Properties

Label 2304.2.k.i.577.3
Level $2304$
Weight $2$
Character 2304.577
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 577.3
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2304.577
Dual form 2304.2.k.i.1729.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.44949 - 2.44949i) q^{5} -1.41421i q^{7} +O(q^{10})\) \(q+(2.44949 - 2.44949i) q^{5} -1.41421i q^{7} +(3.46410 - 3.46410i) q^{11} +(1.00000 + 1.00000i) q^{13} -4.89898 q^{17} +(-4.24264 - 4.24264i) q^{19} -6.92820i q^{23} -7.00000i q^{25} +(2.44949 + 2.44949i) q^{29} -1.41421 q^{31} +(-3.46410 - 3.46410i) q^{35} +(-7.00000 + 7.00000i) q^{37} +4.89898i q^{41} +(4.24264 - 4.24264i) q^{43} +6.92820 q^{47} +5.00000 q^{49} +(-2.44949 + 2.44949i) q^{53} -16.9706i q^{55} +(-6.92820 + 6.92820i) q^{59} +(5.00000 + 5.00000i) q^{61} +4.89898 q^{65} +(-5.65685 - 5.65685i) q^{67} +13.8564i q^{71} -12.0000i q^{73} +(-4.89898 - 4.89898i) q^{77} -15.5563 q^{79} +(10.3923 + 10.3923i) q^{83} +(-12.0000 + 12.0000i) q^{85} -9.79796i q^{89} +(1.41421 - 1.41421i) q^{91} -20.7846 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} - 56 q^{37} + 40 q^{49} + 40 q^{61} - 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) 2.44949 2.44949i 1.09545 1.09545i 0.100509 0.994936i \(-0.467953\pi\)
0.994936 0.100509i \(-0.0320471\pi\)
\(6\) 0 0
\(7\) 1.41421i 0.534522i −0.963624 0.267261i \(-0.913881\pi\)
0.963624 0.267261i \(-0.0861187\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 3.46410i 1.04447 1.04447i 0.0455017 0.998964i \(-0.485511\pi\)
0.998964 0.0455017i \(-0.0144886\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.00000i 0.277350 + 0.277350i 0.832050 0.554700i \(-0.187167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 −1.18818 −0.594089 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 0 0
\(19\) −4.24264 4.24264i −0.973329 0.973329i 0.0263249 0.999653i \(-0.491620\pi\)
−0.999653 + 0.0263249i \(0.991620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.92820i 1.44463i −0.691564 0.722315i \(-0.743078\pi\)
0.691564 0.722315i \(-0.256922\pi\)
\(24\) 0 0
\(25\) 7.00000i 1.40000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.44949 + 2.44949i 0.454859 + 0.454859i 0.896963 0.442105i \(-0.145768\pi\)
−0.442105 + 0.896963i \(0.645768\pi\)
\(30\) 0 0
\(31\) −1.41421 −0.254000 −0.127000 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.46410 3.46410i −0.585540 0.585540i
\(36\) 0 0
\(37\) −7.00000 + 7.00000i −1.15079 + 1.15079i −0.164399 + 0.986394i \(0.552568\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.89898i 0.765092i 0.923936 + 0.382546i \(0.124953\pi\)
−0.923936 + 0.382546i \(0.875047\pi\)
\(42\) 0 0
\(43\) 4.24264 4.24264i 0.646997 0.646997i −0.305269 0.952266i \(-0.598747\pi\)
0.952266 + 0.305269i \(0.0987465\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.44949 + 2.44949i −0.336463 + 0.336463i −0.855034 0.518571i \(-0.826464\pi\)
0.518571 + 0.855034i \(0.326464\pi\)
\(54\) 0 0
\(55\) 16.9706i 2.28831i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.92820 + 6.92820i −0.901975 + 0.901975i −0.995607 0.0936317i \(-0.970152\pi\)
0.0936317 + 0.995607i \(0.470152\pi\)
\(60\) 0 0
\(61\) 5.00000 + 5.00000i 0.640184 + 0.640184i 0.950601 0.310416i \(-0.100468\pi\)
−0.310416 + 0.950601i \(0.600468\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.89898 0.607644
\(66\) 0 0
\(67\) −5.65685 5.65685i −0.691095 0.691095i 0.271378 0.962473i \(-0.412521\pi\)
−0.962473 + 0.271378i \(0.912521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564i 1.64445i 0.569160 + 0.822226i \(0.307268\pi\)
−0.569160 + 0.822226i \(0.692732\pi\)
\(72\) 0 0
\(73\) 12.0000i 1.40449i −0.711934 0.702247i \(-0.752180\pi\)
0.711934 0.702247i \(-0.247820\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.89898 4.89898i −0.558291 0.558291i
\(78\) 0 0
\(79\) −15.5563 −1.75023 −0.875113 0.483919i \(-0.839213\pi\)
−0.875113 + 0.483919i \(0.839213\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.3923 + 10.3923i 1.14070 + 1.14070i 0.988322 + 0.152382i \(0.0486944\pi\)
0.152382 + 0.988322i \(0.451306\pi\)
\(84\) 0 0
\(85\) −12.0000 + 12.0000i −1.30158 + 1.30158i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.79796i 1.03858i −0.854598 0.519291i \(-0.826196\pi\)
0.854598 0.519291i \(-0.173804\pi\)
\(90\) 0 0
\(91\) 1.41421 1.41421i 0.148250 0.148250i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −20.7846 −2.13246
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.34847 + 7.34847i −0.731200 + 0.731200i −0.970858 0.239657i \(-0.922965\pi\)
0.239657 + 0.970858i \(0.422965\pi\)
\(102\) 0 0
\(103\) 7.07107i 0.696733i −0.937358 0.348367i \(-0.886736\pi\)
0.937358 0.348367i \(-0.113264\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) −1.00000 1.00000i −0.0957826 0.0957826i 0.657592 0.753374i \(-0.271575\pi\)
−0.753374 + 0.657592i \(0.771575\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 19.5959 1.84343 0.921714 0.387869i \(-0.126789\pi\)
0.921714 + 0.387869i \(0.126789\pi\)
\(114\) 0 0
\(115\) −16.9706 16.9706i −1.58251 1.58251i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.92820i 0.635107i
\(120\) 0 0
\(121\) 13.0000i 1.18182i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.89898 4.89898i −0.438178 0.438178i
\(126\) 0 0
\(127\) 9.89949 0.878438 0.439219 0.898380i \(-0.355255\pi\)
0.439219 + 0.898380i \(0.355255\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.92820 6.92820i −0.605320 0.605320i 0.336399 0.941719i \(-0.390791\pi\)
−0.941719 + 0.336399i \(0.890791\pi\)
\(132\) 0 0
\(133\) −6.00000 + 6.00000i −0.520266 + 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.6969i 1.25564i −0.778357 0.627822i \(-0.783947\pi\)
0.778357 0.627822i \(-0.216053\pi\)
\(138\) 0 0
\(139\) 11.3137 11.3137i 0.959616 0.959616i −0.0395994 0.999216i \(-0.512608\pi\)
0.999216 + 0.0395994i \(0.0126082\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.92820 0.579365
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.44949 2.44949i 0.200670 0.200670i −0.599617 0.800287i \(-0.704680\pi\)
0.800287 + 0.599617i \(0.204680\pi\)
\(150\) 0 0
\(151\) 15.5563i 1.26596i −0.774169 0.632979i \(-0.781832\pi\)
0.774169 0.632979i \(-0.218168\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.46410 + 3.46410i −0.278243 + 0.278243i
\(156\) 0 0
\(157\) 7.00000 + 7.00000i 0.558661 + 0.558661i 0.928926 0.370265i \(-0.120733\pi\)
−0.370265 + 0.928926i \(0.620733\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.79796 −0.772187
\(162\) 0 0
\(163\) 4.24264 + 4.24264i 0.332309 + 0.332309i 0.853463 0.521154i \(-0.174498\pi\)
−0.521154 + 0.853463i \(0.674498\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.92820i 0.536120i −0.963402 0.268060i \(-0.913617\pi\)
0.963402 0.268060i \(-0.0863826\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.2474 + 12.2474i 0.931156 + 0.931156i 0.997778 0.0666220i \(-0.0212222\pi\)
−0.0666220 + 0.997778i \(0.521222\pi\)
\(174\) 0 0
\(175\) −9.89949 −0.748331
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) 1.00000 1.00000i 0.0743294 0.0743294i −0.668965 0.743294i \(-0.733262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 34.2929i 2.52126i
\(186\) 0 0
\(187\) −16.9706 + 16.9706i −1.24101 + 1.24101i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.44949 2.44949i 0.174519 0.174519i −0.614443 0.788962i \(-0.710619\pi\)
0.788962 + 0.614443i \(0.210619\pi\)
\(198\) 0 0
\(199\) 7.07107i 0.501255i 0.968084 + 0.250627i \(0.0806369\pi\)
−0.968084 + 0.250627i \(0.919363\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.46410 3.46410i 0.243132 0.243132i
\(204\) 0 0
\(205\) 12.0000 + 12.0000i 0.838116 + 0.838116i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −29.3939 −2.03322
\(210\) 0 0
\(211\) −11.3137 11.3137i −0.778868 0.778868i 0.200770 0.979638i \(-0.435655\pi\)
−0.979638 + 0.200770i \(0.935655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 20.7846i 1.41750i
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.89898 4.89898i −0.329541 0.329541i
\(222\) 0 0
\(223\) 7.07107 0.473514 0.236757 0.971569i \(-0.423916\pi\)
0.236757 + 0.971569i \(0.423916\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.46410 + 3.46410i 0.229920 + 0.229920i 0.812659 0.582739i \(-0.198019\pi\)
−0.582739 + 0.812659i \(0.698019\pi\)
\(228\) 0 0
\(229\) 1.00000 1.00000i 0.0660819 0.0660819i −0.673293 0.739375i \(-0.735121\pi\)
0.739375 + 0.673293i \(0.235121\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.79796i 0.641886i 0.947099 + 0.320943i \(0.104000\pi\)
−0.947099 + 0.320943i \(0.896000\pi\)
\(234\) 0 0
\(235\) 16.9706 16.9706i 1.10704 1.10704i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.92820 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.2474 12.2474i 0.782461 0.782461i
\(246\) 0 0
\(247\) 8.48528i 0.539906i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3205 17.3205i 1.09326 1.09326i 0.0980825 0.995178i \(-0.468729\pi\)
0.995178 0.0980825i \(-0.0312709\pi\)
\(252\) 0 0
\(253\) −24.0000 24.0000i −1.50887 1.50887i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.79796 −0.611180 −0.305590 0.952163i \(-0.598854\pi\)
−0.305590 + 0.952163i \(0.598854\pi\)
\(258\) 0 0
\(259\) 9.89949 + 9.89949i 0.615125 + 0.615125i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.8564i 0.854423i −0.904152 0.427211i \(-0.859496\pi\)
0.904152 0.427211i \(-0.140504\pi\)
\(264\) 0 0
\(265\) 12.0000i 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.34847 + 7.34847i 0.448044 + 0.448044i 0.894704 0.446660i \(-0.147387\pi\)
−0.446660 + 0.894704i \(0.647387\pi\)
\(270\) 0 0
\(271\) −7.07107 −0.429537 −0.214768 0.976665i \(-0.568900\pi\)
−0.214768 + 0.976665i \(0.568900\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −24.2487 24.2487i −1.46225 1.46225i
\(276\) 0 0
\(277\) −1.00000 + 1.00000i −0.0600842 + 0.0600842i −0.736510 0.676426i \(-0.763528\pi\)
0.676426 + 0.736510i \(0.263528\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 29.3939i 1.75349i −0.480954 0.876746i \(-0.659710\pi\)
0.480954 0.876746i \(-0.340290\pi\)
\(282\) 0 0
\(283\) −5.65685 + 5.65685i −0.336265 + 0.336265i −0.854960 0.518695i \(-0.826418\pi\)
0.518695 + 0.854960i \(0.326418\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.92820 0.408959
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.2474 12.2474i 0.715504 0.715504i −0.252177 0.967681i \(-0.581147\pi\)
0.967681 + 0.252177i \(0.0811467\pi\)
\(294\) 0 0
\(295\) 33.9411i 1.97613i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.92820 6.92820i 0.400668 0.400668i
\(300\) 0 0
\(301\) −6.00000 6.00000i −0.345834 0.345834i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 24.4949 1.40257
\(306\) 0 0
\(307\) 5.65685 + 5.65685i 0.322854 + 0.322854i 0.849861 0.527007i \(-0.176686\pi\)
−0.527007 + 0.849861i \(0.676686\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.8564i 0.785725i 0.919597 + 0.392862i \(0.128515\pi\)
−0.919597 + 0.392862i \(0.871485\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.34847 + 7.34847i 0.412731 + 0.412731i 0.882689 0.469958i \(-0.155731\pi\)
−0.469958 + 0.882689i \(0.655731\pi\)
\(318\) 0 0
\(319\) 16.9706 0.950169
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.7846 + 20.7846i 1.15649 + 1.15649i
\(324\) 0 0
\(325\) 7.00000 7.00000i 0.388290 0.388290i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.79796i 0.540179i
\(330\) 0 0
\(331\) −11.3137 + 11.3137i −0.621858 + 0.621858i −0.946006 0.324149i \(-0.894922\pi\)
0.324149 + 0.946006i \(0.394922\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −27.7128 −1.51411
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.89898 + 4.89898i −0.265295 + 0.265295i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.46410 3.46410i 0.185963 0.185963i −0.607985 0.793948i \(-0.708022\pi\)
0.793948 + 0.607985i \(0.208022\pi\)
\(348\) 0 0
\(349\) 19.0000 + 19.0000i 1.01705 + 1.01705i 0.999852 + 0.0171945i \(0.00547346\pi\)
0.0171945 + 0.999852i \(0.494527\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.79796 0.521493 0.260746 0.965407i \(-0.416031\pi\)
0.260746 + 0.965407i \(0.416031\pi\)
\(354\) 0 0
\(355\) 33.9411 + 33.9411i 1.80141 + 1.80141i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 17.0000i 0.894737i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −29.3939 29.3939i −1.53855 1.53855i
\(366\) 0 0
\(367\) −32.5269 −1.69789 −0.848945 0.528480i \(-0.822762\pi\)
−0.848945 + 0.528480i \(0.822762\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.46410 + 3.46410i 0.179847 + 0.179847i
\(372\) 0 0
\(373\) 19.0000 19.0000i 0.983783 0.983783i −0.0160879 0.999871i \(-0.505121\pi\)
0.999871 + 0.0160879i \(0.00512115\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.89898i 0.252310i
\(378\) 0 0
\(379\) −4.24264 + 4.24264i −0.217930 + 0.217930i −0.807626 0.589696i \(-0.799248\pi\)
0.589696 + 0.807626i \(0.299248\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.92820 0.354015 0.177007 0.984210i \(-0.443358\pi\)
0.177007 + 0.984210i \(0.443358\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −17.1464 + 17.1464i −0.869358 + 0.869358i −0.992401 0.123043i \(-0.960735\pi\)
0.123043 + 0.992401i \(0.460735\pi\)
\(390\) 0 0
\(391\) 33.9411i 1.71648i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −38.1051 + 38.1051i −1.91728 + 1.91728i
\(396\) 0 0
\(397\) −7.00000 7.00000i −0.351320 0.351320i 0.509281 0.860601i \(-0.329912\pi\)
−0.860601 + 0.509281i \(0.829912\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.6969 −0.733930 −0.366965 0.930235i \(-0.619603\pi\)
−0.366965 + 0.930235i \(0.619603\pi\)
\(402\) 0 0
\(403\) −1.41421 1.41421i −0.0704470 0.0704470i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 48.4974i 2.40393i
\(408\) 0 0
\(409\) 24.0000i 1.18672i 0.804936 + 0.593362i \(0.202200\pi\)
−0.804936 + 0.593362i \(0.797800\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.79796 + 9.79796i 0.482126 + 0.482126i
\(414\) 0 0
\(415\) 50.9117 2.49916
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.3923 + 10.3923i 0.507697 + 0.507697i 0.913819 0.406122i \(-0.133119\pi\)
−0.406122 + 0.913819i \(0.633119\pi\)
\(420\) 0 0
\(421\) 25.0000 25.0000i 1.21843 1.21843i 0.250242 0.968183i \(-0.419490\pi\)
0.968183 0.250242i \(-0.0805102\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 34.2929i 1.66345i
\(426\) 0 0
\(427\) 7.07107 7.07107i 0.342193 0.342193i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −29.3939 + 29.3939i −1.40610 + 1.40610i
\(438\) 0 0
\(439\) 26.8701i 1.28244i 0.767358 + 0.641219i \(0.221571\pi\)
−0.767358 + 0.641219i \(0.778429\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.3923 + 10.3923i −0.493753 + 0.493753i −0.909487 0.415733i \(-0.863525\pi\)
0.415733 + 0.909487i \(0.363525\pi\)
\(444\) 0 0
\(445\) −24.0000 24.0000i −1.13771 1.13771i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 34.2929 1.61838 0.809190 0.587547i \(-0.199906\pi\)
0.809190 + 0.587547i \(0.199906\pi\)
\(450\) 0 0
\(451\) 16.9706 + 16.9706i 0.799113 + 0.799113i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.92820i 0.324799i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.44949 2.44949i −0.114084 0.114084i 0.647760 0.761844i \(-0.275706\pi\)
−0.761844 + 0.647760i \(0.775706\pi\)
\(462\) 0 0
\(463\) 24.0416 1.11731 0.558655 0.829400i \(-0.311318\pi\)
0.558655 + 0.829400i \(0.311318\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.46410 3.46410i −0.160300 0.160300i 0.622400 0.782699i \(-0.286158\pi\)
−0.782699 + 0.622400i \(0.786158\pi\)
\(468\) 0 0
\(469\) −8.00000 + 8.00000i −0.369406 + 0.369406i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 29.3939i 1.35153i
\(474\) 0 0
\(475\) −29.6985 + 29.6985i −1.36266 + 1.36266i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.7128 1.26623 0.633115 0.774057i \(-0.281776\pi\)
0.633115 + 0.774057i \(0.281776\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 35.3553i 1.60210i 0.598595 + 0.801052i \(0.295726\pi\)
−0.598595 + 0.801052i \(0.704274\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −20.7846 + 20.7846i −0.937996 + 0.937996i −0.998187 0.0601906i \(-0.980829\pi\)
0.0601906 + 0.998187i \(0.480829\pi\)
\(492\) 0 0
\(493\) −12.0000 12.0000i −0.540453 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.5959 0.878997
\(498\) 0 0
\(499\) −5.65685 5.65685i −0.253236 0.253236i 0.569060 0.822296i \(-0.307307\pi\)
−0.822296 + 0.569060i \(0.807307\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 36.0000i 1.60198i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.34847 + 7.34847i 0.325715 + 0.325715i 0.850955 0.525239i \(-0.176024\pi\)
−0.525239 + 0.850955i \(0.676024\pi\)
\(510\) 0 0
\(511\) −16.9706 −0.750733
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.3205 17.3205i −0.763233 0.763233i
\(516\) 0 0
\(517\) 24.0000 24.0000i 1.05552 1.05552i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.6969i 0.643885i −0.946759 0.321942i \(-0.895664\pi\)
0.946759 0.321942i \(-0.104336\pi\)
\(522\) 0 0
\(523\) 21.2132 21.2132i 0.927589 0.927589i −0.0699611 0.997550i \(-0.522288\pi\)
0.997550 + 0.0699611i \(0.0222875\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.92820 0.301797
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.89898 + 4.89898i −0.212198 + 0.212198i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17.3205 17.3205i 0.746047 0.746047i
\(540\) 0 0
\(541\) 11.0000 + 11.0000i 0.472927 + 0.472927i 0.902861 0.429934i \(-0.141463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.89898 −0.209849
\(546\) 0 0
\(547\) 12.7279 + 12.7279i 0.544207 + 0.544207i 0.924759 0.380553i \(-0.124266\pi\)
−0.380553 + 0.924759i \(0.624266\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 20.7846i 0.885454i
\(552\) 0 0
\(553\) 22.0000i 0.935535i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.34847 + 7.34847i 0.311365 + 0.311365i 0.845438 0.534073i \(-0.179339\pi\)
−0.534073 + 0.845438i \(0.679339\pi\)
\(558\) 0 0
\(559\) 8.48528 0.358889
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.46410 + 3.46410i 0.145994 + 0.145994i 0.776326 0.630332i \(-0.217081\pi\)
−0.630332 + 0.776326i \(0.717081\pi\)
\(564\) 0 0
\(565\) 48.0000 48.0000i 2.01938 2.01938i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.6969i 0.616128i 0.951366 + 0.308064i \(0.0996810\pi\)
−0.951366 + 0.308064i \(0.900319\pi\)
\(570\) 0 0
\(571\) −28.2843 + 28.2843i −1.18366 + 1.18366i −0.204871 + 0.978789i \(0.565677\pi\)
−0.978789 + 0.204871i \(0.934323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −48.4974 −2.02248
\(576\) 0 0
\(577\) 12.0000 0.499567 0.249783 0.968302i \(-0.419641\pi\)
0.249783 + 0.968302i \(0.419641\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 14.6969 14.6969i 0.609732 0.609732i
\(582\) 0 0
\(583\) 16.9706i 0.702849i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.7128 27.7128i 1.14383 1.14383i 0.156087 0.987743i \(-0.450112\pi\)
0.987743 0.156087i \(-0.0498880\pi\)
\(588\) 0 0
\(589\) 6.00000 + 6.00000i 0.247226 + 0.247226i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −39.1918 −1.60942 −0.804708 0.593671i \(-0.797678\pi\)
−0.804708 + 0.593671i \(0.797678\pi\)
\(594\) 0 0
\(595\) 16.9706 + 16.9706i 0.695725 + 0.695725i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.92820i 0.283079i 0.989933 + 0.141539i \(0.0452052\pi\)
−0.989933 + 0.141539i \(0.954795\pi\)
\(600\) 0 0
\(601\) 12.0000i 0.489490i −0.969587 0.244745i \(-0.921296\pi\)
0.969587 0.244745i \(-0.0787043\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −31.8434 31.8434i −1.29462 1.29462i
\(606\) 0 0
\(607\) 9.89949 0.401808 0.200904 0.979611i \(-0.435612\pi\)
0.200904 + 0.979611i \(0.435612\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.92820 + 6.92820i 0.280285 + 0.280285i
\(612\) 0 0
\(613\) −19.0000 + 19.0000i −0.767403 + 0.767403i −0.977649 0.210246i \(-0.932574\pi\)
0.210246 + 0.977649i \(0.432574\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.5959i 0.788902i −0.918917 0.394451i \(-0.870935\pi\)
0.918917 0.394451i \(-0.129065\pi\)
\(618\) 0 0
\(619\) −11.3137 + 11.3137i −0.454736 + 0.454736i −0.896923 0.442187i \(-0.854203\pi\)
0.442187 + 0.896923i \(0.354203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.8564 −0.555145
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 34.2929 34.2929i 1.36735 1.36735i
\(630\) 0 0
\(631\) 1.41421i 0.0562990i −0.999604 0.0281495i \(-0.991039\pi\)
0.999604 0.0281495i \(-0.00896144\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.2487 24.2487i 0.962281 0.962281i
\(636\) 0 0
\(637\) 5.00000 + 5.00000i 0.198107 + 0.198107i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −14.6969 −0.580494 −0.290247 0.956952i \(-0.593737\pi\)
−0.290247 + 0.956952i \(0.593737\pi\)
\(642\) 0 0
\(643\) 29.6985 + 29.6985i 1.17119 + 1.17119i 0.981925 + 0.189269i \(0.0606117\pi\)
0.189269 + 0.981925i \(0.439388\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.92820i 0.272376i −0.990683 0.136188i \(-0.956515\pi\)
0.990683 0.136188i \(-0.0434851\pi\)
\(648\) 0 0
\(649\) 48.0000i 1.88416i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.9444 + 26.9444i 1.05442 + 1.05442i 0.998432 + 0.0559837i \(0.0178295\pi\)
0.0559837 + 0.998432i \(0.482171\pi\)
\(654\) 0 0
\(655\) −33.9411 −1.32619
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.92820 + 6.92820i 0.269884 + 0.269884i 0.829054 0.559169i \(-0.188880\pi\)
−0.559169 + 0.829054i \(0.688880\pi\)
\(660\) 0 0
\(661\) −19.0000 + 19.0000i −0.739014 + 0.739014i −0.972387 0.233373i \(-0.925024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 29.3939i 1.13985i
\(666\) 0 0
\(667\) 16.9706 16.9706i 0.657103 0.657103i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 34.6410 1.33730
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.9444 26.9444i 1.03556 1.03556i 0.0362128 0.999344i \(-0.488471\pi\)
0.999344 0.0362128i \(-0.0115294\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.3205 17.3205i 0.662751 0.662751i −0.293277 0.956028i \(-0.594746\pi\)
0.956028 + 0.293277i \(0.0947457\pi\)
\(684\) 0 0
\(685\) −36.0000 36.0000i −1.37549 1.37549i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.89898 −0.186636
\(690\) 0 0
\(691\) −4.24264 4.24264i −0.161398 0.161398i 0.621788 0.783186i \(-0.286407\pi\)
−0.783186 + 0.621788i \(0.786407\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 55.4256i 2.10241i
\(696\) 0 0
\(697\) 24.0000i 0.909065i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.44949 + 2.44949i 0.0925160 + 0.0925160i 0.751850 0.659334i \(-0.229162\pi\)
−0.659334 + 0.751850i \(0.729162\pi\)
\(702\) 0 0
\(703\) 59.3970 2.24020
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.3923 + 10.3923i 0.390843 + 0.390843i
\(708\) 0 0
\(709\) −1.00000 + 1.00000i −0.0375558 + 0.0375558i −0.725635 0.688080i \(-0.758454\pi\)
0.688080 + 0.725635i \(0.258454\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.79796i 0.366936i
\(714\) 0 0
\(715\) 16.9706 16.9706i 0.634663 0.634663i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.5692 1.55027 0.775135 0.631795i \(-0.217682\pi\)
0.775135 + 0.631795i \(0.217682\pi\)
\(720\) 0 0
\(721\) −10.0000 −0.372419
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 17.1464 17.1464i 0.636802 0.636802i
\(726\) 0 0
\(727\) 15.5563i 0.576953i 0.957487 + 0.288477i \(0.0931487\pi\)
−0.957487 + 0.288477i \(0.906851\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −20.7846 + 20.7846i −0.768747 + 0.768747i
\(732\) 0 0
\(733\) −11.0000 11.0000i −0.406294 0.406294i 0.474150 0.880444i \(-0.342755\pi\)
−0.880444 + 0.474150i \(0.842755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −39.1918 −1.44365
\(738\) 0 0
\(739\) −5.65685 5.65685i −0.208091 0.208091i 0.595365 0.803456i \(-0.297008\pi\)
−0.803456 + 0.595365i \(0.797008\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.6410i 1.27086i 0.772160 + 0.635428i \(0.219176\pi\)
−0.772160 + 0.635428i \(0.780824\pi\)
\(744\) 0 0
\(745\) 12.0000i 0.439646i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −18.3848 −0.670870 −0.335435 0.942063i \(-0.608883\pi\)
−0.335435 + 0.942063i \(0.608883\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −38.1051 38.1051i −1.38679 1.38679i
\(756\) 0 0
\(757\) 25.0000 25.0000i 0.908640 0.908640i −0.0875221 0.996163i \(-0.527895\pi\)
0.996163 + 0.0875221i \(0.0278948\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.89898i 0.177588i 0.996050 + 0.0887939i \(0.0283013\pi\)
−0.996050 + 0.0887939i \(0.971699\pi\)
\(762\) 0 0
\(763\) −1.41421 + 1.41421i −0.0511980 + 0.0511980i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13.8564 −0.500326
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −26.9444 + 26.9444i −0.969122 + 0.969122i −0.999537 0.0304151i \(-0.990317\pi\)
0.0304151 + 0.999537i \(0.490317\pi\)
\(774\) 0 0
\(775\) 9.89949i 0.355600i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.7846 20.7846i 0.744686 0.744686i
\(780\) 0 0
\(781\) 48.0000 + 48.0000i 1.71758 + 1.71758i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 34.2929 1.22396
\(786\) 0 0
\(787\) −12.7279 12.7279i −0.453701 0.453701i 0.442880 0.896581i \(-0.353957\pi\)
−0.896581 + 0.442880i \(0.853957\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 27.7128i 0.985354i
\(792\) 0 0
\(793\) 10.0000i 0.355110i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 36.7423 + 36.7423i 1.30148 + 1.30148i 0.927394 + 0.374087i \(0.122044\pi\)
0.374087 + 0.927394i \(0.377956\pi\)
\(798\) 0 0
\(799\) −33.9411 −1.20075
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −41.5692 41.5692i −1.46695 1.46695i
\(804\) 0 0
\(805\) −24.0000 + 24.0000i −0.845889 + 0.845889i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.89898i 0.172239i −0.996285 0.0861195i \(-0.972553\pi\)
0.996285 0.0861195i \(-0.0274467\pi\)
\(810\) 0 0
\(811\) −4.24264 + 4.24264i −0.148979 + 0.148979i −0.777662 0.628683i \(-0.783594\pi\)
0.628683 + 0.777662i \(0.283594\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 20.7846 0.728053
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.34847 + 7.34847i −0.256463 + 0.256463i −0.823614 0.567151i \(-0.808046\pi\)
0.567151 + 0.823614i \(0.308046\pi\)
\(822\) 0 0
\(823\) 35.3553i 1.23241i −0.787586 0.616205i \(-0.788669\pi\)
0.787586 0.616205i \(-0.211331\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −34.6410 + 34.6410i −1.20459 + 1.20459i −0.231830 + 0.972756i \(0.574471\pi\)
−0.972756 + 0.231830i \(0.925529\pi\)
\(828\) 0 0
\(829\) −23.0000 23.0000i −0.798823 0.798823i 0.184087 0.982910i \(-0.441067\pi\)
−0.982910 + 0.184087i \(0.941067\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −24.4949 −0.848698
\(834\) 0 0
\(835\) −16.9706 16.9706i −0.587291 0.587291i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 13.8564i 0.478376i −0.970973 0.239188i \(-0.923119\pi\)
0.970973 0.239188i \(-0.0768813\pi\)
\(840\) 0 0
\(841\) 17.0000i 0.586207i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26.9444 26.9444i −0.926915 0.926915i
\(846\) 0 0
\(847\) −18.3848 −0.631708
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 48.4974 + 48.4974i 1.66247 + 1.66247i
\(852\) 0 0
\(853\) −29.0000 + 29.0000i −0.992941 + 0.992941i −0.999975 0.00703417i \(-0.997761\pi\)
0.00703417 + 0.999975i \(0.497761\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.4949i 0.836730i 0.908279 + 0.418365i \(0.137397\pi\)
−0.908279 + 0.418365i \(0.862603\pi\)
\(858\) 0 0
\(859\) −12.7279 + 12.7279i −0.434271 + 0.434271i −0.890078 0.455807i \(-0.849351\pi\)
0.455807 + 0.890078i \(0.349351\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.92820 −0.235839 −0.117919 0.993023i \(-0.537622\pi\)
−0.117919 + 0.993023i \(0.537622\pi\)
\(864\) 0 0
\(865\) 60.0000 2.04006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −53.8888 + 53.8888i −1.82805 + 1.82805i
\(870\) 0 0
\(871\) 11.3137i 0.383350i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −6.92820 + 6.92820i −0.234216 + 0.234216i
\(876\) 0 0
\(877\) −5.00000 5.00000i −0.168838 0.168838i 0.617630 0.786468i \(-0.288093\pi\)
−0.786468 + 0.617630i \(0.788093\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.79796 0.330102 0.165051 0.986285i \(-0.447221\pi\)
0.165051 + 0.986285i \(0.447221\pi\)
\(882\) 0 0
\(883\) 12.7279 + 12.7279i 0.428329 + 0.428329i 0.888059 0.459730i \(-0.152054\pi\)
−0.459730 + 0.888059i \(0.652054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.7128i 0.930505i −0.885178 0.465253i \(-0.845963\pi\)
0.885178 0.465253i \(-0.154037\pi\)
\(888\) 0 0
\(889\) 14.0000i 0.469545i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −29.3939 29.3939i −0.983629 0.983629i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.46410 3.46410i −0.115534 0.115534i
\(900\) 0 0
\(901\) 12.0000 12.0000i 0.399778 0.399778i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.89898i 0.162848i
\(906\) 0 0
\(907\) −21.2132 + 21.2132i −0.704373 + 0.704373i −0.965346 0.260973i \(-0.915957\pi\)
0.260973 + 0.965346i \(0.415957\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −34.6410 −1.14771 −0.573854 0.818958i \(-0.694552\pi\)
−0.573854 + 0.818958i \(0.694552\pi\)
\(912\) 0 0
\(913\) 72.0000 2.38285
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.79796 + 9.79796i −0.323557 + 0.323557i
\(918\) 0 0
\(919\) 35.3553i 1.16627i −0.812377 0.583133i \(-0.801827\pi\)
0.812377 0.583133i \(-0.198173\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −13.8564 + 13.8564i −0.456089 + 0.456089i
\(924\) 0 0
\(925\) 49.0000 + 49.0000i 1.61111 + 1.61111i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.4949 0.803652 0.401826 0.915716i \(-0.368376\pi\)
0.401826 + 0.915716i \(0.368376\pi\)
\(930\) 0 0
\(931\) −21.2132 21.2132i −0.695235 0.695235i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 83.1384i 2.71892i
\(936\) 0 0
\(937\) 22.0000i 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 36.7423 + 36.7423i 1.19777 + 1.19777i 0.974834 + 0.222932i \(0.0715629\pi\)
0.222932 + 0.974834i \(0.428437\pi\)
\(942\) 0 0
\(943\) 33.9411 1.10528
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.7128 + 27.7128i 0.900545 + 0.900545i 0.995483 0.0949378i \(-0.0302652\pi\)
−0.0949378 + 0.995483i \(0.530265\pi\)
\(948\) 0 0
\(949\) 12.0000 12.0000i 0.389536 0.389536i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 44.0908i 1.42824i −0.700022 0.714121i \(-0.746827\pi\)
0.700022 0.714121i \(-0.253173\pi\)
\(954\) 0 0
\(955\) 50.9117 50.9117i 1.64746 1.64746i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −20.7846 −0.671170
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 24.4949 24.4949i 0.788519 0.788519i
\(966\) 0 0
\(967\) 24.0416i 0.773127i −0.922263 0.386563i \(-0.873662\pi\)
0.922263 0.386563i \(-0.126338\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 10.3923 10.3923i 0.333505 0.333505i −0.520411 0.853916i \(-0.674221\pi\)
0.853916 + 0.520411i \(0.174221\pi\)
\(972\) 0 0
\(973\) −16.0000 16.0000i −0.512936 0.512936i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 53.8888 1.72405 0.862027 0.506862i \(-0.169195\pi\)
0.862027 + 0.506862i \(0.169195\pi\)
\(978\) 0 0
\(979\) −33.9411 33.9411i −1.08476 1.08476i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 55.4256i 1.76780i 0.467673 + 0.883901i \(0.345092\pi\)
−0.467673 + 0.883901i \(0.654908\pi\)
\(984\) 0 0
\(985\) 12.0000i 0.382352i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −29.3939 29.3939i −0.934671 0.934671i
\(990\) 0 0
\(991\) −15.5563 −0.494164 −0.247082 0.968995i \(-0.579472\pi\)
−0.247082 + 0.968995i \(0.579472\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17.3205 + 17.3205i 0.549097 + 0.549097i
\(996\) 0 0
\(997\) −7.00000 + 7.00000i −0.221692 + 0.221692i −0.809211 0.587519i \(-0.800105\pi\)
0.587519 + 0.809211i \(0.300105\pi\)
\(998\) 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.i.577.3 yes 8
3.2 odd 2 inner 2304.2.k.i.577.1 yes 8
4.3 odd 2 inner 2304.2.k.i.577.4 yes 8
8.3 odd 2 2304.2.k.h.577.2 yes 8
8.5 even 2 2304.2.k.h.577.1 8
12.11 even 2 inner 2304.2.k.i.577.2 yes 8
16.3 odd 4 2304.2.k.h.1729.1 yes 8
16.5 even 4 inner 2304.2.k.i.1729.4 yes 8
16.11 odd 4 inner 2304.2.k.i.1729.3 yes 8
16.13 even 4 2304.2.k.h.1729.2 yes 8
24.5 odd 2 2304.2.k.h.577.3 yes 8
24.11 even 2 2304.2.k.h.577.4 yes 8
32.5 even 8 9216.2.a.be.1.4 4
32.11 odd 8 9216.2.a.be.1.1 4
32.21 even 8 9216.2.a.bh.1.2 4
32.27 odd 8 9216.2.a.bh.1.3 4
48.5 odd 4 inner 2304.2.k.i.1729.2 yes 8
48.11 even 4 inner 2304.2.k.i.1729.1 yes 8
48.29 odd 4 2304.2.k.h.1729.4 yes 8
48.35 even 4 2304.2.k.h.1729.3 yes 8
96.5 odd 8 9216.2.a.be.1.2 4
96.11 even 8 9216.2.a.be.1.3 4
96.53 odd 8 9216.2.a.bh.1.4 4
96.59 even 8 9216.2.a.bh.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2304.2.k.h.577.1 8 8.5 even 2
2304.2.k.h.577.2 yes 8 8.3 odd 2
2304.2.k.h.577.3 yes 8 24.5 odd 2
2304.2.k.h.577.4 yes 8 24.11 even 2
2304.2.k.h.1729.1 yes 8 16.3 odd 4
2304.2.k.h.1729.2 yes 8 16.13 even 4
2304.2.k.h.1729.3 yes 8 48.35 even 4
2304.2.k.h.1729.4 yes 8 48.29 odd 4
2304.2.k.i.577.1 yes 8 3.2 odd 2 inner
2304.2.k.i.577.2 yes 8 12.11 even 2 inner
2304.2.k.i.577.3 yes 8 1.1 even 1 trivial
2304.2.k.i.577.4 yes 8 4.3 odd 2 inner
2304.2.k.i.1729.1 yes 8 48.11 even 4 inner
2304.2.k.i.1729.2 yes 8 48.5 odd 4 inner
2304.2.k.i.1729.3 yes 8 16.11 odd 4 inner
2304.2.k.i.1729.4 yes 8 16.5 even 4 inner
9216.2.a.be.1.1 4 32.11 odd 8
9216.2.a.be.1.2 4 96.5 odd 8
9216.2.a.be.1.3 4 96.11 even 8
9216.2.a.be.1.4 4 32.5 even 8
9216.2.a.bh.1.1 4 96.59 even 8
9216.2.a.bh.1.2 4 32.21 even 8
9216.2.a.bh.1.3 4 32.27 odd 8
9216.2.a.bh.1.4 4 96.53 odd 8