Properties

Label 4275.2.a.ba.1.1
Level $4275$
Weight $2$
Character 4275.1
Self dual yes
Analytic conductor $34.136$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4275,2,Mod(1,4275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.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: \(34.1360468641\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 475)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.37720\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.37720 q^{2} +3.65109 q^{4} +0.726109 q^{7} -3.92498 q^{8} +O(q^{10})\) \(q-2.37720 q^{2} +3.65109 q^{4} +0.726109 q^{7} -3.92498 q^{8} +0.273891 q^{11} -5.95328 q^{13} -1.72611 q^{14} +2.02830 q^{16} -5.27389 q^{17} +1.00000 q^{19} -0.651093 q^{22} +3.67939 q^{23} +14.1522 q^{26} +2.65109 q^{28} +2.27389 q^{29} +3.19887 q^{31} +3.02830 q^{32} +12.5371 q^{34} +8.12386 q^{37} -2.37720 q^{38} +9.43380 q^{41} +9.81100 q^{43} +1.00000 q^{44} -8.74666 q^{46} -12.1599 q^{47} -6.47277 q^{49} -21.7360 q^{52} -5.69781 q^{53} -2.84997 q^{56} -5.40550 q^{58} +4.20662 q^{59} -0.103312 q^{61} -7.60437 q^{62} -11.2555 q^{64} -11.7827 q^{67} -19.2555 q^{68} -5.75441 q^{71} +6.67939 q^{73} -19.3121 q^{74} +3.65109 q^{76} +0.198875 q^{77} +3.87826 q^{79} -22.4260 q^{82} +0.488265 q^{83} -23.3227 q^{86} -1.07502 q^{88} +16.4338 q^{89} -4.32273 q^{91} +13.4338 q^{92} +28.9066 q^{94} -4.44447 q^{97} +15.3871 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 4 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 4 q^{4} + 4 q^{7} - 3 q^{8} - q^{11} + 3 q^{13} - 7 q^{14} - 6 q^{16} - 14 q^{17} + 3 q^{19} + 5 q^{22} - 8 q^{23} + 11 q^{26} + q^{28} + 5 q^{29} - q^{31} - 3 q^{32} + 5 q^{34} + 5 q^{37} - 2 q^{38} - q^{41} - 5 q^{43} + 3 q^{44} - 12 q^{46} - 9 q^{47} - 7 q^{49} - 22 q^{52} - 31 q^{53} + 9 q^{56} + q^{58} + 6 q^{59} + 3 q^{61} + 5 q^{62} + q^{64} - 13 q^{67} - 23 q^{68} - 7 q^{71} + q^{73} + q^{74} + 4 q^{76} - 10 q^{77} - 18 q^{79} - 34 q^{82} - 3 q^{83} - 40 q^{86} - 12 q^{88} + 20 q^{89} + 17 q^{91} + 11 q^{92} + 45 q^{94} - 13 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37720 −1.68094 −0.840468 0.541861i \(-0.817720\pi\)
−0.840468 + 0.541861i \(0.817720\pi\)
\(3\) 0 0
\(4\) 3.65109 1.82555
\(5\) 0 0
\(6\) 0 0
\(7\) 0.726109 0.274444 0.137222 0.990540i \(-0.456183\pi\)
0.137222 + 0.990540i \(0.456183\pi\)
\(8\) −3.92498 −1.38769
\(9\) 0 0
\(10\) 0 0
\(11\) 0.273891 0.0825811 0.0412906 0.999147i \(-0.486853\pi\)
0.0412906 + 0.999147i \(0.486853\pi\)
\(12\) 0 0
\(13\) −5.95328 −1.65114 −0.825571 0.564298i \(-0.809147\pi\)
−0.825571 + 0.564298i \(0.809147\pi\)
\(14\) −1.72611 −0.461322
\(15\) 0 0
\(16\) 2.02830 0.507074
\(17\) −5.27389 −1.27911 −0.639553 0.768747i \(-0.720881\pi\)
−0.639553 + 0.768747i \(0.720881\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −0.651093 −0.138814
\(23\) 3.67939 0.767206 0.383603 0.923498i \(-0.374683\pi\)
0.383603 + 0.923498i \(0.374683\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 14.1522 2.77547
\(27\) 0 0
\(28\) 2.65109 0.501010
\(29\) 2.27389 0.422251 0.211125 0.977459i \(-0.432287\pi\)
0.211125 + 0.977459i \(0.432287\pi\)
\(30\) 0 0
\(31\) 3.19887 0.574535 0.287267 0.957850i \(-0.407253\pi\)
0.287267 + 0.957850i \(0.407253\pi\)
\(32\) 3.02830 0.535332
\(33\) 0 0
\(34\) 12.5371 2.15010
\(35\) 0 0
\(36\) 0 0
\(37\) 8.12386 1.33555 0.667777 0.744361i \(-0.267246\pi\)
0.667777 + 0.744361i \(0.267246\pi\)
\(38\) −2.37720 −0.385633
\(39\) 0 0
\(40\) 0 0
\(41\) 9.43380 1.47331 0.736656 0.676268i \(-0.236404\pi\)
0.736656 + 0.676268i \(0.236404\pi\)
\(42\) 0 0
\(43\) 9.81100 1.49616 0.748082 0.663607i \(-0.230975\pi\)
0.748082 + 0.663607i \(0.230975\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −8.74666 −1.28962
\(47\) −12.1599 −1.77370 −0.886852 0.462053i \(-0.847113\pi\)
−0.886852 + 0.462053i \(0.847113\pi\)
\(48\) 0 0
\(49\) −6.47277 −0.924681
\(50\) 0 0
\(51\) 0 0
\(52\) −21.7360 −3.01424
\(53\) −5.69781 −0.782655 −0.391327 0.920252i \(-0.627984\pi\)
−0.391327 + 0.920252i \(0.627984\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.84997 −0.380843
\(57\) 0 0
\(58\) −5.40550 −0.709777
\(59\) 4.20662 0.547656 0.273828 0.961779i \(-0.411710\pi\)
0.273828 + 0.961779i \(0.411710\pi\)
\(60\) 0 0
\(61\) −0.103312 −0.0132278 −0.00661389 0.999978i \(-0.502105\pi\)
−0.00661389 + 0.999978i \(0.502105\pi\)
\(62\) −7.60437 −0.965756
\(63\) 0 0
\(64\) −11.2555 −1.40693
\(65\) 0 0
\(66\) 0 0
\(67\) −11.7827 −1.43949 −0.719743 0.694241i \(-0.755740\pi\)
−0.719743 + 0.694241i \(0.755740\pi\)
\(68\) −19.2555 −2.33507
\(69\) 0 0
\(70\) 0 0
\(71\) −5.75441 −0.682922 −0.341461 0.939896i \(-0.610922\pi\)
−0.341461 + 0.939896i \(0.610922\pi\)
\(72\) 0 0
\(73\) 6.67939 0.781763 0.390882 0.920441i \(-0.372170\pi\)
0.390882 + 0.920441i \(0.372170\pi\)
\(74\) −19.3121 −2.24498
\(75\) 0 0
\(76\) 3.65109 0.418809
\(77\) 0.198875 0.0226639
\(78\) 0 0
\(79\) 3.87826 0.436339 0.218169 0.975911i \(-0.429991\pi\)
0.218169 + 0.975911i \(0.429991\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −22.4260 −2.47654
\(83\) 0.488265 0.0535941 0.0267970 0.999641i \(-0.491469\pi\)
0.0267970 + 0.999641i \(0.491469\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −23.3227 −2.51495
\(87\) 0 0
\(88\) −1.07502 −0.114597
\(89\) 16.4338 1.74198 0.870989 0.491302i \(-0.163479\pi\)
0.870989 + 0.491302i \(0.163479\pi\)
\(90\) 0 0
\(91\) −4.32273 −0.453146
\(92\) 13.4338 1.40057
\(93\) 0 0
\(94\) 28.9066 2.98148
\(95\) 0 0
\(96\) 0 0
\(97\) −4.44447 −0.451267 −0.225634 0.974212i \(-0.572445\pi\)
−0.225634 + 0.974212i \(0.572445\pi\)
\(98\) 15.3871 1.55433
\(99\) 0 0
\(100\) 0 0
\(101\) −4.38495 −0.436319 −0.218160 0.975913i \(-0.570005\pi\)
−0.218160 + 0.975913i \(0.570005\pi\)
\(102\) 0 0
\(103\) 3.33048 0.328162 0.164081 0.986447i \(-0.447534\pi\)
0.164081 + 0.986447i \(0.447534\pi\)
\(104\) 23.3665 2.29128
\(105\) 0 0
\(106\) 13.5449 1.31559
\(107\) −16.4904 −1.59419 −0.797093 0.603857i \(-0.793630\pi\)
−0.797093 + 0.603857i \(0.793630\pi\)
\(108\) 0 0
\(109\) 7.79045 0.746190 0.373095 0.927793i \(-0.378297\pi\)
0.373095 + 0.927793i \(0.378297\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.47277 0.139163
\(113\) 0.142282 0.0133848 0.00669238 0.999978i \(-0.497870\pi\)
0.00669238 + 0.999978i \(0.497870\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.30219 0.770839
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) −3.82942 −0.351043
\(120\) 0 0
\(121\) −10.9250 −0.993180
\(122\) 0.245594 0.0222351
\(123\) 0 0
\(124\) 11.6794 1.04884
\(125\) 0 0
\(126\) 0 0
\(127\) −15.1316 −1.34271 −0.671357 0.741135i \(-0.734288\pi\)
−0.671357 + 0.741135i \(0.734288\pi\)
\(128\) 20.6999 1.82963
\(129\) 0 0
\(130\) 0 0
\(131\) −5.58383 −0.487861 −0.243931 0.969793i \(-0.578437\pi\)
−0.243931 + 0.969793i \(0.578437\pi\)
\(132\) 0 0
\(133\) 0.726109 0.0629617
\(134\) 28.0099 2.41968
\(135\) 0 0
\(136\) 20.6999 1.77500
\(137\) −12.8294 −1.09609 −0.548046 0.836448i \(-0.684628\pi\)
−0.548046 + 0.836448i \(0.684628\pi\)
\(138\) 0 0
\(139\) −15.2477 −1.29329 −0.646647 0.762789i \(-0.723829\pi\)
−0.646647 + 0.762789i \(0.723829\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 13.6794 1.14795
\(143\) −1.63055 −0.136353
\(144\) 0 0
\(145\) 0 0
\(146\) −15.8783 −1.31409
\(147\) 0 0
\(148\) 29.6610 2.43812
\(149\) −13.8315 −1.13312 −0.566562 0.824019i \(-0.691727\pi\)
−0.566562 + 0.824019i \(0.691727\pi\)
\(150\) 0 0
\(151\) −11.7077 −0.952758 −0.476379 0.879240i \(-0.658051\pi\)
−0.476379 + 0.879240i \(0.658051\pi\)
\(152\) −3.92498 −0.318358
\(153\) 0 0
\(154\) −0.472765 −0.0380965
\(155\) 0 0
\(156\) 0 0
\(157\) −4.79045 −0.382320 −0.191160 0.981559i \(-0.561225\pi\)
−0.191160 + 0.981559i \(0.561225\pi\)
\(158\) −9.21942 −0.733458
\(159\) 0 0
\(160\) 0 0
\(161\) 2.67164 0.210555
\(162\) 0 0
\(163\) 12.8011 1.00266 0.501331 0.865256i \(-0.332844\pi\)
0.501331 + 0.865256i \(0.332844\pi\)
\(164\) 34.4437 2.68960
\(165\) 0 0
\(166\) −1.16071 −0.0900882
\(167\) 20.9426 1.62059 0.810294 0.586024i \(-0.199308\pi\)
0.810294 + 0.586024i \(0.199308\pi\)
\(168\) 0 0
\(169\) 22.4415 1.72627
\(170\) 0 0
\(171\) 0 0
\(172\) 35.8209 2.73132
\(173\) −15.7282 −1.19580 −0.597898 0.801572i \(-0.703997\pi\)
−0.597898 + 0.801572i \(0.703997\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.555531 0.0418747
\(177\) 0 0
\(178\) −39.0665 −2.92816
\(179\) −3.41325 −0.255118 −0.127559 0.991831i \(-0.540714\pi\)
−0.127559 + 0.991831i \(0.540714\pi\)
\(180\) 0 0
\(181\) 23.5109 1.74755 0.873777 0.486327i \(-0.161664\pi\)
0.873777 + 0.486327i \(0.161664\pi\)
\(182\) 10.2760 0.761709
\(183\) 0 0
\(184\) −14.4415 −1.06464
\(185\) 0 0
\(186\) 0 0
\(187\) −1.44447 −0.105630
\(188\) −44.3969 −3.23798
\(189\) 0 0
\(190\) 0 0
\(191\) −12.4650 −0.901937 −0.450968 0.892540i \(-0.648921\pi\)
−0.450968 + 0.892540i \(0.648921\pi\)
\(192\) 0 0
\(193\) 19.2993 1.38919 0.694596 0.719400i \(-0.255583\pi\)
0.694596 + 0.719400i \(0.255583\pi\)
\(194\) 10.5654 0.758552
\(195\) 0 0
\(196\) −23.6327 −1.68805
\(197\) −6.63055 −0.472407 −0.236203 0.971704i \(-0.575903\pi\)
−0.236203 + 0.971704i \(0.575903\pi\)
\(198\) 0 0
\(199\) −23.0849 −1.63644 −0.818222 0.574902i \(-0.805040\pi\)
−0.818222 + 0.574902i \(0.805040\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.4239 0.733425
\(203\) 1.65109 0.115884
\(204\) 0 0
\(205\) 0 0
\(206\) −7.91723 −0.551620
\(207\) 0 0
\(208\) −12.0750 −0.837252
\(209\) 0.273891 0.0189454
\(210\) 0 0
\(211\) −7.54778 −0.519611 −0.259805 0.965661i \(-0.583658\pi\)
−0.259805 + 0.965661i \(0.583658\pi\)
\(212\) −20.8032 −1.42877
\(213\) 0 0
\(214\) 39.2010 2.67973
\(215\) 0 0
\(216\) 0 0
\(217\) 2.32273 0.157677
\(218\) −18.5195 −1.25430
\(219\) 0 0
\(220\) 0 0
\(221\) 31.3969 2.11199
\(222\) 0 0
\(223\) −1.09344 −0.0732221 −0.0366111 0.999330i \(-0.511656\pi\)
−0.0366111 + 0.999330i \(0.511656\pi\)
\(224\) 2.19887 0.146918
\(225\) 0 0
\(226\) −0.338233 −0.0224989
\(227\) 20.1316 1.33618 0.668091 0.744080i \(-0.267112\pi\)
0.668091 + 0.744080i \(0.267112\pi\)
\(228\) 0 0
\(229\) −5.51656 −0.364545 −0.182272 0.983248i \(-0.558345\pi\)
−0.182272 + 0.983248i \(0.558345\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −8.92498 −0.585954
\(233\) 18.1805 1.19104 0.595520 0.803340i \(-0.296946\pi\)
0.595520 + 0.803340i \(0.296946\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 15.3588 0.999771
\(237\) 0 0
\(238\) 9.10331 0.590080
\(239\) −21.9164 −1.41766 −0.708828 0.705381i \(-0.750776\pi\)
−0.708828 + 0.705381i \(0.750776\pi\)
\(240\) 0 0
\(241\) −28.1882 −1.81576 −0.907881 0.419228i \(-0.862301\pi\)
−0.907881 + 0.419228i \(0.862301\pi\)
\(242\) 25.9709 1.66947
\(243\) 0 0
\(244\) −0.377203 −0.0241479
\(245\) 0 0
\(246\) 0 0
\(247\) −5.95328 −0.378798
\(248\) −12.5555 −0.797277
\(249\) 0 0
\(250\) 0 0
\(251\) −9.00987 −0.568698 −0.284349 0.958721i \(-0.591777\pi\)
−0.284349 + 0.958721i \(0.591777\pi\)
\(252\) 0 0
\(253\) 1.00775 0.0633567
\(254\) 35.9709 2.25702
\(255\) 0 0
\(256\) −26.6970 −1.66856
\(257\) −6.86064 −0.427955 −0.213978 0.976839i \(-0.568642\pi\)
−0.213978 + 0.976839i \(0.568642\pi\)
\(258\) 0 0
\(259\) 5.89881 0.366534
\(260\) 0 0
\(261\) 0 0
\(262\) 13.2739 0.820064
\(263\) 9.25547 0.570717 0.285358 0.958421i \(-0.407887\pi\)
0.285358 + 0.958421i \(0.407887\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.72611 −0.105835
\(267\) 0 0
\(268\) −43.0197 −2.62785
\(269\) −0.498939 −0.0304208 −0.0152104 0.999884i \(-0.504842\pi\)
−0.0152104 + 0.999884i \(0.504842\pi\)
\(270\) 0 0
\(271\) 3.71061 0.225403 0.112702 0.993629i \(-0.464050\pi\)
0.112702 + 0.993629i \(0.464050\pi\)
\(272\) −10.6970 −0.648602
\(273\) 0 0
\(274\) 30.4981 1.84246
\(275\) 0 0
\(276\) 0 0
\(277\) 4.58675 0.275591 0.137796 0.990461i \(-0.455998\pi\)
0.137796 + 0.990461i \(0.455998\pi\)
\(278\) 36.2469 2.17395
\(279\) 0 0
\(280\) 0 0
\(281\) −27.2653 −1.62651 −0.813257 0.581905i \(-0.802308\pi\)
−0.813257 + 0.581905i \(0.802308\pi\)
\(282\) 0 0
\(283\) −10.2661 −0.610259 −0.305129 0.952311i \(-0.598700\pi\)
−0.305129 + 0.952311i \(0.598700\pi\)
\(284\) −21.0099 −1.24671
\(285\) 0 0
\(286\) 3.87614 0.229201
\(287\) 6.84997 0.404341
\(288\) 0 0
\(289\) 10.8139 0.636113
\(290\) 0 0
\(291\) 0 0
\(292\) 24.3871 1.42715
\(293\) 1.87051 0.109277 0.0546383 0.998506i \(-0.482599\pi\)
0.0546383 + 0.998506i \(0.482599\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −31.8860 −1.85334
\(297\) 0 0
\(298\) 32.8804 1.90471
\(299\) −21.9044 −1.26677
\(300\) 0 0
\(301\) 7.12386 0.410612
\(302\) 27.8315 1.60153
\(303\) 0 0
\(304\) 2.02830 0.116331
\(305\) 0 0
\(306\) 0 0
\(307\) 0.227171 0.0129653 0.00648266 0.999979i \(-0.497936\pi\)
0.00648266 + 0.999979i \(0.497936\pi\)
\(308\) 0.726109 0.0413739
\(309\) 0 0
\(310\) 0 0
\(311\) −20.9554 −1.18827 −0.594136 0.804365i \(-0.702506\pi\)
−0.594136 + 0.804365i \(0.702506\pi\)
\(312\) 0 0
\(313\) −11.2349 −0.635035 −0.317518 0.948252i \(-0.602849\pi\)
−0.317518 + 0.948252i \(0.602849\pi\)
\(314\) 11.3879 0.642655
\(315\) 0 0
\(316\) 14.1599 0.796557
\(317\) −18.6228 −1.04596 −0.522980 0.852345i \(-0.675180\pi\)
−0.522980 + 0.852345i \(0.675180\pi\)
\(318\) 0 0
\(319\) 0.622797 0.0348699
\(320\) 0 0
\(321\) 0 0
\(322\) −6.35103 −0.353929
\(323\) −5.27389 −0.293447
\(324\) 0 0
\(325\) 0 0
\(326\) −30.4309 −1.68541
\(327\) 0 0
\(328\) −37.0275 −2.04450
\(329\) −8.82942 −0.486782
\(330\) 0 0
\(331\) −14.1054 −0.775305 −0.387652 0.921806i \(-0.626714\pi\)
−0.387652 + 0.921806i \(0.626714\pi\)
\(332\) 1.78270 0.0978385
\(333\) 0 0
\(334\) −49.7848 −2.72410
\(335\) 0 0
\(336\) 0 0
\(337\) 22.9709 1.25130 0.625652 0.780102i \(-0.284833\pi\)
0.625652 + 0.780102i \(0.284833\pi\)
\(338\) −53.3481 −2.90175
\(339\) 0 0
\(340\) 0 0
\(341\) 0.876142 0.0474457
\(342\) 0 0
\(343\) −9.78270 −0.528216
\(344\) −38.5080 −2.07621
\(345\) 0 0
\(346\) 37.3892 2.01006
\(347\) 3.93273 0.211120 0.105560 0.994413i \(-0.466336\pi\)
0.105560 + 0.994413i \(0.466336\pi\)
\(348\) 0 0
\(349\) −34.4252 −1.84274 −0.921371 0.388685i \(-0.872929\pi\)
−0.921371 + 0.388685i \(0.872929\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.829422 0.0442083
\(353\) 4.25547 0.226496 0.113248 0.993567i \(-0.463875\pi\)
0.113248 + 0.993567i \(0.463875\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 60.0013 3.18006
\(357\) 0 0
\(358\) 8.11399 0.428837
\(359\) 20.2944 1.07110 0.535550 0.844504i \(-0.320104\pi\)
0.535550 + 0.844504i \(0.320104\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −55.8903 −2.93753
\(363\) 0 0
\(364\) −15.7827 −0.827238
\(365\) 0 0
\(366\) 0 0
\(367\) 3.85289 0.201119 0.100560 0.994931i \(-0.467937\pi\)
0.100560 + 0.994931i \(0.467937\pi\)
\(368\) 7.46289 0.389030
\(369\) 0 0
\(370\) 0 0
\(371\) −4.13724 −0.214795
\(372\) 0 0
\(373\) 14.6356 0.757802 0.378901 0.925437i \(-0.376302\pi\)
0.378901 + 0.925437i \(0.376302\pi\)
\(374\) 3.43380 0.177557
\(375\) 0 0
\(376\) 47.7274 2.46135
\(377\) −13.5371 −0.697197
\(378\) 0 0
\(379\) −22.0099 −1.13057 −0.565286 0.824895i \(-0.691234\pi\)
−0.565286 + 0.824895i \(0.691234\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 29.6319 1.51610
\(383\) 3.08569 0.157671 0.0788357 0.996888i \(-0.474880\pi\)
0.0788357 + 0.996888i \(0.474880\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −45.8783 −2.33514
\(387\) 0 0
\(388\) −16.2272 −0.823810
\(389\) −8.77203 −0.444760 −0.222380 0.974960i \(-0.571382\pi\)
−0.222380 + 0.974960i \(0.571382\pi\)
\(390\) 0 0
\(391\) −19.4047 −0.981338
\(392\) 25.4055 1.28317
\(393\) 0 0
\(394\) 15.7622 0.794086
\(395\) 0 0
\(396\) 0 0
\(397\) −1.59450 −0.0800257 −0.0400129 0.999199i \(-0.512740\pi\)
−0.0400129 + 0.999199i \(0.512740\pi\)
\(398\) 54.8775 2.75076
\(399\) 0 0
\(400\) 0 0
\(401\) 17.5526 0.876535 0.438268 0.898844i \(-0.355592\pi\)
0.438268 + 0.898844i \(0.355592\pi\)
\(402\) 0 0
\(403\) −19.0438 −0.948639
\(404\) −16.0099 −0.796521
\(405\) 0 0
\(406\) −3.92498 −0.194794
\(407\) 2.22505 0.110292
\(408\) 0 0
\(409\) 36.6815 1.81378 0.906892 0.421363i \(-0.138448\pi\)
0.906892 + 0.421363i \(0.138448\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 12.1599 0.599076
\(413\) 3.05447 0.150301
\(414\) 0 0
\(415\) 0 0
\(416\) −18.0283 −0.883910
\(417\) 0 0
\(418\) −0.651093 −0.0318460
\(419\) 18.8187 0.919356 0.459678 0.888086i \(-0.347965\pi\)
0.459678 + 0.888086i \(0.347965\pi\)
\(420\) 0 0
\(421\) −33.7819 −1.64643 −0.823215 0.567730i \(-0.807822\pi\)
−0.823215 + 0.567730i \(0.807822\pi\)
\(422\) 17.9426 0.873432
\(423\) 0 0
\(424\) 22.3638 1.08608
\(425\) 0 0
\(426\) 0 0
\(427\) −0.0750160 −0.00363028
\(428\) −60.2079 −2.91026
\(429\) 0 0
\(430\) 0 0
\(431\) 12.7651 0.614872 0.307436 0.951569i \(-0.400529\pi\)
0.307436 + 0.951569i \(0.400529\pi\)
\(432\) 0 0
\(433\) −16.0771 −0.772618 −0.386309 0.922369i \(-0.626250\pi\)
−0.386309 + 0.922369i \(0.626250\pi\)
\(434\) −5.52161 −0.265046
\(435\) 0 0
\(436\) 28.4437 1.36220
\(437\) 3.67939 0.176009
\(438\) 0 0
\(439\) 1.36945 0.0653604 0.0326802 0.999466i \(-0.489596\pi\)
0.0326802 + 0.999466i \(0.489596\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −74.6369 −3.55012
\(443\) −4.62280 −0.219636 −0.109818 0.993952i \(-0.535027\pi\)
−0.109818 + 0.993952i \(0.535027\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.59933 0.123082
\(447\) 0 0
\(448\) −8.17270 −0.386124
\(449\) −23.2555 −1.09749 −0.548747 0.835989i \(-0.684895\pi\)
−0.548747 + 0.835989i \(0.684895\pi\)
\(450\) 0 0
\(451\) 2.58383 0.121668
\(452\) 0.519485 0.0244345
\(453\) 0 0
\(454\) −47.8569 −2.24604
\(455\) 0 0
\(456\) 0 0
\(457\) −35.8443 −1.67673 −0.838364 0.545111i \(-0.816487\pi\)
−0.838364 + 0.545111i \(0.816487\pi\)
\(458\) 13.1140 0.612776
\(459\) 0 0
\(460\) 0 0
\(461\) 14.8812 0.693086 0.346543 0.938034i \(-0.387355\pi\)
0.346543 + 0.938034i \(0.387355\pi\)
\(462\) 0 0
\(463\) −29.9554 −1.39215 −0.696073 0.717971i \(-0.745071\pi\)
−0.696073 + 0.717971i \(0.745071\pi\)
\(464\) 4.61212 0.214112
\(465\) 0 0
\(466\) −43.2186 −2.00206
\(467\) 6.73598 0.311704 0.155852 0.987780i \(-0.450188\pi\)
0.155852 + 0.987780i \(0.450188\pi\)
\(468\) 0 0
\(469\) −8.55553 −0.395058
\(470\) 0 0
\(471\) 0 0
\(472\) −16.5109 −0.759977
\(473\) 2.68714 0.123555
\(474\) 0 0
\(475\) 0 0
\(476\) −13.9816 −0.640845
\(477\) 0 0
\(478\) 52.0998 2.38299
\(479\) −16.6978 −0.762943 −0.381471 0.924381i \(-0.624582\pi\)
−0.381471 + 0.924381i \(0.624582\pi\)
\(480\) 0 0
\(481\) −48.3636 −2.20519
\(482\) 67.0091 3.05218
\(483\) 0 0
\(484\) −39.8881 −1.81310
\(485\) 0 0
\(486\) 0 0
\(487\) 3.64042 0.164963 0.0824816 0.996593i \(-0.473715\pi\)
0.0824816 + 0.996593i \(0.473715\pi\)
\(488\) 0.405499 0.0183561
\(489\) 0 0
\(490\) 0 0
\(491\) 33.3249 1.50393 0.751965 0.659203i \(-0.229106\pi\)
0.751965 + 0.659203i \(0.229106\pi\)
\(492\) 0 0
\(493\) −11.9922 −0.540104
\(494\) 14.1522 0.636736
\(495\) 0 0
\(496\) 6.48827 0.291332
\(497\) −4.17833 −0.187424
\(498\) 0 0
\(499\) −37.9914 −1.70073 −0.850365 0.526193i \(-0.823619\pi\)
−0.850365 + 0.526193i \(0.823619\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 21.4183 0.955945
\(503\) 42.1826 1.88083 0.940414 0.340032i \(-0.110438\pi\)
0.940414 + 0.340032i \(0.110438\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.39563 −0.106499
\(507\) 0 0
\(508\) −55.2469 −2.45119
\(509\) 21.9971 0.975003 0.487502 0.873122i \(-0.337908\pi\)
0.487502 + 0.873122i \(0.337908\pi\)
\(510\) 0 0
\(511\) 4.84997 0.214550
\(512\) 22.0643 0.975115
\(513\) 0 0
\(514\) 16.3091 0.719365
\(515\) 0 0
\(516\) 0 0
\(517\) −3.33048 −0.146474
\(518\) −14.0227 −0.616121
\(519\) 0 0
\(520\) 0 0
\(521\) −20.0977 −0.880496 −0.440248 0.897876i \(-0.645109\pi\)
−0.440248 + 0.897876i \(0.645109\pi\)
\(522\) 0 0
\(523\) −4.64817 −0.203250 −0.101625 0.994823i \(-0.532404\pi\)
−0.101625 + 0.994823i \(0.532404\pi\)
\(524\) −20.3871 −0.890614
\(525\) 0 0
\(526\) −22.0021 −0.959338
\(527\) −16.8705 −0.734891
\(528\) 0 0
\(529\) −9.46209 −0.411395
\(530\) 0 0
\(531\) 0 0
\(532\) 2.65109 0.114939
\(533\) −56.1620 −2.43265
\(534\) 0 0
\(535\) 0 0
\(536\) 46.2469 1.99756
\(537\) 0 0
\(538\) 1.18608 0.0511355
\(539\) −1.77283 −0.0763612
\(540\) 0 0
\(541\) 20.0673 0.862759 0.431380 0.902171i \(-0.358027\pi\)
0.431380 + 0.902171i \(0.358027\pi\)
\(542\) −8.82087 −0.378889
\(543\) 0 0
\(544\) −15.9709 −0.684747
\(545\) 0 0
\(546\) 0 0
\(547\) −37.2010 −1.59060 −0.795300 0.606216i \(-0.792687\pi\)
−0.795300 + 0.606216i \(0.792687\pi\)
\(548\) −46.8414 −2.00097
\(549\) 0 0
\(550\) 0 0
\(551\) 2.27389 0.0968710
\(552\) 0 0
\(553\) 2.81604 0.119750
\(554\) −10.9036 −0.463251
\(555\) 0 0
\(556\) −55.6708 −2.36097
\(557\) −44.8393 −1.89990 −0.949951 0.312399i \(-0.898867\pi\)
−0.949951 + 0.312399i \(0.898867\pi\)
\(558\) 0 0
\(559\) −58.4076 −2.47038
\(560\) 0 0
\(561\) 0 0
\(562\) 64.8152 2.73407
\(563\) 21.9172 0.923701 0.461851 0.886958i \(-0.347186\pi\)
0.461851 + 0.886958i \(0.347186\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 24.4047 1.02581
\(567\) 0 0
\(568\) 22.5860 0.947685
\(569\) −9.90656 −0.415305 −0.207652 0.978203i \(-0.566582\pi\)
−0.207652 + 0.978203i \(0.566582\pi\)
\(570\) 0 0
\(571\) 17.6404 0.738229 0.369114 0.929384i \(-0.379661\pi\)
0.369114 + 0.929384i \(0.379661\pi\)
\(572\) −5.95328 −0.248919
\(573\) 0 0
\(574\) −16.2838 −0.679671
\(575\) 0 0
\(576\) 0 0
\(577\) −12.7048 −0.528906 −0.264453 0.964399i \(-0.585191\pi\)
−0.264453 + 0.964399i \(0.585191\pi\)
\(578\) −25.7069 −1.06927
\(579\) 0 0
\(580\) 0 0
\(581\) 0.354534 0.0147085
\(582\) 0 0
\(583\) −1.56058 −0.0646325
\(584\) −26.2165 −1.08485
\(585\) 0 0
\(586\) −4.44659 −0.183687
\(587\) −15.0438 −0.620924 −0.310462 0.950586i \(-0.600484\pi\)
−0.310462 + 0.950586i \(0.600484\pi\)
\(588\) 0 0
\(589\) 3.19887 0.131807
\(590\) 0 0
\(591\) 0 0
\(592\) 16.4776 0.677225
\(593\) 16.4231 0.674417 0.337208 0.941430i \(-0.390517\pi\)
0.337208 + 0.941430i \(0.390517\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −50.5003 −2.06857
\(597\) 0 0
\(598\) 52.0713 2.12935
\(599\) 19.1260 0.781466 0.390733 0.920504i \(-0.372222\pi\)
0.390733 + 0.920504i \(0.372222\pi\)
\(600\) 0 0
\(601\) 31.4124 1.28134 0.640670 0.767816i \(-0.278657\pi\)
0.640670 + 0.767816i \(0.278657\pi\)
\(602\) −16.9349 −0.690213
\(603\) 0 0
\(604\) −42.7459 −1.73930
\(605\) 0 0
\(606\) 0 0
\(607\) −41.5315 −1.68571 −0.842855 0.538140i \(-0.819127\pi\)
−0.842855 + 0.538140i \(0.819127\pi\)
\(608\) 3.02830 0.122814
\(609\) 0 0
\(610\) 0 0
\(611\) 72.3913 2.92864
\(612\) 0 0
\(613\) 21.7274 0.877563 0.438781 0.898594i \(-0.355410\pi\)
0.438781 + 0.898594i \(0.355410\pi\)
\(614\) −0.540031 −0.0217939
\(615\) 0 0
\(616\) −0.780579 −0.0314504
\(617\) −33.6065 −1.35295 −0.676473 0.736467i \(-0.736493\pi\)
−0.676473 + 0.736467i \(0.736493\pi\)
\(618\) 0 0
\(619\) −27.6036 −1.10948 −0.554741 0.832023i \(-0.687183\pi\)
−0.554741 + 0.832023i \(0.687183\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 49.8152 1.99741
\(623\) 11.9327 0.478075
\(624\) 0 0
\(625\) 0 0
\(626\) 26.7077 1.06745
\(627\) 0 0
\(628\) −17.4904 −0.697942
\(629\) −42.8443 −1.70832
\(630\) 0 0
\(631\) 1.94048 0.0772495 0.0386247 0.999254i \(-0.487702\pi\)
0.0386247 + 0.999254i \(0.487702\pi\)
\(632\) −15.2221 −0.605504
\(633\) 0 0
\(634\) 44.2702 1.75819
\(635\) 0 0
\(636\) 0 0
\(637\) 38.5342 1.52678
\(638\) −1.48052 −0.0586142
\(639\) 0 0
\(640\) 0 0
\(641\) −1.01975 −0.0402775 −0.0201388 0.999797i \(-0.506411\pi\)
−0.0201388 + 0.999797i \(0.506411\pi\)
\(642\) 0 0
\(643\) 36.9866 1.45861 0.729305 0.684189i \(-0.239844\pi\)
0.729305 + 0.684189i \(0.239844\pi\)
\(644\) 9.75441 0.384377
\(645\) 0 0
\(646\) 12.5371 0.493266
\(647\) −24.1182 −0.948186 −0.474093 0.880475i \(-0.657224\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(648\) 0 0
\(649\) 1.15215 0.0452260
\(650\) 0 0
\(651\) 0 0
\(652\) 46.7381 1.83041
\(653\) −37.2603 −1.45811 −0.729054 0.684456i \(-0.760040\pi\)
−0.729054 + 0.684456i \(0.760040\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 19.1345 0.747078
\(657\) 0 0
\(658\) 20.9893 0.818249
\(659\) −21.4386 −0.835130 −0.417565 0.908647i \(-0.637116\pi\)
−0.417565 + 0.908647i \(0.637116\pi\)
\(660\) 0 0
\(661\) 0.783503 0.0304747 0.0152374 0.999884i \(-0.495150\pi\)
0.0152374 + 0.999884i \(0.495150\pi\)
\(662\) 33.5315 1.30324
\(663\) 0 0
\(664\) −1.91643 −0.0743720
\(665\) 0 0
\(666\) 0 0
\(667\) 8.36653 0.323953
\(668\) 76.4634 2.95846
\(669\) 0 0
\(670\) 0 0
\(671\) −0.0282963 −0.00109237
\(672\) 0 0
\(673\) −50.1903 −1.93469 −0.967347 0.253454i \(-0.918433\pi\)
−0.967347 + 0.253454i \(0.918433\pi\)
\(674\) −54.6065 −2.10336
\(675\) 0 0
\(676\) 81.9362 3.15139
\(677\) 29.8804 1.14840 0.574198 0.818716i \(-0.305314\pi\)
0.574198 + 0.818716i \(0.305314\pi\)
\(678\) 0 0
\(679\) −3.22717 −0.123847
\(680\) 0 0
\(681\) 0 0
\(682\) −2.08277 −0.0797532
\(683\) −12.3326 −0.471894 −0.235947 0.971766i \(-0.575819\pi\)
−0.235947 + 0.971766i \(0.575819\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 23.2555 0.887898
\(687\) 0 0
\(688\) 19.8996 0.758666
\(689\) 33.9207 1.29227
\(690\) 0 0
\(691\) 3.62200 0.137787 0.0688936 0.997624i \(-0.478053\pi\)
0.0688936 + 0.997624i \(0.478053\pi\)
\(692\) −57.4252 −2.18298
\(693\) 0 0
\(694\) −9.34891 −0.354880
\(695\) 0 0
\(696\) 0 0
\(697\) −49.7528 −1.88452
\(698\) 81.8358 3.09753
\(699\) 0 0
\(700\) 0 0
\(701\) −34.1209 −1.28873 −0.644365 0.764718i \(-0.722878\pi\)
−0.644365 + 0.764718i \(0.722878\pi\)
\(702\) 0 0
\(703\) 8.12386 0.306397
\(704\) −3.08277 −0.116186
\(705\) 0 0
\(706\) −10.1161 −0.380725
\(707\) −3.18396 −0.119745
\(708\) 0 0
\(709\) −17.1209 −0.642990 −0.321495 0.946911i \(-0.604185\pi\)
−0.321495 + 0.946911i \(0.604185\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −64.5024 −2.41733
\(713\) 11.7699 0.440786
\(714\) 0 0
\(715\) 0 0
\(716\) −12.4621 −0.465730
\(717\) 0 0
\(718\) −48.2440 −1.80045
\(719\) 7.02750 0.262081 0.131041 0.991377i \(-0.458168\pi\)
0.131041 + 0.991377i \(0.458168\pi\)
\(720\) 0 0
\(721\) 2.41830 0.0900620
\(722\) −2.37720 −0.0884703
\(723\) 0 0
\(724\) 85.8406 3.19024
\(725\) 0 0
\(726\) 0 0
\(727\) −11.8938 −0.441115 −0.220558 0.975374i \(-0.570788\pi\)
−0.220558 + 0.975374i \(0.570788\pi\)
\(728\) 16.9667 0.628826
\(729\) 0 0
\(730\) 0 0
\(731\) −51.7421 −1.91375
\(732\) 0 0
\(733\) 20.7154 0.765142 0.382571 0.923926i \(-0.375039\pi\)
0.382571 + 0.923926i \(0.375039\pi\)
\(734\) −9.15910 −0.338069
\(735\) 0 0
\(736\) 11.1423 0.410710
\(737\) −3.22717 −0.118874
\(738\) 0 0
\(739\) 33.8620 1.24563 0.622816 0.782368i \(-0.285988\pi\)
0.622816 + 0.782368i \(0.285988\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.83505 0.361056
\(743\) −42.7381 −1.56791 −0.783955 0.620818i \(-0.786800\pi\)
−0.783955 + 0.620818i \(0.786800\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −34.7918 −1.27382
\(747\) 0 0
\(748\) −5.27389 −0.192833
\(749\) −11.9738 −0.437514
\(750\) 0 0
\(751\) 11.9581 0.436358 0.218179 0.975909i \(-0.429988\pi\)
0.218179 + 0.975909i \(0.429988\pi\)
\(752\) −24.6639 −0.899400
\(753\) 0 0
\(754\) 32.1805 1.17194
\(755\) 0 0
\(756\) 0 0
\(757\) 29.1103 1.05803 0.529015 0.848612i \(-0.322561\pi\)
0.529015 + 0.848612i \(0.322561\pi\)
\(758\) 52.3219 1.90042
\(759\) 0 0
\(760\) 0 0
\(761\) 22.3014 0.808425 0.404212 0.914665i \(-0.367546\pi\)
0.404212 + 0.914665i \(0.367546\pi\)
\(762\) 0 0
\(763\) 5.65672 0.204787
\(764\) −45.5109 −1.64653
\(765\) 0 0
\(766\) −7.33531 −0.265036
\(767\) −25.0432 −0.904258
\(768\) 0 0
\(769\) 3.95891 0.142762 0.0713809 0.997449i \(-0.477259\pi\)
0.0713809 + 0.997449i \(0.477259\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 70.4634 2.53603
\(773\) −27.8139 −1.00040 −0.500199 0.865911i \(-0.666740\pi\)
−0.500199 + 0.865911i \(0.666740\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 17.4445 0.626220
\(777\) 0 0
\(778\) 20.8529 0.747612
\(779\) 9.43380 0.338001
\(780\) 0 0
\(781\) −1.57608 −0.0563965
\(782\) 46.1289 1.64957
\(783\) 0 0
\(784\) −13.1287 −0.468882
\(785\) 0 0
\(786\) 0 0
\(787\) −1.82460 −0.0650398 −0.0325199 0.999471i \(-0.510353\pi\)
−0.0325199 + 0.999471i \(0.510353\pi\)
\(788\) −24.2087 −0.862401
\(789\) 0 0
\(790\) 0 0
\(791\) 0.103312 0.00367336
\(792\) 0 0
\(793\) 0.615047 0.0218410
\(794\) 3.79045 0.134518
\(795\) 0 0
\(796\) −84.2851 −2.98741
\(797\) 21.0360 0.745135 0.372567 0.928005i \(-0.378478\pi\)
0.372567 + 0.928005i \(0.378478\pi\)
\(798\) 0 0
\(799\) 64.1300 2.26876
\(800\) 0 0
\(801\) 0 0
\(802\) −41.7261 −1.47340
\(803\) 1.82942 0.0645589
\(804\) 0 0
\(805\) 0 0
\(806\) 45.2710 1.59460
\(807\) 0 0
\(808\) 17.2109 0.605476
\(809\) 0.0819654 0.00288175 0.00144088 0.999999i \(-0.499541\pi\)
0.00144088 + 0.999999i \(0.499541\pi\)
\(810\) 0 0
\(811\) −6.72531 −0.236158 −0.118079 0.993004i \(-0.537674\pi\)
−0.118079 + 0.993004i \(0.537674\pi\)
\(812\) 6.02830 0.211552
\(813\) 0 0
\(814\) −5.28939 −0.185393
\(815\) 0 0
\(816\) 0 0
\(817\) 9.81100 0.343243
\(818\) −87.1994 −3.04886
\(819\) 0 0
\(820\) 0 0
\(821\) −30.9426 −1.07990 −0.539952 0.841696i \(-0.681558\pi\)
−0.539952 + 0.841696i \(0.681558\pi\)
\(822\) 0 0
\(823\) 26.5908 0.926896 0.463448 0.886124i \(-0.346612\pi\)
0.463448 + 0.886124i \(0.346612\pi\)
\(824\) −13.0721 −0.455388
\(825\) 0 0
\(826\) −7.26109 −0.252646
\(827\) 5.57900 0.194001 0.0970004 0.995284i \(-0.469075\pi\)
0.0970004 + 0.995284i \(0.469075\pi\)
\(828\) 0 0
\(829\) 18.9765 0.659082 0.329541 0.944141i \(-0.393106\pi\)
0.329541 + 0.944141i \(0.393106\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 67.0069 2.32305
\(833\) 34.1367 1.18276
\(834\) 0 0
\(835\) 0 0
\(836\) 1.00000 0.0345857
\(837\) 0 0
\(838\) −44.7360 −1.54538
\(839\) −12.9143 −0.445852 −0.222926 0.974835i \(-0.571561\pi\)
−0.222926 + 0.974835i \(0.571561\pi\)
\(840\) 0 0
\(841\) −23.8294 −0.821704
\(842\) 80.3064 2.76754
\(843\) 0 0
\(844\) −27.5577 −0.948574
\(845\) 0 0
\(846\) 0 0
\(847\) −7.93273 −0.272572
\(848\) −11.5569 −0.396864
\(849\) 0 0
\(850\) 0 0
\(851\) 29.8908 1.02464
\(852\) 0 0
\(853\) 6.46077 0.221213 0.110606 0.993864i \(-0.464721\pi\)
0.110606 + 0.993864i \(0.464721\pi\)
\(854\) 0.178328 0.00610227
\(855\) 0 0
\(856\) 64.7245 2.21224
\(857\) 12.4055 0.423764 0.211882 0.977295i \(-0.432041\pi\)
0.211882 + 0.977295i \(0.432041\pi\)
\(858\) 0 0
\(859\) 40.3425 1.37647 0.688234 0.725489i \(-0.258386\pi\)
0.688234 + 0.725489i \(0.258386\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −30.3452 −1.03356
\(863\) −1.83235 −0.0623738 −0.0311869 0.999514i \(-0.509929\pi\)
−0.0311869 + 0.999514i \(0.509929\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 38.2186 1.29872
\(867\) 0 0
\(868\) 8.48052 0.287847
\(869\) 1.06222 0.0360333
\(870\) 0 0
\(871\) 70.1457 2.37680
\(872\) −30.5774 −1.03548
\(873\) 0 0
\(874\) −8.74666 −0.295860
\(875\) 0 0
\(876\) 0 0
\(877\) −8.14419 −0.275010 −0.137505 0.990501i \(-0.543908\pi\)
−0.137505 + 0.990501i \(0.543908\pi\)
\(878\) −3.25547 −0.109867
\(879\) 0 0
\(880\) 0 0
\(881\) −12.1706 −0.410037 −0.205019 0.978758i \(-0.565725\pi\)
−0.205019 + 0.978758i \(0.565725\pi\)
\(882\) 0 0
\(883\) 46.7614 1.57364 0.786822 0.617179i \(-0.211725\pi\)
0.786822 + 0.617179i \(0.211725\pi\)
\(884\) 114.633 3.85553
\(885\) 0 0
\(886\) 10.9893 0.369194
\(887\) 34.8804 1.17117 0.585584 0.810611i \(-0.300865\pi\)
0.585584 + 0.810611i \(0.300865\pi\)
\(888\) 0 0
\(889\) −10.9872 −0.368499
\(890\) 0 0
\(891\) 0 0
\(892\) −3.99225 −0.133670
\(893\) −12.1599 −0.406916
\(894\) 0 0
\(895\) 0 0
\(896\) 15.0304 0.502131
\(897\) 0 0
\(898\) 55.2830 1.84482
\(899\) 7.27389 0.242598
\(900\) 0 0
\(901\) 30.0496 1.00110
\(902\) −6.14228 −0.204516
\(903\) 0 0
\(904\) −0.558455 −0.0185739
\(905\) 0 0
\(906\) 0 0
\(907\) −25.5080 −0.846980 −0.423490 0.905901i \(-0.639195\pi\)
−0.423490 + 0.905901i \(0.639195\pi\)
\(908\) 73.5024 2.43926
\(909\) 0 0
\(910\) 0 0
\(911\) 21.5032 0.712432 0.356216 0.934404i \(-0.384067\pi\)
0.356216 + 0.934404i \(0.384067\pi\)
\(912\) 0 0
\(913\) 0.133731 0.00442586
\(914\) 85.2093 2.81847
\(915\) 0 0
\(916\) −20.1415 −0.665493
\(917\) −4.05447 −0.133890
\(918\) 0 0
\(919\) 37.1386 1.22509 0.612544 0.790436i \(-0.290146\pi\)
0.612544 + 0.790436i \(0.290146\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −35.3756 −1.16503
\(923\) 34.2576 1.12760
\(924\) 0 0
\(925\) 0 0
\(926\) 71.2101 2.34011
\(927\) 0 0
\(928\) 6.88601 0.226044
\(929\) 3.36170 0.110294 0.0551469 0.998478i \(-0.482437\pi\)
0.0551469 + 0.998478i \(0.482437\pi\)
\(930\) 0 0
\(931\) −6.47277 −0.212136
\(932\) 66.3785 2.17430
\(933\) 0 0
\(934\) −16.0128 −0.523955
\(935\) 0 0
\(936\) 0 0
\(937\) 10.0694 0.328953 0.164476 0.986381i \(-0.447407\pi\)
0.164476 + 0.986381i \(0.447407\pi\)
\(938\) 20.3382 0.664067
\(939\) 0 0
\(940\) 0 0
\(941\) −20.8139 −0.678514 −0.339257 0.940694i \(-0.610176\pi\)
−0.339257 + 0.940694i \(0.610176\pi\)
\(942\) 0 0
\(943\) 34.7106 1.13033
\(944\) 8.53228 0.277702
\(945\) 0 0
\(946\) −6.38788 −0.207688
\(947\) 30.4904 0.990804 0.495402 0.868664i \(-0.335021\pi\)
0.495402 + 0.868664i \(0.335021\pi\)
\(948\) 0 0
\(949\) −39.7643 −1.29080
\(950\) 0 0
\(951\) 0 0
\(952\) 15.0304 0.487139
\(953\) −7.58383 −0.245664 −0.122832 0.992427i \(-0.539198\pi\)
−0.122832 + 0.992427i \(0.539198\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −80.0189 −2.58800
\(957\) 0 0
\(958\) 39.6941 1.28246
\(959\) −9.31556 −0.300815
\(960\) 0 0
\(961\) −20.7672 −0.669910
\(962\) 114.970 3.70678
\(963\) 0 0
\(964\) −102.918 −3.31476
\(965\) 0 0
\(966\) 0 0
\(967\) 54.6687 1.75803 0.879014 0.476797i \(-0.158202\pi\)
0.879014 + 0.476797i \(0.158202\pi\)
\(968\) 42.8804 1.37823
\(969\) 0 0
\(970\) 0 0
\(971\) 39.9632 1.28248 0.641239 0.767341i \(-0.278421\pi\)
0.641239 + 0.767341i \(0.278421\pi\)
\(972\) 0 0
\(973\) −11.0715 −0.354936
\(974\) −8.65402 −0.277293
\(975\) 0 0
\(976\) −0.209548 −0.00670747
\(977\) −15.2400 −0.487570 −0.243785 0.969829i \(-0.578389\pi\)
−0.243785 + 0.969829i \(0.578389\pi\)
\(978\) 0 0
\(979\) 4.50106 0.143855
\(980\) 0 0
\(981\) 0 0
\(982\) −79.2199 −2.52801
\(983\) 45.3609 1.44679 0.723394 0.690435i \(-0.242581\pi\)
0.723394 + 0.690435i \(0.242581\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 28.5080 0.907880
\(987\) 0 0
\(988\) −21.7360 −0.691514
\(989\) 36.0985 1.14787
\(990\) 0 0
\(991\) −16.3537 −0.519493 −0.259747 0.965677i \(-0.583639\pi\)
−0.259747 + 0.965677i \(0.583639\pi\)
\(992\) 9.68714 0.307567
\(993\) 0 0
\(994\) 9.93273 0.315047
\(995\) 0 0
\(996\) 0 0
\(997\) −33.2037 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(998\) 90.3134 2.85882
\(999\) 0 0
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 4275.2.a.ba.1.1 3
3.2 odd 2 475.2.a.g.1.3 yes 3
5.4 even 2 4275.2.a.bm.1.3 3
12.11 even 2 7600.2.a.bh.1.2 3
15.2 even 4 475.2.b.b.324.6 6
15.8 even 4 475.2.b.b.324.1 6
15.14 odd 2 475.2.a.e.1.1 3
57.56 even 2 9025.2.a.y.1.1 3
60.59 even 2 7600.2.a.cc.1.2 3
285.284 even 2 9025.2.a.bc.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.1 3 15.14 odd 2
475.2.a.g.1.3 yes 3 3.2 odd 2
475.2.b.b.324.1 6 15.8 even 4
475.2.b.b.324.6 6 15.2 even 4
4275.2.a.ba.1.1 3 1.1 even 1 trivial
4275.2.a.bm.1.3 3 5.4 even 2
7600.2.a.bh.1.2 3 12.11 even 2
7600.2.a.cc.1.2 3 60.59 even 2
9025.2.a.y.1.1 3 57.56 even 2
9025.2.a.bc.1.3 3 285.284 even 2