Properties

Label 9072.2.a.ca.1.2
Level $9072$
Weight $2$
Character 9072.1
Self dual yes
Analytic conductor $72.440$
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] = [9072,2,Mod(1,9072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9072, 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("9072.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.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: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.34730 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.34730 q^{5} -1.00000 q^{7} -1.65270 q^{11} -3.36959 q^{13} +0.467911 q^{17} +3.22668 q^{19} -8.94356 q^{23} -3.18479 q^{25} +6.26857 q^{29} -9.23442 q^{31} -1.34730 q^{35} +9.23442 q^{37} +3.41147 q^{41} +4.41147 q^{43} -9.35504 q^{47} +1.00000 q^{49} -0.573978 q^{53} -2.22668 q^{55} +10.3969 q^{59} +7.63816 q^{61} -4.53983 q^{65} -0.596267 q^{67} +0.554378 q^{71} +2.04963 q^{73} +1.65270 q^{77} +2.40373 q^{79} +15.0496 q^{83} +0.630415 q^{85} +9.08647 q^{89} +3.36959 q^{91} +4.34730 q^{95} -1.89899 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 3 q^{7} - 6 q^{11} - 3 q^{13} + 6 q^{17} + 3 q^{19} - 12 q^{23} - 6 q^{25} + 9 q^{29} + 3 q^{31} - 3 q^{35} - 3 q^{37} + 3 q^{43} - 3 q^{47} + 3 q^{49} + 6 q^{53} + 3 q^{59} + 6 q^{61} + 15 q^{65} + 12 q^{67} - 9 q^{71} - 21 q^{73} + 6 q^{77} + 21 q^{79} + 18 q^{83} + 9 q^{85} + 12 q^{89} + 3 q^{91} + 12 q^{95} - 3 q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) 1.34730 0.602529 0.301265 0.953541i \(-0.402591\pi\)
0.301265 + 0.953541i \(0.402591\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.65270 −0.498309 −0.249154 0.968464i \(-0.580153\pi\)
−0.249154 + 0.968464i \(0.580153\pi\)
\(12\) 0 0
\(13\) −3.36959 −0.934555 −0.467277 0.884111i \(-0.654765\pi\)
−0.467277 + 0.884111i \(0.654765\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.467911 0.113485 0.0567426 0.998389i \(-0.481929\pi\)
0.0567426 + 0.998389i \(0.481929\pi\)
\(18\) 0 0
\(19\) 3.22668 0.740252 0.370126 0.928982i \(-0.379315\pi\)
0.370126 + 0.928982i \(0.379315\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.94356 −1.86486 −0.932431 0.361348i \(-0.882317\pi\)
−0.932431 + 0.361348i \(0.882317\pi\)
\(24\) 0 0
\(25\) −3.18479 −0.636959
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.26857 1.16404 0.582022 0.813173i \(-0.302262\pi\)
0.582022 + 0.813173i \(0.302262\pi\)
\(30\) 0 0
\(31\) −9.23442 −1.65855 −0.829276 0.558840i \(-0.811247\pi\)
−0.829276 + 0.558840i \(0.811247\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.34730 −0.227735
\(36\) 0 0
\(37\) 9.23442 1.51813 0.759065 0.651015i \(-0.225657\pi\)
0.759065 + 0.651015i \(0.225657\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.41147 0.532783 0.266391 0.963865i \(-0.414169\pi\)
0.266391 + 0.963865i \(0.414169\pi\)
\(42\) 0 0
\(43\) 4.41147 0.672743 0.336372 0.941729i \(-0.390800\pi\)
0.336372 + 0.941729i \(0.390800\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.35504 −1.36457 −0.682286 0.731085i \(-0.739014\pi\)
−0.682286 + 0.731085i \(0.739014\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.573978 −0.0788419 −0.0394210 0.999223i \(-0.512551\pi\)
−0.0394210 + 0.999223i \(0.512551\pi\)
\(54\) 0 0
\(55\) −2.22668 −0.300246
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.3969 1.35356 0.676782 0.736183i \(-0.263374\pi\)
0.676782 + 0.736183i \(0.263374\pi\)
\(60\) 0 0
\(61\) 7.63816 0.977966 0.488983 0.872293i \(-0.337368\pi\)
0.488983 + 0.872293i \(0.337368\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.53983 −0.563097
\(66\) 0 0
\(67\) −0.596267 −0.0728456 −0.0364228 0.999336i \(-0.511596\pi\)
−0.0364228 + 0.999336i \(0.511596\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.554378 0.0657925 0.0328963 0.999459i \(-0.489527\pi\)
0.0328963 + 0.999459i \(0.489527\pi\)
\(72\) 0 0
\(73\) 2.04963 0.239891 0.119946 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.65270 0.188343
\(78\) 0 0
\(79\) 2.40373 0.270441 0.135221 0.990816i \(-0.456826\pi\)
0.135221 + 0.990816i \(0.456826\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.0496 1.65191 0.825956 0.563735i \(-0.190636\pi\)
0.825956 + 0.563735i \(0.190636\pi\)
\(84\) 0 0
\(85\) 0.630415 0.0683781
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.08647 0.963164 0.481582 0.876401i \(-0.340062\pi\)
0.481582 + 0.876401i \(0.340062\pi\)
\(90\) 0 0
\(91\) 3.36959 0.353228
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.34730 0.446023
\(96\) 0 0
\(97\) −1.89899 −0.192813 −0.0964064 0.995342i \(-0.530735\pi\)
−0.0964064 + 0.995342i \(0.530735\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.70914 −0.170066 −0.0850329 0.996378i \(-0.527100\pi\)
−0.0850329 + 0.996378i \(0.527100\pi\)
\(102\) 0 0
\(103\) 3.63816 0.358478 0.179239 0.983806i \(-0.442636\pi\)
0.179239 + 0.983806i \(0.442636\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.12836 −0.689124 −0.344562 0.938764i \(-0.611973\pi\)
−0.344562 + 0.938764i \(0.611973\pi\)
\(108\) 0 0
\(109\) 0.403733 0.0386706 0.0193353 0.999813i \(-0.493845\pi\)
0.0193353 + 0.999813i \(0.493845\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.3696 1.35178 0.675888 0.737004i \(-0.263760\pi\)
0.675888 + 0.737004i \(0.263760\pi\)
\(114\) 0 0
\(115\) −12.0496 −1.12363
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.467911 −0.0428933
\(120\) 0 0
\(121\) −8.26857 −0.751688
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.0273 −0.986315
\(126\) 0 0
\(127\) 20.7716 1.84318 0.921589 0.388167i \(-0.126892\pi\)
0.921589 + 0.388167i \(0.126892\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.16519 0.626026 0.313013 0.949749i \(-0.398662\pi\)
0.313013 + 0.949749i \(0.398662\pi\)
\(132\) 0 0
\(133\) −3.22668 −0.279789
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.56893 0.219478 0.109739 0.993960i \(-0.464998\pi\)
0.109739 + 0.993960i \(0.464998\pi\)
\(138\) 0 0
\(139\) 6.13341 0.520229 0.260114 0.965578i \(-0.416240\pi\)
0.260114 + 0.965578i \(0.416240\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.56893 0.465697
\(144\) 0 0
\(145\) 8.44562 0.701371
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.431074 0.0353150 0.0176575 0.999844i \(-0.494379\pi\)
0.0176575 + 0.999844i \(0.494379\pi\)
\(150\) 0 0
\(151\) 2.47060 0.201055 0.100527 0.994934i \(-0.467947\pi\)
0.100527 + 0.994934i \(0.467947\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −12.4415 −0.999326
\(156\) 0 0
\(157\) 10.1334 0.808734 0.404367 0.914597i \(-0.367492\pi\)
0.404367 + 0.914597i \(0.367492\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.94356 0.704852
\(162\) 0 0
\(163\) 2.59627 0.203355 0.101678 0.994817i \(-0.467579\pi\)
0.101678 + 0.994817i \(0.467579\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 23.1830 1.79396 0.896979 0.442074i \(-0.145757\pi\)
0.896979 + 0.442074i \(0.145757\pi\)
\(168\) 0 0
\(169\) −1.64590 −0.126607
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.75196 −0.361285 −0.180643 0.983549i \(-0.557818\pi\)
−0.180643 + 0.983549i \(0.557818\pi\)
\(174\) 0 0
\(175\) 3.18479 0.240748
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.53209 0.637718 0.318859 0.947802i \(-0.396700\pi\)
0.318859 + 0.947802i \(0.396700\pi\)
\(180\) 0 0
\(181\) −17.2344 −1.28102 −0.640512 0.767948i \(-0.721278\pi\)
−0.640512 + 0.767948i \(0.721278\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.4415 0.914718
\(186\) 0 0
\(187\) −0.773318 −0.0565506
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.9094 −0.934092 −0.467046 0.884233i \(-0.654682\pi\)
−0.467046 + 0.884233i \(0.654682\pi\)
\(192\) 0 0
\(193\) −0.638156 −0.0459355 −0.0229677 0.999736i \(-0.507311\pi\)
−0.0229677 + 0.999736i \(0.507311\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.4456 0.815467 0.407733 0.913101i \(-0.366319\pi\)
0.407733 + 0.913101i \(0.366319\pi\)
\(198\) 0 0
\(199\) 3.63816 0.257902 0.128951 0.991651i \(-0.458839\pi\)
0.128951 + 0.991651i \(0.458839\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.26857 −0.439967
\(204\) 0 0
\(205\) 4.59627 0.321017
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.33275 −0.368874
\(210\) 0 0
\(211\) −5.82295 −0.400868 −0.200434 0.979707i \(-0.564235\pi\)
−0.200434 + 0.979707i \(0.564235\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.94356 0.405348
\(216\) 0 0
\(217\) 9.23442 0.626873
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.57667 −0.106058
\(222\) 0 0
\(223\) −7.08378 −0.474365 −0.237182 0.971465i \(-0.576224\pi\)
−0.237182 + 0.971465i \(0.576224\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.9436 0.792722 0.396361 0.918095i \(-0.370273\pi\)
0.396361 + 0.918095i \(0.370273\pi\)
\(228\) 0 0
\(229\) −17.5526 −1.15991 −0.579955 0.814649i \(-0.696930\pi\)
−0.579955 + 0.814649i \(0.696930\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.2540 1.06484 0.532418 0.846481i \(-0.321283\pi\)
0.532418 + 0.846481i \(0.321283\pi\)
\(234\) 0 0
\(235\) −12.6040 −0.822195
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.0993 0.976690 0.488345 0.872651i \(-0.337601\pi\)
0.488345 + 0.872651i \(0.337601\pi\)
\(240\) 0 0
\(241\) −15.6382 −1.00734 −0.503671 0.863896i \(-0.668018\pi\)
−0.503671 + 0.863896i \(0.668018\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.34730 0.0860756
\(246\) 0 0
\(247\) −10.8726 −0.691806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −19.0651 −1.20338 −0.601690 0.798730i \(-0.705506\pi\)
−0.601690 + 0.798730i \(0.705506\pi\)
\(252\) 0 0
\(253\) 14.7811 0.929277
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.5817 1.65812 0.829061 0.559158i \(-0.188876\pi\)
0.829061 + 0.559158i \(0.188876\pi\)
\(258\) 0 0
\(259\) −9.23442 −0.573799
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.734118 0.0452676 0.0226338 0.999744i \(-0.492795\pi\)
0.0226338 + 0.999744i \(0.492795\pi\)
\(264\) 0 0
\(265\) −0.773318 −0.0475046
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −20.8503 −1.27126 −0.635632 0.771992i \(-0.719261\pi\)
−0.635632 + 0.771992i \(0.719261\pi\)
\(270\) 0 0
\(271\) −6.95811 −0.422675 −0.211338 0.977413i \(-0.567782\pi\)
−0.211338 + 0.977413i \(0.567782\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.26352 0.317402
\(276\) 0 0
\(277\) 17.8726 1.07386 0.536930 0.843627i \(-0.319584\pi\)
0.536930 + 0.843627i \(0.319584\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.3105 1.33093 0.665465 0.746429i \(-0.268233\pi\)
0.665465 + 0.746429i \(0.268233\pi\)
\(282\) 0 0
\(283\) 18.5945 1.10533 0.552665 0.833404i \(-0.313611\pi\)
0.552665 + 0.833404i \(0.313611\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.41147 −0.201373
\(288\) 0 0
\(289\) −16.7811 −0.987121
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.0915 0.764815 0.382407 0.923994i \(-0.375095\pi\)
0.382407 + 0.923994i \(0.375095\pi\)
\(294\) 0 0
\(295\) 14.0077 0.815562
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 30.1361 1.74282
\(300\) 0 0
\(301\) −4.41147 −0.254273
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.2909 0.589253
\(306\) 0 0
\(307\) −6.31046 −0.360157 −0.180078 0.983652i \(-0.557635\pi\)
−0.180078 + 0.983652i \(0.557635\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.52435 0.540076 0.270038 0.962850i \(-0.412964\pi\)
0.270038 + 0.962850i \(0.412964\pi\)
\(312\) 0 0
\(313\) −17.6287 −0.996431 −0.498215 0.867053i \(-0.666011\pi\)
−0.498215 + 0.867053i \(0.666011\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.07697 0.453648 0.226824 0.973936i \(-0.427166\pi\)
0.226824 + 0.973936i \(0.427166\pi\)
\(318\) 0 0
\(319\) −10.3601 −0.580054
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.50980 0.0840075
\(324\) 0 0
\(325\) 10.7314 0.595273
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.35504 0.515760
\(330\) 0 0
\(331\) −23.0496 −1.26692 −0.633461 0.773775i \(-0.718366\pi\)
−0.633461 + 0.773775i \(0.718366\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.803348 −0.0438916
\(336\) 0 0
\(337\) 29.0232 1.58100 0.790498 0.612465i \(-0.209822\pi\)
0.790498 + 0.612465i \(0.209822\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15.2618 0.826471
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.9463 −0.694991 −0.347496 0.937682i \(-0.612968\pi\)
−0.347496 + 0.937682i \(0.612968\pi\)
\(348\) 0 0
\(349\) 1.46286 0.0783050 0.0391525 0.999233i \(-0.487534\pi\)
0.0391525 + 0.999233i \(0.487534\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.3327 0.762855 0.381428 0.924399i \(-0.375433\pi\)
0.381428 + 0.924399i \(0.375433\pi\)
\(354\) 0 0
\(355\) 0.746911 0.0396419
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.9368 1.10500 0.552500 0.833513i \(-0.313674\pi\)
0.552500 + 0.833513i \(0.313674\pi\)
\(360\) 0 0
\(361\) −8.58853 −0.452028
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.76146 0.144541
\(366\) 0 0
\(367\) 12.0574 0.629390 0.314695 0.949193i \(-0.398098\pi\)
0.314695 + 0.949193i \(0.398098\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.573978 0.0297995
\(372\) 0 0
\(373\) −0.781059 −0.0404417 −0.0202209 0.999796i \(-0.506437\pi\)
−0.0202209 + 0.999796i \(0.506437\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −21.1225 −1.08786
\(378\) 0 0
\(379\) 6.92396 0.355660 0.177830 0.984061i \(-0.443092\pi\)
0.177830 + 0.984061i \(0.443092\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.73236 −0.395105 −0.197553 0.980292i \(-0.563299\pi\)
−0.197553 + 0.980292i \(0.563299\pi\)
\(384\) 0 0
\(385\) 2.22668 0.113482
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.39961 0.273771 0.136886 0.990587i \(-0.456291\pi\)
0.136886 + 0.990587i \(0.456291\pi\)
\(390\) 0 0
\(391\) −4.18479 −0.211634
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.23854 0.162949
\(396\) 0 0
\(397\) −29.2344 −1.46723 −0.733617 0.679563i \(-0.762169\pi\)
−0.733617 + 0.679563i \(0.762169\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −27.3979 −1.36818 −0.684092 0.729396i \(-0.739801\pi\)
−0.684092 + 0.729396i \(0.739801\pi\)
\(402\) 0 0
\(403\) 31.1162 1.55001
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −15.2618 −0.756498
\(408\) 0 0
\(409\) −9.02498 −0.446256 −0.223128 0.974789i \(-0.571627\pi\)
−0.223128 + 0.974789i \(0.571627\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −10.3969 −0.511599
\(414\) 0 0
\(415\) 20.2763 0.995325
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.175297 −0.00856382 −0.00428191 0.999991i \(-0.501363\pi\)
−0.00428191 + 0.999991i \(0.501363\pi\)
\(420\) 0 0
\(421\) −24.7050 −1.20405 −0.602025 0.798478i \(-0.705639\pi\)
−0.602025 + 0.798478i \(0.705639\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.49020 −0.0722853
\(426\) 0 0
\(427\) −7.63816 −0.369636
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 29.3191 1.41225 0.706126 0.708086i \(-0.250441\pi\)
0.706126 + 0.708086i \(0.250441\pi\)
\(432\) 0 0
\(433\) 19.6554 0.944578 0.472289 0.881444i \(-0.343428\pi\)
0.472289 + 0.881444i \(0.343428\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −28.8580 −1.38047
\(438\) 0 0
\(439\) 21.9299 1.04666 0.523330 0.852130i \(-0.324690\pi\)
0.523330 + 0.852130i \(0.324690\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.7101 0.888942 0.444471 0.895793i \(-0.353392\pi\)
0.444471 + 0.895793i \(0.353392\pi\)
\(444\) 0 0
\(445\) 12.2422 0.580334
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.68004 0.315251 0.157625 0.987499i \(-0.449616\pi\)
0.157625 + 0.987499i \(0.449616\pi\)
\(450\) 0 0
\(451\) −5.63816 −0.265490
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.53983 0.212830
\(456\) 0 0
\(457\) −19.4287 −0.908837 −0.454418 0.890788i \(-0.650153\pi\)
−0.454418 + 0.890788i \(0.650153\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.965852 −0.0449842 −0.0224921 0.999747i \(-0.507160\pi\)
−0.0224921 + 0.999747i \(0.507160\pi\)
\(462\) 0 0
\(463\) 0.445622 0.0207098 0.0103549 0.999946i \(-0.496704\pi\)
0.0103549 + 0.999946i \(0.496704\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 34.2148 1.58327 0.791637 0.610992i \(-0.209229\pi\)
0.791637 + 0.610992i \(0.209229\pi\)
\(468\) 0 0
\(469\) 0.596267 0.0275330
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.29086 −0.335234
\(474\) 0 0
\(475\) −10.2763 −0.471510
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.7929 0.995744 0.497872 0.867251i \(-0.334115\pi\)
0.497872 + 0.867251i \(0.334115\pi\)
\(480\) 0 0
\(481\) −31.1162 −1.41878
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.55850 −0.116175
\(486\) 0 0
\(487\) −19.3928 −0.878772 −0.439386 0.898298i \(-0.644804\pi\)
−0.439386 + 0.898298i \(0.644804\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −26.1566 −1.18043 −0.590216 0.807245i \(-0.700957\pi\)
−0.590216 + 0.807245i \(0.700957\pi\)
\(492\) 0 0
\(493\) 2.93313 0.132102
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.554378 −0.0248672
\(498\) 0 0
\(499\) 14.3013 0.640214 0.320107 0.947381i \(-0.396281\pi\)
0.320107 + 0.947381i \(0.396281\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −18.7033 −0.833937 −0.416969 0.908921i \(-0.636908\pi\)
−0.416969 + 0.908921i \(0.636908\pi\)
\(504\) 0 0
\(505\) −2.30272 −0.102470
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −25.6091 −1.13510 −0.567551 0.823338i \(-0.692109\pi\)
−0.567551 + 0.823338i \(0.692109\pi\)
\(510\) 0 0
\(511\) −2.04963 −0.0906703
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.90167 0.215994
\(516\) 0 0
\(517\) 15.4611 0.679979
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −21.2121 −0.929320 −0.464660 0.885489i \(-0.653824\pi\)
−0.464660 + 0.885489i \(0.653824\pi\)
\(522\) 0 0
\(523\) −20.8057 −0.909770 −0.454885 0.890550i \(-0.650320\pi\)
−0.454885 + 0.890550i \(0.650320\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.32089 −0.188221
\(528\) 0 0
\(529\) 56.9873 2.47771
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −11.4953 −0.497915
\(534\) 0 0
\(535\) −9.60401 −0.415217
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.65270 −0.0711870
\(540\) 0 0
\(541\) 26.7297 1.14920 0.574599 0.818435i \(-0.305158\pi\)
0.574599 + 0.818435i \(0.305158\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.543948 0.0233002
\(546\) 0 0
\(547\) −36.7624 −1.57185 −0.785923 0.618324i \(-0.787812\pi\)
−0.785923 + 0.618324i \(0.787812\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 20.2267 0.861686
\(552\) 0 0
\(553\) −2.40373 −0.102217
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.3387 1.37024 0.685118 0.728432i \(-0.259751\pi\)
0.685118 + 0.728432i \(0.259751\pi\)
\(558\) 0 0
\(559\) −14.8648 −0.628716
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 17.7419 0.747730 0.373865 0.927483i \(-0.378032\pi\)
0.373865 + 0.927483i \(0.378032\pi\)
\(564\) 0 0
\(565\) 19.3601 0.814485
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.6013 −1.11519 −0.557593 0.830115i \(-0.688275\pi\)
−0.557593 + 0.830115i \(0.688275\pi\)
\(570\) 0 0
\(571\) 10.0172 0.419208 0.209604 0.977786i \(-0.432782\pi\)
0.209604 + 0.977786i \(0.432782\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 28.4834 1.18784
\(576\) 0 0
\(577\) −32.9145 −1.37025 −0.685124 0.728427i \(-0.740252\pi\)
−0.685124 + 0.728427i \(0.740252\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −15.0496 −0.624364
\(582\) 0 0
\(583\) 0.948615 0.0392876
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0729 0.622123 0.311062 0.950390i \(-0.399315\pi\)
0.311062 + 0.950390i \(0.399315\pi\)
\(588\) 0 0
\(589\) −29.7965 −1.22775
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 41.0009 1.68371 0.841853 0.539706i \(-0.181465\pi\)
0.841853 + 0.539706i \(0.181465\pi\)
\(594\) 0 0
\(595\) −0.630415 −0.0258445
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.07367 −0.248164 −0.124082 0.992272i \(-0.539599\pi\)
−0.124082 + 0.992272i \(0.539599\pi\)
\(600\) 0 0
\(601\) −14.1352 −0.576585 −0.288293 0.957542i \(-0.593088\pi\)
−0.288293 + 0.957542i \(0.593088\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.1402 −0.452914
\(606\) 0 0
\(607\) −46.0898 −1.87073 −0.935363 0.353689i \(-0.884927\pi\)
−0.935363 + 0.353689i \(0.884927\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 31.5226 1.27527
\(612\) 0 0
\(613\) −26.4938 −1.07008 −0.535038 0.844828i \(-0.679703\pi\)
−0.535038 + 0.844828i \(0.679703\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.24990 −0.0905777 −0.0452889 0.998974i \(-0.514421\pi\)
−0.0452889 + 0.998974i \(0.514421\pi\)
\(618\) 0 0
\(619\) −6.19078 −0.248828 −0.124414 0.992230i \(-0.539705\pi\)
−0.124414 + 0.992230i \(0.539705\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.08647 −0.364042
\(624\) 0 0
\(625\) 1.06687 0.0426746
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.32089 0.172285
\(630\) 0 0
\(631\) −26.1661 −1.04166 −0.520829 0.853661i \(-0.674377\pi\)
−0.520829 + 0.853661i \(0.674377\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 27.9855 1.11057
\(636\) 0 0
\(637\) −3.36959 −0.133508
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.88888 0.193099 0.0965496 0.995328i \(-0.469219\pi\)
0.0965496 + 0.995328i \(0.469219\pi\)
\(642\) 0 0
\(643\) 40.3678 1.59195 0.795976 0.605328i \(-0.206958\pi\)
0.795976 + 0.605328i \(0.206958\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.28075 0.0896657 0.0448329 0.998995i \(-0.485724\pi\)
0.0448329 + 0.998995i \(0.485724\pi\)
\(648\) 0 0
\(649\) −17.1830 −0.674493
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23.4793 0.918815 0.459407 0.888226i \(-0.348062\pi\)
0.459407 + 0.888226i \(0.348062\pi\)
\(654\) 0 0
\(655\) 9.65364 0.377199
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.9623 1.86835 0.934174 0.356818i \(-0.116138\pi\)
0.934174 + 0.356818i \(0.116138\pi\)
\(660\) 0 0
\(661\) 29.3090 1.13999 0.569995 0.821648i \(-0.306945\pi\)
0.569995 + 0.821648i \(0.306945\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.34730 −0.168581
\(666\) 0 0
\(667\) −56.0634 −2.17078
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.6236 −0.487329
\(672\) 0 0
\(673\) 26.3182 1.01449 0.507246 0.861801i \(-0.330664\pi\)
0.507246 + 0.861801i \(0.330664\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 35.8907 1.37939 0.689697 0.724098i \(-0.257744\pi\)
0.689697 + 0.724098i \(0.257744\pi\)
\(678\) 0 0
\(679\) 1.89899 0.0728764
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −35.0642 −1.34169 −0.670847 0.741596i \(-0.734069\pi\)
−0.670847 + 0.741596i \(0.734069\pi\)
\(684\) 0 0
\(685\) 3.46110 0.132242
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.93407 0.0736821
\(690\) 0 0
\(691\) −2.06687 −0.0786273 −0.0393136 0.999227i \(-0.512517\pi\)
−0.0393136 + 0.999227i \(0.512517\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.26352 0.313453
\(696\) 0 0
\(697\) 1.59627 0.0604629
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −7.36009 −0.277987 −0.138993 0.990293i \(-0.544387\pi\)
−0.138993 + 0.990293i \(0.544387\pi\)
\(702\) 0 0
\(703\) 29.7965 1.12380
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.70914 0.0642788
\(708\) 0 0
\(709\) 9.10876 0.342086 0.171043 0.985264i \(-0.445286\pi\)
0.171043 + 0.985264i \(0.445286\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 82.5886 3.09297
\(714\) 0 0
\(715\) 7.50299 0.280596
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 25.9537 0.967908 0.483954 0.875093i \(-0.339200\pi\)
0.483954 + 0.875093i \(0.339200\pi\)
\(720\) 0 0
\(721\) −3.63816 −0.135492
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −19.9641 −0.741448
\(726\) 0 0
\(727\) 10.1601 0.376819 0.188409 0.982091i \(-0.439667\pi\)
0.188409 + 0.982091i \(0.439667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.06418 0.0763464
\(732\) 0 0
\(733\) 40.6614 1.50186 0.750931 0.660381i \(-0.229605\pi\)
0.750931 + 0.660381i \(0.229605\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.985452 0.0362996
\(738\) 0 0
\(739\) 25.3618 0.932951 0.466475 0.884534i \(-0.345524\pi\)
0.466475 + 0.884534i \(0.345524\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.4442 0.823398 0.411699 0.911320i \(-0.364936\pi\)
0.411699 + 0.911320i \(0.364936\pi\)
\(744\) 0 0
\(745\) 0.580785 0.0212783
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.12836 0.260464
\(750\) 0 0
\(751\) −24.2172 −0.883698 −0.441849 0.897090i \(-0.645677\pi\)
−0.441849 + 0.897090i \(0.645677\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.32863 0.121141
\(756\) 0 0
\(757\) 9.11793 0.331397 0.165698 0.986176i \(-0.447012\pi\)
0.165698 + 0.986176i \(0.447012\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.2722 −0.662366 −0.331183 0.943566i \(-0.607448\pi\)
−0.331183 + 0.943566i \(0.607448\pi\)
\(762\) 0 0
\(763\) −0.403733 −0.0146161
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −35.0333 −1.26498
\(768\) 0 0
\(769\) 18.5294 0.668187 0.334094 0.942540i \(-0.391570\pi\)
0.334094 + 0.942540i \(0.391570\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.96080 −0.106493 −0.0532463 0.998581i \(-0.516957\pi\)
−0.0532463 + 0.998581i \(0.516957\pi\)
\(774\) 0 0
\(775\) 29.4097 1.05643
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 11.0077 0.394393
\(780\) 0 0
\(781\) −0.916222 −0.0327850
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 13.6527 0.487286
\(786\) 0 0
\(787\) 33.4020 1.19065 0.595326 0.803484i \(-0.297023\pi\)
0.595326 + 0.803484i \(0.297023\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.3696 −0.510924
\(792\) 0 0
\(793\) −25.7374 −0.913962
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 49.3509 1.74810 0.874050 0.485837i \(-0.161485\pi\)
0.874050 + 0.485837i \(0.161485\pi\)
\(798\) 0 0
\(799\) −4.37733 −0.154859
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.38743 −0.119540
\(804\) 0 0
\(805\) 12.0496 0.424694
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.8280 0.697115 0.348558 0.937287i \(-0.386672\pi\)
0.348558 + 0.937287i \(0.386672\pi\)
\(810\) 0 0
\(811\) 23.8557 0.837686 0.418843 0.908059i \(-0.362436\pi\)
0.418843 + 0.908059i \(0.362436\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.49794 0.122528
\(816\) 0 0
\(817\) 14.2344 0.497999
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −50.9427 −1.77791 −0.888957 0.457991i \(-0.848569\pi\)
−0.888957 + 0.457991i \(0.848569\pi\)
\(822\) 0 0
\(823\) −13.6149 −0.474587 −0.237293 0.971438i \(-0.576260\pi\)
−0.237293 + 0.971438i \(0.576260\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.2158 −1.25935 −0.629673 0.776861i \(-0.716811\pi\)
−0.629673 + 0.776861i \(0.716811\pi\)
\(828\) 0 0
\(829\) 25.3259 0.879606 0.439803 0.898094i \(-0.355048\pi\)
0.439803 + 0.898094i \(0.355048\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.467911 0.0162122
\(834\) 0 0
\(835\) 31.2344 1.08091
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8.71419 −0.300847 −0.150424 0.988622i \(-0.548064\pi\)
−0.150424 + 0.988622i \(0.548064\pi\)
\(840\) 0 0
\(841\) 10.2950 0.354999
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.21751 −0.0762847
\(846\) 0 0
\(847\) 8.26857 0.284111
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −82.5886 −2.83110
\(852\) 0 0
\(853\) −11.9813 −0.410233 −0.205117 0.978738i \(-0.565757\pi\)
−0.205117 + 0.978738i \(0.565757\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.50030 0.222046 0.111023 0.993818i \(-0.464587\pi\)
0.111023 + 0.993818i \(0.464587\pi\)
\(858\) 0 0
\(859\) 53.5526 1.82719 0.913596 0.406623i \(-0.133294\pi\)
0.913596 + 0.406623i \(0.133294\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.69965 −0.125937 −0.0629687 0.998016i \(-0.520057\pi\)
−0.0629687 + 0.998016i \(0.520057\pi\)
\(864\) 0 0
\(865\) −6.40230 −0.217685
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3.97266 −0.134763
\(870\) 0 0
\(871\) 2.00917 0.0680782
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.0273 0.372792
\(876\) 0 0
\(877\) −11.7888 −0.398079 −0.199040 0.979991i \(-0.563782\pi\)
−0.199040 + 0.979991i \(0.563782\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 49.4858 1.66722 0.833609 0.552355i \(-0.186271\pi\)
0.833609 + 0.552355i \(0.186271\pi\)
\(882\) 0 0
\(883\) 21.5357 0.724734 0.362367 0.932035i \(-0.381969\pi\)
0.362367 + 0.932035i \(0.381969\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −11.8848 −0.399051 −0.199526 0.979893i \(-0.563940\pi\)
−0.199526 + 0.979893i \(0.563940\pi\)
\(888\) 0 0
\(889\) −20.7716 −0.696656
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −30.1857 −1.01013
\(894\) 0 0
\(895\) 11.4953 0.384244
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −57.8866 −1.93063
\(900\) 0 0
\(901\) −0.268571 −0.00894739
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −23.2199 −0.771855
\(906\) 0 0
\(907\) 26.0215 0.864029 0.432014 0.901867i \(-0.357803\pi\)
0.432014 + 0.901867i \(0.357803\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.03272 0.133610 0.0668050 0.997766i \(-0.478719\pi\)
0.0668050 + 0.997766i \(0.478719\pi\)
\(912\) 0 0
\(913\) −24.8726 −0.823162
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.16519 −0.236615
\(918\) 0 0
\(919\) −27.4270 −0.904732 −0.452366 0.891832i \(-0.649420\pi\)
−0.452366 + 0.891832i \(0.649420\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.86802 −0.0614867
\(924\) 0 0
\(925\) −29.4097 −0.966986
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.67675 0.251866 0.125933 0.992039i \(-0.459808\pi\)
0.125933 + 0.992039i \(0.459808\pi\)
\(930\) 0 0
\(931\) 3.22668 0.105750
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.04189 −0.0340734
\(936\) 0 0
\(937\) −2.02465 −0.0661425 −0.0330713 0.999453i \(-0.510529\pi\)
−0.0330713 + 0.999453i \(0.510529\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −6.13928 −0.200135 −0.100067 0.994981i \(-0.531906\pi\)
−0.100067 + 0.994981i \(0.531906\pi\)
\(942\) 0 0
\(943\) −30.5107 −0.993566
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.56448 −0.180821 −0.0904107 0.995905i \(-0.528818\pi\)
−0.0904107 + 0.995905i \(0.528818\pi\)
\(948\) 0 0
\(949\) −6.90640 −0.224191
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −8.72018 −0.282474 −0.141237 0.989976i \(-0.545108\pi\)
−0.141237 + 0.989976i \(0.545108\pi\)
\(954\) 0 0
\(955\) −17.3928 −0.562818
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.56893 −0.0829549
\(960\) 0 0
\(961\) 54.2746 1.75079
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.859785 −0.0276775
\(966\) 0 0
\(967\) 57.7698 1.85775 0.928876 0.370391i \(-0.120776\pi\)
0.928876 + 0.370391i \(0.120776\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 30.7192 0.985828 0.492914 0.870078i \(-0.335932\pi\)
0.492914 + 0.870078i \(0.335932\pi\)
\(972\) 0 0
\(973\) −6.13341 −0.196628
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.3000 0.329527 0.164764 0.986333i \(-0.447314\pi\)
0.164764 + 0.986333i \(0.447314\pi\)
\(978\) 0 0
\(979\) −15.0172 −0.479953
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −13.6963 −0.436846 −0.218423 0.975854i \(-0.570091\pi\)
−0.218423 + 0.975854i \(0.570091\pi\)
\(984\) 0 0
\(985\) 15.4206 0.491343
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −39.4543 −1.25457
\(990\) 0 0
\(991\) −57.9813 −1.84184 −0.920919 0.389754i \(-0.872560\pi\)
−0.920919 + 0.389754i \(0.872560\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.90167 0.155394
\(996\) 0 0
\(997\) 16.2175 0.513614 0.256807 0.966463i \(-0.417330\pi\)
0.256807 + 0.966463i \(0.417330\pi\)
\(998\) 0 0
\(999\) 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 9072.2.a.ca.1.2 3
3.2 odd 2 9072.2.a.bs.1.2 3
4.3 odd 2 567.2.a.h.1.1 3
9.2 odd 6 3024.2.r.k.1009.2 6
9.4 even 3 1008.2.r.h.673.2 6
9.5 odd 6 3024.2.r.k.2017.2 6
9.7 even 3 1008.2.r.h.337.2 6
12.11 even 2 567.2.a.c.1.3 3
28.27 even 2 3969.2.a.q.1.1 3
36.7 odd 6 63.2.f.a.22.3 6
36.11 even 6 189.2.f.b.64.1 6
36.23 even 6 189.2.f.b.127.1 6
36.31 odd 6 63.2.f.a.43.3 yes 6
84.83 odd 2 3969.2.a.l.1.3 3
252.11 even 6 1323.2.h.c.226.3 6
252.23 even 6 1323.2.h.c.802.3 6
252.31 even 6 441.2.g.b.79.3 6
252.47 odd 6 1323.2.g.e.361.1 6
252.59 odd 6 1323.2.g.e.667.1 6
252.67 odd 6 441.2.g.c.79.3 6
252.79 odd 6 441.2.g.c.67.3 6
252.83 odd 6 1323.2.f.d.442.1 6
252.95 even 6 1323.2.g.d.667.1 6
252.103 even 6 441.2.h.e.214.1 6
252.115 even 6 441.2.h.e.373.1 6
252.131 odd 6 1323.2.h.b.802.3 6
252.139 even 6 441.2.f.c.295.3 6
252.151 odd 6 441.2.h.d.373.1 6
252.167 odd 6 1323.2.f.d.883.1 6
252.187 even 6 441.2.g.b.67.3 6
252.191 even 6 1323.2.g.d.361.1 6
252.223 even 6 441.2.f.c.148.3 6
252.227 odd 6 1323.2.h.b.226.3 6
252.247 odd 6 441.2.h.d.214.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 6 36.7 odd 6
63.2.f.a.43.3 yes 6 36.31 odd 6
189.2.f.b.64.1 6 36.11 even 6
189.2.f.b.127.1 6 36.23 even 6
441.2.f.c.148.3 6 252.223 even 6
441.2.f.c.295.3 6 252.139 even 6
441.2.g.b.67.3 6 252.187 even 6
441.2.g.b.79.3 6 252.31 even 6
441.2.g.c.67.3 6 252.79 odd 6
441.2.g.c.79.3 6 252.67 odd 6
441.2.h.d.214.1 6 252.247 odd 6
441.2.h.d.373.1 6 252.151 odd 6
441.2.h.e.214.1 6 252.103 even 6
441.2.h.e.373.1 6 252.115 even 6
567.2.a.c.1.3 3 12.11 even 2
567.2.a.h.1.1 3 4.3 odd 2
1008.2.r.h.337.2 6 9.7 even 3
1008.2.r.h.673.2 6 9.4 even 3
1323.2.f.d.442.1 6 252.83 odd 6
1323.2.f.d.883.1 6 252.167 odd 6
1323.2.g.d.361.1 6 252.191 even 6
1323.2.g.d.667.1 6 252.95 even 6
1323.2.g.e.361.1 6 252.47 odd 6
1323.2.g.e.667.1 6 252.59 odd 6
1323.2.h.b.226.3 6 252.227 odd 6
1323.2.h.b.802.3 6 252.131 odd 6
1323.2.h.c.226.3 6 252.11 even 6
1323.2.h.c.802.3 6 252.23 even 6
3024.2.r.k.1009.2 6 9.2 odd 6
3024.2.r.k.2017.2 6 9.5 odd 6
3969.2.a.l.1.3 3 84.83 odd 2
3969.2.a.q.1.1 3 28.27 even 2
9072.2.a.bs.1.2 3 3.2 odd 2
9072.2.a.ca.1.2 3 1.1 even 1 trivial