Properties

Label 1368.2.s.c.505.1
Level $1368$
Weight $2$
Character 1368.505
Analytic conductor $10.924$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1368,2,Mod(505,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, 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("1368.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.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: \(10.9235349965\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 505.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1368.505
Dual form 1368.2.s.c.577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{5} +3.00000 q^{7} -6.00000 q^{11} +(0.500000 + 0.866025i) q^{13} +(-1.00000 + 1.73205i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(0.500000 + 0.866025i) q^{25} +(-1.00000 - 1.73205i) q^{29} -1.00000 q^{31} +(-3.00000 + 5.19615i) q^{35} -7.00000 q^{37} +(0.500000 - 0.866025i) q^{43} +2.00000 q^{49} +(-2.00000 - 3.46410i) q^{53} +(6.00000 - 10.3923i) q^{55} +(-4.00000 + 6.92820i) q^{59} +(5.50000 + 9.52628i) q^{61} -2.00000 q^{65} +(-7.50000 - 12.9904i) q^{67} +(-3.00000 + 5.19615i) q^{71} +(-4.50000 + 7.79423i) q^{73} -18.0000 q^{77} +(6.50000 - 11.2583i) q^{79} -14.0000 q^{83} +(-2.00000 - 3.46410i) q^{85} +(6.00000 + 10.3923i) q^{89} +(1.50000 + 2.59808i) q^{91} +(1.00000 - 8.66025i) q^{95} +(-5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 6 q^{7} - 12 q^{11} + q^{13} - 2 q^{17} - 8 q^{19} + q^{25} - 2 q^{29} - 2 q^{31} - 6 q^{35} - 14 q^{37} + q^{43} + 4 q^{49} - 4 q^{53} + 12 q^{55} - 8 q^{59} + 11 q^{61} - 4 q^{65} - 15 q^{67}+ \cdots - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) −1.00000 + 1.73205i −0.447214 + 0.774597i −0.998203 0.0599153i \(-0.980917\pi\)
0.550990 + 0.834512i \(0.314250\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.138675 + 0.240192i 0.926995 0.375073i \(-0.122382\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) 0 0
\(19\) −4.00000 + 1.73205i −0.917663 + 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 1.73205i −0.185695 0.321634i 0.758115 0.652121i \(-0.226120\pi\)
−0.943811 + 0.330487i \(0.892787\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 + 5.19615i −0.507093 + 0.878310i
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 0.500000 0.866025i 0.0762493 0.132068i −0.825380 0.564578i \(-0.809039\pi\)
0.901629 + 0.432511i \(0.142372\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 3.46410i −0.274721 0.475831i 0.695344 0.718677i \(-0.255252\pi\)
−0.970065 + 0.242846i \(0.921919\pi\)
\(54\) 0 0
\(55\) 6.00000 10.3923i 0.809040 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 + 6.92820i −0.520756 + 0.901975i 0.478953 + 0.877841i \(0.341016\pi\)
−0.999709 + 0.0241347i \(0.992317\pi\)
\(60\) 0 0
\(61\) 5.50000 + 9.52628i 0.704203 + 1.21972i 0.966978 + 0.254858i \(0.0820288\pi\)
−0.262776 + 0.964857i \(0.584638\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −7.50000 12.9904i −0.916271 1.58703i −0.805030 0.593234i \(-0.797851\pi\)
−0.111241 0.993793i \(-0.535483\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 + 5.19615i −0.356034 + 0.616670i −0.987294 0.158901i \(-0.949205\pi\)
0.631260 + 0.775571i \(0.282538\pi\)
\(72\) 0 0
\(73\) −4.50000 + 7.79423i −0.526685 + 0.912245i 0.472831 + 0.881153i \(0.343232\pi\)
−0.999517 + 0.0310925i \(0.990101\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −18.0000 −2.05129
\(78\) 0 0
\(79\) 6.50000 11.2583i 0.731307 1.26666i −0.225018 0.974355i \(-0.572244\pi\)
0.956325 0.292306i \(-0.0944227\pi\)
\(80\) 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
\(85\) −2.00000 3.46410i −0.216930 0.375735i
\(86\) 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
\(91\) 1.50000 + 2.59808i 0.157243 + 0.272352i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 8.66025i 0.102598 0.888523i
\(96\) 0 0
\(97\) −5.00000 + 8.66025i −0.507673 + 0.879316i 0.492287 + 0.870433i \(0.336161\pi\)
−0.999961 + 0.00888289i \(0.997172\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 + 5.19615i 0.298511 + 0.517036i 0.975796 0.218685i \(-0.0701767\pi\)
−0.677284 + 0.735721i \(0.736843\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) −9.00000 + 15.5885i −0.862044 + 1.49310i 0.00790932 + 0.999969i \(0.497482\pi\)
−0.869953 + 0.493135i \(0.835851\pi\)
\(110\) 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
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 + 5.19615i −0.275010 + 0.476331i
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.00000 15.5885i 0.786334 1.36197i −0.141865 0.989886i \(-0.545310\pi\)
0.928199 0.372084i \(-0.121357\pi\)
\(132\) 0 0
\(133\) −12.0000 + 5.19615i −1.04053 + 0.450564i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 + 15.5885i 0.768922 + 1.33181i 0.938148 + 0.346235i \(0.112540\pi\)
−0.169226 + 0.985577i \(0.554127\pi\)
\(138\) 0 0
\(139\) −2.50000 4.33013i −0.212047 0.367277i 0.740308 0.672268i \(-0.234680\pi\)
−0.952355 + 0.304991i \(0.901346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.00000 5.19615i −0.250873 0.434524i
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.00000 + 1.73205i −0.0819232 + 0.141895i −0.904076 0.427372i \(-0.859440\pi\)
0.822153 + 0.569267i \(0.192773\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.00000 1.73205i 0.0803219 0.139122i
\(156\) 0 0
\(157\) −2.50000 + 4.33013i −0.199522 + 0.345582i −0.948373 0.317156i \(-0.897272\pi\)
0.748852 + 0.662738i \(0.230606\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.00000 + 6.92820i 0.309529 + 0.536120i 0.978259 0.207385i \(-0.0664952\pi\)
−0.668730 + 0.743505i \(0.733162\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.00000 12.1244i 0.532200 0.921798i −0.467093 0.884208i \(-0.654699\pi\)
0.999293 0.0375896i \(-0.0119679\pi\)
\(174\) 0 0
\(175\) 1.50000 + 2.59808i 0.113389 + 0.196396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 5.00000 + 8.66025i 0.371647 + 0.643712i 0.989819 0.142331i \(-0.0454598\pi\)
−0.618172 + 0.786043i \(0.712126\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.00000 12.1244i 0.514650 0.891400i
\(186\) 0 0
\(187\) 6.00000 10.3923i 0.438763 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) 4.50000 7.79423i 0.323917 0.561041i −0.657376 0.753563i \(-0.728333\pi\)
0.981293 + 0.192522i \(0.0616668\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −1.50000 2.59808i −0.106332 0.184173i 0.807950 0.589252i \(-0.200577\pi\)
−0.914282 + 0.405079i \(0.867244\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 5.19615i −0.210559 0.364698i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.0000 10.3923i 1.66011 0.718851i
\(210\) 0 0
\(211\) −4.50000 + 7.79423i −0.309793 + 0.536577i −0.978317 0.207114i \(-0.933593\pi\)
0.668524 + 0.743690i \(0.266926\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00000 + 1.73205i 0.0681994 + 0.118125i
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 2.50000 4.33013i 0.167412 0.289967i −0.770097 0.637927i \(-0.779792\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) −17.0000 −1.12339 −0.561696 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.00000 8.66025i 0.327561 0.567352i −0.654466 0.756091i \(-0.727107\pi\)
0.982027 + 0.188739i \(0.0604400\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) −4.50000 7.79423i −0.289870 0.502070i 0.683908 0.729568i \(-0.260279\pi\)
−0.973779 + 0.227498i \(0.926946\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.00000 + 3.46410i −0.127775 + 0.221313i
\(246\) 0 0
\(247\) −3.50000 2.59808i −0.222700 0.165312i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.00000 + 13.8564i 0.504956 + 0.874609i 0.999984 + 0.00573163i \(0.00182444\pi\)
−0.495028 + 0.868877i \(0.664842\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 10.3923i −0.374270 0.648254i 0.615948 0.787787i \(-0.288773\pi\)
−0.990217 + 0.139533i \(0.955440\pi\)
\(258\) 0 0
\(259\) −21.0000 −1.30488
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5.00000 + 8.66025i −0.308313 + 0.534014i −0.977993 0.208635i \(-0.933098\pi\)
0.669680 + 0.742650i \(0.266431\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 16.0000 27.7128i 0.975537 1.68968i 0.297386 0.954757i \(-0.403885\pi\)
0.678151 0.734923i \(-0.262782\pi\)
\(270\) 0 0
\(271\) −8.00000 + 13.8564i −0.485965 + 0.841717i −0.999870 0.0161307i \(-0.994865\pi\)
0.513905 + 0.857847i \(0.328199\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.00000 5.19615i −0.180907 0.313340i
\(276\) 0 0
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.00000 + 8.66025i 0.298275 + 0.516627i 0.975741 0.218926i \(-0.0702554\pi\)
−0.677466 + 0.735554i \(0.736922\pi\)
\(282\) 0 0
\(283\) −2.00000 + 3.46410i −0.118888 + 0.205919i −0.919327 0.393494i \(-0.871266\pi\)
0.800439 + 0.599414i \(0.204600\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.0000 1.16841 0.584206 0.811605i \(-0.301406\pi\)
0.584206 + 0.811605i \(0.301406\pi\)
\(294\) 0 0
\(295\) −8.00000 13.8564i −0.465778 0.806751i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.50000 2.59808i 0.0864586 0.149751i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −22.0000 −1.25972
\(306\) 0 0
\(307\) 10.0000 17.3205i 0.570730 0.988534i −0.425761 0.904836i \(-0.639994\pi\)
0.996491 0.0836980i \(-0.0266731\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) 5.00000 + 8.66025i 0.282617 + 0.489506i 0.972028 0.234863i \(-0.0754642\pi\)
−0.689412 + 0.724370i \(0.742131\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.0000 + 25.9808i 0.842484 + 1.45922i 0.887788 + 0.460252i \(0.152241\pi\)
−0.0453045 + 0.998973i \(0.514426\pi\)
\(318\) 0 0
\(319\) 6.00000 + 10.3923i 0.335936 + 0.581857i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.00000 8.66025i 0.0556415 0.481869i
\(324\) 0 0
\(325\) −0.500000 + 0.866025i −0.0277350 + 0.0480384i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 30.0000 1.63908
\(336\) 0 0
\(337\) 7.50000 12.9904i 0.408551 0.707631i −0.586177 0.810183i \(-0.699368\pi\)
0.994728 + 0.102552i \(0.0327009\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.00000 15.5885i 0.483145 0.836832i −0.516667 0.856186i \(-0.672828\pi\)
0.999813 + 0.0193540i \(0.00616095\pi\)
\(348\) 0 0
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −6.00000 10.3923i −0.318447 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.0000 17.3205i 0.527780 0.914141i −0.471696 0.881761i \(-0.656358\pi\)
0.999476 0.0323801i \(-0.0103087\pi\)
\(360\) 0 0
\(361\) 13.0000 13.8564i 0.684211 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.00000 15.5885i −0.471082 0.815937i
\(366\) 0 0
\(367\) 17.5000 + 30.3109i 0.913493 + 1.58222i 0.809093 + 0.587680i \(0.199959\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 10.3923i −0.311504 0.539542i
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.00000 1.73205i 0.0515026 0.0892052i
\(378\) 0 0
\(379\) 27.0000 1.38690 0.693448 0.720506i \(-0.256091\pi\)
0.693448 + 0.720506i \(0.256091\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.00000 12.1244i 0.357683 0.619526i −0.629890 0.776684i \(-0.716900\pi\)
0.987573 + 0.157159i \(0.0502334\pi\)
\(384\) 0 0
\(385\) 18.0000 31.1769i 0.917365 1.58892i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.0000 + 27.7128i 0.811232 + 1.40510i 0.912002 + 0.410186i \(0.134536\pi\)
−0.100770 + 0.994910i \(0.532131\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.0000 + 22.5167i 0.654101 + 1.13294i
\(396\) 0 0
\(397\) −12.5000 + 21.6506i −0.627357 + 1.08661i 0.360723 + 0.932673i \(0.382530\pi\)
−0.988080 + 0.153941i \(0.950803\pi\)
\(398\) 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
\(403\) −0.500000 0.866025i −0.0249068 0.0431398i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 42.0000 2.08186
\(408\) 0 0
\(409\) 19.0000 + 32.9090i 0.939490 + 1.62724i 0.766426 + 0.642333i \(0.222033\pi\)
0.173064 + 0.984911i \(0.444633\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −12.0000 + 20.7846i −0.590481 + 1.02274i
\(414\) 0 0
\(415\) 14.0000 24.2487i 0.687233 1.19032i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −13.0000 + 22.5167i −0.633581 + 1.09739i 0.353233 + 0.935536i \(0.385082\pi\)
−0.986814 + 0.161859i \(0.948251\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 16.5000 + 28.5788i 0.798491 + 1.38303i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −18.0000 31.1769i −0.867029 1.50174i −0.865018 0.501741i \(-0.832693\pi\)
−0.00201168 0.999998i \(-0.500640\pi\)
\(432\) 0 0
\(433\) −9.50000 16.4545i −0.456541 0.790752i 0.542234 0.840227i \(-0.317578\pi\)
−0.998775 + 0.0494752i \(0.984245\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −6.50000 + 11.2583i −0.310228 + 0.537331i −0.978412 0.206666i \(-0.933739\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.00000 8.66025i −0.237557 0.411461i 0.722456 0.691417i \(-0.243013\pi\)
−0.960013 + 0.279956i \(0.909680\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) 25.0000 1.16945 0.584725 0.811231i \(-0.301202\pi\)
0.584725 + 0.811231i \(0.301202\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 31.1769i 0.838344 1.45205i −0.0529352 0.998598i \(-0.516858\pi\)
0.891279 0.453456i \(-0.149809\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −22.5000 38.9711i −1.03895 1.79952i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.00000 + 5.19615i −0.137940 + 0.238919i
\(474\) 0 0
\(475\) −3.50000 2.59808i −0.160591 0.119208i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.0000 + 19.0526i 0.502603 + 0.870534i 0.999995 + 0.00300810i \(0.000957509\pi\)
−0.497393 + 0.867526i \(0.665709\pi\)
\(480\) 0 0
\(481\) −3.50000 6.06218i −0.159586 0.276412i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0000 17.3205i −0.454077 0.786484i
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.0000 + 29.4449i −0.767199 + 1.32883i 0.171877 + 0.985118i \(0.445017\pi\)
−0.939076 + 0.343710i \(0.888316\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.00000 + 15.5885i −0.403705 + 0.699238i
\(498\) 0 0
\(499\) −12.5000 + 21.6506i −0.559577 + 0.969216i 0.437955 + 0.898997i \(0.355703\pi\)
−0.997532 + 0.0702185i \(0.977630\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15.0000 + 25.9808i 0.668817 + 1.15842i 0.978235 + 0.207499i \(0.0665323\pi\)
−0.309418 + 0.950926i \(0.600134\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.0000 + 17.3205i 0.443242 + 0.767718i 0.997928 0.0643419i \(-0.0204948\pi\)
−0.554686 + 0.832060i \(0.687161\pi\)
\(510\) 0 0
\(511\) −13.5000 + 23.3827i −0.597205 + 1.03439i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.00000 8.66025i 0.220326 0.381616i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 15.5000 + 26.8468i 0.677768 + 1.17393i 0.975652 + 0.219326i \(0.0703858\pi\)
−0.297884 + 0.954602i \(0.596281\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.00000 1.73205i 0.0435607 0.0754493i
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −10.0000 + 17.3205i −0.432338 + 0.748831i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −15.5000 26.8468i −0.666397 1.15423i −0.978905 0.204318i \(-0.934502\pi\)
0.312507 0.949915i \(-0.398831\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −18.0000 31.1769i −0.771035 1.33547i
\(546\) 0 0
\(547\) 2.50000 + 4.33013i 0.106892 + 0.185143i 0.914510 0.404564i \(-0.132577\pi\)
−0.807617 + 0.589707i \(0.799243\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.00000 + 5.19615i 0.298210 + 0.221364i
\(552\) 0 0
\(553\) 19.5000 33.7750i 0.829224 1.43626i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.00000 + 8.66025i 0.211857 + 0.366947i 0.952296 0.305177i \(-0.0987156\pi\)
−0.740439 + 0.672124i \(0.765382\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −46.0000 −1.93867 −0.969334 0.245745i \(-0.920967\pi\)
−0.969334 + 0.245745i \(0.920967\pi\)
\(564\) 0 0
\(565\) 16.0000 27.7128i 0.673125 1.16589i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) −19.0000 −0.795125 −0.397563 0.917575i \(-0.630144\pi\)
−0.397563 + 0.917575i \(0.630144\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −42.0000 −1.74245
\(582\) 0 0
\(583\) 12.0000 + 20.7846i 0.496989 + 0.860811i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.0000 + 24.2487i −0.577842 + 1.00085i 0.417885 + 0.908500i \(0.362772\pi\)
−0.995726 + 0.0923513i \(0.970562\pi\)
\(588\) 0 0
\(589\) 4.00000 1.73205i 0.164817 0.0713679i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.0000 + 31.1769i 0.739171 + 1.28028i 0.952869 + 0.303383i \(0.0981160\pi\)
−0.213697 + 0.976900i \(0.568551\pi\)
\(594\) 0 0
\(595\) −6.00000 10.3923i −0.245976 0.426043i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −7.00000 12.1244i −0.286012 0.495388i 0.686842 0.726807i \(-0.258996\pi\)
−0.972854 + 0.231419i \(0.925663\pi\)
\(600\) 0 0
\(601\) −1.00000 −0.0407909 −0.0203954 0.999792i \(-0.506493\pi\)
−0.0203954 + 0.999792i \(0.506493\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −25.0000 + 43.3013i −1.01639 + 1.76045i
\(606\) 0 0
\(607\) 27.0000 1.09590 0.547948 0.836512i \(-0.315409\pi\)
0.547948 + 0.836512i \(0.315409\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −7.00000 + 12.1244i −0.282727 + 0.489698i −0.972056 0.234751i \(-0.924572\pi\)
0.689328 + 0.724449i \(0.257906\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.0000 25.9808i −0.603877 1.04595i −0.992228 0.124434i \(-0.960288\pi\)
0.388351 0.921512i \(-0.373045\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 18.0000 + 31.1769i 0.721155 + 1.24908i
\(624\) 0 0
\(625\) 9.50000 16.4545i 0.380000 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.00000 12.1244i 0.279108 0.483430i
\(630\) 0 0
\(631\) −15.5000 26.8468i −0.617045 1.06875i −0.990022 0.140913i \(-0.954996\pi\)
0.372977 0.927841i \(-0.378337\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.00000 + 1.73205i 0.0396214 + 0.0686264i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12.0000 + 20.7846i −0.473972 + 0.820943i −0.999556 0.0297987i \(-0.990513\pi\)
0.525584 + 0.850741i \(0.323847\pi\)
\(642\) 0 0
\(643\) −2.50000 + 4.33013i −0.0985904 + 0.170764i −0.911101 0.412182i \(-0.864767\pi\)
0.812511 + 0.582946i \(0.198100\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) 0 0
\(649\) 24.0000 41.5692i 0.942082 1.63173i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 18.0000 + 31.1769i 0.703318 + 1.21818i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.00000 + 5.19615i 0.116863 + 0.202413i 0.918523 0.395367i \(-0.129383\pi\)
−0.801660 + 0.597781i \(0.796049\pi\)
\(660\) 0 0
\(661\) −7.00000 12.1244i −0.272268 0.471583i 0.697174 0.716902i \(-0.254441\pi\)
−0.969442 + 0.245319i \(0.921107\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.00000 25.9808i 0.116335 1.00749i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −33.0000 57.1577i −1.27395 2.20655i
\(672\) 0 0
\(673\) 15.0000 0.578208 0.289104 0.957298i \(-0.406643\pi\)
0.289104 + 0.957298i \(0.406643\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.00000 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 0 0
\(679\) −15.0000 + 25.9808i −0.575647 + 0.997050i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −10.0000 −0.382639 −0.191320 0.981528i \(-0.561277\pi\)
−0.191320 + 0.981528i \(0.561277\pi\)
\(684\) 0 0
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.00000 3.46410i 0.0761939 0.131972i
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −15.0000 + 25.9808i −0.566542 + 0.981280i 0.430362 + 0.902656i \(0.358386\pi\)
−0.996904 + 0.0786236i \(0.974947\pi\)
\(702\) 0 0
\(703\) 28.0000 12.1244i 1.05604 0.457279i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.00000 + 15.5885i 0.338480 + 0.586264i
\(708\) 0 0
\(709\) 16.5000 + 28.5788i 0.619671 + 1.07330i 0.989546 + 0.144219i \(0.0460671\pi\)
−0.369875 + 0.929081i \(0.620600\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.00000 + 6.92820i −0.149175 + 0.258378i −0.930923 0.365216i \(-0.880995\pi\)
0.781748 + 0.623595i \(0.214328\pi\)
\(720\) 0 0
\(721\) −15.0000 −0.558629
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.00000 1.73205i 0.0371391 0.0643268i
\(726\) 0 0
\(727\) 8.50000 14.7224i 0.315248 0.546025i −0.664243 0.747517i \(-0.731246\pi\)
0.979490 + 0.201492i \(0.0645791\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.00000 + 1.73205i 0.0369863 + 0.0640622i
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 45.0000 + 77.9423i 1.65760 + 2.87104i
\(738\) 0 0
\(739\) −3.50000 + 6.06218i −0.128750 + 0.223001i −0.923192 0.384338i \(-0.874430\pi\)
0.794443 + 0.607339i \(0.207763\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.00000 + 5.19615i −0.110059 + 0.190628i −0.915794 0.401648i \(-0.868437\pi\)
0.805735 + 0.592277i \(0.201771\pi\)
\(744\) 0 0
\(745\) −2.00000 3.46410i −0.0732743 0.126915i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.0000 1.09618
\(750\) 0 0
\(751\) −5.50000 9.52628i −0.200698 0.347619i 0.748056 0.663636i \(-0.230988\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.00000 6.92820i 0.145575 0.252143i
\(756\) 0 0
\(757\) −23.5000 + 40.7032i −0.854122 + 1.47938i 0.0233351 + 0.999728i \(0.492572\pi\)
−0.877457 + 0.479655i \(0.840762\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −40.0000 −1.45000 −0.724999 0.688749i \(-0.758160\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 0 0
\(763\) −27.0000 + 46.7654i −0.977466 + 1.69302i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 6.50000 + 11.2583i 0.234396 + 0.405986i 0.959097 0.283078i \(-0.0913554\pi\)
−0.724701 + 0.689063i \(0.758022\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.0000 34.6410i −0.719350 1.24595i −0.961258 0.275651i \(-0.911106\pi\)
0.241908 0.970299i \(-0.422227\pi\)
\(774\) 0 0
\(775\) −0.500000 0.866025i −0.0179605 0.0311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 18.0000 31.1769i 0.644091 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5.00000 8.66025i −0.178458 0.309098i
\(786\) 0 0
\(787\) 5.00000 0.178231 0.0891154 0.996021i \(-0.471596\pi\)
0.0891154 + 0.996021i \(0.471596\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) −5.50000 + 9.52628i −0.195311 + 0.338288i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 27.0000 46.7654i 0.952809 1.65031i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 18.0000 + 31.1769i 0.632065 + 1.09477i 0.987129 + 0.159927i \(0.0511260\pi\)
−0.355063 + 0.934842i \(0.615541\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −19.0000 + 32.9090i −0.665541 + 1.15275i
\(816\) 0 0
\(817\) −0.500000 + 4.33013i −0.0174928 + 0.151492i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23.0000 39.8372i −0.802706 1.39033i −0.917829 0.396976i \(-0.870060\pi\)
0.115124 0.993351i \(-0.463274\pi\)
\(822\) 0 0
\(823\) −20.0000 34.6410i −0.697156 1.20751i −0.969448 0.245295i \(-0.921115\pi\)
0.272292 0.962215i \(-0.412218\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.00000 + 8.66025i 0.173867 + 0.301147i 0.939769 0.341811i \(-0.111040\pi\)
−0.765902 + 0.642958i \(0.777707\pi\)
\(828\) 0 0
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.00000 + 3.46410i −0.0692959 + 0.120024i
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −15.0000 + 25.9808i −0.517858 + 0.896956i 0.481927 + 0.876211i \(0.339937\pi\)
−0.999785 + 0.0207443i \(0.993396\pi\)
\(840\) 0 0
\(841\) 12.5000 21.6506i 0.431034 0.746574i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.0000 + 20.7846i 0.412813 + 0.715012i
\(846\) 0 0
\(847\) 75.0000 2.57703
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −4.50000 + 7.79423i −0.154077 + 0.266869i −0.932723 0.360595i \(-0.882574\pi\)
0.778646 + 0.627464i \(0.215907\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.00000 6.92820i 0.136637 0.236663i −0.789584 0.613642i \(-0.789704\pi\)
0.926222 + 0.376979i \(0.123037\pi\)
\(858\) 0 0
\(859\) 6.50000 + 11.2583i 0.221777 + 0.384129i 0.955348 0.295484i \(-0.0954809\pi\)
−0.733571 + 0.679613i \(0.762148\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 14.0000 + 24.2487i 0.476014 + 0.824481i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −39.0000 + 67.5500i −1.32298 + 2.29148i
\(870\) 0 0
\(871\) 7.50000 12.9904i 0.254128 0.440162i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −36.0000 −1.21702
\(876\) 0 0
\(877\) −12.5000 + 21.6506i −0.422095 + 0.731090i −0.996144 0.0877308i \(-0.972038\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) −17.5000 30.3109i −0.588922 1.02004i −0.994374 0.105926i \(-0.966219\pi\)
0.405452 0.914116i \(-0.367114\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.0000 17.3205i −0.335767 0.581566i 0.647865 0.761755i \(-0.275662\pi\)
−0.983632 + 0.180190i \(0.942329\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −12.0000 + 20.7846i −0.401116 + 0.694753i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.00000 + 1.73205i 0.0333519 + 0.0577671i
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) 10.0000 17.3205i 0.332045 0.575118i −0.650868 0.759191i \(-0.725595\pi\)
0.982913 + 0.184073i \(0.0589282\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) 84.0000 2.77999
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 27.0000 46.7654i 0.891619 1.54433i
\(918\) 0 0
\(919\) 25.0000 0.824674 0.412337 0.911031i \(-0.364713\pi\)
0.412337 + 0.911031i \(0.364713\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) −3.50000 6.06218i −0.115079 0.199323i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21.0000 36.3731i 0.688988 1.19336i −0.283178 0.959067i \(-0.591389\pi\)
0.972166 0.234294i \(-0.0752779\pi\)
\(930\) 0 0
\(931\) −8.00000 + 3.46410i −0.262189 + 0.113531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12.0000 + 20.7846i 0.392442 + 0.679729i
\(936\) 0 0
\(937\) −11.5000 19.9186i −0.375689 0.650712i 0.614741 0.788729i \(-0.289260\pi\)
−0.990430 + 0.138017i \(0.955927\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −28.0000 48.4974i −0.912774 1.58097i −0.810128 0.586253i \(-0.800603\pi\)
−0.102646 0.994718i \(-0.532731\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25.0000 + 43.3013i −0.812391 + 1.40710i 0.0987955 + 0.995108i \(0.468501\pi\)
−0.911186 + 0.411994i \(0.864832\pi\)
\(948\) 0 0
\(949\) −9.00000 −0.292152
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.00000 + 8.66025i −0.161966 + 0.280533i −0.935574 0.353132i \(-0.885117\pi\)
0.773608 + 0.633665i \(0.218450\pi\)
\(954\) 0 0
\(955\) −6.00000 + 10.3923i −0.194155 + 0.336287i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 27.0000 + 46.7654i 0.871875 + 1.51013i
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.00000 + 15.5885i 0.289720 + 0.501810i
\(966\) 0 0
\(967\) 7.50000 12.9904i 0.241184 0.417742i −0.719868 0.694111i \(-0.755798\pi\)
0.961052 + 0.276368i \(0.0891310\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 14.0000 24.2487i 0.449281 0.778178i −0.549058 0.835784i \(-0.685013\pi\)
0.998339 + 0.0576061i \(0.0183467\pi\)
\(972\) 0 0
\(973\) −7.50000 12.9904i −0.240439 0.416452i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.0000 0.895799 0.447900 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) 0 0
\(979\) −36.0000 62.3538i −1.15056 1.99284i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7.00000 + 12.1244i −0.223265 + 0.386707i −0.955798 0.294025i \(-0.905005\pi\)
0.732532 + 0.680732i \(0.238338\pi\)
\(984\) 0 0
\(985\) −12.0000 + 20.7846i −0.382352 + 0.662253i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −2.50000 + 4.33013i −0.0794151 + 0.137551i −0.902998 0.429645i \(-0.858639\pi\)
0.823583 + 0.567196i \(0.191972\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.00000 0.190213
\(996\) 0 0
\(997\) −12.5000 21.6506i −0.395879 0.685682i 0.597334 0.801993i \(-0.296227\pi\)
−0.993213 + 0.116310i \(0.962893\pi\)
\(998\) 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 1368.2.s.c.505.1 2
3.2 odd 2 1368.2.s.f.505.1 yes 2
4.3 odd 2 2736.2.s.e.1873.1 2
12.11 even 2 2736.2.s.n.1873.1 2
19.7 even 3 inner 1368.2.s.c.577.1 yes 2
57.26 odd 6 1368.2.s.f.577.1 yes 2
76.7 odd 6 2736.2.s.e.577.1 2
228.83 even 6 2736.2.s.n.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.s.c.505.1 2 1.1 even 1 trivial
1368.2.s.c.577.1 yes 2 19.7 even 3 inner
1368.2.s.f.505.1 yes 2 3.2 odd 2
1368.2.s.f.577.1 yes 2 57.26 odd 6
2736.2.s.e.577.1 2 76.7 odd 6
2736.2.s.e.1873.1 2 4.3 odd 2
2736.2.s.n.577.1 2 228.83 even 6
2736.2.s.n.1873.1 2 12.11 even 2