Properties

Label 1440.2.bj.a.593.2
Level $1440$
Weight $2$
Character 1440.593
Analytic conductor $11.498$
Analytic rank $0$
Dimension $48$
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(17,1440)]
 
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([0, 2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.bj (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: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.2
Character \(\chi\) \(=\) 1440.593
Dual form 1440.2.bj.a.17.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.23604 + 0.0113158i) q^{5} +(-0.471963 + 0.471963i) q^{7} +O(q^{10})\) \(q+(-2.23604 + 0.0113158i) q^{5} +(-0.471963 + 0.471963i) q^{7} -0.335652 q^{11} +(-3.50404 + 3.50404i) q^{13} +(-2.53299 - 2.53299i) q^{17} +4.07474 q^{19} +(6.20627 - 6.20627i) q^{23} +(4.99974 - 0.0506050i) q^{25} -2.42367i q^{29} +6.41004 q^{31} +(1.04999 - 1.06067i) q^{35} +(-2.24893 - 2.24893i) q^{37} -5.80736i q^{41} +(4.87603 - 4.87603i) q^{43} +(-1.68276 - 1.68276i) q^{47} +6.55450i q^{49} +(-3.05444 - 3.05444i) q^{53} +(0.750530 - 0.00379815i) q^{55} +12.2950i q^{59} -7.49787i q^{61} +(7.79553 - 7.87483i) q^{65} +(-5.55519 - 5.55519i) q^{67} -13.4793i q^{71} +(5.05035 + 5.05035i) q^{73} +(0.158415 - 0.158415i) q^{77} -8.85503i q^{79} +(-4.78510 - 4.78510i) q^{83} +(5.69252 + 5.63520i) q^{85} +8.33405 q^{89} -3.30755i q^{91} +(-9.11128 + 0.0461088i) q^{95} +(10.1367 - 10.1367i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) \) 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
\(5\) −2.23604 + 0.0113158i −0.999987 + 0.00506057i
\(6\) 0 0
\(7\) −0.471963 + 0.471963i −0.178385 + 0.178385i −0.790652 0.612266i \(-0.790258\pi\)
0.612266 + 0.790652i \(0.290258\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.335652 −0.101203 −0.0506014 0.998719i \(-0.516114\pi\)
−0.0506014 + 0.998719i \(0.516114\pi\)
\(12\) 0 0
\(13\) −3.50404 + 3.50404i −0.971846 + 0.971846i −0.999614 0.0277680i \(-0.991160\pi\)
0.0277680 + 0.999614i \(0.491160\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.53299 2.53299i −0.614340 0.614340i 0.329734 0.944074i \(-0.393041\pi\)
−0.944074 + 0.329734i \(0.893041\pi\)
\(18\) 0 0
\(19\) 4.07474 0.934809 0.467405 0.884044i \(-0.345189\pi\)
0.467405 + 0.884044i \(0.345189\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.20627 6.20627i 1.29410 1.29410i 0.361868 0.932229i \(-0.382139\pi\)
0.932229 0.361868i \(-0.117861\pi\)
\(24\) 0 0
\(25\) 4.99974 0.0506050i 0.999949 0.0101210i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.42367i 0.450065i −0.974351 0.225032i \(-0.927751\pi\)
0.974351 0.225032i \(-0.0722488\pi\)
\(30\) 0 0
\(31\) 6.41004 1.15128 0.575639 0.817704i \(-0.304753\pi\)
0.575639 + 0.817704i \(0.304753\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.04999 1.06067i 0.177480 0.179286i
\(36\) 0 0
\(37\) −2.24893 2.24893i −0.369721 0.369721i 0.497654 0.867375i \(-0.334195\pi\)
−0.867375 + 0.497654i \(0.834195\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.80736i 0.906957i −0.891267 0.453479i \(-0.850183\pi\)
0.891267 0.453479i \(-0.149817\pi\)
\(42\) 0 0
\(43\) 4.87603 4.87603i 0.743588 0.743588i −0.229678 0.973267i \(-0.573767\pi\)
0.973267 + 0.229678i \(0.0737675\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.68276 1.68276i −0.245455 0.245455i 0.573647 0.819102i \(-0.305528\pi\)
−0.819102 + 0.573647i \(0.805528\pi\)
\(48\) 0 0
\(49\) 6.55450i 0.936358i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.05444 3.05444i −0.419560 0.419560i 0.465492 0.885052i \(-0.345877\pi\)
−0.885052 + 0.465492i \(0.845877\pi\)
\(54\) 0 0
\(55\) 0.750530 0.00379815i 0.101201 0.000512143i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.2950i 1.60067i 0.599553 + 0.800335i \(0.295345\pi\)
−0.599553 + 0.800335i \(0.704655\pi\)
\(60\) 0 0
\(61\) 7.49787i 0.960004i −0.877267 0.480002i \(-0.840636\pi\)
0.877267 0.480002i \(-0.159364\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.79553 7.87483i 0.966916 0.976752i
\(66\) 0 0
\(67\) −5.55519 5.55519i −0.678674 0.678674i 0.281026 0.959700i \(-0.409325\pi\)
−0.959700 + 0.281026i \(0.909325\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4793i 1.59970i −0.600199 0.799851i \(-0.704912\pi\)
0.600199 0.799851i \(-0.295088\pi\)
\(72\) 0 0
\(73\) 5.05035 + 5.05035i 0.591099 + 0.591099i 0.937928 0.346830i \(-0.112742\pi\)
−0.346830 + 0.937928i \(0.612742\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.158415 0.158415i 0.0180531 0.0180531i
\(78\) 0 0
\(79\) 8.85503i 0.996268i −0.867100 0.498134i \(-0.834019\pi\)
0.867100 0.498134i \(-0.165981\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.78510 4.78510i −0.525233 0.525233i 0.393914 0.919147i \(-0.371121\pi\)
−0.919147 + 0.393914i \(0.871121\pi\)
\(84\) 0 0
\(85\) 5.69252 + 5.63520i 0.617441 + 0.611223i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.33405 0.883408 0.441704 0.897161i \(-0.354374\pi\)
0.441704 + 0.897161i \(0.354374\pi\)
\(90\) 0 0
\(91\) 3.30755i 0.346726i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −9.11128 + 0.0461088i −0.934797 + 0.00473066i
\(96\) 0 0
\(97\) 10.1367 10.1367i 1.02922 1.02922i 0.0296647 0.999560i \(-0.490556\pi\)
0.999560 0.0296647i \(-0.00944397\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.88830 0.983922 0.491961 0.870617i \(-0.336280\pi\)
0.491961 + 0.870617i \(0.336280\pi\)
\(102\) 0 0
\(103\) 10.0033 + 10.0033i 0.985653 + 0.985653i 0.999899 0.0142455i \(-0.00453463\pi\)
−0.0142455 + 0.999899i \(0.504535\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.2013 10.2013i 0.986198 0.986198i −0.0137076 0.999906i \(-0.504363\pi\)
0.999906 + 0.0137076i \(0.00436339\pi\)
\(108\) 0 0
\(109\) 11.0144 1.05498 0.527492 0.849560i \(-0.323133\pi\)
0.527492 + 0.849560i \(0.323133\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.45369 + 2.45369i −0.230823 + 0.230823i −0.813036 0.582213i \(-0.802187\pi\)
0.582213 + 0.813036i \(0.302187\pi\)
\(114\) 0 0
\(115\) −13.8072 + 13.9477i −1.28753 + 1.30063i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.39095 0.219178
\(120\) 0 0
\(121\) −10.8873 −0.989758
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.1791 + 0.169731i −0.999885 + 0.0151812i
\(126\) 0 0
\(127\) −6.00329 + 6.00329i −0.532705 + 0.532705i −0.921377 0.388671i \(-0.872934\pi\)
0.388671 + 0.921377i \(0.372934\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −11.3345 −0.990301 −0.495151 0.868807i \(-0.664887\pi\)
−0.495151 + 0.868807i \(0.664887\pi\)
\(132\) 0 0
\(133\) −1.92312 + 1.92312i −0.166756 + 0.166756i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.8670 10.8670i −0.928434 0.928434i 0.0691711 0.997605i \(-0.477965\pi\)
−0.997605 + 0.0691711i \(0.977965\pi\)
\(138\) 0 0
\(139\) 13.3432 1.13176 0.565878 0.824489i \(-0.308537\pi\)
0.565878 + 0.824489i \(0.308537\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.17614 1.17614i 0.0983535 0.0983535i
\(144\) 0 0
\(145\) 0.0274257 + 5.41943i 0.00227758 + 0.450059i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.5961i 1.11384i 0.830566 + 0.556919i \(0.188017\pi\)
−0.830566 + 0.556919i \(0.811983\pi\)
\(150\) 0 0
\(151\) 9.64289 0.784727 0.392364 0.919810i \(-0.371657\pi\)
0.392364 + 0.919810i \(0.371657\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −14.3331 + 0.0725345i −1.15126 + 0.00582611i
\(156\) 0 0
\(157\) −7.62403 7.62403i −0.608464 0.608464i 0.334080 0.942545i \(-0.391574\pi\)
−0.942545 + 0.334080i \(0.891574\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.85826i 0.461695i
\(162\) 0 0
\(163\) −16.2490 + 16.2490i −1.27272 + 1.27272i −0.328065 + 0.944655i \(0.606397\pi\)
−0.944655 + 0.328065i \(0.893603\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.7067 + 13.7067i 1.06066 + 1.06066i 0.998037 + 0.0626204i \(0.0199458\pi\)
0.0626204 + 0.998037i \(0.480054\pi\)
\(168\) 0 0
\(169\) 11.5566i 0.888971i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.35678 + 3.35678i 0.255211 + 0.255211i 0.823103 0.567892i \(-0.192241\pi\)
−0.567892 + 0.823103i \(0.692241\pi\)
\(174\) 0 0
\(175\) −2.33581 + 2.38358i −0.176571 + 0.180181i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.7537i 0.803772i −0.915690 0.401886i \(-0.868355\pi\)
0.915690 0.401886i \(-0.131645\pi\)
\(180\) 0 0
\(181\) 10.1290i 0.752883i −0.926440 0.376441i \(-0.877148\pi\)
0.926440 0.376441i \(-0.122852\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.05414 + 5.00324i 0.371587 + 0.367845i
\(186\) 0 0
\(187\) 0.850201 + 0.850201i 0.0621728 + 0.0621728i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.32194i 0.312725i 0.987700 + 0.156362i \(0.0499768\pi\)
−0.987700 + 0.156362i \(0.950023\pi\)
\(192\) 0 0
\(193\) −7.97419 7.97419i −0.573995 0.573995i 0.359248 0.933242i \(-0.383033\pi\)
−0.933242 + 0.359248i \(0.883033\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.30206 + 4.30206i −0.306509 + 0.306509i −0.843554 0.537045i \(-0.819541\pi\)
0.537045 + 0.843554i \(0.319541\pi\)
\(198\) 0 0
\(199\) 3.02113i 0.214162i 0.994250 + 0.107081i \(0.0341505\pi\)
−0.994250 + 0.107081i \(0.965850\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.14388 + 1.14388i 0.0802849 + 0.0802849i
\(204\) 0 0
\(205\) 0.0657147 + 12.9855i 0.00458972 + 0.906946i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.36769 −0.0946052
\(210\) 0 0
\(211\) 8.87754i 0.611155i −0.952167 0.305577i \(-0.901151\pi\)
0.952167 0.305577i \(-0.0988495\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.8478 + 10.9582i −0.739816 + 0.747342i
\(216\) 0 0
\(217\) −3.02530 + 3.02530i −0.205371 + 0.205371i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 17.7514 1.19409
\(222\) 0 0
\(223\) −9.93808 9.93808i −0.665503 0.665503i 0.291169 0.956672i \(-0.405956\pi\)
−0.956672 + 0.291169i \(0.905956\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.21442 7.21442i 0.478838 0.478838i −0.425922 0.904760i \(-0.640050\pi\)
0.904760 + 0.425922i \(0.140050\pi\)
\(228\) 0 0
\(229\) 21.8687 1.44512 0.722562 0.691306i \(-0.242965\pi\)
0.722562 + 0.691306i \(0.242965\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.78300 + 6.78300i −0.444369 + 0.444369i −0.893477 0.449108i \(-0.851742\pi\)
0.449108 + 0.893477i \(0.351742\pi\)
\(234\) 0 0
\(235\) 3.78175 + 3.74367i 0.246694 + 0.244210i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.63287 −0.299676 −0.149838 0.988711i \(-0.547875\pi\)
−0.149838 + 0.988711i \(0.547875\pi\)
\(240\) 0 0
\(241\) −9.72150 −0.626217 −0.313109 0.949717i \(-0.601370\pi\)
−0.313109 + 0.949717i \(0.601370\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.0741692 14.6561i −0.00473850 0.936346i
\(246\) 0 0
\(247\) −14.2781 + 14.2781i −0.908491 + 0.908491i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 30.6190 1.93265 0.966327 0.257316i \(-0.0828382\pi\)
0.966327 + 0.257316i \(0.0828382\pi\)
\(252\) 0 0
\(253\) −2.08314 + 2.08314i −0.130966 + 0.130966i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.63773 1.63773i −0.102158 0.102158i 0.654180 0.756339i \(-0.273014\pi\)
−0.756339 + 0.654180i \(0.773014\pi\)
\(258\) 0 0
\(259\) 2.12282 0.131905
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.77099 + 7.77099i −0.479180 + 0.479180i −0.904869 0.425689i \(-0.860032\pi\)
0.425689 + 0.904869i \(0.360032\pi\)
\(264\) 0 0
\(265\) 6.86441 + 6.79529i 0.421678 + 0.417431i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 16.1341i 0.983714i 0.870676 + 0.491857i \(0.163682\pi\)
−0.870676 + 0.491857i \(0.836318\pi\)
\(270\) 0 0
\(271\) −16.8474 −1.02341 −0.511704 0.859162i \(-0.670986\pi\)
−0.511704 + 0.859162i \(0.670986\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.67817 + 0.0169856i −0.101198 + 0.00102427i
\(276\) 0 0
\(277\) −16.3206 16.3206i −0.980611 0.980611i 0.0192042 0.999816i \(-0.493887\pi\)
−0.999816 + 0.0192042i \(0.993887\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 31.2898i 1.86659i 0.359106 + 0.933297i \(0.383082\pi\)
−0.359106 + 0.933297i \(0.616918\pi\)
\(282\) 0 0
\(283\) 9.97554 9.97554i 0.592984 0.592984i −0.345452 0.938436i \(-0.612274\pi\)
0.938436 + 0.345452i \(0.112274\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.74086 + 2.74086i 0.161788 + 0.161788i
\(288\) 0 0
\(289\) 4.16795i 0.245174i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.26253 5.26253i −0.307440 0.307440i 0.536476 0.843916i \(-0.319755\pi\)
−0.843916 + 0.536476i \(0.819755\pi\)
\(294\) 0 0
\(295\) −0.139127 27.4921i −0.00810030 1.60065i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 43.4941i 2.51533i
\(300\) 0 0
\(301\) 4.60261i 0.265290i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.0848442 + 16.7655i 0.00485816 + 0.959992i
\(306\) 0 0
\(307\) −9.54586 9.54586i −0.544811 0.544811i 0.380124 0.924935i \(-0.375881\pi\)
−0.924935 + 0.380124i \(0.875881\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.08587i 0.231688i 0.993267 + 0.115844i \(0.0369573\pi\)
−0.993267 + 0.115844i \(0.963043\pi\)
\(312\) 0 0
\(313\) −14.4004 14.4004i −0.813958 0.813958i 0.171266 0.985225i \(-0.445214\pi\)
−0.985225 + 0.171266i \(0.945214\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0841 12.0841i 0.678710 0.678710i −0.280998 0.959708i \(-0.590666\pi\)
0.959708 + 0.280998i \(0.0906656\pi\)
\(318\) 0 0
\(319\) 0.813510i 0.0455478i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10.3213 10.3213i −0.574290 0.574290i
\(324\) 0 0
\(325\) −17.3420 + 17.6966i −0.961961 + 0.981633i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.58840 0.0875711
\(330\) 0 0
\(331\) 33.6119i 1.84747i −0.383027 0.923737i \(-0.625118\pi\)
0.383027 0.923737i \(-0.374882\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.4845 + 12.3588i 0.682100 + 0.675231i
\(336\) 0 0
\(337\) −20.2142 + 20.2142i −1.10114 + 1.10114i −0.106864 + 0.994274i \(0.534081\pi\)
−0.994274 + 0.106864i \(0.965919\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.15154 −0.116512
\(342\) 0 0
\(343\) −6.39722 6.39722i −0.345417 0.345417i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.443163 + 0.443163i −0.0237903 + 0.0237903i −0.718902 0.695112i \(-0.755355\pi\)
0.695112 + 0.718902i \(0.255355\pi\)
\(348\) 0 0
\(349\) 10.8766 0.582212 0.291106 0.956691i \(-0.405977\pi\)
0.291106 + 0.956691i \(0.405977\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0437 + 10.0437i −0.534572 + 0.534572i −0.921930 0.387357i \(-0.873388\pi\)
0.387357 + 0.921930i \(0.373388\pi\)
\(354\) 0 0
\(355\) 0.152529 + 30.1403i 0.00809540 + 1.59968i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.6724 −1.03827 −0.519136 0.854692i \(-0.673746\pi\)
−0.519136 + 0.854692i \(0.673746\pi\)
\(360\) 0 0
\(361\) −2.39651 −0.126132
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.3499 11.2356i −0.594082 0.588100i
\(366\) 0 0
\(367\) 9.86005 9.86005i 0.514690 0.514690i −0.401270 0.915960i \(-0.631431\pi\)
0.915960 + 0.401270i \(0.131431\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.88316 0.149686
\(372\) 0 0
\(373\) −12.6526 + 12.6526i −0.655127 + 0.655127i −0.954223 0.299096i \(-0.903315\pi\)
0.299096 + 0.954223i \(0.403315\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.49265 + 8.49265i 0.437394 + 0.437394i
\(378\) 0 0
\(379\) 37.9577 1.94975 0.974877 0.222743i \(-0.0715010\pi\)
0.974877 + 0.222743i \(0.0715010\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.18320 + 9.18320i −0.469240 + 0.469240i −0.901668 0.432428i \(-0.857657\pi\)
0.432428 + 0.901668i \(0.357657\pi\)
\(384\) 0 0
\(385\) −0.352429 + 0.356015i −0.0179615 + 0.0181442i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4.22258i 0.214093i −0.994254 0.107047i \(-0.965861\pi\)
0.994254 0.107047i \(-0.0341394\pi\)
\(390\) 0 0
\(391\) −31.4408 −1.59003
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.100201 + 19.8002i 0.00504168 + 0.996256i
\(396\) 0 0
\(397\) 1.27679 + 1.27679i 0.0640801 + 0.0640801i 0.738421 0.674340i \(-0.235572\pi\)
−0.674340 + 0.738421i \(0.735572\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.899990i 0.0449434i 0.999747 + 0.0224717i \(0.00715356\pi\)
−0.999747 + 0.0224717i \(0.992846\pi\)
\(402\) 0 0
\(403\) −22.4611 + 22.4611i −1.11886 + 1.11886i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.754855 + 0.754855i 0.0374168 + 0.0374168i
\(408\) 0 0
\(409\) 24.1722i 1.19524i 0.801781 + 0.597618i \(0.203886\pi\)
−0.801781 + 0.597618i \(0.796114\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.80277 5.80277i −0.285536 0.285536i
\(414\) 0 0
\(415\) 10.7538 + 10.6455i 0.527884 + 0.522568i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.45513i 0.413060i −0.978440 0.206530i \(-0.933783\pi\)
0.978440 0.206530i \(-0.0662171\pi\)
\(420\) 0 0
\(421\) 13.8561i 0.675307i 0.941270 + 0.337654i \(0.109633\pi\)
−0.941270 + 0.337654i \(0.890367\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.7925 12.5361i −0.620526 0.608090i
\(426\) 0 0
\(427\) 3.53872 + 3.53872i 0.171250 + 0.171250i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.84351i 0.281472i 0.990047 + 0.140736i \(0.0449469\pi\)
−0.990047 + 0.140736i \(0.955053\pi\)
\(432\) 0 0
\(433\) 10.0632 + 10.0632i 0.483605 + 0.483605i 0.906281 0.422676i \(-0.138909\pi\)
−0.422676 + 0.906281i \(0.638909\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.2889 25.2889i 1.20973 1.20973i
\(438\) 0 0
\(439\) 34.0222i 1.62379i −0.583802 0.811896i \(-0.698436\pi\)
0.583802 0.811896i \(-0.301564\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.45259 + 6.45259i 0.306572 + 0.306572i 0.843578 0.537006i \(-0.180445\pi\)
−0.537006 + 0.843578i \(0.680445\pi\)
\(444\) 0 0
\(445\) −18.6353 + 0.0943062i −0.883396 + 0.00447054i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.72541 0.270199 0.135099 0.990832i \(-0.456865\pi\)
0.135099 + 0.990832i \(0.456865\pi\)
\(450\) 0 0
\(451\) 1.94925i 0.0917866i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.0374275 + 7.39582i 0.00175463 + 0.346721i
\(456\) 0 0
\(457\) 14.9718 14.9718i 0.700352 0.700352i −0.264134 0.964486i \(-0.585086\pi\)
0.964486 + 0.264134i \(0.0850862\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −25.8187 −1.20250 −0.601249 0.799062i \(-0.705330\pi\)
−0.601249 + 0.799062i \(0.705330\pi\)
\(462\) 0 0
\(463\) 20.5140 + 20.5140i 0.953368 + 0.953368i 0.998960 0.0455925i \(-0.0145176\pi\)
−0.0455925 + 0.998960i \(0.514518\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.07112 3.07112i 0.142114 0.142114i −0.632470 0.774585i \(-0.717959\pi\)
0.774585 + 0.632470i \(0.217959\pi\)
\(468\) 0 0
\(469\) 5.24368 0.242131
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.63665 + 1.63665i −0.0752532 + 0.0752532i
\(474\) 0 0
\(475\) 20.3726 0.206202i 0.934761 0.00946120i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 33.2621 1.51979 0.759893 0.650049i \(-0.225252\pi\)
0.759893 + 0.650049i \(0.225252\pi\)
\(480\) 0 0
\(481\) 15.7607 0.718624
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −22.5513 + 22.7807i −1.02400 + 1.03442i
\(486\) 0 0
\(487\) −1.39322 + 1.39322i −0.0631329 + 0.0631329i −0.737968 0.674835i \(-0.764215\pi\)
0.674835 + 0.737968i \(0.264215\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.4709 0.653061 0.326530 0.945187i \(-0.394120\pi\)
0.326530 + 0.945187i \(0.394120\pi\)
\(492\) 0 0
\(493\) −6.13913 + 6.13913i −0.276493 + 0.276493i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.36174 + 6.36174i 0.285363 + 0.285363i
\(498\) 0 0
\(499\) −20.4107 −0.913710 −0.456855 0.889541i \(-0.651024\pi\)
−0.456855 + 0.889541i \(0.651024\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.54697 2.54697i 0.113564 0.113564i −0.648041 0.761605i \(-0.724412\pi\)
0.761605 + 0.648041i \(0.224412\pi\)
\(504\) 0 0
\(505\) −22.1106 + 0.111894i −0.983910 + 0.00497920i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.5045i 0.598575i 0.954163 + 0.299288i \(0.0967490\pi\)
−0.954163 + 0.299288i \(0.903251\pi\)
\(510\) 0 0
\(511\) −4.76715 −0.210886
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −22.4809 22.2545i −0.990628 0.980652i
\(516\) 0 0
\(517\) 0.564820 + 0.564820i 0.0248407 + 0.0248407i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13.5151i 0.592108i −0.955171 0.296054i \(-0.904329\pi\)
0.955171 0.296054i \(-0.0956708\pi\)
\(522\) 0 0
\(523\) 12.6641 12.6641i 0.553760 0.553760i −0.373764 0.927524i \(-0.621933\pi\)
0.927524 + 0.373764i \(0.121933\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.2365 16.2365i −0.707275 0.707275i
\(528\) 0 0
\(529\) 54.0356i 2.34938i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20.3492 + 20.3492i 0.881423 + 0.881423i
\(534\) 0 0
\(535\) −22.6951 + 22.9260i −0.981195 + 0.991177i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.20003i 0.0947619i
\(540\) 0 0
\(541\) 17.1804i 0.738644i −0.929301 0.369322i \(-0.879590\pi\)
0.929301 0.369322i \(-0.120410\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −24.6285 + 0.124636i −1.05497 + 0.00533881i
\(546\) 0 0
\(547\) −17.4105 17.4105i −0.744417 0.744417i 0.229007 0.973425i \(-0.426452\pi\)
−0.973425 + 0.229007i \(0.926452\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.87584i 0.420725i
\(552\) 0 0
\(553\) 4.17924 + 4.17924i 0.177719 + 0.177719i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.22732 + 9.22732i −0.390974 + 0.390974i −0.875035 0.484060i \(-0.839162\pi\)
0.484060 + 0.875035i \(0.339162\pi\)
\(558\) 0 0
\(559\) 34.1717i 1.44531i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.8727 19.8727i −0.837535 0.837535i 0.150999 0.988534i \(-0.451751\pi\)
−0.988534 + 0.150999i \(0.951751\pi\)
\(564\) 0 0
\(565\) 5.45877 5.51430i 0.229652 0.231988i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.73031 −0.198305 −0.0991524 0.995072i \(-0.531613\pi\)
−0.0991524 + 0.995072i \(0.531613\pi\)
\(570\) 0 0
\(571\) 5.54039i 0.231858i −0.993257 0.115929i \(-0.963015\pi\)
0.993257 0.115929i \(-0.0369845\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 30.7157 31.3438i 1.28093 1.30713i
\(576\) 0 0
\(577\) −12.9105 + 12.9105i −0.537472 + 0.537472i −0.922786 0.385314i \(-0.874093\pi\)
0.385314 + 0.922786i \(0.374093\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.51677 0.187387
\(582\) 0 0
\(583\) 1.02523 + 1.02523i 0.0424606 + 0.0424606i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.90309 + 1.90309i −0.0785487 + 0.0785487i −0.745290 0.666741i \(-0.767689\pi\)
0.666741 + 0.745290i \(0.267689\pi\)
\(588\) 0 0
\(589\) 26.1192 1.07622
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 11.7312 11.7312i 0.481745 0.481745i −0.423944 0.905688i \(-0.639355\pi\)
0.905688 + 0.423944i \(0.139355\pi\)
\(594\) 0 0
\(595\) −5.34626 + 0.0270554i −0.219175 + 0.00110916i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.61594 −0.106884 −0.0534422 0.998571i \(-0.517019\pi\)
−0.0534422 + 0.998571i \(0.517019\pi\)
\(600\) 0 0
\(601\) 20.1722 0.822840 0.411420 0.911446i \(-0.365033\pi\)
0.411420 + 0.911446i \(0.365033\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 24.3445 0.123199i 0.989745 0.00500873i
\(606\) 0 0
\(607\) 17.3218 17.3218i 0.703069 0.703069i −0.261999 0.965068i \(-0.584382\pi\)
0.965068 + 0.261999i \(0.0843819\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11.7929 0.477090
\(612\) 0 0
\(613\) 16.8299 16.8299i 0.679754 0.679754i −0.280191 0.959944i \(-0.590398\pi\)
0.959944 + 0.280191i \(0.0903977\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.56139 2.56139i −0.103118 0.103118i 0.653666 0.756783i \(-0.273230\pi\)
−0.756783 + 0.653666i \(0.773230\pi\)
\(618\) 0 0
\(619\) 16.9662 0.681931 0.340965 0.940076i \(-0.389246\pi\)
0.340965 + 0.940076i \(0.389246\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.93336 + 3.93336i −0.157587 + 0.157587i
\(624\) 0 0
\(625\) 24.9949 0.506024i 0.999795 0.0202410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.3930i 0.454269i
\(630\) 0 0
\(631\) 39.5630 1.57498 0.787488 0.616330i \(-0.211381\pi\)
0.787488 + 0.616330i \(0.211381\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 13.3557 13.4915i 0.530003 0.535394i
\(636\) 0 0
\(637\) −22.9673 22.9673i −0.909996 0.909996i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 47.3441i 1.86998i −0.354676 0.934989i \(-0.615409\pi\)
0.354676 0.934989i \(-0.384591\pi\)
\(642\) 0 0
\(643\) −14.8250 + 14.8250i −0.584639 + 0.584639i −0.936175 0.351535i \(-0.885660\pi\)
0.351535 + 0.936175i \(0.385660\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.86574 4.86574i −0.191292 0.191292i 0.604962 0.796254i \(-0.293188\pi\)
−0.796254 + 0.604962i \(0.793188\pi\)
\(648\) 0 0
\(649\) 4.12683i 0.161992i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.8214 11.8214i −0.462609 0.462609i 0.436901 0.899510i \(-0.356076\pi\)
−0.899510 + 0.436901i \(0.856076\pi\)
\(654\) 0 0
\(655\) 25.3444 0.128259i 0.990289 0.00501149i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.02230i 0.234595i 0.993097 + 0.117298i \(0.0374232\pi\)
−0.993097 + 0.117298i \(0.962577\pi\)
\(660\) 0 0
\(661\) 37.9790i 1.47721i −0.674137 0.738606i \(-0.735484\pi\)
0.674137 0.738606i \(-0.264516\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.27842 4.32194i 0.165910 0.167598i
\(666\) 0 0
\(667\) −15.0420 15.0420i −0.582428 0.582428i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.51667i 0.0971551i
\(672\) 0 0
\(673\) 7.99686 + 7.99686i 0.308256 + 0.308256i 0.844233 0.535977i \(-0.180057\pi\)
−0.535977 + 0.844233i \(0.680057\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.11103 + 7.11103i −0.273299 + 0.273299i −0.830427 0.557128i \(-0.811903\pi\)
0.557128 + 0.830427i \(0.311903\pi\)
\(678\) 0 0
\(679\) 9.56827i 0.367197i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.99061 + 7.99061i 0.305752 + 0.305752i 0.843259 0.537507i \(-0.180634\pi\)
−0.537507 + 0.843259i \(0.680634\pi\)
\(684\) 0 0
\(685\) 24.4221 + 24.1762i 0.933120 + 0.923723i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 21.4058 0.815495
\(690\) 0 0
\(691\) 32.5931i 1.23990i 0.784641 + 0.619951i \(0.212848\pi\)
−0.784641 + 0.619951i \(0.787152\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −29.8359 + 0.150989i −1.13174 + 0.00572732i
\(696\) 0 0
\(697\) −14.7100 + 14.7100i −0.557180 + 0.557180i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.9064 1.62055 0.810277 0.586047i \(-0.199317\pi\)
0.810277 + 0.586047i \(0.199317\pi\)
\(702\) 0 0
\(703\) −9.16379 9.16379i −0.345619 0.345619i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.66691 + 4.66691i −0.175517 + 0.175517i
\(708\) 0 0
\(709\) 10.3299 0.387949 0.193975 0.981007i \(-0.437862\pi\)
0.193975 + 0.981007i \(0.437862\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 39.7825 39.7825i 1.48986 1.48986i
\(714\) 0 0
\(715\) −2.61658 + 2.64320i −0.0978545 + 0.0988500i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.49219 0.242118 0.121059 0.992645i \(-0.461371\pi\)
0.121059 + 0.992645i \(0.461371\pi\)
\(720\) 0 0
\(721\) −9.44235 −0.351652
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.122650 12.1177i −0.00455511 0.450042i
\(726\) 0 0
\(727\) 11.0479 11.0479i 0.409744 0.409744i −0.471905 0.881649i \(-0.656434\pi\)
0.881649 + 0.471905i \(0.156434\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24.7019 −0.913631
\(732\) 0 0
\(733\) 19.5376 19.5376i 0.721636 0.721636i −0.247302 0.968938i \(-0.579544\pi\)
0.968938 + 0.247302i \(0.0795441\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.86461 + 1.86461i 0.0686837 + 0.0686837i
\(738\) 0 0
\(739\) −24.8980 −0.915890 −0.457945 0.888981i \(-0.651414\pi\)
−0.457945 + 0.888981i \(0.651414\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.8961 15.8961i 0.583171 0.583171i −0.352602 0.935773i \(-0.614703\pi\)
0.935773 + 0.352602i \(0.114703\pi\)
\(744\) 0 0
\(745\) −0.153851 30.4015i −0.00563665 1.11382i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.62928i 0.351846i
\(750\) 0 0
\(751\) −11.5535 −0.421594 −0.210797 0.977530i \(-0.567606\pi\)
−0.210797 + 0.977530i \(0.567606\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −21.5619 + 0.109117i −0.784717 + 0.00397116i
\(756\) 0 0
\(757\) 18.8612 + 18.8612i 0.685524 + 0.685524i 0.961239 0.275716i \(-0.0889148\pi\)
−0.275716 + 0.961239i \(0.588915\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33.0567i 1.19830i −0.800636 0.599151i \(-0.795505\pi\)
0.800636 0.599151i \(-0.204495\pi\)
\(762\) 0 0
\(763\) −5.19836 + 5.19836i −0.188193 + 0.188193i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −43.0821 43.0821i −1.55561 1.55561i
\(768\) 0 0
\(769\) 44.9240i 1.62000i 0.586429 + 0.810001i \(0.300533\pi\)
−0.586429 + 0.810001i \(0.699467\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −28.3672 28.3672i −1.02030 1.02030i −0.999790 0.0205060i \(-0.993472\pi\)
−0.0205060 0.999790i \(-0.506528\pi\)
\(774\) 0 0
\(775\) 32.0486 0.324380i 1.15122 0.0116521i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23.6635i 0.847832i
\(780\) 0 0
\(781\) 4.52436i 0.161894i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.1339 + 16.9614i 0.611536 + 0.605377i
\(786\) 0 0
\(787\) 4.79302 + 4.79302i 0.170853 + 0.170853i 0.787354 0.616501i \(-0.211450\pi\)
−0.616501 + 0.787354i \(0.711450\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.31610i 0.0823509i
\(792\) 0 0
\(793\) 26.2729 + 26.2729i 0.932977 + 0.932977i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.30136 7.30136i 0.258627 0.258627i −0.565868 0.824496i \(-0.691459\pi\)
0.824496 + 0.565868i \(0.191459\pi\)
\(798\) 0 0
\(799\) 8.52480i 0.301586i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.69516 1.69516i −0.0598208 0.0598208i
\(804\) 0 0
\(805\) −0.0662907 13.0993i −0.00233644 0.461689i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −26.9078 −0.946026 −0.473013 0.881055i \(-0.656834\pi\)
−0.473013 + 0.881055i \(0.656834\pi\)
\(810\) 0 0
\(811\) 4.38227i 0.153882i −0.997036 0.0769412i \(-0.975485\pi\)
0.997036 0.0769412i \(-0.0245154\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 36.1495 36.5173i 1.26626 1.27914i
\(816\) 0 0
\(817\) 19.8686 19.8686i 0.695113 0.695113i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −34.5890 −1.20716 −0.603582 0.797301i \(-0.706261\pi\)
−0.603582 + 0.797301i \(0.706261\pi\)
\(822\) 0 0
\(823\) 5.63496 + 5.63496i 0.196422 + 0.196422i 0.798464 0.602042i \(-0.205646\pi\)
−0.602042 + 0.798464i \(0.705646\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −29.8906 + 29.8906i −1.03940 + 1.03940i −0.0402073 + 0.999191i \(0.512802\pi\)
−0.999191 + 0.0402073i \(0.987198\pi\)
\(828\) 0 0
\(829\) −45.0207 −1.56363 −0.781817 0.623508i \(-0.785707\pi\)
−0.781817 + 0.623508i \(0.785707\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 16.6025 16.6025i 0.575241 0.575241i
\(834\) 0 0
\(835\) −30.8039 30.4937i −1.06601 1.05528i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 6.14853 0.212271 0.106135 0.994352i \(-0.466152\pi\)
0.106135 + 0.994352i \(0.466152\pi\)
\(840\) 0 0
\(841\) 23.1258 0.797442
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.130772 + 25.8411i 0.00449870 + 0.888960i
\(846\) 0 0
\(847\) 5.13842 5.13842i 0.176558 0.176558i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −27.9149 −0.956910
\(852\) 0 0
\(853\) −11.5962 + 11.5962i −0.397046 + 0.397046i −0.877190 0.480144i \(-0.840585\pi\)
0.480144 + 0.877190i \(0.340585\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23.8521 + 23.8521i 0.814772 + 0.814772i 0.985345 0.170573i \(-0.0545619\pi\)
−0.170573 + 0.985345i \(0.554562\pi\)
\(858\) 0 0
\(859\) 31.3817 1.07073 0.535365 0.844621i \(-0.320174\pi\)
0.535365 + 0.844621i \(0.320174\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −12.4212 + 12.4212i −0.422823 + 0.422823i −0.886175 0.463351i \(-0.846647\pi\)
0.463351 + 0.886175i \(0.346647\pi\)
\(864\) 0 0
\(865\) −7.54388 7.46791i −0.256500 0.253917i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.97220i 0.100825i
\(870\) 0 0
\(871\) 38.9312 1.31913
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5.19599 5.35620i 0.175656 0.181073i
\(876\) 0 0
\(877\) 38.0074 + 38.0074i 1.28342 + 1.28342i 0.938710 + 0.344708i \(0.112022\pi\)
0.344708 + 0.938710i \(0.387978\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10.9118i 0.367626i −0.982961 0.183813i \(-0.941156\pi\)
0.982961 0.183813i \(-0.0588442\pi\)
\(882\) 0 0
\(883\) −1.85770 + 1.85770i −0.0625165 + 0.0625165i −0.737674 0.675157i \(-0.764076\pi\)
0.675157 + 0.737674i \(0.264076\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.17163 + 3.17163i 0.106493 + 0.106493i 0.758346 0.651853i \(-0.226008\pi\)
−0.651853 + 0.758346i \(0.726008\pi\)
\(888\) 0 0
\(889\) 5.66665i 0.190053i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.85679 6.85679i −0.229454 0.229454i
\(894\) 0 0
\(895\) 0.121687 + 24.0458i 0.00406754 + 0.803762i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.5358i 0.518150i
\(900\) 0 0
\(901\) 15.4737i 0.515504i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.114617 + 22.6488i 0.00381001 + 0.752873i
\(906\) 0 0
\(907\) 5.88501 + 5.88501i 0.195408 + 0.195408i 0.798028 0.602620i \(-0.205876\pi\)
−0.602620 + 0.798028i \(0.705876\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 10.3886i 0.344188i 0.985080 + 0.172094i \(0.0550534\pi\)
−0.985080 + 0.172094i \(0.944947\pi\)
\(912\) 0 0
\(913\) 1.60613 + 1.60613i 0.0531550 + 0.0531550i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.34947 5.34947i 0.176655 0.176655i
\(918\) 0 0
\(919\) 14.5186i 0.478924i 0.970906 + 0.239462i \(0.0769710\pi\)
−0.970906 + 0.239462i \(0.923029\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 47.2321 + 47.2321i 1.55466 + 1.55466i
\(924\) 0 0
\(925\) −11.3579 11.1302i −0.373444 0.365960i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −44.3689 −1.45570 −0.727848 0.685738i \(-0.759480\pi\)
−0.727848 + 0.685738i \(0.759480\pi\)
\(930\) 0 0
\(931\) 26.7079i 0.875316i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.91070 1.89146i −0.0624867 0.0618574i
\(936\) 0 0
\(937\) −2.66665 + 2.66665i −0.0871157 + 0.0871157i −0.749322 0.662206i \(-0.769620\pi\)
0.662206 + 0.749322i \(0.269620\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −41.4675 −1.35180 −0.675901 0.736992i \(-0.736245\pi\)
−0.675901 + 0.736992i \(0.736245\pi\)
\(942\) 0 0
\(943\) −36.0421 36.0421i −1.17369 1.17369i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 34.3858 34.3858i 1.11739 1.11739i 0.125264 0.992123i \(-0.460022\pi\)
0.992123 0.125264i \(-0.0399779\pi\)
\(948\) 0 0
\(949\) −35.3933 −1.14891
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.82947 + 5.82947i −0.188835 + 0.188835i −0.795192 0.606357i \(-0.792630\pi\)
0.606357 + 0.795192i \(0.292630\pi\)
\(954\) 0 0
\(955\) −0.0489061 9.66403i −0.00158256 0.312721i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.2577 0.331237
\(960\) 0 0
\(961\) 10.0886 0.325439
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 17.9208 + 17.7404i 0.576892 + 0.571083i
\(966\) 0 0
\(967\) −18.1550 + 18.1550i −0.583826 + 0.583826i −0.935952 0.352126i \(-0.885459\pi\)
0.352126 + 0.935952i \(0.385459\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.87264 −0.220554 −0.110277 0.993901i \(-0.535174\pi\)
−0.110277 + 0.993901i \(0.535174\pi\)
\(972\) 0 0
\(973\) −6.29749 + 6.29749i −0.201888 + 0.201888i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −14.8533 14.8533i −0.475200 0.475200i 0.428393 0.903593i \(-0.359080\pi\)
−0.903593 + 0.428393i \(0.859080\pi\)
\(978\) 0 0
\(979\) −2.79734 −0.0894033
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 21.3770 21.3770i 0.681822 0.681822i −0.278589 0.960410i \(-0.589867\pi\)
0.960410 + 0.278589i \(0.0898667\pi\)
\(984\) 0 0
\(985\) 9.57090 9.66827i 0.304954 0.308057i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 60.5240i 1.92455i
\(990\) 0 0
\(991\) −27.1515 −0.862497 −0.431249 0.902233i \(-0.641927\pi\)
−0.431249 + 0.902233i \(0.641927\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.0341864 6.75537i −0.00108378 0.214160i
\(996\) 0 0
\(997\) 16.8850 + 16.8850i 0.534754 + 0.534754i 0.921983 0.387230i \(-0.126568\pi\)
−0.387230 + 0.921983i \(0.626568\pi\)
\(998\) 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.bj.a.593.2 48
3.2 odd 2 inner 1440.2.bj.a.593.23 48
4.3 odd 2 360.2.x.a.53.5 48
5.2 odd 4 inner 1440.2.bj.a.17.1 48
8.3 odd 2 360.2.x.a.53.17 yes 48
8.5 even 2 inner 1440.2.bj.a.593.24 48
12.11 even 2 360.2.x.a.53.20 yes 48
15.2 even 4 inner 1440.2.bj.a.17.24 48
20.7 even 4 360.2.x.a.197.8 yes 48
24.5 odd 2 inner 1440.2.bj.a.593.1 48
24.11 even 2 360.2.x.a.53.8 yes 48
40.27 even 4 360.2.x.a.197.20 yes 48
40.37 odd 4 inner 1440.2.bj.a.17.23 48
60.47 odd 4 360.2.x.a.197.17 yes 48
120.77 even 4 inner 1440.2.bj.a.17.2 48
120.107 odd 4 360.2.x.a.197.5 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.x.a.53.5 48 4.3 odd 2
360.2.x.a.53.8 yes 48 24.11 even 2
360.2.x.a.53.17 yes 48 8.3 odd 2
360.2.x.a.53.20 yes 48 12.11 even 2
360.2.x.a.197.5 yes 48 120.107 odd 4
360.2.x.a.197.8 yes 48 20.7 even 4
360.2.x.a.197.17 yes 48 60.47 odd 4
360.2.x.a.197.20 yes 48 40.27 even 4
1440.2.bj.a.17.1 48 5.2 odd 4 inner
1440.2.bj.a.17.2 48 120.77 even 4 inner
1440.2.bj.a.17.23 48 40.37 odd 4 inner
1440.2.bj.a.17.24 48 15.2 even 4 inner
1440.2.bj.a.593.1 48 24.5 odd 2 inner
1440.2.bj.a.593.2 48 1.1 even 1 trivial
1440.2.bj.a.593.23 48 3.2 odd 2 inner
1440.2.bj.a.593.24 48 8.5 even 2 inner