Properties

Label 1200.2.v.l.593.2
Level $1200$
Weight $2$
Character 1200.593
Analytic conductor $9.582$
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] = [1200,2,Mod(257,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.58204824255\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.2
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1200.593
Dual form 1200.2.v.l.257.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.448288 + 1.67303i) q^{3} +(-2.44949 + 2.44949i) q^{7} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\) \(q+(-0.448288 + 1.67303i) q^{3} +(-2.44949 + 2.44949i) q^{7} +(-2.59808 - 1.50000i) q^{9} -5.19615i q^{11} +(-2.12132 - 2.12132i) q^{17} -1.00000i q^{19} +(-3.00000 - 5.19615i) q^{21} +(4.24264 - 4.24264i) q^{23} +(3.67423 - 3.67423i) q^{27} +2.00000 q^{31} +(8.69333 + 2.32937i) q^{33} +(-2.44949 + 2.44949i) q^{37} -5.19615i q^{41} +(2.44949 + 2.44949i) q^{43} -5.00000i q^{49} +(4.50000 - 2.59808i) q^{51} +(4.24264 - 4.24264i) q^{53} +(1.67303 + 0.448288i) q^{57} -10.3923 q^{59} +14.0000 q^{61} +(10.0382 - 2.68973i) q^{63} +(-3.67423 + 3.67423i) q^{67} +(5.19615 + 9.00000i) q^{69} +(-6.12372 - 6.12372i) q^{73} +(12.7279 + 12.7279i) q^{77} -14.0000i q^{79} +(4.50000 + 7.79423i) q^{81} +(-2.12132 + 2.12132i) q^{83} -15.5885 q^{89} +(-0.896575 + 3.34607i) q^{93} +(4.89898 - 4.89898i) q^{97} +(-7.79423 + 13.5000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{21} + 16 q^{31} + 36 q^{51} + 112 q^{61} + 36 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) −0.448288 + 1.67303i −0.258819 + 0.965926i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.44949 + 2.44949i −0.925820 + 0.925820i −0.997433 0.0716124i \(-0.977186\pi\)
0.0716124 + 0.997433i \(0.477186\pi\)
\(8\) 0 0
\(9\) −2.59808 1.50000i −0.866025 0.500000i
\(10\) 0 0
\(11\) 5.19615i 1.56670i −0.621582 0.783349i \(-0.713510\pi\)
0.621582 0.783349i \(-0.286490\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.12132 2.12132i −0.514496 0.514496i 0.401405 0.915901i \(-0.368522\pi\)
−0.915901 + 0.401405i \(0.868522\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i −0.993399 0.114708i \(-0.963407\pi\)
0.993399 0.114708i \(-0.0365932\pi\)
\(20\) 0 0
\(21\) −3.00000 5.19615i −0.654654 1.13389i
\(22\) 0 0
\(23\) 4.24264 4.24264i 0.884652 0.884652i −0.109351 0.994003i \(-0.534877\pi\)
0.994003 + 0.109351i \(0.0348774\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 3.67423i 0.707107 0.707107i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 8.69333 + 2.32937i 1.51331 + 0.405492i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.44949 + 2.44949i −0.402694 + 0.402694i −0.879181 0.476488i \(-0.841910\pi\)
0.476488 + 0.879181i \(0.341910\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.19615i 0.811503i −0.913984 0.405751i \(-0.867010\pi\)
0.913984 0.405751i \(-0.132990\pi\)
\(42\) 0 0
\(43\) 2.44949 + 2.44949i 0.373544 + 0.373544i 0.868766 0.495222i \(-0.164913\pi\)
−0.495222 + 0.868766i \(0.664913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) 4.50000 2.59808i 0.630126 0.363803i
\(52\) 0 0
\(53\) 4.24264 4.24264i 0.582772 0.582772i −0.352892 0.935664i \(-0.614802\pi\)
0.935664 + 0.352892i \(0.114802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.67303 + 0.448288i 0.221599 + 0.0593772i
\(58\) 0 0
\(59\) −10.3923 −1.35296 −0.676481 0.736460i \(-0.736496\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 10.0382 2.68973i 1.26469 0.338874i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.67423 + 3.67423i −0.448879 + 0.448879i −0.894982 0.446103i \(-0.852812\pi\)
0.446103 + 0.894982i \(0.352812\pi\)
\(68\) 0 0
\(69\) 5.19615 + 9.00000i 0.625543 + 1.08347i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −6.12372 6.12372i −0.716728 0.716728i 0.251206 0.967934i \(-0.419173\pi\)
−0.967934 + 0.251206i \(0.919173\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.7279 + 12.7279i 1.45048 + 1.45048i
\(78\) 0 0
\(79\) 14.0000i 1.57512i −0.616236 0.787562i \(-0.711343\pi\)
0.616236 0.787562i \(-0.288657\pi\)
\(80\) 0 0
\(81\) 4.50000 + 7.79423i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) −2.12132 + 2.12132i −0.232845 + 0.232845i −0.813879 0.581034i \(-0.802648\pi\)
0.581034 + 0.813879i \(0.302648\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.5885 −1.65237 −0.826187 0.563397i \(-0.809494\pi\)
−0.826187 + 0.563397i \(0.809494\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −0.896575 + 3.34607i −0.0929705 + 0.346971i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.89898 4.89898i 0.497416 0.497416i −0.413217 0.910633i \(-0.635595\pi\)
0.910633 + 0.413217i \(0.135595\pi\)
\(98\) 0 0
\(99\) −7.79423 + 13.5000i −0.783349 + 1.35680i
\(100\) 0 0
\(101\) 10.3923i 1.03407i −0.855963 0.517036i \(-0.827035\pi\)
0.855963 0.517036i \(-0.172965\pi\)
\(102\) 0 0
\(103\) −9.79796 9.79796i −0.965422 0.965422i 0.0340002 0.999422i \(-0.489175\pi\)
−0.999422 + 0.0340002i \(0.989175\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.36396 + 6.36396i 0.615227 + 0.615227i 0.944303 0.329076i \(-0.106737\pi\)
−0.329076 + 0.944303i \(0.606737\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.957826i −0.877862 0.478913i \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) 0 0
\(111\) −3.00000 5.19615i −0.284747 0.493197i
\(112\) 0 0
\(113\) 6.36396 6.36396i 0.598671 0.598671i −0.341288 0.939959i \(-0.610863\pi\)
0.939959 + 0.341288i \(0.110863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.3923 0.952661
\(120\) 0 0
\(121\) −16.0000 −1.45455
\(122\) 0 0
\(123\) 8.69333 + 2.32937i 0.783851 + 0.210032i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.34847 7.34847i 0.652071 0.652071i −0.301420 0.953491i \(-0.597461\pi\)
0.953491 + 0.301420i \(0.0974607\pi\)
\(128\) 0 0
\(129\) −5.19615 + 3.00000i −0.457496 + 0.264135i
\(130\) 0 0
\(131\) 10.3923i 0.907980i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(132\) 0 0
\(133\) 2.44949 + 2.44949i 0.212398 + 0.212398i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.8492 14.8492i −1.26866 1.26866i −0.946783 0.321874i \(-0.895687\pi\)
−0.321874 0.946783i \(-0.604313\pi\)
\(138\) 0 0
\(139\) 7.00000i 0.593732i −0.954919 0.296866i \(-0.904058\pi\)
0.954919 0.296866i \(-0.0959415\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 8.36516 + 2.24144i 0.689947 + 0.184871i
\(148\) 0 0
\(149\) 20.7846 1.70274 0.851371 0.524564i \(-0.175772\pi\)
0.851371 + 0.524564i \(0.175772\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 0 0
\(153\) 2.32937 + 8.69333i 0.188319 + 0.702814i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.2474 + 12.2474i −0.977453 + 0.977453i −0.999751 0.0222985i \(-0.992902\pi\)
0.0222985 + 0.999751i \(0.492902\pi\)
\(158\) 0 0
\(159\) 5.19615 + 9.00000i 0.412082 + 0.713746i
\(160\) 0 0
\(161\) 20.7846i 1.63806i
\(162\) 0 0
\(163\) −3.67423 3.67423i −0.287788 0.287788i 0.548417 0.836205i \(-0.315231\pi\)
−0.836205 + 0.548417i \(0.815231\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.48528 8.48528i −0.656611 0.656611i 0.297966 0.954577i \(-0.403692\pi\)
−0.954577 + 0.297966i \(0.903692\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) −1.50000 + 2.59808i −0.114708 + 0.198680i
\(172\) 0 0
\(173\) −8.48528 + 8.48528i −0.645124 + 0.645124i −0.951811 0.306687i \(-0.900780\pi\)
0.306687 + 0.951811i \(0.400780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.65874 17.3867i 0.350173 1.30686i
\(178\) 0 0
\(179\) 15.5885 1.16514 0.582568 0.812782i \(-0.302048\pi\)
0.582568 + 0.812782i \(0.302048\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −6.27603 + 23.4225i −0.463937 + 1.73144i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −11.0227 + 11.0227i −0.806060 + 0.806060i
\(188\) 0 0
\(189\) 18.0000i 1.30931i
\(190\) 0 0
\(191\) 10.3923i 0.751961i −0.926628 0.375980i \(-0.877306\pi\)
0.926628 0.375980i \(-0.122694\pi\)
\(192\) 0 0
\(193\) −6.12372 6.12372i −0.440795 0.440795i 0.451484 0.892279i \(-0.350895\pi\)
−0.892279 + 0.451484i \(0.850895\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.24264 4.24264i −0.302276 0.302276i 0.539628 0.841904i \(-0.318565\pi\)
−0.841904 + 0.539628i \(0.818565\pi\)
\(198\) 0 0
\(199\) 16.0000i 1.13421i 0.823646 + 0.567105i \(0.191937\pi\)
−0.823646 + 0.567105i \(0.808063\pi\)
\(200\) 0 0
\(201\) −4.50000 7.79423i −0.317406 0.549762i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −17.3867 + 4.65874i −1.20846 + 0.323805i
\(208\) 0 0
\(209\) −5.19615 −0.359425
\(210\) 0 0
\(211\) −23.0000 −1.58339 −0.791693 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.89898 + 4.89898i −0.332564 + 0.332564i
\(218\) 0 0
\(219\) 12.9904 7.50000i 0.877809 0.506803i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.79796 9.79796i −0.656120 0.656120i 0.298340 0.954460i \(-0.403567\pi\)
−0.954460 + 0.298340i \(0.903567\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.48528 8.48528i −0.563188 0.563188i 0.367024 0.930212i \(-0.380377\pi\)
−0.930212 + 0.367024i \(0.880377\pi\)
\(228\) 0 0
\(229\) 16.0000i 1.05731i 0.848837 + 0.528655i \(0.177303\pi\)
−0.848837 + 0.528655i \(0.822697\pi\)
\(230\) 0 0
\(231\) −27.0000 + 15.5885i −1.77647 + 1.02565i
\(232\) 0 0
\(233\) −12.7279 + 12.7279i −0.833834 + 0.833834i −0.988039 0.154205i \(-0.950718\pi\)
0.154205 + 0.988039i \(0.450718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 23.4225 + 6.27603i 1.52145 + 0.407672i
\(238\) 0 0
\(239\) −20.7846 −1.34444 −0.672222 0.740349i \(-0.734660\pi\)
−0.672222 + 0.740349i \(0.734660\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) −15.0573 + 4.03459i −0.965926 + 0.258819i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −2.59808 4.50000i −0.164646 0.285176i
\(250\) 0 0
\(251\) 5.19615i 0.327978i −0.986462 0.163989i \(-0.947564\pi\)
0.986462 0.163989i \(-0.0524362\pi\)
\(252\) 0 0
\(253\) −22.0454 22.0454i −1.38598 1.38598i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.24264 + 4.24264i 0.264649 + 0.264649i 0.826940 0.562291i \(-0.190080\pi\)
−0.562291 + 0.826940i \(0.690080\pi\)
\(258\) 0 0
\(259\) 12.0000i 0.745644i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.7279 + 12.7279i −0.784837 + 0.784837i −0.980643 0.195805i \(-0.937268\pi\)
0.195805 + 0.980643i \(0.437268\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.98811 26.0800i 0.427666 1.59607i
\(268\) 0 0
\(269\) 20.7846 1.26726 0.633630 0.773636i \(-0.281564\pi\)
0.633630 + 0.773636i \(0.281564\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −9.79796 + 9.79796i −0.588702 + 0.588702i −0.937280 0.348578i \(-0.886665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(278\) 0 0
\(279\) −5.19615 3.00000i −0.311086 0.179605i
\(280\) 0 0
\(281\) 20.7846i 1.23991i 0.784639 + 0.619953i \(0.212848\pi\)
−0.784639 + 0.619953i \(0.787152\pi\)
\(282\) 0 0
\(283\) −6.12372 6.12372i −0.364018 0.364018i 0.501272 0.865290i \(-0.332866\pi\)
−0.865290 + 0.501272i \(0.832866\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.7279 + 12.7279i 0.751305 + 0.751305i
\(288\) 0 0
\(289\) 8.00000i 0.470588i
\(290\) 0 0
\(291\) 6.00000 + 10.3923i 0.351726 + 0.609208i
\(292\) 0 0
\(293\) −21.2132 + 21.2132i −1.23929 + 1.23929i −0.278996 + 0.960292i \(0.590002\pi\)
−0.960292 + 0.278996i \(0.909998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −19.0919 19.0919i −1.10782 1.10782i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 17.3867 + 4.65874i 0.998838 + 0.267638i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.22474 1.22474i 0.0698999 0.0698999i −0.671293 0.741192i \(-0.734261\pi\)
0.741192 + 0.671293i \(0.234261\pi\)
\(308\) 0 0
\(309\) 20.7846 12.0000i 1.18240 0.682656i
\(310\) 0 0
\(311\) 31.1769i 1.76788i 0.467600 + 0.883940i \(0.345119\pi\)
−0.467600 + 0.883940i \(0.654881\pi\)
\(312\) 0 0
\(313\) 14.6969 + 14.6969i 0.830720 + 0.830720i 0.987615 0.156895i \(-0.0501485\pi\)
−0.156895 + 0.987615i \(0.550148\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.48528 + 8.48528i 0.476581 + 0.476581i 0.904036 0.427456i \(-0.140590\pi\)
−0.427456 + 0.904036i \(0.640590\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −13.5000 + 7.79423i −0.753497 + 0.435031i
\(322\) 0 0
\(323\) −2.12132 + 2.12132i −0.118033 + 0.118033i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 16.7303 + 4.48288i 0.925189 + 0.247904i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.0000 0.714545 0.357272 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) 0 0
\(333\) 10.0382 2.68973i 0.550090 0.147396i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.67423 3.67423i 0.200148 0.200148i −0.599915 0.800064i \(-0.704799\pi\)
0.800064 + 0.599915i \(0.204799\pi\)
\(338\) 0 0
\(339\) 7.79423 + 13.5000i 0.423324 + 0.733219i
\(340\) 0 0
\(341\) 10.3923i 0.562775i
\(342\) 0 0
\(343\) −4.89898 4.89898i −0.264520 0.264520i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.36396 6.36396i −0.341635 0.341635i 0.515347 0.856982i \(-0.327663\pi\)
−0.856982 + 0.515347i \(0.827663\pi\)
\(348\) 0 0
\(349\) 22.0000i 1.17763i 0.808267 + 0.588817i \(0.200406\pi\)
−0.808267 + 0.588817i \(0.799594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.7279 12.7279i 0.677439 0.677439i −0.281981 0.959420i \(-0.590992\pi\)
0.959420 + 0.281981i \(0.0909915\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4.65874 + 17.3867i −0.246567 + 0.920200i
\(358\) 0 0
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) 18.0000 0.947368
\(362\) 0 0
\(363\) 7.17260 26.7685i 0.376464 1.40498i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 14.6969 14.6969i 0.767174 0.767174i −0.210434 0.977608i \(-0.567488\pi\)
0.977608 + 0.210434i \(0.0674877\pi\)
\(368\) 0 0
\(369\) −7.79423 + 13.5000i −0.405751 + 0.702782i
\(370\) 0 0
\(371\) 20.7846i 1.07908i
\(372\) 0 0
\(373\) 2.44949 + 2.44949i 0.126830 + 0.126830i 0.767672 0.640843i \(-0.221415\pi\)
−0.640843 + 0.767672i \(0.721415\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11.0000i 0.565032i 0.959263 + 0.282516i \(0.0911690\pi\)
−0.959263 + 0.282516i \(0.908831\pi\)
\(380\) 0 0
\(381\) 9.00000 + 15.5885i 0.461084 + 0.798621i
\(382\) 0 0
\(383\) −4.24264 + 4.24264i −0.216789 + 0.216789i −0.807144 0.590355i \(-0.798988\pi\)
0.590355 + 0.807144i \(0.298988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.68973 10.0382i −0.136726 0.510270i
\(388\) 0 0
\(389\) −10.3923 −0.526911 −0.263455 0.964672i \(-0.584862\pi\)
−0.263455 + 0.964672i \(0.584862\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) −17.3867 4.65874i −0.877041 0.235002i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 19.5959 19.5959i 0.983491 0.983491i −0.0163750 0.999866i \(-0.505213\pi\)
0.999866 + 0.0163750i \(0.00521255\pi\)
\(398\) 0 0
\(399\) −5.19615 + 3.00000i −0.260133 + 0.150188i
\(400\) 0 0
\(401\) 5.19615i 0.259483i 0.991548 + 0.129742i \(0.0414148\pi\)
−0.991548 + 0.129742i \(0.958585\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.7279 + 12.7279i 0.630900 + 0.630900i
\(408\) 0 0
\(409\) 5.00000i 0.247234i −0.992330 0.123617i \(-0.960551\pi\)
0.992330 0.123617i \(-0.0394494\pi\)
\(410\) 0 0
\(411\) 31.5000 18.1865i 1.55378 0.897076i
\(412\) 0 0
\(413\) 25.4558 25.4558i 1.25260 1.25260i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 11.7112 + 3.13801i 0.573501 + 0.153669i
\(418\) 0 0
\(419\) 25.9808 1.26924 0.634622 0.772823i \(-0.281156\pi\)
0.634622 + 0.772823i \(0.281156\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −34.2929 + 34.2929i −1.65955 + 1.65955i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.3923i 0.500580i −0.968171 0.250290i \(-0.919474\pi\)
0.968171 0.250290i \(-0.0805259\pi\)
\(432\) 0 0
\(433\) −15.9217 15.9217i −0.765147 0.765147i 0.212101 0.977248i \(-0.431970\pi\)
−0.977248 + 0.212101i \(0.931970\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.24264 4.24264i −0.202953 0.202953i
\(438\) 0 0
\(439\) 4.00000i 0.190910i 0.995434 + 0.0954548i \(0.0304305\pi\)
−0.995434 + 0.0954548i \(0.969569\pi\)
\(440\) 0 0
\(441\) −7.50000 + 12.9904i −0.357143 + 0.618590i
\(442\) 0 0
\(443\) 14.8492 14.8492i 0.705509 0.705509i −0.260079 0.965587i \(-0.583748\pi\)
0.965587 + 0.260079i \(0.0837485\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −9.31749 + 34.7733i −0.440702 + 1.64472i
\(448\) 0 0
\(449\) −25.9808 −1.22611 −0.613054 0.790041i \(-0.710059\pi\)
−0.613054 + 0.790041i \(0.710059\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) 0 0
\(453\) −6.27603 + 23.4225i −0.294874 + 1.10048i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.3712 18.3712i 0.859367 0.859367i −0.131896 0.991264i \(-0.542107\pi\)
0.991264 + 0.131896i \(0.0421066\pi\)
\(458\) 0 0
\(459\) −15.5885 −0.727607
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 24.4949 + 24.4949i 1.13837 + 1.13837i 0.988742 + 0.149633i \(0.0478091\pi\)
0.149633 + 0.988742i \(0.452191\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.48528 8.48528i −0.392652 0.392652i 0.482980 0.875632i \(-0.339555\pi\)
−0.875632 + 0.482980i \(0.839555\pi\)
\(468\) 0 0
\(469\) 18.0000i 0.831163i
\(470\) 0 0
\(471\) −15.0000 25.9808i −0.691164 1.19713i
\(472\) 0 0
\(473\) 12.7279 12.7279i 0.585230 0.585230i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −17.3867 + 4.65874i −0.796081 + 0.213309i
\(478\) 0 0
\(479\) −10.3923 −0.474837 −0.237418 0.971408i \(-0.576301\pi\)
−0.237418 + 0.971408i \(0.576301\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −34.7733 9.31749i −1.58224 0.423960i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 22.0454 22.0454i 0.998973 0.998973i −0.00102669 0.999999i \(-0.500327\pi\)
0.999999 + 0.00102669i \(0.000326807\pi\)
\(488\) 0 0
\(489\) 7.79423 4.50000i 0.352467 0.203497i
\(490\) 0 0
\(491\) 31.1769i 1.40699i −0.710698 0.703497i \(-0.751621\pi\)
0.710698 0.703497i \(-0.248379\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 20.0000i 0.895323i −0.894203 0.447661i \(-0.852257\pi\)
0.894203 0.447661i \(-0.147743\pi\)
\(500\) 0 0
\(501\) 18.0000 10.3923i 0.804181 0.464294i
\(502\) 0 0
\(503\) −8.48528 + 8.48528i −0.378340 + 0.378340i −0.870503 0.492163i \(-0.836206\pi\)
0.492163 + 0.870503i \(0.336206\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 21.7494 + 5.82774i 0.965926 + 0.258819i
\(508\) 0 0
\(509\) −10.3923 −0.460631 −0.230315 0.973116i \(-0.573976\pi\)
−0.230315 + 0.973116i \(0.573976\pi\)
\(510\) 0 0
\(511\) 30.0000 1.32712
\(512\) 0 0
\(513\) −3.67423 3.67423i −0.162221 0.162221i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −10.3923 18.0000i −0.456172 0.790112i
\(520\) 0 0
\(521\) 36.3731i 1.59353i −0.604287 0.796766i \(-0.706542\pi\)
0.604287 0.796766i \(-0.293458\pi\)
\(522\) 0 0
\(523\) 30.6186 + 30.6186i 1.33886 + 1.33886i 0.897167 + 0.441692i \(0.145622\pi\)
0.441692 + 0.897167i \(0.354378\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.24264 4.24264i −0.184812 0.184812i
\(528\) 0 0
\(529\) 13.0000i 0.565217i
\(530\) 0 0
\(531\) 27.0000 + 15.5885i 1.17170 + 0.676481i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −6.98811 + 26.0800i −0.301559 + 1.12543i
\(538\) 0 0
\(539\) −25.9808 −1.11907
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 0 0
\(543\) 0.896575 3.34607i 0.0384757 0.143593i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.57321 8.57321i 0.366564 0.366564i −0.499658 0.866223i \(-0.666541\pi\)
0.866223 + 0.499658i \(0.166541\pi\)
\(548\) 0 0
\(549\) −36.3731 21.0000i −1.55236 0.896258i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 34.2929 + 34.2929i 1.45828 + 1.45828i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −13.5000 23.3827i −0.569970 0.987218i
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −30.1146 8.06918i −1.26469 0.338874i
\(568\) 0 0
\(569\) 25.9808 1.08917 0.544585 0.838706i \(-0.316687\pi\)
0.544585 + 0.838706i \(0.316687\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 17.3867 + 4.65874i 0.726338 + 0.194622i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −13.4722 + 13.4722i −0.560855 + 0.560855i −0.929550 0.368695i \(-0.879805\pi\)
0.368695 + 0.929550i \(0.379805\pi\)
\(578\) 0 0
\(579\) 12.9904 7.50000i 0.539862 0.311689i
\(580\) 0 0
\(581\) 10.3923i 0.431145i
\(582\) 0 0
\(583\) −22.0454 22.0454i −0.913027 0.913027i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.8492 + 14.8492i 0.612894 + 0.612894i 0.943699 0.330805i \(-0.107320\pi\)
−0.330805 + 0.943699i \(0.607320\pi\)
\(588\) 0 0
\(589\) 2.00000i 0.0824086i
\(590\) 0 0
\(591\) 9.00000 5.19615i 0.370211 0.213741i
\(592\) 0 0
\(593\) −23.3345 + 23.3345i −0.958234 + 0.958234i −0.999162 0.0409281i \(-0.986969\pi\)
0.0409281 + 0.999162i \(0.486969\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −26.7685 7.17260i −1.09556 0.293555i
\(598\) 0 0
\(599\) 10.3923 0.424618 0.212309 0.977203i \(-0.431902\pi\)
0.212309 + 0.977203i \(0.431902\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 0 0
\(603\) 15.0573 4.03459i 0.613180 0.164301i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −4.89898 + 4.89898i −0.198843 + 0.198843i −0.799504 0.600661i \(-0.794904\pi\)
0.600661 + 0.799504i \(0.294904\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 17.1464 + 17.1464i 0.692538 + 0.692538i 0.962790 0.270252i \(-0.0871070\pi\)
−0.270252 + 0.962790i \(0.587107\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21.2132 21.2132i −0.854011 0.854011i 0.136613 0.990624i \(-0.456378\pi\)
−0.990624 + 0.136613i \(0.956378\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i 0.996764 + 0.0803868i \(0.0256155\pi\)
−0.996764 + 0.0803868i \(0.974384\pi\)
\(620\) 0 0
\(621\) 31.1769i 1.25109i
\(622\) 0 0
\(623\) 38.1838 38.1838i 1.52980 1.52980i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.32937 8.69333i 0.0930261 0.347178i
\(628\) 0 0
\(629\) 10.3923 0.414368
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) 10.3106 38.4797i 0.409810 1.52943i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.7846i 0.820943i −0.911873 0.410471i \(-0.865364\pi\)
0.911873 0.410471i \(-0.134636\pi\)
\(642\) 0 0
\(643\) 22.0454 + 22.0454i 0.869386 + 0.869386i 0.992404 0.123018i \(-0.0392574\pi\)
−0.123018 + 0.992404i \(0.539257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −33.9411 33.9411i −1.33436 1.33436i −0.901422 0.432941i \(-0.857476\pi\)
−0.432941 0.901422i \(-0.642524\pi\)
\(648\) 0 0
\(649\) 54.0000i 2.11969i
\(650\) 0 0
\(651\) −6.00000 10.3923i −0.235159 0.407307i
\(652\) 0 0
\(653\) 4.24264 4.24264i 0.166027 0.166027i −0.619203 0.785231i \(-0.712544\pi\)
0.785231 + 0.619203i \(0.212544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.72432 + 25.0955i 0.262341 + 0.979068i
\(658\) 0 0
\(659\) −25.9808 −1.01207 −0.506033 0.862514i \(-0.668889\pi\)
−0.506033 + 0.862514i \(0.668889\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 20.7846 12.0000i 0.803579 0.463947i
\(670\) 0 0
\(671\) 72.7461i 2.80833i
\(672\) 0 0
\(673\) 4.89898 + 4.89898i 0.188842 + 0.188842i 0.795195 0.606353i \(-0.207368\pi\)
−0.606353 + 0.795195i \(0.707368\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.4558 + 25.4558i 0.978348 + 0.978348i 0.999771 0.0214229i \(-0.00681965\pi\)
−0.0214229 + 0.999771i \(0.506820\pi\)
\(678\) 0 0
\(679\) 24.0000i 0.921035i
\(680\) 0 0
\(681\) 18.0000 10.3923i 0.689761 0.398234i
\(682\) 0 0
\(683\) 14.8492 14.8492i 0.568190 0.568190i −0.363431 0.931621i \(-0.618395\pi\)
0.931621 + 0.363431i \(0.118395\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −26.7685 7.17260i −1.02128 0.273652i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −37.0000 −1.40755 −0.703773 0.710425i \(-0.748503\pi\)
−0.703773 + 0.710425i \(0.748503\pi\)
\(692\) 0 0
\(693\) −13.9762 52.1600i −0.530913 1.98139i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −11.0227 + 11.0227i −0.417515 + 0.417515i
\(698\) 0 0
\(699\) −15.5885 27.0000i −0.589610 1.02123i
\(700\) 0 0
\(701\) 20.7846i 0.785024i 0.919747 + 0.392512i \(0.128394\pi\)
−0.919747 + 0.392512i \(0.871606\pi\)
\(702\) 0 0
\(703\) 2.44949 + 2.44949i 0.0923843 + 0.0923843i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 25.4558 + 25.4558i 0.957366 + 0.957366i
\(708\) 0 0
\(709\) 40.0000i 1.50223i 0.660171 + 0.751116i \(0.270484\pi\)
−0.660171 + 0.751116i \(0.729516\pi\)
\(710\) 0 0
\(711\) −21.0000 + 36.3731i −0.787562 + 1.36410i
\(712\) 0 0
\(713\) 8.48528 8.48528i 0.317776 0.317776i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 9.31749 34.7733i 0.347968 1.29863i
\(718\) 0 0
\(719\) −31.1769 −1.16270 −0.581351 0.813653i \(-0.697476\pi\)
−0.581351 + 0.813653i \(0.697476\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) 0.448288 1.67303i 0.0166720 0.0622208i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 4.89898 4.89898i 0.181693 0.181693i −0.610400 0.792093i \(-0.708991\pi\)
0.792093 + 0.610400i \(0.208991\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 10.3923i 0.384373i
\(732\) 0 0
\(733\) 14.6969 + 14.6969i 0.542844 + 0.542844i 0.924362 0.381518i \(-0.124598\pi\)
−0.381518 + 0.924362i \(0.624598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.0919 + 19.0919i 0.703259 + 0.703259i
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12.7279 + 12.7279i −0.466942 + 0.466942i −0.900922 0.433980i \(-0.857109\pi\)
0.433980 + 0.900922i \(0.357109\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 8.69333 2.32937i 0.318072 0.0852272i
\(748\) 0 0
\(749\) −31.1769 −1.13918
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 8.69333 + 2.32937i 0.316803 + 0.0848870i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 24.4949 24.4949i 0.890282 0.890282i −0.104267 0.994549i \(-0.533250\pi\)
0.994549 + 0.104267i \(0.0332497\pi\)
\(758\) 0 0
\(759\) 46.7654 27.0000i 1.69748 0.980038i
\(760\) 0 0
\(761\) 5.19615i 0.188360i −0.995555 0.0941802i \(-0.969977\pi\)
0.995555 0.0941802i \(-0.0300230\pi\)
\(762\) 0 0
\(763\) 24.4949 + 24.4949i 0.886775 + 0.886775i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 13.0000i 0.468792i −0.972141 0.234396i \(-0.924689\pi\)
0.972141 0.234396i \(-0.0753112\pi\)
\(770\) 0 0
\(771\) −9.00000 + 5.19615i −0.324127 + 0.187135i
\(772\) 0 0
\(773\) 8.48528 8.48528i 0.305194 0.305194i −0.537848 0.843042i \(-0.680762\pi\)
0.843042 + 0.537848i \(0.180762\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 20.0764 + 5.37945i 0.720237 + 0.192987i
\(778\) 0 0
\(779\) −5.19615 −0.186171
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −17.1464 + 17.1464i −0.611204 + 0.611204i −0.943260 0.332056i \(-0.892258\pi\)
0.332056 + 0.943260i \(0.392258\pi\)
\(788\) 0 0
\(789\) −15.5885 27.0000i −0.554964 0.961225i
\(790\) 0 0
\(791\) 31.1769i 1.10852i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.6985 29.6985i −1.05197 1.05197i −0.998573 0.0534012i \(-0.982994\pi\)
−0.0534012 0.998573i \(-0.517006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 40.5000 + 23.3827i 1.43100 + 0.826187i
\(802\) 0 0
\(803\) −31.8198 + 31.8198i −1.12290 + 1.12290i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9.31749 + 34.7733i −0.327991 + 1.22408i
\(808\) 0 0
\(809\) 20.7846 0.730748 0.365374 0.930861i \(-0.380941\pi\)
0.365374 + 0.930861i \(0.380941\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) −4.48288 + 16.7303i −0.157221 + 0.586758i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.44949 2.44949i 0.0856968 0.0856968i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 31.1769i 1.08808i 0.839059 + 0.544041i \(0.183106\pi\)
−0.839059 + 0.544041i \(0.816894\pi\)
\(822\) 0 0
\(823\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.12132 + 2.12132i 0.0737655 + 0.0737655i 0.743027 0.669261i \(-0.233389\pi\)
−0.669261 + 0.743027i \(0.733389\pi\)
\(828\) 0 0
\(829\) 44.0000i 1.52818i −0.645108 0.764092i \(-0.723188\pi\)
0.645108 0.764092i \(-0.276812\pi\)
\(830\) 0 0
\(831\) −12.0000 20.7846i −0.416275 0.721010i
\(832\) 0 0
\(833\) −10.6066 + 10.6066i −0.367497 + 0.367497i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.34847 7.34847i 0.254000 0.254000i
\(838\) 0 0
\(839\) 20.7846 0.717564 0.358782 0.933421i \(-0.383192\pi\)
0.358782 + 0.933421i \(0.383192\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −34.7733 9.31749i −1.19766 0.320911i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 39.1918 39.1918i 1.34665 1.34665i
\(848\) 0 0
\(849\) 12.9904 7.50000i 0.445829 0.257399i
\(850\) 0 0
\(851\) 20.7846i 0.712487i
\(852\) 0 0
\(853\) −7.34847 7.34847i −0.251607 0.251607i 0.570022 0.821629i \(-0.306935\pi\)
−0.821629 + 0.570022i \(0.806935\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 36.0624 + 36.0624i 1.23187 + 1.23187i 0.963245 + 0.268625i \(0.0865692\pi\)
0.268625 + 0.963245i \(0.413431\pi\)
\(858\) 0 0
\(859\) 7.00000i 0.238837i 0.992844 + 0.119418i \(0.0381030\pi\)
−0.992844 + 0.119418i \(0.961897\pi\)
\(860\) 0 0
\(861\) −27.0000 + 15.5885i −0.920158 + 0.531253i
\(862\) 0 0
\(863\) 33.9411 33.9411i 1.15537 1.15537i 0.169910 0.985460i \(-0.445652\pi\)
0.985460 0.169910i \(-0.0543476\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.3843 + 3.58630i 0.454553 + 0.121797i
\(868\) 0 0
\(869\) −72.7461 −2.46774
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −20.0764 + 5.37945i −0.679483 + 0.182067i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −34.2929 + 34.2929i −1.15799 + 1.15799i −0.173080 + 0.984908i \(0.555372\pi\)
−0.984908 + 0.173080i \(0.944628\pi\)
\(878\) 0 0
\(879\) −25.9808 45.0000i −0.876309 1.51781i
\(880\) 0 0
\(881\) 20.7846i 0.700251i −0.936703 0.350126i \(-0.886139\pi\)
0.936703 0.350126i \(-0.113861\pi\)
\(882\) 0 0
\(883\) −20.8207 20.8207i −0.700671 0.700671i 0.263883 0.964555i \(-0.414997\pi\)
−0.964555 + 0.263883i \(0.914997\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33.9411 + 33.9411i 1.13963 + 1.13963i 0.988516 + 0.151115i \(0.0482865\pi\)
0.151115 + 0.988516i \(0.451714\pi\)
\(888\) 0 0
\(889\) 36.0000i 1.20740i
\(890\) 0 0
\(891\) 40.5000 23.3827i 1.35680 0.783349i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) 5.37945 20.0764i 0.179017 0.668100i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 17.1464 17.1464i 0.569338 0.569338i −0.362605 0.931943i \(-0.618113\pi\)
0.931943 + 0.362605i \(0.118113\pi\)
\(908\) 0 0
\(909\) −15.5885 + 27.0000i −0.517036 + 0.895533i
\(910\) 0 0
\(911\) 41.5692i 1.37725i 0.725118 + 0.688625i \(0.241785\pi\)
−0.725118 + 0.688625i \(0.758215\pi\)
\(912\) 0 0
\(913\) 11.0227 + 11.0227i 0.364798 + 0.364798i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −25.4558 25.4558i −0.840626 0.840626i
\(918\) 0 0
\(919\) 26.0000i 0.857661i 0.903385 + 0.428830i \(0.141074\pi\)
−0.903385 + 0.428830i \(0.858926\pi\)
\(920\) 0 0
\(921\) 1.50000 + 2.59808i 0.0494267 + 0.0856095i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 10.7589 + 40.1528i 0.353369 + 1.31879i
\(928\) 0 0
\(929\) 41.5692 1.36384 0.681921 0.731426i \(-0.261145\pi\)
0.681921 + 0.731426i \(0.261145\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) 0 0
\(933\) −52.1600 13.9762i −1.70764 0.457561i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 6.12372 6.12372i 0.200053 0.200053i −0.599970 0.800023i \(-0.704821\pi\)
0.800023 + 0.599970i \(0.204821\pi\)
\(938\) 0 0
\(939\) −31.1769 + 18.0000i −1.01742 + 0.587408i
\(940\) 0 0
\(941\) 20.7846i 0.677559i 0.940866 + 0.338779i \(0.110014\pi\)
−0.940866 + 0.338779i \(0.889986\pi\)
\(942\) 0 0
\(943\) −22.0454 22.0454i −0.717897 0.717897i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.9706 16.9706i −0.551469 0.551469i 0.375396 0.926865i \(-0.377507\pi\)
−0.926865 + 0.375396i \(0.877507\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −18.0000 + 10.3923i −0.583690 + 0.336994i
\(952\) 0 0
\(953\) 6.36396 6.36396i 0.206149 0.206149i −0.596479 0.802628i \(-0.703434\pi\)
0.802628 + 0.596479i \(0.203434\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 72.7461 2.34910
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −6.98811 26.0800i −0.225189 0.840416i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 9.79796 9.79796i 0.315081 0.315081i −0.531793 0.846874i \(-0.678482\pi\)
0.846874 + 0.531793i \(0.178482\pi\)
\(968\) 0 0
\(969\) −2.59808 4.50000i −0.0834622 0.144561i
\(970\) 0 0
\(971\) 5.19615i 0.166752i 0.996518 + 0.0833762i \(0.0265703\pi\)
−0.996518 + 0.0833762i \(0.973430\pi\)
\(972\) 0 0
\(973\) 17.1464 + 17.1464i 0.549689 + 0.549689i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −6.36396 6.36396i −0.203601 0.203601i 0.597940 0.801541i \(-0.295986\pi\)
−0.801541 + 0.597940i \(0.795986\pi\)
\(978\) 0 0
\(979\) 81.0000i 2.58877i
\(980\) 0 0
\(981\) −15.0000 + 25.9808i −0.478913 + 0.829502i
\(982\) 0 0
\(983\) −21.2132 + 21.2132i −0.676596 + 0.676596i −0.959228 0.282632i \(-0.908792\pi\)
0.282632 + 0.959228i \(0.408792\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 20.7846 0.660912
\(990\) 0 0
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) 0 0
\(993\) −5.82774 + 21.7494i −0.184938 + 0.690197i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 44.0908 44.0908i 1.39637 1.39637i 0.586214 0.810157i \(-0.300618\pi\)
0.810157 0.586214i \(-0.199382\pi\)
\(998\) 0 0
\(999\) 18.0000i 0.569495i
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 1200.2.v.l.593.2 8
3.2 odd 2 inner 1200.2.v.l.593.4 8
4.3 odd 2 150.2.e.b.143.3 yes 8
5.2 odd 4 inner 1200.2.v.l.257.4 8
5.3 odd 4 inner 1200.2.v.l.257.1 8
5.4 even 2 inner 1200.2.v.l.593.3 8
12.11 even 2 150.2.e.b.143.1 yes 8
15.2 even 4 inner 1200.2.v.l.257.2 8
15.8 even 4 inner 1200.2.v.l.257.3 8
15.14 odd 2 inner 1200.2.v.l.593.1 8
20.3 even 4 150.2.e.b.107.4 yes 8
20.7 even 4 150.2.e.b.107.1 8
20.19 odd 2 150.2.e.b.143.2 yes 8
60.23 odd 4 150.2.e.b.107.2 yes 8
60.47 odd 4 150.2.e.b.107.3 yes 8
60.59 even 2 150.2.e.b.143.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.2.e.b.107.1 8 20.7 even 4
150.2.e.b.107.2 yes 8 60.23 odd 4
150.2.e.b.107.3 yes 8 60.47 odd 4
150.2.e.b.107.4 yes 8 20.3 even 4
150.2.e.b.143.1 yes 8 12.11 even 2
150.2.e.b.143.2 yes 8 20.19 odd 2
150.2.e.b.143.3 yes 8 4.3 odd 2
150.2.e.b.143.4 yes 8 60.59 even 2
1200.2.v.l.257.1 8 5.3 odd 4 inner
1200.2.v.l.257.2 8 15.2 even 4 inner
1200.2.v.l.257.3 8 15.8 even 4 inner
1200.2.v.l.257.4 8 5.2 odd 4 inner
1200.2.v.l.593.1 8 15.14 odd 2 inner
1200.2.v.l.593.2 8 1.1 even 1 trivial
1200.2.v.l.593.3 8 5.4 even 2 inner
1200.2.v.l.593.4 8 3.2 odd 2 inner