Properties

Label 576.2.c.c.575.3
Level $576$
Weight $2$
Character 576.575
Analytic conductor $4.599$
Analytic rank $0$
Dimension $4$
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] = [576,2,Mod(575,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("576.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.c (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: \(4.59938315643\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.3
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 576.575
Dual form 576.2.c.c.575.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{5} -4.00000i q^{7} +O(q^{10})\) \(q+1.41421i q^{5} -4.00000i q^{7} +5.65685 q^{11} -4.00000 q^{13} -4.24264i q^{17} +5.65685 q^{23} +3.00000 q^{25} -1.41421i q^{29} -4.00000i q^{31} +5.65685 q^{35} +6.00000 q^{37} +9.89949i q^{41} -8.00000i q^{43} +5.65685 q^{47} -9.00000 q^{49} +4.24264i q^{53} +8.00000i q^{55} -11.3137 q^{59} +2.00000 q^{61} -5.65685i q^{65} -8.00000i q^{67} -5.65685 q^{71} -22.6274i q^{77} +4.00000i q^{79} -5.65685 q^{83} +6.00000 q^{85} +4.24264i q^{89} +16.0000i q^{91} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{13} + 12 q^{25} + 24 q^{37} - 36 q^{49} + 8 q^{61} + 24 q^{85} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.24264i − 1.02899i −0.857493 0.514496i \(-0.827979\pi\)
0.857493 0.514496i \(-0.172021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 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
\(31\) − 4.00000i − 0.718421i −0.933257 0.359211i \(-0.883046\pi\)
0.933257 0.359211i \(-0.116954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.65685 0.956183
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.89949i 1.54604i 0.634381 + 0.773021i \(0.281255\pi\)
−0.634381 + 0.773021i \(0.718745\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.65685 0.825137 0.412568 0.910927i \(-0.364632\pi\)
0.412568 + 0.910927i \(0.364632\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.24264i 0.582772i 0.956606 + 0.291386i \(0.0941163\pi\)
−0.956606 + 0.291386i \(0.905884\pi\)
\(54\) 0 0
\(55\) 8.00000i 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 5.65685i − 0.701646i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 22.6274i − 2.57863i
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.65685 −0.620920 −0.310460 0.950586i \(-0.600483\pi\)
−0.310460 + 0.950586i \(0.600483\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.24264i 0.449719i 0.974391 + 0.224860i \(0.0721923\pi\)
−0.974391 + 0.224860i \(0.927808\pi\)
\(90\) 0 0
\(91\) 16.0000i 1.67726i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.3848i 1.82935i 0.404186 + 0.914677i \(0.367555\pi\)
−0.404186 + 0.914677i \(0.632445\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 1.41421i − 0.133038i −0.997785 0.0665190i \(-0.978811\pi\)
0.997785 0.0665190i \(-0.0211893\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −16.9706 −1.55569
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 12.0000i 1.06483i 0.846484 + 0.532414i \(0.178715\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.41421i 0.120824i 0.998174 + 0.0604122i \(0.0192415\pi\)
−0.998174 + 0.0604122i \(0.980758\pi\)
\(138\) 0 0
\(139\) 8.00000i 0.678551i 0.940687 + 0.339276i \(0.110182\pi\)
−0.940687 + 0.339276i \(0.889818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −22.6274 −1.89220
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 1.41421i − 0.115857i −0.998321 0.0579284i \(-0.981550\pi\)
0.998321 0.0579284i \(-0.0184495\pi\)
\(150\) 0 0
\(151\) − 20.0000i − 1.62758i −0.581161 0.813788i \(-0.697401\pi\)
0.581161 0.813788i \(-0.302599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.65685 0.454369
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 22.6274i − 1.78329i
\(162\) 0 0
\(163\) 24.0000i 1.87983i 0.341415 + 0.939913i \(0.389094\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 15.5563i − 1.18273i −0.806405 0.591364i \(-0.798590\pi\)
0.806405 0.591364i \(-0.201410\pi\)
\(174\) 0 0
\(175\) − 12.0000i − 0.907115i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.48528i 0.623850i
\(186\) 0 0
\(187\) − 24.0000i − 1.75505i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.3137 −0.818631 −0.409316 0.912393i \(-0.634232\pi\)
−0.409316 + 0.912393i \(0.634232\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.07107i 0.503793i 0.967754 + 0.251896i \(0.0810542\pi\)
−0.967754 + 0.251896i \(0.918946\pi\)
\(198\) 0 0
\(199\) 20.0000i 1.41776i 0.705328 + 0.708881i \(0.250800\pi\)
−0.705328 + 0.708881i \(0.749200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.65685 −0.397033
\(204\) 0 0
\(205\) −14.0000 −0.977802
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 8.00000i − 0.550743i −0.961338 0.275371i \(-0.911199\pi\)
0.961338 0.275371i \(-0.0888008\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.3137 0.771589
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 16.9706i 1.14156i
\(222\) 0 0
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.65685 0.375459 0.187729 0.982221i \(-0.439887\pi\)
0.187729 + 0.982221i \(0.439887\pi\)
\(228\) 0 0
\(229\) 28.0000 1.85029 0.925146 0.379611i \(-0.123942\pi\)
0.925146 + 0.379611i \(0.123942\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.07107i 0.463241i 0.972806 + 0.231621i \(0.0744028\pi\)
−0.972806 + 0.231621i \(0.925597\pi\)
\(234\) 0 0
\(235\) 8.00000i 0.521862i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.6274 1.46365 0.731823 0.681495i \(-0.238670\pi\)
0.731823 + 0.681495i \(0.238670\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 12.7279i − 0.813157i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.9706 1.07117 0.535586 0.844481i \(-0.320091\pi\)
0.535586 + 0.844481i \(0.320091\pi\)
\(252\) 0 0
\(253\) 32.0000 2.01182
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0416i 1.49968i 0.661622 + 0.749838i \(0.269869\pi\)
−0.661622 + 0.749838i \(0.730131\pi\)
\(258\) 0 0
\(259\) − 24.0000i − 1.49129i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 11.3137 0.697633 0.348817 0.937191i \(-0.386584\pi\)
0.348817 + 0.937191i \(0.386584\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 21.2132i − 1.29339i −0.762748 0.646696i \(-0.776150\pi\)
0.762748 0.646696i \(-0.223850\pi\)
\(270\) 0 0
\(271\) 12.0000i 0.728948i 0.931214 + 0.364474i \(0.118751\pi\)
−0.931214 + 0.364474i \(0.881249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.9706 1.02336
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 7.07107i − 0.421825i −0.977505 0.210912i \(-0.932357\pi\)
0.977505 0.210912i \(-0.0676434\pi\)
\(282\) 0 0
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 39.5980 2.33739
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 7.07107i − 0.413096i −0.978436 0.206548i \(-0.933777\pi\)
0.978436 0.206548i \(-0.0662230\pi\)
\(294\) 0 0
\(295\) − 16.0000i − 0.931556i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −22.6274 −1.30858
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.82843i 0.161955i
\(306\) 0 0
\(307\) − 24.0000i − 1.36975i −0.728659 0.684876i \(-0.759856\pi\)
0.728659 0.684876i \(-0.240144\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.3137 −0.641542 −0.320771 0.947157i \(-0.603942\pi\)
−0.320771 + 0.947157i \(0.603942\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.0416i 1.35031i 0.737675 + 0.675156i \(0.235924\pi\)
−0.737675 + 0.675156i \(0.764076\pi\)
\(318\) 0 0
\(319\) − 8.00000i − 0.447914i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 22.6274i − 1.24749i
\(330\) 0 0
\(331\) 8.00000i 0.439720i 0.975531 + 0.219860i \(0.0705600\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 22.6274i − 1.22534i
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.2843 1.51838 0.759190 0.650870i \(-0.225596\pi\)
0.759190 + 0.650870i \(0.225596\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.89949i 0.526897i 0.964673 + 0.263448i \(0.0848599\pi\)
−0.964673 + 0.263448i \(0.915140\pi\)
\(354\) 0 0
\(355\) − 8.00000i − 0.424596i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.9706 −0.895672 −0.447836 0.894116i \(-0.647805\pi\)
−0.447836 + 0.894116i \(0.647805\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 20.0000i − 1.04399i −0.852948 0.521996i \(-0.825188\pi\)
0.852948 0.521996i \(-0.174812\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 16.9706 0.881068
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.65685i 0.291343i
\(378\) 0 0
\(379\) 16.0000i 0.821865i 0.911666 + 0.410932i \(0.134797\pi\)
−0.911666 + 0.410932i \(0.865203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −22.6274 −1.15621 −0.578103 0.815963i \(-0.696207\pi\)
−0.578103 + 0.815963i \(0.696207\pi\)
\(384\) 0 0
\(385\) 32.0000 1.63087
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.89949i 0.501924i 0.967997 + 0.250962i \(0.0807470\pi\)
−0.967997 + 0.250962i \(0.919253\pi\)
\(390\) 0 0
\(391\) − 24.0000i − 1.21373i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.65685 −0.284627
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 15.5563i − 0.776847i −0.921481 0.388424i \(-0.873020\pi\)
0.921481 0.388424i \(-0.126980\pi\)
\(402\) 0 0
\(403\) 16.0000i 0.797017i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.9411 1.68240
\(408\) 0 0
\(409\) 8.00000 0.395575 0.197787 0.980245i \(-0.436624\pi\)
0.197787 + 0.980245i \(0.436624\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 45.2548i 2.22684i
\(414\) 0 0
\(415\) − 8.00000i − 0.392705i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.65685 −0.276355 −0.138178 0.990407i \(-0.544125\pi\)
−0.138178 + 0.990407i \(0.544125\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 12.7279i − 0.617395i
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.9706 −0.817443 −0.408722 0.912659i \(-0.634025\pi\)
−0.408722 + 0.912659i \(0.634025\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 28.0000i 1.33637i 0.743996 + 0.668184i \(0.232928\pi\)
−0.743996 + 0.668184i \(0.767072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.65685 −0.268765 −0.134383 0.990930i \(-0.542905\pi\)
−0.134383 + 0.990930i \(0.542905\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.1838i 1.80200i 0.433816 + 0.901002i \(0.357167\pi\)
−0.433816 + 0.901002i \(0.642833\pi\)
\(450\) 0 0
\(451\) 56.0000i 2.63694i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −22.6274 −1.06079
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 18.3848i − 0.856264i −0.903716 0.428132i \(-0.859172\pi\)
0.903716 0.428132i \(-0.140828\pi\)
\(462\) 0 0
\(463\) 20.0000i 0.929479i 0.885448 + 0.464739i \(0.153852\pi\)
−0.885448 + 0.464739i \(0.846148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.9706 −0.785304 −0.392652 0.919687i \(-0.628442\pi\)
−0.392652 + 0.919687i \(0.628442\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 45.2548i − 2.08082i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −39.5980 −1.80928 −0.904639 0.426179i \(-0.859859\pi\)
−0.904639 + 0.426179i \(0.859859\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 11.3137i − 0.513729i
\(486\) 0 0
\(487\) − 12.0000i − 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.6274 1.02116 0.510581 0.859830i \(-0.329431\pi\)
0.510581 + 0.859830i \(0.329431\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.6274i 1.01498i
\(498\) 0 0
\(499\) 32.0000i 1.43252i 0.697835 + 0.716258i \(0.254147\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.65685 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(504\) 0 0
\(505\) −26.0000 −1.15698
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 32.5269i 1.44173i 0.693075 + 0.720865i \(0.256255\pi\)
−0.693075 + 0.720865i \(0.743745\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.65685 −0.249271
\(516\) 0 0
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 32.5269i − 1.42503i −0.701657 0.712515i \(-0.747556\pi\)
0.701657 0.712515i \(-0.252444\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.9706 −0.739249
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 39.5980i − 1.71518i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −50.9117 −2.19292
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 16.9706i − 0.726939i
\(546\) 0 0
\(547\) − 40.0000i − 1.71028i −0.518400 0.855138i \(-0.673472\pi\)
0.518400 0.855138i \(-0.326528\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 29.6985i − 1.25837i −0.777258 0.629183i \(-0.783390\pi\)
0.777258 0.629183i \(-0.216610\pi\)
\(558\) 0 0
\(559\) 32.0000i 1.35346i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −39.5980 −1.66886 −0.834428 0.551117i \(-0.814202\pi\)
−0.834428 + 0.551117i \(0.814202\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0416i 1.00788i 0.863739 + 0.503939i \(0.168116\pi\)
−0.863739 + 0.503939i \(0.831884\pi\)
\(570\) 0 0
\(571\) 16.0000i 0.669579i 0.942293 + 0.334790i \(0.108665\pi\)
−0.942293 + 0.334790i \(0.891335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.9706 0.707721
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 22.6274i 0.938743i
\(582\) 0 0
\(583\) 24.0000i 0.993978i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.3137 0.466967 0.233483 0.972361i \(-0.424988\pi\)
0.233483 + 0.972361i \(0.424988\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 46.6690i 1.91647i 0.285985 + 0.958234i \(0.407679\pi\)
−0.285985 + 0.958234i \(0.592321\pi\)
\(594\) 0 0
\(595\) − 24.0000i − 0.983904i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −39.5980 −1.61793 −0.808965 0.587857i \(-0.799972\pi\)
−0.808965 + 0.587857i \(0.799972\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 29.6985i 1.20742i
\(606\) 0 0
\(607\) 28.0000i 1.13648i 0.822861 + 0.568242i \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −22.6274 −0.915407
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.8701i 1.08175i 0.841104 + 0.540874i \(0.181906\pi\)
−0.841104 + 0.540874i \(0.818094\pi\)
\(618\) 0 0
\(619\) 32.0000i 1.28619i 0.765787 + 0.643094i \(0.222350\pi\)
−0.765787 + 0.643094i \(0.777650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 16.9706 0.679911
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 25.4558i − 1.01499i
\(630\) 0 0
\(631\) − 12.0000i − 0.477712i −0.971055 0.238856i \(-0.923228\pi\)
0.971055 0.238856i \(-0.0767725\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.9706 −0.673456
\(636\) 0 0
\(637\) 36.0000 1.42637
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7.07107i 0.279290i 0.990202 + 0.139645i \(0.0445962\pi\)
−0.990202 + 0.139645i \(0.955404\pi\)
\(642\) 0 0
\(643\) − 8.00000i − 0.315489i −0.987480 0.157745i \(-0.949578\pi\)
0.987480 0.157745i \(-0.0504223\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.9706 0.667182 0.333591 0.942718i \(-0.391740\pi\)
0.333591 + 0.942718i \(0.391740\pi\)
\(648\) 0 0
\(649\) −64.0000 −2.51222
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.3848i 0.719452i 0.933058 + 0.359726i \(0.117130\pi\)
−0.933058 + 0.359726i \(0.882870\pi\)
\(654\) 0 0
\(655\) − 16.0000i − 0.625172i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −11.3137 −0.440720 −0.220360 0.975419i \(-0.570723\pi\)
−0.220360 + 0.975419i \(0.570723\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 8.00000i − 0.309761i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.3137 0.436761
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 32.5269i − 1.25011i −0.780580 0.625055i \(-0.785076\pi\)
0.780580 0.625055i \(-0.214924\pi\)
\(678\) 0 0
\(679\) 32.0000i 1.22805i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.9706 0.649361 0.324680 0.945824i \(-0.394743\pi\)
0.324680 + 0.945824i \(0.394743\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 16.9706i − 0.646527i
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.3137 −0.429153
\(696\) 0 0
\(697\) 42.0000 1.59086
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 12.7279i − 0.480727i −0.970683 0.240363i \(-0.922733\pi\)
0.970683 0.240363i \(-0.0772666\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 73.5391 2.76572
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 22.6274i − 0.847403i
\(714\) 0 0
\(715\) − 32.0000i − 1.19673i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.9706 0.632895 0.316448 0.948610i \(-0.397510\pi\)
0.316448 + 0.948610i \(0.397510\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 4.24264i − 0.157568i
\(726\) 0 0
\(727\) 4.00000i 0.148352i 0.997245 + 0.0741759i \(0.0236326\pi\)
−0.997245 + 0.0741759i \(0.976367\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −33.9411 −1.25536
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 45.2548i − 1.66698i
\(738\) 0 0
\(739\) − 16.0000i − 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 11.3137 0.415060 0.207530 0.978229i \(-0.433458\pi\)
0.207530 + 0.978229i \(0.433458\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 20.0000i − 0.729810i −0.931045 0.364905i \(-0.881101\pi\)
0.931045 0.364905i \(-0.118899\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 28.2843 1.02937
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.89949i 0.358856i 0.983771 + 0.179428i \(0.0574248\pi\)
−0.983771 + 0.179428i \(0.942575\pi\)
\(762\) 0 0
\(763\) 48.0000i 1.73772i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 45.2548 1.63406
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.24264i 0.152597i 0.997085 + 0.0762986i \(0.0243102\pi\)
−0.997085 + 0.0762986i \(0.975690\pi\)
\(774\) 0 0
\(775\) − 12.0000i − 0.431053i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 2.82843i − 0.100951i
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.65685 −0.201135
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.07107i 0.250470i 0.992127 + 0.125235i \(0.0399685\pi\)
−0.992127 + 0.125235i \(0.960032\pi\)
\(798\) 0 0
\(799\) − 24.0000i − 0.849059i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21.2132i 0.745817i 0.927868 + 0.372908i \(0.121639\pi\)
−0.927868 + 0.372908i \(0.878361\pi\)
\(810\) 0 0
\(811\) − 48.0000i − 1.68551i −0.538299 0.842754i \(-0.680933\pi\)
0.538299 0.842754i \(-0.319067\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −33.9411 −1.18891
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 41.0122i 1.43134i 0.698441 + 0.715668i \(0.253877\pi\)
−0.698441 + 0.715668i \(0.746123\pi\)
\(822\) 0 0
\(823\) − 52.0000i − 1.81261i −0.422628 0.906303i \(-0.638892\pi\)
0.422628 0.906303i \(-0.361108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33.9411 1.18025 0.590124 0.807312i \(-0.299079\pi\)
0.590124 + 0.807312i \(0.299079\pi\)
\(828\) 0 0
\(829\) −12.0000 −0.416777 −0.208389 0.978046i \(-0.566822\pi\)
−0.208389 + 0.978046i \(0.566822\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 38.1838i 1.32299i
\(834\) 0 0
\(835\) 16.0000i 0.553703i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5.65685 0.195296 0.0976481 0.995221i \(-0.468868\pi\)
0.0976481 + 0.995221i \(0.468868\pi\)
\(840\) 0 0
\(841\) 27.0000 0.931034
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.24264i 0.145951i
\(846\) 0 0
\(847\) − 84.0000i − 2.88627i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 33.9411 1.16349
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 24.0416i − 0.821246i −0.911805 0.410623i \(-0.865311\pi\)
0.911805 0.410623i \(-0.134689\pi\)
\(858\) 0 0
\(859\) − 8.00000i − 0.272956i −0.990643 0.136478i \(-0.956422\pi\)
0.990643 0.136478i \(-0.0435784\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −33.9411 −1.15537 −0.577685 0.816260i \(-0.696044\pi\)
−0.577685 + 0.816260i \(0.696044\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 22.6274i 0.767583i
\(870\) 0 0
\(871\) 32.0000i 1.08428i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 45.2548 1.52989
\(876\) 0 0
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 12.7279i 0.428815i 0.976744 + 0.214407i \(0.0687820\pi\)
−0.976744 + 0.214407i \(0.931218\pi\)
\(882\) 0 0
\(883\) − 24.0000i − 0.807664i −0.914833 0.403832i \(-0.867678\pi\)
0.914833 0.403832i \(-0.132322\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 56.5685 1.89939 0.949693 0.313183i \(-0.101395\pi\)
0.949693 + 0.313183i \(0.101395\pi\)
\(888\) 0 0
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −5.65685 −0.188667
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 16.9706i − 0.564121i
\(906\) 0 0
\(907\) − 8.00000i − 0.265636i −0.991140 0.132818i \(-0.957597\pi\)
0.991140 0.132818i \(-0.0424025\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 11.3137 0.374840 0.187420 0.982280i \(-0.439987\pi\)
0.187420 + 0.982280i \(0.439987\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 45.2548i 1.49445i
\(918\) 0 0
\(919\) − 36.0000i − 1.18753i −0.804638 0.593765i \(-0.797641\pi\)
0.804638 0.593765i \(-0.202359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 22.6274 0.744791
\(924\) 0 0
\(925\) 18.0000 0.591836
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 49.4975i − 1.62396i −0.583686 0.811980i \(-0.698390\pi\)
0.583686 0.811980i \(-0.301610\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 33.9411 1.10999
\(936\) 0 0
\(937\) 6.00000 0.196011 0.0980057 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 41.0122i − 1.33696i −0.743730 0.668480i \(-0.766945\pi\)
0.743730 0.668480i \(-0.233055\pi\)
\(942\) 0 0
\(943\) 56.0000i 1.82361i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.3137 −0.367646 −0.183823 0.982959i \(-0.558847\pi\)
−0.183823 + 0.982959i \(0.558847\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 21.2132i − 0.687163i −0.939123 0.343582i \(-0.888360\pi\)
0.939123 0.343582i \(-0.111640\pi\)
\(954\) 0 0
\(955\) − 16.0000i − 0.517748i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.65685 0.182669
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 2.82843i − 0.0910503i
\(966\) 0 0
\(967\) − 28.0000i − 0.900419i −0.892923 0.450210i \(-0.851349\pi\)
0.892923 0.450210i \(-0.148651\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −5.65685 −0.181537 −0.0907685 0.995872i \(-0.528932\pi\)
−0.0907685 + 0.995872i \(0.528932\pi\)
\(972\) 0 0
\(973\) 32.0000 1.02587
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 52.3259i − 1.67405i −0.547162 0.837027i \(-0.684292\pi\)
0.547162 0.837027i \(-0.315708\pi\)
\(978\) 0 0
\(979\) 24.0000i 0.767043i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −56.5685 −1.80426 −0.902128 0.431468i \(-0.857996\pi\)
−0.902128 + 0.431468i \(0.857996\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 45.2548i − 1.43902i
\(990\) 0 0
\(991\) 12.0000i 0.381193i 0.981669 + 0.190596i \(0.0610421\pi\)
−0.981669 + 0.190596i \(0.938958\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −28.2843 −0.896672
\(996\) 0 0
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 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 576.2.c.c.575.3 4
3.2 odd 2 inner 576.2.c.c.575.1 4
4.3 odd 2 inner 576.2.c.c.575.4 4
8.3 odd 2 288.2.c.a.287.2 yes 4
8.5 even 2 288.2.c.a.287.1 4
12.11 even 2 inner 576.2.c.c.575.2 4
16.3 odd 4 2304.2.f.e.1151.3 4
16.5 even 4 2304.2.f.e.1151.2 4
16.11 odd 4 2304.2.f.c.1151.1 4
16.13 even 4 2304.2.f.c.1151.4 4
24.5 odd 2 288.2.c.a.287.3 yes 4
24.11 even 2 288.2.c.a.287.4 yes 4
40.3 even 4 7200.2.o.n.7199.4 4
40.13 odd 4 7200.2.o.a.7199.2 4
40.19 odd 2 7200.2.h.d.1151.2 4
40.27 even 4 7200.2.o.a.7199.3 4
40.29 even 2 7200.2.h.d.1151.3 4
40.37 odd 4 7200.2.o.n.7199.1 4
48.5 odd 4 2304.2.f.e.1151.4 4
48.11 even 4 2304.2.f.c.1151.3 4
48.29 odd 4 2304.2.f.c.1151.2 4
48.35 even 4 2304.2.f.e.1151.1 4
72.5 odd 6 2592.2.s.d.1727.4 8
72.11 even 6 2592.2.s.d.863.2 8
72.13 even 6 2592.2.s.d.1727.2 8
72.29 odd 6 2592.2.s.d.863.1 8
72.43 odd 6 2592.2.s.d.863.4 8
72.59 even 6 2592.2.s.d.1727.3 8
72.61 even 6 2592.2.s.d.863.3 8
72.67 odd 6 2592.2.s.d.1727.1 8
120.29 odd 2 7200.2.h.d.1151.4 4
120.53 even 4 7200.2.o.a.7199.4 4
120.59 even 2 7200.2.h.d.1151.1 4
120.77 even 4 7200.2.o.n.7199.3 4
120.83 odd 4 7200.2.o.n.7199.2 4
120.107 odd 4 7200.2.o.a.7199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.c.a.287.1 4 8.5 even 2
288.2.c.a.287.2 yes 4 8.3 odd 2
288.2.c.a.287.3 yes 4 24.5 odd 2
288.2.c.a.287.4 yes 4 24.11 even 2
576.2.c.c.575.1 4 3.2 odd 2 inner
576.2.c.c.575.2 4 12.11 even 2 inner
576.2.c.c.575.3 4 1.1 even 1 trivial
576.2.c.c.575.4 4 4.3 odd 2 inner
2304.2.f.c.1151.1 4 16.11 odd 4
2304.2.f.c.1151.2 4 48.29 odd 4
2304.2.f.c.1151.3 4 48.11 even 4
2304.2.f.c.1151.4 4 16.13 even 4
2304.2.f.e.1151.1 4 48.35 even 4
2304.2.f.e.1151.2 4 16.5 even 4
2304.2.f.e.1151.3 4 16.3 odd 4
2304.2.f.e.1151.4 4 48.5 odd 4
2592.2.s.d.863.1 8 72.29 odd 6
2592.2.s.d.863.2 8 72.11 even 6
2592.2.s.d.863.3 8 72.61 even 6
2592.2.s.d.863.4 8 72.43 odd 6
2592.2.s.d.1727.1 8 72.67 odd 6
2592.2.s.d.1727.2 8 72.13 even 6
2592.2.s.d.1727.3 8 72.59 even 6
2592.2.s.d.1727.4 8 72.5 odd 6
7200.2.h.d.1151.1 4 120.59 even 2
7200.2.h.d.1151.2 4 40.19 odd 2
7200.2.h.d.1151.3 4 40.29 even 2
7200.2.h.d.1151.4 4 120.29 odd 2
7200.2.o.a.7199.1 4 120.107 odd 4
7200.2.o.a.7199.2 4 40.13 odd 4
7200.2.o.a.7199.3 4 40.27 even 4
7200.2.o.a.7199.4 4 120.53 even 4
7200.2.o.n.7199.1 4 40.37 odd 4
7200.2.o.n.7199.2 4 120.83 odd 4
7200.2.o.n.7199.3 4 120.77 even 4
7200.2.o.n.7199.4 4 40.3 even 4