Properties

Label 6080.2.a.cl.1.1
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
Analytic rank $0$
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] = [6080,2,Mod(1,6080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6080, 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("6080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.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: \(48.5490444289\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.387268.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3040)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.160536\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.13476 q^{3} +1.00000 q^{5} +0.878290 q^{7} +1.55722 q^{9} +O(q^{10})\) \(q-2.13476 q^{3} +1.00000 q^{5} +0.878290 q^{7} +1.55722 q^{9} +4.50981 q^{11} +2.05398 q^{13} -2.13476 q^{15} -4.31693 q^{17} -1.00000 q^{19} -1.87494 q^{21} -2.15430 q^{23} +1.00000 q^{25} +3.08000 q^{27} -9.01549 q^{29} +4.35137 q^{31} -9.62738 q^{33} +0.878290 q^{35} +8.67901 q^{37} -4.38475 q^{39} +4.26953 q^{41} +12.3806 q^{43} +1.55722 q^{45} +2.76620 q^{47} -6.22861 q^{49} +9.21564 q^{51} +3.98510 q^{53} +4.50981 q^{55} +2.13476 q^{57} +11.5800 q^{59} +1.51630 q^{61} +1.36769 q^{63} +2.05398 q^{65} +1.10217 q^{67} +4.59893 q^{69} -14.4854 q^{71} -6.03444 q^{73} -2.13476 q^{75} +3.96092 q^{77} +7.34489 q^{79} -11.2467 q^{81} -1.34175 q^{83} -4.31693 q^{85} +19.2459 q^{87} -15.5455 q^{89} +1.80399 q^{91} -9.28915 q^{93} -1.00000 q^{95} +7.55630 q^{97} +7.02276 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{15} - 12 q^{17} - 5 q^{19} + 10 q^{21} + 8 q^{23} + 5 q^{25} + 16 q^{27} + 6 q^{29} + 10 q^{31} - 18 q^{33} + 4 q^{35} + 6 q^{37} + 18 q^{39} - 8 q^{41} + 12 q^{43} + 7 q^{45} + 16 q^{47} + 7 q^{49} - 14 q^{51} + 18 q^{53} + 2 q^{55} - 4 q^{57} + 8 q^{59} - 2 q^{61} + 36 q^{63} + 4 q^{65} + 10 q^{67} + 22 q^{69} - 18 q^{71} - 28 q^{73} + 4 q^{75} + 28 q^{77} + 14 q^{79} + 25 q^{81} + 8 q^{83} - 12 q^{85} + 24 q^{87} - 30 q^{89} + 28 q^{91} + 24 q^{93} - 5 q^{95} - 18 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) −2.13476 −1.23251 −0.616253 0.787548i \(-0.711350\pi\)
−0.616253 + 0.787548i \(0.711350\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.878290 0.331962 0.165981 0.986129i \(-0.446921\pi\)
0.165981 + 0.986129i \(0.446921\pi\)
\(8\) 0 0
\(9\) 1.55722 0.519073
\(10\) 0 0
\(11\) 4.50981 1.35976 0.679880 0.733324i \(-0.262032\pi\)
0.679880 + 0.733324i \(0.262032\pi\)
\(12\) 0 0
\(13\) 2.05398 0.569670 0.284835 0.958577i \(-0.408061\pi\)
0.284835 + 0.958577i \(0.408061\pi\)
\(14\) 0 0
\(15\) −2.13476 −0.551194
\(16\) 0 0
\(17\) −4.31693 −1.04701 −0.523505 0.852022i \(-0.675376\pi\)
−0.523505 + 0.852022i \(0.675376\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.87494 −0.409146
\(22\) 0 0
\(23\) −2.15430 −0.449203 −0.224602 0.974451i \(-0.572108\pi\)
−0.224602 + 0.974451i \(0.572108\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.08000 0.592746
\(28\) 0 0
\(29\) −9.01549 −1.67413 −0.837067 0.547101i \(-0.815732\pi\)
−0.837067 + 0.547101i \(0.815732\pi\)
\(30\) 0 0
\(31\) 4.35137 0.781529 0.390765 0.920491i \(-0.372211\pi\)
0.390765 + 0.920491i \(0.372211\pi\)
\(32\) 0 0
\(33\) −9.62738 −1.67591
\(34\) 0 0
\(35\) 0.878290 0.148458
\(36\) 0 0
\(37\) 8.67901 1.42682 0.713410 0.700746i \(-0.247150\pi\)
0.713410 + 0.700746i \(0.247150\pi\)
\(38\) 0 0
\(39\) −4.38475 −0.702123
\(40\) 0 0
\(41\) 4.26953 0.666788 0.333394 0.942788i \(-0.391806\pi\)
0.333394 + 0.942788i \(0.391806\pi\)
\(42\) 0 0
\(43\) 12.3806 1.88803 0.944013 0.329908i \(-0.107018\pi\)
0.944013 + 0.329908i \(0.107018\pi\)
\(44\) 0 0
\(45\) 1.55722 0.232136
\(46\) 0 0
\(47\) 2.76620 0.403492 0.201746 0.979438i \(-0.435338\pi\)
0.201746 + 0.979438i \(0.435338\pi\)
\(48\) 0 0
\(49\) −6.22861 −0.889801
\(50\) 0 0
\(51\) 9.21564 1.29045
\(52\) 0 0
\(53\) 3.98510 0.547396 0.273698 0.961816i \(-0.411753\pi\)
0.273698 + 0.961816i \(0.411753\pi\)
\(54\) 0 0
\(55\) 4.50981 0.608103
\(56\) 0 0
\(57\) 2.13476 0.282756
\(58\) 0 0
\(59\) 11.5800 1.50758 0.753792 0.657113i \(-0.228222\pi\)
0.753792 + 0.657113i \(0.228222\pi\)
\(60\) 0 0
\(61\) 1.51630 0.194142 0.0970709 0.995277i \(-0.469053\pi\)
0.0970709 + 0.995277i \(0.469053\pi\)
\(62\) 0 0
\(63\) 1.36769 0.172313
\(64\) 0 0
\(65\) 2.05398 0.254764
\(66\) 0 0
\(67\) 1.10217 0.134652 0.0673258 0.997731i \(-0.478553\pi\)
0.0673258 + 0.997731i \(0.478553\pi\)
\(68\) 0 0
\(69\) 4.59893 0.553646
\(70\) 0 0
\(71\) −14.4854 −1.71910 −0.859552 0.511048i \(-0.829257\pi\)
−0.859552 + 0.511048i \(0.829257\pi\)
\(72\) 0 0
\(73\) −6.03444 −0.706277 −0.353139 0.935571i \(-0.614886\pi\)
−0.353139 + 0.935571i \(0.614886\pi\)
\(74\) 0 0
\(75\) −2.13476 −0.246501
\(76\) 0 0
\(77\) 3.96092 0.451389
\(78\) 0 0
\(79\) 7.34489 0.826364 0.413182 0.910648i \(-0.364417\pi\)
0.413182 + 0.910648i \(0.364417\pi\)
\(80\) 0 0
\(81\) −11.2467 −1.24964
\(82\) 0 0
\(83\) −1.34175 −0.147276 −0.0736381 0.997285i \(-0.523461\pi\)
−0.0736381 + 0.997285i \(0.523461\pi\)
\(84\) 0 0
\(85\) −4.31693 −0.468237
\(86\) 0 0
\(87\) 19.2459 2.06338
\(88\) 0 0
\(89\) −15.5455 −1.64782 −0.823912 0.566718i \(-0.808213\pi\)
−0.823912 + 0.566718i \(0.808213\pi\)
\(90\) 0 0
\(91\) 1.80399 0.189109
\(92\) 0 0
\(93\) −9.28915 −0.963240
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 7.55630 0.767226 0.383613 0.923494i \(-0.374680\pi\)
0.383613 + 0.923494i \(0.374680\pi\)
\(98\) 0 0
\(99\) 7.02276 0.705814
\(100\) 0 0
\(101\) 8.50981 0.846758 0.423379 0.905953i \(-0.360844\pi\)
0.423379 + 0.905953i \(0.360844\pi\)
\(102\) 0 0
\(103\) −0.763993 −0.0752784 −0.0376392 0.999291i \(-0.511984\pi\)
−0.0376392 + 0.999291i \(0.511984\pi\)
\(104\) 0 0
\(105\) −1.87494 −0.182976
\(106\) 0 0
\(107\) −16.4092 −1.58633 −0.793167 0.609005i \(-0.791569\pi\)
−0.793167 + 0.609005i \(0.791569\pi\)
\(108\) 0 0
\(109\) −19.0286 −1.82261 −0.911306 0.411730i \(-0.864925\pi\)
−0.911306 + 0.411730i \(0.864925\pi\)
\(110\) 0 0
\(111\) −18.5276 −1.75857
\(112\) 0 0
\(113\) 6.40121 0.602175 0.301088 0.953596i \(-0.402650\pi\)
0.301088 + 0.953596i \(0.402650\pi\)
\(114\) 0 0
\(115\) −2.15430 −0.200890
\(116\) 0 0
\(117\) 3.19849 0.295700
\(118\) 0 0
\(119\) −3.79152 −0.347568
\(120\) 0 0
\(121\) 9.33840 0.848946
\(122\) 0 0
\(123\) −9.11444 −0.821821
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.1135 1.25237 0.626184 0.779675i \(-0.284616\pi\)
0.626184 + 0.779675i \(0.284616\pi\)
\(128\) 0 0
\(129\) −26.4297 −2.32700
\(130\) 0 0
\(131\) −10.7289 −0.937384 −0.468692 0.883362i \(-0.655275\pi\)
−0.468692 + 0.883362i \(0.655275\pi\)
\(132\) 0 0
\(133\) −0.878290 −0.0761574
\(134\) 0 0
\(135\) 3.08000 0.265084
\(136\) 0 0
\(137\) −21.3366 −1.82291 −0.911453 0.411405i \(-0.865038\pi\)
−0.911453 + 0.411405i \(0.865038\pi\)
\(138\) 0 0
\(139\) −6.49684 −0.551055 −0.275527 0.961293i \(-0.588853\pi\)
−0.275527 + 0.961293i \(0.588853\pi\)
\(140\) 0 0
\(141\) −5.90519 −0.497306
\(142\) 0 0
\(143\) 9.26304 0.774615
\(144\) 0 0
\(145\) −9.01549 −0.748695
\(146\) 0 0
\(147\) 13.2966 1.09669
\(148\) 0 0
\(149\) 13.4904 1.10517 0.552587 0.833456i \(-0.313641\pi\)
0.552587 + 0.833456i \(0.313641\pi\)
\(150\) 0 0
\(151\) 19.7890 1.61040 0.805201 0.593001i \(-0.202057\pi\)
0.805201 + 0.593001i \(0.202057\pi\)
\(152\) 0 0
\(153\) −6.72241 −0.543475
\(154\) 0 0
\(155\) 4.35137 0.349511
\(156\) 0 0
\(157\) −2.37748 −0.189744 −0.0948718 0.995490i \(-0.530244\pi\)
−0.0948718 + 0.995490i \(0.530244\pi\)
\(158\) 0 0
\(159\) −8.50725 −0.674669
\(160\) 0 0
\(161\) −1.89210 −0.149119
\(162\) 0 0
\(163\) 8.86118 0.694061 0.347031 0.937854i \(-0.387190\pi\)
0.347031 + 0.937854i \(0.387190\pi\)
\(164\) 0 0
\(165\) −9.62738 −0.749491
\(166\) 0 0
\(167\) 6.26190 0.484561 0.242280 0.970206i \(-0.422105\pi\)
0.242280 + 0.970206i \(0.422105\pi\)
\(168\) 0 0
\(169\) −8.78118 −0.675476
\(170\) 0 0
\(171\) −1.55722 −0.119083
\(172\) 0 0
\(173\) 18.2673 1.38884 0.694419 0.719571i \(-0.255661\pi\)
0.694419 + 0.719571i \(0.255661\pi\)
\(174\) 0 0
\(175\) 0.878290 0.0663925
\(176\) 0 0
\(177\) −24.7205 −1.85811
\(178\) 0 0
\(179\) 14.1211 1.05546 0.527730 0.849412i \(-0.323043\pi\)
0.527730 + 0.849412i \(0.323043\pi\)
\(180\) 0 0
\(181\) 3.34489 0.248623 0.124312 0.992243i \(-0.460328\pi\)
0.124312 + 0.992243i \(0.460328\pi\)
\(182\) 0 0
\(183\) −3.23694 −0.239281
\(184\) 0 0
\(185\) 8.67901 0.638094
\(186\) 0 0
\(187\) −19.4686 −1.42368
\(188\) 0 0
\(189\) 2.70513 0.196769
\(190\) 0 0
\(191\) 3.76558 0.272468 0.136234 0.990677i \(-0.456500\pi\)
0.136234 + 0.990677i \(0.456500\pi\)
\(192\) 0 0
\(193\) 14.7505 1.06176 0.530881 0.847446i \(-0.321861\pi\)
0.530881 + 0.847446i \(0.321861\pi\)
\(194\) 0 0
\(195\) −4.38475 −0.313999
\(196\) 0 0
\(197\) 22.0976 1.57439 0.787193 0.616706i \(-0.211533\pi\)
0.787193 + 0.616706i \(0.211533\pi\)
\(198\) 0 0
\(199\) 13.2198 0.937129 0.468564 0.883429i \(-0.344771\pi\)
0.468564 + 0.883429i \(0.344771\pi\)
\(200\) 0 0
\(201\) −2.35288 −0.165959
\(202\) 0 0
\(203\) −7.91821 −0.555749
\(204\) 0 0
\(205\) 4.26953 0.298197
\(206\) 0 0
\(207\) −3.35472 −0.233169
\(208\) 0 0
\(209\) −4.50981 −0.311950
\(210\) 0 0
\(211\) −4.41192 −0.303729 −0.151864 0.988401i \(-0.548528\pi\)
−0.151864 + 0.988401i \(0.548528\pi\)
\(212\) 0 0
\(213\) 30.9230 2.11881
\(214\) 0 0
\(215\) 12.3806 0.844351
\(216\) 0 0
\(217\) 3.82177 0.259438
\(218\) 0 0
\(219\) 12.8821 0.870492
\(220\) 0 0
\(221\) −8.86688 −0.596451
\(222\) 0 0
\(223\) 26.0319 1.74322 0.871612 0.490196i \(-0.163075\pi\)
0.871612 + 0.490196i \(0.163075\pi\)
\(224\) 0 0
\(225\) 1.55722 0.103815
\(226\) 0 0
\(227\) 3.44727 0.228803 0.114402 0.993435i \(-0.463505\pi\)
0.114402 + 0.993435i \(0.463505\pi\)
\(228\) 0 0
\(229\) −17.1177 −1.13117 −0.565586 0.824689i \(-0.691350\pi\)
−0.565586 + 0.824689i \(0.691350\pi\)
\(230\) 0 0
\(231\) −8.45564 −0.556340
\(232\) 0 0
\(233\) 0.838212 0.0549131 0.0274565 0.999623i \(-0.491259\pi\)
0.0274565 + 0.999623i \(0.491259\pi\)
\(234\) 0 0
\(235\) 2.76620 0.180447
\(236\) 0 0
\(237\) −15.6796 −1.01850
\(238\) 0 0
\(239\) −9.10560 −0.588993 −0.294496 0.955653i \(-0.595152\pi\)
−0.294496 + 0.955653i \(0.595152\pi\)
\(240\) 0 0
\(241\) −10.4099 −0.670558 −0.335279 0.942119i \(-0.608831\pi\)
−0.335279 + 0.942119i \(0.608831\pi\)
\(242\) 0 0
\(243\) 14.7691 0.947439
\(244\) 0 0
\(245\) −6.22861 −0.397931
\(246\) 0 0
\(247\) −2.05398 −0.130691
\(248\) 0 0
\(249\) 2.86432 0.181519
\(250\) 0 0
\(251\) −18.1237 −1.14396 −0.571978 0.820269i \(-0.693824\pi\)
−0.571978 + 0.820269i \(0.693824\pi\)
\(252\) 0 0
\(253\) −9.71550 −0.610808
\(254\) 0 0
\(255\) 9.21564 0.577106
\(256\) 0 0
\(257\) 3.34726 0.208797 0.104398 0.994536i \(-0.466708\pi\)
0.104398 + 0.994536i \(0.466708\pi\)
\(258\) 0 0
\(259\) 7.62269 0.473651
\(260\) 0 0
\(261\) −14.0391 −0.868997
\(262\) 0 0
\(263\) −10.3379 −0.637459 −0.318730 0.947846i \(-0.603256\pi\)
−0.318730 + 0.947846i \(0.603256\pi\)
\(264\) 0 0
\(265\) 3.98510 0.244803
\(266\) 0 0
\(267\) 33.1861 2.03095
\(268\) 0 0
\(269\) −3.82804 −0.233400 −0.116700 0.993167i \(-0.537232\pi\)
−0.116700 + 0.993167i \(0.537232\pi\)
\(270\) 0 0
\(271\) 0.727573 0.0441969 0.0220985 0.999756i \(-0.492965\pi\)
0.0220985 + 0.999756i \(0.492965\pi\)
\(272\) 0 0
\(273\) −3.85109 −0.233078
\(274\) 0 0
\(275\) 4.50981 0.271952
\(276\) 0 0
\(277\) 20.8347 1.25184 0.625918 0.779889i \(-0.284725\pi\)
0.625918 + 0.779889i \(0.284725\pi\)
\(278\) 0 0
\(279\) 6.77603 0.405671
\(280\) 0 0
\(281\) 27.8475 1.66124 0.830620 0.556840i \(-0.187986\pi\)
0.830620 + 0.556840i \(0.187986\pi\)
\(282\) 0 0
\(283\) 29.1356 1.73193 0.865965 0.500104i \(-0.166705\pi\)
0.865965 + 0.500104i \(0.166705\pi\)
\(284\) 0 0
\(285\) 2.13476 0.126453
\(286\) 0 0
\(287\) 3.74988 0.221349
\(288\) 0 0
\(289\) 1.63592 0.0962309
\(290\) 0 0
\(291\) −16.1309 −0.945611
\(292\) 0 0
\(293\) −10.3105 −0.602348 −0.301174 0.953569i \(-0.597378\pi\)
−0.301174 + 0.953569i \(0.597378\pi\)
\(294\) 0 0
\(295\) 11.5800 0.674212
\(296\) 0 0
\(297\) 13.8902 0.805992
\(298\) 0 0
\(299\) −4.42489 −0.255898
\(300\) 0 0
\(301\) 10.8738 0.626754
\(302\) 0 0
\(303\) −18.1664 −1.04363
\(304\) 0 0
\(305\) 1.51630 0.0868229
\(306\) 0 0
\(307\) −15.6872 −0.895317 −0.447659 0.894205i \(-0.647742\pi\)
−0.447659 + 0.894205i \(0.647742\pi\)
\(308\) 0 0
\(309\) 1.63094 0.0927812
\(310\) 0 0
\(311\) 34.6721 1.96608 0.983038 0.183404i \(-0.0587116\pi\)
0.983038 + 0.183404i \(0.0587116\pi\)
\(312\) 0 0
\(313\) 21.9918 1.24305 0.621526 0.783393i \(-0.286513\pi\)
0.621526 + 0.783393i \(0.286513\pi\)
\(314\) 0 0
\(315\) 1.36769 0.0770605
\(316\) 0 0
\(317\) 20.3496 1.14295 0.571474 0.820620i \(-0.306372\pi\)
0.571474 + 0.820620i \(0.306372\pi\)
\(318\) 0 0
\(319\) −40.6581 −2.27642
\(320\) 0 0
\(321\) 35.0297 1.95517
\(322\) 0 0
\(323\) 4.31693 0.240201
\(324\) 0 0
\(325\) 2.05398 0.113934
\(326\) 0 0
\(327\) 40.6216 2.24638
\(328\) 0 0
\(329\) 2.42953 0.133944
\(330\) 0 0
\(331\) 9.45748 0.519830 0.259915 0.965631i \(-0.416305\pi\)
0.259915 + 0.965631i \(0.416305\pi\)
\(332\) 0 0
\(333\) 13.5151 0.740624
\(334\) 0 0
\(335\) 1.10217 0.0602180
\(336\) 0 0
\(337\) 2.96821 0.161689 0.0808443 0.996727i \(-0.474238\pi\)
0.0808443 + 0.996727i \(0.474238\pi\)
\(338\) 0 0
\(339\) −13.6651 −0.742185
\(340\) 0 0
\(341\) 19.6239 1.06269
\(342\) 0 0
\(343\) −11.6186 −0.627343
\(344\) 0 0
\(345\) 4.59893 0.247598
\(346\) 0 0
\(347\) −12.4197 −0.666724 −0.333362 0.942799i \(-0.608183\pi\)
−0.333362 + 0.942799i \(0.608183\pi\)
\(348\) 0 0
\(349\) 26.9381 1.44196 0.720981 0.692955i \(-0.243692\pi\)
0.720981 + 0.692955i \(0.243692\pi\)
\(350\) 0 0
\(351\) 6.32624 0.337670
\(352\) 0 0
\(353\) −12.3858 −0.659230 −0.329615 0.944115i \(-0.606919\pi\)
−0.329615 + 0.944115i \(0.606919\pi\)
\(354\) 0 0
\(355\) −14.4854 −0.768807
\(356\) 0 0
\(357\) 8.09400 0.428380
\(358\) 0 0
\(359\) −1.35216 −0.0713643 −0.0356822 0.999363i \(-0.511360\pi\)
−0.0356822 + 0.999363i \(0.511360\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −19.9353 −1.04633
\(364\) 0 0
\(365\) −6.03444 −0.315857
\(366\) 0 0
\(367\) 0.362613 0.0189283 0.00946413 0.999955i \(-0.496987\pi\)
0.00946413 + 0.999955i \(0.496987\pi\)
\(368\) 0 0
\(369\) 6.64859 0.346112
\(370\) 0 0
\(371\) 3.50008 0.181715
\(372\) 0 0
\(373\) 6.79352 0.351755 0.175878 0.984412i \(-0.443724\pi\)
0.175878 + 0.984412i \(0.443724\pi\)
\(374\) 0 0
\(375\) −2.13476 −0.110239
\(376\) 0 0
\(377\) −18.5176 −0.953704
\(378\) 0 0
\(379\) −3.29889 −0.169453 −0.0847263 0.996404i \(-0.527002\pi\)
−0.0847263 + 0.996404i \(0.527002\pi\)
\(380\) 0 0
\(381\) −30.1289 −1.54355
\(382\) 0 0
\(383\) 4.62625 0.236390 0.118195 0.992990i \(-0.462289\pi\)
0.118195 + 0.992990i \(0.462289\pi\)
\(384\) 0 0
\(385\) 3.96092 0.201867
\(386\) 0 0
\(387\) 19.2793 0.980023
\(388\) 0 0
\(389\) 18.6792 0.947074 0.473537 0.880774i \(-0.342977\pi\)
0.473537 + 0.880774i \(0.342977\pi\)
\(390\) 0 0
\(391\) 9.29999 0.470320
\(392\) 0 0
\(393\) 22.9036 1.15533
\(394\) 0 0
\(395\) 7.34489 0.369561
\(396\) 0 0
\(397\) −14.1602 −0.710678 −0.355339 0.934737i \(-0.615635\pi\)
−0.355339 + 0.934737i \(0.615635\pi\)
\(398\) 0 0
\(399\) 1.87494 0.0938645
\(400\) 0 0
\(401\) 30.5975 1.52797 0.763984 0.645235i \(-0.223240\pi\)
0.763984 + 0.645235i \(0.223240\pi\)
\(402\) 0 0
\(403\) 8.93761 0.445214
\(404\) 0 0
\(405\) −11.2467 −0.558854
\(406\) 0 0
\(407\) 39.1407 1.94013
\(408\) 0 0
\(409\) −4.69447 −0.232127 −0.116063 0.993242i \(-0.537028\pi\)
−0.116063 + 0.993242i \(0.537028\pi\)
\(410\) 0 0
\(411\) 45.5485 2.24674
\(412\) 0 0
\(413\) 10.1706 0.500461
\(414\) 0 0
\(415\) −1.34175 −0.0658639
\(416\) 0 0
\(417\) 13.8692 0.679179
\(418\) 0 0
\(419\) 4.93761 0.241218 0.120609 0.992700i \(-0.461515\pi\)
0.120609 + 0.992700i \(0.461515\pi\)
\(420\) 0 0
\(421\) −24.4531 −1.19177 −0.595885 0.803070i \(-0.703199\pi\)
−0.595885 + 0.803070i \(0.703199\pi\)
\(422\) 0 0
\(423\) 4.30758 0.209442
\(424\) 0 0
\(425\) −4.31693 −0.209402
\(426\) 0 0
\(427\) 1.33175 0.0644478
\(428\) 0 0
\(429\) −19.7744 −0.954718
\(430\) 0 0
\(431\) 25.8908 1.24712 0.623558 0.781777i \(-0.285686\pi\)
0.623558 + 0.781777i \(0.285686\pi\)
\(432\) 0 0
\(433\) −23.5184 −1.13022 −0.565111 0.825015i \(-0.691167\pi\)
−0.565111 + 0.825015i \(0.691167\pi\)
\(434\) 0 0
\(435\) 19.2459 0.922772
\(436\) 0 0
\(437\) 2.15430 0.103054
\(438\) 0 0
\(439\) −6.16158 −0.294076 −0.147038 0.989131i \(-0.546974\pi\)
−0.147038 + 0.989131i \(0.546974\pi\)
\(440\) 0 0
\(441\) −9.69930 −0.461871
\(442\) 0 0
\(443\) −28.8889 −1.37255 −0.686276 0.727341i \(-0.740756\pi\)
−0.686276 + 0.727341i \(0.740756\pi\)
\(444\) 0 0
\(445\) −15.5455 −0.736929
\(446\) 0 0
\(447\) −28.7987 −1.36213
\(448\) 0 0
\(449\) −23.7011 −1.11853 −0.559263 0.828991i \(-0.688916\pi\)
−0.559263 + 0.828991i \(0.688916\pi\)
\(450\) 0 0
\(451\) 19.2548 0.906672
\(452\) 0 0
\(453\) −42.2448 −1.98483
\(454\) 0 0
\(455\) 1.80399 0.0845722
\(456\) 0 0
\(457\) 12.0585 0.564075 0.282037 0.959403i \(-0.408990\pi\)
0.282037 + 0.959403i \(0.408990\pi\)
\(458\) 0 0
\(459\) −13.2962 −0.620611
\(460\) 0 0
\(461\) 18.6778 0.869912 0.434956 0.900452i \(-0.356764\pi\)
0.434956 + 0.900452i \(0.356764\pi\)
\(462\) 0 0
\(463\) −20.3186 −0.944286 −0.472143 0.881522i \(-0.656519\pi\)
−0.472143 + 0.881522i \(0.656519\pi\)
\(464\) 0 0
\(465\) −9.28915 −0.430774
\(466\) 0 0
\(467\) 27.7921 1.28606 0.643032 0.765839i \(-0.277676\pi\)
0.643032 + 0.765839i \(0.277676\pi\)
\(468\) 0 0
\(469\) 0.968026 0.0446993
\(470\) 0 0
\(471\) 5.07536 0.233860
\(472\) 0 0
\(473\) 55.8342 2.56726
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 6.20567 0.284138
\(478\) 0 0
\(479\) −24.4130 −1.11546 −0.557729 0.830023i \(-0.688327\pi\)
−0.557729 + 0.830023i \(0.688327\pi\)
\(480\) 0 0
\(481\) 17.8265 0.812818
\(482\) 0 0
\(483\) 4.03919 0.183790
\(484\) 0 0
\(485\) 7.55630 0.343114
\(486\) 0 0
\(487\) 11.1955 0.507317 0.253659 0.967294i \(-0.418366\pi\)
0.253659 + 0.967294i \(0.418366\pi\)
\(488\) 0 0
\(489\) −18.9165 −0.855435
\(490\) 0 0
\(491\) 26.8953 1.21377 0.606884 0.794791i \(-0.292419\pi\)
0.606884 + 0.794791i \(0.292419\pi\)
\(492\) 0 0
\(493\) 38.9193 1.75284
\(494\) 0 0
\(495\) 7.02276 0.315650
\(496\) 0 0
\(497\) −12.7224 −0.570678
\(498\) 0 0
\(499\) −11.1629 −0.499721 −0.249861 0.968282i \(-0.580385\pi\)
−0.249861 + 0.968282i \(0.580385\pi\)
\(500\) 0 0
\(501\) −13.3677 −0.597224
\(502\) 0 0
\(503\) 0.979268 0.0436634 0.0218317 0.999762i \(-0.493050\pi\)
0.0218317 + 0.999762i \(0.493050\pi\)
\(504\) 0 0
\(505\) 8.50981 0.378682
\(506\) 0 0
\(507\) 18.7458 0.832528
\(508\) 0 0
\(509\) 21.5263 0.954136 0.477068 0.878866i \(-0.341699\pi\)
0.477068 + 0.878866i \(0.341699\pi\)
\(510\) 0 0
\(511\) −5.29999 −0.234458
\(512\) 0 0
\(513\) −3.08000 −0.135985
\(514\) 0 0
\(515\) −0.763993 −0.0336655
\(516\) 0 0
\(517\) 12.4750 0.548652
\(518\) 0 0
\(519\) −38.9964 −1.71175
\(520\) 0 0
\(521\) 29.2184 1.28008 0.640042 0.768340i \(-0.278917\pi\)
0.640042 + 0.768340i \(0.278917\pi\)
\(522\) 0 0
\(523\) 2.90627 0.127082 0.0635412 0.997979i \(-0.479761\pi\)
0.0635412 + 0.997979i \(0.479761\pi\)
\(524\) 0 0
\(525\) −1.87494 −0.0818292
\(526\) 0 0
\(527\) −18.7846 −0.818269
\(528\) 0 0
\(529\) −18.3590 −0.798216
\(530\) 0 0
\(531\) 18.0326 0.782546
\(532\) 0 0
\(533\) 8.76951 0.379850
\(534\) 0 0
\(535\) −16.4092 −0.709430
\(536\) 0 0
\(537\) −30.1452 −1.30086
\(538\) 0 0
\(539\) −28.0898 −1.20992
\(540\) 0 0
\(541\) −31.0618 −1.33545 −0.667726 0.744407i \(-0.732732\pi\)
−0.667726 + 0.744407i \(0.732732\pi\)
\(542\) 0 0
\(543\) −7.14054 −0.306430
\(544\) 0 0
\(545\) −19.0286 −0.815097
\(546\) 0 0
\(547\) −8.26218 −0.353265 −0.176633 0.984277i \(-0.556520\pi\)
−0.176633 + 0.984277i \(0.556520\pi\)
\(548\) 0 0
\(549\) 2.36120 0.100774
\(550\) 0 0
\(551\) 9.01549 0.384073
\(552\) 0 0
\(553\) 6.45094 0.274322
\(554\) 0 0
\(555\) −18.5276 −0.786455
\(556\) 0 0
\(557\) −8.83469 −0.374338 −0.187169 0.982328i \(-0.559931\pi\)
−0.187169 + 0.982328i \(0.559931\pi\)
\(558\) 0 0
\(559\) 25.4295 1.07555
\(560\) 0 0
\(561\) 41.5608 1.75470
\(562\) 0 0
\(563\) 46.0855 1.94227 0.971137 0.238524i \(-0.0766635\pi\)
0.971137 + 0.238524i \(0.0766635\pi\)
\(564\) 0 0
\(565\) 6.40121 0.269301
\(566\) 0 0
\(567\) −9.87789 −0.414832
\(568\) 0 0
\(569\) −13.1041 −0.549350 −0.274675 0.961537i \(-0.588570\pi\)
−0.274675 + 0.961537i \(0.588570\pi\)
\(570\) 0 0
\(571\) −27.3020 −1.14255 −0.571277 0.820757i \(-0.693552\pi\)
−0.571277 + 0.820757i \(0.693552\pi\)
\(572\) 0 0
\(573\) −8.03863 −0.335819
\(574\) 0 0
\(575\) −2.15430 −0.0898406
\(576\) 0 0
\(577\) 5.56014 0.231472 0.115736 0.993280i \(-0.463077\pi\)
0.115736 + 0.993280i \(0.463077\pi\)
\(578\) 0 0
\(579\) −31.4888 −1.30863
\(580\) 0 0
\(581\) −1.17845 −0.0488902
\(582\) 0 0
\(583\) 17.9721 0.744327
\(584\) 0 0
\(585\) 3.19849 0.132241
\(586\) 0 0
\(587\) −9.47739 −0.391174 −0.195587 0.980686i \(-0.562661\pi\)
−0.195587 + 0.980686i \(0.562661\pi\)
\(588\) 0 0
\(589\) −4.35137 −0.179295
\(590\) 0 0
\(591\) −47.1731 −1.94044
\(592\) 0 0
\(593\) −15.5002 −0.636517 −0.318258 0.948004i \(-0.603098\pi\)
−0.318258 + 0.948004i \(0.603098\pi\)
\(594\) 0 0
\(595\) −3.79152 −0.155437
\(596\) 0 0
\(597\) −28.2212 −1.15502
\(598\) 0 0
\(599\) 2.67436 0.109271 0.0546356 0.998506i \(-0.482600\pi\)
0.0546356 + 0.998506i \(0.482600\pi\)
\(600\) 0 0
\(601\) 39.0353 1.59229 0.796143 0.605109i \(-0.206871\pi\)
0.796143 + 0.605109i \(0.206871\pi\)
\(602\) 0 0
\(603\) 1.71632 0.0698940
\(604\) 0 0
\(605\) 9.33840 0.379660
\(606\) 0 0
\(607\) 5.07601 0.206029 0.103014 0.994680i \(-0.467151\pi\)
0.103014 + 0.994680i \(0.467151\pi\)
\(608\) 0 0
\(609\) 16.9035 0.684965
\(610\) 0 0
\(611\) 5.68171 0.229857
\(612\) 0 0
\(613\) 29.4734 1.19042 0.595210 0.803570i \(-0.297069\pi\)
0.595210 + 0.803570i \(0.297069\pi\)
\(614\) 0 0
\(615\) −9.11444 −0.367530
\(616\) 0 0
\(617\) −34.5806 −1.39216 −0.696081 0.717963i \(-0.745075\pi\)
−0.696081 + 0.717963i \(0.745075\pi\)
\(618\) 0 0
\(619\) 10.1975 0.409873 0.204936 0.978775i \(-0.434301\pi\)
0.204936 + 0.978775i \(0.434301\pi\)
\(620\) 0 0
\(621\) −6.63525 −0.266263
\(622\) 0 0
\(623\) −13.6535 −0.547016
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 9.62738 0.384481
\(628\) 0 0
\(629\) −37.4667 −1.49390
\(630\) 0 0
\(631\) −35.5500 −1.41522 −0.707612 0.706601i \(-0.750228\pi\)
−0.707612 + 0.706601i \(0.750228\pi\)
\(632\) 0 0
\(633\) 9.41840 0.374348
\(634\) 0 0
\(635\) 14.1135 0.560076
\(636\) 0 0
\(637\) −12.7934 −0.506893
\(638\) 0 0
\(639\) −22.5570 −0.892340
\(640\) 0 0
\(641\) 22.9448 0.906263 0.453132 0.891444i \(-0.350307\pi\)
0.453132 + 0.891444i \(0.350307\pi\)
\(642\) 0 0
\(643\) −21.5175 −0.848566 −0.424283 0.905530i \(-0.639474\pi\)
−0.424283 + 0.905530i \(0.639474\pi\)
\(644\) 0 0
\(645\) −26.4297 −1.04067
\(646\) 0 0
\(647\) −2.75737 −0.108403 −0.0542017 0.998530i \(-0.517261\pi\)
−0.0542017 + 0.998530i \(0.517261\pi\)
\(648\) 0 0
\(649\) 52.2235 2.04995
\(650\) 0 0
\(651\) −8.15857 −0.319760
\(652\) 0 0
\(653\) −3.04942 −0.119333 −0.0596665 0.998218i \(-0.519004\pi\)
−0.0596665 + 0.998218i \(0.519004\pi\)
\(654\) 0 0
\(655\) −10.7289 −0.419211
\(656\) 0 0
\(657\) −9.39693 −0.366609
\(658\) 0 0
\(659\) 35.9817 1.40165 0.700825 0.713334i \(-0.252816\pi\)
0.700825 + 0.713334i \(0.252816\pi\)
\(660\) 0 0
\(661\) −14.3411 −0.557805 −0.278903 0.960319i \(-0.589971\pi\)
−0.278903 + 0.960319i \(0.589971\pi\)
\(662\) 0 0
\(663\) 18.9287 0.735130
\(664\) 0 0
\(665\) −0.878290 −0.0340586
\(666\) 0 0
\(667\) 19.4221 0.752026
\(668\) 0 0
\(669\) −55.5720 −2.14854
\(670\) 0 0
\(671\) 6.83821 0.263986
\(672\) 0 0
\(673\) 41.5033 1.59984 0.799918 0.600110i \(-0.204876\pi\)
0.799918 + 0.600110i \(0.204876\pi\)
\(674\) 0 0
\(675\) 3.08000 0.118549
\(676\) 0 0
\(677\) 32.4315 1.24644 0.623221 0.782046i \(-0.285824\pi\)
0.623221 + 0.782046i \(0.285824\pi\)
\(678\) 0 0
\(679\) 6.63662 0.254690
\(680\) 0 0
\(681\) −7.35911 −0.282002
\(682\) 0 0
\(683\) −21.8723 −0.836921 −0.418461 0.908235i \(-0.637430\pi\)
−0.418461 + 0.908235i \(0.637430\pi\)
\(684\) 0 0
\(685\) −21.3366 −0.815228
\(686\) 0 0
\(687\) 36.5423 1.39418
\(688\) 0 0
\(689\) 8.18530 0.311835
\(690\) 0 0
\(691\) 40.6915 1.54798 0.773989 0.633199i \(-0.218259\pi\)
0.773989 + 0.633199i \(0.218259\pi\)
\(692\) 0 0
\(693\) 6.16802 0.234304
\(694\) 0 0
\(695\) −6.49684 −0.246439
\(696\) 0 0
\(697\) −18.4313 −0.698134
\(698\) 0 0
\(699\) −1.78938 −0.0676807
\(700\) 0 0
\(701\) 33.7482 1.27465 0.637326 0.770594i \(-0.280040\pi\)
0.637326 + 0.770594i \(0.280040\pi\)
\(702\) 0 0
\(703\) −8.67901 −0.327335
\(704\) 0 0
\(705\) −5.90519 −0.222402
\(706\) 0 0
\(707\) 7.47408 0.281092
\(708\) 0 0
\(709\) 4.17013 0.156612 0.0783062 0.996929i \(-0.475049\pi\)
0.0783062 + 0.996929i \(0.475049\pi\)
\(710\) 0 0
\(711\) 11.4376 0.428943
\(712\) 0 0
\(713\) −9.37417 −0.351066
\(714\) 0 0
\(715\) 9.26304 0.346418
\(716\) 0 0
\(717\) 19.4383 0.725937
\(718\) 0 0
\(719\) 6.29675 0.234829 0.117414 0.993083i \(-0.462539\pi\)
0.117414 + 0.993083i \(0.462539\pi\)
\(720\) 0 0
\(721\) −0.671007 −0.0249896
\(722\) 0 0
\(723\) 22.2226 0.826467
\(724\) 0 0
\(725\) −9.01549 −0.334827
\(726\) 0 0
\(727\) 32.6776 1.21195 0.605973 0.795485i \(-0.292784\pi\)
0.605973 + 0.795485i \(0.292784\pi\)
\(728\) 0 0
\(729\) 2.21161 0.0819115
\(730\) 0 0
\(731\) −53.4463 −1.97678
\(732\) 0 0
\(733\) 48.1735 1.77933 0.889664 0.456616i \(-0.150939\pi\)
0.889664 + 0.456616i \(0.150939\pi\)
\(734\) 0 0
\(735\) 13.2966 0.490453
\(736\) 0 0
\(737\) 4.97058 0.183094
\(738\) 0 0
\(739\) 2.86274 0.105308 0.0526538 0.998613i \(-0.483232\pi\)
0.0526538 + 0.998613i \(0.483232\pi\)
\(740\) 0 0
\(741\) 4.38475 0.161078
\(742\) 0 0
\(743\) −28.1718 −1.03352 −0.516761 0.856130i \(-0.672863\pi\)
−0.516761 + 0.856130i \(0.672863\pi\)
\(744\) 0 0
\(745\) 13.4904 0.494248
\(746\) 0 0
\(747\) −2.08940 −0.0764471
\(748\) 0 0
\(749\) −14.4120 −0.526603
\(750\) 0 0
\(751\) 33.1632 1.21014 0.605072 0.796171i \(-0.293144\pi\)
0.605072 + 0.796171i \(0.293144\pi\)
\(752\) 0 0
\(753\) 38.6898 1.40993
\(754\) 0 0
\(755\) 19.7890 0.720194
\(756\) 0 0
\(757\) 10.0716 0.366060 0.183030 0.983107i \(-0.441410\pi\)
0.183030 + 0.983107i \(0.441410\pi\)
\(758\) 0 0
\(759\) 20.7403 0.752825
\(760\) 0 0
\(761\) −51.0524 −1.85065 −0.925324 0.379178i \(-0.876207\pi\)
−0.925324 + 0.379178i \(0.876207\pi\)
\(762\) 0 0
\(763\) −16.7127 −0.605039
\(764\) 0 0
\(765\) −6.72241 −0.243049
\(766\) 0 0
\(767\) 23.7850 0.858826
\(768\) 0 0
\(769\) 31.1271 1.12247 0.561236 0.827656i \(-0.310326\pi\)
0.561236 + 0.827656i \(0.310326\pi\)
\(770\) 0 0
\(771\) −7.14562 −0.257343
\(772\) 0 0
\(773\) −16.2291 −0.583721 −0.291860 0.956461i \(-0.594274\pi\)
−0.291860 + 0.956461i \(0.594274\pi\)
\(774\) 0 0
\(775\) 4.35137 0.156306
\(776\) 0 0
\(777\) −16.2726 −0.583778
\(778\) 0 0
\(779\) −4.26953 −0.152972
\(780\) 0 0
\(781\) −65.3266 −2.33757
\(782\) 0 0
\(783\) −27.7677 −0.992336
\(784\) 0 0
\(785\) −2.37748 −0.0848559
\(786\) 0 0
\(787\) 36.1329 1.28800 0.644000 0.765026i \(-0.277274\pi\)
0.644000 + 0.765026i \(0.277274\pi\)
\(788\) 0 0
\(789\) 22.0689 0.785673
\(790\) 0 0
\(791\) 5.62212 0.199900
\(792\) 0 0
\(793\) 3.11444 0.110597
\(794\) 0 0
\(795\) −8.50725 −0.301721
\(796\) 0 0
\(797\) −13.3349 −0.472345 −0.236172 0.971711i \(-0.575893\pi\)
−0.236172 + 0.971711i \(0.575893\pi\)
\(798\) 0 0
\(799\) −11.9415 −0.422460
\(800\) 0 0
\(801\) −24.2078 −0.855340
\(802\) 0 0
\(803\) −27.2142 −0.960367
\(804\) 0 0
\(805\) −1.89210 −0.0666879
\(806\) 0 0
\(807\) 8.17196 0.287667
\(808\) 0 0
\(809\) −18.5912 −0.653630 −0.326815 0.945088i \(-0.605975\pi\)
−0.326815 + 0.945088i \(0.605975\pi\)
\(810\) 0 0
\(811\) 4.56880 0.160432 0.0802161 0.996777i \(-0.474439\pi\)
0.0802161 + 0.996777i \(0.474439\pi\)
\(812\) 0 0
\(813\) −1.55320 −0.0544730
\(814\) 0 0
\(815\) 8.86118 0.310394
\(816\) 0 0
\(817\) −12.3806 −0.433143
\(818\) 0 0
\(819\) 2.80920 0.0981614
\(820\) 0 0
\(821\) −2.15671 −0.0752698 −0.0376349 0.999292i \(-0.511982\pi\)
−0.0376349 + 0.999292i \(0.511982\pi\)
\(822\) 0 0
\(823\) 8.69218 0.302990 0.151495 0.988458i \(-0.451591\pi\)
0.151495 + 0.988458i \(0.451591\pi\)
\(824\) 0 0
\(825\) −9.62738 −0.335183
\(826\) 0 0
\(827\) 14.6572 0.509680 0.254840 0.966983i \(-0.417977\pi\)
0.254840 + 0.966983i \(0.417977\pi\)
\(828\) 0 0
\(829\) −37.7966 −1.31273 −0.656364 0.754444i \(-0.727907\pi\)
−0.656364 + 0.754444i \(0.727907\pi\)
\(830\) 0 0
\(831\) −44.4772 −1.54290
\(832\) 0 0
\(833\) 26.8885 0.931631
\(834\) 0 0
\(835\) 6.26190 0.216702
\(836\) 0 0
\(837\) 13.4022 0.463249
\(838\) 0 0
\(839\) 46.1806 1.59433 0.797165 0.603761i \(-0.206332\pi\)
0.797165 + 0.603761i \(0.206332\pi\)
\(840\) 0 0
\(841\) 52.2790 1.80272
\(842\) 0 0
\(843\) −59.4477 −2.04749
\(844\) 0 0
\(845\) −8.78118 −0.302082
\(846\) 0 0
\(847\) 8.20182 0.281818
\(848\) 0 0
\(849\) −62.1976 −2.13462
\(850\) 0 0
\(851\) −18.6972 −0.640933
\(852\) 0 0
\(853\) −8.42055 −0.288314 −0.144157 0.989555i \(-0.546047\pi\)
−0.144157 + 0.989555i \(0.546047\pi\)
\(854\) 0 0
\(855\) −1.55722 −0.0532557
\(856\) 0 0
\(857\) 36.8489 1.25873 0.629367 0.777108i \(-0.283314\pi\)
0.629367 + 0.777108i \(0.283314\pi\)
\(858\) 0 0
\(859\) −7.62728 −0.260239 −0.130120 0.991498i \(-0.541536\pi\)
−0.130120 + 0.991498i \(0.541536\pi\)
\(860\) 0 0
\(861\) −8.00512 −0.272814
\(862\) 0 0
\(863\) 56.8931 1.93666 0.968332 0.249665i \(-0.0803205\pi\)
0.968332 + 0.249665i \(0.0803205\pi\)
\(864\) 0 0
\(865\) 18.2673 0.621107
\(866\) 0 0
\(867\) −3.49231 −0.118605
\(868\) 0 0
\(869\) 33.1241 1.12366
\(870\) 0 0
\(871\) 2.26383 0.0767070
\(872\) 0 0
\(873\) 11.7668 0.398246
\(874\) 0 0
\(875\) 0.878290 0.0296916
\(876\) 0 0
\(877\) −47.2084 −1.59411 −0.797057 0.603904i \(-0.793611\pi\)
−0.797057 + 0.603904i \(0.793611\pi\)
\(878\) 0 0
\(879\) 22.0106 0.742398
\(880\) 0 0
\(881\) −45.2188 −1.52346 −0.761729 0.647895i \(-0.775649\pi\)
−0.761729 + 0.647895i \(0.775649\pi\)
\(882\) 0 0
\(883\) 12.9525 0.435886 0.217943 0.975962i \(-0.430065\pi\)
0.217943 + 0.975962i \(0.430065\pi\)
\(884\) 0 0
\(885\) −24.7205 −0.830971
\(886\) 0 0
\(887\) −5.58397 −0.187491 −0.0937457 0.995596i \(-0.529884\pi\)
−0.0937457 + 0.995596i \(0.529884\pi\)
\(888\) 0 0
\(889\) 12.3957 0.415739
\(890\) 0 0
\(891\) −50.7206 −1.69920
\(892\) 0 0
\(893\) −2.76620 −0.0925674
\(894\) 0 0
\(895\) 14.1211 0.472016
\(896\) 0 0
\(897\) 9.44609 0.315396
\(898\) 0 0
\(899\) −39.2297 −1.30838
\(900\) 0 0
\(901\) −17.2034 −0.573129
\(902\) 0 0
\(903\) −23.2129 −0.772478
\(904\) 0 0
\(905\) 3.34489 0.111188
\(906\) 0 0
\(907\) 53.7305 1.78409 0.892046 0.451944i \(-0.149269\pi\)
0.892046 + 0.451944i \(0.149269\pi\)
\(908\) 0 0
\(909\) 13.2516 0.439529
\(910\) 0 0
\(911\) −23.6398 −0.783222 −0.391611 0.920131i \(-0.628082\pi\)
−0.391611 + 0.920131i \(0.628082\pi\)
\(912\) 0 0
\(913\) −6.05104 −0.200260
\(914\) 0 0
\(915\) −3.23694 −0.107010
\(916\) 0 0
\(917\) −9.42304 −0.311176
\(918\) 0 0
\(919\) −5.86850 −0.193584 −0.0967920 0.995305i \(-0.530858\pi\)
−0.0967920 + 0.995305i \(0.530858\pi\)
\(920\) 0 0
\(921\) 33.4885 1.10348
\(922\) 0 0
\(923\) −29.7527 −0.979323
\(924\) 0 0
\(925\) 8.67901 0.285364
\(926\) 0 0
\(927\) −1.18970 −0.0390750
\(928\) 0 0
\(929\) 42.1417 1.38263 0.691313 0.722556i \(-0.257033\pi\)
0.691313 + 0.722556i \(0.257033\pi\)
\(930\) 0 0
\(931\) 6.22861 0.204134
\(932\) 0 0
\(933\) −74.0168 −2.42320
\(934\) 0 0
\(935\) −19.4686 −0.636690
\(936\) 0 0
\(937\) 13.0515 0.426373 0.213187 0.977012i \(-0.431616\pi\)
0.213187 + 0.977012i \(0.431616\pi\)
\(938\) 0 0
\(939\) −46.9474 −1.53207
\(940\) 0 0
\(941\) −3.91219 −0.127534 −0.0637668 0.997965i \(-0.520311\pi\)
−0.0637668 + 0.997965i \(0.520311\pi\)
\(942\) 0 0
\(943\) −9.19786 −0.299523
\(944\) 0 0
\(945\) 2.70513 0.0879980
\(946\) 0 0
\(947\) −46.8887 −1.52368 −0.761840 0.647766i \(-0.775704\pi\)
−0.761840 + 0.647766i \(0.775704\pi\)
\(948\) 0 0
\(949\) −12.3946 −0.402345
\(950\) 0 0
\(951\) −43.4416 −1.40869
\(952\) 0 0
\(953\) −36.0745 −1.16857 −0.584284 0.811549i \(-0.698625\pi\)
−0.584284 + 0.811549i \(0.698625\pi\)
\(954\) 0 0
\(955\) 3.76558 0.121851
\(956\) 0 0
\(957\) 86.7956 2.80570
\(958\) 0 0
\(959\) −18.7397 −0.605136
\(960\) 0 0
\(961\) −12.0656 −0.389212
\(962\) 0 0
\(963\) −25.5526 −0.823422
\(964\) 0 0
\(965\) 14.7505 0.474834
\(966\) 0 0
\(967\) −5.14385 −0.165415 −0.0827075 0.996574i \(-0.526357\pi\)
−0.0827075 + 0.996574i \(0.526357\pi\)
\(968\) 0 0
\(969\) −9.21564 −0.296049
\(970\) 0 0
\(971\) 1.55134 0.0497848 0.0248924 0.999690i \(-0.492076\pi\)
0.0248924 + 0.999690i \(0.492076\pi\)
\(972\) 0 0
\(973\) −5.70611 −0.182930
\(974\) 0 0
\(975\) −4.38475 −0.140425
\(976\) 0 0
\(977\) −33.4684 −1.07075 −0.535374 0.844615i \(-0.679829\pi\)
−0.535374 + 0.844615i \(0.679829\pi\)
\(978\) 0 0
\(979\) −70.1075 −2.24064
\(980\) 0 0
\(981\) −29.6317 −0.946068
\(982\) 0 0
\(983\) −11.5436 −0.368184 −0.184092 0.982909i \(-0.558934\pi\)
−0.184092 + 0.982909i \(0.558934\pi\)
\(984\) 0 0
\(985\) 22.0976 0.704087
\(986\) 0 0
\(987\) −5.18647 −0.165087
\(988\) 0 0
\(989\) −26.6716 −0.848108
\(990\) 0 0
\(991\) 0.624895 0.0198504 0.00992522 0.999951i \(-0.496841\pi\)
0.00992522 + 0.999951i \(0.496841\pi\)
\(992\) 0 0
\(993\) −20.1895 −0.640694
\(994\) 0 0
\(995\) 13.2198 0.419097
\(996\) 0 0
\(997\) 19.6074 0.620974 0.310487 0.950578i \(-0.399508\pi\)
0.310487 + 0.950578i \(0.399508\pi\)
\(998\) 0 0
\(999\) 26.7314 0.845743
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 6080.2.a.cl.1.1 5
4.3 odd 2 6080.2.a.ci.1.5 5
8.3 odd 2 3040.2.a.x.1.1 yes 5
8.5 even 2 3040.2.a.u.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.u.1.5 5 8.5 even 2
3040.2.a.x.1.1 yes 5 8.3 odd 2
6080.2.a.ci.1.5 5 4.3 odd 2
6080.2.a.cl.1.1 5 1.1 even 1 trivial