Properties

Label 1440.2.bi.d.1423.4
Level $1440$
Weight $2$
Character 1440.1423
Analytic conductor $11.498$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1440,2,Mod(847,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.847");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.bi (of order \(4\), 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: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{12} + 28x^{8} + 80x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1423.4
Root \(-1.17431 - 0.788026i\) of defining polynomial
Character \(\chi\) \(=\) 1440.1423
Dual form 1440.2.bi.d.847.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.386289 + 2.20245i) q^{5} +(1.51606 + 1.51606i) q^{7} +O(q^{10})\) \(q+(-0.386289 + 2.20245i) q^{5} +(1.51606 + 1.51606i) q^{7} +3.92468 q^{11} +(-3.56393 + 3.56393i) q^{13} +(1.37670 - 1.37670i) q^{17} +4.00000i q^{19} +(5.17748 - 5.17748i) q^{23} +(-4.70156 - 1.70156i) q^{25} -5.95005 q^{29} +7.12785i q^{31} +(-3.92468 + 2.75341i) q^{35} +(3.56393 + 3.56393i) q^{37} -2.75341 q^{41} +(-5.40312 - 5.40312i) q^{43} +(1.54515 + 1.54515i) q^{47} -2.40312i q^{49} +(-1.81616 + 1.81616i) q^{53} +(-1.51606 + 8.64391i) q^{55} +3.92468i q^{59} +13.1921i q^{61} +(-6.47266 - 9.22607i) q^{65} +(5.40312 - 5.40312i) q^{67} +(5.00000 + 5.00000i) q^{73} +(5.95005 + 5.95005i) q^{77} -7.12785 q^{79} +(-6.67809 - 6.67809i) q^{83} +(2.50031 + 3.56393i) q^{85} +18.4521i q^{89} -10.8062 q^{91} +(-8.80980 - 1.54515i) q^{95} +(-10.4031 + 10.4031i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{25} + 16 q^{43} - 16 q^{67} + 80 q^{73} + 32 q^{91} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.386289 + 2.20245i −0.172754 + 0.984965i
\(6\) 0 0
\(7\) 1.51606 + 1.51606i 0.573017 + 0.573017i 0.932970 0.359953i \(-0.117207\pi\)
−0.359953 + 0.932970i \(0.617207\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.92468 1.18334 0.591668 0.806182i \(-0.298470\pi\)
0.591668 + 0.806182i \(0.298470\pi\)
\(12\) 0 0
\(13\) −3.56393 + 3.56393i −0.988455 + 0.988455i −0.999934 0.0114791i \(-0.996346\pi\)
0.0114791 + 0.999934i \(0.496346\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.37670 1.37670i 0.333900 0.333900i −0.520166 0.854065i \(-0.674130\pi\)
0.854065 + 0.520166i \(0.174130\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.17748 5.17748i 1.07958 1.07958i 0.0830312 0.996547i \(-0.473540\pi\)
0.996547 0.0830312i \(-0.0264601\pi\)
\(24\) 0 0
\(25\) −4.70156 1.70156i −0.940312 0.340312i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.95005 −1.10490 −0.552449 0.833547i \(-0.686306\pi\)
−0.552449 + 0.833547i \(0.686306\pi\)
\(30\) 0 0
\(31\) 7.12785i 1.28020i 0.768292 + 0.640100i \(0.221107\pi\)
−0.768292 + 0.640100i \(0.778893\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.92468 + 2.75341i −0.663392 + 0.465411i
\(36\) 0 0
\(37\) 3.56393 + 3.56393i 0.585906 + 0.585906i 0.936520 0.350614i \(-0.114027\pi\)
−0.350614 + 0.936520i \(0.614027\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.75341 −0.430010 −0.215005 0.976613i \(-0.568977\pi\)
−0.215005 + 0.976613i \(0.568977\pi\)
\(42\) 0 0
\(43\) −5.40312 5.40312i −0.823969 0.823969i 0.162706 0.986675i \(-0.447978\pi\)
−0.986675 + 0.162706i \(0.947978\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.54515 + 1.54515i 0.225384 + 0.225384i 0.810761 0.585377i \(-0.199054\pi\)
−0.585377 + 0.810761i \(0.699054\pi\)
\(48\) 0 0
\(49\) 2.40312i 0.343303i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.81616 + 1.81616i −0.249469 + 0.249469i −0.820753 0.571284i \(-0.806446\pi\)
0.571284 + 0.820753i \(0.306446\pi\)
\(54\) 0 0
\(55\) −1.51606 + 8.64391i −0.204425 + 1.16554i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.92468i 0.510950i 0.966816 + 0.255475i \(0.0822318\pi\)
−0.966816 + 0.255475i \(0.917768\pi\)
\(60\) 0 0
\(61\) 13.1921i 1.68907i 0.535497 + 0.844537i \(0.320124\pi\)
−0.535497 + 0.844537i \(0.679876\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.47266 9.22607i −0.802835 1.14435i
\(66\) 0 0
\(67\) 5.40312 5.40312i 0.660097 0.660097i −0.295306 0.955403i \(-0.595422\pi\)
0.955403 + 0.295306i \(0.0954216\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 5.00000 + 5.00000i 0.585206 + 0.585206i 0.936329 0.351123i \(-0.114200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.95005 + 5.95005i 0.678071 + 0.678071i
\(78\) 0 0
\(79\) −7.12785 −0.801946 −0.400973 0.916090i \(-0.631328\pi\)
−0.400973 + 0.916090i \(0.631328\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.67809 6.67809i −0.733016 0.733016i 0.238200 0.971216i \(-0.423442\pi\)
−0.971216 + 0.238200i \(0.923442\pi\)
\(84\) 0 0
\(85\) 2.50031 + 3.56393i 0.271197 + 0.386562i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.4521i 1.95592i 0.208787 + 0.977961i \(0.433048\pi\)
−0.208787 + 0.977961i \(0.566952\pi\)
\(90\) 0 0
\(91\) −10.8062 −1.13280
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.80980 1.54515i −0.903866 0.158530i
\(96\) 0 0
\(97\) −10.4031 + 10.4031i −1.05628 + 1.05628i −0.0579582 + 0.998319i \(0.518459\pi\)
−0.998319 + 0.0579582i \(0.981541\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.03722i 0.799733i −0.916573 0.399867i \(-0.869056\pi\)
0.916573 0.399867i \(-0.130944\pi\)
\(102\) 0 0
\(103\) −5.61179 + 5.61179i −0.552946 + 0.552946i −0.927290 0.374344i \(-0.877868\pi\)
0.374344 + 0.927290i \(0.377868\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.92468 3.92468i 0.379413 0.379413i −0.491477 0.870890i \(-0.663543\pi\)
0.870890 + 0.491477i \(0.163543\pi\)
\(108\) 0 0
\(109\) 6.06424 0.580849 0.290424 0.956898i \(-0.406204\pi\)
0.290424 + 0.956898i \(0.406204\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.37670 + 1.37670i 0.129509 + 0.129509i 0.768890 0.639381i \(-0.220809\pi\)
−0.639381 + 0.768890i \(0.720809\pi\)
\(114\) 0 0
\(115\) 9.40312 + 13.4031i 0.876846 + 1.24985i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.17433 0.382660
\(120\) 0 0
\(121\) 4.40312 0.400284
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.56376 9.69766i 0.497638 0.867385i
\(126\) 0 0
\(127\) 11.6760 + 11.6760i 1.03608 + 1.03608i 0.999324 + 0.0367559i \(0.0117024\pi\)
0.0367559 + 0.999324i \(0.488298\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.2809 1.50984 0.754918 0.655819i \(-0.227677\pi\)
0.754918 + 0.655819i \(0.227677\pi\)
\(132\) 0 0
\(133\) −6.06424 + 6.06424i −0.525836 + 0.525836i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.3220 14.3220i 1.22361 1.22361i 0.257275 0.966338i \(-0.417176\pi\)
0.966338 0.257275i \(-0.0828245\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −13.9873 + 13.9873i −1.16967 + 1.16967i
\(144\) 0 0
\(145\) 2.29844 13.1047i 0.190875 1.08828i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.03722 0.658435 0.329217 0.944254i \(-0.393215\pi\)
0.329217 + 0.944254i \(0.393215\pi\)
\(150\) 0 0
\(151\) 10.1600i 0.826807i −0.910548 0.413403i \(-0.864340\pi\)
0.910548 0.413403i \(-0.135660\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −15.6987 2.75341i −1.26095 0.221159i
\(156\) 0 0
\(157\) −9.62817 9.62817i −0.768411 0.768411i 0.209416 0.977827i \(-0.432844\pi\)
−0.977827 + 0.209416i \(0.932844\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 15.6987 1.23723
\(162\) 0 0
\(163\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.54515 1.54515i −0.119568 0.119568i 0.644791 0.764359i \(-0.276944\pi\)
−0.764359 + 0.644791i \(0.776944\pi\)
\(168\) 0 0
\(169\) 12.4031i 0.954086i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.13389 4.13389i 0.314294 0.314294i −0.532277 0.846571i \(-0.678663\pi\)
0.846571 + 0.532277i \(0.178663\pi\)
\(174\) 0 0
\(175\) −4.54818 9.70752i −0.343810 0.733820i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.1166i 1.05512i 0.849517 + 0.527562i \(0.176894\pi\)
−0.849517 + 0.527562i \(0.823106\pi\)
\(180\) 0 0
\(181\) 6.06424i 0.450751i −0.974272 0.225376i \(-0.927639\pi\)
0.974272 0.225376i \(-0.0723610\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.22607 + 6.47266i −0.678314 + 0.475879i
\(186\) 0 0
\(187\) 5.40312 5.40312i 0.395116 0.395116i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 23.8002i 1.72212i 0.508501 + 0.861061i \(0.330200\pi\)
−0.508501 + 0.861061i \(0.669800\pi\)
\(192\) 0 0
\(193\) 0.403124 + 0.403124i 0.0290175 + 0.0290175i 0.721467 0.692449i \(-0.243468\pi\)
−0.692449 + 0.721467i \(0.743468\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.44848 5.44848i −0.388188 0.388188i 0.485853 0.874041i \(-0.338509\pi\)
−0.874041 + 0.485853i \(0.838509\pi\)
\(198\) 0 0
\(199\) 18.3514 1.30090 0.650449 0.759550i \(-0.274581\pi\)
0.650449 + 0.759550i \(0.274581\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.02064 9.02064i −0.633125 0.633125i
\(204\) 0 0
\(205\) 1.06361 6.06424i 0.0742858 0.423545i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 15.6987i 1.08590i
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 13.9873 9.81294i 0.953924 0.669237i
\(216\) 0 0
\(217\) −10.8062 + 10.8062i −0.733576 + 0.733576i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.81294i 0.660090i
\(222\) 0 0
\(223\) 11.6760 11.6760i 0.781885 0.781885i −0.198264 0.980149i \(-0.563530\pi\)
0.980149 + 0.198264i \(0.0635303\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.9453 12.9453i 0.859211 0.859211i −0.132034 0.991245i \(-0.542151\pi\)
0.991245 + 0.132034i \(0.0421509\pi\)
\(228\) 0 0
\(229\) −13.1921 −0.871758 −0.435879 0.900005i \(-0.643562\pi\)
−0.435879 + 0.900005i \(0.643562\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.3220 14.3220i −0.938267 0.938267i 0.0599354 0.998202i \(-0.480911\pi\)
−0.998202 + 0.0599354i \(0.980911\pi\)
\(234\) 0 0
\(235\) −4.00000 + 2.80625i −0.260931 + 0.183059i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −27.9745 −1.80952 −0.904761 0.425919i \(-0.859951\pi\)
−0.904761 + 0.425919i \(0.859951\pi\)
\(240\) 0 0
\(241\) 18.8062 1.21142 0.605708 0.795687i \(-0.292890\pi\)
0.605708 + 0.795687i \(0.292890\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.29276 + 0.928300i 0.338142 + 0.0593069i
\(246\) 0 0
\(247\) −14.2557 14.2557i −0.907068 0.907068i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.58213 −0.0998634 −0.0499317 0.998753i \(-0.515900\pi\)
−0.0499317 + 0.998753i \(0.515900\pi\)
\(252\) 0 0
\(253\) 20.3199 20.3199i 1.27750 1.27750i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.13011 + 4.13011i −0.257629 + 0.257629i −0.824089 0.566460i \(-0.808313\pi\)
0.566460 + 0.824089i \(0.308313\pi\)
\(258\) 0 0
\(259\) 10.8062i 0.671468i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.1647 19.1647i 1.18175 1.18175i 0.202458 0.979291i \(-0.435107\pi\)
0.979291 0.202458i \(-0.0648929\pi\)
\(264\) 0 0
\(265\) −3.29844 4.70156i −0.202621 0.288815i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.86289 −0.235524 −0.117762 0.993042i \(-0.537572\pi\)
−0.117762 + 0.993042i \(0.537572\pi\)
\(270\) 0 0
\(271\) 16.2242i 0.985551i −0.870157 0.492775i \(-0.835982\pi\)
0.870157 0.492775i \(-0.164018\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.4521 6.67809i −1.11271 0.402704i
\(276\) 0 0
\(277\) −10.6918 10.6918i −0.642407 0.642407i 0.308740 0.951146i \(-0.400093\pi\)
−0.951146 + 0.308740i \(0.900093\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.26022 0.492764 0.246382 0.969173i \(-0.420758\pi\)
0.246382 + 0.969173i \(0.420758\pi\)
\(282\) 0 0
\(283\) 5.40312 + 5.40312i 0.321182 + 0.321182i 0.849221 0.528038i \(-0.177072\pi\)
−0.528038 + 0.849221i \(0.677072\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.17433 4.17433i −0.246403 0.246403i
\(288\) 0 0
\(289\) 13.2094i 0.777022i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.81616 1.81616i 0.106101 0.106101i −0.652063 0.758165i \(-0.726096\pi\)
0.758165 + 0.652063i \(0.226096\pi\)
\(294\) 0 0
\(295\) −8.64391 1.51606i −0.503268 0.0882684i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 36.9043i 2.13423i
\(300\) 0 0
\(301\) 16.3829i 0.944296i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −29.0549 5.09596i −1.66368 0.291794i
\(306\) 0 0
\(307\) 20.0000 20.0000i 1.14146 1.14146i 0.153277 0.988183i \(-0.451017\pi\)
0.988183 0.153277i \(-0.0489827\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.9745i 1.58629i −0.609032 0.793145i \(-0.708442\pi\)
0.609032 0.793145i \(-0.291558\pi\)
\(312\) 0 0
\(313\) 5.80625 + 5.80625i 0.328189 + 0.328189i 0.851897 0.523709i \(-0.175452\pi\)
−0.523709 + 0.851897i \(0.675452\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.3985 11.3985i −0.640205 0.640205i 0.310400 0.950606i \(-0.399537\pi\)
−0.950606 + 0.310400i \(0.899537\pi\)
\(318\) 0 0
\(319\) −23.3521 −1.30746
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.50682 + 5.50682i 0.306407 + 0.306407i
\(324\) 0 0
\(325\) 22.8203 10.6918i 1.26584 0.593073i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.68509i 0.258298i
\(330\) 0 0
\(331\) 22.8062 1.25354 0.626772 0.779202i \(-0.284376\pi\)
0.626772 + 0.779202i \(0.284376\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.81294 + 13.9873i 0.536138 + 0.764206i
\(336\) 0 0
\(337\) 10.4031 10.4031i 0.566694 0.566694i −0.364507 0.931201i \(-0.618762\pi\)
0.931201 + 0.364507i \(0.118762\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 27.9745i 1.51491i
\(342\) 0 0
\(343\) 14.2557 14.2557i 0.769735 0.769735i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.4521 18.4521i 0.990562 0.990562i −0.00939345 0.999956i \(-0.502990\pi\)
0.999956 + 0.00939345i \(0.00299007\pi\)
\(348\) 0 0
\(349\) 13.1921 0.706156 0.353078 0.935594i \(-0.385135\pi\)
0.353078 + 0.935594i \(0.385135\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.13011 4.13011i −0.219824 0.219824i 0.588600 0.808424i \(-0.299679\pi\)
−0.808424 + 0.588600i \(0.799679\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 32.1489 1.69675 0.848376 0.529394i \(-0.177581\pi\)
0.848376 + 0.529394i \(0.177581\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.9437 + 9.08080i −0.677504 + 0.475311i
\(366\) 0 0
\(367\) −7.58030 7.58030i −0.395688 0.395688i 0.481021 0.876709i \(-0.340266\pi\)
−0.876709 + 0.481021i \(0.840266\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.50682 −0.285900
\(372\) 0 0
\(373\) −2.50031 + 2.50031i −0.129461 + 0.129461i −0.768868 0.639407i \(-0.779180\pi\)
0.639407 + 0.768868i \(0.279180\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 21.2055 21.2055i 1.09214 1.09214i
\(378\) 0 0
\(379\) 6.80625i 0.349614i 0.984603 + 0.174807i \(0.0559301\pi\)
−0.984603 + 0.174807i \(0.944070\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.35181 + 9.35181i −0.477855 + 0.477855i −0.904445 0.426590i \(-0.859715\pi\)
0.426590 + 0.904445i \(0.359715\pi\)
\(384\) 0 0
\(385\) −15.4031 + 10.8062i −0.785016 + 0.550737i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −27.6631 −1.40257 −0.701287 0.712879i \(-0.747391\pi\)
−0.701287 + 0.712879i \(0.747391\pi\)
\(390\) 0 0
\(391\) 14.2557i 0.720942i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.75341 15.6987i 0.138539 0.789889i
\(396\) 0 0
\(397\) 4.62754 + 4.62754i 0.232249 + 0.232249i 0.813631 0.581382i \(-0.197488\pi\)
−0.581382 + 0.813631i \(0.697488\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 31.3975 1.56791 0.783957 0.620815i \(-0.213198\pi\)
0.783957 + 0.620815i \(0.213198\pi\)
\(402\) 0 0
\(403\) −25.4031 25.4031i −1.26542 1.26542i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.9873 + 13.9873i 0.693323 + 0.693323i
\(408\) 0 0
\(409\) 9.40312i 0.464955i 0.972602 + 0.232477i \(0.0746831\pi\)
−0.972602 + 0.232477i \(0.925317\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.95005 + 5.95005i −0.292783 + 0.292783i
\(414\) 0 0
\(415\) 17.2878 12.1285i 0.848626 0.595364i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.7877i 1.11325i 0.830764 + 0.556625i \(0.187904\pi\)
−0.830764 + 0.556625i \(0.812096\pi\)
\(420\) 0 0
\(421\) 25.3206i 1.23405i −0.786944 0.617025i \(-0.788338\pi\)
0.786944 0.617025i \(-0.211662\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −8.81521 + 4.13011i −0.427600 + 0.200340i
\(426\) 0 0
\(427\) −20.0000 + 20.0000i −0.967868 + 0.967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.17433i 0.201070i −0.994933 0.100535i \(-0.967944\pi\)
0.994933 0.100535i \(-0.0320555\pi\)
\(432\) 0 0
\(433\) −0.403124 0.403124i −0.0193729 0.0193729i 0.697354 0.716727i \(-0.254361\pi\)
−0.716727 + 0.697354i \(0.754361\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.7099 + 20.7099i 0.990689 + 0.990689i
\(438\) 0 0
\(439\) −4.09573 −0.195479 −0.0977393 0.995212i \(-0.531161\pi\)
−0.0977393 + 0.995212i \(0.531161\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.9453 + 12.9453i 0.615051 + 0.615051i 0.944258 0.329207i \(-0.106781\pi\)
−0.329207 + 0.944258i \(0.606781\pi\)
\(444\) 0 0
\(445\) −40.6399 7.12785i −1.92652 0.337893i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.6987i 0.740869i 0.928859 + 0.370434i \(0.120791\pi\)
−0.928859 + 0.370434i \(0.879209\pi\)
\(450\) 0 0
\(451\) −10.8062 −0.508846
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.17433 23.8002i 0.195696 1.11577i
\(456\) 0 0
\(457\) 15.8062 15.8062i 0.739385 0.739385i −0.233074 0.972459i \(-0.574878\pi\)
0.972459 + 0.233074i \(0.0748784\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.86289i 0.179913i −0.995946 0.0899563i \(-0.971327\pi\)
0.995946 0.0899563i \(-0.0286727\pi\)
\(462\) 0 0
\(463\) −7.58030 + 7.58030i −0.352286 + 0.352286i −0.860960 0.508673i \(-0.830136\pi\)
0.508673 + 0.860960i \(0.330136\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.17127 + 1.17127i −0.0542001 + 0.0542001i −0.733687 0.679487i \(-0.762202\pi\)
0.679487 + 0.733687i \(0.262202\pi\)
\(468\) 0 0
\(469\) 16.3829 0.756493
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −21.2055 21.2055i −0.975032 0.975032i
\(474\) 0 0
\(475\) 6.80625 18.8062i 0.312292 0.862890i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −27.9745 −1.27819 −0.639095 0.769128i \(-0.720691\pi\)
−0.639095 + 0.769128i \(0.720691\pi\)
\(480\) 0 0
\(481\) −25.4031 −1.15828
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −18.8937 26.9310i −0.857921 1.22287i
\(486\) 0 0
\(487\) 8.64391 + 8.64391i 0.391693 + 0.391693i 0.875290 0.483598i \(-0.160670\pi\)
−0.483598 + 0.875290i \(0.660670\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 27.4728 1.23983 0.619914 0.784669i \(-0.287167\pi\)
0.619914 + 0.784669i \(0.287167\pi\)
\(492\) 0 0
\(493\) −8.19146 + 8.19146i −0.368925 + 0.368925i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.80625i 0.304690i −0.988327 0.152345i \(-0.951318\pi\)
0.988327 0.152345i \(-0.0486824\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.80980 8.80980i 0.392809 0.392809i −0.482878 0.875688i \(-0.660409\pi\)
0.875688 + 0.482878i \(0.160409\pi\)
\(504\) 0 0
\(505\) 17.7016 + 3.10469i 0.787709 + 0.138157i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 22.0245 0.976218 0.488109 0.872783i \(-0.337687\pi\)
0.488109 + 0.872783i \(0.337687\pi\)
\(510\) 0 0
\(511\) 15.1606i 0.670665i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.1919 14.5275i −0.449109 0.640156i
\(516\) 0 0
\(517\) 6.06424 + 6.06424i 0.266705 + 0.266705i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.6987 0.687774 0.343887 0.939011i \(-0.388256\pi\)
0.343887 + 0.939011i \(0.388256\pi\)
\(522\) 0 0
\(523\) 30.8062 + 30.8062i 1.34706 + 1.34706i 0.888834 + 0.458229i \(0.151516\pi\)
0.458229 + 0.888834i \(0.348484\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.81294 + 9.81294i 0.427458 + 0.427458i
\(528\) 0 0
\(529\) 30.6125i 1.33098i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.81294 9.81294i 0.425046 0.425046i
\(534\) 0 0
\(535\) 7.12785 + 10.1600i 0.308164 + 0.439254i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 9.43150i 0.406243i
\(540\) 0 0
\(541\) 34.5756i 1.48652i 0.669001 + 0.743261i \(0.266722\pi\)
−0.669001 + 0.743261i \(0.733278\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.34255 + 13.3562i −0.100344 + 0.572116i
\(546\) 0 0
\(547\) −16.2094 + 16.2094i −0.693063 + 0.693063i −0.962905 0.269842i \(-0.913029\pi\)
0.269842 + 0.962905i \(0.413029\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 23.8002i 1.01392i
\(552\) 0 0
\(553\) −10.8062 10.8062i −0.459528 0.459528i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.2614 + 15.2614i 0.646647 + 0.646647i 0.952181 0.305534i \(-0.0988350\pi\)
−0.305534 + 0.952181i \(0.598835\pi\)
\(558\) 0 0
\(559\) 38.5127 1.62891
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.4521 18.4521i −0.777665 0.777665i 0.201769 0.979433i \(-0.435331\pi\)
−0.979433 + 0.201769i \(0.935331\pi\)
\(564\) 0 0
\(565\) −3.56393 + 2.50031i −0.149935 + 0.105189i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.0136i 0.461715i 0.972988 + 0.230858i \(0.0741532\pi\)
−0.972988 + 0.230858i \(0.925847\pi\)
\(570\) 0 0
\(571\) 33.6125 1.40664 0.703320 0.710874i \(-0.251700\pi\)
0.703320 + 0.710874i \(0.251700\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −33.1520 + 15.5324i −1.38253 + 0.647747i
\(576\) 0 0
\(577\) −0.403124 + 0.403124i −0.0167823 + 0.0167823i −0.715448 0.698666i \(-0.753777\pi\)
0.698666 + 0.715448i \(0.253777\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 20.2488i 0.840060i
\(582\) 0 0
\(583\) −7.12785 + 7.12785i −0.295205 + 0.295205i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.92468 + 3.92468i −0.161989 + 0.161989i −0.783447 0.621458i \(-0.786541\pi\)
0.621458 + 0.783447i \(0.286541\pi\)
\(588\) 0 0
\(589\) −28.5114 −1.17479
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.37670 1.37670i −0.0565345 0.0565345i 0.678274 0.734809i \(-0.262728\pi\)
−0.734809 + 0.678274i \(0.762728\pi\)
\(594\) 0 0
\(595\) −1.61250 + 9.19375i −0.0661059 + 0.376907i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.17433 0.170559 0.0852793 0.996357i \(-0.472822\pi\)
0.0852793 + 0.996357i \(0.472822\pi\)
\(600\) 0 0
\(601\) −18.2094 −0.742776 −0.371388 0.928478i \(-0.621118\pi\)
−0.371388 + 0.928478i \(0.621118\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.70088 + 9.69766i −0.0691505 + 0.394266i
\(606\) 0 0
\(607\) −9.70752 9.70752i −0.394016 0.394016i 0.482100 0.876116i \(-0.339874\pi\)
−0.876116 + 0.482100i \(0.839874\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −11.0136 −0.445564
\(612\) 0 0
\(613\) −4.62754 + 4.62754i −0.186904 + 0.186904i −0.794356 0.607452i \(-0.792192\pi\)
0.607452 + 0.794356i \(0.292192\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19.8288 + 19.8288i −0.798279 + 0.798279i −0.982824 0.184545i \(-0.940919\pi\)
0.184545 + 0.982824i \(0.440919\pi\)
\(618\) 0 0
\(619\) 14.8062i 0.595113i −0.954704 0.297557i \(-0.903828\pi\)
0.954704 0.297557i \(-0.0961717\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −27.9745 + 27.9745i −1.12078 + 1.12078i
\(624\) 0 0
\(625\) 19.2094 + 16.0000i 0.768375 + 0.640000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.81294 0.391268
\(630\) 0 0
\(631\) 35.6393i 1.41878i 0.704818 + 0.709388i \(0.251029\pi\)
−0.704818 + 0.709388i \(0.748971\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −30.2262 + 21.2055i −1.19949 + 0.841516i
\(636\) 0 0
\(637\) 8.56455 + 8.56455i 0.339340 + 0.339340i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.4521 −0.728815 −0.364408 0.931240i \(-0.618729\pi\)
−0.364408 + 0.931240i \(0.618729\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.71949 + 5.71949i 0.224856 + 0.224856i 0.810540 0.585684i \(-0.199174\pi\)
−0.585684 + 0.810540i \(0.699174\pi\)
\(648\) 0 0
\(649\) 15.4031i 0.604626i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.67905 5.67905i 0.222238 0.222238i −0.587202 0.809440i \(-0.699771\pi\)
0.809440 + 0.587202i \(0.199771\pi\)
\(654\) 0 0
\(655\) −6.67540 + 38.0602i −0.260829 + 1.48714i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 11.7740i 0.458652i 0.973350 + 0.229326i \(0.0736521\pi\)
−0.973350 + 0.229326i \(0.926348\pi\)
\(660\) 0 0
\(661\) 15.3193i 0.595852i 0.954589 + 0.297926i \(0.0962949\pi\)
−0.954589 + 0.297926i \(0.903705\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.0136 15.6987i −0.427090 0.608770i
\(666\) 0 0
\(667\) −30.8062 + 30.8062i −1.19282 + 1.19282i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 51.7748i 1.99874i
\(672\) 0 0
\(673\) 15.0000 + 15.0000i 0.578208 + 0.578208i 0.934409 0.356202i \(-0.115928\pi\)
−0.356202 + 0.934409i \(0.615928\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.4857 13.4857i −0.518298 0.518298i 0.398758 0.917056i \(-0.369441\pi\)
−0.917056 + 0.398758i \(0.869441\pi\)
\(678\) 0 0
\(679\) −31.5435 −1.21053
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.3768 22.3768i −0.856225 0.856225i 0.134666 0.990891i \(-0.457004\pi\)
−0.990891 + 0.134666i \(0.957004\pi\)
\(684\) 0 0
\(685\) 26.0111 + 37.0760i 0.993833 + 1.41660i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.9453i 0.493177i
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.80980 + 1.54515i 0.334175 + 0.0586111i
\(696\) 0 0
\(697\) −3.79063 + 3.79063i −0.143580 + 0.143580i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 33.9246i 1.28131i −0.767827 0.640657i \(-0.778662\pi\)
0.767827 0.640657i \(-0.221338\pi\)
\(702\) 0 0
\(703\) −14.2557 + 14.2557i −0.537664 + 0.537664i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.1849 12.1849i 0.458261 0.458261i
\(708\) 0 0
\(709\) −15.3193 −0.575329 −0.287664 0.957731i \(-0.592879\pi\)
−0.287664 + 0.957731i \(0.592879\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 36.9043 + 36.9043i 1.38208 + 1.38208i
\(714\) 0 0
\(715\) −25.4031 36.2094i −0.950023 1.35415i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −27.9745 −1.04327 −0.521637 0.853167i \(-0.674679\pi\)
−0.521637 + 0.853167i \(0.674679\pi\)
\(720\) 0 0
\(721\) −17.0156 −0.633695
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 27.9745 + 10.1244i 1.03895 + 0.376010i
\(726\) 0 0
\(727\) 10.7711 + 10.7711i 0.399479 + 0.399479i 0.878049 0.478570i \(-0.158845\pi\)
−0.478570 + 0.878049i \(0.658845\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −14.8770 −0.550246
\(732\) 0 0
\(733\) −3.56393 + 3.56393i −0.131637 + 0.131637i −0.769855 0.638219i \(-0.779672\pi\)
0.638219 + 0.769855i \(0.279672\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.2055 21.2055i 0.781116 0.781116i
\(738\) 0 0
\(739\) 28.4187i 1.04540i −0.852517 0.522700i \(-0.824925\pi\)
0.852517 0.522700i \(-0.175075\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.7971 22.7971i 0.836343 0.836343i −0.152032 0.988376i \(-0.548582\pi\)
0.988376 + 0.152032i \(0.0485818\pi\)
\(744\) 0 0
\(745\) −3.10469 + 17.7016i −0.113747 + 0.648535i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 11.9001 0.434820
\(750\) 0 0
\(751\) 21.3836i 0.780297i 0.920752 + 0.390148i \(0.127576\pi\)
−0.920752 + 0.390148i \(0.872424\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22.3768 + 3.92468i 0.814376 + 0.142834i
\(756\) 0 0
\(757\) 29.9481 + 29.9481i 1.08848 + 1.08848i 0.995685 + 0.0927974i \(0.0295809\pi\)
0.0927974 + 0.995685i \(0.470419\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.68509 0.169835 0.0849173 0.996388i \(-0.472937\pi\)
0.0849173 + 0.996388i \(0.472937\pi\)
\(762\) 0 0
\(763\) 9.19375 + 9.19375i 0.332836 + 0.332836i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13.9873 13.9873i −0.505051 0.505051i
\(768\) 0 0
\(769\) 11.4031i 0.411207i 0.978635 + 0.205604i \(0.0659157\pi\)
−0.978635 + 0.205604i \(0.934084\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 11.9405 11.9405i 0.429472 0.429472i −0.458977 0.888448i \(-0.651784\pi\)
0.888448 + 0.458977i \(0.151784\pi\)
\(774\) 0 0
\(775\) 12.1285 33.5120i 0.435668 1.20379i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 11.0136i 0.394604i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.9248 17.4863i 0.889604 0.624112i
\(786\) 0 0
\(787\) 20.0000 20.0000i 0.712923 0.712923i −0.254223 0.967146i \(-0.581820\pi\)
0.967146 + 0.254223i \(0.0818196\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.17433i 0.148422i
\(792\) 0 0
\(793\) −47.0156 47.0156i −1.66957 1.66957i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −28.4761 28.4761i −1.00868 1.00868i −0.999962 0.00871364i \(-0.997226\pi\)
−0.00871364 0.999962i \(-0.502774\pi\)
\(798\) 0 0
\(799\) 4.25444 0.150511
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.6234 + 19.6234i 0.692495 + 0.692495i
\(804\) 0 0
\(805\) −6.06424 + 34.5756i −0.213736 + 1.21863i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.4521i 0.648742i 0.945930 + 0.324371i \(0.105153\pi\)
−0.945930 + 0.324371i \(0.894847\pi\)
\(810\) 0 0
\(811\) 17.1938 0.603754 0.301877 0.953347i \(-0.402387\pi\)
0.301877 + 0.953347i \(0.402387\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 21.6125 21.6125i 0.756126 0.756126i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.1988i 0.914345i 0.889378 + 0.457173i \(0.151138\pi\)
−0.889378 + 0.457173i \(0.848862\pi\)
\(822\) 0 0
\(823\) −31.9960 + 31.9960i −1.11531 + 1.11531i −0.122889 + 0.992420i \(0.539216\pi\)
−0.992420 + 0.122889i \(0.960784\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.0136 + 11.0136i −0.382981 + 0.382981i −0.872175 0.489194i \(-0.837291\pi\)
0.489194 + 0.872175i \(0.337291\pi\)
\(828\) 0 0
\(829\) 34.5756 1.20086 0.600431 0.799677i \(-0.294996\pi\)
0.600431 + 0.799677i \(0.294996\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.30839 3.30839i −0.114629 0.114629i
\(834\) 0 0
\(835\) 4.00000 2.80625i 0.138426 0.0971142i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.17433 0.144114 0.0720570 0.997401i \(-0.477044\pi\)
0.0720570 + 0.997401i \(0.477044\pi\)
\(840\) 0 0
\(841\) 6.40312 0.220797
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 27.3172 + 4.79119i 0.939742 + 0.164822i
\(846\) 0 0
\(847\) 6.67540 + 6.67540i 0.229369 + 0.229369i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 36.9043 1.26506
\(852\) 0 0
\(853\) −2.50031 + 2.50031i −0.0856091 + 0.0856091i −0.748615 0.663005i \(-0.769281\pi\)
0.663005 + 0.748615i \(0.269281\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.13011 + 4.13011i −0.141082 + 0.141082i −0.774120 0.633038i \(-0.781807\pi\)
0.633038 + 0.774120i \(0.281807\pi\)
\(858\) 0 0
\(859\) 25.6125i 0.873887i −0.899489 0.436944i \(-0.856061\pi\)
0.899489 0.436944i \(-0.143939\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.63546 + 4.63546i −0.157793 + 0.157793i −0.781588 0.623795i \(-0.785590\pi\)
0.623795 + 0.781588i \(0.285590\pi\)
\(864\) 0 0
\(865\) 7.50781 + 10.7016i 0.255273 + 0.363864i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −27.9745 −0.948971
\(870\) 0 0
\(871\) 38.5127i 1.30495i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 23.1372 6.26723i 0.782181 0.211871i
\(876\) 0 0
\(877\) 2.50031 + 2.50031i 0.0844296 + 0.0844296i 0.748060 0.663631i \(-0.230985\pi\)
−0.663631 + 0.748060i \(0.730985\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −38.8360 −1.30842 −0.654208 0.756314i \(-0.726998\pi\)
−0.654208 + 0.756314i \(0.726998\pi\)
\(882\) 0 0
\(883\) 10.8062 + 10.8062i 0.363659 + 0.363659i 0.865158 0.501499i \(-0.167218\pi\)
−0.501499 + 0.865158i \(0.667218\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −25.3454 25.3454i −0.851014 0.851014i 0.139244 0.990258i \(-0.455533\pi\)
−0.990258 + 0.139244i \(0.955533\pi\)
\(888\) 0 0
\(889\) 35.4031i 1.18738i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.18062 + 6.18062i −0.206827 + 0.206827i
\(894\) 0 0
\(895\) −31.0911 5.45308i −1.03926 0.182276i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 42.4111i 1.41449i
\(900\) 0 0
\(901\) 5.00063i 0.166595i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 13.3562 + 2.34255i 0.443974 + 0.0778689i
\(906\) 0 0
\(907\) −25.4031 + 25.4031i −0.843497 + 0.843497i −0.989312 0.145815i \(-0.953420\pi\)
0.145815 + 0.989312i \(0.453420\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.17433i 0.138302i 0.997606 + 0.0691509i \(0.0220290\pi\)
−0.997606 + 0.0691509i \(0.977971\pi\)
\(912\) 0 0
\(913\) −26.2094 26.2094i −0.867404 0.867404i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 26.1988 + 26.1988i 0.865161 + 0.865161i
\(918\) 0 0
\(919\) 17.1291 0.565037 0.282519 0.959262i \(-0.408830\pi\)
0.282519 + 0.959262i \(0.408830\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −10.6918 22.8203i −0.351543 0.750325i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.68509i 0.153713i 0.997042 + 0.0768565i \(0.0244883\pi\)
−0.997042 + 0.0768565i \(0.975512\pi\)
\(930\) 0 0
\(931\) 9.61250 0.315037
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9.81294 + 13.9873i 0.320917 + 0.457433i
\(936\) 0 0
\(937\) 9.59688 9.59688i 0.313516 0.313516i −0.532754 0.846270i \(-0.678843\pi\)
0.846270 + 0.532754i \(0.178843\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 31.8374i 1.03787i 0.854814 + 0.518935i \(0.173671\pi\)
−0.854814 + 0.518935i \(0.826329\pi\)
\(942\) 0 0
\(943\) −14.2557 + 14.2557i −0.464229 + 0.464229i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.50682 + 5.50682i −0.178947 + 0.178947i −0.790897 0.611949i \(-0.790386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(948\) 0 0
\(949\) −35.6393 −1.15690
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 17.0754 + 17.0754i 0.553127 + 0.553127i 0.927342 0.374215i \(-0.122088\pi\)
−0.374215 + 0.927342i \(0.622088\pi\)
\(954\) 0 0
\(955\) −52.4187 9.19375i −1.69623 0.297503i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 43.4261 1.40230
\(960\) 0 0
\(961\) −19.8062 −0.638911
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.04358 + 0.732138i −0.0335941 + 0.0235684i
\(966\) 0 0
\(967\) 8.64391 + 8.64391i 0.277969 + 0.277969i 0.832298 0.554329i \(-0.187025\pi\)
−0.554329 + 0.832298i \(0.687025\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.08895 0.227495 0.113748 0.993510i \(-0.463714\pi\)
0.113748 + 0.993510i \(0.463714\pi\)
\(972\) 0 0
\(973\) 6.06424 6.06424i 0.194411 0.194411i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 19.8288 19.8288i 0.634381 0.634381i −0.314783 0.949164i \(-0.601932\pi\)
0.949164 + 0.314783i \(0.101932\pi\)
\(978\) 0 0
\(979\) 72.4187i 2.31451i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 22.7971 22.7971i 0.727113 0.727113i −0.242930 0.970044i \(-0.578109\pi\)
0.970044 + 0.242930i \(0.0781086\pi\)
\(984\) 0 0
\(985\) 14.1047 9.89531i 0.449413 0.315291i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −55.9491 −1.77908
\(990\) 0 0
\(991\) 26.5429i 0.843163i −0.906791 0.421581i \(-0.861475\pi\)
0.906791 0.421581i \(-0.138525\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.08895 + 40.4181i −0.224735 + 1.28134i
\(996\) 0 0
\(997\) 3.56393 + 3.56393i 0.112871 + 0.112871i 0.761286 0.648416i \(-0.224568\pi\)
−0.648416 + 0.761286i \(0.724568\pi\)
\(998\) 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 1440.2.bi.d.1423.4 16
3.2 odd 2 inner 1440.2.bi.d.1423.6 16
4.3 odd 2 360.2.w.d.163.3 yes 16
5.2 odd 4 inner 1440.2.bi.d.847.5 16
8.3 odd 2 inner 1440.2.bi.d.1423.5 16
8.5 even 2 360.2.w.d.163.2 16
12.11 even 2 360.2.w.d.163.6 yes 16
15.2 even 4 inner 1440.2.bi.d.847.3 16
20.7 even 4 360.2.w.d.307.2 yes 16
24.5 odd 2 360.2.w.d.163.7 yes 16
24.11 even 2 inner 1440.2.bi.d.1423.3 16
40.27 even 4 inner 1440.2.bi.d.847.4 16
40.37 odd 4 360.2.w.d.307.3 yes 16
60.47 odd 4 360.2.w.d.307.7 yes 16
120.77 even 4 360.2.w.d.307.6 yes 16
120.107 odd 4 inner 1440.2.bi.d.847.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.w.d.163.2 16 8.5 even 2
360.2.w.d.163.3 yes 16 4.3 odd 2
360.2.w.d.163.6 yes 16 12.11 even 2
360.2.w.d.163.7 yes 16 24.5 odd 2
360.2.w.d.307.2 yes 16 20.7 even 4
360.2.w.d.307.3 yes 16 40.37 odd 4
360.2.w.d.307.6 yes 16 120.77 even 4
360.2.w.d.307.7 yes 16 60.47 odd 4
1440.2.bi.d.847.3 16 15.2 even 4 inner
1440.2.bi.d.847.4 16 40.27 even 4 inner
1440.2.bi.d.847.5 16 5.2 odd 4 inner
1440.2.bi.d.847.6 16 120.107 odd 4 inner
1440.2.bi.d.1423.3 16 24.11 even 2 inner
1440.2.bi.d.1423.4 16 1.1 even 1 trivial
1440.2.bi.d.1423.5 16 8.3 odd 2 inner
1440.2.bi.d.1423.6 16 3.2 odd 2 inner