Properties

Label 5292.2.x.a.4409.1
Level $5292$
Weight $2$
Character 5292.4409
Analytic conductor $42.257$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5292,2,Mod(881,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.x (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: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 4409.1
Root \(-0.213160 - 1.71888i\) of defining polynomial
Character \(\chi\) \(=\) 5292.4409
Dual form 5292.2.x.a.881.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.43402 - 2.48379i) q^{5} +(-2.34941 - 1.35643i) q^{11} +(-3.18987 + 1.84167i) q^{13} +6.44383 q^{17} +3.16234i q^{19} +(2.59068 - 1.49573i) q^{23} +(-1.61282 + 2.79348i) q^{25} +(2.48332 + 1.43375i) q^{29} +(8.26739 - 4.77318i) q^{31} +3.41279 q^{37} +(0.794538 + 1.37618i) q^{41} +(-4.67828 + 8.10302i) q^{43} +(5.65372 - 9.79254i) q^{47} +2.49899i q^{53} +7.78058i q^{55} +(-4.33680 - 7.51156i) q^{59} +(-0.566915 - 0.327308i) q^{61} +(9.14867 + 5.28199i) q^{65} +(-3.86146 - 6.68825i) q^{67} +7.86582i q^{71} -12.7905i q^{73} +(-2.59566 + 4.49581i) q^{79} +(7.92948 - 13.7343i) q^{83} +(-9.24057 - 16.0051i) q^{85} +6.29653 q^{89} +(7.85460 - 4.53486i) q^{95} +(-13.2065 - 7.62477i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{11} - 3 q^{13} + 18 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{29} + 6 q^{31} - 2 q^{37} + 6 q^{41} - 2 q^{43} - 18 q^{47} - 15 q^{59} + 3 q^{61} - 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} + 42 q^{89}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

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
\(4\) 0 0
\(5\) −1.43402 2.48379i −0.641312 1.11079i −0.985140 0.171753i \(-0.945057\pi\)
0.343828 0.939033i \(-0.388276\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.34941 1.35643i −0.708373 0.408979i 0.102086 0.994776i \(-0.467448\pi\)
−0.810458 + 0.585797i \(0.800782\pi\)
\(12\) 0 0
\(13\) −3.18987 + 1.84167i −0.884712 + 0.510789i −0.872209 0.489133i \(-0.837313\pi\)
−0.0125026 + 0.999922i \(0.503980\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.44383 1.56286 0.781429 0.623994i \(-0.214491\pi\)
0.781429 + 0.623994i \(0.214491\pi\)
\(18\) 0 0
\(19\) 3.16234i 0.725491i 0.931888 + 0.362746i \(0.118161\pi\)
−0.931888 + 0.362746i \(0.881839\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.59068 1.49573i 0.540195 0.311882i −0.204963 0.978770i \(-0.565707\pi\)
0.745158 + 0.666888i \(0.232374\pi\)
\(24\) 0 0
\(25\) −1.61282 + 2.79348i −0.322563 + 0.558696i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.48332 + 1.43375i 0.461142 + 0.266240i 0.712524 0.701648i \(-0.247552\pi\)
−0.251383 + 0.967888i \(0.580885\pi\)
\(30\) 0 0
\(31\) 8.26739 4.77318i 1.48487 0.857289i 0.485016 0.874506i \(-0.338814\pi\)
0.999852 + 0.0172169i \(0.00548059\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.41279 0.561059 0.280530 0.959845i \(-0.409490\pi\)
0.280530 + 0.959845i \(0.409490\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.794538 + 1.37618i 0.124086 + 0.214923i 0.921375 0.388674i \(-0.127067\pi\)
−0.797289 + 0.603597i \(0.793733\pi\)
\(42\) 0 0
\(43\) −4.67828 + 8.10302i −0.713431 + 1.23570i 0.250131 + 0.968212i \(0.419526\pi\)
−0.963562 + 0.267487i \(0.913807\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.65372 9.79254i 0.824680 1.42839i −0.0774831 0.996994i \(-0.524688\pi\)
0.902163 0.431394i \(-0.141978\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.49899i 0.343263i 0.985161 + 0.171632i \(0.0549039\pi\)
−0.985161 + 0.171632i \(0.945096\pi\)
\(54\) 0 0
\(55\) 7.78058i 1.04913i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.33680 7.51156i −0.564604 0.977922i −0.997086 0.0762801i \(-0.975696\pi\)
0.432483 0.901642i \(-0.357638\pi\)
\(60\) 0 0
\(61\) −0.566915 0.327308i −0.0725860 0.0419075i 0.463268 0.886218i \(-0.346677\pi\)
−0.535854 + 0.844311i \(0.680010\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.14867 + 5.28199i 1.13475 + 0.655150i
\(66\) 0 0
\(67\) −3.86146 6.68825i −0.471752 0.817099i 0.527725 0.849415i \(-0.323045\pi\)
−0.999478 + 0.0323159i \(0.989712\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.86582i 0.933501i 0.884389 + 0.466750i \(0.154575\pi\)
−0.884389 + 0.466750i \(0.845425\pi\)
\(72\) 0 0
\(73\) 12.7905i 1.49702i −0.663123 0.748510i \(-0.730769\pi\)
0.663123 0.748510i \(-0.269231\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.59566 + 4.49581i −0.292034 + 0.505819i −0.974291 0.225295i \(-0.927666\pi\)
0.682256 + 0.731113i \(0.260999\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.92948 13.7343i 0.870373 1.50753i 0.00876173 0.999962i \(-0.497211\pi\)
0.861611 0.507569i \(-0.169456\pi\)
\(84\) 0 0
\(85\) −9.24057 16.0051i −1.00228 1.73600i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.29653 0.667430 0.333715 0.942674i \(-0.391698\pi\)
0.333715 + 0.942674i \(0.391698\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.85460 4.53486i 0.805865 0.465267i
\(96\) 0 0
\(97\) −13.2065 7.62477i −1.34092 0.774178i −0.353974 0.935255i \(-0.615170\pi\)
−0.986942 + 0.161077i \(0.948503\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.74451 3.02158i 0.173585 0.300658i −0.766086 0.642739i \(-0.777798\pi\)
0.939671 + 0.342080i \(0.111131\pi\)
\(102\) 0 0
\(103\) 2.89161 1.66947i 0.284919 0.164498i −0.350729 0.936477i \(-0.614066\pi\)
0.635648 + 0.771979i \(0.280733\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.58853i 0.346917i 0.984841 + 0.173458i \(0.0554942\pi\)
−0.984841 + 0.173458i \(0.944506\pi\)
\(108\) 0 0
\(109\) −13.7935 −1.32117 −0.660587 0.750750i \(-0.729692\pi\)
−0.660587 + 0.750750i \(0.729692\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.28607 3.05191i 0.497271 0.287100i −0.230315 0.973116i \(-0.573976\pi\)
0.727586 + 0.686016i \(0.240642\pi\)
\(114\) 0 0
\(115\) −7.43018 4.28981i −0.692867 0.400027i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.82019 3.15267i −0.165472 0.286606i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.08895 −0.455170
\(126\) 0 0
\(127\) −13.3819 −1.18745 −0.593727 0.804666i \(-0.702344\pi\)
−0.593727 + 0.804666i \(0.702344\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.388964 + 0.673705i 0.0339839 + 0.0588619i 0.882517 0.470280i \(-0.155847\pi\)
−0.848533 + 0.529142i \(0.822514\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.3082 8.26083i −1.22243 0.705771i −0.256995 0.966413i \(-0.582732\pi\)
−0.965435 + 0.260642i \(0.916066\pi\)
\(138\) 0 0
\(139\) 9.91826 5.72631i 0.841256 0.485699i −0.0164348 0.999865i \(-0.505232\pi\)
0.857691 + 0.514165i \(0.171898\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.99241 0.835607
\(144\) 0 0
\(145\) 8.22408i 0.682973i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.24781 + 2.45247i −0.347994 + 0.200914i −0.663801 0.747909i \(-0.731058\pi\)
0.315807 + 0.948823i \(0.397725\pi\)
\(150\) 0 0
\(151\) −4.92814 + 8.53579i −0.401047 + 0.694633i −0.993852 0.110712i \(-0.964687\pi\)
0.592806 + 0.805345i \(0.298020\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −23.7112 13.6897i −1.90453 1.09958i
\(156\) 0 0
\(157\) −13.3514 + 7.70843i −1.06556 + 0.615200i −0.926964 0.375149i \(-0.877591\pi\)
−0.138593 + 0.990349i \(0.544258\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.4411 0.896133 0.448066 0.894000i \(-0.352113\pi\)
0.448066 + 0.894000i \(0.352113\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.49103 + 11.2428i 0.502291 + 0.869993i 0.999996 + 0.00264735i \(0.000842678\pi\)
−0.497706 + 0.867346i \(0.665824\pi\)
\(168\) 0 0
\(169\) 0.283528 0.491084i 0.0218098 0.0377757i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.79984 + 16.9738i −0.745068 + 1.29050i 0.205095 + 0.978742i \(0.434250\pi\)
−0.950163 + 0.311754i \(0.899084\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.7788i 1.40360i −0.712375 0.701799i \(-0.752381\pi\)
0.712375 0.701799i \(-0.247619\pi\)
\(180\) 0 0
\(181\) 4.47775i 0.332829i −0.986056 0.166414i \(-0.946781\pi\)
0.986056 0.166414i \(-0.0532190\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.89400 8.47666i −0.359814 0.623217i
\(186\) 0 0
\(187\) −15.1392 8.74061i −1.10709 0.639177i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.90050 3.40665i −0.426945 0.246497i 0.271099 0.962551i \(-0.412613\pi\)
−0.698044 + 0.716055i \(0.745946\pi\)
\(192\) 0 0
\(193\) 7.97694 + 13.8165i 0.574193 + 0.994531i 0.996129 + 0.0879053i \(0.0280173\pi\)
−0.421936 + 0.906626i \(0.638649\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.9511i 1.84894i −0.381254 0.924470i \(-0.624508\pi\)
0.381254 0.924470i \(-0.375492\pi\)
\(198\) 0 0
\(199\) 3.18358i 0.225678i −0.993613 0.112839i \(-0.964006\pi\)
0.993613 0.112839i \(-0.0359944\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.27876 3.94693i 0.159156 0.275666i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.28950 7.42963i 0.296711 0.513918i
\(210\) 0 0
\(211\) −0.0552411 0.0956804i −0.00380295 0.00658691i 0.864118 0.503290i \(-0.167877\pi\)
−0.867921 + 0.496703i \(0.834544\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 26.8349 1.83013
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −20.5550 + 11.8674i −1.38268 + 0.798290i
\(222\) 0 0
\(223\) 11.3064 + 6.52775i 0.757132 + 0.437130i 0.828265 0.560336i \(-0.189328\pi\)
−0.0711331 + 0.997467i \(0.522661\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.63392 + 8.02618i −0.307564 + 0.532716i −0.977829 0.209406i \(-0.932847\pi\)
0.670265 + 0.742122i \(0.266180\pi\)
\(228\) 0 0
\(229\) −11.6204 + 6.70902i −0.767895 + 0.443344i −0.832123 0.554591i \(-0.812875\pi\)
0.0642281 + 0.997935i \(0.479541\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.1789i 1.38748i 0.720227 + 0.693738i \(0.244037\pi\)
−0.720227 + 0.693738i \(0.755963\pi\)
\(234\) 0 0
\(235\) −32.4302 −2.11551
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.73342 4.46489i 0.500233 0.288810i −0.228577 0.973526i \(-0.573407\pi\)
0.728810 + 0.684716i \(0.240074\pi\)
\(240\) 0 0
\(241\) −15.9430 9.20469i −1.02698 0.592926i −0.110860 0.993836i \(-0.535361\pi\)
−0.916117 + 0.400910i \(0.868694\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.82401 10.0875i −0.370573 0.641851i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.33194 0.399669 0.199834 0.979830i \(-0.435960\pi\)
0.199834 + 0.979830i \(0.435960\pi\)
\(252\) 0 0
\(253\) −8.11542 −0.510212
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.19283 14.1904i −0.511054 0.885172i −0.999918 0.0128120i \(-0.995922\pi\)
0.488863 0.872360i \(-0.337412\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.4663 6.04270i −0.645377 0.372609i 0.141306 0.989966i \(-0.454870\pi\)
−0.786683 + 0.617357i \(0.788203\pi\)
\(264\) 0 0
\(265\) 6.20698 3.58360i 0.381292 0.220139i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 25.3304 1.54442 0.772212 0.635365i \(-0.219150\pi\)
0.772212 + 0.635365i \(0.219150\pi\)
\(270\) 0 0
\(271\) 0.225849i 0.0137193i −0.999976 0.00685967i \(-0.997816\pi\)
0.999976 0.00685967i \(-0.00218352\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.57832 4.37534i 0.456990 0.263843i
\(276\) 0 0
\(277\) 10.2170 17.6963i 0.613878 1.06327i −0.376702 0.926335i \(-0.622942\pi\)
0.990580 0.136934i \(-0.0437248\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.96635 5.17672i −0.534887 0.308817i 0.208117 0.978104i \(-0.433267\pi\)
−0.743004 + 0.669287i \(0.766600\pi\)
\(282\) 0 0
\(283\) 11.8781 6.85783i 0.706080 0.407656i −0.103528 0.994627i \(-0.533013\pi\)
0.809608 + 0.586971i \(0.199680\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 24.5230 1.44253
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.21527 7.30105i −0.246258 0.426532i 0.716226 0.697868i \(-0.245868\pi\)
−0.962485 + 0.271336i \(0.912535\pi\)
\(294\) 0 0
\(295\) −12.4381 + 21.5434i −0.724175 + 1.25431i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.50930 + 9.54239i −0.318611 + 0.551851i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.87746i 0.107503i
\(306\) 0 0
\(307\) 5.34345i 0.304967i 0.988306 + 0.152484i \(0.0487271\pi\)
−0.988306 + 0.152484i \(0.951273\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.70867 8.15565i −0.267004 0.462465i 0.701083 0.713080i \(-0.252700\pi\)
−0.968087 + 0.250615i \(0.919367\pi\)
\(312\) 0 0
\(313\) 14.3347 + 8.27614i 0.810245 + 0.467795i 0.847041 0.531528i \(-0.178382\pi\)
−0.0367961 + 0.999323i \(0.511715\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.9725 13.2632i −1.29026 0.744934i −0.311563 0.950225i \(-0.600853\pi\)
−0.978701 + 0.205291i \(0.934186\pi\)
\(318\) 0 0
\(319\) −3.88956 6.73691i −0.217773 0.377194i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.3776i 1.13384i
\(324\) 0 0
\(325\) 11.8811i 0.659046i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.82000 15.2767i 0.484791 0.839682i −0.515056 0.857156i \(-0.672229\pi\)
0.999847 + 0.0174739i \(0.00556238\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.0748 + 19.1821i −0.605081 + 1.04803i
\(336\) 0 0
\(337\) 7.31169 + 12.6642i 0.398293 + 0.689864i 0.993515 0.113697i \(-0.0362694\pi\)
−0.595222 + 0.803561i \(0.702936\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −25.8979 −1.40245
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.05563 0.609467i 0.0566691 0.0327179i −0.471398 0.881921i \(-0.656250\pi\)
0.528067 + 0.849203i \(0.322917\pi\)
\(348\) 0 0
\(349\) −10.6857 6.16942i −0.571995 0.330241i 0.185951 0.982559i \(-0.440463\pi\)
−0.757946 + 0.652318i \(0.773797\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.1484 19.3097i 0.593372 1.02775i −0.400402 0.916339i \(-0.631130\pi\)
0.993774 0.111411i \(-0.0355370\pi\)
\(354\) 0 0
\(355\) 19.5371 11.2797i 1.03692 0.598666i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.1035i 0.638796i −0.947621 0.319398i \(-0.896519\pi\)
0.947621 0.319398i \(-0.103481\pi\)
\(360\) 0 0
\(361\) 8.99958 0.473662
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −31.7691 + 18.3419i −1.66287 + 0.960058i
\(366\) 0 0
\(367\) −12.7544 7.36375i −0.665774 0.384385i 0.128700 0.991684i \(-0.458920\pi\)
−0.794473 + 0.607299i \(0.792253\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.54279 + 7.86834i 0.235217 + 0.407407i 0.959336 0.282268i \(-0.0910867\pi\)
−0.724119 + 0.689675i \(0.757753\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.5620 −0.543970
\(378\) 0 0
\(379\) 21.2298 1.09050 0.545250 0.838273i \(-0.316435\pi\)
0.545250 + 0.838273i \(0.316435\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.35227 5.80630i −0.171293 0.296688i 0.767579 0.640954i \(-0.221461\pi\)
−0.938872 + 0.344266i \(0.888128\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.66661 3.84897i −0.338011 0.195151i 0.321381 0.946950i \(-0.395853\pi\)
−0.659392 + 0.751799i \(0.729186\pi\)
\(390\) 0 0
\(391\) 16.6939 9.63825i 0.844249 0.487427i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.8889 0.749141
\(396\) 0 0
\(397\) 0.0494341i 0.00248102i −0.999999 0.00124051i \(-0.999605\pi\)
0.999999 0.00124051i \(-0.000394867\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.6039 10.1636i 0.879096 0.507546i 0.00873572 0.999962i \(-0.497219\pi\)
0.870360 + 0.492416i \(0.163886\pi\)
\(402\) 0 0
\(403\) −17.5813 + 30.4517i −0.875786 + 1.51691i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.01803 4.62921i −0.397439 0.229461i
\(408\) 0 0
\(409\) 12.1144 6.99428i 0.599021 0.345845i −0.169636 0.985507i \(-0.554259\pi\)
0.768656 + 0.639662i \(0.220926\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −45.4840 −2.23272
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10.6718 18.4842i −0.521353 0.903010i −0.999692 0.0248344i \(-0.992094\pi\)
0.478339 0.878176i \(-0.341239\pi\)
\(420\) 0 0
\(421\) 3.97287 6.88121i 0.193626 0.335370i −0.752823 0.658223i \(-0.771309\pi\)
0.946449 + 0.322853i \(0.104642\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −10.3927 + 18.0007i −0.504121 + 0.873163i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 31.9293i 1.53798i −0.639262 0.768989i \(-0.720760\pi\)
0.639262 0.768989i \(-0.279240\pi\)
\(432\) 0 0
\(433\) 18.5300i 0.890493i −0.895408 0.445247i \(-0.853116\pi\)
0.895408 0.445247i \(-0.146884\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.73002 + 8.19263i 0.226267 + 0.391907i
\(438\) 0 0
\(439\) 1.80316 + 1.04106i 0.0860603 + 0.0496869i 0.542413 0.840112i \(-0.317511\pi\)
−0.456352 + 0.889799i \(0.650844\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.13895 + 1.23493i 0.101625 + 0.0586731i 0.549951 0.835197i \(-0.314646\pi\)
−0.448326 + 0.893870i \(0.647980\pi\)
\(444\) 0 0
\(445\) −9.02933 15.6393i −0.428031 0.741372i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 37.5094i 1.77018i −0.465424 0.885088i \(-0.654098\pi\)
0.465424 0.885088i \(-0.345902\pi\)
\(450\) 0 0
\(451\) 4.31094i 0.202994i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.92345 + 5.06356i −0.136753 + 0.236864i −0.926266 0.376871i \(-0.877000\pi\)
0.789513 + 0.613734i \(0.210333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.82830 + 6.63081i −0.178302 + 0.308827i −0.941299 0.337574i \(-0.890394\pi\)
0.762997 + 0.646402i \(0.223727\pi\)
\(462\) 0 0
\(463\) 4.89449 + 8.47751i 0.227466 + 0.393983i 0.957057 0.289901i \(-0.0936225\pi\)
−0.729590 + 0.683885i \(0.760289\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.1612 1.30314 0.651572 0.758587i \(-0.274110\pi\)
0.651572 + 0.758587i \(0.274110\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.9824 12.6915i 1.01075 0.583557i
\(474\) 0 0
\(475\) −8.83394 5.10028i −0.405329 0.234017i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.8053 25.6435i 0.676470 1.17168i −0.299567 0.954075i \(-0.596842\pi\)
0.976037 0.217605i \(-0.0698245\pi\)
\(480\) 0 0
\(481\) −10.8864 + 6.28525i −0.496376 + 0.286583i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 43.7362i 1.98596i
\(486\) 0 0
\(487\) 29.3403 1.32953 0.664767 0.747051i \(-0.268531\pi\)
0.664767 + 0.747051i \(0.268531\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.63745 4.98683i 0.389803 0.225053i −0.292272 0.956335i \(-0.594411\pi\)
0.682075 + 0.731283i \(0.261078\pi\)
\(492\) 0 0
\(493\) 16.0021 + 9.23883i 0.720699 + 0.416096i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9.79784 + 16.9704i 0.438611 + 0.759697i 0.997583 0.0694898i \(-0.0221371\pi\)
−0.558971 + 0.829187i \(0.688804\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −21.2907 −0.949304 −0.474652 0.880174i \(-0.657426\pi\)
−0.474652 + 0.880174i \(0.657426\pi\)
\(504\) 0 0
\(505\) −10.0066 −0.445289
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21.8307 37.8119i −0.967630 1.67598i −0.702378 0.711804i \(-0.747878\pi\)
−0.265252 0.964179i \(-0.585455\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.29324 4.78810i −0.365444 0.210989i
\(516\) 0 0
\(517\) −26.5658 + 15.3378i −1.16836 + 0.674554i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.20087 0.227854 0.113927 0.993489i \(-0.463657\pi\)
0.113927 + 0.993489i \(0.463657\pi\)
\(522\) 0 0
\(523\) 40.0731i 1.75227i −0.482062 0.876137i \(-0.660112\pi\)
0.482062 0.876137i \(-0.339888\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 53.2737 30.7576i 2.32064 1.33982i
\(528\) 0 0
\(529\) −7.02557 + 12.1686i −0.305460 + 0.529072i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.06895 2.92656i −0.219561 0.126763i
\(534\) 0 0
\(535\) 8.91317 5.14602i 0.385350 0.222482i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.24192 0.354348 0.177174 0.984180i \(-0.443305\pi\)
0.177174 + 0.984180i \(0.443305\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 19.7801 + 34.2601i 0.847285 + 1.46754i
\(546\) 0 0
\(547\) −2.53756 + 4.39518i −0.108498 + 0.187925i −0.915162 0.403086i \(-0.867938\pi\)
0.806664 + 0.591011i \(0.201271\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.53400 + 7.85312i −0.193155 + 0.334554i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 43.4285i 1.84012i 0.391773 + 0.920062i \(0.371862\pi\)
−0.391773 + 0.920062i \(0.628138\pi\)
\(558\) 0 0
\(559\) 34.4635i 1.45765i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.99118 8.64498i −0.210353 0.364343i 0.741472 0.670984i \(-0.234128\pi\)
−0.951825 + 0.306641i \(0.900795\pi\)
\(564\) 0 0
\(565\) −15.1606 8.75300i −0.637813 0.368241i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.0597 8.11739i −0.589415 0.340299i 0.175451 0.984488i \(-0.443862\pi\)
−0.764866 + 0.644189i \(0.777195\pi\)
\(570\) 0 0
\(571\) 6.31028 + 10.9297i 0.264077 + 0.457395i 0.967321 0.253553i \(-0.0815994\pi\)
−0.703244 + 0.710948i \(0.748266\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.64936i 0.402406i
\(576\) 0 0
\(577\) 4.82678i 0.200942i 0.994940 + 0.100471i \(0.0320349\pi\)
−0.994940 + 0.100471i \(0.967965\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3.38971 5.87115i 0.140387 0.243158i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.26032 + 9.11114i −0.217117 + 0.376057i −0.953925 0.300044i \(-0.902999\pi\)
0.736809 + 0.676101i \(0.236332\pi\)
\(588\) 0 0
\(589\) 15.0944 + 26.1443i 0.621955 + 1.07726i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 29.4685 1.21013 0.605063 0.796178i \(-0.293148\pi\)
0.605063 + 0.796178i \(0.293148\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −7.11658 + 4.10876i −0.290776 + 0.167879i −0.638292 0.769795i \(-0.720359\pi\)
0.347516 + 0.937674i \(0.387025\pi\)
\(600\) 0 0
\(601\) 32.7131 + 18.8869i 1.33439 + 0.770413i 0.985970 0.166924i \(-0.0533833\pi\)
0.348425 + 0.937337i \(0.386717\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.22038 + 9.04197i −0.212239 + 0.367608i
\(606\) 0 0
\(607\) 30.8497 17.8111i 1.25215 0.722929i 0.280613 0.959821i \(-0.409462\pi\)
0.971536 + 0.236892i \(0.0761289\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 41.6493i 1.68495i
\(612\) 0 0
\(613\) −23.9319 −0.966601 −0.483301 0.875455i \(-0.660562\pi\)
−0.483301 + 0.875455i \(0.660562\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.98622 1.14675i 0.0799623 0.0461663i −0.459486 0.888185i \(-0.651966\pi\)
0.539448 + 0.842019i \(0.318633\pi\)
\(618\) 0 0
\(619\) 9.10806 + 5.25854i 0.366084 + 0.211359i 0.671746 0.740781i \(-0.265545\pi\)
−0.305662 + 0.952140i \(0.598878\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15.3617 + 26.6073i 0.614469 + 1.06429i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 21.9914 0.876856
\(630\) 0 0
\(631\) −2.02836 −0.0807477 −0.0403739 0.999185i \(-0.512855\pi\)
−0.0403739 + 0.999185i \(0.512855\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 19.1899 + 33.2380i 0.761530 + 1.31901i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.7778 + 6.22257i 0.425698 + 0.245777i 0.697512 0.716573i \(-0.254290\pi\)
−0.271814 + 0.962350i \(0.587624\pi\)
\(642\) 0 0
\(643\) −12.3358 + 7.12209i −0.486477 + 0.280868i −0.723112 0.690731i \(-0.757289\pi\)
0.236635 + 0.971599i \(0.423956\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.3820 −0.801299 −0.400649 0.916231i \(-0.631215\pi\)
−0.400649 + 0.916231i \(0.631215\pi\)
\(648\) 0 0
\(649\) 23.5303i 0.923644i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.55335 4.36093i 0.295585 0.170656i −0.344873 0.938650i \(-0.612078\pi\)
0.640458 + 0.767993i \(0.278745\pi\)
\(654\) 0 0
\(655\) 1.11556 1.93221i 0.0435887 0.0754978i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.7524 + 9.67200i 0.652581 + 0.376768i 0.789444 0.613822i \(-0.210369\pi\)
−0.136864 + 0.990590i \(0.543702\pi\)
\(660\) 0 0
\(661\) −31.8948 + 18.4145i −1.24056 + 0.716240i −0.969209 0.246239i \(-0.920805\pi\)
−0.271355 + 0.962479i \(0.587472\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.57801 0.332142
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.887942 + 1.53796i 0.0342786 + 0.0593723i
\(672\) 0 0
\(673\) 8.79204 15.2283i 0.338908 0.587006i −0.645319 0.763913i \(-0.723276\pi\)
0.984228 + 0.176907i \(0.0566091\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20.4146 35.3590i 0.784595 1.35896i −0.144646 0.989484i \(-0.546204\pi\)
0.929241 0.369475i \(-0.120462\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.88755i 0.378337i −0.981945 0.189168i \(-0.939421\pi\)
0.981945 0.189168i \(-0.0605792\pi\)
\(684\) 0 0
\(685\) 47.3847i 1.81048i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.60233 7.97148i −0.175335 0.303689i
\(690\) 0 0
\(691\) 37.9217 + 21.8941i 1.44261 + 0.832891i 0.998023 0.0628444i \(-0.0200172\pi\)
0.444587 + 0.895736i \(0.353351\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −28.4459 16.4233i −1.07902 0.622970i
\(696\) 0 0
\(697\) 5.11987 + 8.86787i 0.193929 + 0.335894i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.6742i 1.12078i 0.828229 + 0.560389i \(0.189349\pi\)
−0.828229 + 0.560389i \(0.810651\pi\)
\(702\) 0 0
\(703\) 10.7924i 0.407044i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −23.5269 + 40.7498i −0.883572 + 1.53039i −0.0362296 + 0.999343i \(0.511535\pi\)
−0.847342 + 0.531048i \(0.821799\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14.2788 24.7316i 0.534745 0.926206i
\(714\) 0 0
\(715\) −14.3293 24.8191i −0.535885 0.928180i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.81830 0.0678110 0.0339055 0.999425i \(-0.489205\pi\)
0.0339055 + 0.999425i \(0.489205\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.01029 + 4.62474i −0.297495 + 0.171759i
\(726\) 0 0
\(727\) 21.7854 + 12.5778i 0.807976 + 0.466485i 0.846252 0.532782i \(-0.178853\pi\)
−0.0382766 + 0.999267i \(0.512187\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −30.1460 + 52.2145i −1.11499 + 1.93122i
\(732\) 0 0
\(733\) 3.84543 2.22016i 0.142034 0.0820034i −0.427299 0.904110i \(-0.640535\pi\)
0.569333 + 0.822107i \(0.307201\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 20.9512i 0.771747i
\(738\) 0 0
\(739\) 17.9522 0.660381 0.330191 0.943914i \(-0.392887\pi\)
0.330191 + 0.943914i \(0.392887\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.3712 18.1122i 1.15090 0.664472i 0.201793 0.979428i \(-0.435323\pi\)
0.949106 + 0.314956i \(0.101990\pi\)
\(744\) 0 0
\(745\) 12.1829 + 7.03378i 0.446345 + 0.257698i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −5.98210 10.3613i −0.218290 0.378089i 0.735995 0.676986i \(-0.236714\pi\)
−0.954285 + 0.298897i \(0.903381\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 28.2682 1.02878
\(756\) 0 0
\(757\) −49.4440 −1.79707 −0.898537 0.438898i \(-0.855369\pi\)
−0.898537 + 0.438898i \(0.855369\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.6192 25.3212i −0.529945 0.917892i −0.999390 0.0349300i \(-0.988879\pi\)
0.469445 0.882962i \(-0.344454\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27.6677 + 15.9740i 0.999023 + 0.576786i
\(768\) 0 0
\(769\) −4.54689 + 2.62515i −0.163965 + 0.0946653i −0.579737 0.814804i \(-0.696845\pi\)
0.415772 + 0.909469i \(0.363511\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −31.3657 −1.12815 −0.564073 0.825725i \(-0.690766\pi\)
−0.564073 + 0.825725i \(0.690766\pi\)
\(774\) 0 0
\(775\) 30.7930i 1.10612i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.35195 + 2.51260i −0.155925 + 0.0900233i
\(780\) 0 0
\(781\) 10.6694 18.4800i 0.381782 0.661266i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 38.2923 + 22.1081i 1.36671 + 0.789071i
\(786\) 0 0
\(787\) −1.59324 + 0.919855i −0.0567927 + 0.0327893i −0.528128 0.849165i \(-0.677106\pi\)
0.471335 + 0.881954i \(0.343772\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.41118 0.0856236
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.39659 + 11.0792i 0.226579 + 0.392446i 0.956792 0.290773i \(-0.0939126\pi\)
−0.730213 + 0.683219i \(0.760579\pi\)
\(798\) 0 0
\(799\) 36.4316 63.1015i 1.28886 2.23237i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −17.3495 + 30.0502i −0.612250 + 1.06045i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.9206i 0.524582i 0.964989 + 0.262291i \(0.0844780\pi\)
−0.964989 + 0.262291i \(0.915522\pi\)
\(810\) 0 0
\(811\) 37.5478i 1.31848i −0.751933 0.659240i \(-0.770878\pi\)
0.751933 0.659240i \(-0.229122\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16.4067 28.4172i −0.574701 0.995411i
\(816\) 0 0
\(817\) −25.6245 14.7943i −0.896489 0.517588i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.88164 1.66371i −0.100570 0.0580640i 0.448872 0.893596i \(-0.351826\pi\)
−0.549441 + 0.835532i \(0.685159\pi\)
\(822\) 0 0
\(823\) −25.4654 44.1073i −0.887667 1.53748i −0.842626 0.538499i \(-0.818992\pi\)
−0.0450407 0.998985i \(-0.514342\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.9198i 0.588360i −0.955750 0.294180i \(-0.904954\pi\)
0.955750 0.294180i \(-0.0950465\pi\)
\(828\) 0 0
\(829\) 5.37475i 0.186673i 0.995635 + 0.0933364i \(0.0297532\pi\)
−0.995635 + 0.0933364i \(0.970247\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 18.6165 32.2447i 0.644251 1.11588i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 11.8714 20.5618i 0.409846 0.709874i −0.585026 0.811014i \(-0.698916\pi\)
0.994872 + 0.101140i \(0.0322492\pi\)
\(840\) 0 0
\(841\) −10.3887 17.9938i −0.358232 0.620477i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.62634 −0.0559476
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.84146 5.10462i 0.303081 0.174984i
\(852\) 0 0
\(853\) −10.3810 5.99345i −0.355437 0.205212i 0.311640 0.950200i \(-0.399122\pi\)
−0.667077 + 0.744988i \(0.732455\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.5318 + 47.6864i −0.940467 + 1.62894i −0.175884 + 0.984411i \(0.556278\pi\)
−0.764583 + 0.644526i \(0.777055\pi\)
\(858\) 0 0
\(859\) −33.8798 + 19.5605i −1.15596 + 0.667395i −0.950333 0.311235i \(-0.899258\pi\)
−0.205630 + 0.978630i \(0.565924\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 45.3011i 1.54207i −0.636794 0.771034i \(-0.719740\pi\)
0.636794 0.771034i \(-0.280260\pi\)
\(864\) 0 0
\(865\) 56.2126 1.91129
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12.1965 7.04166i 0.413738 0.238872i
\(870\) 0 0
\(871\) 24.6351 + 14.2231i 0.834730 + 0.481931i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.02825 + 3.51304i 0.0684893 + 0.118627i 0.898236 0.439512i \(-0.144849\pi\)
−0.829747 + 0.558139i \(0.811515\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −54.3727 −1.83186 −0.915931 0.401336i \(-0.868546\pi\)
−0.915931 + 0.401336i \(0.868546\pi\)
\(882\) 0 0
\(883\) 23.1175 0.777965 0.388982 0.921245i \(-0.372827\pi\)
0.388982 + 0.921245i \(0.372827\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.7818 22.1387i −0.429171 0.743346i 0.567629 0.823285i \(-0.307861\pi\)
−0.996800 + 0.0799384i \(0.974528\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 30.9674 + 17.8790i 1.03628 + 0.598298i
\(894\) 0 0
\(895\) −46.6428 + 26.9292i −1.55910 + 0.900144i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 27.3741 0.912979
\(900\) 0 0
\(901\) 16.1031i 0.536472i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.1218 + 6.42118i −0.369702 + 0.213447i
\(906\) 0 0
\(907\) 18.5065 32.0542i 0.614498 1.06434i −0.375974 0.926630i \(-0.622692\pi\)
0.990472 0.137712i \(-0.0439748\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.16266 1.82596i −0.104784 0.0604969i 0.446692 0.894688i \(-0.352602\pi\)
−0.551476 + 0.834191i \(0.685935\pi\)
\(912\) 0 0
\(913\) −37.2591 + 21.5116i −1.23310 + 0.711929i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −34.7988 −1.14791 −0.573954 0.818888i \(-0.694591\pi\)
−0.573954 + 0.818888i \(0.694591\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −14.4863 25.0910i −0.476822 0.825879i
\(924\) 0 0
\(925\) −5.50420 + 9.53356i −0.180977 + 0.313461i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −25.1736 + 43.6019i −0.825917 + 1.43053i 0.0752987 + 0.997161i \(0.476009\pi\)
−0.901216 + 0.433370i \(0.857324\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 50.1368i 1.63965i
\(936\) 0 0
\(937\) 6.48087i 0.211721i −0.994381 0.105860i \(-0.966240\pi\)
0.994381 0.105860i \(-0.0337596\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.334024 + 0.578547i 0.0108889 + 0.0188601i 0.871418 0.490540i \(-0.163201\pi\)
−0.860530 + 0.509400i \(0.829867\pi\)
\(942\) 0 0
\(943\) 4.11679 + 2.37683i 0.134061 + 0.0774003i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −50.7461 29.2983i −1.64903 0.952067i −0.977459 0.211125i \(-0.932287\pi\)
−0.671569 0.740942i \(-0.734379\pi\)
\(948\) 0 0
\(949\) 23.5560 + 40.8002i 0.764661 + 1.32443i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48.3707i 1.56688i 0.621467 + 0.783441i \(0.286537\pi\)
−0.621467 + 0.783441i \(0.713463\pi\)
\(954\) 0 0
\(955\) 19.5408i 0.632326i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 30.0665 52.0767i 0.969887 1.67989i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 22.8781 39.6261i 0.736474 1.27561i
\(966\) 0 0
\(967\) 8.51390 + 14.7465i 0.273788 + 0.474216i 0.969829 0.243787i \(-0.0783898\pi\)
−0.696040 + 0.718003i \(0.745057\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.1302 0.870651 0.435325 0.900273i \(-0.356633\pi\)
0.435325 + 0.900273i \(0.356633\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.49838 3.17449i 0.175909 0.101561i −0.409460 0.912328i \(-0.634283\pi\)
0.585369 + 0.810767i \(0.300950\pi\)
\(978\) 0 0
\(979\) −14.7931 8.54080i −0.472789 0.272965i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9.98300 + 17.2911i −0.318408 + 0.551499i −0.980156 0.198227i \(-0.936482\pi\)
0.661748 + 0.749727i \(0.269815\pi\)
\(984\) 0 0
\(985\) −64.4572 + 37.2144i −2.05378 + 1.18575i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 27.9898i 0.890024i
\(990\) 0 0
\(991\) −12.7761 −0.405845 −0.202922 0.979195i \(-0.565044\pi\)
−0.202922 + 0.979195i \(0.565044\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.90734 + 4.56530i −0.250679 + 0.144730i
\(996\) 0 0
\(997\) 17.9846 + 10.3834i 0.569579 + 0.328847i 0.756981 0.653436i \(-0.226673\pi\)
−0.187402 + 0.982283i \(0.560007\pi\)
\(998\) 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 5292.2.x.a.4409.1 16
3.2 odd 2 1764.2.x.a.1469.1 16
7.2 even 3 756.2.w.a.521.1 16
7.3 odd 6 756.2.bm.a.89.1 16
7.4 even 3 5292.2.bm.a.4625.8 16
7.5 odd 6 5292.2.w.b.521.8 16
7.6 odd 2 5292.2.x.b.4409.8 16
9.4 even 3 1764.2.x.b.293.8 16
9.5 odd 6 5292.2.x.b.881.8 16
21.2 odd 6 252.2.w.a.101.5 yes 16
21.5 even 6 1764.2.w.b.1109.4 16
21.11 odd 6 1764.2.bm.a.1685.7 16
21.17 even 6 252.2.bm.a.173.2 yes 16
21.20 even 2 1764.2.x.b.1469.8 16
28.3 even 6 3024.2.df.d.1601.1 16
28.23 odd 6 3024.2.ca.d.2033.1 16
63.2 odd 6 2268.2.t.b.1781.8 16
63.4 even 3 1764.2.w.b.509.4 16
63.5 even 6 5292.2.bm.a.2285.8 16
63.13 odd 6 1764.2.x.a.293.1 16
63.16 even 3 2268.2.t.a.1781.1 16
63.23 odd 6 756.2.bm.a.17.1 16
63.31 odd 6 252.2.w.a.5.5 16
63.32 odd 6 5292.2.w.b.1097.8 16
63.38 even 6 2268.2.t.a.2105.1 16
63.40 odd 6 1764.2.bm.a.1697.7 16
63.41 even 6 inner 5292.2.x.a.881.1 16
63.52 odd 6 2268.2.t.b.2105.8 16
63.58 even 3 252.2.bm.a.185.2 yes 16
63.59 even 6 756.2.w.a.341.1 16
84.23 even 6 1008.2.ca.d.353.4 16
84.59 odd 6 1008.2.df.d.929.7 16
252.23 even 6 3024.2.df.d.17.1 16
252.31 even 6 1008.2.ca.d.257.4 16
252.59 odd 6 3024.2.ca.d.2609.1 16
252.247 odd 6 1008.2.df.d.689.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.5 16 63.31 odd 6
252.2.w.a.101.5 yes 16 21.2 odd 6
252.2.bm.a.173.2 yes 16 21.17 even 6
252.2.bm.a.185.2 yes 16 63.58 even 3
756.2.w.a.341.1 16 63.59 even 6
756.2.w.a.521.1 16 7.2 even 3
756.2.bm.a.17.1 16 63.23 odd 6
756.2.bm.a.89.1 16 7.3 odd 6
1008.2.ca.d.257.4 16 252.31 even 6
1008.2.ca.d.353.4 16 84.23 even 6
1008.2.df.d.689.7 16 252.247 odd 6
1008.2.df.d.929.7 16 84.59 odd 6
1764.2.w.b.509.4 16 63.4 even 3
1764.2.w.b.1109.4 16 21.5 even 6
1764.2.x.a.293.1 16 63.13 odd 6
1764.2.x.a.1469.1 16 3.2 odd 2
1764.2.x.b.293.8 16 9.4 even 3
1764.2.x.b.1469.8 16 21.20 even 2
1764.2.bm.a.1685.7 16 21.11 odd 6
1764.2.bm.a.1697.7 16 63.40 odd 6
2268.2.t.a.1781.1 16 63.16 even 3
2268.2.t.a.2105.1 16 63.38 even 6
2268.2.t.b.1781.8 16 63.2 odd 6
2268.2.t.b.2105.8 16 63.52 odd 6
3024.2.ca.d.2033.1 16 28.23 odd 6
3024.2.ca.d.2609.1 16 252.59 odd 6
3024.2.df.d.17.1 16 252.23 even 6
3024.2.df.d.1601.1 16 28.3 even 6
5292.2.w.b.521.8 16 7.5 odd 6
5292.2.w.b.1097.8 16 63.32 odd 6
5292.2.x.a.881.1 16 63.41 even 6 inner
5292.2.x.a.4409.1 16 1.1 even 1 trivial
5292.2.x.b.881.8 16 9.5 odd 6
5292.2.x.b.4409.8 16 7.6 odd 2
5292.2.bm.a.2285.8 16 63.5 even 6
5292.2.bm.a.4625.8 16 7.4 even 3