Properties

Label 2304.2.l.e.1727.4
Level $2304$
Weight $2$
Character 2304.1727
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(575,2304)]
 
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([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.l (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: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1727.4
Root \(-1.28897 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1727
Dual form 2304.2.l.e.575.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.57794 + 2.57794i) q^{5} -3.74166 q^{7} +O(q^{10})\) \(q+(2.57794 + 2.57794i) q^{5} -3.74166 q^{7} +(3.64575 - 3.64575i) q^{11} +(-4.64575 - 4.64575i) q^{13} +0.913230i q^{17} +(1.41421 - 1.41421i) q^{19} -6.00000i q^{23} +8.29150i q^{25} +(-1.66471 + 1.66471i) q^{29} -6.57008i q^{31} +(-9.64575 - 9.64575i) q^{35} +(-1.00000 + 1.00000i) q^{37} -0.913230 q^{41} +(-3.74166 - 3.74166i) q^{43} -6.00000 q^{47} +7.00000 q^{49} +(-3.49117 - 3.49117i) q^{53} +18.7970 q^{55} +(-1.29150 + 1.29150i) q^{59} +(0.291503 + 0.291503i) q^{61} -23.9529i q^{65} +(3.32941 - 3.32941i) q^{67} -13.2915i q^{71} -8.58301i q^{73} +(-13.6412 + 13.6412i) q^{77} +8.39655i q^{79} +(4.93725 + 4.93725i) q^{83} +(-2.35425 + 2.35425i) q^{85} +14.5544 q^{89} +(17.3828 + 17.3828i) q^{91} +7.29150 q^{95} +13.2915 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{11} - 16 q^{13} - 56 q^{35} - 8 q^{37} - 48 q^{47} + 56 q^{49} + 32 q^{59} - 40 q^{61} - 24 q^{83} - 40 q^{85} + 16 q^{95} + 64 q^{97}+O(q^{100}) \) 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
\(5\) 2.57794 + 2.57794i 1.15289 + 1.15289i 0.985971 + 0.166917i \(0.0533811\pi\)
0.166917 + 0.985971i \(0.446619\pi\)
\(6\) 0 0
\(7\) −3.74166 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.64575 3.64575i 1.09924 1.09924i 0.104735 0.994500i \(-0.466600\pi\)
0.994500 0.104735i \(-0.0333995\pi\)
\(12\) 0 0
\(13\) −4.64575 4.64575i −1.28850 1.28850i −0.935698 0.352801i \(-0.885229\pi\)
−0.352801 0.935698i \(-0.614771\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.913230i 0.221491i 0.993849 + 0.110745i \(0.0353238\pi\)
−0.993849 + 0.110745i \(0.964676\pi\)
\(18\) 0 0
\(19\) 1.41421 1.41421i 0.324443 0.324443i −0.526026 0.850469i \(-0.676318\pi\)
0.850469 + 0.526026i \(0.176318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 8.29150i 1.65830i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.66471 + 1.66471i −0.309128 + 0.309128i −0.844571 0.535443i \(-0.820145\pi\)
0.535443 + 0.844571i \(0.320145\pi\)
\(30\) 0 0
\(31\) 6.57008i 1.18002i −0.807395 0.590011i \(-0.799123\pi\)
0.807395 0.590011i \(-0.200877\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.64575 9.64575i −1.63043 1.63043i
\(36\) 0 0
\(37\) −1.00000 + 1.00000i −0.164399 + 0.164399i −0.784512 0.620113i \(-0.787087\pi\)
0.620113 + 0.784512i \(0.287087\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.913230 −0.142623 −0.0713113 0.997454i \(-0.522718\pi\)
−0.0713113 + 0.997454i \(0.522718\pi\)
\(42\) 0 0
\(43\) −3.74166 3.74166i −0.570597 0.570597i 0.361698 0.932295i \(-0.382197\pi\)
−0.932295 + 0.361698i \(0.882197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.49117 3.49117i −0.479548 0.479548i 0.425439 0.904987i \(-0.360120\pi\)
−0.904987 + 0.425439i \(0.860120\pi\)
\(54\) 0 0
\(55\) 18.7970 2.53459
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.29150 + 1.29150i −0.168139 + 0.168139i −0.786161 0.618022i \(-0.787934\pi\)
0.618022 + 0.786161i \(0.287934\pi\)
\(60\) 0 0
\(61\) 0.291503 + 0.291503i 0.0373231 + 0.0373231i 0.725522 0.688199i \(-0.241598\pi\)
−0.688199 + 0.725522i \(0.741598\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 23.9529i 2.97099i
\(66\) 0 0
\(67\) 3.32941 3.32941i 0.406752 0.406752i −0.473852 0.880604i \(-0.657137\pi\)
0.880604 + 0.473852i \(0.157137\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.2915i 1.57741i −0.614771 0.788706i \(-0.710752\pi\)
0.614771 0.788706i \(-0.289248\pi\)
\(72\) 0 0
\(73\) 8.58301i 1.00456i −0.864704 0.502282i \(-0.832494\pi\)
0.864704 0.502282i \(-0.167506\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −13.6412 + 13.6412i −1.55455 + 1.55455i
\(78\) 0 0
\(79\) 8.39655i 0.944685i 0.881415 + 0.472343i \(0.156591\pi\)
−0.881415 + 0.472343i \(0.843409\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.93725 + 4.93725i 0.541934 + 0.541934i 0.924096 0.382162i \(-0.124820\pi\)
−0.382162 + 0.924096i \(0.624820\pi\)
\(84\) 0 0
\(85\) −2.35425 + 2.35425i −0.255354 + 0.255354i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.5544 1.54276 0.771381 0.636374i \(-0.219566\pi\)
0.771381 + 0.636374i \(0.219566\pi\)
\(90\) 0 0
\(91\) 17.3828 + 17.3828i 1.82221 + 1.82221i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.29150 0.748092
\(96\) 0 0
\(97\) 13.2915 1.34955 0.674774 0.738025i \(-0.264241\pi\)
0.674774 + 0.738025i \(0.264241\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.82058 + 6.82058i 0.678673 + 0.678673i 0.959700 0.281027i \(-0.0906751\pi\)
−0.281027 + 0.959700i \(0.590675\pi\)
\(102\) 0 0
\(103\) 6.57008 0.647370 0.323685 0.946165i \(-0.395078\pi\)
0.323685 + 0.946165i \(0.395078\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.29150 + 7.29150i −0.704896 + 0.704896i −0.965457 0.260561i \(-0.916092\pi\)
0.260561 + 0.965457i \(0.416092\pi\)
\(108\) 0 0
\(109\) −1.35425 1.35425i −0.129713 0.129713i 0.639269 0.768983i \(-0.279237\pi\)
−0.768983 + 0.639269i \(0.779237\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.7279i 1.19734i −0.800995 0.598671i \(-0.795696\pi\)
0.800995 0.598671i \(-0.204304\pi\)
\(114\) 0 0
\(115\) 15.4676 15.4676i 1.44236 1.44236i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.41699i 0.313235i
\(120\) 0 0
\(121\) 15.5830i 1.41664i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.48528 + 8.48528i −0.758947 + 0.758947i
\(126\) 0 0
\(127\) 1.91520i 0.169946i 0.996383 + 0.0849731i \(0.0270804\pi\)
−0.996383 + 0.0849731i \(0.972920\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 12.0000i −1.04844 1.04844i −0.998765 0.0496797i \(-0.984180\pi\)
−0.0496797 0.998765i \(-0.515820\pi\)
\(132\) 0 0
\(133\) −5.29150 + 5.29150i −0.458831 + 0.458831i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.39851 0.802969 0.401485 0.915866i \(-0.368494\pi\)
0.401485 + 0.915866i \(0.368494\pi\)
\(138\) 0 0
\(139\) −3.32941 3.32941i −0.282397 0.282397i 0.551667 0.834064i \(-0.313992\pi\)
−0.834064 + 0.551667i \(0.813992\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −33.8745 −2.83273
\(144\) 0 0
\(145\) −8.58301 −0.712780
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.73381 + 7.73381i 0.633578 + 0.633578i 0.948964 0.315386i \(-0.102134\pi\)
−0.315386 + 0.948964i \(0.602134\pi\)
\(150\) 0 0
\(151\) 10.4005 0.846379 0.423189 0.906041i \(-0.360910\pi\)
0.423189 + 0.906041i \(0.360910\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.9373 16.9373i 1.36043 1.36043i
\(156\) 0 0
\(157\) −6.29150 6.29150i −0.502117 0.502117i 0.409979 0.912095i \(-0.365536\pi\)
−0.912095 + 0.409979i \(0.865536\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 22.4499i 1.76930i
\(162\) 0 0
\(163\) 1.91520 1.91520i 0.150010 0.150010i −0.628113 0.778122i \(-0.716172\pi\)
0.778122 + 0.628113i \(0.216172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.29150i 0.564233i 0.959380 + 0.282117i \(0.0910365\pi\)
−0.959380 + 0.282117i \(0.908963\pi\)
\(168\) 0 0
\(169\) 30.1660i 2.32046i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.57794 + 2.57794i −0.195997 + 0.195997i −0.798281 0.602285i \(-0.794257\pi\)
0.602285 + 0.798281i \(0.294257\pi\)
\(174\) 0 0
\(175\) 31.0240i 2.34519i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.8745 15.8745i −1.18652 1.18652i −0.978024 0.208492i \(-0.933144\pi\)
−0.208492 0.978024i \(-0.566856\pi\)
\(180\) 0 0
\(181\) 1.35425 1.35425i 0.100661 0.100661i −0.654983 0.755644i \(-0.727324\pi\)
0.755644 + 0.654983i \(0.227324\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.15587 −0.379067
\(186\) 0 0
\(187\) 3.32941 + 3.32941i 0.243471 + 0.243471i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.2915 1.39588 0.697942 0.716154i \(-0.254099\pi\)
0.697942 + 0.716154i \(0.254099\pi\)
\(192\) 0 0
\(193\) −4.58301 −0.329892 −0.164946 0.986303i \(-0.552745\pi\)
−0.164946 + 0.986303i \(0.552745\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.9764 11.9764i −0.853287 0.853287i 0.137250 0.990536i \(-0.456174\pi\)
−0.990536 + 0.137250i \(0.956174\pi\)
\(198\) 0 0
\(199\) −16.8818 −1.19672 −0.598360 0.801227i \(-0.704181\pi\)
−0.598360 + 0.801227i \(0.704181\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.22876 6.22876i 0.437173 0.437173i
\(204\) 0 0
\(205\) −2.35425 2.35425i −0.164428 0.164428i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.3117i 0.713278i
\(210\) 0 0
\(211\) −15.4676 + 15.4676i −1.06483 + 1.06483i −0.0670873 + 0.997747i \(0.521371\pi\)
−0.997747 + 0.0670873i \(0.978629\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 19.2915i 1.31567i
\(216\) 0 0
\(217\) 24.5830i 1.66880i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.24264 4.24264i 0.285391 0.285391i
\(222\) 0 0
\(223\) 15.0554i 1.00818i 0.863651 + 0.504091i \(0.168172\pi\)
−0.863651 + 0.504091i \(0.831828\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.35425 2.35425i −0.156257 0.156257i 0.624649 0.780906i \(-0.285242\pi\)
−0.780906 + 0.624649i \(0.785242\pi\)
\(228\) 0 0
\(229\) −9.35425 + 9.35425i −0.618146 + 0.618146i −0.945056 0.326909i \(-0.893993\pi\)
0.326909 + 0.945056i \(0.393993\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.5544 −0.953489 −0.476745 0.879042i \(-0.658183\pi\)
−0.476745 + 0.879042i \(0.658183\pi\)
\(234\) 0 0
\(235\) −15.4676 15.4676i −1.00900 1.00900i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.87451 0.638729 0.319364 0.947632i \(-0.396531\pi\)
0.319364 + 0.947632i \(0.396531\pi\)
\(240\) 0 0
\(241\) 13.8745 0.893736 0.446868 0.894600i \(-0.352539\pi\)
0.446868 + 0.894600i \(0.352539\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 18.0455 + 18.0455i 1.15289 + 1.15289i
\(246\) 0 0
\(247\) −13.1402 −0.836089
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.35425 8.35425i 0.527316 0.527316i −0.392455 0.919771i \(-0.628374\pi\)
0.919771 + 0.392455i \(0.128374\pi\)
\(252\) 0 0
\(253\) −21.8745 21.8745i −1.37524 1.37524i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.06910i 0.378580i −0.981921 0.189290i \(-0.939381\pi\)
0.981921 0.189290i \(-0.0606186\pi\)
\(258\) 0 0
\(259\) 3.74166 3.74166i 0.232495 0.232495i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.41699i 0.580677i 0.956924 + 0.290338i \(0.0937679\pi\)
−0.956924 + 0.290338i \(0.906232\pi\)
\(264\) 0 0
\(265\) 18.0000i 1.10573i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.9588 18.9588i 1.15594 1.15594i 0.170596 0.985341i \(-0.445431\pi\)
0.985341 0.170596i \(-0.0545693\pi\)
\(270\) 0 0
\(271\) 1.91520i 0.116340i 0.998307 + 0.0581700i \(0.0185265\pi\)
−0.998307 + 0.0581700i \(0.981473\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 30.2288 + 30.2288i 1.82286 + 1.82286i
\(276\) 0 0
\(277\) 10.6458 10.6458i 0.639641 0.639641i −0.310826 0.950467i \(-0.600606\pi\)
0.950467 + 0.310826i \(0.100606\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −26.6926 −1.59235 −0.796173 0.605069i \(-0.793146\pi\)
−0.796173 + 0.605069i \(0.793146\pi\)
\(282\) 0 0
\(283\) −18.7970 18.7970i −1.11737 1.11737i −0.992126 0.125241i \(-0.960030\pi\)
−0.125241 0.992126i \(-0.539970\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.41699 0.201699
\(288\) 0 0
\(289\) 16.1660 0.950942
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.161755 0.161755i −0.00944985 0.00944985i 0.702366 0.711816i \(-0.252127\pi\)
−0.711816 + 0.702366i \(0.752127\pi\)
\(294\) 0 0
\(295\) −6.65882 −0.387692
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −27.8745 + 27.8745i −1.61202 + 1.61202i
\(300\) 0 0
\(301\) 14.0000 + 14.0000i 0.806947 + 0.806947i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.50295i 0.0860587i
\(306\) 0 0
\(307\) −22.1264 + 22.1264i −1.26282 + 1.26282i −0.313103 + 0.949719i \(0.601369\pi\)
−0.949719 + 0.313103i \(0.898631\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 29.1660i 1.65385i −0.562310 0.826926i \(-0.690087\pi\)
0.562310 0.826926i \(-0.309913\pi\)
\(312\) 0 0
\(313\) 27.1660i 1.53551i −0.640741 0.767757i \(-0.721373\pi\)
0.640741 0.767757i \(-0.278627\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.49117 + 3.49117i −0.196083 + 0.196083i −0.798319 0.602235i \(-0.794277\pi\)
0.602235 + 0.798319i \(0.294277\pi\)
\(318\) 0 0
\(319\) 12.1382i 0.679609i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.29150 + 1.29150i 0.0718611 + 0.0718611i
\(324\) 0 0
\(325\) 38.5203 38.5203i 2.13672 2.13672i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 22.4499 1.23771
\(330\) 0 0
\(331\) 15.4676 + 15.4676i 0.850177 + 0.850177i 0.990155 0.139978i \(-0.0447031\pi\)
−0.139978 + 0.990155i \(0.544703\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 17.1660 0.937879
\(336\) 0 0
\(337\) 8.58301 0.467546 0.233773 0.972291i \(-0.424893\pi\)
0.233773 + 0.972291i \(0.424893\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −23.9529 23.9529i −1.29712 1.29712i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.35425 2.35425i 0.126383 0.126383i −0.641086 0.767469i \(-0.721516\pi\)
0.767469 + 0.641086i \(0.221516\pi\)
\(348\) 0 0
\(349\) 21.5830 + 21.5830i 1.15531 + 1.15531i 0.985472 + 0.169840i \(0.0543251\pi\)
0.169840 + 0.985472i \(0.445675\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 23.0397i 1.22628i 0.789975 + 0.613139i \(0.210093\pi\)
−0.789975 + 0.613139i \(0.789907\pi\)
\(354\) 0 0
\(355\) 34.2646 34.2646i 1.81858 1.81858i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 37.2915i 1.96817i 0.177698 + 0.984085i \(0.443135\pi\)
−0.177698 + 0.984085i \(0.556865\pi\)
\(360\) 0 0
\(361\) 15.0000i 0.789474i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 22.1264 22.1264i 1.15815 1.15815i
\(366\) 0 0
\(367\) 3.74166i 0.195313i −0.995220 0.0976565i \(-0.968865\pi\)
0.995220 0.0976565i \(-0.0311346\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.0627 + 13.0627i 0.678184 + 0.678184i
\(372\) 0 0
\(373\) −24.2915 + 24.2915i −1.25777 + 1.25777i −0.305609 + 0.952157i \(0.598860\pi\)
−0.952157 + 0.305609i \(0.901140\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.4676 0.796623
\(378\) 0 0
\(379\) 5.24461 + 5.24461i 0.269397 + 0.269397i 0.828857 0.559460i \(-0.188991\pi\)
−0.559460 + 0.828857i \(0.688991\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.4575 −1.24972 −0.624860 0.780737i \(-0.714844\pi\)
−0.624860 + 0.780737i \(0.714844\pi\)
\(384\) 0 0
\(385\) −70.3320 −3.58445
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −19.5485 19.5485i −0.991148 0.991148i 0.00881296 0.999961i \(-0.497195\pi\)
−0.999961 + 0.00881296i \(0.997195\pi\)
\(390\) 0 0
\(391\) 5.47938 0.277104
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −21.6458 + 21.6458i −1.08912 + 1.08912i
\(396\) 0 0
\(397\) 7.58301 + 7.58301i 0.380580 + 0.380580i 0.871311 0.490731i \(-0.163270\pi\)
−0.490731 + 0.871311i \(0.663270\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.0573i 0.801865i 0.916108 + 0.400932i \(0.131314\pi\)
−0.916108 + 0.400932i \(0.868686\pi\)
\(402\) 0 0
\(403\) −30.5230 + 30.5230i −1.52046 + 1.52046i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.29150i 0.361426i
\(408\) 0 0
\(409\) 22.7085i 1.12286i −0.827523 0.561431i \(-0.810251\pi\)
0.827523 0.561431i \(-0.189749\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.83236 4.83236i 0.237785 0.237785i
\(414\) 0 0
\(415\) 25.4558i 1.24958i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.93725 4.93725i −0.241201 0.241201i 0.576146 0.817347i \(-0.304556\pi\)
−0.817347 + 0.576146i \(0.804556\pi\)
\(420\) 0 0
\(421\) −5.93725 + 5.93725i −0.289364 + 0.289364i −0.836829 0.547465i \(-0.815593\pi\)
0.547465 + 0.836829i \(0.315593\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.57205 −0.367298
\(426\) 0 0
\(427\) −1.09070 1.09070i −0.0527828 0.0527828i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.58301 −0.413429 −0.206714 0.978401i \(-0.566277\pi\)
−0.206714 + 0.978401i \(0.566277\pi\)
\(432\) 0 0
\(433\) −26.5830 −1.27750 −0.638749 0.769415i \(-0.720548\pi\)
−0.638749 + 0.769415i \(0.720548\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.48528 8.48528i −0.405906 0.405906i
\(438\) 0 0
\(439\) 15.0554 0.718553 0.359277 0.933231i \(-0.383023\pi\)
0.359277 + 0.933231i \(0.383023\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.35425 + 2.35425i −0.111854 + 0.111854i −0.760818 0.648965i \(-0.775202\pi\)
0.648965 + 0.760818i \(0.275202\pi\)
\(444\) 0 0
\(445\) 37.5203 + 37.5203i 1.77863 + 1.77863i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.57205i 0.357347i 0.983908 + 0.178674i \(0.0571806\pi\)
−0.983908 + 0.178674i \(0.942819\pi\)
\(450\) 0 0
\(451\) −3.32941 + 3.32941i −0.156776 + 0.156776i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 89.6235i 4.20162i
\(456\) 0 0
\(457\) 30.4575i 1.42474i −0.701803 0.712371i \(-0.747621\pi\)
0.701803 0.712371i \(-0.252379\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −23.2014 + 23.2014i −1.08060 + 1.08060i −0.0841442 + 0.996454i \(0.526816\pi\)
−0.996454 + 0.0841442i \(0.973184\pi\)
\(462\) 0 0
\(463\) 37.6828i 1.75127i −0.482976 0.875634i \(-0.660444\pi\)
0.482976 0.875634i \(-0.339556\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.3542 + 14.3542i 0.664235 + 0.664235i 0.956376 0.292140i \(-0.0943674\pi\)
−0.292140 + 0.956376i \(0.594367\pi\)
\(468\) 0 0
\(469\) −12.4575 + 12.4575i −0.575235 + 0.575235i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −27.2823 −1.25444
\(474\) 0 0
\(475\) 11.7260 + 11.7260i 0.538024 + 0.538024i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 25.2915 1.15560 0.577799 0.816179i \(-0.303912\pi\)
0.577799 + 0.816179i \(0.303912\pi\)
\(480\) 0 0
\(481\) 9.29150 0.423656
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 34.2646 + 34.2646i 1.55588 + 1.55588i
\(486\) 0 0
\(487\) −4.74362 −0.214954 −0.107477 0.994208i \(-0.534277\pi\)
−0.107477 + 0.994208i \(0.534277\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.29150 1.29150i 0.0582847 0.0582847i −0.677364 0.735648i \(-0.736878\pi\)
0.735648 + 0.677364i \(0.236878\pi\)
\(492\) 0 0
\(493\) −1.52026 1.52026i −0.0684690 0.0684690i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 49.7322i 2.23080i
\(498\) 0 0
\(499\) −6.65882 + 6.65882i −0.298090 + 0.298090i −0.840265 0.542176i \(-0.817601\pi\)
0.542176 + 0.840265i \(0.317601\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 27.8745i 1.24286i 0.783469 + 0.621431i \(0.213449\pi\)
−0.783469 + 0.621431i \(0.786551\pi\)
\(504\) 0 0
\(505\) 35.1660i 1.56487i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.49117 3.49117i 0.154743 0.154743i −0.625489 0.780233i \(-0.715101\pi\)
0.780233 + 0.625489i \(0.215101\pi\)
\(510\) 0 0
\(511\) 32.1147i 1.42067i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.9373 + 16.9373i 0.746345 + 0.746345i
\(516\) 0 0
\(517\) −21.8745 + 21.8745i −0.962040 + 0.962040i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.39851 0.411756 0.205878 0.978578i \(-0.433995\pi\)
0.205878 + 0.978578i \(0.433995\pi\)
\(522\) 0 0
\(523\) 13.5524 + 13.5524i 0.592606 + 0.592606i 0.938335 0.345729i \(-0.112368\pi\)
−0.345729 + 0.938335i \(0.612368\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.24264 + 4.24264i 0.183769 + 0.183769i
\(534\) 0 0
\(535\) −37.5940 −1.62533
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 25.5203 25.5203i 1.09924 1.09924i
\(540\) 0 0
\(541\) 3.35425 + 3.35425i 0.144210 + 0.144210i 0.775526 0.631316i \(-0.217485\pi\)
−0.631316 + 0.775526i \(0.717485\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.98233i 0.299090i
\(546\) 0 0
\(547\) −22.5387 + 22.5387i −0.963684 + 0.963684i −0.999363 0.0356789i \(-0.988641\pi\)
0.0356789 + 0.999363i \(0.488641\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.70850i 0.200589i
\(552\) 0 0
\(553\) 31.4170i 1.33599i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.8779 22.8779i 0.969368 0.969368i −0.0301765 0.999545i \(-0.509607\pi\)
0.999545 + 0.0301765i \(0.00960693\pi\)
\(558\) 0 0
\(559\) 34.7656i 1.47043i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.6458 + 15.6458i 0.659390 + 0.659390i 0.955236 0.295846i \(-0.0956015\pi\)
−0.295846 + 0.955236i \(0.595601\pi\)
\(564\) 0 0
\(565\) 32.8118 32.8118i 1.38040 1.38040i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.3632 0.979435 0.489718 0.871881i \(-0.337100\pi\)
0.489718 + 0.871881i \(0.337100\pi\)
\(570\) 0 0
\(571\) −18.7970 18.7970i −0.786631 0.786631i 0.194309 0.980940i \(-0.437753\pi\)
−0.980940 + 0.194309i \(0.937753\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 49.7490 2.07468
\(576\) 0 0
\(577\) −9.41699 −0.392035 −0.196017 0.980600i \(-0.562801\pi\)
−0.196017 + 0.980600i \(0.562801\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −18.4735 18.4735i −0.766410 0.766410i
\(582\) 0 0
\(583\) −25.4558 −1.05427
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.2915 + 13.2915i −0.548599 + 0.548599i −0.926035 0.377436i \(-0.876806\pi\)
0.377436 + 0.926035i \(0.376806\pi\)
\(588\) 0 0
\(589\) −9.29150 9.29150i −0.382850 0.382850i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.24264i 0.174224i −0.996199 0.0871122i \(-0.972236\pi\)
0.996199 0.0871122i \(-0.0277639\pi\)
\(594\) 0 0
\(595\) 8.80879 8.80879i 0.361125 0.361125i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 23.1660i 0.946538i −0.880918 0.473269i \(-0.843074\pi\)
0.880918 0.473269i \(-0.156926\pi\)
\(600\) 0 0
\(601\) 9.41699i 0.384127i 0.981382 + 0.192064i \(0.0615180\pi\)
−0.981382 + 0.192064i \(0.938482\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 40.1720 40.1720i 1.63322 1.63322i
\(606\) 0 0
\(607\) 1.09070i 0.0442703i −0.999755 0.0221351i \(-0.992954\pi\)
0.999755 0.0221351i \(-0.00704641\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 27.8745 + 27.8745i 1.12768 + 1.12768i
\(612\) 0 0
\(613\) 7.58301 7.58301i 0.306275 0.306275i −0.537188 0.843463i \(-0.680513\pi\)
0.843463 + 0.537188i \(0.180513\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.5190 −1.14813 −0.574067 0.818808i \(-0.694635\pi\)
−0.574067 + 0.818808i \(0.694635\pi\)
\(618\) 0 0
\(619\) −12.1382 12.1382i −0.487876 0.487876i 0.419760 0.907635i \(-0.362114\pi\)
−0.907635 + 0.419760i \(0.862114\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −54.4575 −2.18179
\(624\) 0 0
\(625\) −2.29150 −0.0916601
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.913230 0.913230i −0.0364129 0.0364129i
\(630\) 0 0
\(631\) −26.1916 −1.04267 −0.521336 0.853352i \(-0.674566\pi\)
−0.521336 + 0.853352i \(0.674566\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.93725 + 4.93725i −0.195929 + 0.195929i
\(636\) 0 0
\(637\) −32.5203 32.5203i −1.28850 1.28850i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.5426i 0.969375i 0.874687 + 0.484687i \(0.161067\pi\)
−0.874687 + 0.484687i \(0.838933\pi\)
\(642\) 0 0
\(643\) 13.2289 13.2289i 0.521697 0.521697i −0.396387 0.918084i \(-0.629736\pi\)
0.918084 + 0.396387i \(0.129736\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 40.3320i 1.58562i −0.609472 0.792808i \(-0.708618\pi\)
0.609472 0.792808i \(-0.291382\pi\)
\(648\) 0 0
\(649\) 9.41699i 0.369649i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.57794 2.57794i 0.100882 0.100882i −0.654864 0.755747i \(-0.727274\pi\)
0.755747 + 0.654864i \(0.227274\pi\)
\(654\) 0 0
\(655\) 61.8705i 2.41748i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −23.1660 23.1660i −0.902420 0.902420i 0.0932254 0.995645i \(-0.470282\pi\)
−0.995645 + 0.0932254i \(0.970282\pi\)
\(660\) 0 0
\(661\) 23.4575 23.4575i 0.912392 0.912392i −0.0840685 0.996460i \(-0.526791\pi\)
0.996460 + 0.0840685i \(0.0267914\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −27.2823 −1.05796
\(666\) 0 0
\(667\) 9.98823 + 9.98823i 0.386746 + 0.386746i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.12549 0.0820537
\(672\) 0 0
\(673\) 43.7490 1.68640 0.843200 0.537600i \(-0.180669\pi\)
0.843200 + 0.537600i \(0.180669\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.23086 6.23086i −0.239471 0.239471i 0.577160 0.816631i \(-0.304161\pi\)
−0.816631 + 0.577160i \(0.804161\pi\)
\(678\) 0 0
\(679\) −49.7322 −1.90855
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −21.6458 + 21.6458i −0.828252 + 0.828252i −0.987275 0.159023i \(-0.949166\pi\)
0.159023 + 0.987275i \(0.449166\pi\)
\(684\) 0 0
\(685\) 24.2288 + 24.2288i 0.925733 + 0.925733i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 32.4382i 1.23580i
\(690\) 0 0
\(691\) −15.8799 + 15.8799i −0.604098 + 0.604098i −0.941398 0.337299i \(-0.890487\pi\)
0.337299 + 0.941398i \(0.390487\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.1660i 0.651144i
\(696\) 0 0
\(697\) 0.833990i 0.0315896i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.98822 1.98822i 0.0750939 0.0750939i −0.668562 0.743656i \(-0.733090\pi\)
0.743656 + 0.668562i \(0.233090\pi\)
\(702\) 0 0
\(703\) 2.82843i 0.106676i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −25.5203 25.5203i −0.959788 0.959788i
\(708\) 0 0
\(709\) 28.6458 28.6458i 1.07581 1.07581i 0.0789339 0.996880i \(-0.474848\pi\)
0.996880 0.0789339i \(-0.0251516\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −39.4205 −1.47631
\(714\) 0 0
\(715\) −87.3263 87.3263i −3.26582 3.26582i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.4575 0.688349 0.344175 0.938906i \(-0.388159\pi\)
0.344175 + 0.938906i \(0.388159\pi\)
\(720\) 0 0
\(721\) −24.5830 −0.915519
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −13.8029 13.8029i −0.512627 0.512627i
\(726\) 0 0
\(727\) 45.9906 1.70570 0.852848 0.522159i \(-0.174873\pi\)
0.852848 + 0.522159i \(0.174873\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.41699 3.41699i 0.126382 0.126382i
\(732\) 0 0
\(733\) 21.9373 + 21.9373i 0.810271 + 0.810271i 0.984674 0.174403i \(-0.0557997\pi\)
−0.174403 + 0.984674i \(0.555800\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.2764i 0.894233i
\(738\) 0 0
\(739\) −18.7970 + 18.7970i −0.691460 + 0.691460i −0.962553 0.271093i \(-0.912615\pi\)
0.271093 + 0.962553i \(0.412615\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 39.0405i 1.43226i −0.697968 0.716129i \(-0.745912\pi\)
0.697968 0.716129i \(-0.254088\pi\)
\(744\) 0 0
\(745\) 39.8745i 1.46089i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 27.2823 27.2823i 0.996874 0.996874i
\(750\) 0 0
\(751\) 17.7063i 0.646113i 0.946380 + 0.323056i \(0.104710\pi\)
−0.946380 + 0.323056i \(0.895290\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 26.8118 + 26.8118i 0.975780 + 0.975780i
\(756\) 0 0
\(757\) 14.6458 14.6458i 0.532309 0.532309i −0.388950 0.921259i \(-0.627162\pi\)
0.921259 + 0.388950i \(0.127162\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 45.1661 1.63727 0.818635 0.574314i \(-0.194731\pi\)
0.818635 + 0.574314i \(0.194731\pi\)
\(762\) 0 0
\(763\) 5.06713 + 5.06713i 0.183443 + 0.183443i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 31.1660 1.12388 0.561938 0.827180i \(-0.310056\pi\)
0.561938 + 0.827180i \(0.310056\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.49117 + 3.49117i 0.125569 + 0.125569i 0.767098 0.641530i \(-0.221700\pi\)
−0.641530 + 0.767098i \(0.721700\pi\)
\(774\) 0 0
\(775\) 54.4759 1.95683
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.29150 + 1.29150i −0.0462729 + 0.0462729i
\(780\) 0 0
\(781\) −48.4575 48.4575i −1.73395 1.73395i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 32.4382i 1.15777i
\(786\) 0 0
\(787\) 8.89753 8.89753i 0.317163 0.317163i −0.530514 0.847676i \(-0.678001\pi\)
0.847676 + 0.530514i \(0.178001\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 47.6235i 1.69330i
\(792\) 0 0
\(793\) 2.70850i 0.0961816i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.8897 12.8897i 0.456576 0.456576i −0.440954 0.897530i \(-0.645360\pi\)
0.897530 + 0.440954i \(0.145360\pi\)
\(798\) 0 0
\(799\) 5.47938i 0.193847i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −31.2915 31.2915i −1.10425 1.10425i
\(804\) 0 0
\(805\) −57.8745 + 57.8745i −2.03981 + 2.03981i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −40.3337 −1.41806 −0.709029 0.705179i \(-0.750867\pi\)
−0.709029 + 0.705179i \(0.750867\pi\)
\(810\) 0 0
\(811\) 24.0416 + 24.0416i 0.844216 + 0.844216i 0.989404 0.145188i \(-0.0463788\pi\)
−0.145188 + 0.989404i \(0.546379\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.87451 0.345889
\(816\) 0 0
\(817\) −10.5830 −0.370252
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32.5999 32.5999i −1.13775 1.13775i −0.988853 0.148892i \(-0.952429\pi\)
−0.148892 0.988853i \(-0.547571\pi\)
\(822\) 0 0
\(823\) −6.74756 −0.235205 −0.117603 0.993061i \(-0.537521\pi\)
−0.117603 + 0.993061i \(0.537521\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33.8745 33.8745i 1.17793 1.17793i 0.197662 0.980270i \(-0.436665\pi\)
0.980270 0.197662i \(-0.0633348\pi\)
\(828\) 0 0
\(829\) −8.06275 8.06275i −0.280031 0.280031i 0.553090 0.833121i \(-0.313448\pi\)
−0.833121 + 0.553090i \(0.813448\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.39261i 0.221491i
\(834\) 0 0
\(835\) −18.7970 + 18.7970i −0.650498 + 0.650498i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.87451i 0.133763i 0.997761 + 0.0668814i \(0.0213049\pi\)
−0.997761 + 0.0668814i \(0.978695\pi\)
\(840\) 0 0
\(841\) 23.4575i 0.808880i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −77.7660 + 77.7660i −2.67523 + 2.67523i
\(846\) 0 0
\(847\) 58.3063i 2.00343i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.00000 + 6.00000i 0.205677 + 0.205677i
\(852\) 0 0
\(853\) −18.2915 + 18.2915i −0.626289 + 0.626289i −0.947132 0.320843i \(-0.896034\pi\)
0.320843 + 0.947132i \(0.396034\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.1955 0.963141 0.481571 0.876407i \(-0.340067\pi\)
0.481571 + 0.876407i \(0.340067\pi\)
\(858\) 0 0
\(859\) 18.7083 + 18.7083i 0.638319 + 0.638319i 0.950141 0.311822i \(-0.100939\pi\)
−0.311822 + 0.950141i \(0.600939\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.0405 0.920470 0.460235 0.887797i \(-0.347765\pi\)
0.460235 + 0.887797i \(0.347765\pi\)
\(864\) 0 0
\(865\) −13.2915 −0.451925
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 30.6117 + 30.6117i 1.03843 + 1.03843i
\(870\) 0 0
\(871\) −30.9352 −1.04820
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 31.7490 31.7490i 1.07331 1.07331i
\(876\) 0 0
\(877\) 33.5830 + 33.5830i 1.13402 + 1.13402i 0.989503 + 0.144515i \(0.0461622\pi\)
0.144515 + 0.989503i \(0.453838\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 52.1484i 1.75693i −0.477811 0.878463i \(-0.658570\pi\)
0.477811 0.878463i \(-0.341430\pi\)
\(882\) 0 0
\(883\) 4.74362 4.74362i 0.159636 0.159636i −0.622770 0.782405i \(-0.713993\pi\)
0.782405 + 0.622770i \(0.213993\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.83399i 0.229463i −0.993397 0.114731i \(-0.963399\pi\)
0.993397 0.114731i \(-0.0366008\pi\)
\(888\) 0 0
\(889\) 7.16601i 0.240340i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.48528 + 8.48528i −0.283949 + 0.283949i
\(894\) 0 0
\(895\) 81.8469i 2.73584i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 10.9373 + 10.9373i 0.364778 + 0.364778i
\(900\) 0 0
\(901\) 3.18824 3.18824i 0.106216 0.106216i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.98233 0.232101
\(906\) 0 0
\(907\) 29.1975 + 29.1975i 0.969487 + 0.969487i 0.999548 0.0300609i \(-0.00957014\pi\)
−0.0300609 + 0.999548i \(0.509570\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −41.6235 −1.37905 −0.689524 0.724262i \(-0.742180\pi\)
−0.689524 + 0.724262i \(0.742180\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 44.8999 + 44.8999i 1.48272 + 1.48272i
\(918\) 0 0
\(919\) 54.4759 1.79699 0.898497 0.438980i \(-0.144660\pi\)
0.898497 + 0.438980i \(0.144660\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −61.7490 + 61.7490i −2.03249 + 2.03249i
\(924\) 0 0
\(925\) −8.29150 8.29150i −0.272623 0.272623i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 22.7162i 0.745293i −0.927973 0.372646i \(-0.878450\pi\)
0.927973 0.372646i \(-0.121550\pi\)
\(930\) 0 0
\(931\) 9.89949 9.89949i 0.324443 0.324443i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 17.1660i 0.561389i
\(936\) 0 0
\(937\) 32.3320i 1.05624i −0.849169 0.528121i \(-0.822897\pi\)
0.849169 0.528121i \(-0.177103\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14.7161 + 14.7161i −0.479732 + 0.479732i −0.905046 0.425314i \(-0.860164\pi\)
0.425314 + 0.905046i \(0.360164\pi\)
\(942\) 0 0
\(943\) 5.47938i 0.178433i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.58301 + 8.58301i 0.278910 + 0.278910i 0.832674 0.553764i \(-0.186809\pi\)
−0.553764 + 0.832674i \(0.686809\pi\)
\(948\) 0 0
\(949\) −39.8745 + 39.8745i −1.29438 + 1.29438i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −26.3691 −0.854178 −0.427089 0.904210i \(-0.640461\pi\)
−0.427089 + 0.904210i \(0.640461\pi\)
\(954\) 0 0
\(955\) 49.7322 + 49.7322i 1.60930 + 1.60930i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −35.1660 −1.13557
\(960\) 0 0
\(961\) −12.1660 −0.392452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −11.8147 11.8147i −0.380328 0.380328i
\(966\) 0 0
\(967\) −11.0475 −0.355264 −0.177632 0.984097i \(-0.556844\pi\)
−0.177632 + 0.984097i \(0.556844\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 19.0627 19.0627i 0.611753 0.611753i −0.331650 0.943403i \(-0.607605\pi\)
0.943403 + 0.331650i \(0.107605\pi\)
\(972\) 0 0
\(973\) 12.4575 + 12.4575i 0.399370 + 0.399370i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 31.8485i 1.01892i 0.860494 + 0.509461i \(0.170155\pi\)
−0.860494 + 0.509461i \(0.829845\pi\)
\(978\) 0 0
\(979\) 53.0617 53.0617i 1.69586 1.69586i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.5830i 0.465126i 0.972581 + 0.232563i \(0.0747111\pi\)
−0.972581 + 0.232563i \(0.925289\pi\)
\(984\) 0 0
\(985\) 61.7490i 1.96749i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −22.4499 + 22.4499i −0.713867 + 0.713867i
\(990\) 0 0
\(991\) 23.5406i 0.747793i 0.927470 + 0.373897i \(0.121979\pi\)
−0.927470 + 0.373897i \(0.878021\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −43.5203 43.5203i −1.37968 1.37968i
\(996\) 0 0
\(997\) 0.291503 0.291503i 0.00923198 0.00923198i −0.702476 0.711708i \(-0.747922\pi\)
0.711708 + 0.702476i \(0.247922\pi\)
\(998\) 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.l.e.1727.4 yes 8
3.2 odd 2 2304.2.l.a.1727.1 yes 8
4.3 odd 2 2304.2.l.a.1727.4 yes 8
8.3 odd 2 2304.2.l.h.1727.1 yes 8
8.5 even 2 2304.2.l.d.1727.1 yes 8
12.11 even 2 inner 2304.2.l.e.1727.1 yes 8
16.3 odd 4 2304.2.l.h.575.4 yes 8
16.5 even 4 inner 2304.2.l.e.575.1 yes 8
16.11 odd 4 2304.2.l.a.575.1 8
16.13 even 4 2304.2.l.d.575.4 yes 8
24.5 odd 2 2304.2.l.h.1727.4 yes 8
24.11 even 2 2304.2.l.d.1727.4 yes 8
48.5 odd 4 2304.2.l.a.575.4 yes 8
48.11 even 4 inner 2304.2.l.e.575.4 yes 8
48.29 odd 4 2304.2.l.h.575.1 yes 8
48.35 even 4 2304.2.l.d.575.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2304.2.l.a.575.1 8 16.11 odd 4
2304.2.l.a.575.4 yes 8 48.5 odd 4
2304.2.l.a.1727.1 yes 8 3.2 odd 2
2304.2.l.a.1727.4 yes 8 4.3 odd 2
2304.2.l.d.575.1 yes 8 48.35 even 4
2304.2.l.d.575.4 yes 8 16.13 even 4
2304.2.l.d.1727.1 yes 8 8.5 even 2
2304.2.l.d.1727.4 yes 8 24.11 even 2
2304.2.l.e.575.1 yes 8 16.5 even 4 inner
2304.2.l.e.575.4 yes 8 48.11 even 4 inner
2304.2.l.e.1727.1 yes 8 12.11 even 2 inner
2304.2.l.e.1727.4 yes 8 1.1 even 1 trivial
2304.2.l.h.575.1 yes 8 48.29 odd 4
2304.2.l.h.575.4 yes 8 16.3 odd 4
2304.2.l.h.1727.1 yes 8 8.3 odd 2
2304.2.l.h.1727.4 yes 8 24.5 odd 2