Properties

Label 7200.2.k.u.3601.1
Level $7200$
Weight $2$
Character 7200.3601
Analytic conductor $57.492$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7200,2,Mod(3601,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.3601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.k (of order \(2\), degree \(1\), 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: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.180227832610816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{53}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3601.1
Root \(1.37729 + 0.321037i\) of defining polynomial
Character \(\chi\) \(=\) 7200.3601
Dual form 7200.2.k.u.3601.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.05705 q^{7} +O(q^{10})\) \(q-4.05705 q^{7} -0.985939i q^{11} -4.94567i q^{13} +4.52323 q^{17} -2.60492i q^{19} -3.53729 q^{23} -7.59434i q^{29} +3.28415 q^{31} -0.945668i q^{37} -0.568295 q^{41} +8.45963i q^{43} -2.60492 q^{47} +9.45963 q^{49} +0.229815i q^{53} -9.10003i q^{59} +11.0183i q^{61} +8.45963i q^{67} +1.43171 q^{71} -11.9507 q^{73} +4.00000i q^{77} +3.28415 q^{79} -9.89134i q^{83} +12.3510 q^{89} +20.0648i q^{91} +3.23797 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 32 q^{31} + 8 q^{41} + 12 q^{49} + 32 q^{71} + 32 q^{79} - 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 0
\(6\) 0 0
\(7\) −4.05705 −1.53342 −0.766710 0.641994i \(-0.778107\pi\)
−0.766710 + 0.641994i \(0.778107\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 0.985939i − 0.297272i −0.988892 0.148636i \(-0.952512\pi\)
0.988892 0.148636i \(-0.0474882\pi\)
\(12\) 0 0
\(13\) − 4.94567i − 1.37168i −0.727752 0.685841i \(-0.759435\pi\)
0.727752 0.685841i \(-0.240565\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.52323 1.09704 0.548522 0.836136i \(-0.315191\pi\)
0.548522 + 0.836136i \(0.315191\pi\)
\(18\) 0 0
\(19\) − 2.60492i − 0.597610i −0.954314 0.298805i \(-0.903412\pi\)
0.954314 0.298805i \(-0.0965881\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.53729 −0.737577 −0.368788 0.929513i \(-0.620227\pi\)
−0.368788 + 0.929513i \(0.620227\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 7.59434i − 1.41023i −0.709091 0.705117i \(-0.750895\pi\)
0.709091 0.705117i \(-0.249105\pi\)
\(30\) 0 0
\(31\) 3.28415 0.589850 0.294925 0.955520i \(-0.404705\pi\)
0.294925 + 0.955520i \(0.404705\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 0.945668i − 0.155467i −0.996974 0.0777334i \(-0.975232\pi\)
0.996974 0.0777334i \(-0.0247683\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.568295 −0.0887527 −0.0443763 0.999015i \(-0.514130\pi\)
−0.0443763 + 0.999015i \(0.514130\pi\)
\(42\) 0 0
\(43\) 8.45963i 1.29008i 0.764148 + 0.645041i \(0.223160\pi\)
−0.764148 + 0.645041i \(0.776840\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.60492 −0.379967 −0.189984 0.981787i \(-0.560843\pi\)
−0.189984 + 0.981787i \(0.560843\pi\)
\(48\) 0 0
\(49\) 9.45963 1.35138
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.229815i 0.0315675i 0.999875 + 0.0157838i \(0.00502434\pi\)
−0.999875 + 0.0157838i \(0.994976\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 9.10003i − 1.18472i −0.805672 0.592362i \(-0.798196\pi\)
0.805672 0.592362i \(-0.201804\pi\)
\(60\) 0 0
\(61\) 11.0183i 1.41075i 0.708832 + 0.705377i \(0.249222\pi\)
−0.708832 + 0.705377i \(0.750778\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.45963i 1.03351i 0.856134 + 0.516754i \(0.172860\pi\)
−0.856134 + 0.516754i \(0.827140\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.43171 0.169912 0.0849561 0.996385i \(-0.472925\pi\)
0.0849561 + 0.996385i \(0.472925\pi\)
\(72\) 0 0
\(73\) −11.9507 −1.39873 −0.699363 0.714767i \(-0.746533\pi\)
−0.699363 + 0.714767i \(0.746533\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) 3.28415 0.369495 0.184748 0.982786i \(-0.440853\pi\)
0.184748 + 0.982786i \(0.440853\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 9.89134i − 1.08572i −0.839825 0.542858i \(-0.817342\pi\)
0.839825 0.542858i \(-0.182658\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.3510 1.30920 0.654600 0.755976i \(-0.272837\pi\)
0.654600 + 0.755976i \(0.272837\pi\)
\(90\) 0 0
\(91\) 20.0648i 2.10336i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.23797 0.328766 0.164383 0.986397i \(-0.447437\pi\)
0.164383 + 0.986397i \(0.447437\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 4.35637i − 0.433475i −0.976230 0.216738i \(-0.930458\pi\)
0.976230 0.216738i \(-0.0695416\pi\)
\(102\) 0 0
\(103\) −15.0754 −1.48542 −0.742711 0.669612i \(-0.766460\pi\)
−0.742711 + 0.669612i \(0.766460\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) − 4.17034i − 0.399446i −0.979852 0.199723i \(-0.935996\pi\)
0.979852 0.199723i \(-0.0640042\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.28526 −0.120907 −0.0604537 0.998171i \(-0.519255\pi\)
−0.0604537 + 0.998171i \(0.519255\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −18.3510 −1.68223
\(120\) 0 0
\(121\) 10.0279 0.911630
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.15280 −0.102294 −0.0511472 0.998691i \(-0.516288\pi\)
−0.0511472 + 0.998691i \(0.516288\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 3.89019i − 0.339887i −0.985454 0.169944i \(-0.945641\pi\)
0.985454 0.169944i \(-0.0543586\pi\)
\(132\) 0 0
\(133\) 10.5683i 0.916387i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −17.5135 −1.49628 −0.748138 0.663544i \(-0.769052\pi\)
−0.748138 + 0.663544i \(0.769052\pi\)
\(138\) 0 0
\(139\) 16.8612i 1.43015i 0.699047 + 0.715076i \(0.253608\pi\)
−0.699047 + 0.715076i \(0.746392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.87613 −0.407762
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 10.4986i − 0.860078i −0.902810 0.430039i \(-0.858500\pi\)
0.902810 0.430039i \(-0.141500\pi\)
\(150\) 0 0
\(151\) −4.71585 −0.383771 −0.191885 0.981417i \(-0.561460\pi\)
−0.191885 + 0.981417i \(0.561460\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 8.94567i − 0.713942i −0.934115 0.356971i \(-0.883809\pi\)
0.934115 0.356971i \(-0.116191\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 14.3510 1.13101
\(162\) 0 0
\(163\) − 15.7827i − 1.23619i −0.786102 0.618097i \(-0.787904\pi\)
0.786102 0.618097i \(-0.212096\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.50917 −0.426312 −0.213156 0.977018i \(-0.568374\pi\)
−0.213156 + 0.977018i \(0.568374\pi\)
\(168\) 0 0
\(169\) −11.4596 −0.881510
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.3385i 0.786020i 0.919534 + 0.393010i \(0.128566\pi\)
−0.919534 + 0.393010i \(0.871434\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.1746i 1.20895i 0.796625 + 0.604474i \(0.206617\pi\)
−0.796625 + 0.604474i \(0.793383\pi\)
\(180\) 0 0
\(181\) 8.11409i 0.603116i 0.953448 + 0.301558i \(0.0975067\pi\)
−0.953448 + 0.301558i \(0.902493\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 4.45963i − 0.326120i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.9193 −1.80309 −0.901547 0.432681i \(-0.857568\pi\)
−0.901547 + 0.432681i \(0.857568\pi\)
\(192\) 0 0
\(193\) −1.03951 −0.0748254 −0.0374127 0.999300i \(-0.511912\pi\)
−0.0374127 + 0.999300i \(0.511912\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 9.66152i − 0.688355i −0.938905 0.344177i \(-0.888158\pi\)
0.938905 0.344177i \(-0.111842\pi\)
\(198\) 0 0
\(199\) −23.0668 −1.63516 −0.817582 0.575813i \(-0.804686\pi\)
−0.817582 + 0.575813i \(0.804686\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 30.8106i 2.16248i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.56829 −0.177653
\(210\) 0 0
\(211\) − 6.44154i − 0.443454i −0.975109 0.221727i \(-0.928831\pi\)
0.975109 0.221727i \(-0.0711694\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −13.3239 −0.904488
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 22.3704i − 1.50480i
\(222\) 0 0
\(223\) −17.9796 −1.20401 −0.602003 0.798494i \(-0.705630\pi\)
−0.602003 + 0.798494i \(0.705630\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.02792i 0.466460i 0.972422 + 0.233230i \(0.0749295\pi\)
−0.972422 + 0.233230i \(0.925071\pi\)
\(228\) 0 0
\(229\) 4.17034i 0.275584i 0.990461 + 0.137792i \(0.0440005\pi\)
−0.990461 + 0.137792i \(0.955999\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −23.9894 −1.57160 −0.785799 0.618483i \(-0.787748\pi\)
−0.785799 + 0.618483i \(0.787748\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.91926 −0.576939 −0.288469 0.957489i \(-0.593146\pi\)
−0.288469 + 0.957489i \(0.593146\pi\)
\(240\) 0 0
\(241\) −16.3510 −1.05326 −0.526629 0.850095i \(-0.676544\pi\)
−0.526629 + 0.850095i \(0.676544\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.8831 −0.819731
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 4.22391i − 0.266611i −0.991075 0.133305i \(-0.957441\pi\)
0.991075 0.133305i \(-0.0425591\pi\)
\(252\) 0 0
\(253\) 3.48755i 0.219261i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.6952 1.54044 0.770221 0.637777i \(-0.220146\pi\)
0.770221 + 0.637777i \(0.220146\pi\)
\(258\) 0 0
\(259\) 3.83662i 0.238396i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −14.6628 −0.904145 −0.452073 0.891981i \(-0.649315\pi\)
−0.452073 + 0.891981i \(0.649315\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.5381i 0.703490i 0.936096 + 0.351745i \(0.114412\pi\)
−0.936096 + 0.351745i \(0.885588\pi\)
\(270\) 0 0
\(271\) −5.63511 −0.342309 −0.171154 0.985244i \(-0.554750\pi\)
−0.171154 + 0.985244i \(0.554750\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.4053i 1.04578i 0.852399 + 0.522892i \(0.175147\pi\)
−0.852399 + 0.522892i \(0.824853\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −21.7827 −1.29945 −0.649723 0.760171i \(-0.725115\pi\)
−0.649723 + 0.760171i \(0.725115\pi\)
\(282\) 0 0
\(283\) 21.5962i 1.28376i 0.766804 + 0.641881i \(0.221846\pi\)
−0.766804 + 0.641881i \(0.778154\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.30560 0.136095
\(288\) 0 0
\(289\) 3.45963 0.203508
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 32.0125i − 1.87019i −0.354398 0.935095i \(-0.615314\pi\)
0.354398 0.935095i \(-0.384686\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 17.4943i 1.01172i
\(300\) 0 0
\(301\) − 34.3211i − 1.97824i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.13659i 0.0648686i 0.999474 + 0.0324343i \(0.0103260\pi\)
−0.999474 + 0.0324343i \(0.989674\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.13659 0.291269 0.145635 0.989338i \(-0.453478\pi\)
0.145635 + 0.989338i \(0.453478\pi\)
\(312\) 0 0
\(313\) −23.0762 −1.30434 −0.652172 0.758071i \(-0.726142\pi\)
−0.652172 + 0.758071i \(0.726142\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.66152i − 0.0933203i −0.998911 0.0466601i \(-0.985142\pi\)
0.998911 0.0466601i \(-0.0148578\pi\)
\(318\) 0 0
\(319\) −7.48755 −0.419223
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 11.7827i − 0.655605i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.5683 0.582649
\(330\) 0 0
\(331\) − 25.9077i − 1.42402i −0.702171 0.712008i \(-0.747786\pi\)
0.702171 0.712008i \(-0.252214\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.00696 −0.436167 −0.218083 0.975930i \(-0.569980\pi\)
−0.218083 + 0.975930i \(0.569980\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 3.23797i − 0.175346i
\(342\) 0 0
\(343\) −9.97884 −0.538806
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 23.0279i − 1.23620i −0.786098 0.618102i \(-0.787902\pi\)
0.786098 0.618102i \(-0.212098\pi\)
\(348\) 0 0
\(349\) 21.4380i 1.14755i 0.819012 + 0.573776i \(0.194522\pi\)
−0.819012 + 0.573776i \(0.805478\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.52323 −0.240747 −0.120374 0.992729i \(-0.538409\pi\)
−0.120374 + 0.992729i \(0.538409\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.3510 0.546303 0.273152 0.961971i \(-0.411934\pi\)
0.273152 + 0.961971i \(0.411934\pi\)
\(360\) 0 0
\(361\) 12.2144 0.642862
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.485359 −0.0253355 −0.0126678 0.999920i \(-0.504032\pi\)
−0.0126678 + 0.999920i \(0.504032\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 0.932371i − 0.0484063i
\(372\) 0 0
\(373\) 30.0823i 1.55760i 0.627272 + 0.778800i \(0.284171\pi\)
−0.627272 + 0.778800i \(0.715829\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −37.5591 −1.93439
\(378\) 0 0
\(379\) − 33.6881i − 1.73044i −0.501392 0.865220i \(-0.667179\pi\)
0.501392 0.865220i \(-0.332821\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.17545 0.264453 0.132227 0.991220i \(-0.457787\pi\)
0.132227 + 0.991220i \(0.457787\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.6408i 0.843722i 0.906660 + 0.421861i \(0.138623\pi\)
−0.906660 + 0.421861i \(0.861377\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.4332i 1.02551i 0.858534 + 0.512757i \(0.171376\pi\)
−0.858534 + 0.512757i \(0.828624\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.56829 0.228130 0.114065 0.993473i \(-0.463613\pi\)
0.114065 + 0.993473i \(0.463613\pi\)
\(402\) 0 0
\(403\) − 16.2423i − 0.809087i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.932371 −0.0462159
\(408\) 0 0
\(409\) 19.8913 0.983563 0.491782 0.870719i \(-0.336346\pi\)
0.491782 + 0.870719i \(0.336346\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 36.9193i 1.81668i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 0.387288i − 0.0189203i −0.999955 0.00946013i \(-0.996989\pi\)
0.999955 0.00946013i \(-0.00301130\pi\)
\(420\) 0 0
\(421\) − 12.0578i − 0.587664i −0.955857 0.293832i \(-0.905069\pi\)
0.955857 0.293832i \(-0.0949306\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 44.7019i − 2.16328i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 40.4068 1.94633 0.973164 0.230113i \(-0.0739096\pi\)
0.973164 + 0.230113i \(0.0739096\pi\)
\(432\) 0 0
\(433\) −36.1859 −1.73898 −0.869491 0.493949i \(-0.835553\pi\)
−0.869491 + 0.493949i \(0.835553\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.21438i 0.440783i
\(438\) 0 0
\(439\) 25.4178 1.21312 0.606562 0.795036i \(-0.292548\pi\)
0.606562 + 0.795036i \(0.292548\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.02792i 0.333907i 0.985965 + 0.166953i \(0.0533929\pi\)
−0.985965 + 0.166953i \(0.946607\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 0.560304i 0.0263837i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −25.2747 −1.18230 −0.591149 0.806562i \(-0.701326\pi\)
−0.591149 + 0.806562i \(0.701326\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 41.0902i 1.91376i 0.290479 + 0.956881i \(0.406185\pi\)
−0.290479 + 0.956881i \(0.593815\pi\)
\(462\) 0 0
\(463\) −13.2106 −0.613951 −0.306975 0.951717i \(-0.599317\pi\)
−0.306975 + 0.951717i \(0.599317\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 1.89134i − 0.0875206i −0.999042 0.0437603i \(-0.986066\pi\)
0.999042 0.0437603i \(-0.0139338\pi\)
\(468\) 0 0
\(469\) − 34.3211i − 1.58480i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.34068 0.383505
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 31.4876 1.43870 0.719352 0.694646i \(-0.244439\pi\)
0.719352 + 0.694646i \(0.244439\pi\)
\(480\) 0 0
\(481\) −4.67696 −0.213251
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.9964 0.588922 0.294461 0.955664i \(-0.404860\pi\)
0.294461 + 0.955664i \(0.404860\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.9085i 0.672812i 0.941717 + 0.336406i \(0.109212\pi\)
−0.941717 + 0.336406i \(0.890788\pi\)
\(492\) 0 0
\(493\) − 34.3510i − 1.54709i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.80850 −0.260547
\(498\) 0 0
\(499\) 35.6599i 1.59636i 0.602420 + 0.798179i \(0.294203\pi\)
−0.602420 + 0.798179i \(0.705797\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.3090 1.12847 0.564237 0.825613i \(-0.309170\pi\)
0.564237 + 0.825613i \(0.309170\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.7366i 0.608862i 0.952534 + 0.304431i \(0.0984663\pi\)
−0.952534 + 0.304431i \(0.901534\pi\)
\(510\) 0 0
\(511\) 48.4846 2.14483
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.56829i 0.112953i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.9193 1.35460 0.677299 0.735708i \(-0.263150\pi\)
0.677299 + 0.735708i \(0.263150\pi\)
\(522\) 0 0
\(523\) 21.8385i 0.954932i 0.878650 + 0.477466i \(0.158445\pi\)
−0.878650 + 0.477466i \(0.841555\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.8550 0.647092
\(528\) 0 0
\(529\) −10.4876 −0.455981
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.81060i 0.121740i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 9.32662i − 0.401726i
\(540\) 0 0
\(541\) 24.3423i 1.04656i 0.852162 + 0.523278i \(0.175291\pi\)
−0.852162 + 0.523278i \(0.824709\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 33.3789i 1.42718i 0.700564 + 0.713589i \(0.252932\pi\)
−0.700564 + 0.713589i \(0.747068\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −19.7827 −0.842770
\(552\) 0 0
\(553\) −13.3239 −0.566592
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.5808i 1.29575i 0.761747 + 0.647875i \(0.224342\pi\)
−0.761747 + 0.647875i \(0.775658\pi\)
\(558\) 0 0
\(559\) 41.8385 1.76958
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 7.02792i − 0.296192i −0.988973 0.148096i \(-0.952686\pi\)
0.988973 0.148096i \(-0.0473144\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.3510 −1.02085 −0.510423 0.859924i \(-0.670511\pi\)
−0.510423 + 0.859924i \(0.670511\pi\)
\(570\) 0 0
\(571\) − 20.0992i − 0.841125i −0.907263 0.420563i \(-0.861833\pi\)
0.907263 0.420563i \(-0.138167\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.76899 0.198536 0.0992678 0.995061i \(-0.468350\pi\)
0.0992678 + 0.995061i \(0.468350\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 40.1296i 1.66486i
\(582\) 0 0
\(583\) 0.226584 0.00938413
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.21733i 0.339165i 0.985516 + 0.169583i \(0.0542420\pi\)
−0.985516 + 0.169583i \(0.945758\pi\)
\(588\) 0 0
\(589\) − 8.55495i − 0.352501i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −7.76120 −0.318714 −0.159357 0.987221i \(-0.550942\pi\)
−0.159357 + 0.987221i \(0.550942\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.64903 −0.230813 −0.115407 0.993318i \(-0.536817\pi\)
−0.115407 + 0.993318i \(0.536817\pi\)
\(600\) 0 0
\(601\) −37.7299 −1.53903 −0.769516 0.638627i \(-0.779503\pi\)
−0.769516 + 0.638627i \(0.779503\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.113292 −0.00459837 −0.00229919 0.999997i \(-0.500732\pi\)
−0.00229919 + 0.999997i \(0.500732\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.8831i 0.521194i
\(612\) 0 0
\(613\) 0.703366i 0.0284087i 0.999899 + 0.0142044i \(0.00452154\pi\)
−0.999899 + 0.0142044i \(0.995478\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.4809 0.985564 0.492782 0.870153i \(-0.335980\pi\)
0.492782 + 0.870153i \(0.335980\pi\)
\(618\) 0 0
\(619\) − 39.4966i − 1.58750i −0.608243 0.793751i \(-0.708126\pi\)
0.608243 0.793751i \(-0.291874\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −50.1084 −2.00755
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 4.27748i − 0.170554i
\(630\) 0 0
\(631\) −17.3400 −0.690294 −0.345147 0.938549i \(-0.612171\pi\)
−0.345147 + 0.938549i \(0.612171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 46.7842i − 1.85366i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −38.7019 −1.52863 −0.764317 0.644840i \(-0.776924\pi\)
−0.764317 + 0.644840i \(0.776924\pi\)
\(642\) 0 0
\(643\) − 1.13659i − 0.0448227i −0.999749 0.0224113i \(-0.992866\pi\)
0.999749 0.0224113i \(-0.00713435\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.54868 −0.257455 −0.128728 0.991680i \(-0.541089\pi\)
−0.128728 + 0.991680i \(0.541089\pi\)
\(648\) 0 0
\(649\) −8.97208 −0.352185
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.74226i 0.342111i 0.985261 + 0.171056i \(0.0547178\pi\)
−0.985261 + 0.171056i \(0.945282\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 35.5336i − 1.38419i −0.721804 0.692097i \(-0.756687\pi\)
0.721804 0.692097i \(-0.243313\pi\)
\(660\) 0 0
\(661\) − 23.3028i − 0.906373i −0.891416 0.453186i \(-0.850287\pi\)
0.891416 0.453186i \(-0.149713\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 26.8634i 1.04016i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.8634 0.419377
\(672\) 0 0
\(673\) 14.3634 0.553670 0.276835 0.960917i \(-0.410714\pi\)
0.276835 + 0.960917i \(0.410714\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 24.3076i − 0.934217i −0.884200 0.467109i \(-0.845296\pi\)
0.884200 0.467109i \(-0.154704\pi\)
\(678\) 0 0
\(679\) −13.1366 −0.504136
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 38.5933i − 1.47673i −0.674401 0.738365i \(-0.735598\pi\)
0.674401 0.738365i \(-0.264402\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.13659 0.0433006
\(690\) 0 0
\(691\) − 13.4090i − 0.510102i −0.966928 0.255051i \(-0.917908\pi\)
0.966928 0.255051i \(-0.0820923\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.57053 −0.0973657
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.6013i 0.589253i 0.955613 + 0.294626i \(0.0951952\pi\)
−0.955613 + 0.294626i \(0.904805\pi\)
\(702\) 0 0
\(703\) −2.46339 −0.0929086
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 17.6740i 0.664699i
\(708\) 0 0
\(709\) 15.7873i 0.592906i 0.955047 + 0.296453i \(0.0958038\pi\)
−0.955047 + 0.296453i \(0.904196\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −11.6170 −0.435060
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22.5683 0.841655 0.420828 0.907141i \(-0.361740\pi\)
0.420828 + 0.907141i \(0.361740\pi\)
\(720\) 0 0
\(721\) 61.1616 2.27778
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.79096 0.103511 0.0517554 0.998660i \(-0.483518\pi\)
0.0517554 + 0.998660i \(0.483518\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 38.2649i 1.41528i
\(732\) 0 0
\(733\) − 16.4860i − 0.608926i −0.952524 0.304463i \(-0.901523\pi\)
0.952524 0.304463i \(-0.0984769\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.34068 0.307233
\(738\) 0 0
\(739\) − 14.8894i − 0.547714i −0.961770 0.273857i \(-0.911701\pi\)
0.961770 0.273857i \(-0.0882995\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −41.4301 −1.51992 −0.759961 0.649968i \(-0.774782\pi\)
−0.759961 + 0.649968i \(0.774782\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 16.2282i − 0.592965i
\(750\) 0 0
\(751\) −30.5544 −1.11494 −0.557472 0.830195i \(-0.688229\pi\)
−0.557472 + 0.830195i \(0.688229\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 0.433223i − 0.0157457i −0.999969 0.00787287i \(-0.997494\pi\)
0.999969 0.00787287i \(-0.00250604\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 14.9193 0.540823 0.270411 0.962745i \(-0.412840\pi\)
0.270411 + 0.962745i \(0.412840\pi\)
\(762\) 0 0
\(763\) 16.9193i 0.612518i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −45.0057 −1.62506
\(768\) 0 0
\(769\) −31.3789 −1.13155 −0.565776 0.824559i \(-0.691423\pi\)
−0.565776 + 0.824559i \(0.691423\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 13.4192i − 0.482656i −0.970444 0.241328i \(-0.922417\pi\)
0.970444 0.241328i \(-0.0775829\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.48036i 0.0530395i
\(780\) 0 0
\(781\) − 1.41157i − 0.0505101i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 36.4846i 1.30054i 0.759705 + 0.650268i \(0.225343\pi\)
−0.759705 + 0.650268i \(0.774657\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.21438 0.185402
\(792\) 0 0
\(793\) 54.4931 1.93511
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 26.0683i − 0.923388i −0.887039 0.461694i \(-0.847242\pi\)
0.887039 0.461694i \(-0.152758\pi\)
\(798\) 0 0
\(799\) −11.7827 −0.416841
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.7827i 0.415801i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −35.6212 −1.25237 −0.626187 0.779673i \(-0.715385\pi\)
−0.626187 + 0.779673i \(0.715385\pi\)
\(810\) 0 0
\(811\) 43.8935i 1.54131i 0.637253 + 0.770654i \(0.280071\pi\)
−0.637253 + 0.770654i \(0.719929\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 22.0367 0.770966
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 28.8058i − 1.00533i −0.864482 0.502665i \(-0.832353\pi\)
0.864482 0.502665i \(-0.167647\pi\)
\(822\) 0 0
\(823\) 27.9585 0.974571 0.487286 0.873243i \(-0.337987\pi\)
0.487286 + 0.873243i \(0.337987\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 14.8634i − 0.516851i −0.966031 0.258426i \(-0.916796\pi\)
0.966031 0.258426i \(-0.0832037\pi\)
\(828\) 0 0
\(829\) − 41.7678i − 1.45065i −0.688404 0.725327i \(-0.741688\pi\)
0.688404 0.725327i \(-0.258312\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 42.7881 1.48252
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.6490 −0.747408 −0.373704 0.927548i \(-0.621912\pi\)
−0.373704 + 0.927548i \(0.621912\pi\)
\(840\) 0 0
\(841\) −28.6740 −0.988759
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −40.6838 −1.39791
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.34510i 0.114669i
\(852\) 0 0
\(853\) 49.1880i 1.68416i 0.539350 + 0.842082i \(0.318670\pi\)
−0.539350 + 0.842082i \(0.681330\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.65849 −0.0908123 −0.0454062 0.998969i \(-0.514458\pi\)
−0.0454062 + 0.998969i \(0.514458\pi\)
\(858\) 0 0
\(859\) 4.57680i 0.156158i 0.996947 + 0.0780792i \(0.0248787\pi\)
−0.996947 + 0.0780792i \(0.975121\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 34.0218 1.15812 0.579058 0.815287i \(-0.303421\pi\)
0.579058 + 0.815287i \(0.303421\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 3.23797i − 0.109841i
\(870\) 0 0
\(871\) 41.8385 1.41764
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11.7563i 0.396981i 0.980103 + 0.198490i \(0.0636039\pi\)
−0.980103 + 0.198490i \(0.936396\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −13.2702 −0.447085 −0.223543 0.974694i \(-0.571762\pi\)
−0.223543 + 0.974694i \(0.571762\pi\)
\(882\) 0 0
\(883\) − 17.1366i − 0.576692i −0.957526 0.288346i \(-0.906895\pi\)
0.957526 0.288346i \(-0.0931054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.0883 1.07742 0.538709 0.842492i \(-0.318912\pi\)
0.538709 + 0.842492i \(0.318912\pi\)
\(888\) 0 0
\(889\) 4.67696 0.156860
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.78562i 0.227072i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 24.9409i − 0.831827i
\(900\) 0 0
\(901\) 1.03951i 0.0346310i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 40.4596i − 1.34344i −0.740805 0.671720i \(-0.765556\pi\)
0.740805 0.671720i \(-0.234444\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −9.75225 −0.322752
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.7827i 0.521190i
\(918\) 0 0
\(919\) −50.8495 −1.67737 −0.838685 0.544617i \(-0.816675\pi\)
−0.838685 + 0.544617i \(0.816675\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 7.08074i − 0.233065i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −12.6461 −0.414904 −0.207452 0.978245i \(-0.566517\pi\)
−0.207452 + 0.978245i \(0.566517\pi\)
\(930\) 0 0
\(931\) − 24.6416i − 0.807596i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 40.7971 1.33278 0.666391 0.745603i \(-0.267838\pi\)
0.666391 + 0.745603i \(0.267838\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 14.1086i − 0.459928i −0.973199 0.229964i \(-0.926139\pi\)
0.973199 0.229964i \(-0.0738608\pi\)
\(942\) 0 0
\(943\) 2.01022 0.0654619
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.8385i 1.48955i 0.667315 + 0.744776i \(0.267444\pi\)
−0.667315 + 0.744776i \(0.732556\pi\)
\(948\) 0 0
\(949\) 59.1043i 1.91861i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −21.9104 −0.709747 −0.354873 0.934914i \(-0.615476\pi\)
−0.354873 + 0.934914i \(0.615476\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 71.0529 2.29442
\(960\) 0 0
\(961\) −20.2144 −0.652077
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0359 0.451364 0.225682 0.974201i \(-0.427539\pi\)
0.225682 + 0.974201i \(0.427539\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 21.6494i − 0.694762i −0.937724 0.347381i \(-0.887071\pi\)
0.937724 0.347381i \(-0.112929\pi\)
\(972\) 0 0
\(973\) − 68.4068i − 2.19302i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.60225 0.211225 0.105612 0.994407i \(-0.466320\pi\)
0.105612 + 0.994407i \(0.466320\pi\)
\(978\) 0 0
\(979\) − 12.1773i − 0.389188i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 53.8600 1.71787 0.858934 0.512087i \(-0.171127\pi\)
0.858934 + 0.512087i \(0.171127\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 29.9242i − 0.951534i
\(990\) 0 0
\(991\) −29.7129 −0.943861 −0.471931 0.881636i \(-0.656443\pi\)
−0.471931 + 0.881636i \(0.656443\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 16.6506i 0.527328i 0.964615 + 0.263664i \(0.0849311\pi\)
−0.964615 + 0.263664i \(0.915069\pi\)
\(998\) 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 7200.2.k.u.3601.1 12
3.2 odd 2 2400.2.k.f.1201.1 12
4.3 odd 2 1800.2.k.u.901.1 12
5.2 odd 4 1440.2.d.e.1009.2 6
5.3 odd 4 1440.2.d.f.1009.6 6
5.4 even 2 inner 7200.2.k.u.3601.11 12
8.3 odd 2 1800.2.k.u.901.2 12
8.5 even 2 inner 7200.2.k.u.3601.2 12
12.11 even 2 600.2.k.f.301.12 12
15.2 even 4 480.2.d.a.49.5 6
15.8 even 4 480.2.d.b.49.1 6
15.14 odd 2 2400.2.k.f.1201.12 12
20.3 even 4 360.2.d.e.109.4 6
20.7 even 4 360.2.d.f.109.3 6
20.19 odd 2 1800.2.k.u.901.12 12
24.5 odd 2 2400.2.k.f.1201.7 12
24.11 even 2 600.2.k.f.301.11 12
40.3 even 4 360.2.d.f.109.4 6
40.13 odd 4 1440.2.d.e.1009.1 6
40.19 odd 2 1800.2.k.u.901.11 12
40.27 even 4 360.2.d.e.109.3 6
40.29 even 2 inner 7200.2.k.u.3601.12 12
40.37 odd 4 1440.2.d.f.1009.5 6
60.23 odd 4 120.2.d.b.109.3 yes 6
60.47 odd 4 120.2.d.a.109.4 yes 6
60.59 even 2 600.2.k.f.301.1 12
120.29 odd 2 2400.2.k.f.1201.6 12
120.53 even 4 480.2.d.a.49.6 6
120.59 even 2 600.2.k.f.301.2 12
120.77 even 4 480.2.d.b.49.2 6
120.83 odd 4 120.2.d.a.109.3 6
120.107 odd 4 120.2.d.b.109.4 yes 6
240.53 even 4 3840.2.f.m.769.4 12
240.77 even 4 3840.2.f.m.769.3 12
240.83 odd 4 3840.2.f.l.769.3 12
240.107 odd 4 3840.2.f.l.769.4 12
240.173 even 4 3840.2.f.m.769.9 12
240.197 even 4 3840.2.f.m.769.10 12
240.203 odd 4 3840.2.f.l.769.10 12
240.227 odd 4 3840.2.f.l.769.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.d.a.109.3 6 120.83 odd 4
120.2.d.a.109.4 yes 6 60.47 odd 4
120.2.d.b.109.3 yes 6 60.23 odd 4
120.2.d.b.109.4 yes 6 120.107 odd 4
360.2.d.e.109.3 6 40.27 even 4
360.2.d.e.109.4 6 20.3 even 4
360.2.d.f.109.3 6 20.7 even 4
360.2.d.f.109.4 6 40.3 even 4
480.2.d.a.49.5 6 15.2 even 4
480.2.d.a.49.6 6 120.53 even 4
480.2.d.b.49.1 6 15.8 even 4
480.2.d.b.49.2 6 120.77 even 4
600.2.k.f.301.1 12 60.59 even 2
600.2.k.f.301.2 12 120.59 even 2
600.2.k.f.301.11 12 24.11 even 2
600.2.k.f.301.12 12 12.11 even 2
1440.2.d.e.1009.1 6 40.13 odd 4
1440.2.d.e.1009.2 6 5.2 odd 4
1440.2.d.f.1009.5 6 40.37 odd 4
1440.2.d.f.1009.6 6 5.3 odd 4
1800.2.k.u.901.1 12 4.3 odd 2
1800.2.k.u.901.2 12 8.3 odd 2
1800.2.k.u.901.11 12 40.19 odd 2
1800.2.k.u.901.12 12 20.19 odd 2
2400.2.k.f.1201.1 12 3.2 odd 2
2400.2.k.f.1201.6 12 120.29 odd 2
2400.2.k.f.1201.7 12 24.5 odd 2
2400.2.k.f.1201.12 12 15.14 odd 2
3840.2.f.l.769.3 12 240.83 odd 4
3840.2.f.l.769.4 12 240.107 odd 4
3840.2.f.l.769.9 12 240.227 odd 4
3840.2.f.l.769.10 12 240.203 odd 4
3840.2.f.m.769.3 12 240.77 even 4
3840.2.f.m.769.4 12 240.53 even 4
3840.2.f.m.769.9 12 240.173 even 4
3840.2.f.m.769.10 12 240.197 even 4
7200.2.k.u.3601.1 12 1.1 even 1 trivial
7200.2.k.u.3601.2 12 8.5 even 2 inner
7200.2.k.u.3601.11 12 5.4 even 2 inner
7200.2.k.u.3601.12 12 40.29 even 2 inner