Properties

Label 4900.2.e.p.2549.3
Level $4900$
Weight $2$
Character 4900.2549
Analytic conductor $39.127$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4900,2,Mod(2549,4900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4900.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.e (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: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2549.3
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.p.2549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843i q^{3} -5.00000 q^{9} +O(q^{10})\) \(q+2.82843i q^{3} -5.00000 q^{9} +4.00000 q^{11} -4.24264i q^{13} +1.41421i q^{17} -2.82843 q^{19} +4.00000i q^{23} -5.65685i q^{27} -8.00000 q^{29} +11.3137i q^{33} -8.00000i q^{37} +12.0000 q^{39} -7.07107 q^{41} +4.00000i q^{43} +5.65685i q^{47} -4.00000 q^{51} -10.0000i q^{53} -8.00000i q^{57} -14.1421 q^{59} -7.07107 q^{61} -11.3137 q^{69} +7.07107i q^{73} -8.00000 q^{79} +1.00000 q^{81} +14.1421i q^{83} -22.6274i q^{87} -7.07107 q^{89} -1.41421i q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{9} + 16 q^{11} - 32 q^{29} + 48 q^{39} - 16 q^{51} - 32 q^{79} + 4 q^{81} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 2.82843i 1.63299i 0.577350 + 0.816497i \(0.304087\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −5.00000 −1.66667
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) − 4.24264i − 1.17670i −0.808608 0.588348i \(-0.799778\pi\)
0.808608 0.588348i \(-0.200222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.65685i − 1.08866i
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 11.3137i 1.96946i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 12.0000 1.92154
\(40\) 0 0
\(41\) −7.07107 −1.10432 −0.552158 0.833740i \(-0.686195\pi\)
−0.552158 + 0.833740i \(0.686195\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.65685i 0.825137i 0.910927 + 0.412568i \(0.135368\pi\)
−0.910927 + 0.412568i \(0.864632\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 8.00000i − 1.05963i
\(58\) 0 0
\(59\) −14.1421 −1.84115 −0.920575 0.390567i \(-0.872279\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) −7.07107 −0.905357 −0.452679 0.891674i \(-0.649532\pi\)
−0.452679 + 0.891674i \(0.649532\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −11.3137 −1.36201
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 7.07107i 0.827606i 0.910366 + 0.413803i \(0.135800\pi\)
−0.910366 + 0.413803i \(0.864200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.1421i 1.55230i 0.630548 + 0.776151i \(0.282830\pi\)
−0.630548 + 0.776151i \(0.717170\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 22.6274i − 2.42591i
\(88\) 0 0
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.41421i − 0.143592i −0.997419 0.0717958i \(-0.977127\pi\)
0.997419 0.0717958i \(-0.0228730\pi\)
\(98\) 0 0
\(99\) −20.0000 −2.01008
\(100\) 0 0
\(101\) −12.7279 −1.26648 −0.633238 0.773957i \(-0.718274\pi\)
−0.633238 + 0.773957i \(0.718274\pi\)
\(102\) 0 0
\(103\) 11.3137i 1.11477i 0.830253 + 0.557386i \(0.188196\pi\)
−0.830253 + 0.557386i \(0.811804\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) 22.6274 2.14770
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 21.2132i 1.96116i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) − 20.0000i − 1.80334i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 20.0000i − 1.77471i −0.461084 0.887357i \(-0.652539\pi\)
0.461084 0.887357i \(-0.347461\pi\)
\(128\) 0 0
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) −8.48528 −0.741362 −0.370681 0.928760i \(-0.620876\pi\)
−0.370681 + 0.928760i \(0.620876\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −2.82843 −0.239904 −0.119952 0.992780i \(-0.538274\pi\)
−0.119952 + 0.992780i \(0.538274\pi\)
\(140\) 0 0
\(141\) −16.0000 −1.34744
\(142\) 0 0
\(143\) − 16.9706i − 1.41915i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) − 7.07107i − 0.571662i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.07107i 0.564333i 0.959366 + 0.282166i \(0.0910530\pi\)
−0.959366 + 0.282166i \(0.908947\pi\)
\(158\) 0 0
\(159\) 28.2843 2.24309
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 5.65685i − 0.437741i −0.975754 0.218870i \(-0.929763\pi\)
0.975754 0.218870i \(-0.0702371\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 14.1421 1.08148
\(172\) 0 0
\(173\) 4.24264i 0.322562i 0.986909 + 0.161281i \(0.0515625\pi\)
−0.986909 + 0.161281i \(0.948437\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 40.0000i − 3.00658i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −21.2132 −1.57676 −0.788382 0.615185i \(-0.789081\pi\)
−0.788382 + 0.615185i \(0.789081\pi\)
\(182\) 0 0
\(183\) − 20.0000i − 1.47844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.65685i 0.413670i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 20.0000i − 1.39010i
\(208\) 0 0
\(209\) −11.3137 −0.782586
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −20.0000 −1.35147
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) − 16.9706i − 1.13643i −0.822879 0.568216i \(-0.807634\pi\)
0.822879 0.568216i \(-0.192366\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.48528i 0.563188i 0.959534 + 0.281594i \(0.0908631\pi\)
−0.959534 + 0.281594i \(0.909137\pi\)
\(228\) 0 0
\(229\) 21.2132 1.40181 0.700904 0.713256i \(-0.252780\pi\)
0.700904 + 0.713256i \(0.252780\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 22.6274i − 1.46981i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 12.7279 0.819878 0.409939 0.912113i \(-0.365550\pi\)
0.409939 + 0.912113i \(0.365550\pi\)
\(242\) 0 0
\(243\) − 14.1421i − 0.907218i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.0000i 0.763542i
\(248\) 0 0
\(249\) −40.0000 −2.53490
\(250\) 0 0
\(251\) 19.7990 1.24970 0.624851 0.780744i \(-0.285160\pi\)
0.624851 + 0.780744i \(0.285160\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.2132i 1.32324i 0.749838 + 0.661622i \(0.230131\pi\)
−0.749838 + 0.661622i \(0.769869\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 40.0000 2.47594
\(262\) 0 0
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 20.0000i − 1.22398i
\(268\) 0 0
\(269\) 18.3848 1.12094 0.560470 0.828175i \(-0.310621\pi\)
0.560470 + 0.828175i \(0.310621\pi\)
\(270\) 0 0
\(271\) 28.2843 1.71815 0.859074 0.511852i \(-0.171040\pi\)
0.859074 + 0.511852i \(0.171040\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) 2.82843i 0.168133i 0.996460 + 0.0840663i \(0.0267907\pi\)
−0.996460 + 0.0840663i \(0.973209\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) − 32.5269i − 1.90024i −0.311881 0.950121i \(-0.600959\pi\)
0.311881 0.950121i \(-0.399041\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 22.6274i − 1.31298i
\(298\) 0 0
\(299\) 16.9706 0.981433
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 36.0000i − 2.06815i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 19.7990i − 1.12999i −0.825095 0.564994i \(-0.808878\pi\)
0.825095 0.564994i \(-0.191122\pi\)
\(308\) 0 0
\(309\) −32.0000 −1.82042
\(310\) 0 0
\(311\) −22.6274 −1.28308 −0.641542 0.767088i \(-0.721705\pi\)
−0.641542 + 0.767088i \(0.721705\pi\)
\(312\) 0 0
\(313\) − 4.24264i − 0.239808i −0.992785 0.119904i \(-0.961741\pi\)
0.992785 0.119904i \(-0.0382587\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) −32.0000 −1.79166
\(320\) 0 0
\(321\) −22.6274 −1.26294
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 22.6274i 1.25130i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 40.0000i 2.19199i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) 16.9706 0.921714
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 4.24264 0.227103 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) 0 0
\(351\) −24.0000 −1.28103
\(352\) 0 0
\(353\) − 9.89949i − 0.526897i −0.964673 0.263448i \(-0.915140\pi\)
0.964673 0.263448i \(-0.0848599\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 14.1421i 0.742270i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.65685i 0.295285i 0.989041 + 0.147643i \(0.0471686\pi\)
−0.989041 + 0.147643i \(0.952831\pi\)
\(368\) 0 0
\(369\) 35.3553 1.84053
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 33.9411i 1.74806i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 56.5685 2.89809
\(382\) 0 0
\(383\) 5.65685i 0.289052i 0.989501 + 0.144526i \(0.0461657\pi\)
−0.989501 + 0.144526i \(0.953834\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 20.0000i − 1.01666i
\(388\) 0 0
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) −5.65685 −0.286079
\(392\) 0 0
\(393\) − 24.0000i − 1.21064i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 15.5563i 0.780751i 0.920656 + 0.390375i \(0.127655\pi\)
−0.920656 + 0.390375i \(0.872345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 32.0000i − 1.58618i
\(408\) 0 0
\(409\) −38.1838 −1.88807 −0.944033 0.329851i \(-0.893001\pi\)
−0.944033 + 0.329851i \(0.893001\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 8.00000i − 0.391762i
\(418\) 0 0
\(419\) −14.1421 −0.690889 −0.345444 0.938439i \(-0.612272\pi\)
−0.345444 + 0.938439i \(0.612272\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) − 28.2843i − 1.37523i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 48.0000 2.31746
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 0 0
\(433\) 21.2132i 1.01944i 0.860340 + 0.509721i \(0.170251\pi\)
−0.860340 + 0.509721i \(0.829749\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 11.3137i − 0.541208i
\(438\) 0 0
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.0000i 0.760183i 0.924949 + 0.380091i \(0.124107\pi\)
−0.924949 + 0.380091i \(0.875893\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 28.2843i − 1.33780i
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −28.2843 −1.33185
\(452\) 0 0
\(453\) − 11.3137i − 0.531564i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 30.0000i 1.40334i 0.712502 + 0.701670i \(0.247562\pi\)
−0.712502 + 0.701670i \(0.752438\pi\)
\(458\) 0 0
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) 7.07107 0.329332 0.164666 0.986349i \(-0.447345\pi\)
0.164666 + 0.986349i \(0.447345\pi\)
\(462\) 0 0
\(463\) − 40.0000i − 1.85896i −0.368875 0.929479i \(-0.620257\pi\)
0.368875 0.929479i \(-0.379743\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 19.7990i − 0.916188i −0.888904 0.458094i \(-0.848532\pi\)
0.888904 0.458094i \(-0.151468\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 50.0000i 2.28934i
\(478\) 0 0
\(479\) 11.3137 0.516937 0.258468 0.966020i \(-0.416782\pi\)
0.258468 + 0.966020i \(0.416782\pi\)
\(480\) 0 0
\(481\) −33.9411 −1.54758
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 12.0000i − 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 11.3137 0.511624
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) − 11.3137i − 0.509544i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 0 0
\(503\) 39.5980i 1.76559i 0.469762 + 0.882793i \(0.344340\pi\)
−0.469762 + 0.882793i \(0.655660\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 14.1421i − 0.628074i
\(508\) 0 0
\(509\) 18.3848 0.814891 0.407445 0.913230i \(-0.366420\pi\)
0.407445 + 0.913230i \(0.366420\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 16.0000i 0.706417i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 22.6274i 0.995153i
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −41.0122 −1.79678 −0.898388 0.439202i \(-0.855261\pi\)
−0.898388 + 0.439202i \(0.855261\pi\)
\(522\) 0 0
\(523\) − 42.4264i − 1.85518i −0.373603 0.927589i \(-0.621878\pi\)
0.373603 0.927589i \(-0.378122\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 70.7107 3.06858
\(532\) 0 0
\(533\) 30.0000i 1.29944i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) − 60.0000i − 2.57485i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) 35.3553 1.50893
\(550\) 0 0
\(551\) 22.6274 0.963960
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) 16.9706 0.717778
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 0 0
\(563\) − 14.1421i − 0.596020i −0.954563 0.298010i \(-0.903677\pi\)
0.954563 0.298010i \(-0.0963229\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −40.0000 −1.67689 −0.838444 0.544988i \(-0.816534\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) − 45.2548i − 1.89055i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 12.7279i 0.529870i 0.964266 + 0.264935i \(0.0853506\pi\)
−0.964266 + 0.264935i \(0.914649\pi\)
\(578\) 0 0
\(579\) −28.2843 −1.17545
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 40.0000i − 1.65663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.4558i 1.05068i 0.850894 + 0.525338i \(0.176061\pi\)
−0.850894 + 0.525338i \(0.823939\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 28.2843 1.16346
\(592\) 0 0
\(593\) 9.89949i 0.406524i 0.979124 + 0.203262i \(0.0651542\pi\)
−0.979124 + 0.203262i \(0.934846\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −29.6985 −1.21143 −0.605713 0.795683i \(-0.707112\pi\)
−0.605713 + 0.795683i \(0.707112\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 33.9411i − 1.37763i −0.724938 0.688814i \(-0.758132\pi\)
0.724938 0.688814i \(-0.241868\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) − 24.0000i − 0.969351i −0.874694 0.484675i \(-0.838938\pi\)
0.874694 0.484675i \(-0.161062\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 8.00000i − 0.322068i −0.986949 0.161034i \(-0.948517\pi\)
0.986949 0.161034i \(-0.0514829\pi\)
\(618\) 0 0
\(619\) 31.1127 1.25052 0.625262 0.780415i \(-0.284992\pi\)
0.625262 + 0.780415i \(0.284992\pi\)
\(620\) 0 0
\(621\) 22.6274 0.908007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 32.0000i − 1.27796i
\(628\) 0 0
\(629\) 11.3137 0.451107
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 67.8823i 2.69808i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 0 0
\(643\) − 2.82843i − 0.111542i −0.998444 0.0557711i \(-0.982238\pi\)
0.998444 0.0557711i \(-0.0177617\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −56.5685 −2.22051
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 35.3553i − 1.37934i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −21.2132 −0.825098 −0.412549 0.910935i \(-0.635361\pi\)
−0.412549 + 0.910935i \(0.635361\pi\)
\(662\) 0 0
\(663\) 16.9706i 0.659082i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 32.0000i − 1.23904i
\(668\) 0 0
\(669\) 48.0000 1.85579
\(670\) 0 0
\(671\) −28.2843 −1.09190
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.7279i 0.489174i 0.969627 + 0.244587i \(0.0786523\pi\)
−0.969627 + 0.244587i \(0.921348\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 60.0000i 2.28914i
\(688\) 0 0
\(689\) −42.4264 −1.61632
\(690\) 0 0
\(691\) −42.4264 −1.61398 −0.806988 0.590567i \(-0.798904\pi\)
−0.806988 + 0.590567i \(0.798904\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 10.0000i − 0.378777i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) 0 0
\(703\) 22.6274i 0.853409i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) 40.0000 1.50012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 33.9411i 1.26755i
\(718\) 0 0
\(719\) −39.5980 −1.47676 −0.738378 0.674387i \(-0.764408\pi\)
−0.738378 + 0.674387i \(0.764408\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 36.0000i 1.33885i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.2843i 1.04901i 0.851409 + 0.524503i \(0.175749\pi\)
−0.851409 + 0.524503i \(0.824251\pi\)
\(728\) 0 0
\(729\) 43.0000 1.59259
\(730\) 0 0
\(731\) −5.65685 −0.209226
\(732\) 0 0
\(733\) 38.1838i 1.41035i 0.709034 + 0.705175i \(0.249131\pi\)
−0.709034 + 0.705175i \(0.750869\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −33.9411 −1.24686
\(742\) 0 0
\(743\) 20.0000i 0.733729i 0.930274 + 0.366864i \(0.119569\pi\)
−0.930274 + 0.366864i \(0.880431\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 70.7107i − 2.58717i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 56.0000i 2.04075i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 40.0000i − 1.45382i −0.686730 0.726912i \(-0.740955\pi\)
0.686730 0.726912i \(-0.259045\pi\)
\(758\) 0 0
\(759\) −45.2548 −1.64265
\(760\) 0 0
\(761\) 41.0122 1.48669 0.743345 0.668908i \(-0.233238\pi\)
0.743345 + 0.668908i \(0.233238\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 60.0000i 2.16647i
\(768\) 0 0
\(769\) −46.6690 −1.68293 −0.841464 0.540312i \(-0.818306\pi\)
−0.841464 + 0.540312i \(0.818306\pi\)
\(770\) 0 0
\(771\) −60.0000 −2.16085
\(772\) 0 0
\(773\) − 24.0416i − 0.864717i −0.901702 0.432359i \(-0.857681\pi\)
0.901702 0.432359i \(-0.142319\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 45.2548i 1.61728i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 48.0833i 1.71398i 0.515330 + 0.856992i \(0.327669\pi\)
−0.515330 + 0.856992i \(0.672331\pi\)
\(788\) 0 0
\(789\) −67.8823 −2.41667
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 30.0000i 1.06533i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 29.6985i − 1.05197i −0.850493 0.525987i \(-0.823696\pi\)
0.850493 0.525987i \(-0.176304\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 35.3553 1.24922
\(802\) 0 0
\(803\) 28.2843i 0.998130i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 52.0000i 1.83049i
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 8.48528 0.297959 0.148979 0.988840i \(-0.452401\pi\)
0.148979 + 0.988840i \(0.452401\pi\)
\(812\) 0 0
\(813\) 80.0000i 2.80572i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 11.3137i − 0.395817i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) − 40.0000i − 1.39431i −0.716919 0.697156i \(-0.754448\pi\)
0.716919 0.697156i \(-0.245552\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) 0 0
\(829\) 7.07107 0.245588 0.122794 0.992432i \(-0.460815\pi\)
0.122794 + 0.992432i \(0.460815\pi\)
\(830\) 0 0
\(831\) −62.2254 −2.15858
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 11.3137 0.390593 0.195296 0.980744i \(-0.437433\pi\)
0.195296 + 0.980744i \(0.437433\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 45.2548i 1.55866i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) − 12.7279i − 0.435796i −0.975972 0.217898i \(-0.930080\pi\)
0.975972 0.217898i \(-0.0699200\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.07107i 0.241543i 0.992680 + 0.120772i \(0.0385368\pi\)
−0.992680 + 0.120772i \(0.961463\pi\)
\(858\) 0 0
\(859\) −14.1421 −0.482523 −0.241262 0.970460i \(-0.577561\pi\)
−0.241262 + 0.970460i \(0.577561\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.0000i 1.36162i 0.732462 + 0.680808i \(0.238371\pi\)
−0.732462 + 0.680808i \(0.761629\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 42.4264i 1.44088i
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 7.07107i 0.239319i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.00000i 0.270141i 0.990836 + 0.135070i \(0.0431261\pi\)
−0.990836 + 0.135070i \(0.956874\pi\)
\(878\) 0 0
\(879\) 92.0000 3.10308
\(880\) 0 0
\(881\) −21.2132 −0.714691 −0.357345 0.933972i \(-0.616318\pi\)
−0.357345 + 0.933972i \(0.616318\pi\)
\(882\) 0 0
\(883\) 40.0000i 1.34611i 0.739594 + 0.673054i \(0.235018\pi\)
−0.739594 + 0.673054i \(0.764982\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.2843i 0.949693i 0.880069 + 0.474846i \(0.157496\pi\)
−0.880069 + 0.474846i \(0.842504\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) − 16.0000i − 0.535420i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 48.0000i 1.60267i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 14.1421 0.471143
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 8.00000i − 0.265636i −0.991140 0.132818i \(-0.957597\pi\)
0.991140 0.132818i \(-0.0424025\pi\)
\(908\) 0 0
\(909\) 63.6396 2.11079
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) 56.5685i 1.87215i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 56.0000 1.84526
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 56.5685i − 1.85795i
\(928\) 0 0
\(929\) −35.3553 −1.15997 −0.579986 0.814627i \(-0.696942\pi\)
−0.579986 + 0.814627i \(0.696942\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 64.0000i − 2.09527i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 15.5563i 0.508204i 0.967177 + 0.254102i \(0.0817799\pi\)
−0.967177 + 0.254102i \(0.918220\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) 0 0
\(941\) −7.07107 −0.230510 −0.115255 0.993336i \(-0.536769\pi\)
−0.115255 + 0.993336i \(0.536769\pi\)
\(942\) 0 0
\(943\) − 28.2843i − 0.921063i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 60.0000i 1.94974i 0.222779 + 0.974869i \(0.428487\pi\)
−0.222779 + 0.974869i \(0.571513\pi\)
\(948\) 0 0
\(949\) 30.0000 0.973841
\(950\) 0 0
\(951\) 5.65685 0.183436
\(952\) 0 0
\(953\) − 10.0000i − 0.323932i −0.986796 0.161966i \(-0.948217\pi\)
0.986796 0.161966i \(-0.0517835\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 90.5097i − 2.92576i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 40.0000i − 1.28898i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 12.0000i 0.385894i 0.981209 + 0.192947i \(0.0618045\pi\)
−0.981209 + 0.192947i \(0.938195\pi\)
\(968\) 0 0
\(969\) 11.3137 0.363449
\(970\) 0 0
\(971\) 14.1421 0.453843 0.226921 0.973913i \(-0.427134\pi\)
0.226921 + 0.973913i \(0.427134\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 48.0000i 1.53566i 0.640656 + 0.767828i \(0.278662\pi\)
−0.640656 + 0.767828i \(0.721338\pi\)
\(978\) 0 0
\(979\) −28.2843 −0.903969
\(980\) 0 0
\(981\) −40.0000 −1.27710
\(982\) 0 0
\(983\) − 45.2548i − 1.44341i −0.692203 0.721703i \(-0.743360\pi\)
0.692203 0.721703i \(-0.256640\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) − 56.5685i − 1.79515i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.41421i 0.0447886i 0.999749 + 0.0223943i \(0.00712892\pi\)
−0.999749 + 0.0223943i \(0.992871\pi\)
\(998\) 0 0
\(999\) −45.2548 −1.43180
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 4900.2.e.p.2549.3 4
5.2 odd 4 4900.2.a.y.1.2 2
5.3 odd 4 196.2.a.c.1.1 2
5.4 even 2 inner 4900.2.e.p.2549.1 4
7.6 odd 2 inner 4900.2.e.p.2549.2 4
15.8 even 4 1764.2.a.l.1.1 2
20.3 even 4 784.2.a.m.1.2 2
35.3 even 12 196.2.e.b.177.1 4
35.13 even 4 196.2.a.c.1.2 yes 2
35.18 odd 12 196.2.e.b.177.2 4
35.23 odd 12 196.2.e.b.165.2 4
35.27 even 4 4900.2.a.y.1.1 2
35.33 even 12 196.2.e.b.165.1 4
35.34 odd 2 inner 4900.2.e.p.2549.4 4
40.3 even 4 3136.2.a.bs.1.1 2
40.13 odd 4 3136.2.a.br.1.2 2
60.23 odd 4 7056.2.a.cr.1.1 2
105.23 even 12 1764.2.k.l.361.2 4
105.38 odd 12 1764.2.k.l.1549.1 4
105.53 even 12 1764.2.k.l.1549.2 4
105.68 odd 12 1764.2.k.l.361.1 4
105.83 odd 4 1764.2.a.l.1.2 2
140.3 odd 12 784.2.i.l.177.2 4
140.23 even 12 784.2.i.l.753.1 4
140.83 odd 4 784.2.a.m.1.1 2
140.103 odd 12 784.2.i.l.753.2 4
140.123 even 12 784.2.i.l.177.1 4
280.13 even 4 3136.2.a.br.1.1 2
280.83 odd 4 3136.2.a.bs.1.2 2
420.83 even 4 7056.2.a.cr.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.2.a.c.1.1 2 5.3 odd 4
196.2.a.c.1.2 yes 2 35.13 even 4
196.2.e.b.165.1 4 35.33 even 12
196.2.e.b.165.2 4 35.23 odd 12
196.2.e.b.177.1 4 35.3 even 12
196.2.e.b.177.2 4 35.18 odd 12
784.2.a.m.1.1 2 140.83 odd 4
784.2.a.m.1.2 2 20.3 even 4
784.2.i.l.177.1 4 140.123 even 12
784.2.i.l.177.2 4 140.3 odd 12
784.2.i.l.753.1 4 140.23 even 12
784.2.i.l.753.2 4 140.103 odd 12
1764.2.a.l.1.1 2 15.8 even 4
1764.2.a.l.1.2 2 105.83 odd 4
1764.2.k.l.361.1 4 105.68 odd 12
1764.2.k.l.361.2 4 105.23 even 12
1764.2.k.l.1549.1 4 105.38 odd 12
1764.2.k.l.1549.2 4 105.53 even 12
3136.2.a.br.1.1 2 280.13 even 4
3136.2.a.br.1.2 2 40.13 odd 4
3136.2.a.bs.1.1 2 40.3 even 4
3136.2.a.bs.1.2 2 280.83 odd 4
4900.2.a.y.1.1 2 35.27 even 4
4900.2.a.y.1.2 2 5.2 odd 4
4900.2.e.p.2549.1 4 5.4 even 2 inner
4900.2.e.p.2549.2 4 7.6 odd 2 inner
4900.2.e.p.2549.3 4 1.1 even 1 trivial
4900.2.e.p.2549.4 4 35.34 odd 2 inner
7056.2.a.cr.1.1 2 60.23 odd 4
7056.2.a.cr.1.2 2 420.83 even 4