Properties

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

Related objects

Downloads

Learn more

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.d.949.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.23205 - 0.133975i) q^{5} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(-2.23205 - 0.133975i) q^{5} +(-1.50000 + 2.59808i) q^{9} -4.00000i q^{13} +(3.46410 - 2.00000i) q^{17} +(2.00000 - 3.46410i) q^{19} +(6.92820 + 4.00000i) q^{23} +(4.96410 + 0.598076i) q^{25} -2.00000 q^{29} +(4.00000 + 6.92820i) q^{31} +(6.92820 + 4.00000i) q^{37} +6.00000 q^{41} -8.00000i q^{43} +(3.69615 - 5.59808i) q^{45} +(-6.92820 - 4.00000i) q^{47} +(-2.00000 - 3.46410i) q^{59} +(3.00000 - 5.19615i) q^{61} +(-0.535898 + 8.92820i) q^{65} +(6.92820 - 4.00000i) q^{67} +12.0000 q^{71} +(3.46410 - 2.00000i) q^{73} +(-2.00000 + 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-8.00000 + 4.00000i) q^{85} +(-5.00000 + 8.66025i) q^{89} +(-4.92820 + 7.46410i) q^{95} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 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
\(5\) −2.23205 0.133975i −0.998203 0.0599153i
\(6\) 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 −0.540068 0.841621i \(-0.681602\pi\)
0.998899 + 0.0469020i \(0.0149348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.92820 + 4.00000i 1.44463 + 0.834058i 0.998154 0.0607377i \(-0.0193453\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(24\) 0 0
\(25\) 4.96410 + 0.598076i 0.992820 + 0.119615i
\(26\) 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 \(0.0884683\pi\)
−0.243204 + 0.969975i \(0.578198\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.92820 + 4.00000i 1.13899 + 0.657596i 0.946180 0.323640i \(-0.104907\pi\)
0.192809 + 0.981236i \(0.438240\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 3.69615 5.59808i 0.550990 0.834512i
\(46\) 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 0.705965 0.708247i \(-0.250514\pi\)
−0.966342 + 0.257260i \(0.917180\pi\)
\(60\) 0 0
\(61\) 3.00000 5.19615i 0.384111 0.665299i −0.607535 0.794293i \(-0.707841\pi\)
0.991645 + 0.128994i \(0.0411748\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.535898 + 8.92820i −0.0664700 + 1.10741i
\(66\) 0 0
\(67\) 6.92820 4.00000i 0.846415 0.488678i −0.0130248 0.999915i \(-0.504146\pi\)
0.859440 + 0.511237i \(0.170813\pi\)
\(68\) 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 \(0.344474\pi\)
−0.999388 + 0.0349934i \(0.988859\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.92820 + 7.46410i −0.505623 + 0.765801i
\(96\) 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 \(0.186539\pi\)
0.0623905 + 0.998052i \(0.480128\pi\)
\(102\) 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 0.126217 0.992003i \(-0.459717\pi\)
−0.795991 + 0.605308i \(0.793050\pi\)
\(108\) 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 \(0.271189\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) −14.9282 9.85641i −1.39206 0.919115i
\(116\) 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 \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00000 3.46410i 0.174741 0.302660i −0.765331 0.643637i \(-0.777425\pi\)
0.940072 + 0.340977i \(0.110758\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.92820 + 4.00000i −0.591916 + 0.341743i −0.765855 0.643013i \(-0.777684\pi\)
0.173939 + 0.984757i \(0.444351\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.46410 + 0.267949i 0.370723 + 0.0222520i
\(146\) 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 \(-0.944240\pi\)
−0.643280 0.765631i \(-0.722427\pi\)
\(164\) 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 \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −14.9282 9.85641i −1.09754 0.724657i
\(186\) 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 0.0899262 0.995948i \(-0.471337\pi\)
0.907480 + 0.420096i \(0.138004\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.0000i 1.13995i 0.821661 + 0.569976i \(0.193048\pi\)
−0.821661 + 0.569976i \(0.806952\pi\)
\(198\) 0 0
\(199\) −12.0000 20.7846i −0.850657 1.47338i −0.880616 0.473831i \(-0.842871\pi\)
0.0299585 0.999551i \(-0.490462\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −13.3923 0.803848i −0.935359 0.0561432i
\(206\) 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
\(215\) −1.07180 + 17.8564i −0.0730959 + 1.21780i
\(216\) 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 −0.999088 0.0426906i \(-0.986407\pi\)
0.536515 + 0.843891i \(0.319740\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.92820 4.00000i −0.453882 0.262049i 0.255586 0.966786i \(-0.417731\pi\)
−0.709468 + 0.704737i \(0.751065\pi\)
\(234\) 0 0
\(235\) 14.9282 + 9.85641i 0.973809 + 0.642961i
\(236\) 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 \(0.0301796\pi\)
−0.415768 + 0.909471i \(0.636487\pi\)
\(242\) 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 \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 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 −0.977147 0.212565i \(-0.931818\pi\)
−0.304487 + 0.952517i \(0.598485\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.00000 8.66025i −0.304855 0.528025i 0.672374 0.740212i \(-0.265275\pi\)
−0.977229 + 0.212187i \(0.931941\pi\)
\(270\) 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 −0.0221745 0.999754i \(-0.507059\pi\)
0.854725 + 0.519081i \(0.173726\pi\)
\(278\) 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
\(305\) −7.39230 + 11.1962i −0.423282 + 0.641090i
\(306\) 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 \(0.0715523\pi\)
−0.294384 + 0.955687i \(0.595114\pi\)
\(312\) 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 0.0575576 0.998342i \(-0.481669\pi\)
−0.835811 + 0.549017i \(0.815002\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 2.39230 19.8564i 0.132701 1.10144i
\(326\) 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 \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 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 0.175457 0.984487i \(-0.443860\pi\)
0.940319 + 0.340293i \(0.110526\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 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
\(355\) −26.7846 1.60770i −1.42158 0.0853276i
\(356\) 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 0.997490 0.0708063i \(-0.0225572\pi\)
0.437425 + 0.899255i \(0.355891\pi\)
\(374\) 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
\(395\) 4.92820 7.46410i 0.247965 0.375560i
\(396\) 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 0.962757 0.270367i \(-0.0871450\pi\)
−0.715523 + 0.698589i \(0.753812\pi\)
\(410\) 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 \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 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
\(425\) 18.3923 7.85641i 0.892158 0.381092i
\(426\) 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 −0.754642 0.656136i \(-0.772190\pi\)
0.945552 + 0.325471i \(0.105523\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.7846 + 12.0000i 0.987507 + 0.570137i 0.904528 0.426414i \(-0.140223\pi\)
0.0829786 + 0.996551i \(0.473557\pi\)
\(444\) 0 0
\(445\) 12.3205 18.6603i 0.584048 0.884581i
\(446\) 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 0.787757 0.615986i \(-0.211242\pi\)
−0.139581 + 0.990211i \(0.544576\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −26.0000 −1.21094 −0.605470 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 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
\(485\) −1.60770 + 26.7846i −0.0730017 + 1.21623i
\(486\) 0 0
\(487\) −6.92820 + 4.00000i −0.313947 + 0.181257i −0.648691 0.761052i \(-0.724683\pi\)
0.334744 + 0.942309i \(0.391350\pi\)
\(488\) 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 −0.999898 0.0142980i \(-0.995449\pi\)
0.512331 + 0.858788i \(0.328782\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.9282 + 9.85641i 0.657815 + 0.434325i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 + 5.19615i 0.131432 + 0.227648i 0.924229 0.381839i \(-0.124709\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(522\) 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
\(535\) 14.9282 + 9.85641i 0.645403 + 0.426130i
\(536\) 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 \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 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 −0.763956 0.645269i \(-0.776745\pi\)
0.176841 + 0.984239i \(0.443412\pi\)
\(558\) 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
\(565\) 2.14359 35.7128i 0.0901817 1.50245i
\(566\) 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
\(585\) −22.3923 14.7846i −0.925808 0.611268i
\(586\) 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 \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) −13.5526 + 20.5263i −0.550990 + 0.834512i
\(606\) 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 −0.0176058 0.999845i \(-0.505604\pi\)
0.857088 + 0.515170i \(0.172271\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000i 0.322068i 0.986949 + 0.161034i \(0.0514829\pi\)
−0.986949 + 0.161034i \(0.948517\pi\)
\(618\) 0 0
\(619\) 2.00000 + 3.46410i 0.0803868 + 0.139234i 0.903416 0.428765i \(-0.141051\pi\)
−0.823029 + 0.567999i \(0.807718\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 24.2846 + 5.93782i 0.971384 + 0.237513i
\(626\) 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
\(635\) −1.07180 + 17.8564i −0.0425330 + 0.708610i
\(636\) 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 0.0347583 0.999396i \(-0.488934\pi\)
−0.848123 + 0.529799i \(0.822267\pi\)
\(654\) 0 0
\(655\) −4.92820 + 7.46410i −0.192561 + 0.291647i
\(656\) 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 \(-0.936704\pi\)
0.319071 0.947731i \(-0.396629\pi\)
\(662\) 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 \(-0.950724\pi\)
0.988041 0.154189i \(-0.0492764\pi\)
\(674\) 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 −0.361559 0.932349i \(-0.617755\pi\)
0.626659 + 0.779294i \(0.284422\pi\)
\(684\) 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 \(0.345446\pi\)
−0.999276 + 0.0380440i \(0.987887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.92820 + 0.535898i 0.338666 + 0.0203278i
\(696\) 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 \(-0.629812\pi\)
0.993305 0.115524i \(-0.0368548\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −9.92820 1.19615i −0.368724 0.0444240i
\(726\) 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 \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) −7.39230 + 11.1962i −0.270833 + 0.410195i
\(746\) 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 \(-0.856343\pi\)
0.899875 0.436147i \(-0.143657\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −5.00000 + 8.66025i −0.181250 + 0.313934i −0.942306 0.334752i \(-0.891348\pi\)
0.761057 + 0.648686i \(0.224681\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.60770 26.7846i 0.0581263 0.968400i
\(766\) 0 0
\(767\) −13.8564 + 8.00000i −0.500326 + 0.288863i
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 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
\(775\) 15.7128 + 36.7846i 0.564421 + 1.32134i
\(776\) 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 \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 44.7846 + 29.5692i 1.56874 + 1.03576i
\(816\) 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 −0.816409 0.577474i \(-0.804038\pi\)
0.0919029 + 0.995768i \(0.470705\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000i 0.278187i 0.990279 + 0.139094i \(0.0444189\pi\)
−0.990279 + 0.139094i \(0.955581\pi\)
\(828\) 0 0
\(829\) 11.0000 + 19.0526i 0.382046 + 0.661723i 0.991355 0.131210i \(-0.0418863\pi\)
−0.609309 + 0.792933i \(0.708553\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3.21539 53.5692i 0.111273 1.85384i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.69615 + 0.401924i 0.230355 + 0.0138266i
\(846\) 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 \(-0.622831\pi\)
0.990533 0.137277i \(-0.0438352\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.92820 4.00000i −0.235839 0.136162i 0.377424 0.926041i \(-0.376810\pi\)
−0.613263 + 0.789879i \(0.710143\pi\)
\(864\) 0 0
\(865\) −22.3923 14.7846i −0.761361 0.502692i
\(866\) 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 −0.216124 0.976366i \(-0.569342\pi\)
−0.953620 + 0.301014i \(0.902675\pi\)
\(878\) 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
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) 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
\(905\) −31.2487 1.87564i −1.03874 0.0623485i
\(906\) 0 0
\(907\) 6.92820 4.00000i 0.230047 0.132818i −0.380547 0.924762i \(-0.624264\pi\)
0.610594 + 0.791944i \(0.290931\pi\)
\(908\) 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 −0.999959 0.00905914i \(-0.997116\pi\)
0.507825 + 0.861460i \(0.330450\pi\)
\(930\) 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 \(-0.954050\pi\)
0.370216 0.928946i \(-0.379284\pi\)
\(942\) 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 0.122714 0.992442i \(-0.460840\pi\)
−0.798123 + 0.602494i \(0.794174\pi\)
\(948\) 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
\(955\) −24.6410 + 37.3205i −0.797365 + 1.20766i
\(956\) 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 \(0.356740\pi\)
−0.435023 + 0.900419i \(0.643260\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −30.0000 + 51.9615i −0.962746 + 1.66752i −0.247193 + 0.968966i \(0.579508\pi\)
−0.715553 + 0.698558i \(0.753825\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20.7846 + 12.0000i −0.664959 + 0.383914i −0.794164 0.607704i \(-0.792091\pi\)
0.129205 + 0.991618i \(0.458757\pi\)
\(978\) 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
\(985\) 2.14359 35.7128i 0.0683006 1.13790i
\(986\) 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.d.569.1 4
5.4 even 2 inner 980.2.q.d.569.2 4
7.2 even 3 140.2.e.b.29.2 yes 2
7.3 odd 6 980.2.q.e.949.1 4
7.4 even 3 inner 980.2.q.d.949.2 4
7.5 odd 6 980.2.e.a.589.1 2
7.6 odd 2 980.2.q.e.569.2 4
21.2 odd 6 1260.2.k.b.1009.1 2
28.23 odd 6 560.2.g.c.449.2 2
35.2 odd 12 700.2.a.f.1.1 1
35.4 even 6 inner 980.2.q.d.949.1 4
35.9 even 6 140.2.e.b.29.1 2
35.12 even 12 4900.2.a.m.1.1 1
35.19 odd 6 980.2.e.a.589.2 2
35.23 odd 12 700.2.a.h.1.1 1
35.24 odd 6 980.2.q.e.949.2 4
35.33 even 12 4900.2.a.l.1.1 1
35.34 odd 2 980.2.q.e.569.1 4
56.37 even 6 2240.2.g.d.449.1 2
56.51 odd 6 2240.2.g.c.449.1 2
84.23 even 6 5040.2.t.g.1009.1 2
105.2 even 12 6300.2.a.g.1.1 1
105.23 even 12 6300.2.a.y.1.1 1
105.44 odd 6 1260.2.k.b.1009.2 2
140.23 even 12 2800.2.a.o.1.1 1
140.79 odd 6 560.2.g.c.449.1 2
140.107 even 12 2800.2.a.s.1.1 1
280.149 even 6 2240.2.g.d.449.2 2
280.219 odd 6 2240.2.g.c.449.2 2
420.359 even 6 5040.2.t.g.1009.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.e.b.29.1 2 35.9 even 6
140.2.e.b.29.2 yes 2 7.2 even 3
560.2.g.c.449.1 2 140.79 odd 6
560.2.g.c.449.2 2 28.23 odd 6
700.2.a.f.1.1 1 35.2 odd 12
700.2.a.h.1.1 1 35.23 odd 12
980.2.e.a.589.1 2 7.5 odd 6
980.2.e.a.589.2 2 35.19 odd 6
980.2.q.d.569.1 4 1.1 even 1 trivial
980.2.q.d.569.2 4 5.4 even 2 inner
980.2.q.d.949.1 4 35.4 even 6 inner
980.2.q.d.949.2 4 7.4 even 3 inner
980.2.q.e.569.1 4 35.34 odd 2
980.2.q.e.569.2 4 7.6 odd 2
980.2.q.e.949.1 4 7.3 odd 6
980.2.q.e.949.2 4 35.24 odd 6
1260.2.k.b.1009.1 2 21.2 odd 6
1260.2.k.b.1009.2 2 105.44 odd 6
2240.2.g.c.449.1 2 56.51 odd 6
2240.2.g.c.449.2 2 280.219 odd 6
2240.2.g.d.449.1 2 56.37 even 6
2240.2.g.d.449.2 2 280.149 even 6
2800.2.a.o.1.1 1 140.23 even 12
2800.2.a.s.1.1 1 140.107 even 12
4900.2.a.l.1.1 1 35.33 even 12
4900.2.a.m.1.1 1 35.12 even 12
5040.2.t.g.1009.1 2 84.23 even 6
5040.2.t.g.1009.2 2 420.359 even 6
6300.2.a.g.1.1 1 105.2 even 12
6300.2.a.y.1.1 1 105.23 even 12