\) 0 0
\(165\) −1.79834 −0.140001
\(166\) 0 0
\(167\) −12.7961 −0.990190 −0.495095 0.868839i \(-0.664867\pi\)
−0.495095 + 0.868839i \(0.664867\pi\)
\(168\) 0 0
\(169\) −11.6913 −0.899330
\(170\) 0 0
\(171\) −22.9114 −1.75208
\(172\) 0 0
\(173\) 18.1233 1.37789 0.688943 0.724816i \(-0.258075\pi\)
0.688943 + 0.724816i \(0.258075\pi\)
\(174\) 0 0
\(175\) 4.67282 0.353232
\(176\) 0 0
\(177\) −28.8066 −2.16524
\(178\) 0 0
\(179\) −6.28797 −0.469985 −0.234993 0.971997i \(-0.575507\pi\)
−0.234993 + 0.971997i \(0.575507\pi\)
\(180\) 0 0
\(181\) 4.56804 0.339540 0.169770 0.985484i \(-0.445698\pi\)
0.169770 + 0.985484i \(0.445698\pi\)
\(182\) 0 0
\(183\) −38.7282 −2.86287
\(184\) 0 0
\(185\) −7.16246 −0.526594
\(186\) 0 0
\(187\) 2.37429 0.173625
\(188\) 0 0
\(189\) 14.2880 1.03930
\(190\) 0 0
\(191\) −21.6521 −1.56669 −0.783345 0.621587i \(-0.786488\pi\)
−0.783345 + 0.621587i \(0.786488\pi\)
\(192\) 0 0
\(193\) −13.2409 −0.953098 −0.476549 0.879148i \(-0.658113\pi\)
−0.476549 + 0.879148i \(0.658113\pi\)
\(194\) 0 0
\(195\) 3.05767 0.218965
\(196\) 0 0
\(197\) 7.34565 0.523356 0.261678 0.965155i \(-0.415724\pi\)
0.261678 + 0.965155i \(0.415724\pi\)
\(198\) 0 0
\(199\) −18.7776 −1.33111 −0.665555 0.746349i \(-0.731805\pi\)
−0.665555 + 0.746349i \(0.731805\pi\)
\(200\) 0 0
\(201\) 20.8930 1.47368
\(202\) 0 0
\(203\) −4.67282 −0.327968
\(204\) 0 0
\(205\) −2.85601 −0.199473
\(206\) 0 0
\(207\) 15.8168 1.09934
\(208\) 0 0
\(209\) −3.71993 −0.257313
\(210\) 0 0
\(211\) −17.6521 −1.21522 −0.607610 0.794235i \(-0.707872\pi\)
−0.607610 + 0.794235i \(0.707872\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 8.96080 0.611121
\(216\) 0 0
\(217\) 7.14399 0.484965
\(218\) 0 0
\(219\) −13.2593 −0.895983
\(220\) 0 0
\(221\) −4.03694 −0.271554
\(222\) 0 0
\(223\) −7.45043 −0.498918 −0.249459 0.968385i \(-0.580253\pi\)
−0.249459 + 0.968385i \(0.580253\pi\)
\(224\) 0 0
\(225\) 4.14399 0.276266
\(226\) 0 0
\(227\) 24.2201 1.60755 0.803773 0.594936i \(-0.202822\pi\)
0.803773 + 0.594936i \(0.202822\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 8.40332 0.552898
\(232\) 0 0
\(233\) −0.287973 −0.0188657 −0.00943287 0.999956i \(-0.503003\pi\)
−0.00943287 + 0.999956i \(0.503003\pi\)
\(234\) 0 0
\(235\) −6.67282 −0.435287
\(236\) 0 0
\(237\) −6.37429 −0.414054
\(238\) 0 0
\(239\) −7.51037 −0.485805 −0.242903 0.970051i \(-0.578100\pi\)
−0.242903 + 0.970051i \(0.578100\pi\)
\(240\) 0 0
\(241\) −21.6706 −1.39592 −0.697962 0.716135i \(-0.745909\pi\)
−0.697962 + 0.716135i \(0.745909\pi\)
\(242\) 0 0
\(243\) −20.5575 −1.31876
\(244\) 0 0
\(245\) −14.8353 −0.947791
\(246\) 0 0
\(247\) 6.32492 0.402445
\(248\) 0 0
\(249\) −37.4689 −2.37450
\(250\) 0 0
\(251\) 12.6728 0.799902 0.399951 0.916537i \(-0.369027\pi\)
0.399951 + 0.916537i \(0.369027\pi\)
\(252\) 0 0
\(253\) 2.56804 0.161451
\(254\) 0 0
\(255\) −9.43196 −0.590652
\(256\) 0 0
\(257\) 16.9714 1.05864 0.529322 0.848421i \(-0.322446\pi\)
0.529322 + 0.848421i \(0.322446\pi\)
\(258\) 0 0
\(259\) 33.4689 2.07966
\(260\) 0 0
\(261\) −4.14399 −0.256506
\(262\) 0 0
\(263\) −5.90312 −0.364002 −0.182001 0.983298i \(-0.558257\pi\)
−0.182001 + 0.983298i \(0.558257\pi\)
\(264\) 0 0
\(265\) −10.4896 −0.644373
\(266\) 0 0
\(267\) −4.36638 −0.267218
\(268\) 0 0
\(269\) −14.8560 −0.905787 −0.452894 0.891565i \(-0.649608\pi\)
−0.452894 + 0.891565i \(0.649608\pi\)
\(270\) 0 0
\(271\) −22.5944 −1.37251 −0.686257 0.727360i \(-0.740747\pi\)
−0.686257 + 0.727360i \(0.740747\pi\)
\(272\) 0 0
\(273\) −14.2880 −0.864747
\(274\) 0 0
\(275\) 0.672824 0.0405728
\(276\) 0 0
\(277\) 27.4689 1.65045 0.825223 0.564807i \(-0.191049\pi\)
0.825223 + 0.564807i \(0.191049\pi\)
\(278\) 0 0
\(279\) 6.33548 0.379295
\(280\) 0 0
\(281\) 17.5473 1.04678 0.523392 0.852092i \(-0.324666\pi\)
0.523392 + 0.852092i \(0.324666\pi\)
\(282\) 0 0
\(283\) −26.5081 −1.57574 −0.787872 0.615839i \(-0.788817\pi\)
−0.787872 + 0.615839i \(0.788817\pi\)
\(284\) 0 0
\(285\) 14.7776 0.875350
\(286\) 0 0
\(287\) 13.3456 0.787769
\(288\) 0 0
\(289\) −4.54731 −0.267489
\(290\) 0 0
\(291\) −24.9299 −1.46142
\(292\) 0 0
\(293\) −22.7098 −1.32672 −0.663359 0.748301i \(-0.730870\pi\)
−0.663359 + 0.748301i \(0.730870\pi\)
\(294\) 0 0
\(295\) 10.7776 0.627497
\(296\) 0 0
\(297\) 2.05728 0.119375
\(298\) 0 0
\(299\) −4.36638 −0.252514
\(300\) 0 0
\(301\) −41.8722 −2.41347
\(302\) 0 0
\(303\) −43.6336 −2.50669
\(304\) 0 0
\(305\) 14.4896 0.829674
\(306\) 0 0
\(307\) −20.9608 −1.19630 −0.598148 0.801386i \(-0.704096\pi\)
−0.598148 + 0.801386i \(0.704096\pi\)
\(308\) 0 0
\(309\) 36.1602 2.05708
\(310\) 0 0
\(311\) 27.4874 1.55867 0.779333 0.626610i \(-0.215558\pi\)
0.779333 + 0.626610i \(0.215558\pi\)
\(312\) 0 0
\(313\) −18.8560 −1.06580 −0.532902 0.846177i \(-0.678899\pi\)
−0.532902 + 0.846177i \(0.678899\pi\)
\(314\) 0 0
\(315\) −19.3641 −1.09104
\(316\) 0 0
\(317\) 29.4504 1.65410 0.827050 0.562128i \(-0.190017\pi\)
0.827050 + 0.562128i \(0.190017\pi\)
\(318\) 0 0
\(319\) −0.672824 −0.0376709
\(320\) 0 0
\(321\) −1.79834 −0.100374
\(322\) 0 0
\(323\) −19.5104 −1.08559
\(324\) 0 0
\(325\) −1.14399 −0.0634570
\(326\) 0 0
\(327\) 36.2096 2.00239
\(328\) 0 0
\(329\) 31.1809 1.71906
\(330\) 0 0
\(331\) −3.60724 −0.198272 −0.0991360 0.995074i \(-0.531608\pi\)
−0.0991360 + 0.995074i \(0.531608\pi\)
\(332\) 0 0
\(333\) 29.6811 1.62652
\(334\) 0 0
\(335\) −7.81681 −0.427078
\(336\) 0 0
\(337\) −18.3064 −0.997216 −0.498608 0.866828i \(-0.666155\pi\)
−0.498608 + 0.866828i \(0.666155\pi\)
\(338\) 0 0
\(339\) 53.5058 2.90604
\(340\) 0 0
\(341\) 1.02864 0.0557039
\(342\) 0 0
\(343\) 36.6129 1.97691
\(344\) 0 0
\(345\) −10.2017 −0.549239
\(346\) 0 0
\(347\) −6.38485 −0.342757 −0.171378 0.985205i \(-0.554822\pi\)
−0.171378 + 0.985205i \(0.554822\pi\)
\(348\) 0 0
\(349\) 17.2672 0.924294 0.462147 0.886803i \(-0.347079\pi\)
0.462147 + 0.886803i \(0.347079\pi\)
\(350\) 0 0
\(351\) −3.49794 −0.186706
\(352\) 0 0
\(353\) −9.91369 −0.527652 −0.263826 0.964570i \(-0.584985\pi\)
−0.263826 + 0.964570i \(0.584985\pi\)
\(354\) 0 0
\(355\) 4.48963 0.238285
\(356\) 0 0
\(357\) 44.0739 2.33264
\(358\) 0 0
\(359\) 3.81681 0.201444 0.100722 0.994915i \(-0.467885\pi\)
0.100722 + 0.994915i \(0.467885\pi\)
\(360\) 0 0
\(361\) 11.5680 0.608844
\(362\) 0 0
\(363\) −28.1911 −1.47965
\(364\) 0 0
\(365\) 4.96080 0.259660
\(366\) 0 0
\(367\) 7.98153 0.416632 0.208316 0.978062i \(-0.433202\pi\)
0.208316 + 0.978062i \(0.433202\pi\)
\(368\) 0 0
\(369\) 11.8353 0.616120
\(370\) 0 0
\(371\) 49.0162 2.54479
\(372\) 0 0
\(373\) −26.0369 −1.34814 −0.674071 0.738667i \(-0.735456\pi\)
−0.674071 + 0.738667i \(0.735456\pi\)
\(374\) 0 0
\(375\) −2.67282 −0.138024
\(376\) 0 0
\(377\) 1.14399 0.0589183
\(378\) 0 0
\(379\) −28.8824 −1.48359 −0.741794 0.670627i \(-0.766025\pi\)
−0.741794 + 0.670627i \(0.766025\pi\)
\(380\) 0 0
\(381\) 8.12325 0.416167
\(382\) 0 0
\(383\) −0.0968776 −0.00495021 −0.00247511 0.999997i \(-0.500788\pi\)
−0.00247511 + 0.999997i \(0.500788\pi\)
\(384\) 0 0
\(385\) −3.14399 −0.160232
\(386\) 0 0
\(387\) −37.1334 −1.88760
\(388\) 0 0
\(389\) −0.740665 −0.0375532 −0.0187766 0.999824i \(-0.505977\pi\)
−0.0187766 + 0.999824i \(0.505977\pi\)
\(390\) 0 0
\(391\) 13.4689 0.681152
\(392\) 0 0
\(393\) 10.2017 0.514606
\(394\) 0 0
\(395\) 2.38485 0.119995
\(396\) 0 0
\(397\) −23.9216 −1.20059 −0.600295 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(398\) 0 0
\(399\) −69.0532 −3.45698
\(400\) 0 0
\(401\) −5.54731 −0.277019 −0.138510 0.990361i \(-0.544231\pi\)
−0.138510 + 0.990361i \(0.544231\pi\)
\(402\) 0 0
\(403\) −1.74897 −0.0871224
\(404\) 0 0
\(405\) 4.25934 0.211648
\(406\) 0 0
\(407\) 4.81907 0.238872
\(408\) 0 0
\(409\) −34.8930 −1.72535 −0.862673 0.505762i \(-0.831211\pi\)
−0.862673 + 0.505762i \(0.831211\pi\)
\(410\) 0 0
\(411\) 43.0452 2.12327
\(412\) 0 0
\(413\) −50.3619 −2.47814
\(414\) 0 0
\(415\) 14.0185 0.688140
\(416\) 0 0
\(417\) −8.17262 −0.400215
\(418\) 0 0
\(419\) −12.1233 −0.592260 −0.296130 0.955148i \(-0.595696\pi\)
−0.296130 + 0.955148i \(0.595696\pi\)
\(420\) 0 0
\(421\) 19.0656 0.929200 0.464600 0.885521i \(-0.346198\pi\)
0.464600 + 0.885521i \(0.346198\pi\)
\(422\) 0 0
\(423\) 27.6521 1.34449
\(424\) 0 0
\(425\) 3.52884 0.171174
\(426\) 0 0
\(427\) −67.7075 −3.27660
\(428\) 0 0
\(429\) −2.05728 −0.0993262
\(430\) 0 0
\(431\) −2.69129 −0.129635 −0.0648176 0.997897i \(-0.520647\pi\)
−0.0648176 + 0.997897i \(0.520647\pi\)
\(432\) 0 0
\(433\) 17.4874 0.840390 0.420195 0.907434i \(-0.361962\pi\)
0.420195 + 0.907434i \(0.361962\pi\)
\(434\) 0 0
\(435\) 2.67282 0.128152
\(436\) 0 0
\(437\) −21.1025 −1.00947
\(438\) 0 0
\(439\) 4.45269 0.212515 0.106258 0.994339i \(-0.466113\pi\)
0.106258 + 0.994339i \(0.466113\pi\)
\(440\) 0 0
\(441\) 61.4772 2.92749
\(442\) 0 0
\(443\) −11.6151 −0.551852 −0.275926 0.961179i \(-0.588985\pi\)
−0.275926 + 0.961179i \(0.588985\pi\)
\(444\) 0 0
\(445\) 1.63362 0.0774410
\(446\) 0 0
\(447\) 48.1108 2.27556
\(448\) 0 0
\(449\) −9.91369 −0.467856 −0.233928 0.972254i \(-0.575158\pi\)
−0.233928 + 0.972254i \(0.575158\pi\)
\(450\) 0 0
\(451\) 1.92159 0.0904843
\(452\) 0 0
\(453\) −18.1153 −0.851133
\(454\) 0 0
\(455\) 5.34565 0.250608
\(456\) 0 0
\(457\) 32.6050 1.52520 0.762598 0.646872i \(-0.223923\pi\)
0.762598 + 0.646872i \(0.223923\pi\)
\(458\) 0 0
\(459\) 10.7900 0.503636
\(460\) 0 0
\(461\) 5.23030 0.243599 0.121800 0.992555i \(-0.461133\pi\)
0.121800 + 0.992555i \(0.461133\pi\)
\(462\) 0 0
\(463\) 35.0841 1.63049 0.815247 0.579113i \(-0.196601\pi\)
0.815247 + 0.579113i \(0.196601\pi\)
\(464\) 0 0
\(465\) −4.08631 −0.189498
\(466\) 0 0
\(467\) −21.3641 −0.988614 −0.494307 0.869288i \(-0.664578\pi\)
−0.494307 + 0.869288i \(0.664578\pi\)
\(468\) 0 0
\(469\) 36.5266 1.68664
\(470\) 0 0
\(471\) 23.9506 1.10359
\(472\) 0 0
\(473\) −6.02904 −0.277215
\(474\) 0 0
\(475\) −5.52884 −0.253680
\(476\) 0 0
\(477\) 43.4689 1.99030
\(478\) 0 0
\(479\) −5.03920 −0.230247 −0.115124 0.993351i \(-0.536726\pi\)
−0.115124 + 0.993351i \(0.536726\pi\)
\(480\) 0 0
\(481\) −8.19376 −0.373603
\(482\) 0 0
\(483\) 47.6706 2.16909
\(484\) 0 0
\(485\) 9.32718 0.423525
\(486\) 0 0
\(487\) −28.8824 −1.30879 −0.654393 0.756155i \(-0.727076\pi\)
−0.654393 + 0.756155i \(0.727076\pi\)
\(488\) 0 0
\(489\) 16.8560 0.762255
\(490\) 0 0
\(491\) −5.16246 −0.232978 −0.116489 0.993192i \(-0.537164\pi\)
−0.116489 + 0.993192i \(0.537164\pi\)
\(492\) 0 0
\(493\) −3.52884 −0.158931
\(494\) 0 0
\(495\) −2.78817 −0.125319
\(496\) 0 0
\(497\) −20.9793 −0.941049
\(498\) 0 0
\(499\) 27.0946 1.21292 0.606461 0.795113i \(-0.292589\pi\)
0.606461 + 0.795113i \(0.292589\pi\)
\(500\) 0 0
\(501\) −34.2017 −1.52802
\(502\) 0 0
\(503\) −25.7305 −1.14727 −0.573633 0.819112i \(-0.694466\pi\)
−0.573633 + 0.819112i \(0.694466\pi\)
\(504\) 0 0
\(505\) 16.3249 0.726449
\(506\) 0 0
\(507\) −31.2488 −1.38781
\(508\) 0 0
\(509\) 11.8353 0.524590 0.262295 0.964988i \(-0.415521\pi\)
0.262295 + 0.964988i \(0.415521\pi\)
\(510\) 0 0
\(511\) −23.1809 −1.02546
\(512\) 0 0
\(513\) −16.9054 −0.746391
\(514\) 0 0
\(515\) −13.5288 −0.596152
\(516\) 0 0
\(517\) 4.48963 0.197454
\(518\) 0 0
\(519\) 48.4403 2.12629
\(520\) 0 0
\(521\) −0.164719 −0.00721646 −0.00360823 0.999993i \(-0.501149\pi\)
−0.00360823 + 0.999993i \(0.501149\pi\)
\(522\) 0 0
\(523\) 11.6072 0.507549 0.253775 0.967263i \(-0.418328\pi\)
0.253775 + 0.967263i \(0.418328\pi\)
\(524\) 0 0
\(525\) 12.4896 0.545092
\(526\) 0 0
\(527\) 5.39502 0.235011
\(528\) 0 0
\(529\) −8.43196 −0.366607
\(530\) 0 0
\(531\) −44.6623 −1.93818
\(532\) 0 0
\(533\) −3.26724 −0.141520
\(534\) 0 0
\(535\) 0.672824 0.0290887
\(536\) 0 0
\(537\) −16.8066 −0.725260
\(538\) 0 0
\(539\) 9.98153 0.429935
\(540\) 0 0
\(541\) −35.5922 −1.53023 −0.765113 0.643896i \(-0.777317\pi\)
−0.765113 + 0.643896i \(0.777317\pi\)
\(542\) 0 0
\(543\) 12.2096 0.523963
\(544\) 0 0
\(545\) −13.5473 −0.580303
\(546\) 0 0
\(547\) 44.3803 1.89757 0.948783 0.315929i \(-0.102316\pi\)
0.948783 + 0.315929i \(0.102316\pi\)
\(548\) 0 0
\(549\) −60.0448 −2.56265
\(550\) 0 0
\(551\) 5.52884 0.235536
\(552\) 0 0
\(553\) −11.1440 −0.473891
\(554\) 0 0
\(555\) −19.1440 −0.812617
\(556\) 0 0
\(557\) −26.0739 −1.10479 −0.552393 0.833584i \(-0.686285\pi\)
−0.552393 + 0.833584i \(0.686285\pi\)
\(558\) 0 0
\(559\) 10.2510 0.433572
\(560\) 0 0
\(561\) 6.34605 0.267930
\(562\) 0 0
\(563\) 16.4218 0.692096 0.346048 0.938217i \(-0.387523\pi\)
0.346048 + 0.938217i \(0.387523\pi\)
\(564\) 0 0
\(565\) −20.0185 −0.842183
\(566\) 0 0
\(567\) −19.9031 −0.835853
\(568\) 0 0
\(569\) −7.79834 −0.326923 −0.163462 0.986550i \(-0.552266\pi\)
−0.163462 + 0.986550i \(0.552266\pi\)
\(570\) 0 0
\(571\) 33.3826 1.39702 0.698509 0.715601i \(-0.253847\pi\)
0.698509 + 0.715601i \(0.253847\pi\)
\(572\) 0 0
\(573\) −57.8722 −2.41765
\(574\) 0 0
\(575\) 3.81681 0.159172
\(576\) 0 0
\(577\) −8.10478 −0.337407 −0.168703 0.985667i \(-0.553958\pi\)
−0.168703 + 0.985667i \(0.553958\pi\)
\(578\) 0 0
\(579\) −35.3905 −1.47078
\(580\) 0 0
\(581\) −65.5058 −2.71764
\(582\) 0 0
\(583\) 7.05767 0.292299
\(584\) 0 0
\(585\) 4.74066 0.196002
\(586\) 0 0
\(587\) −28.5865 −1.17989 −0.589946 0.807443i \(-0.700851\pi\)
−0.589946 + 0.807443i \(0.700851\pi\)
\(588\) 0 0
\(589\) −8.45269 −0.348287
\(590\) 0 0
\(591\) 19.6336 0.807619
\(592\) 0 0
\(593\) 22.4896 0.923539 0.461769 0.887000i \(-0.347215\pi\)
0.461769 + 0.887000i \(0.347215\pi\)
\(594\) 0 0
\(595\) −16.4896 −0.676009
\(596\) 0 0
\(597\) −50.1892 −2.05411
\(598\) 0 0
\(599\) 31.7305 1.29647 0.648237 0.761439i \(-0.275507\pi\)
0.648237 + 0.761439i \(0.275507\pi\)
\(600\) 0 0
\(601\) 4.56804 0.186334 0.0931671 0.995650i \(-0.470301\pi\)
0.0931671 + 0.995650i \(0.470301\pi\)
\(602\) 0 0
\(603\) 32.3928 1.31914
\(604\) 0 0
\(605\) 10.5473 0.428809
\(606\) 0 0
\(607\) 12.7882 0.519056 0.259528 0.965736i \(-0.416433\pi\)
0.259528 + 0.965736i \(0.416433\pi\)
\(608\) 0 0
\(609\) −12.4896 −0.506106
\(610\) 0 0
\(611\) −7.63362 −0.308823
\(612\) 0 0
\(613\) −6.97927 −0.281890 −0.140945 0.990017i \(-0.545014\pi\)
−0.140945 + 0.990017i \(0.545014\pi\)
\(614\) 0 0
\(615\) −7.63362 −0.307817
\(616\) 0 0
\(617\) 30.9193 1.24477 0.622383 0.782713i \(-0.286165\pi\)
0.622383 + 0.782713i \(0.286165\pi\)
\(618\) 0 0
\(619\) 20.7098 0.832396 0.416198 0.909274i \(-0.363362\pi\)
0.416198 + 0.909274i \(0.363362\pi\)
\(620\) 0 0
\(621\) 11.6706 0.468324
\(622\) 0 0
\(623\) −7.63362 −0.305835
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −9.94272 −0.397074
\(628\) 0 0
\(629\) 25.2751 1.00779
\(630\) 0 0
\(631\) −9.67508 −0.385159 −0.192580 0.981281i \(-0.561685\pi\)
−0.192580 + 0.981281i \(0.561685\pi\)
\(632\) 0 0
\(633\) −47.1809 −1.87527
\(634\) 0 0
\(635\) −3.03920 −0.120607
\(636\) 0 0
\(637\) −16.9714 −0.672430
\(638\) 0 0
\(639\) −18.6050 −0.736002
\(640\) 0 0
\(641\) 23.5058 0.928425 0.464213 0.885724i \(-0.346337\pi\)
0.464213 + 0.885724i \(0.346337\pi\)
\(642\) 0 0
\(643\) −40.6235 −1.60203 −0.801016 0.598643i \(-0.795707\pi\)
−0.801016 + 0.598643i \(0.795707\pi\)
\(644\) 0 0
\(645\) 23.9506 0.943055
\(646\) 0 0
\(647\) 10.2280 0.402106 0.201053 0.979580i \(-0.435564\pi\)
0.201053 + 0.979580i \(0.435564\pi\)
\(648\) 0 0
\(649\) −7.25143 −0.284644
\(650\) 0 0
\(651\) 19.0946 0.748377
\(652\) 0 0
\(653\) −28.2280 −1.10465 −0.552324 0.833629i \(-0.686259\pi\)
−0.552324 + 0.833629i \(0.686259\pi\)
\(654\) 0 0
\(655\) −3.81681 −0.149135
\(656\) 0 0
\(657\) −20.5575 −0.802023
\(658\) 0 0
\(659\) 15.0471 0.586152 0.293076 0.956089i \(-0.405321\pi\)
0.293076 + 0.956089i \(0.405321\pi\)
\(660\) 0 0
\(661\) 13.8767 0.539743 0.269871 0.962896i \(-0.413019\pi\)
0.269871 + 0.962896i \(0.413019\pi\)
\(662\) 0 0
\(663\) −10.7900 −0.419050
\(664\) 0 0
\(665\) 25.8353 1.00185
\(666\) 0 0
\(667\) −3.81681 −0.147787
\(668\) 0 0
\(669\) −19.9137 −0.769908
\(670\) 0 0
\(671\) −9.74897 −0.376355
\(672\) 0 0
\(673\) 15.3456 0.591531 0.295766 0.955261i \(-0.404425\pi\)
0.295766 + 0.955261i \(0.404425\pi\)
\(674\) 0 0
\(675\) 3.05767 0.117690
\(676\) 0 0
\(677\) −50.5530 −1.94291 −0.971454 0.237228i \(-0.923761\pi\)
−0.971454 + 0.237228i \(0.923761\pi\)
\(678\) 0 0
\(679\) −43.5843 −1.67261
\(680\) 0 0
\(681\) 64.7361 2.48069
\(682\) 0 0
\(683\) −41.6027 −1.59188 −0.795942 0.605373i \(-0.793024\pi\)
−0.795942 + 0.605373i \(0.793024\pi\)
\(684\) 0 0
\(685\) −16.1048 −0.615332
\(686\) 0 0
\(687\) 5.34565 0.203949
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 41.6627 1.58492 0.792461 0.609922i \(-0.208799\pi\)
0.792461 + 0.609922i \(0.208799\pi\)
\(692\) 0 0
\(693\) 13.0286 0.494917
\(694\) 0 0
\(695\) 3.05767 0.115984
\(696\) 0 0
\(697\) 10.0784 0.381747
\(698\) 0 0
\(699\) −0.769701 −0.0291128
\(700\) 0 0
\(701\) −44.8145 −1.69262 −0.846311 0.532689i \(-0.821182\pi\)
−0.846311 + 0.532689i \(0.821182\pi\)
\(702\) 0 0
\(703\) −39.6001 −1.49354
\(704\) 0 0
\(705\) −17.8353 −0.671716
\(706\) 0 0
\(707\) −76.2835 −2.86893
\(708\) 0 0
\(709\) 23.4320 0.880006 0.440003 0.897996i \(-0.354977\pi\)
0.440003 + 0.897996i \(0.354977\pi\)
\(710\) 0 0
\(711\) −9.88279 −0.370634
\(712\) 0 0
\(713\) 5.83528 0.218533
\(714\) 0 0
\(715\) 0.769701 0.0287852
\(716\) 0 0
\(717\) −20.0739 −0.749673
\(718\) 0 0
\(719\) −32.3328 −1.20581 −0.602905 0.797813i \(-0.705990\pi\)
−0.602905 + 0.797813i \(0.705990\pi\)
\(720\) 0 0
\(721\) 63.2179 2.35436
\(722\) 0 0
\(723\) −57.9216 −2.15413
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −4.01847 −0.149037 −0.0745184 0.997220i \(-0.523742\pi\)
−0.0745184 + 0.997220i \(0.523742\pi\)
\(728\) 0 0
\(729\) −42.1685 −1.56180
\(730\) 0 0
\(731\) −31.6212 −1.16955
\(732\) 0 0
\(733\) −3.49189 −0.128976 −0.0644880 0.997918i \(-0.520541\pi\)
−0.0644880 + 0.997918i \(0.520541\pi\)
\(734\) 0 0
\(735\) −39.6521 −1.46259
\(736\) 0 0
\(737\) 5.25934 0.193730
\(738\) 0 0
\(739\) 45.4090 1.67040 0.835198 0.549949i \(-0.185353\pi\)
0.835198 + 0.549949i \(0.185353\pi\)
\(740\) 0 0
\(741\) 16.9054 0.621035
\(742\) 0 0
\(743\) −40.6683 −1.49198 −0.745988 0.665960i \(-0.768022\pi\)
−0.745988 + 0.665960i \(0.768022\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) −58.0924 −2.12549
\(748\) 0 0
\(749\) −3.14399 −0.114879
\(750\) 0 0
\(751\) −49.0268 −1.78901 −0.894506 0.447055i \(-0.852473\pi\)
−0.894506 + 0.447055i \(0.852473\pi\)
\(752\) 0 0
\(753\) 33.8722 1.23437
\(754\) 0 0
\(755\) 6.77761 0.246662
\(756\) 0 0
\(757\) −51.7260 −1.88001 −0.940006 0.341157i \(-0.889181\pi\)
−0.940006 + 0.341157i \(0.889181\pi\)
\(758\) 0 0
\(759\) 6.86392 0.249144
\(760\) 0 0
\(761\) 34.2307 1.24086 0.620431 0.784261i \(-0.286958\pi\)
0.620431 + 0.784261i \(0.286958\pi\)
\(762\) 0 0
\(763\) 63.3042 2.29177
\(764\) 0 0
\(765\) −14.6235 −0.528712
\(766\) 0 0
\(767\) 12.3294 0.445190
\(768\) 0 0
\(769\) 22.5971 0.814871 0.407436 0.913234i \(-0.366423\pi\)
0.407436 + 0.913234i \(0.366423\pi\)
\(770\) 0 0
\(771\) 45.3615 1.63365
\(772\) 0 0
\(773\) 13.8538 0.498285 0.249142 0.968467i \(-0.419851\pi\)
0.249142 + 0.968467i \(0.419851\pi\)
\(774\) 0 0
\(775\) 1.52884 0.0549175
\(776\) 0 0
\(777\) 89.4565 3.20923
\(778\) 0 0
\(779\) −15.7904 −0.565751
\(780\) 0 0
\(781\) −3.02073 −0.108090
\(782\) 0 0
\(783\) −3.05767 −0.109272
\(784\) 0 0
\(785\) −8.96080 −0.319825
\(786\) 0 0
\(787\) −20.0969 −0.716376 −0.358188 0.933649i \(-0.616605\pi\)
−0.358188 + 0.933649i \(0.616605\pi\)
\(788\) 0 0
\(789\) −15.7780 −0.561712
\(790\) 0 0
\(791\) 93.5428 3.32600
\(792\) 0 0
\(793\) 16.5759 0.588629
\(794\) 0 0
\(795\) −28.0369 −0.994368
\(796\) 0 0
\(797\) 43.6890 1.54754 0.773772 0.633464i \(-0.218367\pi\)
0.773772 + 0.633464i \(0.218367\pi\)
\(798\) 0 0
\(799\) 23.5473 0.833044
\(800\) 0 0
\(801\) −6.76970 −0.239196
\(802\) 0 0
\(803\) −3.33774 −0.117786
\(804\) 0 0
\(805\) −17.8353 −0.628611
\(806\) 0 0
\(807\) −39.7075 −1.39777
\(808\) 0 0
\(809\) −18.4403 −0.648325 −0.324163 0.946001i \(-0.605082\pi\)
−0.324163 + 0.946001i \(0.605082\pi\)
\(810\) 0 0
\(811\) −15.0577 −0.528746 −0.264373 0.964420i \(-0.585165\pi\)
−0.264373 + 0.964420i \(0.585165\pi\)
\(812\) 0 0
\(813\) −60.3909 −2.11800
\(814\) 0 0
\(815\) −6.30644 −0.220905
\(816\) 0 0
\(817\) 49.5428 1.73328
\(818\) 0 0
\(819\) −22.1523 −0.774064
\(820\) 0 0
\(821\) 50.9793 1.77919 0.889594 0.456751i \(-0.150987\pi\)
0.889594 + 0.456751i \(0.150987\pi\)
\(822\) 0 0
\(823\) 2.84545 0.0991861 0.0495930 0.998770i \(-0.484208\pi\)
0.0495930 + 0.998770i \(0.484208\pi\)
\(824\) 0 0
\(825\) 1.79834 0.0626101
\(826\) 0 0
\(827\) 15.0761 0.524249 0.262124 0.965034i \(-0.415577\pi\)
0.262124 + 0.965034i \(0.415577\pi\)
\(828\) 0 0
\(829\) −34.6992 −1.20515 −0.602577 0.798061i \(-0.705859\pi\)
−0.602577 + 0.798061i \(0.705859\pi\)
\(830\) 0 0
\(831\) 73.4195 2.54690
\(832\) 0 0
\(833\) 52.3513 1.81386
\(834\) 0 0
\(835\) 12.7961 0.442827
\(836\) 0 0
\(837\) 4.67469 0.161581
\(838\) 0 0
\(839\) 14.7882 0.510544 0.255272 0.966869i \(-0.417835\pi\)
0.255272 + 0.966869i \(0.417835\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 46.9009 1.61535
\(844\) 0 0
\(845\) 11.6913 0.402193
\(846\) 0 0
\(847\) −49.2857 −1.69348
\(848\) 0 0
\(849\) −70.8515 −2.43162
\(850\) 0 0
\(851\) 27.3377 0.937126
\(852\) 0 0
\(853\) 35.7753 1.22492 0.612462 0.790500i \(-0.290179\pi\)
0.612462 + 0.790500i \(0.290179\pi\)
\(854\) 0 0
\(855\) 22.9114 0.783554
\(856\) 0 0
\(857\) −12.4112 −0.423959 −0.211980 0.977274i \(-0.567991\pi\)
−0.211980 + 0.977274i \(0.567991\pi\)
\(858\) 0 0
\(859\) 0.0599353 0.00204497 0.00102248 0.999999i \(-0.499675\pi\)
0.00102248 + 0.999999i \(0.499675\pi\)
\(860\) 0 0
\(861\) 35.6706 1.21565
\(862\) 0 0
\(863\) 44.5530 1.51660 0.758300 0.651906i \(-0.226030\pi\)
0.758300 + 0.651906i \(0.226030\pi\)
\(864\) 0 0
\(865\) −18.1233 −0.616209
\(866\) 0 0
\(867\) −12.1542 −0.412777
\(868\) 0 0
\(869\) −1.60458 −0.0544318
\(870\) 0 0
\(871\) −8.94233 −0.302999
\(872\) 0 0
\(873\) −38.6517 −1.30816
\(874\) 0 0
\(875\) −4.67282 −0.157970
\(876\) 0 0
\(877\) 29.9137 1.01011 0.505057 0.863086i \(-0.331472\pi\)
0.505057 + 0.863086i \(0.331472\pi\)
\(878\) 0 0
\(879\) −60.6992 −2.04733
\(880\) 0 0
\(881\) −32.2386 −1.08615 −0.543073 0.839685i \(-0.682739\pi\)
−0.543073 + 0.839685i \(0.682739\pi\)
\(882\) 0 0
\(883\) −36.7098 −1.23538 −0.617691 0.786421i \(-0.711932\pi\)
−0.617691 + 0.786421i \(0.711932\pi\)
\(884\) 0 0
\(885\) 28.8066 0.968325
\(886\) 0 0
\(887\) 2.63588 0.0885042 0.0442521 0.999020i \(-0.485910\pi\)
0.0442521 + 0.999020i \(0.485910\pi\)
\(888\) 0 0
\(889\) 14.2017 0.476308
\(890\) 0 0
\(891\) −2.86578 −0.0960073
\(892\) 0 0
\(893\) −36.8930 −1.23458
\(894\) 0 0
\(895\) 6.28797 0.210184
\(896\) 0 0
\(897\) −11.6706 −0.389669
\(898\) 0 0
\(899\) −1.52884 −0.0509896
\(900\) 0 0
\(901\) 37.0162 1.23319
\(902\) 0 0
\(903\) −111.917 −3.72437
\(904\) 0 0
\(905\) −4.56804 −0.151847
\(906\) 0 0
\(907\) −16.0185 −0.531885 −0.265942 0.963989i \(-0.585683\pi\)
−0.265942 + 0.963989i \(0.585683\pi\)
\(908\) 0 0
\(909\) −67.6502 −2.24382
\(910\) 0 0
\(911\) −37.9480 −1.25727 −0.628636 0.777700i \(-0.716387\pi\)
−0.628636 + 0.777700i \(0.716387\pi\)
\(912\) 0 0
\(913\) −9.43196 −0.312152
\(914\) 0 0
\(915\) 38.7282 1.28032
\(916\) 0 0
\(917\) 17.8353 0.588973
\(918\) 0 0
\(919\) 7.58425 0.250181 0.125091 0.992145i \(-0.460078\pi\)
0.125091 + 0.992145i \(0.460078\pi\)
\(920\) 0 0
\(921\) −56.0245 −1.84607
\(922\) 0 0
\(923\) 5.13608 0.169056
\(924\) 0 0
\(925\) 7.16246 0.235500
\(926\) 0 0
\(927\) 56.0633 1.84136
\(928\) 0 0
\(929\) 26.4033 0.866265 0.433132 0.901330i \(-0.357408\pi\)
0.433132 + 0.901330i \(0.357408\pi\)
\(930\) 0 0
\(931\) −82.0219 −2.68816
\(932\) 0 0
\(933\) 73.4689 2.40526
\(934\) 0 0
\(935\) −2.37429 −0.0776474
\(936\) 0 0
\(937\) 44.6050 1.45718 0.728591 0.684949i \(-0.240176\pi\)
0.728591 + 0.684949i \(0.240176\pi\)
\(938\) 0 0
\(939\) −50.3988 −1.64470
\(940\) 0 0
\(941\) −9.26724 −0.302103 −0.151052 0.988526i \(-0.548266\pi\)
−0.151052 + 0.988526i \(0.548266\pi\)
\(942\) 0 0
\(943\) 10.9009 0.354981
\(944\) 0 0
\(945\) −14.2880 −0.464788
\(946\) 0 0
\(947\) 28.4218 0.923584 0.461792 0.886988i \(-0.347207\pi\)
0.461792 + 0.886988i \(0.347207\pi\)
\(948\) 0 0
\(949\) 5.67508 0.184221
\(950\) 0 0
\(951\) 78.7158 2.55254
\(952\) 0 0
\(953\) 55.2100 1.78843 0.894213 0.447642i \(-0.147736\pi\)
0.894213 + 0.447642i \(0.147736\pi\)
\(954\) 0 0
\(955\) 21.6521 0.700645
\(956\) 0 0
\(957\) −1.79834 −0.0581320
\(958\) 0 0
\(959\) 75.2548 2.43010
\(960\) 0 0
\(961\) −28.6627 −0.924602
\(962\) 0 0
\(963\) −2.78817 −0.0898476
\(964\) 0 0
\(965\) 13.2409 0.426238
\(966\) 0 0
\(967\) 34.1496 1.09818 0.549089 0.835764i \(-0.314975\pi\)
0.549089 + 0.835764i \(0.314975\pi\)
\(968\) 0 0
\(969\) −52.1478 −1.67523
\(970\) 0 0
\(971\) 41.4090 1.32888 0.664438 0.747343i \(-0.268671\pi\)
0.664438 + 0.747343i \(0.268671\pi\)
\(972\) 0 0
\(973\) −14.2880 −0.458051
\(974\) 0 0
\(975\) −3.05767 −0.0979239
\(976\) 0 0
\(977\) −13.9506 −0.446320 −0.223160 0.974782i \(-0.571637\pi\)
−0.223160 + 0.974782i \(0.571637\pi\)
\(978\) 0 0
\(979\) −1.09914 −0.0351286
\(980\) 0 0
\(981\) 56.1399 1.79241
\(982\) 0 0
\(983\) 39.6521 1.26471 0.632353 0.774681i \(-0.282089\pi\)
0.632353 + 0.774681i \(0.282089\pi\)
\(984\) 0 0
\(985\) −7.34565 −0.234052
\(986\) 0 0
\(987\) 83.3411 2.65278
\(988\) 0 0
\(989\) −34.2017 −1.08755
\(990\) 0 0
\(991\) −2.93442 −0.0932149 −0.0466075 0.998913i \(-0.514841\pi\)
−0.0466075 + 0.998913i \(0.514841\pi\)
\(992\) 0 0
\(993\) −9.64153 −0.305965
\(994\) 0 0
\(995\) 18.7776 0.595290
\(996\) 0 0
\(997\) 12.9977 0.411643 0.205821 0.978590i \(-0.434013\pi\)
0.205821 + 0.978590i \(0.434013\pi\)
\(998\) 0 0
\(999\) 21.9005 0.692900
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.m.1.3 3
4.3 odd 2 580.2.a.c.1.1 3
8.3 odd 2 9280.2.a.bk.1.3 3
8.5 even 2 9280.2.a.bw.1.1 3
12.11 even 2 5220.2.a.x.1.1 3
20.3 even 4 2900.2.c.f.349.2 6
20.7 even 4 2900.2.c.f.349.5 6
20.19 odd 2 2900.2.a.g.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.a.c.1.1 3 4.3 odd 2
2320.2.a.m.1.3 3 1.1 even 1 trivial
2900.2.a.g.1.3 3 20.19 odd 2
2900.2.c.f.349.2 6 20.3 even 4
2900.2.c.f.349.5 6 20.7 even 4
5220.2.a.x.1.1 3 12.11 even 2
9280.2.a.bk.1.3 3 8.3 odd 2
9280.2.a.bw.1.1 3 8.5 even 2