Properties

Label 2304.2.k.f.1729.1
Level $2304$
Weight $2$
Character 2304.1729
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,2,Mod(577,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.k (of order \(4\), degree \(2\), 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: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 256)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1729.1
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1729
Dual form 2304.2.k.f.577.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.73205 - 1.73205i) q^{5} -1.03528i q^{7} +O(q^{10})\) \(q+(-1.73205 - 1.73205i) q^{5} -1.03528i q^{7} +(-0.896575 - 0.896575i) q^{11} +(-3.73205 + 3.73205i) q^{13} +3.46410 q^{17} +(-0.896575 + 0.896575i) q^{19} +6.69213i q^{23} +1.00000i q^{25} +(-1.73205 + 1.73205i) q^{29} -5.65685 q^{31} +(-1.79315 + 1.79315i) q^{35} +(0.267949 + 0.267949i) q^{37} -6.92820i q^{41} +(5.79555 + 5.79555i) q^{43} +9.79796 q^{47} +5.92820 q^{49} +(-4.26795 - 4.26795i) q^{53} +3.10583i q^{55} +(7.58871 + 7.58871i) q^{59} +(0.267949 - 0.267949i) q^{61} +12.9282 q^{65} +(-2.96713 + 2.96713i) q^{67} +6.69213i q^{71} +9.46410i q^{73} +(-0.928203 + 0.928203i) q^{77} +15.4548 q^{79} +(5.79555 - 5.79555i) q^{83} +(-6.00000 - 6.00000i) q^{85} +9.46410i q^{89} +(3.86370 + 3.86370i) q^{91} +3.10583 q^{95} -3.46410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{13} + 16 q^{37} - 8 q^{49} - 48 q^{53} + 16 q^{61} + 48 q^{65} + 48 q^{77} - 48 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.73205 1.73205i −0.774597 0.774597i 0.204310 0.978906i \(-0.434505\pi\)
−0.978906 + 0.204310i \(0.934505\pi\)
\(6\) 0 0
\(7\) 1.03528i 0.391298i −0.980674 0.195649i \(-0.937319\pi\)
0.980674 0.195649i \(-0.0626813\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.896575 0.896575i −0.270328 0.270328i 0.558904 0.829232i \(-0.311222\pi\)
−0.829232 + 0.558904i \(0.811222\pi\)
\(12\) 0 0
\(13\) −3.73205 + 3.73205i −1.03508 + 1.03508i −0.0357229 + 0.999362i \(0.511373\pi\)
−0.999362 + 0.0357229i \(0.988627\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −0.896575 + 0.896575i −0.205689 + 0.205689i −0.802432 0.596744i \(-0.796461\pi\)
0.596744 + 0.802432i \(0.296461\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.69213i 1.39541i 0.716387 + 0.697703i \(0.245794\pi\)
−0.716387 + 0.697703i \(0.754206\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.73205 + 1.73205i −0.321634 + 0.321634i −0.849394 0.527760i \(-0.823032\pi\)
0.527760 + 0.849394i \(0.323032\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.79315 + 1.79315i −0.303098 + 0.303098i
\(36\) 0 0
\(37\) 0.267949 + 0.267949i 0.0440506 + 0.0440506i 0.728789 0.684738i \(-0.240084\pi\)
−0.684738 + 0.728789i \(0.740084\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 5.79555 + 5.79555i 0.883814 + 0.883814i 0.993920 0.110106i \(-0.0351190\pi\)
−0.110106 + 0.993920i \(0.535119\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.79796 1.42918 0.714590 0.699544i \(-0.246613\pi\)
0.714590 + 0.699544i \(0.246613\pi\)
\(48\) 0 0
\(49\) 5.92820 0.846886
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.26795 4.26795i −0.586248 0.586248i 0.350365 0.936613i \(-0.386057\pi\)
−0.936613 + 0.350365i \(0.886057\pi\)
\(54\) 0 0
\(55\) 3.10583i 0.418790i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.58871 + 7.58871i 0.987965 + 0.987965i 0.999928 0.0119631i \(-0.00380806\pi\)
−0.0119631 + 0.999928i \(0.503808\pi\)
\(60\) 0 0
\(61\) 0.267949 0.267949i 0.0343074 0.0343074i −0.689745 0.724052i \(-0.742277\pi\)
0.724052 + 0.689745i \(0.242277\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.9282 1.60355
\(66\) 0 0
\(67\) −2.96713 + 2.96713i −0.362492 + 0.362492i −0.864730 0.502237i \(-0.832510\pi\)
0.502237 + 0.864730i \(0.332510\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.69213i 0.794210i 0.917773 + 0.397105i \(0.129985\pi\)
−0.917773 + 0.397105i \(0.870015\pi\)
\(72\) 0 0
\(73\) 9.46410i 1.10769i 0.832620 + 0.553845i \(0.186840\pi\)
−0.832620 + 0.553845i \(0.813160\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.928203 + 0.928203i −0.105779 + 0.105779i
\(78\) 0 0
\(79\) 15.4548 1.73880 0.869401 0.494107i \(-0.164505\pi\)
0.869401 + 0.494107i \(0.164505\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.79555 5.79555i 0.636145 0.636145i −0.313457 0.949602i \(-0.601487\pi\)
0.949602 + 0.313457i \(0.101487\pi\)
\(84\) 0 0
\(85\) −6.00000 6.00000i −0.650791 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.46410i 1.00319i 0.865102 + 0.501596i \(0.167254\pi\)
−0.865102 + 0.501596i \(0.832746\pi\)
\(90\) 0 0
\(91\) 3.86370 + 3.86370i 0.405026 + 0.405026i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.10583 0.318651
\(96\) 0 0
\(97\) −3.46410 −0.351726 −0.175863 0.984415i \(-0.556272\pi\)
−0.175863 + 0.984415i \(0.556272\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.1962 11.1962i −1.11406 1.11406i −0.992596 0.121463i \(-0.961241\pi\)
−0.121463 0.992596i \(-0.538759\pi\)
\(102\) 0 0
\(103\) 4.62158i 0.455378i 0.973734 + 0.227689i \(0.0731169\pi\)
−0.973734 + 0.227689i \(0.926883\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.90138 + 8.90138i 0.860529 + 0.860529i 0.991400 0.130870i \(-0.0417771\pi\)
−0.130870 + 0.991400i \(0.541777\pi\)
\(108\) 0 0
\(109\) 2.80385 2.80385i 0.268560 0.268560i −0.559960 0.828520i \(-0.689183\pi\)
0.828520 + 0.559960i \(0.189183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.928203 0.0873180 0.0436590 0.999046i \(-0.486098\pi\)
0.0436590 + 0.999046i \(0.486098\pi\)
\(114\) 0 0
\(115\) 11.5911 11.5911i 1.08088 1.08088i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.58630i 0.328756i
\(120\) 0 0
\(121\) 9.39230i 0.853846i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.92820 + 6.92820i −0.619677 + 0.619677i
\(126\) 0 0
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.4877 + 12.4877i −1.09105 + 1.09105i −0.0956380 + 0.995416i \(0.530489\pi\)
−0.995416 + 0.0956380i \(0.969511\pi\)
\(132\) 0 0
\(133\) 0.928203 + 0.928203i 0.0804854 + 0.0804854i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 0 0
\(139\) 0.138701 + 0.138701i 0.0117644 + 0.0117644i 0.712965 0.701200i \(-0.247352\pi\)
−0.701200 + 0.712965i \(0.747352\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.69213 0.559624
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.2679 + 10.2679i 0.841183 + 0.841183i 0.989013 0.147830i \(-0.0472287\pi\)
−0.147830 + 0.989013i \(0.547229\pi\)
\(150\) 0 0
\(151\) 4.62158i 0.376099i 0.982160 + 0.188049i \(0.0602165\pi\)
−0.982160 + 0.188049i \(0.939784\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.79796 + 9.79796i 0.786991 + 0.786991i
\(156\) 0 0
\(157\) 3.19615 3.19615i 0.255081 0.255081i −0.567969 0.823050i \(-0.692271\pi\)
0.823050 + 0.567969i \(0.192271\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) −9.38186 + 9.38186i −0.734844 + 0.734844i −0.971575 0.236731i \(-0.923924\pi\)
0.236731 + 0.971575i \(0.423924\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 23.6627i 1.83107i 0.402234 + 0.915537i \(0.368234\pi\)
−0.402234 + 0.915537i \(0.631766\pi\)
\(168\) 0 0
\(169\) 14.8564i 1.14280i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.26795 + 4.26795i −0.324486 + 0.324486i −0.850485 0.525999i \(-0.823692\pi\)
0.525999 + 0.850485i \(0.323692\pi\)
\(174\) 0 0
\(175\) 1.03528 0.0782595
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.79555 5.79555i 0.433180 0.433180i −0.456529 0.889709i \(-0.650907\pi\)
0.889709 + 0.456529i \(0.150907\pi\)
\(180\) 0 0
\(181\) −8.12436 8.12436i −0.603879 0.603879i 0.337461 0.941340i \(-0.390432\pi\)
−0.941340 + 0.337461i \(0.890432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.928203i 0.0682429i
\(186\) 0 0
\(187\) −3.10583 3.10583i −0.227121 0.227121i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.9706 −1.22795 −0.613973 0.789327i \(-0.710430\pi\)
−0.613973 + 0.789327i \(0.710430\pi\)
\(192\) 0 0
\(193\) 2.39230 0.172202 0.0861009 0.996286i \(-0.472559\pi\)
0.0861009 + 0.996286i \(0.472559\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.73205 1.73205i −0.123404 0.123404i 0.642708 0.766111i \(-0.277811\pi\)
−0.766111 + 0.642708i \(0.777811\pi\)
\(198\) 0 0
\(199\) 18.5606i 1.31573i 0.753136 + 0.657865i \(0.228540\pi\)
−0.753136 + 0.657865i \(0.771460\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.79315 + 1.79315i 0.125855 + 0.125855i
\(204\) 0 0
\(205\) −12.0000 + 12.0000i −0.838116 + 0.838116i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.60770 0.111207
\(210\) 0 0
\(211\) −0.138701 + 0.138701i −0.00954855 + 0.00954855i −0.711865 0.702316i \(-0.752149\pi\)
0.702316 + 0.711865i \(0.252149\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 20.0764i 1.36920i
\(216\) 0 0
\(217\) 5.85641i 0.397559i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −12.9282 + 12.9282i −0.869645 + 0.869645i
\(222\) 0 0
\(223\) −5.65685 −0.378811 −0.189405 0.981899i \(-0.560656\pi\)
−0.189405 + 0.981899i \(0.560656\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.20925 2.20925i 0.146633 0.146633i −0.629979 0.776612i \(-0.716936\pi\)
0.776612 + 0.629979i \(0.216936\pi\)
\(228\) 0 0
\(229\) −1.19615 1.19615i −0.0790440 0.0790440i 0.666479 0.745523i \(-0.267800\pi\)
−0.745523 + 0.666479i \(0.767800\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.46410i 0.620014i −0.950734 0.310007i \(-0.899669\pi\)
0.950734 0.310007i \(-0.100331\pi\)
\(234\) 0 0
\(235\) −16.9706 16.9706i −1.10704 1.10704i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.17260 0.463957 0.231979 0.972721i \(-0.425480\pi\)
0.231979 + 0.972721i \(0.425480\pi\)
\(240\) 0 0
\(241\) 18.3923 1.18475 0.592376 0.805661i \(-0.298190\pi\)
0.592376 + 0.805661i \(0.298190\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.2679 10.2679i −0.655995 0.655995i
\(246\) 0 0
\(247\) 6.69213i 0.425810i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.20925 2.20925i −0.139447 0.139447i 0.633937 0.773384i \(-0.281438\pi\)
−0.773384 + 0.633937i \(0.781438\pi\)
\(252\) 0 0
\(253\) 6.00000 6.00000i 0.377217 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.928203 0.0578997 0.0289499 0.999581i \(-0.490784\pi\)
0.0289499 + 0.999581i \(0.490784\pi\)
\(258\) 0 0
\(259\) 0.277401 0.277401i 0.0172369 0.0172369i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.9038i 0.795682i −0.917454 0.397841i \(-0.869760\pi\)
0.917454 0.397841i \(-0.130240\pi\)
\(264\) 0 0
\(265\) 14.7846i 0.908211i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.19615 5.19615i 0.316815 0.316815i −0.530728 0.847543i \(-0.678081\pi\)
0.847543 + 0.530728i \(0.178081\pi\)
\(270\) 0 0
\(271\) −21.1117 −1.28244 −0.641221 0.767356i \(-0.721572\pi\)
−0.641221 + 0.767356i \(0.721572\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.896575 0.896575i 0.0540655 0.0540655i
\(276\) 0 0
\(277\) 7.19615 + 7.19615i 0.432375 + 0.432375i 0.889436 0.457061i \(-0.151098\pi\)
−0.457061 + 0.889436i \(0.651098\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.39230i 0.262023i 0.991381 + 0.131011i \(0.0418225\pi\)
−0.991381 + 0.131011i \(0.958178\pi\)
\(282\) 0 0
\(283\) 8.62398 + 8.62398i 0.512643 + 0.512643i 0.915335 0.402693i \(-0.131926\pi\)
−0.402693 + 0.915335i \(0.631926\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.17260 −0.423385
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.1244 + 12.1244i 0.708312 + 0.708312i 0.966180 0.257868i \(-0.0830199\pi\)
−0.257868 + 0.966180i \(0.583020\pi\)
\(294\) 0 0
\(295\) 26.2880i 1.53055i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −24.9754 24.9754i −1.44436 1.44436i
\(300\) 0 0
\(301\) 6.00000 6.00000i 0.345834 0.345834i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.928203 −0.0531488
\(306\) 0 0
\(307\) −14.8356 + 14.8356i −0.846715 + 0.846715i −0.989722 0.143007i \(-0.954323\pi\)
0.143007 + 0.989722i \(0.454323\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.2784i 0.582836i −0.956596 0.291418i \(-0.905873\pi\)
0.956596 0.291418i \(-0.0941271\pi\)
\(312\) 0 0
\(313\) 28.7846i 1.62700i −0.581563 0.813501i \(-0.697559\pi\)
0.581563 0.813501i \(-0.302441\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.1962 17.1962i 0.965832 0.965832i −0.0336031 0.999435i \(-0.510698\pi\)
0.999435 + 0.0336031i \(0.0106982\pi\)
\(318\) 0 0
\(319\) 3.10583 0.173893
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.10583 + 3.10583i −0.172813 + 0.172813i
\(324\) 0 0
\(325\) −3.73205 3.73205i −0.207017 0.207017i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.1436i 0.559234i
\(330\) 0 0
\(331\) −3.24453 3.24453i −0.178335 0.178335i 0.612294 0.790630i \(-0.290247\pi\)
−0.790630 + 0.612294i \(0.790247\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.2784 0.561571
\(336\) 0 0
\(337\) −33.7128 −1.83645 −0.918227 0.396055i \(-0.870379\pi\)
−0.918227 + 0.396055i \(0.870379\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.07180 + 5.07180i 0.274653 + 0.274653i
\(342\) 0 0
\(343\) 13.3843i 0.722682i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.58871 + 7.58871i 0.407383 + 0.407383i 0.880825 0.473442i \(-0.156989\pi\)
−0.473442 + 0.880825i \(0.656989\pi\)
\(348\) 0 0
\(349\) −20.1244 + 20.1244i −1.07723 + 1.07723i −0.0804755 + 0.996757i \(0.525644\pi\)
−0.996757 + 0.0804755i \(0.974356\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 0 0
\(355\) 11.5911 11.5911i 0.615192 0.615192i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.2880i 1.38743i 0.720250 + 0.693715i \(0.244027\pi\)
−0.720250 + 0.693715i \(0.755973\pi\)
\(360\) 0 0
\(361\) 17.3923i 0.915384i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.3923 16.3923i 0.858012 0.858012i
\(366\) 0 0
\(367\) 12.8295 0.669692 0.334846 0.942273i \(-0.391316\pi\)
0.334846 + 0.942273i \(0.391316\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.41851 + 4.41851i −0.229397 + 0.229397i
\(372\) 0 0
\(373\) 10.1244 + 10.1244i 0.524219 + 0.524219i 0.918843 0.394624i \(-0.129125\pi\)
−0.394624 + 0.918843i \(0.629125\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.9282i 0.665836i
\(378\) 0 0
\(379\) 19.1798 + 19.1798i 0.985201 + 0.985201i 0.999892 0.0146911i \(-0.00467650\pi\)
−0.0146911 + 0.999892i \(0.504676\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −19.5959 −1.00130 −0.500652 0.865648i \(-0.666906\pi\)
−0.500652 + 0.865648i \(0.666906\pi\)
\(384\) 0 0
\(385\) 3.21539 0.163871
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.73205 + 7.73205i 0.392031 + 0.392031i 0.875411 0.483380i \(-0.160591\pi\)
−0.483380 + 0.875411i \(0.660591\pi\)
\(390\) 0 0
\(391\) 23.1822i 1.17238i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −26.7685 26.7685i −1.34687 1.34687i
\(396\) 0 0
\(397\) 11.5885 11.5885i 0.581608 0.581608i −0.353737 0.935345i \(-0.615089\pi\)
0.935345 + 0.353737i \(0.115089\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −34.3923 −1.71747 −0.858735 0.512420i \(-0.828749\pi\)
−0.858735 + 0.512420i \(0.828749\pi\)
\(402\) 0 0
\(403\) 21.1117 21.1117i 1.05165 1.05165i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.480473i 0.0238162i
\(408\) 0 0
\(409\) 30.9282i 1.52930i 0.644445 + 0.764651i \(0.277088\pi\)
−0.644445 + 0.764651i \(0.722912\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.85641 7.85641i 0.386588 0.386588i
\(414\) 0 0
\(415\) −20.0764 −0.985511
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.68973 + 2.68973i −0.131402 + 0.131402i −0.769749 0.638347i \(-0.779619\pi\)
0.638347 + 0.769749i \(0.279619\pi\)
\(420\) 0 0
\(421\) 26.5167 + 26.5167i 1.29234 + 1.29234i 0.933334 + 0.359009i \(0.116885\pi\)
0.359009 + 0.933334i \(0.383115\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.46410i 0.168034i
\(426\) 0 0
\(427\) −0.277401 0.277401i −0.0134244 0.0134244i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.17260 0.345492 0.172746 0.984966i \(-0.444736\pi\)
0.172746 + 0.984966i \(0.444736\pi\)
\(432\) 0 0
\(433\) −13.6077 −0.653944 −0.326972 0.945034i \(-0.606028\pi\)
−0.326972 + 0.945034i \(0.606028\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 6.00000i −0.287019 0.287019i
\(438\) 0 0
\(439\) 31.9449i 1.52465i −0.647196 0.762324i \(-0.724059\pi\)
0.647196 0.762324i \(-0.275941\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.68973 + 2.68973i 0.127793 + 0.127793i 0.768110 0.640318i \(-0.221197\pi\)
−0.640318 + 0.768110i \(0.721197\pi\)
\(444\) 0 0
\(445\) 16.3923 16.3923i 0.777070 0.777070i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −24.2487 −1.14437 −0.572184 0.820125i \(-0.693904\pi\)
−0.572184 + 0.820125i \(0.693904\pi\)
\(450\) 0 0
\(451\) −6.21166 + 6.21166i −0.292496 + 0.292496i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 13.3843i 0.627464i
\(456\) 0 0
\(457\) 20.7846i 0.972263i −0.873886 0.486132i \(-0.838408\pi\)
0.873886 0.486132i \(-0.161592\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.124356 0.124356i 0.00579182 0.00579182i −0.704205 0.709997i \(-0.748696\pi\)
0.709997 + 0.704205i \(0.248696\pi\)
\(462\) 0 0
\(463\) 18.4863 0.859132 0.429566 0.903036i \(-0.358667\pi\)
0.429566 + 0.903036i \(0.358667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.8671 17.8671i 0.826793 0.826793i −0.160279 0.987072i \(-0.551239\pi\)
0.987072 + 0.160279i \(0.0512394\pi\)
\(468\) 0 0
\(469\) 3.07180 + 3.07180i 0.141842 + 0.141842i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.3923i 0.477839i
\(474\) 0 0
\(475\) −0.896575 0.896575i −0.0411377 0.0411377i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.62536 −0.119956 −0.0599778 0.998200i \(-0.519103\pi\)
−0.0599778 + 0.998200i \(0.519103\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 + 6.00000i 0.272446 + 0.272446i
\(486\) 0 0
\(487\) 7.24693i 0.328390i 0.986428 + 0.164195i \(0.0525026\pi\)
−0.986428 + 0.164195i \(0.947497\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −22.7661 22.7661i −1.02742 1.02742i −0.999613 0.0278072i \(-0.991148\pi\)
−0.0278072 0.999613i \(-0.508852\pi\)
\(492\) 0 0
\(493\) −6.00000 + 6.00000i −0.270226 + 0.270226i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.92820 0.310772
\(498\) 0 0
\(499\) −11.4524 + 11.4524i −0.512680 + 0.512680i −0.915347 0.402667i \(-0.868083\pi\)
0.402667 + 0.915347i \(0.368083\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 32.4997i 1.44909i −0.689227 0.724545i \(-0.742050\pi\)
0.689227 0.724545i \(-0.257950\pi\)
\(504\) 0 0
\(505\) 38.7846i 1.72589i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 21.5885 21.5885i 0.956892 0.956892i −0.0422169 0.999108i \(-0.513442\pi\)
0.999108 + 0.0422169i \(0.0134421\pi\)
\(510\) 0 0
\(511\) 9.79796 0.433436
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.00481 8.00481i 0.352734 0.352734i
\(516\) 0 0
\(517\) −8.78461 8.78461i −0.386347 0.386347i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.0000i 0.525730i 0.964833 + 0.262865i \(0.0846673\pi\)
−0.964833 + 0.262865i \(0.915333\pi\)
\(522\) 0 0
\(523\) −16.0740 16.0740i −0.702866 0.702866i 0.262158 0.965025i \(-0.415566\pi\)
−0.965025 + 0.262158i \(0.915566\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19.5959 −0.853612
\(528\) 0 0
\(529\) −21.7846 −0.947157
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 25.8564 + 25.8564i 1.11997 + 1.11997i
\(534\) 0 0
\(535\) 30.8353i 1.33313i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.31508 5.31508i −0.228937 0.228937i
\(540\) 0 0
\(541\) 14.1244 14.1244i 0.607253 0.607253i −0.334974 0.942227i \(-0.608728\pi\)
0.942227 + 0.334974i \(0.108728\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.71281 −0.416051
\(546\) 0 0
\(547\) 12.4877 12.4877i 0.533935 0.533935i −0.387806 0.921741i \(-0.626767\pi\)
0.921741 + 0.387806i \(0.126767\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.10583i 0.132313i
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.803848 0.803848i 0.0340601 0.0340601i −0.689872 0.723932i \(-0.742333\pi\)
0.723932 + 0.689872i \(0.242333\pi\)
\(558\) 0 0
\(559\) −43.2586 −1.82964
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 27.6651 27.6651i 1.16594 1.16594i 0.182794 0.983151i \(-0.441486\pi\)
0.983151 0.182794i \(-0.0585140\pi\)
\(564\) 0 0
\(565\) −1.60770 1.60770i −0.0676362 0.0676362i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0000i 0.503066i −0.967849 0.251533i \(-0.919065\pi\)
0.967849 0.251533i \(-0.0809347\pi\)
\(570\) 0 0
\(571\) −14.0034 14.0034i −0.586026 0.586026i 0.350527 0.936553i \(-0.386002\pi\)
−0.936553 + 0.350527i \(0.886002\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.69213 −0.279081
\(576\) 0 0
\(577\) −16.1436 −0.672067 −0.336033 0.941850i \(-0.609085\pi\)
−0.336033 + 0.941850i \(0.609085\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.00000 6.00000i −0.248922 0.248922i
\(582\) 0 0
\(583\) 7.65308i 0.316958i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.6945 10.6945i −0.441411 0.441411i 0.451075 0.892486i \(-0.351041\pi\)
−0.892486 + 0.451075i \(0.851041\pi\)
\(588\) 0 0
\(589\) 5.07180 5.07180i 0.208980 0.208980i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −7.85641 −0.322624 −0.161312 0.986903i \(-0.551573\pi\)
−0.161312 + 0.986903i \(0.551573\pi\)
\(594\) 0 0
\(595\) −6.21166 + 6.21166i −0.254653 + 0.254653i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 29.8744i 1.22063i −0.792158 0.610316i \(-0.791042\pi\)
0.792158 0.610316i \(-0.208958\pi\)
\(600\) 0 0
\(601\) 19.6077i 0.799815i −0.916556 0.399907i \(-0.869042\pi\)
0.916556 0.399907i \(-0.130958\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16.2679 + 16.2679i −0.661386 + 0.661386i
\(606\) 0 0
\(607\) −30.9096 −1.25458 −0.627292 0.778785i \(-0.715837\pi\)
−0.627292 + 0.778785i \(0.715837\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −36.5665 + 36.5665i −1.47932 + 1.47932i
\(612\) 0 0
\(613\) −23.0526 23.0526i −0.931084 0.931084i 0.0666897 0.997774i \(-0.478756\pi\)
−0.997774 + 0.0666897i \(0.978756\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.3205i 0.938848i 0.882973 + 0.469424i \(0.155538\pi\)
−0.882973 + 0.469424i \(0.844462\pi\)
\(618\) 0 0
\(619\) −3.24453 3.24453i −0.130409 0.130409i 0.638890 0.769298i \(-0.279394\pi\)
−0.769298 + 0.638890i \(0.779394\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.79796 0.392547
\(624\) 0 0
\(625\) 29.0000 1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.928203 + 0.928203i 0.0370099 + 0.0370099i
\(630\) 0 0
\(631\) 35.5312i 1.41447i 0.706976 + 0.707237i \(0.250059\pi\)
−0.706976 + 0.707237i \(0.749941\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.79796 9.79796i −0.388820 0.388820i
\(636\) 0 0
\(637\) −22.1244 + 22.1244i −0.876599 + 0.876599i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −34.3923 −1.35841 −0.679207 0.733947i \(-0.737676\pi\)
−0.679207 + 0.733947i \(0.737676\pi\)
\(642\) 0 0
\(643\) −14.2808 + 14.2808i −0.563181 + 0.563181i −0.930210 0.367028i \(-0.880375\pi\)
0.367028 + 0.930210i \(0.380375\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.69213i 0.263095i 0.991310 + 0.131547i \(0.0419946\pi\)
−0.991310 + 0.131547i \(0.958005\pi\)
\(648\) 0 0
\(649\) 13.6077i 0.534149i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −32.6603 + 32.6603i −1.27809 + 1.27809i −0.336362 + 0.941733i \(0.609196\pi\)
−0.941733 + 0.336362i \(0.890804\pi\)
\(654\) 0 0
\(655\) 43.2586 1.69025
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −13.8004 + 13.8004i −0.537586 + 0.537586i −0.922819 0.385233i \(-0.874121\pi\)
0.385233 + 0.922819i \(0.374121\pi\)
\(660\) 0 0
\(661\) 21.7321 + 21.7321i 0.845279 + 0.845279i 0.989540 0.144261i \(-0.0460804\pi\)
−0.144261 + 0.989540i \(0.546080\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.21539i 0.124687i
\(666\) 0 0
\(667\) −11.5911 11.5911i −0.448810 0.448810i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.480473 −0.0185485
\(672\) 0 0
\(673\) 6.67949 0.257475 0.128738 0.991679i \(-0.458907\pi\)
0.128738 + 0.991679i \(0.458907\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.7321 + 31.7321i 1.21956 + 1.21956i 0.967785 + 0.251776i \(0.0810147\pi\)
0.251776 + 0.967785i \(0.418985\pi\)
\(678\) 0 0
\(679\) 3.58630i 0.137630i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 20.9730 + 20.9730i 0.802508 + 0.802508i 0.983487 0.180979i \(-0.0579265\pi\)
−0.180979 + 0.983487i \(0.557927\pi\)
\(684\) 0 0
\(685\) 20.7846 20.7846i 0.794139 0.794139i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 31.8564 1.21363
\(690\) 0 0
\(691\) 23.5983 23.5983i 0.897722 0.897722i −0.0975119 0.995234i \(-0.531088\pi\)
0.995234 + 0.0975119i \(0.0310884\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.480473i 0.0182254i
\(696\) 0 0
\(697\) 24.0000i 0.909065i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.6603 26.6603i 1.00694 1.00694i 0.00696818 0.999976i \(-0.497782\pi\)
0.999976 0.00696818i \(-0.00221806\pi\)
\(702\) 0 0
\(703\) −0.480473 −0.0181214
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −11.5911 + 11.5911i −0.435929 + 0.435929i
\(708\) 0 0
\(709\) 9.05256 + 9.05256i 0.339976 + 0.339976i 0.856358 0.516382i \(-0.172722\pi\)
−0.516382 + 0.856358i \(0.672722\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 37.8564i 1.41773i
\(714\) 0 0
\(715\) −11.5911 11.5911i −0.433483 0.433483i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −46.3644 −1.72910 −0.864551 0.502545i \(-0.832397\pi\)
−0.864551 + 0.502545i \(0.832397\pi\)
\(720\) 0 0
\(721\) 4.78461 0.178188
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.73205 1.73205i −0.0643268 0.0643268i
\(726\) 0 0
\(727\) 20.6312i 0.765169i −0.923921 0.382584i \(-0.875034\pi\)
0.923921 0.382584i \(-0.124966\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 20.0764 + 20.0764i 0.742552 + 0.742552i
\(732\) 0 0
\(733\) −25.1962 + 25.1962i −0.930641 + 0.930641i −0.997746 0.0671048i \(-0.978624\pi\)
0.0671048 + 0.997746i \(0.478624\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.32051 0.195983
\(738\) 0 0
\(739\) −34.6346 + 34.6346i −1.27406 + 1.27406i −0.330115 + 0.943941i \(0.607087\pi\)
−0.943941 + 0.330115i \(0.892913\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.2784i 0.377079i −0.982066 0.188540i \(-0.939625\pi\)
0.982066 0.188540i \(-0.0603754\pi\)
\(744\) 0 0
\(745\) 35.5692i 1.30316i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.21539 9.21539i 0.336723 0.336723i
\(750\) 0 0
\(751\) −15.4548 −0.563954 −0.281977 0.959421i \(-0.590990\pi\)
−0.281977 + 0.959421i \(0.590990\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00481 8.00481i 0.291325 0.291325i
\(756\) 0 0
\(757\) −18.2679 18.2679i −0.663960 0.663960i 0.292351 0.956311i \(-0.405562\pi\)
−0.956311 + 0.292351i \(0.905562\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.6410i 1.25574i 0.778320 + 0.627868i \(0.216072\pi\)
−0.778320 + 0.627868i \(0.783928\pi\)
\(762\) 0 0
\(763\) −2.90276 2.90276i −0.105087 0.105087i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −56.6429 −2.04526
\(768\) 0 0
\(769\) 18.3923 0.663243 0.331622 0.943412i \(-0.392404\pi\)
0.331622 + 0.943412i \(0.392404\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.66025 + 2.66025i 0.0956827 + 0.0956827i 0.753328 0.657645i \(-0.228447\pi\)
−0.657645 + 0.753328i \(0.728447\pi\)
\(774\) 0 0
\(775\) 5.65685i 0.203200i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.21166 + 6.21166i 0.222556 + 0.222556i
\(780\) 0 0
\(781\) 6.00000 6.00000i 0.214697 0.214697i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −11.0718 −0.395169
\(786\) 0 0
\(787\) −12.9682 + 12.9682i −0.462265 + 0.462265i −0.899397 0.437132i \(-0.855994\pi\)
0.437132 + 0.899397i \(0.355994\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.960947i 0.0341673i
\(792\) 0 0
\(793\) 2.00000i 0.0710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.41154 + 2.41154i −0.0854212 + 0.0854212i −0.748526 0.663105i \(-0.769238\pi\)
0.663105 + 0.748526i \(0.269238\pi\)
\(798\) 0 0
\(799\) 33.9411 1.20075
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.48528 8.48528i 0.299439 0.299439i
\(804\) 0 0
\(805\) −12.0000 12.0000i −0.422944 0.422944i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.9282i 1.08738i 0.839287 + 0.543689i \(0.182973\pi\)
−0.839287 + 0.543689i \(0.817027\pi\)
\(810\) 0 0
\(811\) 3.52193 + 3.52193i 0.123672 + 0.123672i 0.766234 0.642562i \(-0.222128\pi\)
−0.642562 + 0.766234i \(0.722128\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 32.4997 1.13842
\(816\) 0 0
\(817\) −10.3923 −0.363581
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −31.9808 31.9808i −1.11614 1.11614i −0.992303 0.123833i \(-0.960481\pi\)
−0.123833 0.992303i \(-0.539519\pi\)
\(822\) 0 0
\(823\) 18.5606i 0.646983i 0.946231 + 0.323492i \(0.104857\pi\)
−0.946231 + 0.323492i \(0.895143\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4.48288 4.48288i −0.155885 0.155885i 0.624856 0.780740i \(-0.285158\pi\)
−0.780740 + 0.624856i \(0.785158\pi\)
\(828\) 0 0
\(829\) 22.1244 22.1244i 0.768411 0.768411i −0.209416 0.977827i \(-0.567156\pi\)
0.977827 + 0.209416i \(0.0671563\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 20.5359 0.711527
\(834\) 0 0
\(835\) 40.9850 40.9850i 1.41834 1.41834i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.06678i 0.140401i 0.997533 + 0.0702003i \(0.0223639\pi\)
−0.997533 + 0.0702003i \(0.977636\pi\)
\(840\) 0 0
\(841\) 23.0000i 0.793103i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −25.7321 + 25.7321i −0.885209 + 0.885209i
\(846\) 0 0
\(847\) −9.72363 −0.334108
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.79315 + 1.79315i −0.0614684 + 0.0614684i
\(852\) 0 0
\(853\) 34.1244 + 34.1244i 1.16840 + 1.16840i 0.982586 + 0.185810i \(0.0594909\pi\)
0.185810 + 0.982586i \(0.440509\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.85641i 0.0634136i −0.999497 0.0317068i \(-0.989906\pi\)
0.999497 0.0317068i \(-0.0100943\pi\)
\(858\) 0 0
\(859\) −27.1846 27.1846i −0.927527 0.927527i 0.0700183 0.997546i \(-0.477694\pi\)
−0.997546 + 0.0700183i \(0.977694\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 36.5665 1.24474 0.622369 0.782724i \(-0.286170\pi\)
0.622369 + 0.782724i \(0.286170\pi\)
\(864\) 0 0
\(865\) 14.7846 0.502692
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13.8564 13.8564i −0.470046 0.470046i
\(870\) 0 0
\(871\) 22.1469i 0.750421i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 7.17260 + 7.17260i 0.242478 + 0.242478i
\(876\) 0 0
\(877\) −8.12436 + 8.12436i −0.274340 + 0.274340i −0.830845 0.556505i \(-0.812142\pi\)
0.556505 + 0.830845i \(0.312142\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.14359 −0.139601 −0.0698006 0.997561i \(-0.522236\pi\)
−0.0698006 + 0.997561i \(0.522236\pi\)
\(882\) 0 0
\(883\) 23.2466 23.2466i 0.782310 0.782310i −0.197910 0.980220i \(-0.563415\pi\)
0.980220 + 0.197910i \(0.0634154\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 10.2784i 0.345116i −0.984999 0.172558i \(-0.944797\pi\)
0.984999 0.172558i \(-0.0552032\pi\)
\(888\) 0 0
\(889\) 5.85641i 0.196418i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.78461 + 8.78461i −0.293966 + 0.293966i
\(894\) 0 0
\(895\) −20.0764 −0.671080
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.79796 9.79796i 0.326780 0.326780i
\(900\) 0 0
\(901\) −14.7846 14.7846i −0.492547 0.492547i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 28.1436i 0.935525i
\(906\) 0 0
\(907\) 25.3915 + 25.3915i 0.843110 + 0.843110i 0.989262 0.146152i \(-0.0466889\pi\)
−0.146152 + 0.989262i \(0.546689\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 43.7391 1.44914 0.724570 0.689201i \(-0.242038\pi\)
0.724570 + 0.689201i \(0.242038\pi\)
\(912\) 0 0
\(913\) −10.3923 −0.343935
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.9282 + 12.9282i 0.426927 + 0.426927i
\(918\) 0 0
\(919\) 55.1271i 1.81848i 0.416277 + 0.909238i \(0.363335\pi\)
−0.416277 + 0.909238i \(0.636665\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −24.9754 24.9754i −0.822074 0.822074i
\(924\) 0 0
\(925\) −0.267949 + 0.267949i −0.00881012 + 0.00881012i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.46410 0.113653 0.0568267 0.998384i \(-0.481902\pi\)
0.0568267 + 0.998384i \(0.481902\pi\)
\(930\) 0 0
\(931\) −5.31508 + 5.31508i −0.174195 + 0.174195i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 10.7589i 0.351854i
\(936\) 0 0
\(937\) 53.1769i 1.73721i 0.495502 + 0.868607i \(0.334984\pi\)
−0.495502 + 0.868607i \(0.665016\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 29.1962 29.1962i 0.951767 0.951767i −0.0471218 0.998889i \(-0.515005\pi\)
0.998889 + 0.0471218i \(0.0150049\pi\)
\(942\) 0 0
\(943\) 46.3644 1.50983
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.0740 + 16.0740i −0.522334 + 0.522334i −0.918276 0.395941i \(-0.870418\pi\)
0.395941 + 0.918276i \(0.370418\pi\)
\(948\) 0 0
\(949\) −35.3205 35.3205i −1.14655 1.14655i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 29.5692i 0.957841i −0.877858 0.478920i \(-0.841028\pi\)
0.877858 0.478920i \(-0.158972\pi\)
\(954\) 0 0
\(955\) 29.3939 + 29.3939i 0.951164 + 0.951164i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12.4233 0.401170
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.14359 4.14359i −0.133387 0.133387i
\(966\) 0 0
\(967\) 7.24693i 0.233046i 0.993188 + 0.116523i \(0.0371748\pi\)
−0.993188 + 0.116523i \(0.962825\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 23.5983 + 23.5983i 0.757306 + 0.757306i 0.975831 0.218525i \(-0.0701246\pi\)
−0.218525 + 0.975831i \(0.570125\pi\)
\(972\) 0 0
\(973\) 0.143594 0.143594i 0.00460340 0.00460340i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 51.4641 1.64648 0.823241 0.567692i \(-0.192163\pi\)
0.823241 + 0.567692i \(0.192163\pi\)
\(978\) 0 0
\(979\) 8.48528 8.48528i 0.271191 0.271191i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 57.6038i 1.83728i 0.395100 + 0.918638i \(0.370710\pi\)
−0.395100 + 0.918638i \(0.629290\pi\)
\(984\) 0 0
\(985\) 6.00000i 0.191176i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −38.7846 + 38.7846i −1.23328 + 1.23328i
\(990\) 0 0
\(991\) −16.5644 −0.526186 −0.263093 0.964770i \(-0.584743\pi\)
−0.263093 + 0.964770i \(0.584743\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 32.1480 32.1480i 1.01916 1.01916i
\(996\) 0 0
\(997\) −4.80385 4.80385i −0.152139 0.152139i 0.626933 0.779073i \(-0.284310\pi\)
−0.779073 + 0.626933i \(0.784310\pi\)
\(998\) 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 2304.2.k.f.1729.1 8
3.2 odd 2 256.2.e.a.193.4 yes 8
4.3 odd 2 inner 2304.2.k.f.1729.2 8
8.3 odd 2 2304.2.k.k.1729.4 8
8.5 even 2 2304.2.k.k.1729.3 8
12.11 even 2 256.2.e.a.193.1 yes 8
16.3 odd 4 inner 2304.2.k.f.577.1 8
16.5 even 4 2304.2.k.k.577.4 8
16.11 odd 4 2304.2.k.k.577.3 8
16.13 even 4 inner 2304.2.k.f.577.2 8
24.5 odd 2 256.2.e.b.193.1 yes 8
24.11 even 2 256.2.e.b.193.4 yes 8
32.3 odd 8 9216.2.a.bk.1.3 4
32.13 even 8 9216.2.a.bk.1.2 4
32.19 odd 8 9216.2.a.bb.1.1 4
32.29 even 8 9216.2.a.bb.1.4 4
48.5 odd 4 256.2.e.b.65.1 yes 8
48.11 even 4 256.2.e.b.65.4 yes 8
48.29 odd 4 256.2.e.a.65.4 yes 8
48.35 even 4 256.2.e.a.65.1 8
96.5 odd 8 1024.2.b.h.513.8 8
96.11 even 8 1024.2.b.h.513.7 8
96.29 odd 8 1024.2.a.j.1.3 4
96.35 even 8 1024.2.a.g.1.1 4
96.53 odd 8 1024.2.b.h.513.1 8
96.59 even 8 1024.2.b.h.513.2 8
96.77 odd 8 1024.2.a.g.1.2 4
96.83 even 8 1024.2.a.j.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
256.2.e.a.65.1 8 48.35 even 4
256.2.e.a.65.4 yes 8 48.29 odd 4
256.2.e.a.193.1 yes 8 12.11 even 2
256.2.e.a.193.4 yes 8 3.2 odd 2
256.2.e.b.65.1 yes 8 48.5 odd 4
256.2.e.b.65.4 yes 8 48.11 even 4
256.2.e.b.193.1 yes 8 24.5 odd 2
256.2.e.b.193.4 yes 8 24.11 even 2
1024.2.a.g.1.1 4 96.35 even 8
1024.2.a.g.1.2 4 96.77 odd 8
1024.2.a.j.1.3 4 96.29 odd 8
1024.2.a.j.1.4 4 96.83 even 8
1024.2.b.h.513.1 8 96.53 odd 8
1024.2.b.h.513.2 8 96.59 even 8
1024.2.b.h.513.7 8 96.11 even 8
1024.2.b.h.513.8 8 96.5 odd 8
2304.2.k.f.577.1 8 16.3 odd 4 inner
2304.2.k.f.577.2 8 16.13 even 4 inner
2304.2.k.f.1729.1 8 1.1 even 1 trivial
2304.2.k.f.1729.2 8 4.3 odd 2 inner
2304.2.k.k.577.3 8 16.11 odd 4
2304.2.k.k.577.4 8 16.5 even 4
2304.2.k.k.1729.3 8 8.5 even 2
2304.2.k.k.1729.4 8 8.3 odd 2
9216.2.a.bb.1.1 4 32.19 odd 8
9216.2.a.bb.1.4 4 32.29 even 8
9216.2.a.bk.1.2 4 32.13 even 8
9216.2.a.bk.1.3 4 32.3 odd 8