Properties

Label 1008.2.s.k.289.1
Level $1008$
Weight $2$
Character 1008.289
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(289,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.s (of order \(3\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.289
Dual form 1008.2.s.k.865.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-2.50000 - 4.33013i) q^{11} +(-2.00000 - 3.46410i) q^{17} +(4.00000 - 6.92820i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.00000 + 3.46410i) q^{25} +5.00000 q^{29} +(1.50000 + 2.59808i) q^{31} +(2.00000 + 1.73205i) q^{35} +(2.00000 - 3.46410i) q^{37} -2.00000 q^{43} +(3.00000 - 5.19615i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-4.50000 - 7.79423i) q^{53} -5.00000 q^{55} +(5.50000 + 9.52628i) q^{59} +(3.00000 - 5.19615i) q^{61} +(-1.00000 - 1.73205i) q^{67} +2.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(12.5000 - 4.33013i) q^{77} +(1.50000 - 2.59808i) q^{79} -7.00000 q^{83} -4.00000 q^{85} +(-3.00000 + 5.19615i) q^{89} +(-4.00000 - 6.92820i) q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} - q^{7} - 5 q^{11} - 4 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{25} + 10 q^{29} + 3 q^{31} + 4 q^{35} + 4 q^{37} - 4 q^{43} + 6 q^{47} - 13 q^{49} - 9 q^{53} - 10 q^{55} + 11 q^{59} + 6 q^{61} - 2 q^{67} + 4 q^{71} - 10 q^{73} + 25 q^{77} + 3 q^{79} - 14 q^{83} - 8 q^{85} - 6 q^{89} - 8 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i −0.732294 0.680989i \(-0.761550\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.50000 4.33013i −0.753778 1.30558i −0.945979 0.324227i \(-0.894896\pi\)
0.192201 0.981356i \(-0.438437\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 3.46410i −0.485071 0.840168i 0.514782 0.857321i \(-0.327873\pi\)
−0.999853 + 0.0171533i \(0.994540\pi\)
\(18\) 0 0
\(19\) 4.00000 6.92820i 0.917663 1.58944i 0.114708 0.993399i \(-0.463407\pi\)
0.802955 0.596040i \(-0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 3.46410i 0.417029 0.722315i −0.578610 0.815604i \(-0.696405\pi\)
0.995639 + 0.0932891i \(0.0297381\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 1.50000 + 2.59808i 0.269408 + 0.466628i 0.968709 0.248199i \(-0.0798387\pi\)
−0.699301 + 0.714827i \(0.746505\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 + 1.73205i 0.338062 + 0.292770i
\(36\) 0 0
\(37\) 2.00000 3.46410i 0.328798 0.569495i −0.653476 0.756948i \(-0.726690\pi\)
0.982274 + 0.187453i \(0.0600231\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 5.19615i 0.437595 0.757937i −0.559908 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 7.79423i −0.618123 1.07062i −0.989828 0.142269i \(-0.954560\pi\)
0.371706 0.928351i \(-0.378773\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.50000 + 9.52628i 0.716039 + 1.24022i 0.962557 + 0.271078i \(0.0873801\pi\)
−0.246518 + 0.969138i \(0.579287\pi\)
\(60\) 0 0
\(61\) 3.00000 5.19615i 0.384111 0.665299i −0.607535 0.794293i \(-0.707841\pi\)
0.991645 + 0.128994i \(0.0411748\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 1.73205i −0.122169 0.211604i 0.798454 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −5.00000 8.66025i −0.585206 1.01361i −0.994850 0.101361i \(-0.967680\pi\)
0.409644 0.912245i \(-0.365653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.5000 4.33013i 1.42451 0.493464i
\(78\) 0 0
\(79\) 1.50000 2.59808i 0.168763 0.292306i −0.769222 0.638982i \(-0.779356\pi\)
0.937985 + 0.346675i \(0.112689\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 + 5.19615i −0.317999 + 0.550791i −0.980071 0.198650i \(-0.936344\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 6.92820i −0.410391 0.710819i
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.00000 + 8.66025i 0.497519 + 0.861727i 0.999996 0.00286291i \(-0.000911295\pi\)
−0.502477 + 0.864590i \(0.667578\pi\)
\(102\) 0 0
\(103\) 4.00000 6.92820i 0.394132 0.682656i −0.598858 0.800855i \(-0.704379\pi\)
0.992990 + 0.118199i \(0.0377120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.50000 + 2.59808i −0.145010 + 0.251166i −0.929377 0.369132i \(-0.879655\pi\)
0.784366 + 0.620298i \(0.212988\pi\)
\(108\) 0 0
\(109\) 1.00000 + 1.73205i 0.0957826 + 0.165900i 0.909935 0.414751i \(-0.136131\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −2.00000 3.46410i −0.186501 0.323029i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.0000 3.46410i 0.916698 0.317554i
\(120\) 0 0
\(121\) −7.00000 + 12.1244i −0.636364 + 1.10221i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −9.00000 −0.798621 −0.399310 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.500000 + 0.866025i −0.0436852 + 0.0756650i −0.887041 0.461690i \(-0.847243\pi\)
0.843356 + 0.537355i \(0.180577\pi\)
\(132\) 0 0
\(133\) 16.0000 + 13.8564i 1.38738 + 1.20150i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.00000 1.73205i −0.0854358 0.147979i 0.820141 0.572161i \(-0.193895\pi\)
−0.905577 + 0.424182i \(0.860562\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.50000 4.33013i 0.207614 0.359597i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 + 15.5885i −0.737309 + 1.27706i 0.216394 + 0.976306i \(0.430570\pi\)
−0.953703 + 0.300750i \(0.902763\pi\)
\(150\) 0 0
\(151\) 9.50000 + 16.4545i 0.773099 + 1.33905i 0.935857 + 0.352381i \(0.114628\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) 2.00000 + 3.46410i 0.159617 + 0.276465i 0.934731 0.355357i \(-0.115641\pi\)
−0.775113 + 0.631822i \(0.782307\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.00000 + 6.92820i 0.630488 + 0.546019i
\(162\) 0 0
\(163\) −2.00000 + 3.46410i −0.156652 + 0.271329i −0.933659 0.358162i \(-0.883403\pi\)
0.777007 + 0.629492i \(0.216737\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.0000 19.0526i 0.836315 1.44854i −0.0566411 0.998395i \(-0.518039\pi\)
0.892956 0.450145i \(-0.148628\pi\)
\(174\) 0 0
\(175\) −10.0000 + 3.46410i −0.755929 + 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 10.3923i −0.448461 0.776757i 0.549825 0.835280i \(-0.314694\pi\)
−0.998286 + 0.0585225i \(0.981361\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 3.46410i −0.147043 0.254686i
\(186\) 0 0
\(187\) −10.0000 + 17.3205i −0.731272 + 1.26660i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 + 20.7846i −0.868290 + 1.50392i −0.00454614 + 0.999990i \(0.501447\pi\)
−0.863743 + 0.503932i \(0.831886\pi\)
\(192\) 0 0
\(193\) −2.50000 4.33013i −0.179954 0.311689i 0.761911 0.647682i \(-0.224262\pi\)
−0.941865 + 0.335993i \(0.890928\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −2.00000 3.46410i −0.141776 0.245564i 0.786389 0.617731i \(-0.211948\pi\)
−0.928166 + 0.372168i \(0.878615\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.50000 + 12.9904i −0.175466 + 0.911746i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −40.0000 −2.76686
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 + 1.73205i −0.0681994 + 0.118125i
\(216\) 0 0
\(217\) −7.50000 + 2.59808i −0.509133 + 0.176369i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.50000 2.59808i −0.0995585 0.172440i 0.811943 0.583736i \(-0.198410\pi\)
−0.911502 + 0.411296i \(0.865076\pi\)
\(228\) 0 0
\(229\) 10.0000 17.3205i 0.660819 1.14457i −0.319582 0.947559i \(-0.603543\pi\)
0.980401 0.197013i \(-0.0631241\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.00000 + 3.46410i −0.131024 + 0.226941i −0.924072 0.382219i \(-0.875160\pi\)
0.793047 + 0.609160i \(0.208493\pi\)
\(234\) 0 0
\(235\) −3.00000 5.19615i −0.195698 0.338960i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 12.5000 + 21.6506i 0.805196 + 1.39464i 0.916159 + 0.400815i \(0.131273\pi\)
−0.110963 + 0.993825i \(0.535394\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.50000 + 4.33013i −0.351382 + 0.276642i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.00000 + 5.19615i −0.187135 + 0.324127i −0.944294 0.329104i \(-0.893253\pi\)
0.757159 + 0.653231i \(0.226587\pi\)
\(258\) 0 0
\(259\) 8.00000 + 6.92820i 0.497096 + 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.0000 + 25.9808i 0.924940 + 1.60204i 0.791658 + 0.610964i \(0.209218\pi\)
0.133281 + 0.991078i \(0.457449\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.5000 + 26.8468i 0.945052 + 1.63688i 0.755648 + 0.654978i \(0.227322\pi\)
0.189404 + 0.981899i \(0.439344\pi\)
\(270\) 0 0
\(271\) 7.50000 12.9904i 0.455593 0.789109i −0.543130 0.839649i \(-0.682761\pi\)
0.998722 + 0.0505395i \(0.0160941\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.0000 17.3205i 0.603023 1.04447i
\(276\) 0 0
\(277\) 8.00000 + 13.8564i 0.480673 + 0.832551i 0.999754 0.0221745i \(-0.00705893\pi\)
−0.519081 + 0.854725i \(0.673726\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 5.00000 + 8.66025i 0.297219 + 0.514799i 0.975499 0.220005i \(-0.0706075\pi\)
−0.678280 + 0.734804i \(0.737274\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.500000 0.866025i 0.0294118 0.0509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.0000 1.22683 0.613417 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) 0 0
\(295\) 11.0000 0.640445
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 5.19615i 0.0576390 0.299501i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.00000 5.19615i −0.171780 0.297531i
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 + 27.7128i 0.907277 + 1.57145i 0.817832 + 0.575458i \(0.195176\pi\)
0.0894452 + 0.995992i \(0.471491\pi\)
\(312\) 0 0
\(313\) −0.500000 + 0.866025i −0.0282617 + 0.0489506i −0.879810 0.475325i \(-0.842331\pi\)
0.851549 + 0.524276i \(0.175664\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.50000 2.59808i 0.0842484 0.145922i −0.820822 0.571184i \(-0.806484\pi\)
0.905071 + 0.425261i \(0.139818\pi\)
\(318\) 0 0
\(319\) −12.5000 21.6506i −0.699866 1.21220i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000 + 10.3923i 0.661581 + 0.572946i
\(330\) 0 0
\(331\) −2.00000 + 3.46410i −0.109930 + 0.190404i −0.915742 0.401768i \(-0.868396\pi\)
0.805812 + 0.592172i \(0.201729\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.50000 12.9904i 0.406148 0.703469i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 10.3923i −0.322097 0.557888i 0.658824 0.752297i \(-0.271054\pi\)
−0.980921 + 0.194409i \(0.937721\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 + 20.7846i 0.638696 + 1.10625i 0.985719 + 0.168397i \(0.0538590\pi\)
−0.347024 + 0.937856i \(0.612808\pi\)
\(354\) 0 0
\(355\) 1.00000 1.73205i 0.0530745 0.0919277i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.00000 + 8.66025i −0.263890 + 0.457071i −0.967272 0.253741i \(-0.918339\pi\)
0.703382 + 0.710812i \(0.251672\pi\)
\(360\) 0 0
\(361\) −22.5000 38.9711i −1.18421 2.05111i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 8.50000 + 14.7224i 0.443696 + 0.768505i 0.997960 0.0638362i \(-0.0203335\pi\)
−0.554264 + 0.832341i \(0.687000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 22.5000 7.79423i 1.16814 0.404656i
\(372\) 0 0
\(373\) 16.0000 27.7128i 0.828449 1.43492i −0.0708063 0.997490i \(-0.522557\pi\)
0.899255 0.437425i \(-0.144109\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.0000 29.4449i 0.868659 1.50456i 0.00529229 0.999986i \(-0.498315\pi\)
0.863367 0.504576i \(-0.168351\pi\)
\(384\) 0 0
\(385\) 2.50000 12.9904i 0.127412 0.662051i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.00000 1.73205i −0.0507020 0.0878185i 0.839561 0.543266i \(-0.182813\pi\)
−0.890263 + 0.455448i \(0.849479\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.50000 2.59808i −0.0754732 0.130723i
\(396\) 0 0
\(397\) −18.0000 + 31.1769i −0.903394 + 1.56472i −0.0803356 + 0.996768i \(0.525599\pi\)
−0.823058 + 0.567957i \(0.807734\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 20.7846i 0.599251 1.03793i −0.393680 0.919247i \(-0.628798\pi\)
0.992932 0.118686i \(-0.0378683\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) 12.5000 + 21.6506i 0.618085 + 1.07056i 0.989835 + 0.142222i \(0.0454247\pi\)
−0.371750 + 0.928333i \(0.621242\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −27.5000 + 9.52628i −1.35319 + 0.468758i
\(414\) 0 0
\(415\) −3.50000 + 6.06218i −0.171808 + 0.297581i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.00000 13.8564i 0.388057 0.672134i
\(426\) 0 0
\(427\) 12.0000 + 10.3923i 0.580721 + 0.502919i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.00000 10.3923i −0.289010 0.500580i 0.684564 0.728953i \(-0.259993\pi\)
−0.973574 + 0.228373i \(0.926659\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −16.0000 27.7128i −0.765384 1.32568i
\(438\) 0 0
\(439\) 7.50000 12.9904i 0.357955 0.619997i −0.629664 0.776868i \(-0.716807\pi\)
0.987619 + 0.156871i \(0.0501406\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.50000 + 14.7224i −0.403847 + 0.699484i −0.994187 0.107671i \(-0.965661\pi\)
0.590339 + 0.807155i \(0.298994\pi\)
\(444\) 0 0
\(445\) 3.00000 + 5.19615i 0.142214 + 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.5000 + 26.8468i −0.725059 + 1.25584i 0.233890 + 0.972263i \(0.424854\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.0000 17.3205i 0.462745 0.801498i −0.536352 0.843995i \(-0.680198\pi\)
0.999097 + 0.0424970i \(0.0135313\pi\)
\(468\) 0 0
\(469\) 5.00000 1.73205i 0.230879 0.0799787i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.00000 + 8.66025i 0.229900 + 0.398199i
\(474\) 0 0
\(475\) 32.0000 1.46826
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.0000 32.9090i −0.868132 1.50365i −0.863903 0.503658i \(-0.831987\pi\)
−0.00422900 0.999991i \(-0.501346\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.50000 6.06218i 0.158927 0.275269i
\(486\) 0 0
\(487\) 2.50000 + 4.33013i 0.113286 + 0.196217i 0.917093 0.398673i \(-0.130529\pi\)
−0.803807 + 0.594890i \(0.797196\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) −10.0000 17.3205i −0.450377 0.780076i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.00000 + 5.19615i −0.0448561 + 0.233079i
\(498\) 0 0
\(499\) 5.00000 8.66025i 0.223831 0.387686i −0.732137 0.681157i \(-0.761477\pi\)
0.955968 + 0.293471i \(0.0948104\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.50000 12.9904i 0.332432 0.575789i −0.650556 0.759458i \(-0.725464\pi\)
0.982988 + 0.183669i \(0.0587976\pi\)
\(510\) 0 0
\(511\) 25.0000 8.66025i 1.10593 0.383107i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.00000 6.92820i −0.176261 0.305293i
\(516\) 0 0
\(517\) −30.0000 −1.31940
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.00000 15.5885i −0.394297 0.682943i 0.598714 0.800963i \(-0.295679\pi\)
−0.993011 + 0.118020i \(0.962345\pi\)
\(522\) 0 0
\(523\) 4.00000 6.92820i 0.174908 0.302949i −0.765222 0.643767i \(-0.777371\pi\)
0.940129 + 0.340818i \(0.110704\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 0 0
\(529\) 3.50000 + 6.06218i 0.152174 + 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.50000 + 2.59808i 0.0648507 + 0.112325i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.00000 + 34.6410i 0.215365 + 1.49209i
\(540\) 0 0
\(541\) 9.00000 15.5885i 0.386940 0.670200i −0.605096 0.796152i \(-0.706865\pi\)
0.992036 + 0.125952i \(0.0401986\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 20.0000 34.6410i 0.852029 1.47576i
\(552\) 0 0
\(553\) 6.00000 + 5.19615i 0.255146 + 0.220963i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.5000 19.9186i −0.487271 0.843978i 0.512622 0.858614i \(-0.328674\pi\)
−0.999893 + 0.0146368i \(0.995341\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.50000 14.7224i −0.358232 0.620477i 0.629433 0.777055i \(-0.283287\pi\)
−0.987666 + 0.156578i \(0.949954\pi\)
\(564\) 0 0
\(565\) −8.00000 + 13.8564i −0.336563 + 0.582943i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0000 20.7846i 0.503066 0.871336i −0.496928 0.867792i \(-0.665539\pi\)
0.999994 0.00354413i \(-0.00112814\pi\)
\(570\) 0 0
\(571\) −15.0000 25.9808i −0.627730 1.08726i −0.988006 0.154415i \(-0.950651\pi\)
0.360276 0.932846i \(-0.382683\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) −15.5000 26.8468i −0.645273 1.11765i −0.984238 0.176847i \(-0.943410\pi\)
0.338965 0.940799i \(-0.389923\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.50000 18.1865i 0.145204 0.754505i
\(582\) 0 0
\(583\) −22.5000 + 38.9711i −0.931855 + 1.61402i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.0000 1.44460 0.722302 0.691577i \(-0.243084\pi\)
0.722302 + 0.691577i \(0.243084\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.0000 31.1769i 0.739171 1.28028i −0.213697 0.976900i \(-0.568551\pi\)
0.952869 0.303383i \(-0.0981160\pi\)
\(594\) 0 0
\(595\) 2.00000 10.3923i 0.0819920 0.426043i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.0000 + 25.9808i 0.612883 + 1.06155i 0.990752 + 0.135686i \(0.0433238\pi\)
−0.377869 + 0.925859i \(0.623343\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.00000 + 12.1244i 0.284590 + 0.492925i
\(606\) 0 0
\(607\) −13.5000 + 23.3827i −0.547948 + 0.949074i 0.450467 + 0.892793i \(0.351258\pi\)
−0.998415 + 0.0562808i \(0.982076\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −6.00000 10.3923i −0.242338 0.419741i 0.719042 0.694967i \(-0.244581\pi\)
−0.961380 + 0.275225i \(0.911248\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) 5.00000 + 8.66025i 0.200967 + 0.348085i 0.948840 0.315757i \(-0.102258\pi\)
−0.747873 + 0.663842i \(0.768925\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 10.3923i −0.480770 0.416359i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 19.0000 0.756378 0.378189 0.925728i \(-0.376547\pi\)
0.378189 + 0.925728i \(0.376547\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.50000 + 7.79423i −0.178577 + 0.309305i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.0000 + 22.5167i 0.513469 + 0.889355i 0.999878 + 0.0156233i \(0.00497325\pi\)
−0.486409 + 0.873731i \(0.661693\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.00000 + 15.5885i 0.353827 + 0.612845i 0.986916 0.161233i \(-0.0515470\pi\)
−0.633090 + 0.774078i \(0.718214\pi\)
\(648\) 0 0
\(649\) 27.5000 47.6314i 1.07947 1.86970i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.5000 + 33.7750i −0.763094 + 1.32172i 0.178154 + 0.984003i \(0.442987\pi\)
−0.941248 + 0.337715i \(0.890346\pi\)
\(654\) 0 0
\(655\) 0.500000 + 0.866025i 0.0195366 + 0.0338384i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) −5.00000 8.66025i −0.194477 0.336845i 0.752252 0.658876i \(-0.228968\pi\)
−0.946729 + 0.322031i \(0.895634\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 20.0000 6.92820i 0.775567 0.268664i
\(666\) 0 0
\(667\) 10.0000 17.3205i 0.387202 0.670653i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.5000 + 23.3827i −0.518847 + 0.898670i 0.480913 + 0.876768i \(0.340305\pi\)
−0.999760 + 0.0219013i \(0.993028\pi\)
\(678\) 0 0
\(679\) −3.50000 + 18.1865i −0.134318 + 0.697935i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.50000 + 7.79423i 0.172188 + 0.298238i 0.939184 0.343413i \(-0.111583\pi\)
−0.766997 + 0.641651i \(0.778250\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 4.00000 6.92820i 0.152167 0.263561i −0.779857 0.625958i \(-0.784708\pi\)
0.932024 + 0.362397i \(0.118041\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.00000 12.1244i 0.265525 0.459903i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.00000 0.188847 0.0944237 0.995532i \(-0.469899\pi\)
0.0944237 + 0.995532i \(0.469899\pi\)
\(702\) 0 0
\(703\) −16.0000 27.7128i −0.603451 1.04521i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −25.0000 + 8.66025i −0.940222 + 0.325702i
\(708\) 0 0
\(709\) −19.0000 + 32.9090i −0.713560 + 1.23592i 0.249952 + 0.968258i \(0.419585\pi\)
−0.963512 + 0.267664i \(0.913748\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.00000 5.19615i 0.111881 0.193784i −0.804648 0.593753i \(-0.797646\pi\)
0.916529 + 0.399969i \(0.130979\pi\)
\(720\) 0 0
\(721\) 16.0000 + 13.8564i 0.595871 + 0.516040i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10.0000 + 17.3205i 0.371391 + 0.643268i
\(726\) 0 0
\(727\) −7.00000 −0.259616 −0.129808 0.991539i \(-0.541436\pi\)
−0.129808 + 0.991539i \(0.541436\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.00000 + 6.92820i 0.147945 + 0.256249i
\(732\) 0 0
\(733\) 3.00000 5.19615i 0.110808 0.191924i −0.805289 0.592883i \(-0.797990\pi\)
0.916096 + 0.400959i \(0.131323\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.00000 + 8.66025i −0.184177 + 0.319005i
\(738\) 0 0
\(739\) −15.0000 25.9808i −0.551784 0.955718i −0.998146 0.0608653i \(-0.980614\pi\)
0.446362 0.894852i \(-0.352719\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) 9.00000 + 15.5885i 0.329734 + 0.571117i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.00000 5.19615i −0.219235 0.189863i
\(750\) 0 0
\(751\) 22.5000 38.9711i 0.821037 1.42208i −0.0838743 0.996476i \(-0.526729\pi\)
0.904911 0.425601i \(-0.139937\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 19.0000 0.691481
\(756\) 0 0
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.00000 6.92820i 0.145000 0.251147i −0.784373 0.620289i \(-0.787015\pi\)
0.929373 + 0.369142i \(0.120348\pi\)
\(762\) 0 0
\(763\) −5.00000 + 1.73205i −0.181012 + 0.0627044i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.00000 + 8.66025i 0.179838 + 0.311488i 0.941825 0.336104i \(-0.109109\pi\)
−0.761987 + 0.647592i \(0.775776\pi\)
\(774\) 0 0
\(775\) −6.00000 + 10.3923i −0.215526 + 0.373303i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −5.00000 8.66025i −0.178914 0.309888i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −9.00000 15.5885i −0.320815 0.555668i 0.659841 0.751405i \(-0.270624\pi\)
−0.980656 + 0.195737i \(0.937290\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.00000 41.5692i 0.284447 1.47803i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.0000 −0.743858 −0.371929 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −25.0000 + 43.3013i −0.882231 + 1.52807i
\(804\) 0 0
\(805\) 10.0000 3.46410i 0.352454 0.122094i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.0000 + 34.6410i 0.703163 + 1.21791i 0.967351 + 0.253442i \(0.0815627\pi\)
−0.264188 + 0.964471i \(0.585104\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.00000 + 3.46410i 0.0700569 + 0.121342i
\(816\) 0 0
\(817\) −8.00000 + 13.8564i −0.279885 + 0.484774i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.5000 + 21.6506i −0.436253 + 0.755612i −0.997397 0.0721058i \(-0.977028\pi\)
0.561144 + 0.827718i \(0.310361\pi\)
\(822\) 0 0
\(823\) 20.0000 + 34.6410i 0.697156 + 1.20751i 0.969448 + 0.245295i \(0.0788849\pi\)
−0.272292 + 0.962215i \(0.587782\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 0 0
\(829\) 16.0000 + 27.7128i 0.555703 + 0.962506i 0.997848 + 0.0655624i \(0.0208842\pi\)
−0.442145 + 0.896943i \(0.645783\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.00000 + 27.7128i 0.138592 + 0.960192i
\(834\) 0 0
\(835\) −7.00000 + 12.1244i −0.242245 + 0.419581i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.0000 −0.966667 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.50000 + 11.2583i −0.223607 + 0.387298i
\(846\) 0 0
\(847\) −28.0000 24.2487i −0.962091 0.833196i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.00000 13.8564i −0.274236 0.474991i
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.00000 15.5885i −0.307434 0.532492i 0.670366 0.742030i \(-0.266137\pi\)
−0.977800 + 0.209539i \(0.932804\pi\)
\(858\) 0 0
\(859\) −17.0000 + 29.4449i −0.580033 + 1.00465i 0.415442 + 0.909620i \(0.363627\pi\)
−0.995475 + 0.0950262i \(0.969707\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.00000 + 8.66025i −0.170202 + 0.294798i −0.938490 0.345305i \(-0.887775\pi\)
0.768288 + 0.640104i \(0.221109\pi\)
\(864\) 0 0
\(865\) −11.0000 19.0526i −0.374011 0.647806i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.50000 + 23.3827i −0.152128 + 0.790479i
\(876\) 0 0
\(877\) 16.0000 27.7128i 0.540282 0.935795i −0.458606 0.888640i \(-0.651651\pi\)
0.998888 0.0471555i \(-0.0150156\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 40.0000 1.34611 0.673054 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.0000 + 31.1769i −0.604381 + 1.04682i 0.387768 + 0.921757i \(0.373246\pi\)
−0.992149 + 0.125061i \(0.960087\pi\)
\(888\) 0 0
\(889\) 4.50000 23.3827i 0.150925 0.784230i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −24.0000 41.5692i −0.803129 1.39106i
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7.50000 + 12.9904i 0.250139 + 0.433253i
\(900\) 0 0
\(901\) −18.0000 + 31.1769i −0.599667 + 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 6.00000 + 10.3923i 0.199227 + 0.345071i 0.948278 0.317441i \(-0.102824\pi\)
−0.749051 + 0.662512i \(0.769490\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 17.5000 + 30.3109i 0.579165 + 1.00314i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.00000 1.73205i −0.0660458 0.0571974i
\(918\) 0 0
\(919\) −16.0000 + 27.7128i −0.527791 + 0.914161i 0.471684 + 0.881768i \(0.343646\pi\)
−0.999475 + 0.0323936i \(0.989687\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 16.0000 0.526077
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.00000 + 5.19615i −0.0984268 + 0.170480i −0.911034 0.412332i \(-0.864714\pi\)
0.812607 + 0.582812i \(0.198048\pi\)
\(930\) 0 0
\(931\) −44.0000 + 34.6410i −1.44204 + 1.13531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 10.0000 + 17.3205i 0.327035 + 0.566441i
\(936\) 0 0
\(937\) 35.0000 1.14340 0.571700 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −5.50000 9.52628i −0.179295 0.310548i 0.762344 0.647172i \(-0.224048\pi\)
−0.941639 + 0.336624i \(0.890715\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.0000 27.7128i 0.519930 0.900545i −0.479801 0.877377i \(-0.659291\pi\)
0.999732 0.0231683i \(-0.00737536\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 0 0
\(955\) 12.0000 + 20.7846i 0.388311 + 0.672574i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.00000 1.73205i 0.161458 0.0559308i
\(960\) 0 0
\(961\) 11.0000 19.0526i 0.354839 0.614599i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −5.00000 −0.160956
\(966\) 0 0
\(967\) 61.0000 1.96163 0.980814 0.194946i \(-0.0624533\pi\)
0.980814 + 0.194946i \(0.0624533\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −7.50000 + 12.9904i −0.240686 + 0.416881i −0.960910 0.276861i \(-0.910706\pi\)
0.720224 + 0.693742i \(0.244039\pi\)
\(972\) 0 0
\(973\) −7.00000 + 36.3731i −0.224410 + 1.16607i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −15.0000 25.9808i −0.479893 0.831198i 0.519841 0.854263i \(-0.325991\pi\)
−0.999734 + 0.0230645i \(0.992658\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 30.0000 + 51.9615i 0.956851 + 1.65732i 0.730073 + 0.683369i \(0.239486\pi\)
0.226778 + 0.973946i \(0.427181\pi\)
\(984\) 0 0
\(985\) −1.00000 + 1.73205i −0.0318626 + 0.0551877i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 + 6.92820i −0.127193 + 0.220304i
\(990\) 0 0
\(991\) 23.5000 + 40.7032i 0.746502 + 1.29298i 0.949490 + 0.313798i \(0.101602\pi\)
−0.202988 + 0.979181i \(0.565065\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −19.0000 32.9090i −0.601736 1.04224i −0.992558 0.121771i \(-0.961143\pi\)
0.390822 0.920466i \(-0.372191\pi\)
\(998\) 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 1008.2.s.k.289.1 2
3.2 odd 2 336.2.q.b.289.1 2
4.3 odd 2 126.2.g.c.37.1 2
7.2 even 3 7056.2.a.w.1.1 1
7.4 even 3 inner 1008.2.s.k.865.1 2
7.5 odd 6 7056.2.a.bl.1.1 1
12.11 even 2 42.2.e.a.37.1 yes 2
21.2 odd 6 2352.2.a.t.1.1 1
21.5 even 6 2352.2.a.f.1.1 1
21.11 odd 6 336.2.q.b.193.1 2
21.17 even 6 2352.2.q.u.1537.1 2
21.20 even 2 2352.2.q.u.961.1 2
24.5 odd 2 1344.2.q.s.961.1 2
24.11 even 2 1344.2.q.g.961.1 2
28.3 even 6 882.2.g.i.361.1 2
28.11 odd 6 126.2.g.c.109.1 2
28.19 even 6 882.2.a.d.1.1 1
28.23 odd 6 882.2.a.c.1.1 1
28.27 even 2 882.2.g.i.667.1 2
36.7 odd 6 1134.2.e.e.919.1 2
36.11 even 6 1134.2.e.l.919.1 2
36.23 even 6 1134.2.h.e.541.1 2
36.31 odd 6 1134.2.h.l.541.1 2
60.23 odd 4 1050.2.o.a.499.2 4
60.47 odd 4 1050.2.o.a.499.1 4
60.59 even 2 1050.2.i.l.751.1 2
84.11 even 6 42.2.e.a.25.1 2
84.23 even 6 294.2.a.e.1.1 1
84.47 odd 6 294.2.a.f.1.1 1
84.59 odd 6 294.2.e.b.67.1 2
84.83 odd 2 294.2.e.b.79.1 2
168.5 even 6 9408.2.a.cr.1.1 1
168.11 even 6 1344.2.q.g.193.1 2
168.53 odd 6 1344.2.q.s.193.1 2
168.107 even 6 9408.2.a.ce.1.1 1
168.131 odd 6 9408.2.a.z.1.1 1
168.149 odd 6 9408.2.a.q.1.1 1
252.11 even 6 1134.2.h.e.109.1 2
252.67 odd 6 1134.2.e.e.865.1 2
252.95 even 6 1134.2.e.l.865.1 2
252.151 odd 6 1134.2.h.l.109.1 2
420.179 even 6 1050.2.i.l.151.1 2
420.263 odd 12 1050.2.o.a.949.1 4
420.299 odd 6 7350.2.a.q.1.1 1
420.347 odd 12 1050.2.o.a.949.2 4
420.359 even 6 7350.2.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.a.25.1 2 84.11 even 6
42.2.e.a.37.1 yes 2 12.11 even 2
126.2.g.c.37.1 2 4.3 odd 2
126.2.g.c.109.1 2 28.11 odd 6
294.2.a.e.1.1 1 84.23 even 6
294.2.a.f.1.1 1 84.47 odd 6
294.2.e.b.67.1 2 84.59 odd 6
294.2.e.b.79.1 2 84.83 odd 2
336.2.q.b.193.1 2 21.11 odd 6
336.2.q.b.289.1 2 3.2 odd 2
882.2.a.c.1.1 1 28.23 odd 6
882.2.a.d.1.1 1 28.19 even 6
882.2.g.i.361.1 2 28.3 even 6
882.2.g.i.667.1 2 28.27 even 2
1008.2.s.k.289.1 2 1.1 even 1 trivial
1008.2.s.k.865.1 2 7.4 even 3 inner
1050.2.i.l.151.1 2 420.179 even 6
1050.2.i.l.751.1 2 60.59 even 2
1050.2.o.a.499.1 4 60.47 odd 4
1050.2.o.a.499.2 4 60.23 odd 4
1050.2.o.a.949.1 4 420.263 odd 12
1050.2.o.a.949.2 4 420.347 odd 12
1134.2.e.e.865.1 2 252.67 odd 6
1134.2.e.e.919.1 2 36.7 odd 6
1134.2.e.l.865.1 2 252.95 even 6
1134.2.e.l.919.1 2 36.11 even 6
1134.2.h.e.109.1 2 252.11 even 6
1134.2.h.e.541.1 2 36.23 even 6
1134.2.h.l.109.1 2 252.151 odd 6
1134.2.h.l.541.1 2 36.31 odd 6
1344.2.q.g.193.1 2 168.11 even 6
1344.2.q.g.961.1 2 24.11 even 2
1344.2.q.s.193.1 2 168.53 odd 6
1344.2.q.s.961.1 2 24.5 odd 2
2352.2.a.f.1.1 1 21.5 even 6
2352.2.a.t.1.1 1 21.2 odd 6
2352.2.q.u.961.1 2 21.20 even 2
2352.2.q.u.1537.1 2 21.17 even 6
7056.2.a.w.1.1 1 7.2 even 3
7056.2.a.bl.1.1 1 7.5 odd 6
7350.2.a.q.1.1 1 420.299 odd 6
7350.2.a.bl.1.1 1 420.359 even 6
9408.2.a.q.1.1 1 168.149 odd 6
9408.2.a.z.1.1 1 168.131 odd 6
9408.2.a.ce.1.1 1 168.107 even 6
9408.2.a.cr.1.1 1 168.5 even 6