Properties

Label 6300.2.d.f.3401.15
Level $6300$
Weight $2$
Character 6300.3401
Analytic conductor $50.306$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6300,2,Mod(3401,6300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6300.3401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.d (of order \(2\), degree \(1\), 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: \(50.3057532734\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.101415451701035401216.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3401.15
Root \(-0.752908 - 0.137538i\) of defining polynomial
Character \(\chi\) \(=\) 6300.3401
Dual form 6300.2.d.f.3401.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.57794 + 0.595188i) q^{7} +O(q^{10})\) \(q+(2.57794 + 0.595188i) q^{7} -3.74166i q^{11} -3.36028i q^{13} -0.841723 q^{17} +5.59388i q^{19} +2.35425i q^{23} -1.41421i q^{29} -8.66259i q^{31} +5.15587 q^{37} +5.74103 q^{41} +3.32941 q^{43} -6.43560 q^{47} +(6.29150 + 3.06871i) q^{49} -9.64575i q^{53} -12.2508 q^{59} +1.82646 q^{67} -3.74166i q^{71} +0.979531i q^{73} +(2.22699 - 9.64575i) q^{77} +6.58301 q^{79} +12.5730 q^{83} +2.16991 q^{89} +(2.00000 - 8.66259i) q^{91} +12.0399i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{49} - 64 q^{79} + 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(7\) 2.57794 + 0.595188i 0.974368 + 0.224960i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.74166i 1.12815i −0.825723 0.564076i \(-0.809232\pi\)
0.825723 0.564076i \(-0.190768\pi\)
\(12\) 0 0
\(13\) 3.36028i 0.931975i −0.884791 0.465987i \(-0.845699\pi\)
0.884791 0.465987i \(-0.154301\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.841723 −0.204148 −0.102074 0.994777i \(-0.532548\pi\)
−0.102074 + 0.994777i \(0.532548\pi\)
\(18\) 0 0
\(19\) 5.59388i 1.28332i 0.766987 + 0.641662i \(0.221755\pi\)
−0.766987 + 0.641662i \(0.778245\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.35425i 0.490895i 0.969410 + 0.245447i \(0.0789349\pi\)
−0.969410 + 0.245447i \(0.921065\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41421i 0.262613i −0.991342 0.131306i \(-0.958083\pi\)
0.991342 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) 8.66259i 1.55585i −0.628359 0.777924i \(-0.716273\pi\)
0.628359 0.777924i \(-0.283727\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.15587 0.847620 0.423810 0.905751i \(-0.360692\pi\)
0.423810 + 0.905751i \(0.360692\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.74103 0.896599 0.448300 0.893883i \(-0.352030\pi\)
0.448300 + 0.893883i \(0.352030\pi\)
\(42\) 0 0
\(43\) 3.32941 0.507730 0.253865 0.967240i \(-0.418298\pi\)
0.253865 + 0.967240i \(0.418298\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.43560 −0.938729 −0.469365 0.883004i \(-0.655517\pi\)
−0.469365 + 0.883004i \(0.655517\pi\)
\(48\) 0 0
\(49\) 6.29150 + 3.06871i 0.898786 + 0.438387i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.64575i 1.32495i −0.749086 0.662473i \(-0.769507\pi\)
0.749086 0.662473i \(-0.230493\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.2508 −1.59491 −0.797456 0.603377i \(-0.793822\pi\)
−0.797456 + 0.603377i \(0.793822\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.82646 0.223138 0.111569 0.993757i \(-0.464412\pi\)
0.111569 + 0.993757i \(0.464412\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.74166i 0.444053i −0.975041 0.222027i \(-0.928733\pi\)
0.975041 0.222027i \(-0.0712672\pi\)
\(72\) 0 0
\(73\) 0.979531i 0.114645i 0.998356 + 0.0573227i \(0.0182564\pi\)
−0.998356 + 0.0573227i \(0.981744\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.22699 9.64575i 0.253789 1.09924i
\(78\) 0 0
\(79\) 6.58301 0.740646 0.370323 0.928903i \(-0.379247\pi\)
0.370323 + 0.928903i \(0.379247\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.5730 1.38007 0.690035 0.723776i \(-0.257595\pi\)
0.690035 + 0.723776i \(0.257595\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.16991 0.230010 0.115005 0.993365i \(-0.463312\pi\)
0.115005 + 0.993365i \(0.463312\pi\)
\(90\) 0 0
\(91\) 2.00000 8.66259i 0.209657 0.908086i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0399i 1.22247i 0.791450 + 0.611234i \(0.209327\pi\)
−0.791450 + 0.611234i \(0.790673\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.74103 −0.571254 −0.285627 0.958341i \(-0.592202\pi\)
−0.285627 + 0.958341i \(0.592202\pi\)
\(102\) 0 0
\(103\) 13.4411i 1.32439i −0.749330 0.662197i \(-0.769624\pi\)
0.749330 0.662197i \(-0.230376\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.9373i 1.63739i −0.574231 0.818693i \(-0.694699\pi\)
0.574231 0.818693i \(-0.305301\pi\)
\(108\) 0 0
\(109\) −17.8745 −1.71207 −0.856034 0.516920i \(-0.827078\pi\)
−0.856034 + 0.516920i \(0.827078\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.93725i 0.464458i −0.972661 0.232229i \(-0.925398\pi\)
0.972661 0.232229i \(-0.0746018\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.16991 0.500983i −0.198915 0.0459251i
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.3117 0.915019 0.457510 0.889205i \(-0.348742\pi\)
0.457510 + 0.889205i \(0.348742\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.91094 0.691182 0.345591 0.938385i \(-0.387678\pi\)
0.345591 + 0.938385i \(0.387678\pi\)
\(132\) 0 0
\(133\) −3.32941 + 14.4207i −0.288696 + 1.25043i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.2288i 1.04477i −0.852709 0.522387i \(-0.825042\pi\)
0.852709 0.522387i \(-0.174958\pi\)
\(138\) 0 0
\(139\) 3.61226i 0.306388i −0.988196 0.153194i \(-0.951044\pi\)
0.988196 0.153194i \(-0.0489559\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.5730 −1.05141
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 17.3828i 1.42406i 0.702151 + 0.712028i \(0.252223\pi\)
−0.702151 + 0.712028i \(0.747777\pi\)
\(150\) 0 0
\(151\) −5.29150 −0.430616 −0.215308 0.976546i \(-0.569076\pi\)
−0.215308 + 0.976546i \(0.569076\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20.3725i 1.62591i −0.582329 0.812953i \(-0.697859\pi\)
0.582329 0.812953i \(-0.302141\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.40122 + 6.06910i −0.110432 + 0.478312i
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −20.6921 −1.60120 −0.800601 0.599198i \(-0.795486\pi\)
−0.800601 + 0.599198i \(0.795486\pi\)
\(168\) 0 0
\(169\) 1.70850 0.131423
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −18.4651 −1.40387 −0.701937 0.712239i \(-0.747681\pi\)
−0.701937 + 0.712239i \(0.747681\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.39655i 0.627587i −0.949491 0.313794i \(-0.898400\pi\)
0.949491 0.313794i \(-0.101600\pi\)
\(180\) 0 0
\(181\) 26.5313i 1.97206i 0.166575 + 0.986029i \(0.446729\pi\)
−0.166575 + 0.986029i \(0.553271\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.14944i 0.230310i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.8818i 1.22153i 0.791813 + 0.610763i \(0.209137\pi\)
−0.791813 + 0.610763i \(0.790863\pi\)
\(192\) 0 0
\(193\) 15.4676 1.11338 0.556692 0.830719i \(-0.312071\pi\)
0.556692 + 0.830719i \(0.312071\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.8118i 1.91026i −0.296189 0.955129i \(-0.595716\pi\)
0.296189 0.955129i \(-0.404284\pi\)
\(198\) 0 0
\(199\) 6.68097i 0.473601i 0.971558 + 0.236801i \(0.0760988\pi\)
−0.971558 + 0.236801i \(0.923901\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.841723 3.64575i 0.0590774 0.255882i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 20.9304 1.44779
\(210\) 0 0
\(211\) −5.29150 −0.364282 −0.182141 0.983272i \(-0.558303\pi\)
−0.182141 + 0.983272i \(0.558303\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.15587 22.3316i 0.350003 1.51597i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.82843i 0.190261i
\(222\) 0 0
\(223\) 14.2098i 0.951560i −0.879564 0.475780i \(-0.842166\pi\)
0.879564 0.475780i \(-0.157834\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.0270 1.13012 0.565061 0.825049i \(-0.308853\pi\)
0.565061 + 0.825049i \(0.308853\pi\)
\(228\) 0 0
\(229\) 25.4442i 1.68140i −0.541499 0.840701i \(-0.682143\pi\)
0.541499 0.840701i \(-0.317857\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.6458i 1.41806i −0.705178 0.709030i \(-0.749133\pi\)
0.705178 0.709030i \(-0.250867\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.0554i 0.973851i −0.873444 0.486925i \(-0.838118\pi\)
0.873444 0.486925i \(-0.161882\pi\)
\(240\) 0 0
\(241\) 8.11905i 0.522994i 0.965204 + 0.261497i \(0.0842161\pi\)
−0.965204 + 0.261497i \(0.915784\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 18.7970 1.19603
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.5906 −1.04719 −0.523594 0.851968i \(-0.675409\pi\)
−0.523594 + 0.851968i \(0.675409\pi\)
\(252\) 0 0
\(253\) 8.80879 0.553804
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.89206 −0.367537 −0.183768 0.982970i \(-0.558830\pi\)
−0.183768 + 0.982970i \(0.558830\pi\)
\(258\) 0 0
\(259\) 13.2915 + 3.06871i 0.825894 + 0.190681i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.64575i 0.594783i 0.954756 + 0.297391i \(0.0961166\pi\)
−0.954756 + 0.297391i \(0.903883\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 25.9027 1.57932 0.789659 0.613546i \(-0.210258\pi\)
0.789659 + 0.613546i \(0.210258\pi\)
\(270\) 0 0
\(271\) 14.8000i 0.899037i 0.893271 + 0.449519i \(0.148404\pi\)
−0.893271 + 0.449519i \(0.851596\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 25.4558 1.52949 0.764747 0.644331i \(-0.222864\pi\)
0.764747 + 0.644331i \(0.222864\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.06713i 0.302280i 0.988512 + 0.151140i \(0.0482944\pi\)
−0.988512 + 0.151140i \(0.951706\pi\)
\(282\) 0 0
\(283\) 27.6510i 1.64368i 0.569719 + 0.821839i \(0.307052\pi\)
−0.569719 + 0.821839i \(0.692948\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.8000 + 3.41699i 0.873617 + 0.201699i
\(288\) 0 0
\(289\) −16.2915 −0.958324
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 19.2540 1.12483 0.562415 0.826855i \(-0.309872\pi\)
0.562415 + 0.826855i \(0.309872\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.91094 0.457502
\(300\) 0 0
\(301\) 8.58301 + 1.98162i 0.494716 + 0.114219i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 18.5496i 1.05868i 0.848409 + 0.529342i \(0.177561\pi\)
−0.848409 + 0.529342i \(0.822439\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.9304 1.18685 0.593427 0.804888i \(-0.297775\pi\)
0.593427 + 0.804888i \(0.297775\pi\)
\(312\) 0 0
\(313\) 4.55066i 0.257218i 0.991695 + 0.128609i \(0.0410513\pi\)
−0.991695 + 0.128609i \(0.958949\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.8118i 0.831911i −0.909385 0.415956i \(-0.863447\pi\)
0.909385 0.415956i \(-0.136553\pi\)
\(318\) 0 0
\(319\) −5.29150 −0.296267
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.70850i 0.261988i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.5906 3.83039i −0.914668 0.211176i
\(330\) 0 0
\(331\) 11.8745 0.652682 0.326341 0.945252i \(-0.394184\pi\)
0.326341 + 0.945252i \(0.394184\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.9529 1.30480 0.652399 0.757876i \(-0.273763\pi\)
0.652399 + 0.757876i \(0.273763\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −32.4125 −1.75523
\(342\) 0 0
\(343\) 14.3926 + 11.6556i 0.777129 + 0.629342i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.9373i 1.55343i −0.629850 0.776717i \(-0.716884\pi\)
0.629850 0.776717i \(-0.283116\pi\)
\(348\) 0 0
\(349\) 29.6000i 1.58445i −0.610227 0.792227i \(-0.708922\pi\)
0.610227 0.792227i \(-0.291078\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.4651 −0.982797 −0.491399 0.870935i \(-0.663514\pi\)
−0.491399 + 0.870935i \(0.663514\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.1975i 1.54099i 0.637449 + 0.770493i \(0.279990\pi\)
−0.637449 + 0.770493i \(0.720010\pi\)
\(360\) 0 0
\(361\) −12.2915 −0.646921
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 27.6510i 1.44337i −0.692223 0.721684i \(-0.743368\pi\)
0.692223 0.721684i \(-0.256632\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.74103 24.8661i 0.298060 1.29098i
\(372\) 0 0
\(373\) 30.9352 1.60177 0.800883 0.598821i \(-0.204364\pi\)
0.800883 + 0.598821i \(0.204364\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.75216 −0.244749
\(378\) 0 0
\(379\) 2.70850 0.139126 0.0695631 0.997578i \(-0.477839\pi\)
0.0695631 + 0.997578i \(0.477839\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 20.6921 1.05732 0.528658 0.848835i \(-0.322695\pi\)
0.528658 + 0.848835i \(0.322695\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.06713i 0.256914i −0.991715 0.128457i \(-0.958998\pi\)
0.991715 0.128457i \(-0.0410024\pi\)
\(390\) 0 0
\(391\) 1.98162i 0.100215i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.0327i 0.804660i −0.915495 0.402330i \(-0.868201\pi\)
0.915495 0.402330i \(-0.131799\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.5504i 0.626740i −0.949631 0.313370i \(-0.898542\pi\)
0.949631 0.313370i \(-0.101458\pi\)
\(402\) 0 0
\(403\) −29.1088 −1.45001
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 19.2915i 0.956244i
\(408\) 0 0
\(409\) 14.2565i 0.704937i 0.935824 + 0.352469i \(0.114658\pi\)
−0.935824 + 0.352469i \(0.885342\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −31.5817 7.29150i −1.55403 0.358791i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7.14226 −0.348922 −0.174461 0.984664i \(-0.555818\pi\)
−0.174461 + 0.984664i \(0.555818\pi\)
\(420\) 0 0
\(421\) 23.8745 1.16357 0.581786 0.813342i \(-0.302354\pi\)
0.581786 + 0.813342i \(0.302354\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.39655i 0.404447i 0.979339 + 0.202224i \(0.0648168\pi\)
−0.979339 + 0.202224i \(0.935183\pi\)
\(432\) 0 0
\(433\) 35.7727i 1.71913i −0.511028 0.859564i \(-0.670735\pi\)
0.511028 0.859564i \(-0.329265\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13.1694 −0.629977
\(438\) 0 0
\(439\) 4.50679i 0.215098i 0.994200 + 0.107549i \(0.0343002\pi\)
−0.994200 + 0.107549i \(0.965700\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.64575i 0.458283i 0.973393 + 0.229142i \(0.0735919\pi\)
−0.973393 + 0.229142i \(0.926408\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.6965i 1.35427i −0.735858 0.677136i \(-0.763221\pi\)
0.735858 0.677136i \(-0.236779\pi\)
\(450\) 0 0
\(451\) 21.4810i 1.01150i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.98823 −0.467230 −0.233615 0.972329i \(-0.575055\pi\)
−0.233615 + 0.972329i \(0.575055\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −31.0112 −1.44434 −0.722169 0.691717i \(-0.756855\pi\)
−0.722169 + 0.691717i \(0.756855\pi\)
\(462\) 0 0
\(463\) −16.9706 −0.788689 −0.394344 0.918963i \(-0.629028\pi\)
−0.394344 + 0.918963i \(0.629028\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.7105 −0.865816 −0.432908 0.901438i \(-0.642513\pi\)
−0.432908 + 0.901438i \(0.642513\pi\)
\(468\) 0 0
\(469\) 4.70850 + 1.08709i 0.217418 + 0.0501970i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.4575i 0.572797i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.9304 −0.956334 −0.478167 0.878269i \(-0.658699\pi\)
−0.478167 + 0.878269i \(0.658699\pi\)
\(480\) 0 0
\(481\) 17.3252i 0.789960i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.98233 0.316400 0.158200 0.987407i \(-0.449431\pi\)
0.158200 + 0.987407i \(0.449431\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.1916i 1.18201i −0.806668 0.591005i \(-0.798731\pi\)
0.806668 0.591005i \(-0.201269\pi\)
\(492\) 0 0
\(493\) 1.19038i 0.0536118i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.22699 9.64575i 0.0998941 0.432671i
\(498\) 0 0
\(499\) −18.7085 −0.837507 −0.418754 0.908100i \(-0.637533\pi\)
−0.418754 + 0.908100i \(0.637533\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.82087 −0.348715 −0.174358 0.984682i \(-0.555785\pi\)
−0.174358 + 0.984682i \(0.555785\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −30.2425 −1.34048 −0.670239 0.742146i \(-0.733808\pi\)
−0.670239 + 0.742146i \(0.733808\pi\)
\(510\) 0 0
\(511\) −0.583005 + 2.52517i −0.0257906 + 0.111707i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 24.0798i 1.05903i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.2231 0.754558 0.377279 0.926100i \(-0.376860\pi\)
0.377279 + 0.926100i \(0.376860\pi\)
\(522\) 0 0
\(523\) 13.4411i 0.587740i −0.955845 0.293870i \(-0.905057\pi\)
0.955845 0.293870i \(-0.0949432\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.29150i 0.317623i
\(528\) 0 0
\(529\) 17.4575 0.759022
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 19.2915i 0.835608i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11.4821 23.5406i 0.494568 1.01397i
\(540\) 0 0
\(541\) −19.8745 −0.854472 −0.427236 0.904140i \(-0.640513\pi\)
−0.427236 + 0.904140i \(0.640513\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −25.4558 −1.08841 −0.544207 0.838951i \(-0.683169\pi\)
−0.544207 + 0.838951i \(0.683169\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.91094 0.337017
\(552\) 0 0
\(553\) 16.9706 + 3.91813i 0.721662 + 0.166616i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.5203i 0.827100i −0.910481 0.413550i \(-0.864289\pi\)
0.910481 0.413550i \(-0.135711\pi\)
\(558\) 0 0
\(559\) 11.1878i 0.473192i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.27980 0.0960823 0.0480411 0.998845i \(-0.484702\pi\)
0.0480411 + 0.998845i \(0.484702\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22.2152i 0.931308i 0.884967 + 0.465654i \(0.154181\pi\)
−0.884967 + 0.465654i \(0.845819\pi\)
\(570\) 0 0
\(571\) 22.5830 0.945069 0.472535 0.881312i \(-0.343339\pi\)
0.472535 + 0.881312i \(0.343339\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 27.0931i 1.12790i 0.825809 + 0.563950i \(0.190719\pi\)
−0.825809 + 0.563950i \(0.809281\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 32.4125 + 7.48331i 1.34470 + 0.310460i
\(582\) 0 0
\(583\) −36.0911 −1.49474
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −32.9669 −1.36069 −0.680345 0.732892i \(-0.738170\pi\)
−0.680345 + 0.732892i \(0.738170\pi\)
\(588\) 0 0
\(589\) 48.4575 1.99666
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 27.3730 1.12408 0.562038 0.827111i \(-0.310017\pi\)
0.562038 + 0.827111i \(0.310017\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26.1916i 1.07016i 0.844801 + 0.535080i \(0.179719\pi\)
−0.844801 + 0.535080i \(0.820281\pi\)
\(600\) 0 0
\(601\) 27.4259i 1.11872i −0.828923 0.559362i \(-0.811046\pi\)
0.828923 0.559362i \(-0.188954\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 34.3715i 1.39510i 0.716538 + 0.697548i \(0.245726\pi\)
−0.716538 + 0.697548i \(0.754274\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.6255i 0.874872i
\(612\) 0 0
\(613\) 27.2823 1.10192 0.550961 0.834531i \(-0.314261\pi\)
0.550961 + 0.834531i \(0.314261\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.52026i 0.302754i 0.988476 + 0.151377i \(0.0483708\pi\)
−0.988476 + 0.151377i \(0.951629\pi\)
\(618\) 0 0
\(619\) 20.9374i 0.841547i 0.907166 + 0.420773i \(0.138241\pi\)
−0.907166 + 0.420773i \(0.861759\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.59388 + 1.29150i 0.224114 + 0.0517430i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.33981 −0.173040
\(630\) 0 0
\(631\) −21.1660 −0.842606 −0.421303 0.906920i \(-0.638427\pi\)
−0.421303 + 0.906920i \(0.638427\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 10.3117 21.1412i 0.408566 0.837646i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.23871i 0.0884236i −0.999022 0.0442118i \(-0.985922\pi\)
0.999022 0.0442118i \(-0.0140776\pi\)
\(642\) 0 0
\(643\) 30.0317i 1.18433i −0.805815 0.592167i \(-0.798272\pi\)
0.805815 0.592167i \(-0.201728\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.1461 0.988593 0.494297 0.869293i \(-0.335426\pi\)
0.494297 + 0.869293i \(0.335426\pi\)
\(648\) 0 0
\(649\) 45.8381i 1.79930i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.1033i 0.864968i −0.901642 0.432484i \(-0.857637\pi\)
0.901642 0.432484i \(-0.142363\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.09070i 0.0424878i −0.999774 0.0212439i \(-0.993237\pi\)
0.999774 0.0212439i \(-0.00676264\pi\)
\(660\) 0 0
\(661\) 6.13742i 0.238718i 0.992851 + 0.119359i \(0.0380840\pi\)
−0.992851 + 0.119359i \(0.961916\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.32941 0.128915
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.16177 0.314613 0.157307 0.987550i \(-0.449719\pi\)
0.157307 + 0.987550i \(0.449719\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 33.8086 1.29937 0.649686 0.760203i \(-0.274900\pi\)
0.649686 + 0.760203i \(0.274900\pi\)
\(678\) 0 0
\(679\) −7.16601 + 31.0381i −0.275006 + 1.19113i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.5203i 0.746922i 0.927646 + 0.373461i \(0.121829\pi\)
−0.927646 + 0.373461i \(0.878171\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −32.4125 −1.23482
\(690\) 0 0
\(691\) 2.52517i 0.0960619i −0.998846 0.0480310i \(-0.984705\pi\)
0.998846 0.0480310i \(-0.0152946\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.83236 −0.183039
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.7299i 0.518571i −0.965801 0.259285i \(-0.916513\pi\)
0.965801 0.259285i \(-0.0834870\pi\)
\(702\) 0 0
\(703\) 28.8413i 1.08777i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.8000 3.41699i −0.556612 0.128509i
\(708\) 0 0
\(709\) −49.1660 −1.84647 −0.923234 0.384238i \(-0.874464\pi\)
−0.923234 + 0.384238i \(0.874464\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.3939 0.763757
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 38.2896 1.42796 0.713981 0.700165i \(-0.246890\pi\)
0.713981 + 0.700165i \(0.246890\pi\)
\(720\) 0 0
\(721\) 8.00000 34.6504i 0.297936 1.29045i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.91813i 0.145315i −0.997357 0.0726576i \(-0.976852\pi\)
0.997357 0.0726576i \(-0.0231480\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.80244 −0.103652
\(732\) 0 0
\(733\) 13.9990i 0.517064i 0.966003 + 0.258532i \(0.0832387\pi\)
−0.966003 + 0.258532i \(0.916761\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.83399i 0.251733i
\(738\) 0 0
\(739\) 9.16601 0.337177 0.168589 0.985687i \(-0.446079\pi\)
0.168589 + 0.985687i \(0.446079\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.9373i 1.06160i 0.847496 + 0.530802i \(0.178109\pi\)
−0.847496 + 0.530802i \(0.821891\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.0808 43.6631i 0.368346 1.59542i
\(750\) 0 0
\(751\) −29.2915 −1.06886 −0.534431 0.845212i \(-0.679474\pi\)
−0.534431 + 0.845212i \(0.679474\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13.6412 0.495796 0.247898 0.968786i \(-0.420260\pi\)
0.247898 + 0.968786i \(0.420260\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 51.1729 1.85502 0.927509 0.373802i \(-0.121946\pi\)
0.927509 + 0.373802i \(0.121946\pi\)
\(762\) 0 0
\(763\) −46.0793 10.6387i −1.66818 0.385146i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 41.1660i 1.48642i
\(768\) 0 0
\(769\) 27.4259i 0.989002i 0.869177 + 0.494501i \(0.164649\pi\)
−0.869177 + 0.494501i \(0.835351\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 11.1349 0.400496 0.200248 0.979745i \(-0.435825\pi\)
0.200248 + 0.979745i \(0.435825\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.1147i 1.15063i
\(780\) 0 0
\(781\) −14.0000 −0.500959
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 24.0798i 0.858353i −0.903221 0.429177i \(-0.858804\pi\)
0.903221 0.429177i \(-0.141196\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.93859 12.7279i 0.104484 0.452553i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.2860 −0.931096 −0.465548 0.885023i \(-0.654143\pi\)
−0.465548 + 0.885023i \(0.654143\pi\)
\(798\) 0 0
\(799\) 5.41699 0.191639
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.66507 0.129338
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.1445i 1.76299i 0.472197 + 0.881493i \(0.343461\pi\)
−0.472197 + 0.881493i \(0.656539\pi\)
\(810\) 0 0
\(811\) 14.8000i 0.519699i 0.965649 + 0.259849i \(0.0836729\pi\)
−0.965649 + 0.259849i \(0.916327\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 18.6243i 0.651583i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.89949i 0.345495i 0.984966 + 0.172747i \(0.0552644\pi\)
−0.984966 + 0.172747i \(0.944736\pi\)
\(822\) 0 0
\(823\) 27.2823 0.951001 0.475501 0.879715i \(-0.342267\pi\)
0.475501 + 0.879715i \(0.342267\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 52.9373i 1.84081i 0.390968 + 0.920404i \(0.372140\pi\)
−0.390968 + 0.920404i \(0.627860\pi\)
\(828\) 0 0
\(829\) 15.3436i 0.532904i −0.963848 0.266452i \(-0.914149\pi\)
0.963848 0.266452i \(-0.0858514\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.29570 2.58301i −0.183485 0.0894958i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.33981 0.149827 0.0749135 0.997190i \(-0.476132\pi\)
0.0749135 + 0.997190i \(0.476132\pi\)
\(840\) 0 0
\(841\) 27.0000 0.931034
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −7.73381 1.78556i −0.265737 0.0613527i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.1382i 0.416092i
\(852\) 0 0
\(853\) 0.210845i 0.00721918i 0.999993 + 0.00360959i \(0.00114897\pi\)
−0.999993 + 0.00360959i \(0.998851\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −16.1853 −0.552879 −0.276439 0.961031i \(-0.589155\pi\)
−0.276439 + 0.961031i \(0.589155\pi\)
\(858\) 0 0
\(859\) 39.1572i 1.33603i −0.744150 0.668013i \(-0.767145\pi\)
0.744150 0.668013i \(-0.232855\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 48.6863i 1.65730i 0.559767 + 0.828650i \(0.310891\pi\)
−0.559767 + 0.828650i \(0.689109\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.6314i 0.835561i
\(870\) 0 0
\(871\) 6.13742i 0.207959i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 16.9706 0.573055 0.286528 0.958072i \(-0.407499\pi\)
0.286528 + 0.958072i \(0.407499\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 15.1894 0.511742 0.255871 0.966711i \(-0.417638\pi\)
0.255871 + 0.966711i \(0.417638\pi\)
\(882\) 0 0
\(883\) 1.50295 0.0505783 0.0252891 0.999680i \(-0.491949\pi\)
0.0252891 + 0.999680i \(0.491949\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.66507 0.123061 0.0615305 0.998105i \(-0.480402\pi\)
0.0615305 + 0.998105i \(0.480402\pi\)
\(888\) 0 0
\(889\) 26.5830 + 6.13742i 0.891565 + 0.205843i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 36.0000i 1.20469i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.2508 −0.408586
\(900\) 0 0
\(901\) 8.11905i 0.270485i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −29.4323 −0.977283 −0.488641 0.872485i \(-0.662507\pi\)
−0.488641 + 0.872485i \(0.662507\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 49.6435i 1.64476i 0.568936 + 0.822381i \(0.307355\pi\)
−0.568936 + 0.822381i \(0.692645\pi\)
\(912\) 0 0
\(913\) 47.0440i 1.55693i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 20.3939 + 4.70850i 0.673466 + 0.155488i
\(918\) 0 0
\(919\) −47.8745 −1.57923 −0.789617 0.613600i \(-0.789721\pi\)
−0.789617 + 0.613600i \(0.789721\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.5730 −0.413846
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.70728 −0.121632 −0.0608160 0.998149i \(-0.519370\pi\)
−0.0608160 + 0.998149i \(0.519370\pi\)
\(930\) 0 0
\(931\) −17.1660 + 35.1939i −0.562593 + 1.15343i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.7533i 0.743318i 0.928369 + 0.371659i \(0.121211\pi\)
−0.928369 + 0.371659i \(0.878789\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −6.50972 −0.212211 −0.106105 0.994355i \(-0.533838\pi\)
−0.106105 + 0.994355i \(0.533838\pi\)
\(942\) 0 0
\(943\) 13.5158i 0.440136i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.9373i 0.940334i 0.882577 + 0.470167i \(0.155806\pi\)
−0.882577 + 0.470167i \(0.844194\pi\)
\(948\) 0 0
\(949\) 3.29150 0.106847
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.81176i 0.0910819i −0.998962 0.0455409i \(-0.985499\pi\)
0.998962 0.0455409i \(-0.0145011\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 7.27841 31.5249i 0.235032 1.01799i
\(960\) 0 0
\(961\) −44.0405 −1.42066
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −5.47938 −0.176205 −0.0881025 0.996111i \(-0.528080\pi\)
−0.0881025 + 0.996111i \(0.528080\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −41.8608 −1.34338 −0.671688 0.740834i \(-0.734430\pi\)
−0.671688 + 0.740834i \(0.734430\pi\)
\(972\) 0 0
\(973\) 2.14997 9.31216i 0.0689249 0.298534i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 60.6863i 1.94153i 0.240040 + 0.970763i \(0.422839\pi\)
−0.240040 + 0.970763i \(0.577161\pi\)
\(978\) 0 0
\(979\) 8.11905i 0.259486i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −11.9767 −0.381996 −0.190998 0.981590i \(-0.561172\pi\)
−0.190998 + 0.981590i \(0.561172\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.83826i 0.249242i
\(990\) 0 0
\(991\) 14.4575 0.459258 0.229629 0.973278i \(-0.426249\pi\)
0.229629 + 0.973278i \(0.426249\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.46878i 0.268209i 0.990967 + 0.134105i \(0.0428158\pi\)
−0.990967 + 0.134105i \(0.957184\pi\)
\(998\) 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 6300.2.d.f.3401.15 16
3.2 odd 2 inner 6300.2.d.f.3401.16 16
5.2 odd 4 1260.2.f.b.629.5 yes 8
5.3 odd 4 1260.2.f.a.629.6 yes 8
5.4 even 2 inner 6300.2.d.f.3401.1 16
7.6 odd 2 inner 6300.2.d.f.3401.13 16
15.2 even 4 1260.2.f.a.629.4 yes 8
15.8 even 4 1260.2.f.b.629.3 yes 8
15.14 odd 2 inner 6300.2.d.f.3401.2 16
20.3 even 4 5040.2.k.f.1889.6 8
20.7 even 4 5040.2.k.e.1889.5 8
21.20 even 2 inner 6300.2.d.f.3401.14 16
35.13 even 4 1260.2.f.a.629.3 8
35.27 even 4 1260.2.f.b.629.4 yes 8
35.34 odd 2 inner 6300.2.d.f.3401.3 16
60.23 odd 4 5040.2.k.e.1889.3 8
60.47 odd 4 5040.2.k.f.1889.4 8
105.62 odd 4 1260.2.f.a.629.5 yes 8
105.83 odd 4 1260.2.f.b.629.6 yes 8
105.104 even 2 inner 6300.2.d.f.3401.4 16
140.27 odd 4 5040.2.k.e.1889.4 8
140.83 odd 4 5040.2.k.f.1889.3 8
420.83 even 4 5040.2.k.e.1889.6 8
420.167 even 4 5040.2.k.f.1889.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.2.f.a.629.3 8 35.13 even 4
1260.2.f.a.629.4 yes 8 15.2 even 4
1260.2.f.a.629.5 yes 8 105.62 odd 4
1260.2.f.a.629.6 yes 8 5.3 odd 4
1260.2.f.b.629.3 yes 8 15.8 even 4
1260.2.f.b.629.4 yes 8 35.27 even 4
1260.2.f.b.629.5 yes 8 5.2 odd 4
1260.2.f.b.629.6 yes 8 105.83 odd 4
5040.2.k.e.1889.3 8 60.23 odd 4
5040.2.k.e.1889.4 8 140.27 odd 4
5040.2.k.e.1889.5 8 20.7 even 4
5040.2.k.e.1889.6 8 420.83 even 4
5040.2.k.f.1889.3 8 140.83 odd 4
5040.2.k.f.1889.4 8 60.47 odd 4
5040.2.k.f.1889.5 8 420.167 even 4
5040.2.k.f.1889.6 8 20.3 even 4
6300.2.d.f.3401.1 16 5.4 even 2 inner
6300.2.d.f.3401.2 16 15.14 odd 2 inner
6300.2.d.f.3401.3 16 35.34 odd 2 inner
6300.2.d.f.3401.4 16 105.104 even 2 inner
6300.2.d.f.3401.13 16 7.6 odd 2 inner
6300.2.d.f.3401.14 16 21.20 even 2 inner
6300.2.d.f.3401.15 16 1.1 even 1 trivial
6300.2.d.f.3401.16 16 3.2 odd 2 inner