Properties

Label 2304.3.b.q.127.1
Level $2304$
Weight $3$
Character 2304.127
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(127,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2304.127
Dual form 2304.3.b.q.127.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.29253i q^{5} +8.55583i q^{7} +O(q^{10})\) \(q-8.29253i q^{5} +8.55583i q^{7} -13.7980 q^{11} +17.0693i q^{13} -20.3246 q^{17} +20.4440 q^{19} +5.51575i q^{23} -43.7660 q^{25} -41.0586i q^{29} -22.2953i q^{31} +70.9495 q^{35} +11.6326i q^{37} +35.9527 q^{41} +66.8648 q^{43} +19.9366i q^{47} -24.2023 q^{49} +17.6329i q^{53} +114.420i q^{55} +62.1572 q^{59} -47.4836i q^{61} +141.548 q^{65} +74.8105 q^{67} -16.9150i q^{71} -101.712 q^{73} -118.053i q^{77} -0.879320i q^{79} -23.2131 q^{83} +168.542i q^{85} -16.3089 q^{89} -146.042 q^{91} -169.532i q^{95} +188.307 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{11} - 16 q^{17} + 96 q^{19} + 8 q^{25} + 96 q^{35} + 80 q^{41} + 224 q^{43} - 88 q^{49} + 512 q^{59} + 160 q^{65} + 16 q^{73} + 544 q^{83} - 240 q^{89} - 32 q^{91} + 400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) − 8.29253i − 1.65851i −0.558874 0.829253i \(-0.688766\pi\)
0.558874 0.829253i \(-0.311234\pi\)
\(6\) 0 0
\(7\) 8.55583i 1.22226i 0.791529 + 0.611131i \(0.209285\pi\)
−0.791529 + 0.611131i \(0.790715\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −13.7980 −1.25436 −0.627180 0.778874i \(-0.715791\pi\)
−0.627180 + 0.778874i \(0.715791\pi\)
\(12\) 0 0
\(13\) 17.0693i 1.31302i 0.754316 + 0.656512i \(0.227969\pi\)
−0.754316 + 0.656512i \(0.772031\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −20.3246 −1.19556 −0.597781 0.801659i \(-0.703951\pi\)
−0.597781 + 0.801659i \(0.703951\pi\)
\(18\) 0 0
\(19\) 20.4440 1.07600 0.537999 0.842946i \(-0.319181\pi\)
0.537999 + 0.842946i \(0.319181\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.51575i 0.239815i 0.992785 + 0.119908i \(0.0382598\pi\)
−0.992785 + 0.119908i \(0.961740\pi\)
\(24\) 0 0
\(25\) −43.7660 −1.75064
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 41.0586i − 1.41581i −0.706306 0.707906i \(-0.749640\pi\)
0.706306 0.707906i \(-0.250360\pi\)
\(30\) 0 0
\(31\) − 22.2953i − 0.719205i −0.933106 0.359602i \(-0.882912\pi\)
0.933106 0.359602i \(-0.117088\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 70.9495 2.02713
\(36\) 0 0
\(37\) 11.6326i 0.314396i 0.987567 + 0.157198i \(0.0502461\pi\)
−0.987567 + 0.157198i \(0.949754\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 35.9527 0.876894 0.438447 0.898757i \(-0.355529\pi\)
0.438447 + 0.898757i \(0.355529\pi\)
\(42\) 0 0
\(43\) 66.8648 1.55499 0.777497 0.628886i \(-0.216489\pi\)
0.777497 + 0.628886i \(0.216489\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 19.9366i 0.424182i 0.977250 + 0.212091i \(0.0680274\pi\)
−0.977250 + 0.212091i \(0.931973\pi\)
\(48\) 0 0
\(49\) −24.2023 −0.493924
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 17.6329i 0.332697i 0.986067 + 0.166348i \(0.0531977\pi\)
−0.986067 + 0.166348i \(0.946802\pi\)
\(54\) 0 0
\(55\) 114.420i 2.08036i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 62.1572 1.05351 0.526756 0.850017i \(-0.323408\pi\)
0.526756 + 0.850017i \(0.323408\pi\)
\(60\) 0 0
\(61\) − 47.4836i − 0.778419i −0.921149 0.389210i \(-0.872748\pi\)
0.921149 0.389210i \(-0.127252\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 141.548 2.17766
\(66\) 0 0
\(67\) 74.8105 1.11657 0.558287 0.829648i \(-0.311459\pi\)
0.558287 + 0.829648i \(0.311459\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 16.9150i − 0.238240i −0.992880 0.119120i \(-0.961993\pi\)
0.992880 0.119120i \(-0.0380073\pi\)
\(72\) 0 0
\(73\) −101.712 −1.39331 −0.696657 0.717404i \(-0.745330\pi\)
−0.696657 + 0.717404i \(0.745330\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 118.053i − 1.53316i
\(78\) 0 0
\(79\) − 0.879320i − 0.0111306i −0.999985 0.00556532i \(-0.998228\pi\)
0.999985 0.00556532i \(-0.00177151\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −23.2131 −0.279676 −0.139838 0.990174i \(-0.544658\pi\)
−0.139838 + 0.990174i \(0.544658\pi\)
\(84\) 0 0
\(85\) 168.542i 1.98285i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −16.3089 −0.183246 −0.0916231 0.995794i \(-0.529205\pi\)
−0.0916231 + 0.995794i \(0.529205\pi\)
\(90\) 0 0
\(91\) −146.042 −1.60486
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 169.532i − 1.78455i
\(96\) 0 0
\(97\) 188.307 1.94131 0.970656 0.240474i \(-0.0773028\pi\)
0.970656 + 0.240474i \(0.0773028\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 66.8468i − 0.661849i −0.943657 0.330925i \(-0.892639\pi\)
0.943657 0.330925i \(-0.107361\pi\)
\(102\) 0 0
\(103\) − 166.313i − 1.61469i −0.590083 0.807343i \(-0.700905\pi\)
0.590083 0.807343i \(-0.299095\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 80.1467 0.749035 0.374517 0.927220i \(-0.377808\pi\)
0.374517 + 0.927220i \(0.377808\pi\)
\(108\) 0 0
\(109\) 26.8781i 0.246589i 0.992370 + 0.123294i \(0.0393459\pi\)
−0.992370 + 0.123294i \(0.960654\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 79.0958 0.699963 0.349981 0.936757i \(-0.386188\pi\)
0.349981 + 0.936757i \(0.386188\pi\)
\(114\) 0 0
\(115\) 45.7395 0.397735
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 173.894i − 1.46129i
\(120\) 0 0
\(121\) 69.3837 0.573419
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 155.618i 1.24494i
\(126\) 0 0
\(127\) − 170.725i − 1.34429i −0.740419 0.672145i \(-0.765373\pi\)
0.740419 0.672145i \(-0.234627\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 178.980 1.36626 0.683129 0.730297i \(-0.260619\pi\)
0.683129 + 0.730297i \(0.260619\pi\)
\(132\) 0 0
\(133\) 174.915i 1.31515i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −93.2075 −0.680347 −0.340173 0.940363i \(-0.610486\pi\)
−0.340173 + 0.940363i \(0.610486\pi\)
\(138\) 0 0
\(139\) −222.476 −1.60055 −0.800274 0.599635i \(-0.795313\pi\)
−0.800274 + 0.599635i \(0.795313\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 235.522i − 1.64700i
\(144\) 0 0
\(145\) −340.479 −2.34813
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 94.0043i 0.630901i 0.948942 + 0.315451i \(0.102156\pi\)
−0.948942 + 0.315451i \(0.897844\pi\)
\(150\) 0 0
\(151\) − 82.0081i − 0.543100i −0.962424 0.271550i \(-0.912464\pi\)
0.962424 0.271550i \(-0.0875362\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −184.885 −1.19281
\(156\) 0 0
\(157\) − 56.7180i − 0.361261i −0.983551 0.180631i \(-0.942186\pi\)
0.983551 0.180631i \(-0.0578139\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −47.1918 −0.293117
\(162\) 0 0
\(163\) 189.588 1.16312 0.581559 0.813504i \(-0.302443\pi\)
0.581559 + 0.813504i \(0.302443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 312.777i 1.87291i 0.350783 + 0.936457i \(0.385916\pi\)
−0.350783 + 0.936457i \(0.614084\pi\)
\(168\) 0 0
\(169\) −122.361 −0.724031
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 182.538i 1.05513i 0.849514 + 0.527567i \(0.176896\pi\)
−0.849514 + 0.527567i \(0.823104\pi\)
\(174\) 0 0
\(175\) − 374.455i − 2.13974i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −19.7781 −0.110492 −0.0552461 0.998473i \(-0.517594\pi\)
−0.0552461 + 0.998473i \(0.517594\pi\)
\(180\) 0 0
\(181\) − 265.750i − 1.46823i −0.679023 0.734117i \(-0.737597\pi\)
0.679023 0.734117i \(-0.262403\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 96.4640 0.521427
\(186\) 0 0
\(187\) 280.438 1.49967
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 288.025i − 1.50799i −0.656882 0.753993i \(-0.728125\pi\)
0.656882 0.753993i \(-0.271875\pi\)
\(192\) 0 0
\(193\) 281.010 1.45601 0.728005 0.685572i \(-0.240448\pi\)
0.728005 + 0.685572i \(0.240448\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 33.8330i − 0.171741i −0.996306 0.0858705i \(-0.972633\pi\)
0.996306 0.0858705i \(-0.0273671\pi\)
\(198\) 0 0
\(199\) − 84.9700i − 0.426985i −0.976945 0.213493i \(-0.931516\pi\)
0.976945 0.213493i \(-0.0684839\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 351.290 1.73049
\(204\) 0 0
\(205\) − 298.138i − 1.45433i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −282.085 −1.34969
\(210\) 0 0
\(211\) −140.700 −0.666826 −0.333413 0.942781i \(-0.608200\pi\)
−0.333413 + 0.942781i \(0.608200\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 554.478i − 2.57897i
\(216\) 0 0
\(217\) 190.755 0.879057
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 346.926i − 1.56980i
\(222\) 0 0
\(223\) 247.529i 1.11000i 0.831851 + 0.554999i \(0.187281\pi\)
−0.831851 + 0.554999i \(0.812719\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 129.752 0.571593 0.285797 0.958290i \(-0.407742\pi\)
0.285797 + 0.958290i \(0.407742\pi\)
\(228\) 0 0
\(229\) − 294.304i − 1.28517i −0.766214 0.642586i \(-0.777862\pi\)
0.766214 0.642586i \(-0.222138\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −239.948 −1.02982 −0.514911 0.857244i \(-0.672175\pi\)
−0.514911 + 0.857244i \(0.672175\pi\)
\(234\) 0 0
\(235\) 165.325 0.703509
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 425.346i 1.77969i 0.456261 + 0.889846i \(0.349188\pi\)
−0.456261 + 0.889846i \(0.650812\pi\)
\(240\) 0 0
\(241\) 264.493 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 200.698i 0.819176i
\(246\) 0 0
\(247\) 348.964i 1.41281i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 221.125 0.880976 0.440488 0.897759i \(-0.354805\pi\)
0.440488 + 0.897759i \(0.354805\pi\)
\(252\) 0 0
\(253\) − 76.1061i − 0.300815i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −87.0655 −0.338776 −0.169388 0.985549i \(-0.554179\pi\)
−0.169388 + 0.985549i \(0.554179\pi\)
\(258\) 0 0
\(259\) −99.5270 −0.384274
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 277.324i − 1.05447i −0.849721 0.527233i \(-0.823230\pi\)
0.849721 0.527233i \(-0.176770\pi\)
\(264\) 0 0
\(265\) 146.222 0.551780
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 49.8005i 0.185132i 0.995707 + 0.0925659i \(0.0295069\pi\)
−0.995707 + 0.0925659i \(0.970493\pi\)
\(270\) 0 0
\(271\) − 31.3145i − 0.115552i −0.998330 0.0577758i \(-0.981599\pi\)
0.998330 0.0577758i \(-0.0184009\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 603.882 2.19593
\(276\) 0 0
\(277\) 79.1431i 0.285715i 0.989743 + 0.142858i \(0.0456291\pi\)
−0.989743 + 0.142858i \(0.954371\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −106.190 −0.377902 −0.188951 0.981987i \(-0.560509\pi\)
−0.188951 + 0.981987i \(0.560509\pi\)
\(282\) 0 0
\(283\) 270.251 0.954949 0.477475 0.878646i \(-0.341552\pi\)
0.477475 + 0.878646i \(0.341552\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 307.605i 1.07179i
\(288\) 0 0
\(289\) 124.088 0.429371
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.2487i 0.0827600i 0.999143 + 0.0413800i \(0.0131754\pi\)
−0.999143 + 0.0413800i \(0.986825\pi\)
\(294\) 0 0
\(295\) − 515.440i − 1.74726i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −94.1500 −0.314883
\(300\) 0 0
\(301\) 572.084i 1.90061i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −393.759 −1.29101
\(306\) 0 0
\(307\) 122.865 0.400211 0.200106 0.979774i \(-0.435871\pi\)
0.200106 + 0.979774i \(0.435871\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 108.808i − 0.349865i −0.984580 0.174933i \(-0.944029\pi\)
0.984580 0.174933i \(-0.0559707\pi\)
\(312\) 0 0
\(313\) 52.2216 0.166842 0.0834211 0.996514i \(-0.473415\pi\)
0.0834211 + 0.996514i \(0.473415\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 94.3251i − 0.297555i −0.988871 0.148778i \(-0.952466\pi\)
0.988871 0.148778i \(-0.0475339\pi\)
\(318\) 0 0
\(319\) 566.524i 1.77594i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −415.515 −1.28642
\(324\) 0 0
\(325\) − 747.056i − 2.29863i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −170.574 −0.518462
\(330\) 0 0
\(331\) 406.107 1.22691 0.613454 0.789730i \(-0.289780\pi\)
0.613454 + 0.789730i \(0.289780\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 620.368i − 1.85184i
\(336\) 0 0
\(337\) −22.4140 −0.0665105 −0.0332552 0.999447i \(-0.510587\pi\)
−0.0332552 + 0.999447i \(0.510587\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 307.630i 0.902142i
\(342\) 0 0
\(343\) 212.165i 0.618557i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 458.377 1.32097 0.660486 0.750839i \(-0.270350\pi\)
0.660486 + 0.750839i \(0.270350\pi\)
\(348\) 0 0
\(349\) 282.960i 0.810774i 0.914145 + 0.405387i \(0.132863\pi\)
−0.914145 + 0.405387i \(0.867137\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 175.820 0.498073 0.249036 0.968494i \(-0.419886\pi\)
0.249036 + 0.968494i \(0.419886\pi\)
\(354\) 0 0
\(355\) −140.268 −0.395122
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 53.5791i 0.149246i 0.997212 + 0.0746228i \(0.0237753\pi\)
−0.997212 + 0.0746228i \(0.976225\pi\)
\(360\) 0 0
\(361\) 56.9552 0.157771
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 843.449i 2.31082i
\(366\) 0 0
\(367\) − 62.4258i − 0.170097i −0.996377 0.0850487i \(-0.972895\pi\)
0.996377 0.0850487i \(-0.0271046\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −150.864 −0.406643
\(372\) 0 0
\(373\) 370.552i 0.993437i 0.867912 + 0.496719i \(0.165462\pi\)
−0.867912 + 0.496719i \(0.834538\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 700.841 1.85900
\(378\) 0 0
\(379\) 68.7286 0.181342 0.0906709 0.995881i \(-0.471099\pi\)
0.0906709 + 0.995881i \(0.471099\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 155.858i − 0.406939i −0.979081 0.203470i \(-0.934778\pi\)
0.979081 0.203470i \(-0.0652218\pi\)
\(384\) 0 0
\(385\) −978.958 −2.54275
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 481.792i − 1.23854i −0.785179 0.619269i \(-0.787429\pi\)
0.785179 0.619269i \(-0.212571\pi\)
\(390\) 0 0
\(391\) − 112.105i − 0.286714i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.29179 −0.0184602
\(396\) 0 0
\(397\) − 422.315i − 1.06377i −0.846818 0.531883i \(-0.821485\pi\)
0.846818 0.531883i \(-0.178515\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 641.025 1.59857 0.799283 0.600955i \(-0.205213\pi\)
0.799283 + 0.600955i \(0.205213\pi\)
\(402\) 0 0
\(403\) 380.566 0.944333
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 160.507i − 0.394365i
\(408\) 0 0
\(409\) 278.562 0.681081 0.340541 0.940230i \(-0.389390\pi\)
0.340541 + 0.940230i \(0.389390\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 531.807i 1.28767i
\(414\) 0 0
\(415\) 192.496i 0.463845i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 505.611 1.20671 0.603354 0.797473i \(-0.293831\pi\)
0.603354 + 0.797473i \(0.293831\pi\)
\(420\) 0 0
\(421\) 179.846i 0.427189i 0.976922 + 0.213594i \(0.0685171\pi\)
−0.976922 + 0.213594i \(0.931483\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 889.526 2.09300
\(426\) 0 0
\(427\) 406.262 0.951432
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 580.645i 1.34720i 0.739094 + 0.673602i \(0.235254\pi\)
−0.739094 + 0.673602i \(0.764746\pi\)
\(432\) 0 0
\(433\) −425.621 −0.982959 −0.491479 0.870889i \(-0.663544\pi\)
−0.491479 + 0.870889i \(0.663544\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 112.764i 0.258041i
\(438\) 0 0
\(439\) − 129.991i − 0.296107i −0.988979 0.148053i \(-0.952699\pi\)
0.988979 0.148053i \(-0.0473008\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 205.564 0.464027 0.232014 0.972713i \(-0.425469\pi\)
0.232014 + 0.972713i \(0.425469\pi\)
\(444\) 0 0
\(445\) 135.242i 0.303915i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −415.190 −0.924698 −0.462349 0.886698i \(-0.652993\pi\)
−0.462349 + 0.886698i \(0.652993\pi\)
\(450\) 0 0
\(451\) −496.073 −1.09994
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1211.06i 2.66167i
\(456\) 0 0
\(457\) 598.927 1.31056 0.655281 0.755385i \(-0.272550\pi\)
0.655281 + 0.755385i \(0.272550\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 376.804i − 0.817361i −0.912677 0.408681i \(-0.865989\pi\)
0.912677 0.408681i \(-0.134011\pi\)
\(462\) 0 0
\(463\) 218.577i 0.472088i 0.971742 + 0.236044i \(0.0758510\pi\)
−0.971742 + 0.236044i \(0.924149\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −443.608 −0.949911 −0.474955 0.880010i \(-0.657536\pi\)
−0.474955 + 0.880010i \(0.657536\pi\)
\(468\) 0 0
\(469\) 640.066i 1.36475i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −922.597 −1.95052
\(474\) 0 0
\(475\) −894.751 −1.88369
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 222.133i − 0.463743i −0.972746 0.231871i \(-0.925515\pi\)
0.972746 0.231871i \(-0.0744849\pi\)
\(480\) 0 0
\(481\) −198.561 −0.412809
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 1561.54i − 3.21968i
\(486\) 0 0
\(487\) − 403.795i − 0.829148i −0.910016 0.414574i \(-0.863931\pi\)
0.910016 0.414574i \(-0.136069\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −808.347 −1.64633 −0.823164 0.567803i \(-0.807793\pi\)
−0.823164 + 0.567803i \(0.807793\pi\)
\(492\) 0 0
\(493\) 834.498i 1.69269i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 144.722 0.291192
\(498\) 0 0
\(499\) 132.172 0.264874 0.132437 0.991191i \(-0.457720\pi\)
0.132437 + 0.991191i \(0.457720\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 92.8360i 0.184565i 0.995733 + 0.0922823i \(0.0294162\pi\)
−0.995733 + 0.0922823i \(0.970584\pi\)
\(504\) 0 0
\(505\) −554.329 −1.09768
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 927.417i − 1.82204i −0.412366 0.911018i \(-0.635297\pi\)
0.412366 0.911018i \(-0.364703\pi\)
\(510\) 0 0
\(511\) − 870.231i − 1.70300i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1379.15 −2.67796
\(516\) 0 0
\(517\) − 275.084i − 0.532077i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 527.145 1.01179 0.505897 0.862594i \(-0.331162\pi\)
0.505897 + 0.862594i \(0.331162\pi\)
\(522\) 0 0
\(523\) −972.249 −1.85898 −0.929492 0.368843i \(-0.879754\pi\)
−0.929492 + 0.368843i \(0.879754\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 453.143i 0.859854i
\(528\) 0 0
\(529\) 498.577 0.942489
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 613.687i 1.15138i
\(534\) 0 0
\(535\) − 664.619i − 1.24228i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 333.942 0.619559
\(540\) 0 0
\(541\) 4.66591i 0.00862461i 0.999991 + 0.00431231i \(0.00137265\pi\)
−0.999991 + 0.00431231i \(0.998627\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 222.888 0.408968
\(546\) 0 0
\(547\) −386.796 −0.707123 −0.353561 0.935411i \(-0.615029\pi\)
−0.353561 + 0.935411i \(0.615029\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 839.399i − 1.52341i
\(552\) 0 0
\(553\) 7.52332 0.0136046
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 219.630i − 0.394308i −0.980373 0.197154i \(-0.936830\pi\)
0.980373 0.197154i \(-0.0631699\pi\)
\(558\) 0 0
\(559\) 1141.34i 2.04175i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 266.807 0.473902 0.236951 0.971522i \(-0.423852\pi\)
0.236951 + 0.971522i \(0.423852\pi\)
\(564\) 0 0
\(565\) − 655.904i − 1.16089i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −972.046 −1.70834 −0.854170 0.519993i \(-0.825934\pi\)
−0.854170 + 0.519993i \(0.825934\pi\)
\(570\) 0 0
\(571\) 340.194 0.595786 0.297893 0.954599i \(-0.403716\pi\)
0.297893 + 0.954599i \(0.403716\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 241.402i − 0.419830i
\(576\) 0 0
\(577\) 65.0584 0.112753 0.0563764 0.998410i \(-0.482045\pi\)
0.0563764 + 0.998410i \(0.482045\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 198.608i − 0.341838i
\(582\) 0 0
\(583\) − 243.299i − 0.417322i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −162.144 −0.276226 −0.138113 0.990417i \(-0.544104\pi\)
−0.138113 + 0.990417i \(0.544104\pi\)
\(588\) 0 0
\(589\) − 455.805i − 0.773862i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 793.227 1.33765 0.668825 0.743420i \(-0.266797\pi\)
0.668825 + 0.743420i \(0.266797\pi\)
\(594\) 0 0
\(595\) −1442.02 −2.42356
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 432.194i − 0.721525i −0.932658 0.360763i \(-0.882516\pi\)
0.932658 0.360763i \(-0.117484\pi\)
\(600\) 0 0
\(601\) −936.310 −1.55792 −0.778960 0.627074i \(-0.784252\pi\)
−0.778960 + 0.627074i \(0.784252\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 575.366i − 0.951018i
\(606\) 0 0
\(607\) − 223.972i − 0.368982i −0.982834 0.184491i \(-0.940936\pi\)
0.982834 0.184491i \(-0.0590637\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −340.304 −0.556962
\(612\) 0 0
\(613\) 1010.40i 1.64828i 0.566385 + 0.824141i \(0.308342\pi\)
−0.566385 + 0.824141i \(0.691658\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −400.206 −0.648633 −0.324316 0.945949i \(-0.605134\pi\)
−0.324316 + 0.945949i \(0.605134\pi\)
\(618\) 0 0
\(619\) 417.349 0.674231 0.337116 0.941463i \(-0.390549\pi\)
0.337116 + 0.941463i \(0.390549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 139.536i − 0.223975i
\(624\) 0 0
\(625\) 196.315 0.314104
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 236.428i − 0.375880i
\(630\) 0 0
\(631\) 718.033i 1.13793i 0.822362 + 0.568965i \(0.192656\pi\)
−0.822362 + 0.568965i \(0.807344\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1415.74 −2.22951
\(636\) 0 0
\(637\) − 413.116i − 0.648534i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −139.926 −0.218293 −0.109146 0.994026i \(-0.534812\pi\)
−0.109146 + 0.994026i \(0.534812\pi\)
\(642\) 0 0
\(643\) 777.990 1.20994 0.604969 0.796249i \(-0.293186\pi\)
0.604969 + 0.796249i \(0.293186\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 673.336i − 1.04070i −0.853952 0.520352i \(-0.825801\pi\)
0.853952 0.520352i \(-0.174199\pi\)
\(648\) 0 0
\(649\) −857.642 −1.32148
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 223.410i − 0.342128i −0.985260 0.171064i \(-0.945279\pi\)
0.985260 0.171064i \(-0.0547206\pi\)
\(654\) 0 0
\(655\) − 1484.20i − 2.26595i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 166.367 0.252453 0.126227 0.992001i \(-0.459713\pi\)
0.126227 + 0.992001i \(0.459713\pi\)
\(660\) 0 0
\(661\) 655.773i 0.992092i 0.868296 + 0.496046i \(0.165215\pi\)
−0.868296 + 0.496046i \(0.834785\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1450.49 2.18119
\(666\) 0 0
\(667\) 226.469 0.339533
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 655.176i 0.976418i
\(672\) 0 0
\(673\) −863.477 −1.28303 −0.641513 0.767112i \(-0.721693\pi\)
−0.641513 + 0.767112i \(0.721693\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 831.520i 1.22824i 0.789212 + 0.614121i \(0.210489\pi\)
−0.789212 + 0.614121i \(0.789511\pi\)
\(678\) 0 0
\(679\) 1611.13i 2.37279i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 308.273 0.451351 0.225676 0.974202i \(-0.427541\pi\)
0.225676 + 0.974202i \(0.427541\pi\)
\(684\) 0 0
\(685\) 772.926i 1.12836i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −300.982 −0.436839
\(690\) 0 0
\(691\) 666.811 0.964994 0.482497 0.875898i \(-0.339730\pi\)
0.482497 + 0.875898i \(0.339730\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1844.89i 2.65452i
\(696\) 0 0
\(697\) −730.722 −1.04838
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 242.205i − 0.345514i −0.984964 0.172757i \(-0.944732\pi\)
0.984964 0.172757i \(-0.0552675\pi\)
\(702\) 0 0
\(703\) 237.817i 0.338289i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 571.930 0.808953
\(708\) 0 0
\(709\) − 1380.24i − 1.94674i −0.229237 0.973371i \(-0.573623\pi\)
0.229237 0.973371i \(-0.426377\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 122.976 0.172476
\(714\) 0 0
\(715\) −1953.07 −2.73157
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 968.365i 1.34682i 0.739268 + 0.673411i \(0.235172\pi\)
−0.739268 + 0.673411i \(0.764828\pi\)
\(720\) 0 0
\(721\) 1422.94 1.97357
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1796.97i 2.47858i
\(726\) 0 0
\(727\) 1017.52i 1.39962i 0.714329 + 0.699810i \(0.246732\pi\)
−0.714329 + 0.699810i \(0.753268\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1359.00 −1.85909
\(732\) 0 0
\(733\) − 1026.22i − 1.40003i −0.714127 0.700016i \(-0.753176\pi\)
0.714127 0.700016i \(-0.246824\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1032.23 −1.40059
\(738\) 0 0
\(739\) −923.444 −1.24959 −0.624793 0.780791i \(-0.714817\pi\)
−0.624793 + 0.780791i \(0.714817\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1267.12i 1.70542i 0.522387 + 0.852708i \(0.325042\pi\)
−0.522387 + 0.852708i \(0.674958\pi\)
\(744\) 0 0
\(745\) 779.533 1.04635
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 685.722i 0.915517i
\(750\) 0 0
\(751\) 791.442i 1.05385i 0.849912 + 0.526925i \(0.176655\pi\)
−0.849912 + 0.526925i \(0.823345\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −680.054 −0.900734
\(756\) 0 0
\(757\) 333.990i 0.441202i 0.975364 + 0.220601i \(0.0708019\pi\)
−0.975364 + 0.220601i \(0.929198\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 617.122 0.810936 0.405468 0.914109i \(-0.367109\pi\)
0.405468 + 0.914109i \(0.367109\pi\)
\(762\) 0 0
\(763\) −229.965 −0.301396
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1060.98i 1.38329i
\(768\) 0 0
\(769\) −657.604 −0.855142 −0.427571 0.903982i \(-0.640631\pi\)
−0.427571 + 0.903982i \(0.640631\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1314.59i − 1.70064i −0.526268 0.850319i \(-0.676409\pi\)
0.526268 0.850319i \(-0.323591\pi\)
\(774\) 0 0
\(775\) 975.779i 1.25907i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 735.015 0.943536
\(780\) 0 0
\(781\) 233.393i 0.298839i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −470.336 −0.599154
\(786\) 0 0
\(787\) −301.018 −0.382488 −0.191244 0.981542i \(-0.561252\pi\)
−0.191244 + 0.981542i \(0.561252\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 676.731i 0.855538i
\(792\) 0 0
\(793\) 810.512 1.02208
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 476.193i 0.597481i 0.954334 + 0.298741i \(0.0965665\pi\)
−0.954334 + 0.298741i \(0.903433\pi\)
\(798\) 0 0
\(799\) − 405.202i − 0.507137i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1403.42 1.74772
\(804\) 0 0
\(805\) 391.340i 0.486136i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −692.874 −0.856457 −0.428228 0.903671i \(-0.640862\pi\)
−0.428228 + 0.903671i \(0.640862\pi\)
\(810\) 0 0
\(811\) 785.074 0.968032 0.484016 0.875059i \(-0.339178\pi\)
0.484016 + 0.875059i \(0.339178\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1572.17i − 1.92904i
\(816\) 0 0
\(817\) 1366.98 1.67317
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 527.681i 0.642729i 0.946956 + 0.321365i \(0.104141\pi\)
−0.946956 + 0.321365i \(0.895859\pi\)
\(822\) 0 0
\(823\) − 228.552i − 0.277706i −0.990313 0.138853i \(-0.955658\pi\)
0.990313 0.138853i \(-0.0443416\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −608.409 −0.735683 −0.367841 0.929889i \(-0.619903\pi\)
−0.367841 + 0.929889i \(0.619903\pi\)
\(828\) 0 0
\(829\) − 151.637i − 0.182915i −0.995809 0.0914577i \(-0.970847\pi\)
0.995809 0.0914577i \(-0.0291526\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 491.901 0.590518
\(834\) 0 0
\(835\) 2593.71 3.10624
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 653.351i 0.778725i 0.921085 + 0.389363i \(0.127305\pi\)
−0.921085 + 0.389363i \(0.872695\pi\)
\(840\) 0 0
\(841\) −844.805 −1.00452
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1014.68i 1.20081i
\(846\) 0 0
\(847\) 593.635i 0.700868i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −64.1628 −0.0753969
\(852\) 0 0
\(853\) 1372.25i 1.60873i 0.594132 + 0.804367i \(0.297495\pi\)
−0.594132 + 0.804367i \(0.702505\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1453.61 −1.69617 −0.848083 0.529863i \(-0.822243\pi\)
−0.848083 + 0.529863i \(0.822243\pi\)
\(858\) 0 0
\(859\) 962.466 1.12045 0.560225 0.828341i \(-0.310715\pi\)
0.560225 + 0.828341i \(0.310715\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 695.890i 0.806362i 0.915120 + 0.403181i \(0.132095\pi\)
−0.915120 + 0.403181i \(0.867905\pi\)
\(864\) 0 0
\(865\) 1513.70 1.74994
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12.1328i 0.0139618i
\(870\) 0 0
\(871\) 1276.96i 1.46609i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1331.44 −1.52165
\(876\) 0 0
\(877\) − 338.191i − 0.385622i −0.981236 0.192811i \(-0.938240\pi\)
0.981236 0.192811i \(-0.0617605\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 459.985 0.522117 0.261058 0.965323i \(-0.415928\pi\)
0.261058 + 0.965323i \(0.415928\pi\)
\(882\) 0 0
\(883\) 1208.31 1.36842 0.684209 0.729286i \(-0.260147\pi\)
0.684209 + 0.729286i \(0.260147\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1500.73i 1.69192i 0.533250 + 0.845958i \(0.320971\pi\)
−0.533250 + 0.845958i \(0.679029\pi\)
\(888\) 0 0
\(889\) 1460.69 1.64307
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 407.582i 0.456419i
\(894\) 0 0
\(895\) 164.010i 0.183252i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −915.415 −1.01826
\(900\) 0 0
\(901\) − 358.382i − 0.397760i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2203.74 −2.43507
\(906\) 0 0
\(907\) 665.128 0.733328 0.366664 0.930353i \(-0.380500\pi\)
0.366664 + 0.930353i \(0.380500\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 345.687i 0.379459i 0.981836 + 0.189729i \(0.0607611\pi\)
−0.981836 + 0.189729i \(0.939239\pi\)
\(912\) 0 0
\(913\) 320.294 0.350815
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1531.32i 1.66993i
\(918\) 0 0
\(919\) − 1102.20i − 1.19934i −0.800246 0.599671i \(-0.795298\pi\)
0.800246 0.599671i \(-0.204702\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 288.728 0.312815
\(924\) 0 0
\(925\) − 509.115i − 0.550394i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1256.03 1.35202 0.676010 0.736893i \(-0.263708\pi\)
0.676010 + 0.736893i \(0.263708\pi\)
\(930\) 0 0
\(931\) −494.790 −0.531461
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 2325.54i − 2.48720i
\(936\) 0 0
\(937\) 822.214 0.877496 0.438748 0.898610i \(-0.355422\pi\)
0.438748 + 0.898610i \(0.355422\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1185.72i − 1.26006i −0.776570 0.630031i \(-0.783042\pi\)
0.776570 0.630031i \(-0.216958\pi\)
\(942\) 0 0
\(943\) 198.306i 0.210293i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1335.34 −1.41007 −0.705037 0.709170i \(-0.749070\pi\)
−0.705037 + 0.709170i \(0.749070\pi\)
\(948\) 0 0
\(949\) − 1736.15i − 1.82945i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 663.428 0.696147 0.348073 0.937467i \(-0.386836\pi\)
0.348073 + 0.937467i \(0.386836\pi\)
\(954\) 0 0
\(955\) −2388.46 −2.50100
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 797.468i − 0.831562i
\(960\) 0 0
\(961\) 463.918 0.482745
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 2330.28i − 2.41480i
\(966\) 0 0
\(967\) − 155.996i − 0.161320i −0.996742 0.0806600i \(-0.974297\pi\)
0.996742 0.0806600i \(-0.0257028\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1280.67 1.31892 0.659460 0.751740i \(-0.270785\pi\)
0.659460 + 0.751740i \(0.270785\pi\)
\(972\) 0 0
\(973\) − 1903.47i − 1.95629i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 217.470 0.222590 0.111295 0.993787i \(-0.464500\pi\)
0.111295 + 0.993787i \(0.464500\pi\)
\(978\) 0 0
\(979\) 225.030 0.229857
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1027.63i 1.04540i 0.852515 + 0.522702i \(0.175076\pi\)
−0.852515 + 0.522702i \(0.824924\pi\)
\(984\) 0 0
\(985\) −280.561 −0.284833
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 368.809i 0.372911i
\(990\) 0 0
\(991\) 797.792i 0.805038i 0.915412 + 0.402519i \(0.131865\pi\)
−0.915412 + 0.402519i \(0.868135\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −704.616 −0.708157
\(996\) 0 0
\(997\) − 1237.58i − 1.24131i −0.784085 0.620654i \(-0.786867\pi\)
0.784085 0.620654i \(-0.213133\pi\)
\(998\) 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 2304.3.b.q.127.1 8
3.2 odd 2 768.3.b.f.127.4 8
4.3 odd 2 2304.3.b.t.127.1 8
8.3 odd 2 inner 2304.3.b.q.127.8 8
8.5 even 2 2304.3.b.t.127.8 8
12.11 even 2 768.3.b.e.127.8 8
16.3 odd 4 1152.3.g.c.127.2 8
16.5 even 4 1152.3.g.f.127.7 8
16.11 odd 4 1152.3.g.f.127.8 8
16.13 even 4 1152.3.g.c.127.1 8
24.5 odd 2 768.3.b.e.127.5 8
24.11 even 2 768.3.b.f.127.1 8
48.5 odd 4 384.3.g.a.127.5 yes 8
48.11 even 4 384.3.g.a.127.1 8
48.29 odd 4 384.3.g.b.127.4 yes 8
48.35 even 4 384.3.g.b.127.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.1 8 48.11 even 4
384.3.g.a.127.5 yes 8 48.5 odd 4
384.3.g.b.127.4 yes 8 48.29 odd 4
384.3.g.b.127.8 yes 8 48.35 even 4
768.3.b.e.127.5 8 24.5 odd 2
768.3.b.e.127.8 8 12.11 even 2
768.3.b.f.127.1 8 24.11 even 2
768.3.b.f.127.4 8 3.2 odd 2
1152.3.g.c.127.1 8 16.13 even 4
1152.3.g.c.127.2 8 16.3 odd 4
1152.3.g.f.127.7 8 16.5 even 4
1152.3.g.f.127.8 8 16.11 odd 4
2304.3.b.q.127.1 8 1.1 even 1 trivial
2304.3.b.q.127.8 8 8.3 odd 2 inner
2304.3.b.t.127.1 8 4.3 odd 2
2304.3.b.t.127.8 8 8.5 even 2