Properties

Label 1512.2.s.a.865.1
Level $1512$
Weight $2$
Character 1512.865
Analytic conductor $12.073$
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] = [1512,2,Mod(865,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.s (of order \(3\), 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: \(12.0733807856\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1512.865
Dual form 1512.2.s.a.1297.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.00000 - 3.46410i) q^{5} +(2.00000 - 1.73205i) q^{7} +O(q^{10})\) \(q+(-2.00000 - 3.46410i) q^{5} +(2.00000 - 1.73205i) q^{7} +(1.00000 - 1.73205i) q^{11} +5.00000 q^{13} +(3.00000 - 5.19615i) q^{17} +(2.00000 + 3.46410i) q^{19} +(3.00000 + 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} -6.00000 q^{29} +(3.50000 - 6.06218i) q^{31} +(-10.0000 - 3.46410i) q^{35} +(-3.50000 - 6.06218i) q^{37} -2.00000 q^{41} -7.00000 q^{43} +(-1.00000 - 1.73205i) q^{47} +(1.00000 - 6.92820i) q^{49} +(3.00000 - 5.19615i) q^{53} -8.00000 q^{55} +(3.00000 - 5.19615i) q^{59} +(4.50000 + 7.79423i) q^{61} +(-10.0000 - 17.3205i) q^{65} +(3.50000 - 6.06218i) q^{67} +8.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(-1.00000 - 5.19615i) q^{77} +(-0.500000 - 0.866025i) q^{79} -14.0000 q^{83} -24.0000 q^{85} +(6.00000 + 10.3923i) q^{89} +(10.0000 - 8.66025i) q^{91} +(8.00000 - 13.8564i) q^{95} -15.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} + 4 q^{7} + 2 q^{11} + 10 q^{13} + 6 q^{17} + 4 q^{19} + 6 q^{23} - 11 q^{25} - 12 q^{29} + 7 q^{31} - 20 q^{35} - 7 q^{37} - 4 q^{41} - 14 q^{43} - 2 q^{47} + 2 q^{49} + 6 q^{53} - 16 q^{55} + 6 q^{59} + 9 q^{61} - 20 q^{65} + 7 q^{67} + 16 q^{71} - 10 q^{73} - 2 q^{77} - q^{79} - 28 q^{83} - 48 q^{85} + 12 q^{89} + 20 q^{91} + 16 q^{95} - 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) −2.00000 3.46410i −0.894427 1.54919i −0.834512 0.550990i \(-0.814250\pi\)
−0.0599153 0.998203i \(-0.519083\pi\)
\(6\) 0 0
\(7\) 2.00000 1.73205i 0.755929 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 1.73205i 0.301511 0.522233i −0.674967 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 5.19615i 0.727607 1.26025i −0.230285 0.973123i \(-0.573966\pi\)
0.957892 0.287129i \(-0.0927008\pi\)
\(18\) 0 0
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) 0 0
\(25\) −5.50000 + 9.52628i −1.10000 + 1.90526i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 3.50000 6.06218i 0.628619 1.08880i −0.359211 0.933257i \(-0.616954\pi\)
0.987829 0.155543i \(-0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −10.0000 3.46410i −1.69031 0.585540i
\(36\) 0 0
\(37\) −3.50000 6.06218i −0.575396 0.996616i −0.995998 0.0893706i \(-0.971514\pi\)
0.420602 0.907245i \(-0.361819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 1.73205i −0.145865 0.252646i 0.783830 0.620975i \(-0.213263\pi\)
−0.929695 + 0.368329i \(0.879930\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000 5.19615i 0.412082 0.713746i −0.583036 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 5.19615i 0.390567 0.676481i −0.601958 0.798528i \(-0.705612\pi\)
0.992524 + 0.122047i \(0.0389457\pi\)
\(60\) 0 0
\(61\) 4.50000 + 7.79423i 0.576166 + 0.997949i 0.995914 + 0.0903080i \(0.0287851\pi\)
−0.419748 + 0.907641i \(0.637882\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.0000 17.3205i −1.24035 2.14834i
\(66\) 0 0
\(67\) 3.50000 6.06218i 0.427593 0.740613i −0.569066 0.822292i \(-0.692695\pi\)
0.996659 + 0.0816792i \(0.0260283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −5.00000 + 8.66025i −0.585206 + 1.01361i 0.409644 + 0.912245i \(0.365653\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 5.19615i −0.113961 0.592157i
\(78\) 0 0
\(79\) −0.500000 0.866025i −0.0562544 0.0974355i 0.836527 0.547926i \(-0.184582\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) −24.0000 −2.60317
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 + 10.3923i 0.635999 + 1.10158i 0.986303 + 0.164946i \(0.0527450\pi\)
−0.350304 + 0.936636i \(0.613922\pi\)
\(90\) 0 0
\(91\) 10.0000 8.66025i 1.04828 0.907841i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 13.8564i 0.820783 1.42164i
\(96\) 0 0
\(97\) −15.0000 −1.52302 −0.761510 0.648154i \(-0.775541\pi\)
−0.761510 + 0.648154i \(0.775541\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.00000 + 13.8564i −0.796030 + 1.37876i 0.126153 + 0.992011i \(0.459737\pi\)
−0.922183 + 0.386753i \(0.873597\pi\)
\(102\) 0 0
\(103\) 6.50000 + 11.2583i 0.640464 + 1.10932i 0.985329 + 0.170664i \(0.0545913\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.00000 5.19615i −0.290021 0.502331i 0.683793 0.729676i \(-0.260329\pi\)
−0.973814 + 0.227345i \(0.926996\pi\)
\(108\) 0 0
\(109\) −2.50000 + 4.33013i −0.239457 + 0.414751i −0.960558 0.278078i \(-0.910303\pi\)
0.721102 + 0.692829i \(0.243636\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) 12.0000 20.7846i 1.11901 1.93817i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 15.5885i −0.275010 1.42899i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.00000 3.46410i −0.174741 0.302660i 0.765331 0.643637i \(-0.222575\pi\)
−0.940072 + 0.340977i \(0.889242\pi\)
\(132\) 0 0
\(133\) 10.0000 + 3.46410i 0.867110 + 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000 5.19615i 0.256307 0.443937i −0.708942 0.705266i \(-0.750827\pi\)
0.965250 + 0.261329i \(0.0841608\pi\)
\(138\) 0 0
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.00000 8.66025i 0.418121 0.724207i
\(144\) 0 0
\(145\) 12.0000 + 20.7846i 0.996546 + 1.72607i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 4.50000 7.79423i 0.366205 0.634285i −0.622764 0.782410i \(-0.713990\pi\)
0.988969 + 0.148124i \(0.0473236\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −28.0000 −2.24901
\(156\) 0 0
\(157\) 3.00000 5.19615i 0.239426 0.414698i −0.721124 0.692806i \(-0.756374\pi\)
0.960550 + 0.278108i \(0.0897074\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 15.0000 + 5.19615i 1.18217 + 0.409514i
\(162\) 0 0
\(163\) −10.5000 18.1865i −0.822423 1.42448i −0.903873 0.427802i \(-0.859288\pi\)
0.0814491 0.996678i \(-0.474045\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 5.50000 + 28.5788i 0.415761 + 2.16036i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.00000 + 3.46410i −0.149487 + 0.258919i −0.931038 0.364922i \(-0.881096\pi\)
0.781551 + 0.623841i \(0.214429\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −14.0000 + 24.2487i −1.02930 + 1.78280i
\(186\) 0 0
\(187\) −6.00000 10.3923i −0.438763 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 + 17.3205i 0.723575 + 1.25327i 0.959558 + 0.281511i \(0.0908356\pi\)
−0.235983 + 0.971757i \(0.575831\pi\)
\(192\) 0 0
\(193\) −3.50000 + 6.06218i −0.251936 + 0.436365i −0.964059 0.265689i \(-0.914400\pi\)
0.712123 + 0.702055i \(0.247734\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 1.50000 2.59808i 0.106332 0.184173i −0.807950 0.589252i \(-0.799423\pi\)
0.914282 + 0.405079i \(0.132756\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.0000 + 10.3923i −0.842235 + 0.729397i
\(204\) 0 0
\(205\) 4.00000 + 6.92820i 0.279372 + 0.483887i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −27.0000 −1.85876 −0.929378 0.369129i \(-0.879656\pi\)
−0.929378 + 0.369129i \(0.879656\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14.0000 + 24.2487i 0.954792 + 1.65375i
\(216\) 0 0
\(217\) −3.50000 18.1865i −0.237595 1.23458i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.0000 25.9808i 1.00901 1.74766i
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.00000 8.66025i 0.331862 0.574801i −0.651015 0.759065i \(-0.725657\pi\)
0.982877 + 0.184263i \(0.0589899\pi\)
\(228\) 0 0
\(229\) −6.50000 11.2583i −0.429532 0.743971i 0.567300 0.823511i \(-0.307988\pi\)
−0.996832 + 0.0795401i \(0.974655\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.00000 + 6.92820i 0.262049 + 0.453882i 0.966786 0.255586i \(-0.0822686\pi\)
−0.704737 + 0.709468i \(0.748935\pi\)
\(234\) 0 0
\(235\) −4.00000 + 6.92820i −0.260931 + 0.451946i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) −0.500000 + 0.866025i −0.0322078 + 0.0557856i −0.881680 0.471848i \(-0.843587\pi\)
0.849472 + 0.527633i \(0.176921\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −26.0000 + 10.3923i −1.66108 + 0.663940i
\(246\) 0 0
\(247\) 10.0000 + 17.3205i 0.636285 + 1.10208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.00000 + 12.1244i 0.436648 + 0.756297i 0.997429 0.0716680i \(-0.0228322\pi\)
−0.560781 + 0.827964i \(0.689499\pi\)
\(258\) 0 0
\(259\) −17.5000 6.06218i −1.08740 0.376685i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.00000 + 5.19615i −0.184988 + 0.320408i −0.943572 0.331166i \(-0.892558\pi\)
0.758585 + 0.651575i \(0.225891\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.0000 + 27.7128i −0.975537 + 1.68968i −0.297386 + 0.954757i \(0.596115\pi\)
−0.678151 + 0.734923i \(0.737218\pi\)
\(270\) 0 0
\(271\) −7.50000 12.9904i −0.455593 0.789109i 0.543130 0.839649i \(-0.317239\pi\)
−0.998722 + 0.0505395i \(0.983906\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11.0000 + 19.0526i 0.663325 + 1.14891i
\(276\) 0 0
\(277\) −0.500000 + 0.866025i −0.0300421 + 0.0520344i −0.880656 0.473757i \(-0.842897\pi\)
0.850613 + 0.525792i \(0.176231\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) 5.50000 9.52628i 0.326941 0.566279i −0.654962 0.755662i \(-0.727315\pi\)
0.981903 + 0.189383i \(0.0606488\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.00000 + 3.46410i −0.236113 + 0.204479i
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.0000 + 25.9808i 0.867472 + 1.50251i
\(300\) 0 0
\(301\) −14.0000 + 12.1244i −0.806947 + 0.698836i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 18.0000 31.1769i 1.03068 1.78518i
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.00000 + 13.8564i −0.453638 + 0.785725i −0.998609 0.0527306i \(-0.983208\pi\)
0.544970 + 0.838455i \(0.316541\pi\)
\(312\) 0 0
\(313\) −5.00000 8.66025i −0.282617 0.489506i 0.689412 0.724370i \(-0.257869\pi\)
−0.972028 + 0.234863i \(0.924536\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.00000 5.19615i −0.168497 0.291845i 0.769395 0.638774i \(-0.220558\pi\)
−0.937892 + 0.346929i \(0.887225\pi\)
\(318\) 0 0
\(319\) −6.00000 + 10.3923i −0.335936 + 0.581857i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) −27.5000 + 47.6314i −1.52543 + 2.64211i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.00000 1.73205i −0.275659 0.0954911i
\(330\) 0 0
\(331\) 2.00000 + 3.46410i 0.109930 + 0.190404i 0.915742 0.401768i \(-0.131604\pi\)
−0.805812 + 0.592172i \(0.798271\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −28.0000 −1.52980
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.00000 12.1244i −0.379071 0.656571i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(348\) 0 0
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 24.2487i 0.745145 1.29063i −0.204982 0.978766i \(-0.565714\pi\)
0.950127 0.311863i \(-0.100953\pi\)
\(354\) 0 0
\(355\) −16.0000 27.7128i −0.849192 1.47084i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 + 5.19615i 0.158334 + 0.274242i 0.934268 0.356572i \(-0.116054\pi\)
−0.775934 + 0.630814i \(0.782721\pi\)
\(360\) 0 0
\(361\) 1.50000 2.59808i 0.0789474 0.136741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 40.0000 2.09370
\(366\) 0 0
\(367\) −14.0000 + 24.2487i −0.730794 + 1.26577i 0.225750 + 0.974185i \(0.427517\pi\)
−0.956544 + 0.291587i \(0.905817\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.00000 15.5885i −0.155752 0.809312i
\(372\) 0 0
\(373\) −5.00000 8.66025i −0.258890 0.448411i 0.707055 0.707159i \(-0.250023\pi\)
−0.965945 + 0.258748i \(0.916690\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −30.0000 −1.54508
\(378\) 0 0
\(379\) 17.0000 0.873231 0.436616 0.899648i \(-0.356177\pi\)
0.436616 + 0.899648i \(0.356177\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.0000 + 25.9808i 0.766464 + 1.32755i 0.939469 + 0.342634i \(0.111319\pi\)
−0.173005 + 0.984921i \(0.555348\pi\)
\(384\) 0 0
\(385\) −16.0000 + 13.8564i −0.815436 + 0.706188i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.0000 + 17.3205i −0.507020 + 0.878185i 0.492947 + 0.870059i \(0.335920\pi\)
−0.999967 + 0.00812520i \(0.997414\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.00000 + 3.46410i −0.100631 + 0.174298i
\(396\) 0 0
\(397\) 2.50000 + 4.33013i 0.125471 + 0.217323i 0.921917 0.387387i \(-0.126622\pi\)
−0.796446 + 0.604710i \(0.793289\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 + 15.5885i 0.449439 + 0.778450i 0.998350 0.0574304i \(-0.0182907\pi\)
−0.548911 + 0.835881i \(0.684957\pi\)
\(402\) 0 0
\(403\) 17.5000 30.3109i 0.871737 1.50989i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −14.0000 −0.693954
\(408\) 0 0
\(409\) 17.5000 30.3109i 0.865319 1.49878i −0.00141047 0.999999i \(-0.500449\pi\)
0.866730 0.498778i \(-0.166218\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.00000 15.5885i −0.147620 0.767058i
\(414\) 0 0
\(415\) 28.0000 + 48.4974i 1.37447 + 2.38064i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 33.0000 + 57.1577i 1.60074 + 2.77255i
\(426\) 0 0
\(427\) 22.5000 + 7.79423i 1.08885 + 0.377189i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.00000 8.66025i 0.240842 0.417150i −0.720113 0.693857i \(-0.755910\pi\)
0.960954 + 0.276707i \(0.0892433\pi\)
\(432\) 0 0
\(433\) 7.00000 0.336399 0.168199 0.985753i \(-0.446205\pi\)
0.168199 + 0.985753i \(0.446205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.0000 + 20.7846i −0.574038 + 0.994263i
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 + 20.7846i 0.570137 + 0.987507i 0.996551 + 0.0829786i \(0.0264433\pi\)
−0.426414 + 0.904528i \(0.640223\pi\)
\(444\) 0 0
\(445\) 24.0000 41.5692i 1.13771 1.97057i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −28.0000 −1.32140 −0.660701 0.750649i \(-0.729741\pi\)
−0.660701 + 0.750649i \(0.729741\pi\)
\(450\) 0 0
\(451\) −2.00000 + 3.46410i −0.0941763 + 0.163118i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −50.0000 17.3205i −2.34404 0.811998i
\(456\) 0 0
\(457\) 5.50000 + 9.52628i 0.257279 + 0.445621i 0.965512 0.260358i \(-0.0838407\pi\)
−0.708233 + 0.705979i \(0.750507\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.0000 + 34.6410i 0.925490 + 1.60300i 0.790772 + 0.612111i \(0.209679\pi\)
0.134718 + 0.990884i \(0.456987\pi\)
\(468\) 0 0
\(469\) −3.50000 18.1865i −0.161615 0.839776i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.00000 + 12.1244i −0.321860 + 0.557478i
\(474\) 0 0
\(475\) −44.0000 −2.01886
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.0000 + 20.7846i −0.548294 + 0.949673i 0.450098 + 0.892979i \(0.351389\pi\)
−0.998392 + 0.0566937i \(0.981944\pi\)
\(480\) 0 0
\(481\) −17.5000 30.3109i −0.797931 1.38206i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 30.0000 + 51.9615i 1.36223 + 2.35945i
\(486\) 0 0
\(487\) −4.00000 + 6.92820i −0.181257 + 0.313947i −0.942309 0.334744i \(-0.891350\pi\)
0.761052 + 0.648691i \(0.224683\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) −18.0000 + 31.1769i −0.810679 + 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 13.8564i 0.717698 0.621545i
\(498\) 0 0
\(499\) 15.5000 + 26.8468i 0.693875 + 1.20183i 0.970558 + 0.240866i \(0.0774314\pi\)
−0.276683 + 0.960961i \(0.589235\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 26.0000 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 0 0
\(505\) 64.0000 2.84796
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.0000 32.9090i −0.842160 1.45866i −0.888065 0.459718i \(-0.847950\pi\)
0.0459045 0.998946i \(-0.485383\pi\)
\(510\) 0 0
\(511\) 5.00000 + 25.9808i 0.221187 + 1.14932i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 26.0000 45.0333i 1.14570 1.98441i
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 5.19615i 0.131432 0.227648i −0.792797 0.609486i \(-0.791376\pi\)
0.924229 + 0.381839i \(0.124709\pi\)
\(522\) 0 0
\(523\) −12.5000 21.6506i −0.546587 0.946716i −0.998505 0.0546569i \(-0.982594\pi\)
0.451918 0.892059i \(-0.350740\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −21.0000 36.3731i −0.914774 1.58444i
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 0 0
\(535\) −12.0000 + 20.7846i −0.518805 + 0.898597i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −11.0000 8.66025i −0.473804 0.373024i
\(540\) 0 0
\(541\) −19.0000 32.9090i −0.816874 1.41487i −0.907975 0.419025i \(-0.862372\pi\)
0.0911008 0.995842i \(-0.470961\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −3.00000 −0.128271 −0.0641354 0.997941i \(-0.520429\pi\)
−0.0641354 + 0.997941i \(0.520429\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 20.7846i −0.511217 0.885454i
\(552\) 0 0
\(553\) −2.50000 0.866025i −0.106311 0.0368271i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.0000 17.3205i 0.423714 0.733893i −0.572586 0.819845i \(-0.694060\pi\)
0.996299 + 0.0859514i \(0.0273930\pi\)
\(558\) 0 0
\(559\) −35.0000 −1.48034
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.00000 12.1244i 0.295015 0.510981i −0.679974 0.733237i \(-0.738009\pi\)
0.974988 + 0.222256i \(0.0713421\pi\)
\(564\) 0 0
\(565\) 16.0000 + 27.7128i 0.673125 + 1.16589i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −20.0000 34.6410i −0.838444 1.45223i −0.891196 0.453619i \(-0.850133\pi\)
0.0527519 0.998608i \(-0.483201\pi\)
\(570\) 0 0
\(571\) −8.00000 + 13.8564i −0.334790 + 0.579873i −0.983444 0.181210i \(-0.941999\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −66.0000 −2.75239
\(576\) 0 0
\(577\) −3.50000 + 6.06218i −0.145707 + 0.252372i −0.929636 0.368478i \(-0.879879\pi\)
0.783930 + 0.620850i \(0.213212\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −28.0000 + 24.2487i −1.16164 + 1.00601i
\(582\) 0 0
\(583\) −6.00000 10.3923i −0.248495 0.430405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.0000 + 34.6410i 0.821302 + 1.42254i 0.904713 + 0.426021i \(0.140085\pi\)
−0.0834118 + 0.996515i \(0.526582\pi\)
\(594\) 0 0
\(595\) −48.0000 + 41.5692i −1.96781 + 1.70417i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.00000 10.3923i 0.245153 0.424618i −0.717021 0.697051i \(-0.754495\pi\)
0.962175 + 0.272433i \(0.0878284\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.0000 24.2487i 0.569181 0.985850i
\(606\) 0 0
\(607\) −4.00000 6.92820i −0.162355 0.281207i 0.773358 0.633970i \(-0.218576\pi\)
−0.935713 + 0.352763i \(0.885242\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.00000 8.66025i −0.202278 0.350356i
\(612\) 0 0
\(613\) 10.5000 18.1865i 0.424091 0.734547i −0.572244 0.820083i \(-0.693927\pi\)
0.996335 + 0.0855362i \(0.0272603\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 0 0
\(619\) 11.5000 19.9186i 0.462224 0.800595i −0.536847 0.843679i \(-0.680385\pi\)
0.999071 + 0.0430838i \(0.0137183\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 30.0000 + 10.3923i 1.20192 + 0.416359i
\(624\) 0 0
\(625\) −20.5000 35.5070i −0.820000 1.42028i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −42.0000 −1.67465
\(630\) 0 0
\(631\) −9.00000 −0.358284 −0.179142 0.983823i \(-0.557332\pi\)
−0.179142 + 0.983823i \(0.557332\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −26.0000 45.0333i −1.03178 1.78709i
\(636\) 0 0
\(637\) 5.00000 34.6410i 0.198107 1.37253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.00000 + 5.19615i −0.118493 + 0.205236i −0.919171 0.393860i \(-0.871140\pi\)
0.800678 + 0.599095i \(0.204473\pi\)
\(642\) 0 0
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.0000 24.2487i 0.550397 0.953315i −0.447849 0.894109i \(-0.647810\pi\)
0.998246 0.0592060i \(-0.0188569\pi\)
\(648\) 0 0
\(649\) −6.00000 10.3923i −0.235521 0.407934i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.00000 + 8.66025i 0.195665 + 0.338902i 0.947118 0.320884i \(-0.103980\pi\)
−0.751453 + 0.659786i \(0.770647\pi\)
\(654\) 0 0
\(655\) −8.00000 + 13.8564i −0.312586 + 0.541415i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −1.00000 + 1.73205i −0.0388955 + 0.0673690i −0.884818 0.465937i \(-0.845717\pi\)
0.845922 + 0.533306i \(0.179051\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 41.5692i −0.310227 1.61199i
\(666\) 0 0
\(667\) −18.0000 31.1769i −0.696963 1.20717i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 18.0000 0.694882
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000 + 31.1769i 0.691796 + 1.19823i 0.971249 + 0.238067i \(0.0765137\pi\)
−0.279453 + 0.960159i \(0.590153\pi\)
\(678\) 0 0
\(679\) −30.0000 + 25.9808i −1.15129 + 0.997050i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 20.7846i 0.459167 0.795301i −0.539750 0.841825i \(-0.681481\pi\)
0.998917 + 0.0465244i \(0.0148145\pi\)
\(684\) 0 0
\(685\) −24.0000 −0.916993
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.0000 25.9808i 0.571454 0.989788i
\(690\) 0 0
\(691\) −15.5000 26.8468i −0.589648 1.02130i −0.994278 0.106820i \(-0.965933\pi\)
0.404631 0.914480i \(-0.367400\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −14.0000 24.2487i −0.531050 0.919806i
\(696\) 0 0
\(697\) −6.00000 + 10.3923i −0.227266 + 0.393637i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) 0 0
\(703\) 14.0000 24.2487i 0.528020 0.914557i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.00000 + 41.5692i 0.300871 + 1.56337i
\(708\) 0 0
\(709\) −4.50000 7.79423i −0.169001 0.292718i 0.769068 0.639167i \(-0.220721\pi\)
−0.938069 + 0.346449i \(0.887387\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 42.0000 1.57291
\(714\) 0 0
\(715\) −40.0000 −1.49592
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0000 34.6410i −0.745874 1.29189i −0.949785 0.312903i \(-0.898699\pi\)
0.203911 0.978989i \(-0.434635\pi\)
\(720\) 0 0
\(721\) 32.5000 + 11.2583i 1.21036 + 0.419282i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 33.0000 57.1577i 1.22559 2.12278i
\(726\) 0 0
\(727\) 19.0000 0.704671 0.352335 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −21.0000 + 36.3731i −0.776713 + 1.34531i
\(732\) 0 0
\(733\) 15.5000 + 26.8468i 0.572506 + 0.991609i 0.996308 + 0.0858539i \(0.0273618\pi\)
−0.423802 + 0.905755i \(0.639305\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.00000 12.1244i −0.257848 0.446606i
\(738\) 0 0
\(739\) 11.5000 19.9186i 0.423034 0.732717i −0.573200 0.819415i \(-0.694298\pi\)
0.996235 + 0.0866983i \(0.0276316\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.0000 1.24734 0.623670 0.781688i \(-0.285641\pi\)
0.623670 + 0.781688i \(0.285641\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −15.0000 5.19615i −0.548088 0.189863i
\(750\) 0 0
\(751\) 16.0000 + 27.7128i 0.583848 + 1.01125i 0.995018 + 0.0996961i \(0.0317870\pi\)
−0.411170 + 0.911559i \(0.634880\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −36.0000 −1.31017
\(756\) 0 0
\(757\) −11.0000 −0.399802 −0.199901 0.979816i \(-0.564062\pi\)
−0.199901 + 0.979816i \(0.564062\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 2.50000 + 12.9904i 0.0905061 + 0.470283i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.0000 25.9808i 0.541619 0.938111i
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8.00000 + 13.8564i −0.287740 + 0.498380i −0.973270 0.229664i \(-0.926237\pi\)
0.685530 + 0.728044i \(0.259571\pi\)
\(774\) 0 0
\(775\) 38.5000 + 66.6840i 1.38296 + 2.39536i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.00000 6.92820i −0.143315 0.248229i
\(780\) 0 0
\(781\) 8.00000 13.8564i 0.286263 0.495821i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) −11.5000 + 19.9186i −0.409931 + 0.710021i −0.994882 0.101048i \(-0.967780\pi\)
0.584951 + 0.811069i \(0.301114\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −16.0000 + 13.8564i −0.568895 + 0.492677i
\(792\) 0 0
\(793\) 22.5000 + 38.9711i 0.798998 + 1.38391i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.0000 + 17.3205i 0.352892 + 0.611227i
\(804\) 0 0
\(805\) −12.0000 62.3538i −0.422944 2.19768i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.00000 + 8.66025i −0.175791 + 0.304478i −0.940435 0.339975i \(-0.889582\pi\)
0.764644 + 0.644453i \(0.222915\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −42.0000 + 72.7461i −1.47120 + 2.54819i
\(816\) 0 0
\(817\) −14.0000 24.2487i −0.489798 0.848355i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −11.0000 19.0526i −0.383903 0.664939i 0.607714 0.794156i \(-0.292087\pi\)
−0.991616 + 0.129217i \(0.958754\pi\)
\(822\) 0 0
\(823\) −7.50000 + 12.9904i −0.261434 + 0.452816i −0.966623 0.256203i \(-0.917529\pi\)
0.705190 + 0.709019i \(0.250862\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) 5.00000 8.66025i 0.173657 0.300783i −0.766039 0.642795i \(-0.777775\pi\)
0.939696 + 0.342012i \(0.111108\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −33.0000 25.9808i −1.14338 0.900180i
\(834\) 0 0
\(835\) −4.00000 6.92820i −0.138426 0.239760i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −24.0000 41.5692i −0.825625 1.43002i
\(846\) 0 0
\(847\) 17.5000 + 6.06218i 0.601307 + 0.208299i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 21.0000 36.3731i 0.719871 1.24685i
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.0000 20.7846i 0.409912 0.709989i −0.584967 0.811057i \(-0.698893\pi\)
0.994880 + 0.101068i \(0.0322260\pi\)
\(858\) 0 0
\(859\) 15.5000 + 26.8468i 0.528853 + 0.916001i 0.999434 + 0.0336436i \(0.0107111\pi\)
−0.470581 + 0.882357i \(0.655956\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −22.0000 38.1051i −0.748889 1.29711i −0.948356 0.317209i \(-0.897254\pi\)
0.199467 0.979905i \(-0.436079\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) 17.5000 30.3109i 0.592965 1.02705i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 48.0000 41.5692i 1.62270 1.40530i
\(876\) 0 0
\(877\) 10.5000 + 18.1865i 0.354560 + 0.614116i 0.987043 0.160459i \(-0.0512974\pi\)
−0.632483 + 0.774574i \(0.717964\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −56.0000 −1.88455 −0.942275 0.334840i \(-0.891318\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.00000 8.66025i −0.167884 0.290783i 0.769792 0.638295i \(-0.220360\pi\)
−0.937676 + 0.347512i \(0.887027\pi\)
\(888\) 0 0
\(889\) 26.0000 22.5167i 0.872012 0.755185i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.00000 6.92820i 0.133855 0.231843i
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −21.0000 + 36.3731i −0.700389 + 1.21311i
\(900\) 0 0
\(901\) −18.0000 31.1769i −0.599667 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.00000 + 6.92820i 0.132964 + 0.230301i
\(906\) 0 0
\(907\) −2.50000 + 4.33013i −0.0830111 + 0.143780i −0.904542 0.426385i \(-0.859787\pi\)
0.821531 + 0.570164i \(0.193120\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 42.0000 1.39152 0.695761 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(912\) 0 0
\(913\) −14.0000 + 24.2487i −0.463332 + 0.802515i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.0000 3.46410i −0.330229 0.114395i
\(918\) 0 0
\(919\) 24.5000 + 42.4352i 0.808180 + 1.39981i 0.914123 + 0.405437i \(0.132881\pi\)
−0.105942 + 0.994372i \(0.533786\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 40.0000 1.31662
\(924\) 0 0
\(925\) 77.0000 2.53174
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.00000 + 3.46410i 0.0656179 + 0.113653i 0.896968 0.442096i \(-0.145765\pi\)
−0.831350 + 0.555749i \(0.812431\pi\)
\(930\) 0 0
\(931\) 26.0000 10.3923i 0.852116 0.340594i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −24.0000 + 41.5692i −0.784884 + 1.35946i
\(936\) 0 0
\(937\) 43.0000 1.40475 0.702374 0.711808i \(-0.252123\pi\)
0.702374 + 0.711808i \(0.252123\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.00000 + 5.19615i −0.0977972 + 0.169390i −0.910773 0.412908i \(-0.864513\pi\)
0.812975 + 0.582298i \(0.197846\pi\)
\(942\) 0 0
\(943\) −6.00000 10.3923i −0.195387 0.338420i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.0000 + 19.0526i 0.357452 + 0.619125i 0.987534 0.157403i \(-0.0503122\pi\)
−0.630082 + 0.776528i \(0.716979\pi\)
\(948\) 0 0
\(949\) −25.0000 + 43.3013i −0.811534 + 1.40562i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 40.0000 69.2820i 1.29437 2.24191i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.00000 15.5885i −0.0968751 0.503378i
\(960\) 0 0
\(961\) −9.00000 15.5885i −0.290323 0.502853i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) −43.0000 −1.38279 −0.691393 0.722478i \(-0.743003\pi\)
−0.691393 + 0.722478i \(0.743003\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 10.3923i −0.192549 0.333505i 0.753545 0.657396i \(-0.228342\pi\)
−0.946094 + 0.323891i \(0.895009\pi\)
\(972\) 0 0
\(973\) 14.0000 12.1244i 0.448819 0.388689i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.00000 1.73205i 0.0319928 0.0554132i −0.849586 0.527451i \(-0.823148\pi\)
0.881579 + 0.472037i \(0.156481\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.0000 31.1769i 0.574111 0.994389i −0.422027 0.906583i \(-0.638681\pi\)
0.996138 0.0878058i \(-0.0279855\pi\)
\(984\) 0 0
\(985\) 24.0000 + 41.5692i 0.764704 + 1.32451i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −21.0000 36.3731i −0.667761 1.15660i
\(990\) 0 0
\(991\) −17.5000 + 30.3109i −0.555906 + 0.962857i 0.441927 + 0.897051i \(0.354295\pi\)
−0.997832 + 0.0658059i \(0.979038\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) 8.50000 14.7224i 0.269198 0.466264i −0.699457 0.714675i \(-0.746575\pi\)
0.968655 + 0.248410i \(0.0799082\pi\)
\(998\) 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 1512.2.s.a.865.1 2
3.2 odd 2 1512.2.s.j.865.1 yes 2
7.2 even 3 inner 1512.2.s.a.1297.1 yes 2
21.2 odd 6 1512.2.s.j.1297.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.s.a.865.1 2 1.1 even 1 trivial
1512.2.s.a.1297.1 yes 2 7.2 even 3 inner
1512.2.s.j.865.1 yes 2 3.2 odd 2
1512.2.s.j.1297.1 yes 2 21.2 odd 6