Properties

Label 980.2.q.e.569.1
Level $980$
Weight $2$
Character 980.569
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(569,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.569");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.q (of order \(6\), 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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 569.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 980.569
Dual form 980.2.q.e.949.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.23205 - 1.86603i) q^{5} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(-1.23205 - 1.86603i) q^{5} +(-1.50000 + 2.59808i) q^{9} -4.00000i q^{13} +(3.46410 - 2.00000i) q^{17} +(-2.00000 + 3.46410i) q^{19} +(-6.92820 - 4.00000i) q^{23} +(-1.96410 + 4.59808i) q^{25} -2.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(-6.92820 - 4.00000i) q^{37} -6.00000 q^{41} +8.00000i q^{43} +(6.69615 - 0.401924i) q^{45} +(-6.92820 - 4.00000i) q^{47} +(2.00000 + 3.46410i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(-7.46410 + 4.92820i) q^{65} +(-6.92820 + 4.00000i) q^{67} +12.0000 q^{71} +(3.46410 - 2.00000i) q^{73} +(-2.00000 + 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-8.00000 - 4.00000i) q^{85} +(5.00000 - 8.66025i) q^{89} +(8.92820 - 0.535898i) q^{95} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 6 q^{9} - 8 q^{19} + 6 q^{25} - 8 q^{29} - 16 q^{31} - 24 q^{41} + 6 q^{45} + 8 q^{59} - 12 q^{61} - 16 q^{65} + 48 q^{71} - 8 q^{79} - 18 q^{81} - 32 q^{85} + 20 q^{89} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\)

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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) −1.23205 1.86603i −0.550990 0.834512i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.50000 + 2.59808i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 2.00000i 0.840168 0.485071i −0.0171533 0.999853i \(-0.505460\pi\)
0.857321 + 0.514782i \(0.172127\pi\)
\(18\) 0 0
\(19\) −2.00000 + 3.46410i −0.458831 + 0.794719i −0.998899 0.0469020i \(-0.985065\pi\)
0.540068 + 0.841621i \(0.318398\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 4.00000i −1.44463 0.834058i −0.446476 0.894795i \(-0.647321\pi\)
−0.998154 + 0.0607377i \(0.980655\pi\)
\(24\) 0 0
\(25\) −1.96410 + 4.59808i −0.392820 + 0.919615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 6.92820i −0.718421 1.24434i −0.961625 0.274367i \(-0.911532\pi\)
0.243204 0.969975i \(-0.421802\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.92820 4.00000i −1.13899 0.657596i −0.192809 0.981236i \(-0.561760\pi\)
−0.946180 + 0.323640i \(0.895093\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 6.69615 0.401924i 0.998203 0.0599153i
\(46\) 0 0
\(47\) −6.92820 4.00000i −1.01058 0.583460i −0.0992202 0.995066i \(-0.531635\pi\)
−0.911362 + 0.411606i \(0.864968\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 + 3.46410i 0.260378 + 0.450988i 0.966342 0.257260i \(-0.0828195\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(60\) 0 0
\(61\) −3.00000 + 5.19615i −0.384111 + 0.665299i −0.991645 0.128994i \(-0.958825\pi\)
0.607535 + 0.794293i \(0.292159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.46410 + 4.92820i −0.925808 + 0.611268i
\(66\) 0 0
\(67\) −6.92820 + 4.00000i −0.846415 + 0.488678i −0.859440 0.511237i \(-0.829187\pi\)
0.0130248 + 0.999915i \(0.495854\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 3.46410 2.00000i 0.405442 0.234082i −0.283387 0.959006i \(-0.591458\pi\)
0.688830 + 0.724923i \(0.258125\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −8.00000 4.00000i −0.867722 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.00000 8.66025i 0.529999 0.917985i −0.469389 0.882992i \(-0.655526\pi\)
0.999388 0.0349934i \(-0.0111410\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.92820 0.535898i 0.916014 0.0549820i
\(96\) 0 0
\(97\) 12.0000i 1.21842i −0.793011 0.609208i \(-0.791488\pi\)
0.793011 0.609208i \(-0.208512\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.00000 15.5885i −0.895533 1.55111i −0.833143 0.553058i \(-0.813461\pi\)
−0.0623905 0.998052i \(-0.519872\pi\)
\(102\) 0 0
\(103\) −6.92820 4.00000i −0.682656 0.394132i 0.118199 0.992990i \(-0.462288\pi\)
−0.800855 + 0.598858i \(0.795621\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.92820 + 4.00000i 0.669775 + 0.386695i 0.795991 0.605308i \(-0.206950\pi\)
−0.126217 + 0.992003i \(0.540283\pi\)
\(108\) 0 0
\(109\) 7.00000 + 12.1244i 0.670478 + 1.16130i 0.977769 + 0.209687i \(0.0672444\pi\)
−0.307290 + 0.951616i \(0.599422\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.0000i 1.50515i −0.658505 0.752577i \(-0.728811\pi\)
0.658505 0.752577i \(-0.271189\pi\)
\(114\) 0 0
\(115\) 1.07180 + 17.8564i 0.0999456 + 1.66512i
\(116\) 0 0
\(117\) 10.3923 + 6.00000i 0.960769 + 0.554700i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.00000 + 3.46410i −0.174741 + 0.302660i −0.940072 0.340977i \(-0.889242\pi\)
0.765331 + 0.643637i \(0.222575\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.92820 4.00000i 0.591916 0.341743i −0.173939 0.984757i \(-0.555649\pi\)
0.765855 + 0.643013i \(0.222316\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.46410 + 3.73205i 0.204633 + 0.309930i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 5.19615i 0.245770 0.425685i −0.716578 0.697507i \(-0.754293\pi\)
0.962348 + 0.271821i \(0.0876260\pi\)
\(150\) 0 0
\(151\) −6.00000 10.3923i −0.488273 0.845714i 0.511636 0.859202i \(-0.329040\pi\)
−0.999909 + 0.0134886i \(0.995706\pi\)
\(152\) 0 0
\(153\) 12.0000i 0.970143i
\(154\) 0 0
\(155\) −8.00000 + 16.0000i −0.642575 + 1.28515i
\(156\) 0 0
\(157\) −10.3923 + 6.00000i −0.829396 + 0.478852i −0.853646 0.520854i \(-0.825614\pi\)
0.0242497 + 0.999706i \(0.492280\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.7846 + 12.0000i 1.62798 + 0.939913i 0.984696 + 0.174282i \(0.0557604\pi\)
0.643280 + 0.765631i \(0.277573\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.0000i 1.85718i 0.371113 + 0.928588i \(0.378976\pi\)
−0.371113 + 0.928588i \(0.621024\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −6.00000 10.3923i −0.458831 0.794719i
\(172\) 0 0
\(173\) 10.3923 + 6.00000i 0.790112 + 0.456172i 0.840002 0.542583i \(-0.182554\pi\)
−0.0498898 + 0.998755i \(0.515887\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.00000 6.92820i −0.298974 0.517838i 0.676927 0.736050i \(-0.263311\pi\)
−0.975901 + 0.218212i \(0.929978\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.07180 + 17.8564i 0.0788001 + 1.31283i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 17.3205i 0.723575 1.25327i −0.235983 0.971757i \(-0.575831\pi\)
0.959558 0.281511i \(-0.0908356\pi\)
\(192\) 0 0
\(193\) −13.8564 + 8.00000i −0.997406 + 0.575853i −0.907480 0.420096i \(-0.861996\pi\)
−0.0899262 + 0.995948i \(0.528663\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.0000i 1.13995i −0.821661 0.569976i \(-0.806952\pi\)
0.821661 0.569976i \(-0.193048\pi\)
\(198\) 0 0
\(199\) 12.0000 + 20.7846i 0.850657 + 1.47338i 0.880616 + 0.473831i \(0.157129\pi\)
−0.0299585 + 0.999551i \(0.509538\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 7.39230 + 11.1962i 0.516301 + 0.781973i
\(206\) 0 0
\(207\) 20.7846 12.0000i 1.44463 0.834058i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14.9282 9.85641i 1.01810 0.672201i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.00000 13.8564i −0.538138 0.932083i
\(222\) 0 0
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) −9.00000 12.0000i −0.600000 0.800000i
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) 7.00000 12.1244i 0.462573 0.801200i −0.536515 0.843891i \(-0.680260\pi\)
0.999088 + 0.0426906i \(0.0135930\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.92820 + 4.00000i 0.453882 + 0.262049i 0.709468 0.704737i \(-0.248935\pi\)
−0.255586 + 0.966786i \(0.582269\pi\)
\(234\) 0 0
\(235\) 1.07180 + 17.8564i 0.0699163 + 1.16482i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −9.00000 15.5885i −0.579741 1.00414i −0.995509 0.0946700i \(-0.969820\pi\)
0.415768 0.909471i \(-0.363513\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.8564 + 8.00000i 0.881662 + 0.509028i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.3923 + 6.00000i 0.648254 + 0.374270i 0.787787 0.615948i \(-0.211227\pi\)
−0.139533 + 0.990217i \(0.544560\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 5.19615i 0.185695 0.321634i
\(262\) 0 0
\(263\) 20.7846 12.0000i 1.28163 0.739952i 0.304487 0.952517i \(-0.401515\pi\)
0.977147 + 0.212565i \(0.0681817\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.00000 + 8.66025i 0.304855 + 0.528025i 0.977229 0.212187i \(-0.0680585\pi\)
−0.672374 + 0.740212i \(0.734725\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.8564 + 8.00000i −0.832551 + 0.480673i −0.854725 0.519081i \(-0.826274\pi\)
0.0221745 + 0.999754i \(0.492941\pi\)
\(278\) 0 0
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −27.7128 + 16.0000i −1.64736 + 0.951101i −0.669238 + 0.743048i \(0.733379\pi\)
−0.978117 + 0.208053i \(0.933287\pi\)
\(284\) 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
\(293\) 12.0000i 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 4.00000 8.00000i 0.232889 0.465778i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.0000 + 27.7128i −0.925304 + 1.60267i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 13.3923 0.803848i 0.766841 0.0460282i
\(306\) 0 0
\(307\) 16.0000i 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) −3.46410 2.00000i −0.195803 0.113047i 0.398894 0.916997i \(-0.369394\pi\)
−0.594696 + 0.803951i \(0.702728\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.8564 + 8.00000i 0.778253 + 0.449325i 0.835811 0.549017i \(-0.184998\pi\)
−0.0575576 + 0.998342i \(0.518331\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 18.3923 + 7.85641i 1.02022 + 0.435795i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 13.8564i 0.439720 0.761617i −0.557948 0.829876i \(-0.688411\pi\)
0.997668 + 0.0682590i \(0.0217444\pi\)
\(332\) 0 0
\(333\) 20.7846 12.0000i 1.13899 0.657596i
\(334\) 0 0
\(335\) 16.0000 + 8.00000i 0.874173 + 0.437087i
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.7846 + 12.0000i −1.11578 + 0.644194i −0.940319 0.340293i \(-0.889474\pi\)
−0.175457 + 0.984487i \(0.556140\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.3923 6.00000i 0.553127 0.319348i −0.197256 0.980352i \(-0.563203\pi\)
0.750382 + 0.661004i \(0.229870\pi\)
\(354\) 0 0
\(355\) −14.7846 22.3923i −0.784686 1.18846i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.0000 + 17.3205i −0.527780 + 0.914141i 0.471696 + 0.881761i \(0.343642\pi\)
−0.999476 + 0.0323801i \(0.989691\pi\)
\(360\) 0 0
\(361\) 1.50000 + 2.59808i 0.0789474 + 0.136741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.00000 4.00000i −0.418739 0.209370i
\(366\) 0 0
\(367\) −6.92820 + 4.00000i −0.361649 + 0.208798i −0.669804 0.742538i \(-0.733622\pi\)
0.308155 + 0.951336i \(0.400289\pi\)
\(368\) 0 0
\(369\) 9.00000 15.5885i 0.468521 0.811503i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −27.7128 16.0000i −1.43492 0.828449i −0.437425 0.899255i \(-0.644109\pi\)
−0.997490 + 0.0708063i \(0.977443\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000i 0.412021i
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 20.7846 + 12.0000i 1.06204 + 0.613171i 0.925997 0.377531i \(-0.123227\pi\)
0.136047 + 0.990702i \(0.456560\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −20.7846 12.0000i −1.05654 0.609994i
\(388\) 0 0
\(389\) 3.00000 + 5.19615i 0.152106 + 0.263455i 0.932002 0.362454i \(-0.118061\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.92820 0.535898i 0.449227 0.0269640i
\(396\) 0 0
\(397\) −24.2487 14.0000i −1.21701 0.702640i −0.252731 0.967537i \(-0.581329\pi\)
−0.964277 + 0.264897i \(0.914662\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.00000 + 15.5885i −0.449439 + 0.778450i −0.998350 0.0574304i \(-0.981709\pi\)
0.548911 + 0.835881i \(0.315043\pi\)
\(402\) 0 0
\(403\) −27.7128 + 16.0000i −1.38047 + 0.797017i
\(404\) 0 0
\(405\) −9.00000 + 18.0000i −0.447214 + 0.894427i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.00000 8.66025i −0.247234 0.428222i 0.715523 0.698589i \(-0.246188\pi\)
−0.962757 + 0.270367i \(0.912855\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 20.7846 12.0000i 1.01058 0.583460i
\(424\) 0 0
\(425\) 2.39230 + 19.8564i 0.116044 + 0.963177i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.0000 17.3205i −0.481683 0.834300i 0.518096 0.855323i \(-0.326641\pi\)
−0.999779 + 0.0210230i \(0.993308\pi\)
\(432\) 0 0
\(433\) 28.0000i 1.34559i −0.739827 0.672797i \(-0.765093\pi\)
0.739827 0.672797i \(-0.234907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.7128 16.0000i 1.32568 0.765384i
\(438\) 0 0
\(439\) −4.00000 + 6.92820i −0.190910 + 0.330665i −0.945552 0.325471i \(-0.894477\pi\)
0.754642 + 0.656136i \(0.227810\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −20.7846 12.0000i −0.987507 0.570137i −0.0829786 0.996551i \(-0.526443\pi\)
−0.904528 + 0.426414i \(0.859777\pi\)
\(444\) 0 0
\(445\) −22.3205 + 1.33975i −1.05809 + 0.0635100i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13.8564 8.00000i −0.648175 0.374224i 0.139581 0.990211i \(-0.455424\pi\)
−0.787757 + 0.615986i \(0.788758\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13.8564 8.00000i −0.641198 0.370196i 0.143878 0.989595i \(-0.454043\pi\)
−0.785076 + 0.619400i \(0.787376\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −12.0000 16.0000i −0.550598 0.734130i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) −16.0000 + 27.7128i −0.729537 + 1.26360i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −22.3923 + 14.7846i −1.01678 + 0.671335i
\(486\) 0 0
\(487\) 6.92820 4.00000i 0.313947 0.181257i −0.334744 0.942309i \(-0.608650\pi\)
0.648691 + 0.761052i \(0.275317\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −6.92820 + 4.00000i −0.312031 + 0.180151i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 + 6.92820i −0.179065 + 0.310149i −0.941560 0.336844i \(-0.890640\pi\)
0.762496 + 0.646993i \(0.223974\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.00000i 0.356702i 0.983967 + 0.178351i \(0.0570763\pi\)
−0.983967 + 0.178351i \(0.942924\pi\)
\(504\) 0 0
\(505\) −18.0000 + 36.0000i −0.800989 + 1.60198i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.0000 19.0526i 0.487566 0.844490i −0.512331 0.858788i \(-0.671218\pi\)
0.999898 + 0.0142980i \(0.00455136\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.07180 + 17.8564i 0.0472290 + 0.786847i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.00000 5.19615i −0.131432 0.227648i 0.792797 0.609486i \(-0.208624\pi\)
−0.924229 + 0.381839i \(0.875291\pi\)
\(522\) 0 0
\(523\) −27.7128 16.0000i −1.21180 0.699631i −0.248646 0.968594i \(-0.579986\pi\)
−0.963150 + 0.268963i \(0.913319\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −27.7128 16.0000i −1.20719 0.696971i
\(528\) 0 0
\(529\) 20.5000 + 35.5070i 0.891304 + 1.54378i
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 24.0000i 1.03956i
\(534\) 0 0
\(535\) −1.07180 17.8564i −0.0463378 0.772000i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 + 12.1244i −0.300954 + 0.521267i −0.976352 0.216186i \(-0.930638\pi\)
0.675399 + 0.737453i \(0.263972\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.0000 28.0000i 0.599694 1.19939i
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 0 0
\(549\) −9.00000 15.5885i −0.384111 0.665299i
\(550\) 0 0
\(551\) 4.00000 6.92820i 0.170406 0.295151i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.8564 8.00000i 0.587115 0.338971i −0.176841 0.984239i \(-0.556588\pi\)
0.763956 + 0.645269i \(0.223255\pi\)
\(558\) 0 0
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 13.8564 8.00000i 0.583978 0.337160i −0.178735 0.983897i \(-0.557200\pi\)
0.762713 + 0.646737i \(0.223867\pi\)
\(564\) 0 0
\(565\) −29.8564 + 19.7128i −1.25607 + 0.829324i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.0000 36.3731i 0.880366 1.52484i 0.0294311 0.999567i \(-0.490630\pi\)
0.850935 0.525271i \(-0.176036\pi\)
\(570\) 0 0
\(571\) 20.0000 + 34.6410i 0.836974 + 1.44968i 0.892413 + 0.451219i \(0.149011\pi\)
−0.0554391 + 0.998462i \(0.517656\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 32.0000 24.0000i 1.33449 1.00087i
\(576\) 0 0
\(577\) −3.46410 + 2.00000i −0.144212 + 0.0832611i −0.570370 0.821388i \(-0.693200\pi\)
0.426158 + 0.904649i \(0.359867\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.60770 26.7846i −0.0664700 1.10741i
\(586\) 0 0
\(587\) 32.0000i 1.32078i −0.750922 0.660391i \(-0.770391\pi\)
0.750922 0.660391i \(-0.229609\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −31.1769 18.0000i −1.28028 0.739171i −0.303383 0.952869i \(-0.598116\pi\)
−0.976900 + 0.213697i \(0.931449\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −18.0000 31.1769i −0.735460 1.27385i −0.954521 0.298143i \(-0.903633\pi\)
0.219061 0.975711i \(-0.429701\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) −24.5526 + 1.47372i −0.998203 + 0.0599153i
\(606\) 0 0
\(607\) 34.6410 + 20.0000i 1.40604 + 0.811775i 0.995003 0.0998457i \(-0.0318349\pi\)
0.411033 + 0.911621i \(0.365168\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 + 27.7128i −0.647291 + 1.12114i
\(612\) 0 0
\(613\) −20.7846 + 12.0000i −0.839482 + 0.484675i −0.857088 0.515170i \(-0.827729\pi\)
0.0176058 + 0.999845i \(0.494396\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000i 0.322068i −0.986949 0.161034i \(-0.948517\pi\)
0.986949 0.161034i \(-0.0514829\pi\)
\(618\) 0 0
\(619\) −2.00000 3.46410i −0.0803868 0.139234i 0.823029 0.567999i \(-0.192282\pi\)
−0.903416 + 0.428765i \(0.858949\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −17.2846 18.0622i −0.691384 0.722487i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −32.0000 −1.27592
\(630\) 0 0
\(631\) −36.0000 −1.43314 −0.716569 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 14.9282 9.85641i 0.592408 0.391140i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −18.0000 + 31.1769i −0.712069 + 1.23334i
\(640\) 0 0
\(641\) 1.00000 + 1.73205i 0.0394976 + 0.0684119i 0.885098 0.465404i \(-0.154091\pi\)
−0.845601 + 0.533816i \(0.820758\pi\)
\(642\) 0 0
\(643\) 16.0000i 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.7846 12.0000i 0.817127 0.471769i −0.0322975 0.999478i \(-0.510282\pi\)
0.849425 + 0.527710i \(0.176949\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.7846 + 12.0000i 0.813365 + 0.469596i 0.848123 0.529799i \(-0.177733\pi\)
−0.0347583 + 0.999396i \(0.511066\pi\)
\(654\) 0 0
\(655\) 8.92820 0.535898i 0.348854 0.0209393i
\(656\) 0 0
\(657\) 12.0000i 0.468165i
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 17.0000 + 29.4449i 0.661223 + 1.14527i 0.980294 + 0.197542i \(0.0632958\pi\)
−0.319071 + 0.947731i \(0.603371\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.8564 + 8.00000i 0.536522 + 0.309761i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.00000i 0.308377i 0.988041 + 0.154189i \(0.0492764\pi\)
−0.988041 + 0.154189i \(0.950724\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.46410 2.00000i −0.133136 0.0768662i 0.431953 0.901896i \(-0.357825\pi\)
−0.565089 + 0.825030i \(0.691158\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.92820 + 4.00000i −0.265100 + 0.153056i −0.626659 0.779294i \(-0.715578\pi\)
0.361559 + 0.932349i \(0.382245\pi\)
\(684\) 0 0
\(685\) −16.0000 8.00000i −0.611329 0.305664i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 14.0000 24.2487i 0.532585 0.922464i −0.466691 0.884420i \(-0.654554\pi\)
0.999276 0.0380440i \(-0.0121127\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.92820 7.46410i −0.186937 0.283130i
\(696\) 0 0
\(697\) −20.7846 + 12.0000i −0.787273 + 0.454532i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 27.7128 16.0000i 1.04521 0.603451i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.00000 + 5.19615i −0.112667 + 0.195146i −0.916845 0.399244i \(-0.869273\pi\)
0.804178 + 0.594389i \(0.202606\pi\)
\(710\) 0 0
\(711\) −6.00000 10.3923i −0.225018 0.389742i
\(712\) 0 0
\(713\) 64.0000i 2.39682i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16.0000 + 27.7128i −0.596699 + 1.03351i 0.396605 + 0.917989i \(0.370188\pi\)
−0.993305 + 0.115524i \(0.963145\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.92820 9.19615i 0.145890 0.341537i
\(726\) 0 0
\(727\) 8.00000i 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 16.0000 + 27.7128i 0.591781 + 1.02500i
\(732\) 0 0
\(733\) 17.3205 + 10.0000i 0.639748 + 0.369358i 0.784517 0.620107i \(-0.212911\pi\)
−0.144770 + 0.989465i \(0.546244\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −4.00000 6.92820i −0.147142 0.254858i 0.783028 0.621987i \(-0.213674\pi\)
−0.930170 + 0.367129i \(0.880341\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000i 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) −13.3923 + 0.803848i −0.490656 + 0.0294507i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 10.0000 17.3205i 0.364905 0.632034i −0.623856 0.781540i \(-0.714435\pi\)
0.988761 + 0.149505i \(0.0477681\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 + 24.0000i −0.436725 + 0.873449i
\(756\) 0 0
\(757\) 24.0000i 0.872295i 0.899875 + 0.436147i \(0.143657\pi\)
−0.899875 + 0.436147i \(0.856343\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.00000 8.66025i 0.181250 0.313934i −0.761057 0.648686i \(-0.775319\pi\)
0.942306 + 0.334752i \(0.108652\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 22.3923 14.7846i 0.809595 0.534539i
\(766\) 0 0
\(767\) 13.8564 8.00000i 0.500326 0.288863i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −31.1769 + 18.0000i −1.12136 + 0.647415i −0.941747 0.336323i \(-0.890817\pi\)
−0.179609 + 0.983738i \(0.557483\pi\)
\(774\) 0 0
\(775\) 39.7128 4.78461i 1.42653 0.171868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 20.7846i 0.429945 0.744686i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.0000 + 12.0000i 0.856597 + 0.428298i
\(786\) 0 0
\(787\) −41.5692 + 24.0000i −1.48178 + 0.855508i −0.999786 0.0206657i \(-0.993421\pi\)
−0.481996 + 0.876173i \(0.660088\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 20.7846 + 12.0000i 0.738083 + 0.426132i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.0000i 0.991811i 0.868377 + 0.495905i \(0.165164\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 0 0
\(801\) 15.0000 + 25.9808i 0.529999 + 0.917985i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −13.0000 22.5167i −0.457056 0.791644i 0.541748 0.840541i \(-0.317763\pi\)
−0.998804 + 0.0488972i \(0.984429\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.21539 53.5692i −0.112630 1.87645i
\(816\) 0 0
\(817\) −27.7128 16.0000i −0.969549 0.559769i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.00000 5.19615i 0.104701 0.181347i −0.808915 0.587925i \(-0.799945\pi\)
0.913616 + 0.406578i \(0.133278\pi\)
\(822\) 0 0
\(823\) 20.7846 12.0000i 0.724506 0.418294i −0.0919029 0.995768i \(-0.529295\pi\)
0.816409 + 0.577474i \(0.195962\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000i 0.278187i −0.990279 0.139094i \(-0.955581\pi\)
0.990279 0.139094i \(-0.0444189\pi\)
\(828\) 0 0
\(829\) −11.0000 19.0526i −0.382046 0.661723i 0.609309 0.792933i \(-0.291447\pi\)
−0.991355 + 0.131210i \(0.958114\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 44.7846 29.5692i 1.54984 1.02329i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.69615 + 5.59808i 0.127152 + 0.192580i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 32.0000 + 55.4256i 1.09695 + 1.89997i
\(852\) 0 0
\(853\) 4.00000i 0.136957i 0.997653 + 0.0684787i \(0.0218145\pi\)
−0.997653 + 0.0684787i \(0.978185\pi\)
\(854\) 0 0
\(855\) −12.0000 + 24.0000i −0.410391 + 0.820783i
\(856\) 0 0
\(857\) −45.0333 + 26.0000i −1.53831 + 0.888143i −0.539371 + 0.842068i \(0.681338\pi\)
−0.998938 + 0.0460748i \(0.985329\pi\)
\(858\) 0 0
\(859\) −18.0000 + 31.1769i −0.614152 + 1.06374i 0.376381 + 0.926465i \(0.377169\pi\)
−0.990533 + 0.137277i \(0.956165\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.92820 + 4.00000i 0.235839 + 0.136162i 0.613263 0.789879i \(-0.289857\pi\)
−0.377424 + 0.926041i \(0.623190\pi\)
\(864\) 0 0
\(865\) −1.60770 26.7846i −0.0546633 0.910704i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 16.0000 + 27.7128i 0.542139 + 0.939013i
\(872\) 0 0
\(873\) 31.1769 + 18.0000i 1.05518 + 0.609208i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.6410 + 20.0000i 1.16974 + 0.675352i 0.953620 0.301014i \(-0.0973250\pi\)
0.216124 + 0.976366i \(0.430658\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 8.00000i 0.269221i −0.990899 0.134611i \(-0.957022\pi\)
0.990899 0.134611i \(-0.0429784\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20.7846 + 12.0000i 0.697879 + 0.402921i 0.806557 0.591156i \(-0.201328\pi\)
−0.108678 + 0.994077i \(0.534662\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.7128 16.0000i 0.927374 0.535420i
\(894\) 0 0
\(895\) −8.00000 + 16.0000i −0.267411 + 0.534821i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.00000 + 13.8564i 0.266815 + 0.462137i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17.2487 + 26.1244i 0.573367 + 0.868403i
\(906\) 0 0
\(907\) −6.92820 + 4.00000i −0.230047 + 0.132818i −0.610594 0.791944i \(-0.709069\pi\)
0.380547 + 0.924762i \(0.375736\pi\)
\(908\) 0 0
\(909\) 54.0000 1.79107
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −2.00000 + 3.46410i −0.0659739 + 0.114270i −0.897126 0.441776i \(-0.854349\pi\)
0.831152 + 0.556046i \(0.187682\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) 32.0000 24.0000i 1.05215 0.789115i
\(926\) 0 0
\(927\) 20.7846 12.0000i 0.682656 0.394132i
\(928\) 0 0
\(929\) 15.0000 25.9808i 0.492134 0.852401i −0.507825 0.861460i \(-0.669550\pi\)
0.999959 + 0.00905914i \(0.00288365\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.00000i 0.130674i 0.997863 + 0.0653372i \(0.0208123\pi\)
−0.997863 + 0.0653372i \(0.979188\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.0000 + 32.9090i 0.619382 + 1.07280i 0.989599 + 0.143856i \(0.0459502\pi\)
−0.370216 + 0.928946i \(0.620716\pi\)
\(942\) 0 0
\(943\) 41.5692 + 24.0000i 1.35368 + 0.781548i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.7846 + 12.0000i 0.675409 + 0.389948i 0.798123 0.602494i \(-0.205826\pi\)
−0.122714 + 0.992442i \(0.539160\pi\)
\(948\) 0 0
\(949\) −8.00000 13.8564i −0.259691 0.449798i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −44.6410 + 2.67949i −1.44455 + 0.0867063i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16.5000 + 28.5788i −0.532258 + 0.921898i
\(962\) 0 0
\(963\) −20.7846 + 12.0000i −0.669775 + 0.386695i
\(964\) 0 0
\(965\) 32.0000 + 16.0000i 1.03012 + 0.515058i
\(966\) 0 0
\(967\) 56.0000i 1.80084i −0.435023 0.900419i \(-0.643260\pi\)
0.435023 0.900419i \(-0.356740\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 30.0000 51.9615i 0.962746 1.66752i 0.247193 0.968966i \(-0.420492\pi\)
0.715553 0.698558i \(-0.246175\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20.7846 12.0000i 0.664959 0.383914i −0.129205 0.991618i \(-0.541243\pi\)
0.794164 + 0.607704i \(0.207909\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −42.0000 −1.34096
\(982\) 0 0
\(983\) −34.6410 + 20.0000i −1.10488 + 0.637901i −0.937498 0.347992i \(-0.886864\pi\)
−0.167379 + 0.985893i \(0.553530\pi\)
\(984\) 0 0
\(985\) −29.8564 + 19.7128i −0.951304 + 0.628102i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000 55.4256i 1.01754 1.76243i
\(990\) 0 0
\(991\) −2.00000 3.46410i −0.0635321 0.110041i 0.832510 0.554010i \(-0.186903\pi\)
−0.896042 + 0.443969i \(0.853570\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 24.0000 48.0000i 0.760851 1.52170i
\(996\) 0 0
\(997\) 24.2487 14.0000i 0.767964 0.443384i −0.0641836 0.997938i \(-0.520444\pi\)
0.832148 + 0.554554i \(0.187111\pi\)
\(998\) 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 980.2.q.e.569.1 4
5.4 even 2 inner 980.2.q.e.569.2 4
7.2 even 3 980.2.e.a.589.2 2
7.3 odd 6 980.2.q.d.949.1 4
7.4 even 3 inner 980.2.q.e.949.2 4
7.5 odd 6 140.2.e.b.29.1 2
7.6 odd 2 980.2.q.d.569.2 4
21.5 even 6 1260.2.k.b.1009.2 2
28.19 even 6 560.2.g.c.449.1 2
35.2 odd 12 4900.2.a.l.1.1 1
35.4 even 6 inner 980.2.q.e.949.1 4
35.9 even 6 980.2.e.a.589.1 2
35.12 even 12 700.2.a.h.1.1 1
35.19 odd 6 140.2.e.b.29.2 yes 2
35.23 odd 12 4900.2.a.m.1.1 1
35.24 odd 6 980.2.q.d.949.2 4
35.33 even 12 700.2.a.f.1.1 1
35.34 odd 2 980.2.q.d.569.1 4
56.5 odd 6 2240.2.g.d.449.2 2
56.19 even 6 2240.2.g.c.449.2 2
84.47 odd 6 5040.2.t.g.1009.2 2
105.47 odd 12 6300.2.a.y.1.1 1
105.68 odd 12 6300.2.a.g.1.1 1
105.89 even 6 1260.2.k.b.1009.1 2
140.19 even 6 560.2.g.c.449.2 2
140.47 odd 12 2800.2.a.o.1.1 1
140.103 odd 12 2800.2.a.s.1.1 1
280.19 even 6 2240.2.g.c.449.1 2
280.229 odd 6 2240.2.g.d.449.1 2
420.299 odd 6 5040.2.t.g.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.e.b.29.1 2 7.5 odd 6
140.2.e.b.29.2 yes 2 35.19 odd 6
560.2.g.c.449.1 2 28.19 even 6
560.2.g.c.449.2 2 140.19 even 6
700.2.a.f.1.1 1 35.33 even 12
700.2.a.h.1.1 1 35.12 even 12
980.2.e.a.589.1 2 35.9 even 6
980.2.e.a.589.2 2 7.2 even 3
980.2.q.d.569.1 4 35.34 odd 2
980.2.q.d.569.2 4 7.6 odd 2
980.2.q.d.949.1 4 7.3 odd 6
980.2.q.d.949.2 4 35.24 odd 6
980.2.q.e.569.1 4 1.1 even 1 trivial
980.2.q.e.569.2 4 5.4 even 2 inner
980.2.q.e.949.1 4 35.4 even 6 inner
980.2.q.e.949.2 4 7.4 even 3 inner
1260.2.k.b.1009.1 2 105.89 even 6
1260.2.k.b.1009.2 2 21.5 even 6
2240.2.g.c.449.1 2 280.19 even 6
2240.2.g.c.449.2 2 56.19 even 6
2240.2.g.d.449.1 2 280.229 odd 6
2240.2.g.d.449.2 2 56.5 odd 6
2800.2.a.o.1.1 1 140.47 odd 12
2800.2.a.s.1.1 1 140.103 odd 12
4900.2.a.l.1.1 1 35.2 odd 12
4900.2.a.m.1.1 1 35.23 odd 12
5040.2.t.g.1009.1 2 420.299 odd 6
5040.2.t.g.1009.2 2 84.47 odd 6
6300.2.a.g.1.1 1 105.68 odd 12
6300.2.a.y.1.1 1 105.47 odd 12