Properties

Label 605.2.m.a.282.1
Level $605$
Weight $2$
Character 605.282
Analytic conductor $4.831$
Analytic rank $0$
Dimension $16$
CM discriminant -11
Inner twists $16$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(112,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.112");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.m (of order \(20\), degree \(8\), not minimal)

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: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: 16.0.3429742096000000000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 16x^{12} + 35x^{10} + 31x^{8} + 315x^{6} + 1296x^{4} + 3645x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label 282.1
Root \(0.0369185 - 1.73166i\) of defining polynomial
Character \(\chi\) \(=\) 605.282
Dual form 605.2.m.a.118.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.743682 - 1.45956i) q^{3} +(-1.90211 + 0.618034i) q^{4} +(2.22328 + 0.238794i) q^{5} +(0.186107 - 0.256155i) q^{9} +O(q^{10})\) \(q+(-0.743682 - 1.45956i) q^{3} +(-1.90211 + 0.618034i) q^{4} +(2.22328 + 0.238794i) q^{5} +(0.186107 - 0.256155i) q^{9} +(2.31662 + 2.31662i) q^{12} +(-1.30488 - 3.42260i) q^{15} +(3.23607 - 2.35114i) q^{16} +(-4.37651 + 0.919850i) q^{20} +(6.15831 - 6.15831i) q^{23} +(4.88595 + 1.06181i) q^{25} +(-5.36608 - 0.849903i) q^{27} +(-8.04962 - 5.84839i) q^{31} +(-0.195685 + 0.602256i) q^{36} +(5.44124 - 10.6790i) q^{37} +(0.474937 - 0.525063i) q^{45} +(-3.38125 + 1.72283i) q^{47} +(-5.83824 - 2.97473i) q^{48} +(-4.11450 - 5.66312i) q^{49} +(13.4557 - 2.13118i) q^{53} +(-3.15430 + 1.02489i) q^{59} +(4.59731 + 5.70370i) q^{60} +(-4.70228 + 6.47214i) q^{64} +(1.52506 + 1.52506i) q^{67} +(-13.5682 - 4.40859i) q^{69} +(2.42705 - 1.76336i) q^{71} +(-2.08382 - 7.92099i) q^{75} +(7.75613 - 4.45449i) q^{80} +(2.45665 + 7.56078i) q^{81} +9.00000i q^{89} +(-7.90776 + 15.5199i) q^{92} +(-2.54971 + 16.0982i) q^{93} +(2.98108 + 18.8218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71} - 32 q^{75} + 8 q^{81} - 36 q^{92} - 66 q^{93} + 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{7}{10}\right)\)

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 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(3\) −0.743682 1.45956i −0.429365 0.842677i −0.999773 0.0213149i \(-0.993215\pi\)
0.570408 0.821362i \(-0.306785\pi\)
\(4\) −1.90211 + 0.618034i −0.951057 + 0.309017i
\(5\) 2.22328 + 0.238794i 0.994281 + 0.106792i
\(6\) 0 0
\(7\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(8\) 0 0
\(9\) 0.186107 0.256155i 0.0620358 0.0853849i
\(10\) 0 0
\(11\) 0 0
\(12\) 2.31662 + 2.31662i 0.668752 + 0.668752i
\(13\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(14\) 0 0
\(15\) −1.30488 3.42260i −0.336919 0.883710i
\(16\) 3.23607 2.35114i 0.809017 0.587785i
\(17\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(20\) −4.37651 + 0.919850i −0.978618 + 0.205685i
\(21\) 0 0
\(22\) 0 0
\(23\) 6.15831 6.15831i 1.28410 1.28410i 0.345782 0.938315i \(-0.387614\pi\)
0.938315 0.345782i \(-0.112386\pi\)
\(24\) 0 0
\(25\) 4.88595 + 1.06181i 0.977191 + 0.212362i
\(26\) 0 0
\(27\) −5.36608 0.849903i −1.03270 0.163564i
\(28\) 0 0
\(29\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(30\) 0 0
\(31\) −8.04962 5.84839i −1.44575 1.05040i −0.986800 0.161942i \(-0.948224\pi\)
−0.458954 0.888460i \(-0.651776\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.195685 + 0.602256i −0.0326141 + 0.100376i
\(37\) 5.44124 10.6790i 0.894535 1.75562i 0.294497 0.955652i \(-0.404848\pi\)
0.600038 0.799972i \(-0.295152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0.474937 0.525063i 0.0707995 0.0782717i
\(46\) 0 0
\(47\) −3.38125 + 1.72283i −0.493206 + 0.251301i −0.682859 0.730550i \(-0.739264\pi\)
0.189653 + 0.981851i \(0.439264\pi\)
\(48\) −5.83824 2.97473i −0.842677 0.429365i
\(49\) −4.11450 5.66312i −0.587785 0.809017i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.4557 2.13118i 1.84829 0.292740i 0.868940 0.494918i \(-0.164802\pi\)
0.979349 + 0.202178i \(0.0648018\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.15430 + 1.02489i −0.410655 + 0.133430i −0.507057 0.861913i \(-0.669267\pi\)
0.0964021 + 0.995342i \(0.469267\pi\)
\(60\) 4.59731 + 5.70370i 0.593510 + 0.736345i
\(61\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −4.70228 + 6.47214i −0.587785 + 0.809017i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.52506 + 1.52506i 0.186316 + 0.186316i 0.794101 0.607785i \(-0.207942\pi\)
−0.607785 + 0.794101i \(0.707942\pi\)
\(68\) 0 0
\(69\) −13.5682 4.40859i −1.63342 0.530732i
\(70\) 0 0
\(71\) 2.42705 1.76336i 0.288038 0.209272i −0.434378 0.900731i \(-0.643032\pi\)
0.722416 + 0.691459i \(0.243032\pi\)
\(72\) 0 0
\(73\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(74\) 0 0
\(75\) −2.08382 7.92099i −0.240619 0.914637i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(80\) 7.75613 4.45449i 0.867161 0.498027i
\(81\) 2.45665 + 7.56078i 0.272961 + 0.840087i
\(82\) 0 0
\(83\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000i 0.953998i 0.878904 + 0.476999i \(0.158275\pi\)
−0.878904 + 0.476999i \(0.841725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.90776 + 15.5199i −0.824441 + 1.61806i
\(93\) −2.54971 + 16.0982i −0.264393 + 1.66931i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.98108 + 18.8218i 0.302683 + 1.91107i 0.401310 + 0.915942i \(0.368555\pi\)
−0.0986273 + 0.995124i \(0.531445\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −9.94987 + 1.00000i −0.994987 + 0.100000i
\(101\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(102\) 0 0
\(103\) 15.0577 + 7.67229i 1.48368 + 0.755973i 0.993303 0.115536i \(-0.0368587\pi\)
0.490378 + 0.871510i \(0.336859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(108\) 10.7322 1.69981i 1.03270 0.163564i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −19.6332 −1.86351
\(112\) 0 0
\(113\) 5.67671 + 11.1412i 0.534020 + 1.04807i 0.987620 + 0.156868i \(0.0501397\pi\)
−0.453599 + 0.891206i \(0.649860\pi\)
\(114\) 0 0
\(115\) 15.1622 12.2211i 1.41388 1.13962i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 18.9258 + 6.14936i 1.69959 + 0.552229i
\(125\) 10.6093 + 3.52744i 0.948924 + 0.315504i
\(126\) 0 0
\(127\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −11.7273 3.17096i −1.00933 0.272913i
\(136\) 0 0
\(137\) −18.3095 2.89995i −1.56429 0.247759i −0.686617 0.727019i \(-0.740905\pi\)
−0.877673 + 0.479260i \(0.840905\pi\)
\(138\) 0 0
\(139\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(140\) 0 0
\(141\) 5.02915 + 3.65389i 0.423531 + 0.307713i
\(142\) 0 0
\(143\) 0 0
\(144\) 1.26650i 0.105542i
\(145\) 0 0
\(146\) 0 0
\(147\) −5.20578 + 10.2169i −0.429365 + 0.842677i
\(148\) −3.74985 + 23.6756i −0.308236 + 1.94612i
\(149\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(150\) 0 0
\(151\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.5000 14.9248i −1.32531 1.19879i
\(156\) 0 0
\(157\) 8.22206 4.18935i 0.656192 0.334346i −0.0939948 0.995573i \(-0.529964\pi\)
0.750186 + 0.661226i \(0.229964\pi\)
\(158\) 0 0
\(159\) −13.1174 18.0545i −1.04028 1.43182i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.72359 + 0.431374i −0.213328 + 0.0337878i −0.262184 0.965018i \(-0.584443\pi\)
0.0488556 + 0.998806i \(0.484443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(168\) 0 0
\(169\) 12.3637 4.01722i 0.951057 0.309017i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.84169 + 3.84169i 0.288759 + 0.288759i
\(178\) 0 0
\(179\) −19.9722 6.48936i −1.49279 0.485037i −0.554885 0.831927i \(-0.687238\pi\)
−0.937906 + 0.346890i \(0.887238\pi\)
\(180\) −0.578878 + 1.29226i −0.0431470 + 0.0963191i
\(181\) −8.04962 + 5.84839i −0.598323 + 0.434707i −0.845283 0.534318i \(-0.820568\pi\)
0.246960 + 0.969026i \(0.420568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 14.6475 22.4432i 1.07691 1.65006i
\(186\) 0 0
\(187\) 0 0
\(188\) 5.36675 5.36675i 0.391411 0.391411i
\(189\) 0 0
\(190\) 0 0
\(191\) −7.17425 22.0801i −0.519111 1.59766i −0.775676 0.631132i \(-0.782591\pi\)
0.256565 0.966527i \(-0.417409\pi\)
\(192\) 12.9435 + 2.05004i 0.934114 + 0.147949i
\(193\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 11.3262 + 8.22899i 0.809017 + 0.587785i
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 19.8997i 1.41066i 0.708881 + 0.705328i \(0.249200\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 1.09176 3.36008i 0.0770066 0.237002i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.431374 2.72359i −0.0299826 0.189303i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(212\) −24.2772 + 12.3699i −1.66737 + 0.849565i
\(213\) −4.37868 2.23105i −0.300022 0.152869i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 9.90334 + 19.4364i 0.663177 + 1.30156i 0.940177 + 0.340687i \(0.110660\pi\)
−0.277000 + 0.960870i \(0.589340\pi\)
\(224\) 0 0
\(225\) 1.18130 1.05395i 0.0787534 0.0702633i
\(226\) 0 0
\(227\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(228\) 0 0
\(229\) −17.5452 + 24.1489i −1.15942 + 1.59580i −0.446055 + 0.895005i \(0.647172\pi\)
−0.713362 + 0.700796i \(0.752828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(234\) 0 0
\(235\) −7.92887 + 3.02292i −0.517222 + 0.197194i
\(236\) 5.36641 3.89893i 0.349324 0.253798i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(240\) −12.2697 8.00779i −0.792005 0.516901i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −2.31662 + 2.31662i −0.148612 + 0.148612i
\(244\) 0 0
\(245\) −7.79536 13.5732i −0.498027 0.867161i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.8435 15.8702i −1.37875 1.00172i −0.996996 0.0774530i \(-0.975321\pi\)
−0.381751 0.924265i \(-0.624679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.94427 15.2169i 0.309017 0.951057i
\(257\) 2.73926 5.37610i 0.170870 0.335352i −0.789652 0.613555i \(-0.789739\pi\)
0.960522 + 0.278203i \(0.0897388\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 30.4248 1.52506i 1.86898 0.0936839i
\(266\) 0 0
\(267\) 13.1360 6.69314i 0.803912 0.409614i
\(268\) −3.84338 1.95830i −0.234772 0.119622i
\(269\) 7.79785 + 10.7328i 0.475443 + 0.654392i 0.977621 0.210373i \(-0.0674677\pi\)
−0.502178 + 0.864764i \(0.667468\pi\)
\(270\) 0 0
\(271\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 28.5330 1.71748
\(277\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(278\) 0 0
\(279\) −2.99619 + 0.973520i −0.179377 + 0.0582831i
\(280\) 0 0
\(281\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(284\) −3.52671 + 4.85410i −0.209272 + 0.288038i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 16.1680 + 5.25329i 0.951057 + 0.309017i
\(290\) 0 0
\(291\) 25.2546 18.3485i 1.48045 1.07561i
\(292\) 0 0
\(293\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(294\) 0 0
\(295\) −7.25763 + 1.52540i −0.422555 + 0.0888121i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 8.85910 + 13.7787i 0.511481 + 0.795516i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0 0
\(309\) 27.6834i 1.57485i
\(310\) 0 0
\(311\) 3.70820 11.4127i 0.210273 0.647154i −0.789183 0.614159i \(-0.789495\pi\)
0.999456 0.0329949i \(-0.0105045\pi\)
\(312\) 0 0
\(313\) 1.20014 7.57739i 0.0678360 0.428299i −0.930275 0.366863i \(-0.880432\pi\)
0.998111 0.0614365i \(-0.0195682\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.418529 + 2.64249i 0.0235069 + 0.148417i 0.996650 0.0817838i \(-0.0260617\pi\)
−0.973143 + 0.230201i \(0.926062\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −12.0000 + 13.2665i −0.670820 + 0.741620i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −9.34564 12.8632i −0.519202 0.714620i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.94987 0.546895 0.273447 0.961887i \(-0.411836\pi\)
0.273447 + 0.961887i \(0.411836\pi\)
\(332\) 0 0
\(333\) −1.72283 3.38125i −0.0944107 0.185291i
\(334\) 0 0
\(335\) 3.02647 + 3.75482i 0.165354 + 0.205148i
\(336\) 0 0
\(337\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(338\) 0 0
\(339\) 12.0395 16.5710i 0.653898 0.900013i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −29.1133 13.0415i −1.56741 0.702133i
\(346\) 0 0
\(347\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.7414 22.7414i 1.21040 1.21040i 0.239511 0.970894i \(-0.423013\pi\)
0.970894 0.239511i \(-0.0769871\pi\)
\(354\) 0 0
\(355\) 5.81709 3.34087i 0.308739 0.177315i
\(356\) −5.56231 17.1190i −0.294802 0.907306i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(360\) 0 0
\(361\) 15.3713 + 11.1679i 0.809017 + 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 15.0718 29.5801i 0.786744 1.54407i −0.0514358 0.998676i \(-0.516380\pi\)
0.838179 0.545395i \(-0.183620\pi\)
\(368\) 5.44966 34.4078i 0.284083 1.79363i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −5.09942 32.1965i −0.264393 1.66931i
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) −2.74144 18.1082i −0.141567 0.935103i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 17.5452 + 24.1489i 0.901235 + 1.24044i 0.970073 + 0.242815i \(0.0780709\pi\)
−0.0688378 + 0.997628i \(0.521929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −29.5540 + 4.68089i −1.51014 + 0.239182i −0.855914 0.517119i \(-0.827005\pi\)
−0.654224 + 0.756301i \(0.727005\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −17.3029 33.9588i −0.878421 1.72400i
\(389\) −34.6973 + 11.2738i −1.75922 + 0.571606i −0.997119 0.0758507i \(-0.975833\pi\)
−0.762102 + 0.647456i \(0.775833\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 18.8997 + 18.8997i 0.948551 + 0.948551i 0.998740 0.0501886i \(-0.0159822\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 18.3078 8.05147i 0.915388 0.402574i
\(401\) −21.4656 + 15.5957i −1.07194 + 0.778812i −0.976261 0.216600i \(-0.930503\pi\)
−0.0956827 + 0.995412i \(0.530503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.65635 + 17.3964i 0.181685 + 0.864433i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(410\) 0 0
\(411\) 9.38384 + 28.8805i 0.462871 + 1.42457i
\(412\) −33.3832 5.28738i −1.64467 0.260491i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000i 1.17248i 0.810139 + 0.586238i \(0.199392\pi\)
−0.810139 + 0.586238i \(0.800608\pi\)
\(420\) 0 0
\(421\) −12.2987 + 37.8516i −0.599403 + 1.84477i −0.0679432 + 0.997689i \(0.521644\pi\)
−0.531460 + 0.847084i \(0.678356\pi\)
\(422\) 0 0
\(423\) −0.187964 + 1.18676i −0.00913910 + 0.0577020i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(432\) −19.3632 + 9.86606i −0.931614 + 0.474681i
\(433\) −0.535294 0.272746i −0.0257246 0.0131073i 0.441081 0.897467i \(-0.354595\pi\)
−0.466805 + 0.884360i \(0.654595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −2.21637 −0.105542
\(442\) 0 0
\(443\) −4.97031 9.75478i −0.236146 0.463464i 0.742271 0.670099i \(-0.233748\pi\)
−0.978418 + 0.206636i \(0.933748\pi\)
\(444\) 37.3447 12.1340i 1.77230 0.575855i
\(445\) −2.14915 + 20.0095i −0.101879 + 0.948543i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.9236 31.5517i 1.08183 1.48902i 0.224346 0.974510i \(-0.427976\pi\)
0.857487 0.514505i \(-0.172024\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −17.6834 17.6834i −0.831756 0.831756i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −21.2872 + 32.6167i −0.992522 + 1.52076i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −30.4248 + 30.4248i −1.41396 + 1.41396i −0.693615 + 0.720346i \(0.743983\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 0 0
\(465\) −9.51288 + 35.1820i −0.441149 + 1.63153i
\(466\) 0 0
\(467\) 28.0171 + 4.43748i 1.29648 + 0.205342i 0.766267 0.642523i \(-0.222112\pi\)
0.530212 + 0.847865i \(0.322112\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −12.2292 8.88503i −0.563492 0.409401i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.95830 3.84338i 0.0896645 0.175976i
\(478\) 0 0
\(479\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.13325 + 42.5581i 0.0968659 + 1.93246i
\(486\) 0 0
\(487\) 20.8228 10.6097i 0.943571 0.480773i 0.0866600 0.996238i \(-0.472381\pi\)
0.856911 + 0.515465i \(0.172381\pi\)
\(488\) 0 0
\(489\) 2.65510 + 3.65443i 0.120068 + 0.165259i
\(490\) 0 0
\(491\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −39.7995 −1.78705
\(497\) 0 0
\(498\) 0 0
\(499\) −18.9258 + 6.14936i −0.847235 + 0.275283i −0.700287 0.713861i \(-0.746945\pi\)
−0.146947 + 0.989144i \(0.546945\pi\)
\(500\) −22.3602 0.152689i −0.999977 0.00682845i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −15.0581 15.0581i −0.668752 0.668752i
\(508\) 0 0
\(509\) 3.15430 + 1.02489i 0.139812 + 0.0454276i 0.378087 0.925770i \(-0.376582\pi\)
−0.238275 + 0.971198i \(0.576582\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 31.6454 + 20.6534i 1.39446 + 0.910095i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13.3236 + 41.0059i 0.583718 + 1.79650i 0.604358 + 0.796713i \(0.293430\pi\)
−0.0206400 + 0.999787i \(0.506570\pi\)
\(522\) 0 0
\(523\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 52.8496i 2.29781i
\(530\) 0 0
\(531\) −0.324507 + 0.998729i −0.0140824 + 0.0433411i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 5.38136 + 33.9766i 0.232223 + 1.46620i
\(538\) 0 0
\(539\) 0 0
\(540\) 24.2665 1.21637i 1.04426 0.0523444i
\(541\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(542\) 0 0
\(543\) 14.5224 + 7.39955i 0.623217 + 0.317545i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(548\) 36.6191 5.79989i 1.56429 0.247759i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −43.6502 4.68830i −1.85285 0.199007i
\(556\) 0 0
\(557\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(564\) −11.8242 3.84193i −0.497891 0.161774i
\(565\) 9.96048 + 26.1255i 0.419041 + 1.09911i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −26.8918 + 26.8918i −1.12342 + 1.12342i
\(574\) 0 0
\(575\) 36.6282 23.5503i 1.52750 0.982114i
\(576\) 0.782740 + 2.40902i 0.0326141 + 0.100376i
\(577\) 39.7739 + 6.29956i 1.65581 + 0.262254i 0.913212 0.407486i \(-0.133594\pi\)
0.742596 + 0.669740i \(0.233594\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.5384 + 34.4210i −0.723885 + 1.42070i 0.175916 + 0.984405i \(0.443711\pi\)
−0.899801 + 0.436300i \(0.856289\pi\)
\(588\) 3.58758 22.6511i 0.147949 0.934114i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −7.49970 47.3512i −0.308236 1.94612i
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 29.0449 14.7991i 1.18873 0.605686i
\(598\) 0 0
\(599\) 21.1603 + 29.1246i 0.864585 + 1.19000i 0.980457 + 0.196735i \(0.0630339\pi\)
−0.115872 + 0.993264i \(0.536966\pi\)
\(600\) 0 0
\(601\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(602\) 0 0
\(603\) 0.674478 0.106827i 0.0274669 0.00435032i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −34.2665 34.2665i −1.37952 1.37952i −0.845428 0.534089i \(-0.820655\pi\)
−0.534089 0.845428i \(-0.679345\pi\)
\(618\) 0 0
\(619\) −0.951057 0.309017i −0.0382262 0.0124204i 0.289841 0.957075i \(-0.406397\pi\)
−0.328068 + 0.944654i \(0.606397\pi\)
\(620\) 40.6089 + 18.1911i 1.63089 + 0.730573i
\(621\) −38.2800 + 27.8120i −1.53612 + 1.11606i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 22.7451 + 10.3759i 0.909804 + 0.415037i
\(626\) 0 0
\(627\) 0 0
\(628\) −13.0501 + 13.0501i −0.520757 + 0.520757i
\(629\) 0 0
\(630\) 0 0
\(631\) 2.16312 + 6.65740i 0.0861124 + 0.265027i 0.984836 0.173489i \(-0.0555042\pi\)
−0.898723 + 0.438516i \(0.855504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 36.1091 + 26.2348i 1.43182 + 1.04028i
\(637\) 0 0
\(638\) 0 0
\(639\) 0.949874i 0.0375764i
\(640\) 0 0
\(641\) −7.17425 + 22.0801i −0.283366 + 0.872111i 0.703518 + 0.710678i \(0.251612\pi\)
−0.986884 + 0.161433i \(0.948388\pi\)
\(642\) 0 0
\(643\) 7.83709 49.4815i 0.309065 1.95136i 0.000979141 1.00000i \(-0.499688\pi\)
0.308086 0.951359i \(-0.400312\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.75596 + 48.9692i 0.304918 + 1.92518i 0.373709 + 0.927546i \(0.378086\pi\)
−0.0687910 + 0.997631i \(0.521914\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.91397 2.50379i 0.192446 0.0980561i
\(653\) −30.0423 15.3073i −1.17565 0.599021i −0.246647 0.969105i \(-0.579329\pi\)
−0.928998 + 0.370084i \(0.879329\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 21.0036 28.9090i 0.812047 1.11769i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(674\) 0 0
\(675\) −25.3160 9.85035i −0.974412 0.379140i
\(676\) −21.0344 + 15.2824i −0.809017 + 0.587785i
\(677\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −11.2164 + 11.2164i −0.429183 + 0.429183i −0.888350 0.459167i \(-0.848148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −40.0148 10.8196i −1.52889 0.413396i
\(686\) 0 0
\(687\) 48.2947 + 7.64913i 1.84256 + 0.291832i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −13.7533 9.99235i −0.523200 0.380127i 0.294608 0.955618i \(-0.404811\pi\)
−0.817808 + 0.575491i \(0.804811\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 10.3087 + 9.32456i 0.388248 + 0.351183i
\(706\) 0 0
\(707\) 0 0
\(708\) −9.68162 4.93303i −0.363857 0.185395i
\(709\) −11.1679 15.3713i −0.419420 0.577282i 0.546064 0.837743i \(-0.316125\pi\)
−0.965484 + 0.260461i \(0.916125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −85.5883 + 13.5558i −3.20531 + 0.507671i
\(714\) 0 0
\(715\) 0 0
\(716\) 42.0000 1.56961
\(717\) 0 0
\(718\) 0 0
\(719\) 48.5039 15.7599i 1.80889 0.587744i 0.808890 0.587961i \(-0.200069\pi\)
1.00000 0.000216702i \(6.89785e-5\pi\)
\(720\) 0.302432 2.81578i 0.0112710 0.104938i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 11.6968 16.0992i 0.434707 0.598323i
\(725\) 0 0
\(726\) 0 0
\(727\) 21.5251 + 21.5251i 0.798320 + 0.798320i 0.982831 0.184510i \(-0.0590699\pi\)
−0.184510 + 0.982831i \(0.559070\pi\)
\(728\) 0 0
\(729\) 27.7864 + 9.02836i 1.02913 + 0.334384i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(734\) 0 0
\(735\) −14.0136 + 21.4720i −0.516901 + 0.792005i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(740\) −13.9906 + 51.7421i −0.514303 + 1.90208i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −7.10739 + 21.8743i −0.259352 + 0.798205i 0.733588 + 0.679594i \(0.237844\pi\)
−0.992941 + 0.118611i \(0.962156\pi\)
\(752\) −6.89133 + 13.5250i −0.251301 + 0.493206i
\(753\) −6.91890 + 43.6842i −0.252139 + 1.59194i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −8.60586 54.3353i −0.312785 1.97485i −0.179232 0.983807i \(-0.557361\pi\)
−0.133554 0.991042i \(-0.542639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 27.2925 + 37.5649i 0.987407 + 1.35905i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −25.8869 + 4.10009i −0.934114 + 0.147949i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −9.88388 −0.355959
\(772\) 0 0
\(773\) −13.0763 25.6636i −0.470320 0.923056i −0.997318 0.0731890i \(-0.976682\pi\)
0.526998 0.849867i \(-0.323318\pi\)
\(774\) 0 0
\(775\) −33.1202 37.1221i −1.18971 1.33347i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −26.6296 8.65248i −0.951057 0.309017i
\(785\) 19.2803 7.35072i 0.688145 0.262358i
\(786\) 0 0
\(787\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −24.8523 43.2726i −0.881421 1.53472i
\(796\) −12.2987 37.8516i −0.435917 1.34161i
\(797\) −41.4729 6.56866i −1.46905 0.232674i −0.629940 0.776644i \(-0.716921\pi\)
−0.839105 + 0.543970i \(0.816921\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 2.30539 + 1.67497i 0.0814571 + 0.0591820i
\(802\) 0 0
\(803\) 0 0
\(804\) 7.06600i 0.249198i
\(805\) 0 0
\(806\) 0 0
\(807\) 9.86606 19.3632i 0.347302 0.681618i
\(808\) 0 0
\(809\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(810\) 0 0
\(811\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.15831 + 0.308689i −0.215716 + 0.0108129i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(822\) 0 0
\(823\) −13.3746 + 2.11834i −0.466211 + 0.0738405i −0.385121 0.922866i \(-0.625840\pi\)
−0.0810902 + 0.996707i \(0.525840\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(828\) 2.50379 + 4.91397i 0.0870128 + 0.170772i
\(829\) −27.5806 + 8.96149i −0.957915 + 0.311246i −0.745928 0.666027i \(-0.767994\pi\)
−0.211987 + 0.977272i \(0.567994\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 38.2243 + 38.2243i 1.32123 + 1.32123i
\(838\) 0 0
\(839\) 34.6973 + 11.2738i 1.19788 + 0.389216i 0.838982 0.544160i \(-0.183151\pi\)
0.358901 + 0.933376i \(0.383151\pi\)
\(840\) 0 0
\(841\) 23.4615 17.0458i 0.809017 0.587785i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 28.4473 5.97903i 0.978618 0.205685i
\(846\) 0 0
\(847\) 0 0
\(848\) 38.5330 38.5330i 1.32323 1.32323i
\(849\) 0 0
\(850\) 0 0
\(851\) −32.2560 99.2738i −1.10572 3.40306i
\(852\) 9.70760 + 1.53753i 0.332577 + 0.0526750i
\(853\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 31.0000i 1.05771i −0.848713 0.528853i \(-0.822622\pi\)
0.848713 0.528853i \(-0.177378\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.11837 57.5711i 0.310393 1.95974i 0.0311832 0.999514i \(-0.490072\pi\)
0.279210 0.960230i \(-0.409928\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4.35634 27.5049i −0.147949 0.934114i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 5.37610 + 2.73926i 0.181954 + 0.0927099i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 57.0000 1.92038 0.960189 0.279350i \(-0.0901189\pi\)
0.960189 + 0.279350i \(0.0901189\pi\)
\(882\) 0 0
\(883\) 11.5889 + 22.7445i 0.389997 + 0.765413i 0.999628 0.0272727i \(-0.00868225\pi\)
−0.609631 + 0.792686i \(0.708682\pi\)
\(884\) 0 0
\(885\) 7.62378 + 9.45852i 0.256271 + 0.317945i
\(886\) 0 0
\(887\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −30.8496 30.8496i −1.03292 1.03292i
\(893\) 0 0
\(894\) 0 0
\(895\) −42.8542 19.1969i −1.43246 0.641682i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.59559 + 2.73482i −0.0531863 + 0.0911605i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −19.2931 + 11.0804i −0.641325 + 0.368325i
\(906\) 0 0
\(907\) 36.1068 + 5.71876i 1.19891 + 0.189888i 0.723781 0.690030i \(-0.242403\pi\)
0.475126 + 0.879918i \(0.342403\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.36641 + 3.89893i 0.177797 + 0.129177i 0.673124 0.739529i \(-0.264952\pi\)
−0.495327 + 0.868706i \(0.664952\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 18.4481 56.7774i 0.609542 1.87598i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 37.9248 46.3997i 1.24696 1.52561i
\(926\) 0 0
\(927\) 4.76765 2.42924i 0.156590 0.0797866i
\(928\) 0 0
\(929\) −31.1914 42.9313i −1.02336 1.40853i −0.909823 0.414996i \(-0.863783\pi\)
−0.113534 0.993534i \(-0.536217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −19.4152 + 3.07507i −0.635625 + 0.100673i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(938\) 0 0
\(939\) −11.9522 + 3.88349i −0.390044 + 0.126733i
\(940\) 13.2133 10.6502i 0.430972 0.347373i
\(941\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −7.79785 + 10.7328i −0.253798 + 0.349324i
\(945\) 0 0
\(946\) 0 0
\(947\) −16.8918 16.8918i −0.548910 0.548910i 0.377215 0.926126i \(-0.376882\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 3.54562 2.57604i 0.114974 0.0835338i
\(952\) 0 0
\(953\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(954\) 0 0
\(955\) −10.6778 50.8034i −0.345525 1.64396i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 28.2874 + 7.64864i 0.912973 + 0.246859i
\(961\) 21.0132 + 64.6718i 0.677844 + 2.08619i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 13.3236 41.0059i 0.427575 1.31594i −0.472932 0.881099i \(-0.656804\pi\)
0.900507 0.434842i \(-0.143196\pi\)
\(972\) 2.97473 5.83824i 0.0954145 0.187261i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.25018 20.5209i −0.103983 0.656520i −0.983535 0.180718i \(-0.942158\pi\)
0.879552 0.475802i \(-0.157842\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 23.2164 + 21.0000i 0.741620 + 0.670820i
\(981\) 0 0
\(982\) 0 0
\(983\) −9.14632 4.66028i −0.291722 0.148640i 0.302005 0.953306i \(-0.402344\pi\)
−0.593727 + 0.804666i \(0.702344\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 59.6992 1.89641 0.948205 0.317660i \(-0.102897\pi\)
0.948205 + 0.317660i \(0.102897\pi\)
\(992\) 0 0
\(993\) −7.39955 14.5224i −0.234817 0.460855i
\(994\) 0 0
\(995\) −4.75194 + 44.2427i −0.150647 + 1.40259i
\(996\) 0 0
\(997\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(998\) 0 0
\(999\) −38.2743 + 52.6800i −1.21095 + 1.66672i
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 605.2.m.a.282.1 16
5.3 odd 4 inner 605.2.m.a.403.2 16
11.2 odd 10 inner 605.2.m.a.457.1 16
11.3 even 5 inner 605.2.m.a.602.2 16
11.4 even 5 inner 605.2.m.a.112.1 16
11.5 even 5 55.2.e.b.32.2 4
11.6 odd 10 55.2.e.b.32.2 4
11.7 odd 10 inner 605.2.m.a.112.1 16
11.8 odd 10 inner 605.2.m.a.602.2 16
11.9 even 5 inner 605.2.m.a.457.1 16
11.10 odd 2 CM 605.2.m.a.282.1 16
33.5 odd 10 495.2.k.a.307.2 4
33.17 even 10 495.2.k.a.307.2 4
44.27 odd 10 880.2.bd.d.417.1 4
44.39 even 10 880.2.bd.d.417.1 4
55.3 odd 20 inner 605.2.m.a.118.1 16
55.8 even 20 inner 605.2.m.a.118.1 16
55.13 even 20 inner 605.2.m.a.578.1 16
55.17 even 20 275.2.e.a.43.1 4
55.18 even 20 inner 605.2.m.a.233.1 16
55.27 odd 20 275.2.e.a.43.1 4
55.28 even 20 55.2.e.b.43.2 yes 4
55.38 odd 20 55.2.e.b.43.2 yes 4
55.39 odd 10 275.2.e.a.32.1 4
55.43 even 4 inner 605.2.m.a.403.2 16
55.48 odd 20 inner 605.2.m.a.233.1 16
55.49 even 10 275.2.e.a.32.1 4
55.53 odd 20 inner 605.2.m.a.578.1 16
165.38 even 20 495.2.k.a.208.2 4
165.83 odd 20 495.2.k.a.208.2 4
220.83 odd 20 880.2.bd.d.593.1 4
220.203 even 20 880.2.bd.d.593.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.e.b.32.2 4 11.5 even 5
55.2.e.b.32.2 4 11.6 odd 10
55.2.e.b.43.2 yes 4 55.28 even 20
55.2.e.b.43.2 yes 4 55.38 odd 20
275.2.e.a.32.1 4 55.39 odd 10
275.2.e.a.32.1 4 55.49 even 10
275.2.e.a.43.1 4 55.17 even 20
275.2.e.a.43.1 4 55.27 odd 20
495.2.k.a.208.2 4 165.38 even 20
495.2.k.a.208.2 4 165.83 odd 20
495.2.k.a.307.2 4 33.5 odd 10
495.2.k.a.307.2 4 33.17 even 10
605.2.m.a.112.1 16 11.4 even 5 inner
605.2.m.a.112.1 16 11.7 odd 10 inner
605.2.m.a.118.1 16 55.3 odd 20 inner
605.2.m.a.118.1 16 55.8 even 20 inner
605.2.m.a.233.1 16 55.18 even 20 inner
605.2.m.a.233.1 16 55.48 odd 20 inner
605.2.m.a.282.1 16 1.1 even 1 trivial
605.2.m.a.282.1 16 11.10 odd 2 CM
605.2.m.a.403.2 16 5.3 odd 4 inner
605.2.m.a.403.2 16 55.43 even 4 inner
605.2.m.a.457.1 16 11.2 odd 10 inner
605.2.m.a.457.1 16 11.9 even 5 inner
605.2.m.a.578.1 16 55.13 even 20 inner
605.2.m.a.578.1 16 55.53 odd 20 inner
605.2.m.a.602.2 16 11.3 even 5 inner
605.2.m.a.602.2 16 11.8 odd 10 inner
880.2.bd.d.417.1 4 44.27 odd 10
880.2.bd.d.417.1 4 44.39 even 10
880.2.bd.d.593.1 4 220.83 odd 20
880.2.bd.d.593.1 4 220.203 even 20