Properties

Label 2576.2.a.bd.1.1
Level $2576$
Weight $2$
Character 2576.1
Self dual yes
Analytic conductor $20.569$
Analytic rank $1$
Dimension $5$
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] = [2576,2,Mod(1,2576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2576, 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("2576.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2576.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: \(20.5694635607\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.2147108.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.23828\) of defining polynomial
Character \(\chi\) \(=\) 2576.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.68857 q^{3} -1.86253 q^{5} -1.00000 q^{7} +4.22838 q^{9} +O(q^{10})\) \(q-2.68857 q^{3} -1.86253 q^{5} -1.00000 q^{7} +4.22838 q^{9} +0.846153 q^{11} +2.55110 q^{13} +5.00754 q^{15} -7.07080 q^{17} +0.476559 q^{19} +2.68857 q^{21} +1.00000 q^{23} -1.53098 q^{25} -3.30259 q^{27} +8.63827 q^{29} -3.31143 q^{31} -2.27494 q^{33} +1.86253 q^{35} +7.85369 q^{37} -6.85879 q^{39} +2.82603 q^{41} +0.274938 q^{43} -7.87550 q^{45} +13.4756 q^{47} +1.00000 q^{49} +19.0103 q^{51} +8.93333 q^{53} -1.57599 q^{55} -1.28126 q^{57} -1.66091 q^{59} -11.7162 q^{61} -4.22838 q^{63} -4.75150 q^{65} +2.82636 q^{67} -2.68857 q^{69} -9.92823 q^{71} +7.31556 q^{73} +4.11614 q^{75} -0.846153 q^{77} -11.7795 q^{79} -3.80592 q^{81} -3.72506 q^{83} +13.1696 q^{85} -23.2246 q^{87} -8.76310 q^{89} -2.55110 q^{91} +8.90301 q^{93} -0.887605 q^{95} +1.82229 q^{97} +3.57786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9} + 4 q^{11} - 6 q^{13} - 10 q^{15} - 12 q^{17} - 6 q^{19} + 5 q^{23} + 19 q^{25} - 4 q^{29} - 30 q^{31} - 22 q^{33} + 4 q^{35} + 4 q^{37} - 16 q^{39} + 6 q^{41} + 12 q^{43} - 12 q^{45} - 10 q^{47} + 5 q^{49} + 4 q^{51} + 16 q^{53} - 18 q^{55} + 6 q^{57} - 22 q^{59} - 18 q^{61} - 11 q^{63} - 26 q^{65} + 2 q^{67} - 4 q^{71} - 2 q^{73} + 30 q^{75} - 4 q^{77} - 30 q^{79} - 3 q^{81} - 8 q^{83} - 12 q^{85} + 12 q^{87} - 20 q^{89} + 6 q^{91} - 26 q^{93} - 8 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) \) 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
\(3\) −2.68857 −1.55224 −0.776122 0.630583i \(-0.782816\pi\)
−0.776122 + 0.630583i \(0.782816\pi\)
\(4\) 0 0
\(5\) −1.86253 −0.832949 −0.416475 0.909147i \(-0.636735\pi\)
−0.416475 + 0.909147i \(0.636735\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 4.22838 1.40946
\(10\) 0 0
\(11\) 0.846153 0.255125 0.127562 0.991831i \(-0.459285\pi\)
0.127562 + 0.991831i \(0.459285\pi\)
\(12\) 0 0
\(13\) 2.55110 0.707547 0.353773 0.935331i \(-0.384898\pi\)
0.353773 + 0.935331i \(0.384898\pi\)
\(14\) 0 0
\(15\) 5.00754 1.29294
\(16\) 0 0
\(17\) −7.07080 −1.71492 −0.857460 0.514550i \(-0.827959\pi\)
−0.857460 + 0.514550i \(0.827959\pi\)
\(18\) 0 0
\(19\) 0.476559 0.109330 0.0546650 0.998505i \(-0.482591\pi\)
0.0546650 + 0.998505i \(0.482591\pi\)
\(20\) 0 0
\(21\) 2.68857 0.586693
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.53098 −0.306196
\(26\) 0 0
\(27\) −3.30259 −0.635584
\(28\) 0 0
\(29\) 8.63827 1.60409 0.802043 0.597266i \(-0.203746\pi\)
0.802043 + 0.597266i \(0.203746\pi\)
\(30\) 0 0
\(31\) −3.31143 −0.594751 −0.297376 0.954761i \(-0.596111\pi\)
−0.297376 + 0.954761i \(0.596111\pi\)
\(32\) 0 0
\(33\) −2.27494 −0.396016
\(34\) 0 0
\(35\) 1.86253 0.314825
\(36\) 0 0
\(37\) 7.85369 1.29114 0.645569 0.763702i \(-0.276620\pi\)
0.645569 + 0.763702i \(0.276620\pi\)
\(38\) 0 0
\(39\) −6.85879 −1.09829
\(40\) 0 0
\(41\) 2.82603 0.441352 0.220676 0.975347i \(-0.429174\pi\)
0.220676 + 0.975347i \(0.429174\pi\)
\(42\) 0 0
\(43\) 0.274938 0.0419276 0.0209638 0.999780i \(-0.493327\pi\)
0.0209638 + 0.999780i \(0.493327\pi\)
\(44\) 0 0
\(45\) −7.87550 −1.17401
\(46\) 0 0
\(47\) 13.4756 1.96562 0.982808 0.184631i \(-0.0591089\pi\)
0.982808 + 0.184631i \(0.0591089\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 19.0103 2.66197
\(52\) 0 0
\(53\) 8.93333 1.22709 0.613544 0.789661i \(-0.289744\pi\)
0.613544 + 0.789661i \(0.289744\pi\)
\(54\) 0 0
\(55\) −1.57599 −0.212506
\(56\) 0 0
\(57\) −1.28126 −0.169707
\(58\) 0 0
\(59\) −1.66091 −0.216232 −0.108116 0.994138i \(-0.534482\pi\)
−0.108116 + 0.994138i \(0.534482\pi\)
\(60\) 0 0
\(61\) −11.7162 −1.50011 −0.750054 0.661376i \(-0.769973\pi\)
−0.750054 + 0.661376i \(0.769973\pi\)
\(62\) 0 0
\(63\) −4.22838 −0.532726
\(64\) 0 0
\(65\) −4.75150 −0.589351
\(66\) 0 0
\(67\) 2.82636 0.345295 0.172648 0.984984i \(-0.444768\pi\)
0.172648 + 0.984984i \(0.444768\pi\)
\(68\) 0 0
\(69\) −2.68857 −0.323665
\(70\) 0 0
\(71\) −9.92823 −1.17826 −0.589132 0.808037i \(-0.700530\pi\)
−0.589132 + 0.808037i \(0.700530\pi\)
\(72\) 0 0
\(73\) 7.31556 0.856222 0.428111 0.903726i \(-0.359179\pi\)
0.428111 + 0.903726i \(0.359179\pi\)
\(74\) 0 0
\(75\) 4.11614 0.475290
\(76\) 0 0
\(77\) −0.846153 −0.0964281
\(78\) 0 0
\(79\) −11.7795 −1.32530 −0.662648 0.748931i \(-0.730567\pi\)
−0.662648 + 0.748931i \(0.730567\pi\)
\(80\) 0 0
\(81\) −3.80592 −0.422880
\(82\) 0 0
\(83\) −3.72506 −0.408879 −0.204439 0.978879i \(-0.565537\pi\)
−0.204439 + 0.978879i \(0.565537\pi\)
\(84\) 0 0
\(85\) 13.1696 1.42844
\(86\) 0 0
\(87\) −23.2246 −2.48993
\(88\) 0 0
\(89\) −8.76310 −0.928887 −0.464444 0.885603i \(-0.653746\pi\)
−0.464444 + 0.885603i \(0.653746\pi\)
\(90\) 0 0
\(91\) −2.55110 −0.267428
\(92\) 0 0
\(93\) 8.90301 0.923199
\(94\) 0 0
\(95\) −0.887605 −0.0910664
\(96\) 0 0
\(97\) 1.82229 0.185026 0.0925130 0.995711i \(-0.470510\pi\)
0.0925130 + 0.995711i \(0.470510\pi\)
\(98\) 0 0
\(99\) 3.57786 0.359589
\(100\) 0 0
\(101\) −14.7870 −1.47136 −0.735682 0.677327i \(-0.763138\pi\)
−0.735682 + 0.677327i \(0.763138\pi\)
\(102\) 0 0
\(103\) −10.9333 −1.07729 −0.538646 0.842532i \(-0.681064\pi\)
−0.538646 + 0.842532i \(0.681064\pi\)
\(104\) 0 0
\(105\) −5.00754 −0.488686
\(106\) 0 0
\(107\) 17.2636 1.66893 0.834466 0.551059i \(-0.185776\pi\)
0.834466 + 0.551059i \(0.185776\pi\)
\(108\) 0 0
\(109\) −13.2438 −1.26852 −0.634262 0.773118i \(-0.718696\pi\)
−0.634262 + 0.773118i \(0.718696\pi\)
\(110\) 0 0
\(111\) −21.1152 −2.00416
\(112\) 0 0
\(113\) −7.10219 −0.668118 −0.334059 0.942552i \(-0.608419\pi\)
−0.334059 + 0.942552i \(0.608419\pi\)
\(114\) 0 0
\(115\) −1.86253 −0.173682
\(116\) 0 0
\(117\) 10.7870 0.997260
\(118\) 0 0
\(119\) 7.07080 0.648179
\(120\) 0 0
\(121\) −10.2840 −0.934911
\(122\) 0 0
\(123\) −7.59798 −0.685087
\(124\) 0 0
\(125\) 12.1641 1.08799
\(126\) 0 0
\(127\) −16.1101 −1.42954 −0.714768 0.699361i \(-0.753468\pi\)
−0.714768 + 0.699361i \(0.753468\pi\)
\(128\) 0 0
\(129\) −0.739189 −0.0650819
\(130\) 0 0
\(131\) 0.854665 0.0746724 0.0373362 0.999303i \(-0.488113\pi\)
0.0373362 + 0.999303i \(0.488113\pi\)
\(132\) 0 0
\(133\) −0.476559 −0.0413229
\(134\) 0 0
\(135\) 6.15118 0.529409
\(136\) 0 0
\(137\) 7.11516 0.607889 0.303945 0.952690i \(-0.401696\pi\)
0.303945 + 0.952690i \(0.401696\pi\)
\(138\) 0 0
\(139\) −10.7111 −0.908505 −0.454253 0.890873i \(-0.650094\pi\)
−0.454253 + 0.890873i \(0.650094\pi\)
\(140\) 0 0
\(141\) −36.2300 −3.05112
\(142\) 0 0
\(143\) 2.15862 0.180513
\(144\) 0 0
\(145\) −16.0890 −1.33612
\(146\) 0 0
\(147\) −2.68857 −0.221749
\(148\) 0 0
\(149\) −6.43632 −0.527284 −0.263642 0.964621i \(-0.584924\pi\)
−0.263642 + 0.964621i \(0.584924\pi\)
\(150\) 0 0
\(151\) −0.803480 −0.0653863 −0.0326931 0.999465i \(-0.510408\pi\)
−0.0326931 + 0.999465i \(0.510408\pi\)
\(152\) 0 0
\(153\) −29.8981 −2.41711
\(154\) 0 0
\(155\) 6.16765 0.495398
\(156\) 0 0
\(157\) −14.7959 −1.18084 −0.590419 0.807097i \(-0.701038\pi\)
−0.590419 + 0.807097i \(0.701038\pi\)
\(158\) 0 0
\(159\) −24.0178 −1.90474
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −3.13373 −0.245453 −0.122726 0.992441i \(-0.539164\pi\)
−0.122726 + 0.992441i \(0.539164\pi\)
\(164\) 0 0
\(165\) 4.23714 0.329861
\(166\) 0 0
\(167\) 23.4168 1.81204 0.906022 0.423230i \(-0.139104\pi\)
0.906022 + 0.423230i \(0.139104\pi\)
\(168\) 0 0
\(169\) −6.49191 −0.499377
\(170\) 0 0
\(171\) 2.01507 0.154097
\(172\) 0 0
\(173\) −0.645424 −0.0490706 −0.0245353 0.999699i \(-0.507811\pi\)
−0.0245353 + 0.999699i \(0.507811\pi\)
\(174\) 0 0
\(175\) 1.53098 0.115731
\(176\) 0 0
\(177\) 4.46547 0.335645
\(178\) 0 0
\(179\) 16.9141 1.26422 0.632110 0.774879i \(-0.282189\pi\)
0.632110 + 0.774879i \(0.282189\pi\)
\(180\) 0 0
\(181\) −4.61403 −0.342958 −0.171479 0.985188i \(-0.554855\pi\)
−0.171479 + 0.985188i \(0.554855\pi\)
\(182\) 0 0
\(183\) 31.4998 2.32853
\(184\) 0 0
\(185\) −14.6277 −1.07545
\(186\) 0 0
\(187\) −5.98298 −0.437519
\(188\) 0 0
\(189\) 3.30259 0.240228
\(190\) 0 0
\(191\) −6.44380 −0.466257 −0.233129 0.972446i \(-0.574896\pi\)
−0.233129 + 0.972446i \(0.574896\pi\)
\(192\) 0 0
\(193\) 10.1400 0.729897 0.364948 0.931028i \(-0.381087\pi\)
0.364948 + 0.931028i \(0.381087\pi\)
\(194\) 0 0
\(195\) 12.7747 0.914816
\(196\) 0 0
\(197\) −2.37475 −0.169194 −0.0845970 0.996415i \(-0.526960\pi\)
−0.0845970 + 0.996415i \(0.526960\pi\)
\(198\) 0 0
\(199\) −18.6052 −1.31889 −0.659443 0.751754i \(-0.729208\pi\)
−0.659443 + 0.751754i \(0.729208\pi\)
\(200\) 0 0
\(201\) −7.59887 −0.535983
\(202\) 0 0
\(203\) −8.63827 −0.606288
\(204\) 0 0
\(205\) −5.26358 −0.367624
\(206\) 0 0
\(207\) 4.22838 0.293893
\(208\) 0 0
\(209\) 0.403242 0.0278928
\(210\) 0 0
\(211\) 21.1247 1.45429 0.727144 0.686485i \(-0.240847\pi\)
0.727144 + 0.686485i \(0.240847\pi\)
\(212\) 0 0
\(213\) 26.6927 1.82895
\(214\) 0 0
\(215\) −0.512080 −0.0349236
\(216\) 0 0
\(217\) 3.31143 0.224795
\(218\) 0 0
\(219\) −19.6684 −1.32906
\(220\) 0 0
\(221\) −18.0383 −1.21339
\(222\) 0 0
\(223\) 21.5426 1.44260 0.721301 0.692622i \(-0.243544\pi\)
0.721301 + 0.692622i \(0.243544\pi\)
\(224\) 0 0
\(225\) −6.47356 −0.431571
\(226\) 0 0
\(227\) −12.9135 −0.857102 −0.428551 0.903518i \(-0.640976\pi\)
−0.428551 + 0.903518i \(0.640976\pi\)
\(228\) 0 0
\(229\) 28.4835 1.88224 0.941120 0.338074i \(-0.109775\pi\)
0.941120 + 0.338074i \(0.109775\pi\)
\(230\) 0 0
\(231\) 2.27494 0.149680
\(232\) 0 0
\(233\) 2.58296 0.169215 0.0846077 0.996414i \(-0.473036\pi\)
0.0846077 + 0.996414i \(0.473036\pi\)
\(234\) 0 0
\(235\) −25.0987 −1.63726
\(236\) 0 0
\(237\) 31.6699 2.05718
\(238\) 0 0
\(239\) −11.9610 −0.773692 −0.386846 0.922144i \(-0.626435\pi\)
−0.386846 + 0.922144i \(0.626435\pi\)
\(240\) 0 0
\(241\) 2.03140 0.130854 0.0654269 0.997857i \(-0.479159\pi\)
0.0654269 + 0.997857i \(0.479159\pi\)
\(242\) 0 0
\(243\) 20.1402 1.29200
\(244\) 0 0
\(245\) −1.86253 −0.118993
\(246\) 0 0
\(247\) 1.21575 0.0773562
\(248\) 0 0
\(249\) 10.0151 0.634680
\(250\) 0 0
\(251\) −27.4454 −1.73234 −0.866169 0.499750i \(-0.833425\pi\)
−0.866169 + 0.499750i \(0.833425\pi\)
\(252\) 0 0
\(253\) 0.846153 0.0531972
\(254\) 0 0
\(255\) −35.4073 −2.21729
\(256\) 0 0
\(257\) −25.1645 −1.56972 −0.784860 0.619673i \(-0.787265\pi\)
−0.784860 + 0.619673i \(0.787265\pi\)
\(258\) 0 0
\(259\) −7.85369 −0.488005
\(260\) 0 0
\(261\) 36.5259 2.26090
\(262\) 0 0
\(263\) −10.8612 −0.669732 −0.334866 0.942266i \(-0.608691\pi\)
−0.334866 + 0.942266i \(0.608691\pi\)
\(264\) 0 0
\(265\) −16.6386 −1.02210
\(266\) 0 0
\(267\) 23.5602 1.44186
\(268\) 0 0
\(269\) −2.02133 −0.123243 −0.0616215 0.998100i \(-0.519627\pi\)
−0.0616215 + 0.998100i \(0.519627\pi\)
\(270\) 0 0
\(271\) 4.05733 0.246465 0.123233 0.992378i \(-0.460674\pi\)
0.123233 + 0.992378i \(0.460674\pi\)
\(272\) 0 0
\(273\) 6.85879 0.415113
\(274\) 0 0
\(275\) −1.29544 −0.0781181
\(276\) 0 0
\(277\) −31.6071 −1.89909 −0.949544 0.313634i \(-0.898454\pi\)
−0.949544 + 0.313634i \(0.898454\pi\)
\(278\) 0 0
\(279\) −14.0020 −0.838279
\(280\) 0 0
\(281\) 25.4680 1.51929 0.759646 0.650337i \(-0.225372\pi\)
0.759646 + 0.650337i \(0.225372\pi\)
\(282\) 0 0
\(283\) 8.06860 0.479629 0.239814 0.970819i \(-0.422913\pi\)
0.239814 + 0.970819i \(0.422913\pi\)
\(284\) 0 0
\(285\) 2.38639 0.141357
\(286\) 0 0
\(287\) −2.82603 −0.166816
\(288\) 0 0
\(289\) 32.9962 1.94095
\(290\) 0 0
\(291\) −4.89936 −0.287205
\(292\) 0 0
\(293\) −12.5766 −0.734730 −0.367365 0.930077i \(-0.619740\pi\)
−0.367365 + 0.930077i \(0.619740\pi\)
\(294\) 0 0
\(295\) 3.09350 0.180110
\(296\) 0 0
\(297\) −2.79450 −0.162153
\(298\) 0 0
\(299\) 2.55110 0.147534
\(300\) 0 0
\(301\) −0.274938 −0.0158472
\(302\) 0 0
\(303\) 39.7559 2.28391
\(304\) 0 0
\(305\) 21.8218 1.24951
\(306\) 0 0
\(307\) −0.588753 −0.0336019 −0.0168009 0.999859i \(-0.505348\pi\)
−0.0168009 + 0.999859i \(0.505348\pi\)
\(308\) 0 0
\(309\) 29.3950 1.67222
\(310\) 0 0
\(311\) −2.32490 −0.131833 −0.0659165 0.997825i \(-0.520997\pi\)
−0.0659165 + 0.997825i \(0.520997\pi\)
\(312\) 0 0
\(313\) −8.02863 −0.453805 −0.226903 0.973917i \(-0.572860\pi\)
−0.226903 + 0.973917i \(0.572860\pi\)
\(314\) 0 0
\(315\) 7.87550 0.443734
\(316\) 0 0
\(317\) 27.0431 1.51889 0.759445 0.650572i \(-0.225471\pi\)
0.759445 + 0.650572i \(0.225471\pi\)
\(318\) 0 0
\(319\) 7.30930 0.409242
\(320\) 0 0
\(321\) −46.4143 −2.59059
\(322\) 0 0
\(323\) −3.36965 −0.187492
\(324\) 0 0
\(325\) −3.90567 −0.216648
\(326\) 0 0
\(327\) 35.6068 1.96906
\(328\) 0 0
\(329\) −13.4756 −0.742933
\(330\) 0 0
\(331\) −22.7555 −1.25075 −0.625377 0.780323i \(-0.715055\pi\)
−0.625377 + 0.780323i \(0.715055\pi\)
\(332\) 0 0
\(333\) 33.2084 1.81981
\(334\) 0 0
\(335\) −5.26419 −0.287613
\(336\) 0 0
\(337\) −1.05919 −0.0576978 −0.0288489 0.999584i \(-0.509184\pi\)
−0.0288489 + 0.999584i \(0.509184\pi\)
\(338\) 0 0
\(339\) 19.0947 1.03708
\(340\) 0 0
\(341\) −2.80198 −0.151736
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 5.00754 0.269597
\(346\) 0 0
\(347\) −14.2245 −0.763611 −0.381806 0.924243i \(-0.624698\pi\)
−0.381806 + 0.924243i \(0.624698\pi\)
\(348\) 0 0
\(349\) −23.4865 −1.25721 −0.628603 0.777727i \(-0.716373\pi\)
−0.628603 + 0.777727i \(0.716373\pi\)
\(350\) 0 0
\(351\) −8.42523 −0.449706
\(352\) 0 0
\(353\) −1.19652 −0.0636843 −0.0318422 0.999493i \(-0.510137\pi\)
−0.0318422 + 0.999493i \(0.510137\pi\)
\(354\) 0 0
\(355\) 18.4916 0.981434
\(356\) 0 0
\(357\) −19.0103 −1.00613
\(358\) 0 0
\(359\) −8.62376 −0.455145 −0.227572 0.973761i \(-0.573079\pi\)
−0.227572 + 0.973761i \(0.573079\pi\)
\(360\) 0 0
\(361\) −18.7729 −0.988047
\(362\) 0 0
\(363\) 27.6493 1.45121
\(364\) 0 0
\(365\) −13.6255 −0.713189
\(366\) 0 0
\(367\) −1.40241 −0.0732050 −0.0366025 0.999330i \(-0.511654\pi\)
−0.0366025 + 0.999330i \(0.511654\pi\)
\(368\) 0 0
\(369\) 11.9496 0.622069
\(370\) 0 0
\(371\) −8.93333 −0.463795
\(372\) 0 0
\(373\) 5.38461 0.278805 0.139402 0.990236i \(-0.455482\pi\)
0.139402 + 0.990236i \(0.455482\pi\)
\(374\) 0 0
\(375\) −32.7041 −1.68883
\(376\) 0 0
\(377\) 22.0371 1.13497
\(378\) 0 0
\(379\) −26.3413 −1.35306 −0.676532 0.736413i \(-0.736518\pi\)
−0.676532 + 0.736413i \(0.736518\pi\)
\(380\) 0 0
\(381\) 43.3130 2.21899
\(382\) 0 0
\(383\) −23.4277 −1.19710 −0.598550 0.801085i \(-0.704256\pi\)
−0.598550 + 0.801085i \(0.704256\pi\)
\(384\) 0 0
\(385\) 1.57599 0.0803197
\(386\) 0 0
\(387\) 1.16254 0.0590954
\(388\) 0 0
\(389\) −13.5760 −0.688330 −0.344165 0.938909i \(-0.611838\pi\)
−0.344165 + 0.938909i \(0.611838\pi\)
\(390\) 0 0
\(391\) −7.07080 −0.357586
\(392\) 0 0
\(393\) −2.29782 −0.115910
\(394\) 0 0
\(395\) 21.9396 1.10390
\(396\) 0 0
\(397\) 3.24696 0.162960 0.0814801 0.996675i \(-0.474035\pi\)
0.0814801 + 0.996675i \(0.474035\pi\)
\(398\) 0 0
\(399\) 1.28126 0.0641432
\(400\) 0 0
\(401\) 30.2805 1.51213 0.756067 0.654494i \(-0.227118\pi\)
0.756067 + 0.654494i \(0.227118\pi\)
\(402\) 0 0
\(403\) −8.44779 −0.420814
\(404\) 0 0
\(405\) 7.08864 0.352237
\(406\) 0 0
\(407\) 6.64542 0.329401
\(408\) 0 0
\(409\) 12.3775 0.612029 0.306014 0.952027i \(-0.401004\pi\)
0.306014 + 0.952027i \(0.401004\pi\)
\(410\) 0 0
\(411\) −19.1296 −0.943592
\(412\) 0 0
\(413\) 1.66091 0.0817280
\(414\) 0 0
\(415\) 6.93804 0.340575
\(416\) 0 0
\(417\) 28.7975 1.41022
\(418\) 0 0
\(419\) −5.46520 −0.266992 −0.133496 0.991049i \(-0.542620\pi\)
−0.133496 + 0.991049i \(0.542620\pi\)
\(420\) 0 0
\(421\) −6.55232 −0.319340 −0.159670 0.987170i \(-0.551043\pi\)
−0.159670 + 0.987170i \(0.551043\pi\)
\(422\) 0 0
\(423\) 56.9800 2.77046
\(424\) 0 0
\(425\) 10.8252 0.525101
\(426\) 0 0
\(427\) 11.7162 0.566988
\(428\) 0 0
\(429\) −5.80359 −0.280200
\(430\) 0 0
\(431\) 7.88645 0.379877 0.189938 0.981796i \(-0.439171\pi\)
0.189938 + 0.981796i \(0.439171\pi\)
\(432\) 0 0
\(433\) −33.7313 −1.62102 −0.810511 0.585723i \(-0.800811\pi\)
−0.810511 + 0.585723i \(0.800811\pi\)
\(434\) 0 0
\(435\) 43.2565 2.07399
\(436\) 0 0
\(437\) 0.476559 0.0227969
\(438\) 0 0
\(439\) 13.6800 0.652909 0.326455 0.945213i \(-0.394146\pi\)
0.326455 + 0.945213i \(0.394146\pi\)
\(440\) 0 0
\(441\) 4.22838 0.201352
\(442\) 0 0
\(443\) 17.2726 0.820646 0.410323 0.911940i \(-0.365416\pi\)
0.410323 + 0.911940i \(0.365416\pi\)
\(444\) 0 0
\(445\) 16.3216 0.773716
\(446\) 0 0
\(447\) 17.3045 0.818473
\(448\) 0 0
\(449\) 8.26286 0.389949 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(450\) 0 0
\(451\) 2.39126 0.112600
\(452\) 0 0
\(453\) 2.16021 0.101495
\(454\) 0 0
\(455\) 4.75150 0.222754
\(456\) 0 0
\(457\) 2.80779 0.131343 0.0656714 0.997841i \(-0.479081\pi\)
0.0656714 + 0.997841i \(0.479081\pi\)
\(458\) 0 0
\(459\) 23.3520 1.08998
\(460\) 0 0
\(461\) −0.220189 −0.0102552 −0.00512761 0.999987i \(-0.501632\pi\)
−0.00512761 + 0.999987i \(0.501632\pi\)
\(462\) 0 0
\(463\) −25.5262 −1.18630 −0.593152 0.805091i \(-0.702117\pi\)
−0.593152 + 0.805091i \(0.702117\pi\)
\(464\) 0 0
\(465\) −16.5821 −0.768978
\(466\) 0 0
\(467\) −6.43084 −0.297584 −0.148792 0.988869i \(-0.547538\pi\)
−0.148792 + 0.988869i \(0.547538\pi\)
\(468\) 0 0
\(469\) −2.82636 −0.130509
\(470\) 0 0
\(471\) 39.7796 1.83295
\(472\) 0 0
\(473\) 0.232640 0.0106968
\(474\) 0 0
\(475\) −0.729601 −0.0334764
\(476\) 0 0
\(477\) 37.7736 1.72953
\(478\) 0 0
\(479\) −25.1810 −1.15055 −0.575276 0.817960i \(-0.695105\pi\)
−0.575276 + 0.817960i \(0.695105\pi\)
\(480\) 0 0
\(481\) 20.0355 0.913541
\(482\) 0 0
\(483\) 2.68857 0.122334
\(484\) 0 0
\(485\) −3.39408 −0.154117
\(486\) 0 0
\(487\) −37.4074 −1.69509 −0.847545 0.530724i \(-0.821920\pi\)
−0.847545 + 0.530724i \(0.821920\pi\)
\(488\) 0 0
\(489\) 8.42523 0.381002
\(490\) 0 0
\(491\) 8.61145 0.388629 0.194315 0.980939i \(-0.437752\pi\)
0.194315 + 0.980939i \(0.437752\pi\)
\(492\) 0 0
\(493\) −61.0795 −2.75088
\(494\) 0 0
\(495\) −6.66388 −0.299519
\(496\) 0 0
\(497\) 9.92823 0.445342
\(498\) 0 0
\(499\) 31.2249 1.39782 0.698909 0.715211i \(-0.253669\pi\)
0.698909 + 0.715211i \(0.253669\pi\)
\(500\) 0 0
\(501\) −62.9575 −2.81274
\(502\) 0 0
\(503\) −12.5421 −0.559223 −0.279612 0.960113i \(-0.590206\pi\)
−0.279612 + 0.960113i \(0.590206\pi\)
\(504\) 0 0
\(505\) 27.5413 1.22557
\(506\) 0 0
\(507\) 17.4539 0.775156
\(508\) 0 0
\(509\) 12.5436 0.555986 0.277993 0.960583i \(-0.410331\pi\)
0.277993 + 0.960583i \(0.410331\pi\)
\(510\) 0 0
\(511\) −7.31556 −0.323621
\(512\) 0 0
\(513\) −1.57388 −0.0694885
\(514\) 0 0
\(515\) 20.3637 0.897330
\(516\) 0 0
\(517\) 11.4024 0.501477
\(518\) 0 0
\(519\) 1.73526 0.0761696
\(520\) 0 0
\(521\) 18.3842 0.805426 0.402713 0.915326i \(-0.368067\pi\)
0.402713 + 0.915326i \(0.368067\pi\)
\(522\) 0 0
\(523\) −21.0572 −0.920769 −0.460384 0.887720i \(-0.652288\pi\)
−0.460384 + 0.887720i \(0.652288\pi\)
\(524\) 0 0
\(525\) −4.11614 −0.179643
\(526\) 0 0
\(527\) 23.4145 1.01995
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −7.02297 −0.304771
\(532\) 0 0
\(533\) 7.20949 0.312278
\(534\) 0 0
\(535\) −32.1539 −1.39014
\(536\) 0 0
\(537\) −45.4747 −1.96238
\(538\) 0 0
\(539\) 0.846153 0.0364464
\(540\) 0 0
\(541\) 30.0230 1.29079 0.645395 0.763849i \(-0.276693\pi\)
0.645395 + 0.763849i \(0.276693\pi\)
\(542\) 0 0
\(543\) 12.4051 0.532354
\(544\) 0 0
\(545\) 24.6670 1.05662
\(546\) 0 0
\(547\) 9.39847 0.401849 0.200925 0.979607i \(-0.435605\pi\)
0.200925 + 0.979607i \(0.435605\pi\)
\(548\) 0 0
\(549\) −49.5407 −2.11435
\(550\) 0 0
\(551\) 4.11664 0.175375
\(552\) 0 0
\(553\) 11.7795 0.500915
\(554\) 0 0
\(555\) 39.3276 1.66937
\(556\) 0 0
\(557\) −22.6305 −0.958885 −0.479443 0.877573i \(-0.659161\pi\)
−0.479443 + 0.877573i \(0.659161\pi\)
\(558\) 0 0
\(559\) 0.701393 0.0296658
\(560\) 0 0
\(561\) 16.0856 0.679136
\(562\) 0 0
\(563\) 21.0667 0.887854 0.443927 0.896063i \(-0.353585\pi\)
0.443927 + 0.896063i \(0.353585\pi\)
\(564\) 0 0
\(565\) 13.2281 0.556508
\(566\) 0 0
\(567\) 3.80592 0.159833
\(568\) 0 0
\(569\) −40.2625 −1.68789 −0.843945 0.536430i \(-0.819773\pi\)
−0.843945 + 0.536430i \(0.819773\pi\)
\(570\) 0 0
\(571\) −7.61151 −0.318532 −0.159266 0.987236i \(-0.550913\pi\)
−0.159266 + 0.987236i \(0.550913\pi\)
\(572\) 0 0
\(573\) 17.3246 0.723745
\(574\) 0 0
\(575\) −1.53098 −0.0638462
\(576\) 0 0
\(577\) 18.6364 0.775845 0.387923 0.921692i \(-0.373193\pi\)
0.387923 + 0.921692i \(0.373193\pi\)
\(578\) 0 0
\(579\) −27.2622 −1.13298
\(580\) 0 0
\(581\) 3.72506 0.154542
\(582\) 0 0
\(583\) 7.55896 0.313060
\(584\) 0 0
\(585\) −20.0912 −0.830667
\(586\) 0 0
\(587\) −40.9445 −1.68996 −0.844981 0.534797i \(-0.820388\pi\)
−0.844981 + 0.534797i \(0.820388\pi\)
\(588\) 0 0
\(589\) −1.57809 −0.0650242
\(590\) 0 0
\(591\) 6.38467 0.262630
\(592\) 0 0
\(593\) 20.5649 0.844501 0.422251 0.906479i \(-0.361240\pi\)
0.422251 + 0.906479i \(0.361240\pi\)
\(594\) 0 0
\(595\) −13.1696 −0.539900
\(596\) 0 0
\(597\) 50.0213 2.04723
\(598\) 0 0
\(599\) −2.94630 −0.120382 −0.0601912 0.998187i \(-0.519171\pi\)
−0.0601912 + 0.998187i \(0.519171\pi\)
\(600\) 0 0
\(601\) 37.8860 1.54540 0.772702 0.634769i \(-0.218905\pi\)
0.772702 + 0.634769i \(0.218905\pi\)
\(602\) 0 0
\(603\) 11.9510 0.486681
\(604\) 0 0
\(605\) 19.1543 0.778734
\(606\) 0 0
\(607\) 7.00707 0.284408 0.142204 0.989837i \(-0.454581\pi\)
0.142204 + 0.989837i \(0.454581\pi\)
\(608\) 0 0
\(609\) 23.2246 0.941107
\(610\) 0 0
\(611\) 34.3775 1.39077
\(612\) 0 0
\(613\) −18.9408 −0.765012 −0.382506 0.923953i \(-0.624939\pi\)
−0.382506 + 0.923953i \(0.624939\pi\)
\(614\) 0 0
\(615\) 14.1515 0.570642
\(616\) 0 0
\(617\) −2.22511 −0.0895797 −0.0447898 0.998996i \(-0.514262\pi\)
−0.0447898 + 0.998996i \(0.514262\pi\)
\(618\) 0 0
\(619\) 15.2083 0.611272 0.305636 0.952148i \(-0.401131\pi\)
0.305636 + 0.952148i \(0.401131\pi\)
\(620\) 0 0
\(621\) −3.30259 −0.132529
\(622\) 0 0
\(623\) 8.76310 0.351086
\(624\) 0 0
\(625\) −15.0012 −0.600049
\(626\) 0 0
\(627\) −1.08414 −0.0432964
\(628\) 0 0
\(629\) −55.5319 −2.21420
\(630\) 0 0
\(631\) −6.16049 −0.245245 −0.122623 0.992453i \(-0.539130\pi\)
−0.122623 + 0.992453i \(0.539130\pi\)
\(632\) 0 0
\(633\) −56.7953 −2.25741
\(634\) 0 0
\(635\) 30.0055 1.19073
\(636\) 0 0
\(637\) 2.55110 0.101078
\(638\) 0 0
\(639\) −41.9804 −1.66072
\(640\) 0 0
\(641\) 16.3586 0.646126 0.323063 0.946377i \(-0.395287\pi\)
0.323063 + 0.946377i \(0.395287\pi\)
\(642\) 0 0
\(643\) 30.8553 1.21681 0.608407 0.793625i \(-0.291809\pi\)
0.608407 + 0.793625i \(0.291809\pi\)
\(644\) 0 0
\(645\) 1.37676 0.0542099
\(646\) 0 0
\(647\) −10.3205 −0.405742 −0.202871 0.979206i \(-0.565027\pi\)
−0.202871 + 0.979206i \(0.565027\pi\)
\(648\) 0 0
\(649\) −1.40538 −0.0551662
\(650\) 0 0
\(651\) −8.90301 −0.348936
\(652\) 0 0
\(653\) 1.58679 0.0620961 0.0310480 0.999518i \(-0.490116\pi\)
0.0310480 + 0.999518i \(0.490116\pi\)
\(654\) 0 0
\(655\) −1.59184 −0.0621983
\(656\) 0 0
\(657\) 30.9330 1.20681
\(658\) 0 0
\(659\) 43.3271 1.68779 0.843893 0.536512i \(-0.180258\pi\)
0.843893 + 0.536512i \(0.180258\pi\)
\(660\) 0 0
\(661\) 18.2262 0.708917 0.354458 0.935072i \(-0.384665\pi\)
0.354458 + 0.935072i \(0.384665\pi\)
\(662\) 0 0
\(663\) 48.4971 1.88347
\(664\) 0 0
\(665\) 0.887605 0.0344199
\(666\) 0 0
\(667\) 8.63827 0.334475
\(668\) 0 0
\(669\) −57.9188 −2.23927
\(670\) 0 0
\(671\) −9.91372 −0.382715
\(672\) 0 0
\(673\) 6.47655 0.249653 0.124826 0.992179i \(-0.460163\pi\)
0.124826 + 0.992179i \(0.460163\pi\)
\(674\) 0 0
\(675\) 5.05620 0.194613
\(676\) 0 0
\(677\) −8.19344 −0.314899 −0.157450 0.987527i \(-0.550327\pi\)
−0.157450 + 0.987527i \(0.550327\pi\)
\(678\) 0 0
\(679\) −1.82229 −0.0699332
\(680\) 0 0
\(681\) 34.7189 1.33043
\(682\) 0 0
\(683\) 7.95826 0.304514 0.152257 0.988341i \(-0.451346\pi\)
0.152257 + 0.988341i \(0.451346\pi\)
\(684\) 0 0
\(685\) −13.2522 −0.506341
\(686\) 0 0
\(687\) −76.5796 −2.92169
\(688\) 0 0
\(689\) 22.7898 0.868222
\(690\) 0 0
\(691\) 3.66850 0.139556 0.0697782 0.997563i \(-0.477771\pi\)
0.0697782 + 0.997563i \(0.477771\pi\)
\(692\) 0 0
\(693\) −3.57786 −0.135912
\(694\) 0 0
\(695\) 19.9498 0.756739
\(696\) 0 0
\(697\) −19.9823 −0.756884
\(698\) 0 0
\(699\) −6.94446 −0.262664
\(700\) 0 0
\(701\) 40.6182 1.53413 0.767063 0.641571i \(-0.221717\pi\)
0.767063 + 0.641571i \(0.221717\pi\)
\(702\) 0 0
\(703\) 3.74274 0.141160
\(704\) 0 0
\(705\) 67.4795 2.54142
\(706\) 0 0
\(707\) 14.7870 0.556123
\(708\) 0 0
\(709\) −38.7523 −1.45537 −0.727687 0.685910i \(-0.759405\pi\)
−0.727687 + 0.685910i \(0.759405\pi\)
\(710\) 0 0
\(711\) −49.8082 −1.86795
\(712\) 0 0
\(713\) −3.31143 −0.124014
\(714\) 0 0
\(715\) −4.02049 −0.150358
\(716\) 0 0
\(717\) 32.1579 1.20096
\(718\) 0 0
\(719\) −34.5902 −1.29000 −0.644998 0.764184i \(-0.723142\pi\)
−0.644998 + 0.764184i \(0.723142\pi\)
\(720\) 0 0
\(721\) 10.9333 0.407178
\(722\) 0 0
\(723\) −5.46154 −0.203117
\(724\) 0 0
\(725\) −13.2250 −0.491164
\(726\) 0 0
\(727\) 8.51989 0.315985 0.157993 0.987440i \(-0.449498\pi\)
0.157993 + 0.987440i \(0.449498\pi\)
\(728\) 0 0
\(729\) −42.7306 −1.58261
\(730\) 0 0
\(731\) −1.94403 −0.0719026
\(732\) 0 0
\(733\) −35.1739 −1.29918 −0.649590 0.760285i \(-0.725059\pi\)
−0.649590 + 0.760285i \(0.725059\pi\)
\(734\) 0 0
\(735\) 5.00754 0.184706
\(736\) 0 0
\(737\) 2.39154 0.0880934
\(738\) 0 0
\(739\) −15.1355 −0.556769 −0.278384 0.960470i \(-0.589799\pi\)
−0.278384 + 0.960470i \(0.589799\pi\)
\(740\) 0 0
\(741\) −3.26862 −0.120076
\(742\) 0 0
\(743\) 20.2165 0.741670 0.370835 0.928699i \(-0.379072\pi\)
0.370835 + 0.928699i \(0.379072\pi\)
\(744\) 0 0
\(745\) 11.9878 0.439201
\(746\) 0 0
\(747\) −15.7510 −0.576299
\(748\) 0 0
\(749\) −17.2636 −0.630797
\(750\) 0 0
\(751\) −31.1017 −1.13492 −0.567459 0.823401i \(-0.692074\pi\)
−0.567459 + 0.823401i \(0.692074\pi\)
\(752\) 0 0
\(753\) 73.7888 2.68901
\(754\) 0 0
\(755\) 1.49651 0.0544634
\(756\) 0 0
\(757\) −11.9728 −0.435158 −0.217579 0.976043i \(-0.569816\pi\)
−0.217579 + 0.976043i \(0.569816\pi\)
\(758\) 0 0
\(759\) −2.27494 −0.0825750
\(760\) 0 0
\(761\) 17.8372 0.646597 0.323299 0.946297i \(-0.395208\pi\)
0.323299 + 0.946297i \(0.395208\pi\)
\(762\) 0 0
\(763\) 13.2438 0.479457
\(764\) 0 0
\(765\) 55.6861 2.01333
\(766\) 0 0
\(767\) −4.23714 −0.152994
\(768\) 0 0
\(769\) −47.9412 −1.72880 −0.864401 0.502802i \(-0.832302\pi\)
−0.864401 + 0.502802i \(0.832302\pi\)
\(770\) 0 0
\(771\) 67.6565 2.43659
\(772\) 0 0
\(773\) 22.7079 0.816748 0.408374 0.912815i \(-0.366096\pi\)
0.408374 + 0.912815i \(0.366096\pi\)
\(774\) 0 0
\(775\) 5.06973 0.182110
\(776\) 0 0
\(777\) 21.1152 0.757502
\(778\) 0 0
\(779\) 1.34677 0.0482531
\(780\) 0 0
\(781\) −8.40080 −0.300604
\(782\) 0 0
\(783\) −28.5287 −1.01953
\(784\) 0 0
\(785\) 27.5577 0.983578
\(786\) 0 0
\(787\) 20.2521 0.721910 0.360955 0.932583i \(-0.382451\pi\)
0.360955 + 0.932583i \(0.382451\pi\)
\(788\) 0 0
\(789\) 29.2011 1.03959
\(790\) 0 0
\(791\) 7.10219 0.252525
\(792\) 0 0
\(793\) −29.8892 −1.06140
\(794\) 0 0
\(795\) 44.7340 1.58655
\(796\) 0 0
\(797\) −22.1099 −0.783172 −0.391586 0.920142i \(-0.628073\pi\)
−0.391586 + 0.920142i \(0.628073\pi\)
\(798\) 0 0
\(799\) −95.2831 −3.37087
\(800\) 0 0
\(801\) −37.0538 −1.30923
\(802\) 0 0
\(803\) 6.19008 0.218443
\(804\) 0 0
\(805\) 1.86253 0.0656456
\(806\) 0 0
\(807\) 5.43449 0.191303
\(808\) 0 0
\(809\) −35.3424 −1.24257 −0.621287 0.783583i \(-0.713390\pi\)
−0.621287 + 0.783583i \(0.713390\pi\)
\(810\) 0 0
\(811\) 15.3968 0.540654 0.270327 0.962769i \(-0.412868\pi\)
0.270327 + 0.962769i \(0.412868\pi\)
\(812\) 0 0
\(813\) −10.9084 −0.382574
\(814\) 0 0
\(815\) 5.83667 0.204450
\(816\) 0 0
\(817\) 0.131024 0.00458395
\(818\) 0 0
\(819\) −10.7870 −0.376929
\(820\) 0 0
\(821\) −47.6244 −1.66210 −0.831052 0.556195i \(-0.812261\pi\)
−0.831052 + 0.556195i \(0.812261\pi\)
\(822\) 0 0
\(823\) −24.1554 −0.842006 −0.421003 0.907059i \(-0.638322\pi\)
−0.421003 + 0.907059i \(0.638322\pi\)
\(824\) 0 0
\(825\) 3.48288 0.121258
\(826\) 0 0
\(827\) 14.8594 0.516710 0.258355 0.966050i \(-0.416820\pi\)
0.258355 + 0.966050i \(0.416820\pi\)
\(828\) 0 0
\(829\) 20.2699 0.704003 0.352001 0.935999i \(-0.385501\pi\)
0.352001 + 0.935999i \(0.385501\pi\)
\(830\) 0 0
\(831\) 84.9778 2.94785
\(832\) 0 0
\(833\) −7.07080 −0.244989
\(834\) 0 0
\(835\) −43.6145 −1.50934
\(836\) 0 0
\(837\) 10.9363 0.378015
\(838\) 0 0
\(839\) 20.9307 0.722609 0.361304 0.932448i \(-0.382332\pi\)
0.361304 + 0.932448i \(0.382332\pi\)
\(840\) 0 0
\(841\) 45.6197 1.57309
\(842\) 0 0
\(843\) −68.4723 −2.35831
\(844\) 0 0
\(845\) 12.0914 0.415956
\(846\) 0 0
\(847\) 10.2840 0.353363
\(848\) 0 0
\(849\) −21.6930 −0.744501
\(850\) 0 0
\(851\) 7.85369 0.269221
\(852\) 0 0
\(853\) −28.3729 −0.971470 −0.485735 0.874106i \(-0.661448\pi\)
−0.485735 + 0.874106i \(0.661448\pi\)
\(854\) 0 0
\(855\) −3.75314 −0.128355
\(856\) 0 0
\(857\) −32.4441 −1.10827 −0.554134 0.832428i \(-0.686950\pi\)
−0.554134 + 0.832428i \(0.686950\pi\)
\(858\) 0 0
\(859\) −32.7552 −1.11759 −0.558797 0.829305i \(-0.688737\pi\)
−0.558797 + 0.829305i \(0.688737\pi\)
\(860\) 0 0
\(861\) 7.59798 0.258938
\(862\) 0 0
\(863\) −5.88556 −0.200347 −0.100173 0.994970i \(-0.531940\pi\)
−0.100173 + 0.994970i \(0.531940\pi\)
\(864\) 0 0
\(865\) 1.20212 0.0408734
\(866\) 0 0
\(867\) −88.7124 −3.01283
\(868\) 0 0
\(869\) −9.96724 −0.338116
\(870\) 0 0
\(871\) 7.21033 0.244313
\(872\) 0 0
\(873\) 7.70536 0.260787
\(874\) 0 0
\(875\) −12.1641 −0.411223
\(876\) 0 0
\(877\) −53.7879 −1.81629 −0.908144 0.418658i \(-0.862501\pi\)
−0.908144 + 0.418658i \(0.862501\pi\)
\(878\) 0 0
\(879\) 33.8129 1.14048
\(880\) 0 0
\(881\) 6.04127 0.203536 0.101768 0.994808i \(-0.467550\pi\)
0.101768 + 0.994808i \(0.467550\pi\)
\(882\) 0 0
\(883\) −44.1976 −1.48737 −0.743683 0.668532i \(-0.766923\pi\)
−0.743683 + 0.668532i \(0.766923\pi\)
\(884\) 0 0
\(885\) −8.31707 −0.279575
\(886\) 0 0
\(887\) 36.4986 1.22550 0.612752 0.790275i \(-0.290063\pi\)
0.612752 + 0.790275i \(0.290063\pi\)
\(888\) 0 0
\(889\) 16.1101 0.540314
\(890\) 0 0
\(891\) −3.22039 −0.107887
\(892\) 0 0
\(893\) 6.42191 0.214901
\(894\) 0 0
\(895\) −31.5030 −1.05303
\(896\) 0 0
\(897\) −6.85879 −0.229008
\(898\) 0 0
\(899\) −28.6051 −0.954033
\(900\) 0 0
\(901\) −63.1658 −2.10436
\(902\) 0 0
\(903\) 0.739189 0.0245987
\(904\) 0 0
\(905\) 8.59377 0.285667
\(906\) 0 0
\(907\) −8.22173 −0.272998 −0.136499 0.990640i \(-0.543585\pi\)
−0.136499 + 0.990640i \(0.543585\pi\)
\(908\) 0 0
\(909\) −62.5252 −2.07383
\(910\) 0 0
\(911\) 21.5853 0.715154 0.357577 0.933884i \(-0.383603\pi\)
0.357577 + 0.933884i \(0.383603\pi\)
\(912\) 0 0
\(913\) −3.15197 −0.104315
\(914\) 0 0
\(915\) −58.6694 −1.93955
\(916\) 0 0
\(917\) −0.854665 −0.0282235
\(918\) 0 0
\(919\) 38.0934 1.25658 0.628292 0.777978i \(-0.283754\pi\)
0.628292 + 0.777978i \(0.283754\pi\)
\(920\) 0 0
\(921\) 1.58290 0.0521583
\(922\) 0 0
\(923\) −25.3279 −0.833677
\(924\) 0 0
\(925\) −12.0238 −0.395341
\(926\) 0 0
\(927\) −46.2303 −1.51840
\(928\) 0 0
\(929\) −9.15900 −0.300497 −0.150249 0.988648i \(-0.548007\pi\)
−0.150249 + 0.988648i \(0.548007\pi\)
\(930\) 0 0
\(931\) 0.476559 0.0156186
\(932\) 0 0
\(933\) 6.25065 0.204637
\(934\) 0 0
\(935\) 11.1435 0.364431
\(936\) 0 0
\(937\) −18.8314 −0.615194 −0.307597 0.951517i \(-0.599525\pi\)
−0.307597 + 0.951517i \(0.599525\pi\)
\(938\) 0 0
\(939\) 21.5855 0.704416
\(940\) 0 0
\(941\) 25.4216 0.828721 0.414360 0.910113i \(-0.364005\pi\)
0.414360 + 0.910113i \(0.364005\pi\)
\(942\) 0 0
\(943\) 2.82603 0.0920283
\(944\) 0 0
\(945\) −6.15118 −0.200098
\(946\) 0 0
\(947\) −33.3211 −1.08279 −0.541395 0.840768i \(-0.682104\pi\)
−0.541395 + 0.840768i \(0.682104\pi\)
\(948\) 0 0
\(949\) 18.6627 0.605817
\(950\) 0 0
\(951\) −72.7070 −2.35769
\(952\) 0 0
\(953\) 30.5121 0.988383 0.494192 0.869353i \(-0.335464\pi\)
0.494192 + 0.869353i \(0.335464\pi\)
\(954\) 0 0
\(955\) 12.0018 0.388369
\(956\) 0 0
\(957\) −19.6515 −0.635244
\(958\) 0 0
\(959\) −7.11516 −0.229761
\(960\) 0 0
\(961\) −20.0344 −0.646271
\(962\) 0 0
\(963\) 72.9970 2.35230
\(964\) 0 0
\(965\) −18.8862 −0.607967
\(966\) 0 0
\(967\) 23.0298 0.740587 0.370294 0.928915i \(-0.379257\pi\)
0.370294 + 0.928915i \(0.379257\pi\)
\(968\) 0 0
\(969\) 9.05953 0.291034
\(970\) 0 0
\(971\) −6.95588 −0.223225 −0.111612 0.993752i \(-0.535602\pi\)
−0.111612 + 0.993752i \(0.535602\pi\)
\(972\) 0 0
\(973\) 10.7111 0.343383
\(974\) 0 0
\(975\) 10.5007 0.336290
\(976\) 0 0
\(977\) −23.1596 −0.740942 −0.370471 0.928844i \(-0.620804\pi\)
−0.370471 + 0.928844i \(0.620804\pi\)
\(978\) 0 0
\(979\) −7.41493 −0.236982
\(980\) 0 0
\(981\) −55.9998 −1.78794
\(982\) 0 0
\(983\) −11.7384 −0.374397 −0.187198 0.982322i \(-0.559941\pi\)
−0.187198 + 0.982322i \(0.559941\pi\)
\(984\) 0 0
\(985\) 4.42305 0.140930
\(986\) 0 0
\(987\) 36.2300 1.15321
\(988\) 0 0
\(989\) 0.274938 0.00874252
\(990\) 0 0
\(991\) 14.8198 0.470766 0.235383 0.971903i \(-0.424366\pi\)
0.235383 + 0.971903i \(0.424366\pi\)
\(992\) 0 0
\(993\) 61.1796 1.94148
\(994\) 0 0
\(995\) 34.6527 1.09857
\(996\) 0 0
\(997\) −8.94358 −0.283246 −0.141623 0.989921i \(-0.545232\pi\)
−0.141623 + 0.989921i \(0.545232\pi\)
\(998\) 0 0
\(999\) −25.9375 −0.820628
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 2576.2.a.bd.1.1 5
4.3 odd 2 161.2.a.d.1.3 5
12.11 even 2 1449.2.a.r.1.3 5
20.19 odd 2 4025.2.a.p.1.3 5
28.27 even 2 1127.2.a.h.1.3 5
92.91 even 2 3703.2.a.j.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.d.1.3 5 4.3 odd 2
1127.2.a.h.1.3 5 28.27 even 2
1449.2.a.r.1.3 5 12.11 even 2
2576.2.a.bd.1.1 5 1.1 even 1 trivial
3703.2.a.j.1.3 5 92.91 even 2
4025.2.a.p.1.3 5 20.19 odd 2