Properties

Label 7360.2.a.cc.1.2
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
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] = [7360,2,Mod(1,7360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7360, 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("7360.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.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: \(58.7698958877\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2597.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.878468\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.878468 q^{3} -1.00000 q^{5} -0.121532 q^{7} -2.22829 q^{9} +O(q^{10})\) \(q+0.878468 q^{3} -1.00000 q^{5} -0.121532 q^{7} -2.22829 q^{9} +2.87847 q^{11} -5.22829 q^{13} -0.878468 q^{15} -2.22829 q^{17} +1.22829 q^{19} -0.106762 q^{21} -1.00000 q^{23} +1.00000 q^{25} -4.59289 q^{27} -9.34983 q^{29} +2.12153 q^{31} +2.52864 q^{33} +0.121532 q^{35} -5.59289 q^{37} -4.59289 q^{39} +8.22829 q^{41} +8.00000 q^{43} +2.22829 q^{45} +10.4566 q^{47} -6.98523 q^{49} -1.95749 q^{51} +3.59289 q^{53} -2.87847 q^{55} +1.07902 q^{57} -0.650174 q^{59} +7.33506 q^{61} +0.270809 q^{63} +5.22829 q^{65} +5.59289 q^{67} -0.878468 q^{69} +13.9852 q^{71} +12.9427 q^{73} +0.878468 q^{75} -0.349826 q^{77} +3.51387 q^{79} +2.65017 q^{81} -11.1068 q^{83} +2.22829 q^{85} -8.21352 q^{87} -0.486128 q^{89} +0.635404 q^{91} +1.86370 q^{93} -1.22829 q^{95} +0.635404 q^{97} -6.41407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{5} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 3 q^{5} - 2 q^{7} + 10 q^{9} + 7 q^{11} + q^{13} - q^{15} + 10 q^{17} - 13 q^{19} + 18 q^{21} - 3 q^{23} + 3 q^{25} - 2 q^{27} - 13 q^{29} + 8 q^{31} + 21 q^{33} + 2 q^{35} - 5 q^{37} - 2 q^{39} + 8 q^{41} + 24 q^{43} - 10 q^{45} - 2 q^{47} - q^{49} + q^{51} - q^{53} - 7 q^{55} - 2 q^{57} - 17 q^{59} - 13 q^{61} - 9 q^{63} - q^{65} + 5 q^{67} - q^{69} + 22 q^{71} + 12 q^{73} + q^{75} + 14 q^{77} + 4 q^{79} + 23 q^{81} - 15 q^{83} - 10 q^{85} + 12 q^{87} - 8 q^{89} - 3 q^{91} - 16 q^{93} + 13 q^{95} - 3 q^{97} + 21 q^{99}+O(q^{100}) \) 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
\(3\) 0.878468 0.507184 0.253592 0.967311i \(-0.418388\pi\)
0.253592 + 0.967311i \(0.418388\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.121532 −0.0459347 −0.0229674 0.999736i \(-0.507311\pi\)
−0.0229674 + 0.999736i \(0.507311\pi\)
\(8\) 0 0
\(9\) −2.22829 −0.742765
\(10\) 0 0
\(11\) 2.87847 0.867891 0.433945 0.900939i \(-0.357121\pi\)
0.433945 + 0.900939i \(0.357121\pi\)
\(12\) 0 0
\(13\) −5.22829 −1.45007 −0.725034 0.688713i \(-0.758176\pi\)
−0.725034 + 0.688713i \(0.758176\pi\)
\(14\) 0 0
\(15\) −0.878468 −0.226819
\(16\) 0 0
\(17\) −2.22829 −0.540441 −0.270220 0.962799i \(-0.587097\pi\)
−0.270220 + 0.962799i \(0.587097\pi\)
\(18\) 0 0
\(19\) 1.22829 0.281790 0.140895 0.990025i \(-0.455002\pi\)
0.140895 + 0.990025i \(0.455002\pi\)
\(20\) 0 0
\(21\) −0.106762 −0.0232974
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.59289 −0.883902
\(28\) 0 0
\(29\) −9.34983 −1.73622 −0.868110 0.496373i \(-0.834665\pi\)
−0.868110 + 0.496373i \(0.834665\pi\)
\(30\) 0 0
\(31\) 2.12153 0.381038 0.190519 0.981683i \(-0.438983\pi\)
0.190519 + 0.981683i \(0.438983\pi\)
\(32\) 0 0
\(33\) 2.52864 0.440180
\(34\) 0 0
\(35\) 0.121532 0.0205426
\(36\) 0 0
\(37\) −5.59289 −0.919465 −0.459733 0.888057i \(-0.652055\pi\)
−0.459733 + 0.888057i \(0.652055\pi\)
\(38\) 0 0
\(39\) −4.59289 −0.735451
\(40\) 0 0
\(41\) 8.22829 1.28504 0.642522 0.766267i \(-0.277888\pi\)
0.642522 + 0.766267i \(0.277888\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 2.22829 0.332174
\(46\) 0 0
\(47\) 10.4566 1.52525 0.762625 0.646841i \(-0.223910\pi\)
0.762625 + 0.646841i \(0.223910\pi\)
\(48\) 0 0
\(49\) −6.98523 −0.997890
\(50\) 0 0
\(51\) −1.95749 −0.274103
\(52\) 0 0
\(53\) 3.59289 0.493521 0.246761 0.969076i \(-0.420634\pi\)
0.246761 + 0.969076i \(0.420634\pi\)
\(54\) 0 0
\(55\) −2.87847 −0.388133
\(56\) 0 0
\(57\) 1.07902 0.142919
\(58\) 0 0
\(59\) −0.650174 −0.0846455 −0.0423227 0.999104i \(-0.513476\pi\)
−0.0423227 + 0.999104i \(0.513476\pi\)
\(60\) 0 0
\(61\) 7.33506 0.939158 0.469579 0.882891i \(-0.344406\pi\)
0.469579 + 0.882891i \(0.344406\pi\)
\(62\) 0 0
\(63\) 0.270809 0.0341187
\(64\) 0 0
\(65\) 5.22829 0.648490
\(66\) 0 0
\(67\) 5.59289 0.683280 0.341640 0.939831i \(-0.389018\pi\)
0.341640 + 0.939831i \(0.389018\pi\)
\(68\) 0 0
\(69\) −0.878468 −0.105755
\(70\) 0 0
\(71\) 13.9852 1.65974 0.829871 0.557956i \(-0.188414\pi\)
0.829871 + 0.557956i \(0.188414\pi\)
\(72\) 0 0
\(73\) 12.9427 1.51483 0.757415 0.652934i \(-0.226462\pi\)
0.757415 + 0.652934i \(0.226462\pi\)
\(74\) 0 0
\(75\) 0.878468 0.101437
\(76\) 0 0
\(77\) −0.349826 −0.0398663
\(78\) 0 0
\(79\) 3.51387 0.395342 0.197671 0.980268i \(-0.436662\pi\)
0.197671 + 0.980268i \(0.436662\pi\)
\(80\) 0 0
\(81\) 2.65017 0.294464
\(82\) 0 0
\(83\) −11.1068 −1.21913 −0.609563 0.792738i \(-0.708655\pi\)
−0.609563 + 0.792738i \(0.708655\pi\)
\(84\) 0 0
\(85\) 2.22829 0.241692
\(86\) 0 0
\(87\) −8.21352 −0.880582
\(88\) 0 0
\(89\) −0.486128 −0.0515294 −0.0257647 0.999668i \(-0.508202\pi\)
−0.0257647 + 0.999668i \(0.508202\pi\)
\(90\) 0 0
\(91\) 0.635404 0.0666085
\(92\) 0 0
\(93\) 1.86370 0.193256
\(94\) 0 0
\(95\) −1.22829 −0.126020
\(96\) 0 0
\(97\) 0.635404 0.0645155 0.0322578 0.999480i \(-0.489730\pi\)
0.0322578 + 0.999480i \(0.489730\pi\)
\(98\) 0 0
\(99\) −6.41407 −0.644639
\(100\) 0 0
\(101\) −13.3498 −1.32836 −0.664179 0.747574i \(-0.731219\pi\)
−0.664179 + 0.747574i \(0.731219\pi\)
\(102\) 0 0
\(103\) 7.33506 0.722745 0.361372 0.932422i \(-0.382308\pi\)
0.361372 + 0.932422i \(0.382308\pi\)
\(104\) 0 0
\(105\) 0.106762 0.0104189
\(106\) 0 0
\(107\) 16.0495 1.55156 0.775781 0.631003i \(-0.217356\pi\)
0.775781 + 0.631003i \(0.217356\pi\)
\(108\) 0 0
\(109\) −1.01477 −0.0971973 −0.0485987 0.998818i \(-0.515476\pi\)
−0.0485987 + 0.998818i \(0.515476\pi\)
\(110\) 0 0
\(111\) −4.91318 −0.466338
\(112\) 0 0
\(113\) 17.3203 1.62936 0.814678 0.579914i \(-0.196914\pi\)
0.814678 + 0.579914i \(0.196914\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 11.6502 1.07706
\(118\) 0 0
\(119\) 0.270809 0.0248250
\(120\) 0 0
\(121\) −2.71442 −0.246766
\(122\) 0 0
\(123\) 7.22829 0.651753
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.75694 −0.155903 −0.0779514 0.996957i \(-0.524838\pi\)
−0.0779514 + 0.996957i \(0.524838\pi\)
\(128\) 0 0
\(129\) 7.02774 0.618758
\(130\) 0 0
\(131\) −20.2135 −1.76606 −0.883032 0.469313i \(-0.844502\pi\)
−0.883032 + 0.469313i \(0.844502\pi\)
\(132\) 0 0
\(133\) −0.149277 −0.0129439
\(134\) 0 0
\(135\) 4.59289 0.395293
\(136\) 0 0
\(137\) −17.9279 −1.53169 −0.765844 0.643027i \(-0.777678\pi\)
−0.765844 + 0.643027i \(0.777678\pi\)
\(138\) 0 0
\(139\) −10.8637 −0.921447 −0.460723 0.887544i \(-0.652410\pi\)
−0.460723 + 0.887544i \(0.652410\pi\)
\(140\) 0 0
\(141\) 9.18578 0.773582
\(142\) 0 0
\(143\) −15.0495 −1.25850
\(144\) 0 0
\(145\) 9.34983 0.776461
\(146\) 0 0
\(147\) −6.13630 −0.506114
\(148\) 0 0
\(149\) 15.7144 1.28738 0.643688 0.765288i \(-0.277404\pi\)
0.643688 + 0.765288i \(0.277404\pi\)
\(150\) 0 0
\(151\) 8.39234 0.682959 0.341479 0.939889i \(-0.389072\pi\)
0.341479 + 0.939889i \(0.389072\pi\)
\(152\) 0 0
\(153\) 4.96529 0.401420
\(154\) 0 0
\(155\) −2.12153 −0.170406
\(156\) 0 0
\(157\) 19.5633 1.56133 0.780663 0.624953i \(-0.214882\pi\)
0.780663 + 0.624953i \(0.214882\pi\)
\(158\) 0 0
\(159\) 3.15624 0.250306
\(160\) 0 0
\(161\) 0.121532 0.00957805
\(162\) 0 0
\(163\) 1.01477 0.0794829 0.0397415 0.999210i \(-0.487347\pi\)
0.0397415 + 0.999210i \(0.487347\pi\)
\(164\) 0 0
\(165\) −2.52864 −0.196855
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 14.3351 1.10270
\(170\) 0 0
\(171\) −2.73700 −0.209304
\(172\) 0 0
\(173\) 2.63540 0.200366 0.100183 0.994969i \(-0.468057\pi\)
0.100183 + 0.994969i \(0.468057\pi\)
\(174\) 0 0
\(175\) −0.121532 −0.00918695
\(176\) 0 0
\(177\) −0.571157 −0.0429308
\(178\) 0 0
\(179\) −14.4566 −1.08054 −0.540268 0.841493i \(-0.681677\pi\)
−0.540268 + 0.841493i \(0.681677\pi\)
\(180\) 0 0
\(181\) −10.7422 −0.798459 −0.399229 0.916851i \(-0.630722\pi\)
−0.399229 + 0.916851i \(0.630722\pi\)
\(182\) 0 0
\(183\) 6.44361 0.476326
\(184\) 0 0
\(185\) 5.59289 0.411197
\(186\) 0 0
\(187\) −6.41407 −0.469043
\(188\) 0 0
\(189\) 0.558182 0.0406018
\(190\) 0 0
\(191\) 22.6701 1.64035 0.820176 0.572112i \(-0.193876\pi\)
0.820176 + 0.572112i \(0.193876\pi\)
\(192\) 0 0
\(193\) 2.24306 0.161459 0.0807296 0.996736i \(-0.474275\pi\)
0.0807296 + 0.996736i \(0.474275\pi\)
\(194\) 0 0
\(195\) 4.59289 0.328904
\(196\) 0 0
\(197\) 7.57812 0.539919 0.269959 0.962872i \(-0.412990\pi\)
0.269959 + 0.962872i \(0.412990\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 4.91318 0.346549
\(202\) 0 0
\(203\) 1.13630 0.0797528
\(204\) 0 0
\(205\) −8.22829 −0.574689
\(206\) 0 0
\(207\) 2.22829 0.154877
\(208\) 0 0
\(209\) 3.53560 0.244563
\(210\) 0 0
\(211\) −3.10676 −0.213878 −0.106939 0.994266i \(-0.534105\pi\)
−0.106939 + 0.994266i \(0.534105\pi\)
\(212\) 0 0
\(213\) 12.2856 0.841794
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −0.257834 −0.0175029
\(218\) 0 0
\(219\) 11.3698 0.768297
\(220\) 0 0
\(221\) 11.6502 0.783676
\(222\) 0 0
\(223\) −18.2135 −1.21967 −0.609834 0.792529i \(-0.708764\pi\)
−0.609834 + 0.792529i \(0.708764\pi\)
\(224\) 0 0
\(225\) −2.22829 −0.148553
\(226\) 0 0
\(227\) 13.9705 0.927252 0.463626 0.886031i \(-0.346548\pi\)
0.463626 + 0.886031i \(0.346548\pi\)
\(228\) 0 0
\(229\) −12.2135 −0.807092 −0.403546 0.914959i \(-0.632223\pi\)
−0.403546 + 0.914959i \(0.632223\pi\)
\(230\) 0 0
\(231\) −0.307311 −0.0202196
\(232\) 0 0
\(233\) 9.75694 0.639198 0.319599 0.947553i \(-0.396452\pi\)
0.319599 + 0.947553i \(0.396452\pi\)
\(234\) 0 0
\(235\) −10.4566 −0.682113
\(236\) 0 0
\(237\) 3.08682 0.200511
\(238\) 0 0
\(239\) −8.62063 −0.557622 −0.278811 0.960346i \(-0.589940\pi\)
−0.278811 + 0.960346i \(0.589940\pi\)
\(240\) 0 0
\(241\) 3.18578 0.205214 0.102607 0.994722i \(-0.467282\pi\)
0.102607 + 0.994722i \(0.467282\pi\)
\(242\) 0 0
\(243\) 16.1068 1.03325
\(244\) 0 0
\(245\) 6.98523 0.446270
\(246\) 0 0
\(247\) −6.42188 −0.408614
\(248\) 0 0
\(249\) −9.75694 −0.618321
\(250\) 0 0
\(251\) −25.2283 −1.59240 −0.796198 0.605036i \(-0.793159\pi\)
−0.796198 + 0.605036i \(0.793159\pi\)
\(252\) 0 0
\(253\) −2.87847 −0.180968
\(254\) 0 0
\(255\) 1.95749 0.122582
\(256\) 0 0
\(257\) −10.6701 −0.665583 −0.332792 0.943000i \(-0.607991\pi\)
−0.332792 + 0.943000i \(0.607991\pi\)
\(258\) 0 0
\(259\) 0.679714 0.0422354
\(260\) 0 0
\(261\) 20.8342 1.28960
\(262\) 0 0
\(263\) 18.6849 1.15216 0.576080 0.817394i \(-0.304582\pi\)
0.576080 + 0.817394i \(0.304582\pi\)
\(264\) 0 0
\(265\) −3.59289 −0.220709
\(266\) 0 0
\(267\) −0.427048 −0.0261349
\(268\) 0 0
\(269\) 24.8637 1.51597 0.757983 0.652274i \(-0.226185\pi\)
0.757983 + 0.652274i \(0.226185\pi\)
\(270\) 0 0
\(271\) −0.578119 −0.0351183 −0.0175591 0.999846i \(-0.505590\pi\)
−0.0175591 + 0.999846i \(0.505590\pi\)
\(272\) 0 0
\(273\) 0.558182 0.0337827
\(274\) 0 0
\(275\) 2.87847 0.173578
\(276\) 0 0
\(277\) 9.51387 0.571633 0.285817 0.958284i \(-0.407735\pi\)
0.285817 + 0.958284i \(0.407735\pi\)
\(278\) 0 0
\(279\) −4.72740 −0.283022
\(280\) 0 0
\(281\) −13.1562 −0.784835 −0.392418 0.919787i \(-0.628361\pi\)
−0.392418 + 0.919787i \(0.628361\pi\)
\(282\) 0 0
\(283\) 24.5061 1.45673 0.728367 0.685187i \(-0.240280\pi\)
0.728367 + 0.685187i \(0.240280\pi\)
\(284\) 0 0
\(285\) −1.07902 −0.0639154
\(286\) 0 0
\(287\) −1.00000 −0.0590281
\(288\) 0 0
\(289\) −12.0347 −0.707924
\(290\) 0 0
\(291\) 0.558182 0.0327212
\(292\) 0 0
\(293\) 6.29254 0.367614 0.183807 0.982962i \(-0.441158\pi\)
0.183807 + 0.982962i \(0.441158\pi\)
\(294\) 0 0
\(295\) 0.650174 0.0378546
\(296\) 0 0
\(297\) −13.2205 −0.767130
\(298\) 0 0
\(299\) 5.22829 0.302360
\(300\) 0 0
\(301\) −0.972255 −0.0560398
\(302\) 0 0
\(303\) −11.7274 −0.673721
\(304\) 0 0
\(305\) −7.33506 −0.420004
\(306\) 0 0
\(307\) 13.3351 0.761072 0.380536 0.924766i \(-0.375740\pi\)
0.380536 + 0.924766i \(0.375740\pi\)
\(308\) 0 0
\(309\) 6.44361 0.366564
\(310\) 0 0
\(311\) 24.4270 1.38513 0.692565 0.721355i \(-0.256480\pi\)
0.692565 + 0.721355i \(0.256480\pi\)
\(312\) 0 0
\(313\) 26.5486 1.50061 0.750307 0.661089i \(-0.229906\pi\)
0.750307 + 0.661089i \(0.229906\pi\)
\(314\) 0 0
\(315\) −0.270809 −0.0152583
\(316\) 0 0
\(317\) 16.9852 0.953986 0.476993 0.878907i \(-0.341727\pi\)
0.476993 + 0.878907i \(0.341727\pi\)
\(318\) 0 0
\(319\) −26.9132 −1.50685
\(320\) 0 0
\(321\) 14.0990 0.786927
\(322\) 0 0
\(323\) −2.73700 −0.152291
\(324\) 0 0
\(325\) −5.22829 −0.290014
\(326\) 0 0
\(327\) −0.891443 −0.0492969
\(328\) 0 0
\(329\) −1.27081 −0.0700620
\(330\) 0 0
\(331\) 17.8914 0.983403 0.491701 0.870764i \(-0.336375\pi\)
0.491701 + 0.870764i \(0.336375\pi\)
\(332\) 0 0
\(333\) 12.4626 0.682946
\(334\) 0 0
\(335\) −5.59289 −0.305572
\(336\) 0 0
\(337\) −4.52864 −0.246691 −0.123345 0.992364i \(-0.539362\pi\)
−0.123345 + 0.992364i \(0.539362\pi\)
\(338\) 0 0
\(339\) 15.2153 0.826383
\(340\) 0 0
\(341\) 6.10676 0.330700
\(342\) 0 0
\(343\) 1.69965 0.0917725
\(344\) 0 0
\(345\) 0.878468 0.0472951
\(346\) 0 0
\(347\) −0.0937869 −0.00503475 −0.00251737 0.999997i \(-0.500801\pi\)
−0.00251737 + 0.999997i \(0.500801\pi\)
\(348\) 0 0
\(349\) −8.07902 −0.432460 −0.216230 0.976342i \(-0.569376\pi\)
−0.216230 + 0.976342i \(0.569376\pi\)
\(350\) 0 0
\(351\) 24.0130 1.28172
\(352\) 0 0
\(353\) −8.00000 −0.425797 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(354\) 0 0
\(355\) −13.9852 −0.742259
\(356\) 0 0
\(357\) 0.237897 0.0125908
\(358\) 0 0
\(359\) −8.69965 −0.459150 −0.229575 0.973291i \(-0.573734\pi\)
−0.229575 + 0.973291i \(0.573734\pi\)
\(360\) 0 0
\(361\) −17.4913 −0.920594
\(362\) 0 0
\(363\) −2.38453 −0.125156
\(364\) 0 0
\(365\) −12.9427 −0.677453
\(366\) 0 0
\(367\) −31.8064 −1.66028 −0.830141 0.557554i \(-0.811740\pi\)
−0.830141 + 0.557554i \(0.811740\pi\)
\(368\) 0 0
\(369\) −18.3351 −0.954485
\(370\) 0 0
\(371\) −0.436651 −0.0226698
\(372\) 0 0
\(373\) 13.4288 0.695319 0.347660 0.937621i \(-0.386977\pi\)
0.347660 + 0.937621i \(0.386977\pi\)
\(374\) 0 0
\(375\) −0.878468 −0.0453639
\(376\) 0 0
\(377\) 48.8836 2.51764
\(378\) 0 0
\(379\) 31.8212 1.63454 0.817272 0.576252i \(-0.195485\pi\)
0.817272 + 0.576252i \(0.195485\pi\)
\(380\) 0 0
\(381\) −1.54341 −0.0790714
\(382\) 0 0
\(383\) −34.3480 −1.75510 −0.877551 0.479483i \(-0.840824\pi\)
−0.877551 + 0.479483i \(0.840824\pi\)
\(384\) 0 0
\(385\) 0.349826 0.0178288
\(386\) 0 0
\(387\) −17.8264 −0.906164
\(388\) 0 0
\(389\) −8.14928 −0.413185 −0.206592 0.978427i \(-0.566237\pi\)
−0.206592 + 0.978427i \(0.566237\pi\)
\(390\) 0 0
\(391\) 2.22829 0.112690
\(392\) 0 0
\(393\) −17.7569 −0.895719
\(394\) 0 0
\(395\) −3.51387 −0.176802
\(396\) 0 0
\(397\) −19.0625 −0.956717 −0.478359 0.878165i \(-0.658768\pi\)
−0.478359 + 0.878165i \(0.658768\pi\)
\(398\) 0 0
\(399\) −0.131135 −0.00656496
\(400\) 0 0
\(401\) 19.9705 0.997277 0.498639 0.866810i \(-0.333833\pi\)
0.498639 + 0.866810i \(0.333833\pi\)
\(402\) 0 0
\(403\) −11.0920 −0.552531
\(404\) 0 0
\(405\) −2.65017 −0.131688
\(406\) 0 0
\(407\) −16.0990 −0.797996
\(408\) 0 0
\(409\) −10.3351 −0.511036 −0.255518 0.966804i \(-0.582246\pi\)
−0.255518 + 0.966804i \(0.582246\pi\)
\(410\) 0 0
\(411\) −15.7491 −0.776847
\(412\) 0 0
\(413\) 0.0790169 0.00388817
\(414\) 0 0
\(415\) 11.1068 0.545209
\(416\) 0 0
\(417\) −9.54341 −0.467343
\(418\) 0 0
\(419\) 5.72740 0.279802 0.139901 0.990166i \(-0.455322\pi\)
0.139901 + 0.990166i \(0.455322\pi\)
\(420\) 0 0
\(421\) 8.44361 0.411516 0.205758 0.978603i \(-0.434034\pi\)
0.205758 + 0.978603i \(0.434034\pi\)
\(422\) 0 0
\(423\) −23.3003 −1.13290
\(424\) 0 0
\(425\) −2.22829 −0.108088
\(426\) 0 0
\(427\) −0.891443 −0.0431400
\(428\) 0 0
\(429\) −13.2205 −0.638291
\(430\) 0 0
\(431\) −35.3698 −1.70370 −0.851851 0.523785i \(-0.824520\pi\)
−0.851851 + 0.523785i \(0.824520\pi\)
\(432\) 0 0
\(433\) 13.3923 0.643595 0.321797 0.946809i \(-0.395713\pi\)
0.321797 + 0.946809i \(0.395713\pi\)
\(434\) 0 0
\(435\) 8.21352 0.393808
\(436\) 0 0
\(437\) −1.22829 −0.0587573
\(438\) 0 0
\(439\) 21.1137 1.00770 0.503852 0.863790i \(-0.331916\pi\)
0.503852 + 0.863790i \(0.331916\pi\)
\(440\) 0 0
\(441\) 15.5651 0.741197
\(442\) 0 0
\(443\) −31.1692 −1.48089 −0.740447 0.672115i \(-0.765386\pi\)
−0.740447 + 0.672115i \(0.765386\pi\)
\(444\) 0 0
\(445\) 0.486128 0.0230447
\(446\) 0 0
\(447\) 13.8046 0.652936
\(448\) 0 0
\(449\) −6.09199 −0.287499 −0.143749 0.989614i \(-0.545916\pi\)
−0.143749 + 0.989614i \(0.545916\pi\)
\(450\) 0 0
\(451\) 23.6849 1.11528
\(452\) 0 0
\(453\) 7.37240 0.346386
\(454\) 0 0
\(455\) −0.635404 −0.0297882
\(456\) 0 0
\(457\) 22.1345 1.03541 0.517704 0.855560i \(-0.326787\pi\)
0.517704 + 0.855560i \(0.326787\pi\)
\(458\) 0 0
\(459\) 10.2343 0.477697
\(460\) 0 0
\(461\) 11.5434 0.537630 0.268815 0.963192i \(-0.413368\pi\)
0.268815 + 0.963192i \(0.413368\pi\)
\(462\) 0 0
\(463\) −30.4270 −1.41406 −0.707032 0.707181i \(-0.749967\pi\)
−0.707032 + 0.707181i \(0.749967\pi\)
\(464\) 0 0
\(465\) −1.86370 −0.0864269
\(466\) 0 0
\(467\) 10.3776 0.480217 0.240108 0.970746i \(-0.422817\pi\)
0.240108 + 0.970746i \(0.422817\pi\)
\(468\) 0 0
\(469\) −0.679714 −0.0313863
\(470\) 0 0
\(471\) 17.1858 0.791879
\(472\) 0 0
\(473\) 23.0277 1.05882
\(474\) 0 0
\(475\) 1.22829 0.0563580
\(476\) 0 0
\(477\) −8.00601 −0.366570
\(478\) 0 0
\(479\) 29.6128 1.35304 0.676522 0.736422i \(-0.263486\pi\)
0.676522 + 0.736422i \(0.263486\pi\)
\(480\) 0 0
\(481\) 29.2413 1.33329
\(482\) 0 0
\(483\) 0.106762 0.00485783
\(484\) 0 0
\(485\) −0.635404 −0.0288522
\(486\) 0 0
\(487\) −11.1562 −0.505537 −0.252769 0.967527i \(-0.581341\pi\)
−0.252769 + 0.967527i \(0.581341\pi\)
\(488\) 0 0
\(489\) 0.891443 0.0403125
\(490\) 0 0
\(491\) 38.2630 1.72679 0.863393 0.504533i \(-0.168335\pi\)
0.863393 + 0.504533i \(0.168335\pi\)
\(492\) 0 0
\(493\) 20.8342 0.938323
\(494\) 0 0
\(495\) 6.41407 0.288291
\(496\) 0 0
\(497\) −1.69965 −0.0762398
\(498\) 0 0
\(499\) 6.40711 0.286822 0.143411 0.989663i \(-0.454193\pi\)
0.143411 + 0.989663i \(0.454193\pi\)
\(500\) 0 0
\(501\) 7.02774 0.313976
\(502\) 0 0
\(503\) 18.0920 0.806682 0.403341 0.915050i \(-0.367849\pi\)
0.403341 + 0.915050i \(0.367849\pi\)
\(504\) 0 0
\(505\) 13.3498 0.594059
\(506\) 0 0
\(507\) 12.5929 0.559270
\(508\) 0 0
\(509\) 34.1285 1.51272 0.756359 0.654156i \(-0.226976\pi\)
0.756359 + 0.654156i \(0.226976\pi\)
\(510\) 0 0
\(511\) −1.57295 −0.0695833
\(512\) 0 0
\(513\) −5.64142 −0.249075
\(514\) 0 0
\(515\) −7.33506 −0.323221
\(516\) 0 0
\(517\) 30.0990 1.32375
\(518\) 0 0
\(519\) 2.31512 0.101622
\(520\) 0 0
\(521\) 35.8854 1.57217 0.786085 0.618119i \(-0.212105\pi\)
0.786085 + 0.618119i \(0.212105\pi\)
\(522\) 0 0
\(523\) 11.7865 0.515387 0.257693 0.966227i \(-0.417038\pi\)
0.257693 + 0.966227i \(0.417038\pi\)
\(524\) 0 0
\(525\) −0.106762 −0.00465947
\(526\) 0 0
\(527\) −4.72740 −0.205929
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.44878 0.0628717
\(532\) 0 0
\(533\) −43.0199 −1.86340
\(534\) 0 0
\(535\) −16.0495 −0.693879
\(536\) 0 0
\(537\) −12.6997 −0.548030
\(538\) 0 0
\(539\) −20.1068 −0.866060
\(540\) 0 0
\(541\) 17.2413 0.741260 0.370630 0.928781i \(-0.379142\pi\)
0.370630 + 0.928781i \(0.379142\pi\)
\(542\) 0 0
\(543\) −9.43665 −0.404965
\(544\) 0 0
\(545\) 1.01477 0.0434680
\(546\) 0 0
\(547\) 14.6571 0.626694 0.313347 0.949639i \(-0.398550\pi\)
0.313347 + 0.949639i \(0.398550\pi\)
\(548\) 0 0
\(549\) −16.3447 −0.697573
\(550\) 0 0
\(551\) −11.4843 −0.489249
\(552\) 0 0
\(553\) −0.427048 −0.0181599
\(554\) 0 0
\(555\) 4.91318 0.208553
\(556\) 0 0
\(557\) −2.07902 −0.0880908 −0.0440454 0.999030i \(-0.514025\pi\)
−0.0440454 + 0.999030i \(0.514025\pi\)
\(558\) 0 0
\(559\) −41.8264 −1.76907
\(560\) 0 0
\(561\) −5.63456 −0.237891
\(562\) 0 0
\(563\) −5.10676 −0.215224 −0.107612 0.994193i \(-0.534320\pi\)
−0.107612 + 0.994193i \(0.534320\pi\)
\(564\) 0 0
\(565\) −17.3203 −0.728670
\(566\) 0 0
\(567\) −0.322081 −0.0135261
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 10.0642 0.421176 0.210588 0.977575i \(-0.432462\pi\)
0.210588 + 0.977575i \(0.432462\pi\)
\(572\) 0 0
\(573\) 19.9150 0.831960
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −15.7274 −0.654740 −0.327370 0.944896i \(-0.606162\pi\)
−0.327370 + 0.944896i \(0.606162\pi\)
\(578\) 0 0
\(579\) 1.97046 0.0818895
\(580\) 0 0
\(581\) 1.34983 0.0560002
\(582\) 0 0
\(583\) 10.3420 0.428323
\(584\) 0 0
\(585\) −11.6502 −0.481675
\(586\) 0 0
\(587\) −18.7344 −0.773250 −0.386625 0.922237i \(-0.626359\pi\)
−0.386625 + 0.922237i \(0.626359\pi\)
\(588\) 0 0
\(589\) 2.60586 0.107373
\(590\) 0 0
\(591\) 6.65714 0.273838
\(592\) 0 0
\(593\) −3.21532 −0.132037 −0.0660187 0.997818i \(-0.521030\pi\)
−0.0660187 + 0.997818i \(0.521030\pi\)
\(594\) 0 0
\(595\) −0.270809 −0.0111021
\(596\) 0 0
\(597\) 12.2986 0.503346
\(598\) 0 0
\(599\) −22.7639 −0.930108 −0.465054 0.885282i \(-0.653965\pi\)
−0.465054 + 0.885282i \(0.653965\pi\)
\(600\) 0 0
\(601\) 4.30552 0.175626 0.0878128 0.996137i \(-0.472012\pi\)
0.0878128 + 0.996137i \(0.472012\pi\)
\(602\) 0 0
\(603\) −12.4626 −0.507516
\(604\) 0 0
\(605\) 2.71442 0.110357
\(606\) 0 0
\(607\) −11.0868 −0.450000 −0.225000 0.974359i \(-0.572238\pi\)
−0.225000 + 0.974359i \(0.572238\pi\)
\(608\) 0 0
\(609\) 0.998205 0.0404493
\(610\) 0 0
\(611\) −54.6701 −2.21172
\(612\) 0 0
\(613\) −29.9409 −1.20930 −0.604651 0.796490i \(-0.706687\pi\)
−0.604651 + 0.796490i \(0.706687\pi\)
\(614\) 0 0
\(615\) −7.22829 −0.291473
\(616\) 0 0
\(617\) 35.0130 1.40957 0.704785 0.709421i \(-0.251044\pi\)
0.704785 + 0.709421i \(0.251044\pi\)
\(618\) 0 0
\(619\) 4.98523 0.200373 0.100187 0.994969i \(-0.468056\pi\)
0.100187 + 0.994969i \(0.468056\pi\)
\(620\) 0 0
\(621\) 4.59289 0.184306
\(622\) 0 0
\(623\) 0.0590800 0.00236699
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.10592 0.124038
\(628\) 0 0
\(629\) 12.4626 0.496916
\(630\) 0 0
\(631\) 3.64237 0.145000 0.0725002 0.997368i \(-0.476902\pi\)
0.0725002 + 0.997368i \(0.476902\pi\)
\(632\) 0 0
\(633\) −2.72919 −0.108476
\(634\) 0 0
\(635\) 1.75694 0.0697219
\(636\) 0 0
\(637\) 36.5208 1.44701
\(638\) 0 0
\(639\) −31.1632 −1.23280
\(640\) 0 0
\(641\) 38.6997 1.52854 0.764272 0.644894i \(-0.223098\pi\)
0.764272 + 0.644894i \(0.223098\pi\)
\(642\) 0 0
\(643\) −28.8637 −1.13827 −0.569137 0.822243i \(-0.692722\pi\)
−0.569137 + 0.822243i \(0.692722\pi\)
\(644\) 0 0
\(645\) −7.02774 −0.276717
\(646\) 0 0
\(647\) 19.8854 0.781777 0.390888 0.920438i \(-0.372168\pi\)
0.390888 + 0.920438i \(0.372168\pi\)
\(648\) 0 0
\(649\) −1.87151 −0.0734630
\(650\) 0 0
\(651\) −0.226499 −0.00887719
\(652\) 0 0
\(653\) 27.1988 1.06437 0.532185 0.846628i \(-0.321371\pi\)
0.532185 + 0.846628i \(0.321371\pi\)
\(654\) 0 0
\(655\) 20.2135 0.789808
\(656\) 0 0
\(657\) −28.8402 −1.12516
\(658\) 0 0
\(659\) −17.7274 −0.690561 −0.345281 0.938499i \(-0.612216\pi\)
−0.345281 + 0.938499i \(0.612216\pi\)
\(660\) 0 0
\(661\) −40.7048 −1.58323 −0.791617 0.611018i \(-0.790760\pi\)
−0.791617 + 0.611018i \(0.790760\pi\)
\(662\) 0 0
\(663\) 10.2343 0.397468
\(664\) 0 0
\(665\) 0.149277 0.00578871
\(666\) 0 0
\(667\) 9.34983 0.362027
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 21.1137 0.815086
\(672\) 0 0
\(673\) 12.2135 0.470797 0.235398 0.971899i \(-0.424361\pi\)
0.235398 + 0.971899i \(0.424361\pi\)
\(674\) 0 0
\(675\) −4.59289 −0.176780
\(676\) 0 0
\(677\) −16.5061 −0.634380 −0.317190 0.948362i \(-0.602739\pi\)
−0.317190 + 0.948362i \(0.602739\pi\)
\(678\) 0 0
\(679\) −0.0772219 −0.00296350
\(680\) 0 0
\(681\) 12.2726 0.470287
\(682\) 0 0
\(683\) −29.1988 −1.11726 −0.558630 0.829417i \(-0.688673\pi\)
−0.558630 + 0.829417i \(0.688673\pi\)
\(684\) 0 0
\(685\) 17.9279 0.684992
\(686\) 0 0
\(687\) −10.7292 −0.409344
\(688\) 0 0
\(689\) −18.7847 −0.715639
\(690\) 0 0
\(691\) −15.9150 −0.605434 −0.302717 0.953080i \(-0.597894\pi\)
−0.302717 + 0.953080i \(0.597894\pi\)
\(692\) 0 0
\(693\) 0.779514 0.0296113
\(694\) 0 0
\(695\) 10.8637 0.412084
\(696\) 0 0
\(697\) −18.3351 −0.694490
\(698\) 0 0
\(699\) 8.57116 0.324191
\(700\) 0 0
\(701\) −34.5981 −1.30675 −0.653375 0.757034i \(-0.726648\pi\)
−0.653375 + 0.757034i \(0.726648\pi\)
\(702\) 0 0
\(703\) −6.86971 −0.259096
\(704\) 0 0
\(705\) −9.18578 −0.345956
\(706\) 0 0
\(707\) 1.62243 0.0610177
\(708\) 0 0
\(709\) −35.8212 −1.34529 −0.672646 0.739964i \(-0.734842\pi\)
−0.672646 + 0.739964i \(0.734842\pi\)
\(710\) 0 0
\(711\) −7.82994 −0.293646
\(712\) 0 0
\(713\) −2.12153 −0.0794520
\(714\) 0 0
\(715\) 15.0495 0.562819
\(716\) 0 0
\(717\) −7.57295 −0.282817
\(718\) 0 0
\(719\) 32.5781 1.21496 0.607479 0.794335i \(-0.292181\pi\)
0.607479 + 0.794335i \(0.292181\pi\)
\(720\) 0 0
\(721\) −0.891443 −0.0331991
\(722\) 0 0
\(723\) 2.79861 0.104081
\(724\) 0 0
\(725\) −9.34983 −0.347244
\(726\) 0 0
\(727\) 31.9557 1.18517 0.592585 0.805508i \(-0.298107\pi\)
0.592585 + 0.805508i \(0.298107\pi\)
\(728\) 0 0
\(729\) 6.19875 0.229583
\(730\) 0 0
\(731\) −17.8264 −0.659331
\(732\) 0 0
\(733\) 2.19359 0.0810220 0.0405110 0.999179i \(-0.487101\pi\)
0.0405110 + 0.999179i \(0.487101\pi\)
\(734\) 0 0
\(735\) 6.13630 0.226341
\(736\) 0 0
\(737\) 16.0990 0.593013
\(738\) 0 0
\(739\) −51.7473 −1.90356 −0.951778 0.306787i \(-0.900746\pi\)
−0.951778 + 0.306787i \(0.900746\pi\)
\(740\) 0 0
\(741\) −5.64142 −0.207243
\(742\) 0 0
\(743\) 14.4436 0.529885 0.264942 0.964264i \(-0.414647\pi\)
0.264942 + 0.964264i \(0.414647\pi\)
\(744\) 0 0
\(745\) −15.7144 −0.575732
\(746\) 0 0
\(747\) 24.7491 0.905523
\(748\) 0 0
\(749\) −1.95052 −0.0712706
\(750\) 0 0
\(751\) −17.4288 −0.635987 −0.317994 0.948093i \(-0.603009\pi\)
−0.317994 + 0.948093i \(0.603009\pi\)
\(752\) 0 0
\(753\) −22.1623 −0.807637
\(754\) 0 0
\(755\) −8.39234 −0.305429
\(756\) 0 0
\(757\) 34.9331 1.26967 0.634833 0.772650i \(-0.281069\pi\)
0.634833 + 0.772650i \(0.281069\pi\)
\(758\) 0 0
\(759\) −2.52864 −0.0917839
\(760\) 0 0
\(761\) −23.2482 −0.842748 −0.421374 0.906887i \(-0.638452\pi\)
−0.421374 + 0.906887i \(0.638452\pi\)
\(762\) 0 0
\(763\) 0.123327 0.00446473
\(764\) 0 0
\(765\) −4.96529 −0.179521
\(766\) 0 0
\(767\) 3.39930 0.122742
\(768\) 0 0
\(769\) −48.6997 −1.75615 −0.878077 0.478519i \(-0.841174\pi\)
−0.878077 + 0.478519i \(0.841174\pi\)
\(770\) 0 0
\(771\) −9.37335 −0.337573
\(772\) 0 0
\(773\) 15.6128 0.561554 0.280777 0.959773i \(-0.409408\pi\)
0.280777 + 0.959773i \(0.409408\pi\)
\(774\) 0 0
\(775\) 2.12153 0.0762077
\(776\) 0 0
\(777\) 0.597107 0.0214211
\(778\) 0 0
\(779\) 10.1068 0.362112
\(780\) 0 0
\(781\) 40.2560 1.44047
\(782\) 0 0
\(783\) 42.9427 1.53465
\(784\) 0 0
\(785\) −19.5633 −0.698246
\(786\) 0 0
\(787\) 24.2630 0.864883 0.432441 0.901662i \(-0.357652\pi\)
0.432441 + 0.901662i \(0.357652\pi\)
\(788\) 0 0
\(789\) 16.4141 0.584356
\(790\) 0 0
\(791\) −2.10497 −0.0748440
\(792\) 0 0
\(793\) −38.3498 −1.36184
\(794\) 0 0
\(795\) −3.15624 −0.111940
\(796\) 0 0
\(797\) −44.5911 −1.57950 −0.789749 0.613430i \(-0.789789\pi\)
−0.789749 + 0.613430i \(0.789789\pi\)
\(798\) 0 0
\(799\) −23.3003 −0.824307
\(800\) 0 0
\(801\) 1.08323 0.0382742
\(802\) 0 0
\(803\) 37.2552 1.31471
\(804\) 0 0
\(805\) −0.121532 −0.00428344
\(806\) 0 0
\(807\) 21.8420 0.768874
\(808\) 0 0
\(809\) −44.7144 −1.57208 −0.786038 0.618179i \(-0.787871\pi\)
−0.786038 + 0.618179i \(0.787871\pi\)
\(810\) 0 0
\(811\) 1.19179 0.0418495 0.0209247 0.999781i \(-0.493339\pi\)
0.0209247 + 0.999781i \(0.493339\pi\)
\(812\) 0 0
\(813\) −0.507859 −0.0178114
\(814\) 0 0
\(815\) −1.01477 −0.0355458
\(816\) 0 0
\(817\) 9.82635 0.343780
\(818\) 0 0
\(819\) −1.41587 −0.0494744
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) −26.7551 −0.932626 −0.466313 0.884620i \(-0.654418\pi\)
−0.466313 + 0.884620i \(0.654418\pi\)
\(824\) 0 0
\(825\) 2.52864 0.0880360
\(826\) 0 0
\(827\) −27.2907 −0.948992 −0.474496 0.880258i \(-0.657370\pi\)
−0.474496 + 0.880258i \(0.657370\pi\)
\(828\) 0 0
\(829\) −34.2925 −1.19103 −0.595515 0.803344i \(-0.703052\pi\)
−0.595515 + 0.803344i \(0.703052\pi\)
\(830\) 0 0
\(831\) 8.35763 0.289923
\(832\) 0 0
\(833\) 15.5651 0.539300
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) −9.74396 −0.336801
\(838\) 0 0
\(839\) 40.6701 1.40409 0.702044 0.712133i \(-0.252271\pi\)
0.702044 + 0.712133i \(0.252271\pi\)
\(840\) 0 0
\(841\) 58.4192 2.01446
\(842\) 0 0
\(843\) −11.5573 −0.398056
\(844\) 0 0
\(845\) −14.3351 −0.493141
\(846\) 0 0
\(847\) 0.329889 0.0113351
\(848\) 0 0
\(849\) 21.5278 0.738832
\(850\) 0 0
\(851\) 5.59289 0.191722
\(852\) 0 0
\(853\) 34.6276 1.18563 0.592813 0.805340i \(-0.298017\pi\)
0.592813 + 0.805340i \(0.298017\pi\)
\(854\) 0 0
\(855\) 2.73700 0.0936034
\(856\) 0 0
\(857\) 26.4861 0.904749 0.452374 0.891828i \(-0.350577\pi\)
0.452374 + 0.891828i \(0.350577\pi\)
\(858\) 0 0
\(859\) 40.9627 1.39763 0.698814 0.715304i \(-0.253712\pi\)
0.698814 + 0.715304i \(0.253712\pi\)
\(860\) 0 0
\(861\) −0.878468 −0.0299381
\(862\) 0 0
\(863\) 18.5712 0.632170 0.316085 0.948731i \(-0.397632\pi\)
0.316085 + 0.948731i \(0.397632\pi\)
\(864\) 0 0
\(865\) −2.63540 −0.0896064
\(866\) 0 0
\(867\) −10.5721 −0.359048
\(868\) 0 0
\(869\) 10.1146 0.343113
\(870\) 0 0
\(871\) −29.2413 −0.990803
\(872\) 0 0
\(873\) −1.41587 −0.0479199
\(874\) 0 0
\(875\) 0.121532 0.00410853
\(876\) 0 0
\(877\) 28.4913 0.962083 0.481041 0.876698i \(-0.340259\pi\)
0.481041 + 0.876698i \(0.340259\pi\)
\(878\) 0 0
\(879\) 5.52780 0.186448
\(880\) 0 0
\(881\) −4.91318 −0.165529 −0.0827645 0.996569i \(-0.526375\pi\)
−0.0827645 + 0.996569i \(0.526375\pi\)
\(882\) 0 0
\(883\) 30.1710 1.01534 0.507668 0.861553i \(-0.330508\pi\)
0.507668 + 0.861553i \(0.330508\pi\)
\(884\) 0 0
\(885\) 0.571157 0.0191992
\(886\) 0 0
\(887\) −25.7865 −0.865825 −0.432913 0.901436i \(-0.642514\pi\)
−0.432913 + 0.901436i \(0.642514\pi\)
\(888\) 0 0
\(889\) 0.213524 0.00716136
\(890\) 0 0
\(891\) 7.62844 0.255562
\(892\) 0 0
\(893\) 12.8438 0.429800
\(894\) 0 0
\(895\) 14.4566 0.483230
\(896\) 0 0
\(897\) 4.59289 0.153352
\(898\) 0 0
\(899\) −19.8360 −0.661566
\(900\) 0 0
\(901\) −8.00601 −0.266719
\(902\) 0 0
\(903\) −0.854095 −0.0284225
\(904\) 0 0
\(905\) 10.7422 0.357082
\(906\) 0 0
\(907\) −44.3776 −1.47353 −0.736767 0.676147i \(-0.763648\pi\)
−0.736767 + 0.676147i \(0.763648\pi\)
\(908\) 0 0
\(909\) 29.7473 0.986657
\(910\) 0 0
\(911\) −47.7968 −1.58358 −0.791789 0.610794i \(-0.790850\pi\)
−0.791789 + 0.610794i \(0.790850\pi\)
\(912\) 0 0
\(913\) −31.9705 −1.05807
\(914\) 0 0
\(915\) −6.44361 −0.213019
\(916\) 0 0
\(917\) 2.45659 0.0811237
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 11.7144 0.386003
\(922\) 0 0
\(923\) −73.1189 −2.40674
\(924\) 0 0
\(925\) −5.59289 −0.183893
\(926\) 0 0
\(927\) −16.3447 −0.536829
\(928\) 0 0
\(929\) 24.0199 0.788069 0.394034 0.919096i \(-0.371079\pi\)
0.394034 + 0.919096i \(0.371079\pi\)
\(930\) 0 0
\(931\) −8.57991 −0.281195
\(932\) 0 0
\(933\) 21.4584 0.702516
\(934\) 0 0
\(935\) 6.41407 0.209763
\(936\) 0 0
\(937\) 0.384533 0.0125621 0.00628107 0.999980i \(-0.498001\pi\)
0.00628107 + 0.999980i \(0.498001\pi\)
\(938\) 0 0
\(939\) 23.3221 0.761087
\(940\) 0 0
\(941\) −25.0920 −0.817976 −0.408988 0.912540i \(-0.634118\pi\)
−0.408988 + 0.912540i \(0.634118\pi\)
\(942\) 0 0
\(943\) −8.22829 −0.267950
\(944\) 0 0
\(945\) −0.558182 −0.0181577
\(946\) 0 0
\(947\) 44.7126 1.45297 0.726483 0.687185i \(-0.241154\pi\)
0.726483 + 0.687185i \(0.241154\pi\)
\(948\) 0 0
\(949\) −67.6683 −2.19661
\(950\) 0 0
\(951\) 14.9210 0.483846
\(952\) 0 0
\(953\) −21.2500 −0.688356 −0.344178 0.938904i \(-0.611842\pi\)
−0.344178 + 0.938904i \(0.611842\pi\)
\(954\) 0 0
\(955\) −22.6701 −0.733588
\(956\) 0 0
\(957\) −23.6424 −0.764249
\(958\) 0 0
\(959\) 2.17882 0.0703577
\(960\) 0 0
\(961\) −26.4991 −0.854810
\(962\) 0 0
\(963\) −35.7629 −1.15244
\(964\) 0 0
\(965\) −2.24306 −0.0722068
\(966\) 0 0
\(967\) 41.0417 1.31981 0.659906 0.751349i \(-0.270596\pi\)
0.659906 + 0.751349i \(0.270596\pi\)
\(968\) 0 0
\(969\) −2.40437 −0.0772394
\(970\) 0 0
\(971\) 35.2760 1.13206 0.566030 0.824385i \(-0.308479\pi\)
0.566030 + 0.824385i \(0.308479\pi\)
\(972\) 0 0
\(973\) 1.32029 0.0423264
\(974\) 0 0
\(975\) −4.59289 −0.147090
\(976\) 0 0
\(977\) −6.79164 −0.217284 −0.108642 0.994081i \(-0.534650\pi\)
−0.108642 + 0.994081i \(0.534650\pi\)
\(978\) 0 0
\(979\) −1.39930 −0.0447219
\(980\) 0 0
\(981\) 2.26121 0.0721947
\(982\) 0 0
\(983\) −19.5981 −0.625081 −0.312540 0.949904i \(-0.601180\pi\)
−0.312540 + 0.949904i \(0.601180\pi\)
\(984\) 0 0
\(985\) −7.57812 −0.241459
\(986\) 0 0
\(987\) −1.11636 −0.0355343
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 37.2995 1.18486 0.592429 0.805623i \(-0.298169\pi\)
0.592429 + 0.805623i \(0.298169\pi\)
\(992\) 0 0
\(993\) 15.7171 0.498766
\(994\) 0 0
\(995\) −14.0000 −0.443830
\(996\) 0 0
\(997\) 19.7274 0.624773 0.312386 0.949955i \(-0.398872\pi\)
0.312386 + 0.949955i \(0.398872\pi\)
\(998\) 0 0
\(999\) 25.6875 0.812717
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 7360.2.a.cc.1.2 3
4.3 odd 2 7360.2.a.by.1.2 3
8.3 odd 2 920.2.a.h.1.2 3
8.5 even 2 1840.2.a.s.1.2 3
24.11 even 2 8280.2.a.bj.1.2 3
40.3 even 4 4600.2.e.p.4049.4 6
40.19 odd 2 4600.2.a.x.1.2 3
40.27 even 4 4600.2.e.p.4049.3 6
40.29 even 2 9200.2.a.ce.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.h.1.2 3 8.3 odd 2
1840.2.a.s.1.2 3 8.5 even 2
4600.2.a.x.1.2 3 40.19 odd 2
4600.2.e.p.4049.3 6 40.27 even 4
4600.2.e.p.4049.4 6 40.3 even 4
7360.2.a.by.1.2 3 4.3 odd 2
7360.2.a.cc.1.2 3 1.1 even 1 trivial
8280.2.a.bj.1.2 3 24.11 even 2
9200.2.a.ce.1.2 3 40.29 even 2