Properties

Label 384.7.l.a.31.6
Level $384$
Weight $7$
Character 384.31
Analytic conductor $88.341$
Analytic rank $0$
Dimension $48$
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] = [384,7,Mod(31,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.l (of order \(4\), degree \(2\), 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: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 31.6
Character \(\chi\) \(=\) 384.31
Dual form 384.7.l.a.223.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-11.0227 - 11.0227i) q^{3} +(55.7840 + 55.7840i) q^{5} -496.753 q^{7} +243.000i q^{9} +O(q^{10})\) \(q+(-11.0227 - 11.0227i) q^{3} +(55.7840 + 55.7840i) q^{5} -496.753 q^{7} +243.000i q^{9} +(694.522 - 694.522i) q^{11} +(-214.555 + 214.555i) q^{13} -1229.78i q^{15} -4131.33 q^{17} +(-7221.52 - 7221.52i) q^{19} +(5475.56 + 5475.56i) q^{21} +20205.1 q^{23} -9401.30i q^{25} +(2678.52 - 2678.52i) q^{27} +(-25794.6 + 25794.6i) q^{29} -35543.8i q^{31} -15311.0 q^{33} +(-27710.9 - 27710.9i) q^{35} +(12024.7 + 12024.7i) q^{37} +4729.95 q^{39} -5598.69i q^{41} +(-12263.5 + 12263.5i) q^{43} +(-13555.5 + 13555.5i) q^{45} +126271. i q^{47} +129115. q^{49} +(45538.4 + 45538.4i) q^{51} +(69107.7 + 69107.7i) q^{53} +77486.4 q^{55} +159201. i q^{57} +(-101232. + 101232. i) q^{59} +(-123707. + 123707. i) q^{61} -120711. i q^{63} -23937.4 q^{65} +(-394365. - 394365. i) q^{67} +(-222715. - 222715. i) q^{69} +264238. q^{71} -230983. i q^{73} +(-103628. + 103628. i) q^{75} +(-345006. + 345006. i) q^{77} -278466. i q^{79} -59049.0 q^{81} +(605066. + 605066. i) q^{83} +(-230462. - 230462. i) q^{85} +568652. q^{87} +526452. i q^{89} +(106581. - 106581. i) q^{91} +(-391789. + 391789. i) q^{93} -805690. i q^{95} +840241. q^{97} +(168769. + 168769. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 2720 q^{11} - 3936 q^{19} - 26240 q^{23} - 66400 q^{29} + 162336 q^{35} + 7200 q^{37} + 340704 q^{43} + 806736 q^{49} + 80352 q^{51} - 443680 q^{53} - 232704 q^{55} - 886144 q^{59} + 326496 q^{61} - 372832 q^{65} - 962112 q^{67} - 541728 q^{69} - 534016 q^{71} - 1073088 q^{75} + 932960 q^{77} - 2834352 q^{81} - 2497760 q^{83} + 372000 q^{85} + 775008 q^{91} - 660960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) −11.0227 11.0227i −0.408248 0.408248i
\(4\) 0 0
\(5\) 55.7840 + 55.7840i 0.446272 + 0.446272i 0.894113 0.447841i \(-0.147807\pi\)
−0.447841 + 0.894113i \(0.647807\pi\)
\(6\) 0 0
\(7\) −496.753 −1.44826 −0.724130 0.689664i \(-0.757758\pi\)
−0.724130 + 0.689664i \(0.757758\pi\)
\(8\) 0 0
\(9\) 243.000i 0.333333i
\(10\) 0 0
\(11\) 694.522 694.522i 0.521805 0.521805i −0.396311 0.918116i \(-0.629710\pi\)
0.918116 + 0.396311i \(0.129710\pi\)
\(12\) 0 0
\(13\) −214.555 + 214.555i −0.0976581 + 0.0976581i −0.754248 0.656590i \(-0.771998\pi\)
0.656590 + 0.754248i \(0.271998\pi\)
\(14\) 0 0
\(15\) 1229.78i 0.364379i
\(16\) 0 0
\(17\) −4131.33 −0.840897 −0.420449 0.907316i \(-0.638127\pi\)
−0.420449 + 0.907316i \(0.638127\pi\)
\(18\) 0 0
\(19\) −7221.52 7221.52i −1.05285 1.05285i −0.998523 0.0543305i \(-0.982698\pi\)
−0.0543305 0.998523i \(-0.517302\pi\)
\(20\) 0 0
\(21\) 5475.56 + 5475.56i 0.591249 + 0.591249i
\(22\) 0 0
\(23\) 20205.1 1.66065 0.830323 0.557283i \(-0.188156\pi\)
0.830323 + 0.557283i \(0.188156\pi\)
\(24\) 0 0
\(25\) 9401.30i 0.601683i
\(26\) 0 0
\(27\) 2678.52 2678.52i 0.136083 0.136083i
\(28\) 0 0
\(29\) −25794.6 + 25794.6i −1.05763 + 1.05763i −0.0593977 + 0.998234i \(0.518918\pi\)
−0.998234 + 0.0593977i \(0.981082\pi\)
\(30\) 0 0
\(31\) 35543.8i 1.19311i −0.802574 0.596553i \(-0.796537\pi\)
0.802574 0.596553i \(-0.203463\pi\)
\(32\) 0 0
\(33\) −15311.0 −0.426052
\(34\) 0 0
\(35\) −27710.9 27710.9i −0.646317 0.646317i
\(36\) 0 0
\(37\) 12024.7 + 12024.7i 0.237394 + 0.237394i 0.815770 0.578376i \(-0.196313\pi\)
−0.578376 + 0.815770i \(0.696313\pi\)
\(38\) 0 0
\(39\) 4729.95 0.0797375
\(40\) 0 0
\(41\) 5598.69i 0.0812335i −0.999175 0.0406167i \(-0.987068\pi\)
0.999175 0.0406167i \(-0.0129323\pi\)
\(42\) 0 0
\(43\) −12263.5 + 12263.5i −0.154244 + 0.154244i −0.780010 0.625766i \(-0.784786\pi\)
0.625766 + 0.780010i \(0.284786\pi\)
\(44\) 0 0
\(45\) −13555.5 + 13555.5i −0.148757 + 0.148757i
\(46\) 0 0
\(47\) 126271.i 1.21621i 0.793857 + 0.608105i \(0.208070\pi\)
−0.793857 + 0.608105i \(0.791930\pi\)
\(48\) 0 0
\(49\) 129115. 1.09746
\(50\) 0 0
\(51\) 45538.4 + 45538.4i 0.343295 + 0.343295i
\(52\) 0 0
\(53\) 69107.7 + 69107.7i 0.464193 + 0.464193i 0.900027 0.435834i \(-0.143546\pi\)
−0.435834 + 0.900027i \(0.643546\pi\)
\(54\) 0 0
\(55\) 77486.4 0.465734
\(56\) 0 0
\(57\) 159201.i 0.859651i
\(58\) 0 0
\(59\) −101232. + 101232.i −0.492903 + 0.492903i −0.909220 0.416316i \(-0.863321\pi\)
0.416316 + 0.909220i \(0.363321\pi\)
\(60\) 0 0
\(61\) −123707. + 123707.i −0.545011 + 0.545011i −0.924994 0.379982i \(-0.875930\pi\)
0.379982 + 0.924994i \(0.375930\pi\)
\(62\) 0 0
\(63\) 120711.i 0.482753i
\(64\) 0 0
\(65\) −23937.4 −0.0871641
\(66\) 0 0
\(67\) −394365. 394365.i −1.31121 1.31121i −0.920521 0.390694i \(-0.872235\pi\)
−0.390694 0.920521i \(-0.627765\pi\)
\(68\) 0 0
\(69\) −222715. 222715.i −0.677956 0.677956i
\(70\) 0 0
\(71\) 264238. 0.738280 0.369140 0.929374i \(-0.379652\pi\)
0.369140 + 0.929374i \(0.379652\pi\)
\(72\) 0 0
\(73\) 230983.i 0.593761i −0.954915 0.296880i \(-0.904054\pi\)
0.954915 0.296880i \(-0.0959462\pi\)
\(74\) 0 0
\(75\) −103628. + 103628.i −0.245636 + 0.245636i
\(76\) 0 0
\(77\) −345006. + 345006.i −0.755709 + 0.755709i
\(78\) 0 0
\(79\) 278466.i 0.564795i −0.959297 0.282398i \(-0.908870\pi\)
0.959297 0.282398i \(-0.0911298\pi\)
\(80\) 0 0
\(81\) −59049.0 −0.111111
\(82\) 0 0
\(83\) 605066. + 605066.i 1.05820 + 1.05820i 0.998198 + 0.0600039i \(0.0191113\pi\)
0.0600039 + 0.998198i \(0.480889\pi\)
\(84\) 0 0
\(85\) −230462. 230462.i −0.375269 0.375269i
\(86\) 0 0
\(87\) 568652. 0.863553
\(88\) 0 0
\(89\) 526452.i 0.746773i 0.927676 + 0.373386i \(0.121803\pi\)
−0.927676 + 0.373386i \(0.878197\pi\)
\(90\) 0 0
\(91\) 106581. 106581.i 0.141434 0.141434i
\(92\) 0 0
\(93\) −391789. + 391789.i −0.487083 + 0.487083i
\(94\) 0 0
\(95\) 805690.i 0.939718i
\(96\) 0 0
\(97\) 840241. 0.920637 0.460319 0.887754i \(-0.347735\pi\)
0.460319 + 0.887754i \(0.347735\pi\)
\(98\) 0 0
\(99\) 168769. + 168769.i 0.173935 + 0.173935i
\(100\) 0 0
\(101\) 222739. + 222739.i 0.216188 + 0.216188i 0.806890 0.590702i \(-0.201149\pi\)
−0.590702 + 0.806890i \(0.701149\pi\)
\(102\) 0 0
\(103\) 660421. 0.604379 0.302189 0.953248i \(-0.402283\pi\)
0.302189 + 0.953248i \(0.402283\pi\)
\(104\) 0 0
\(105\) 610897.i 0.527716i
\(106\) 0 0
\(107\) −79564.2 + 79564.2i −0.0649481 + 0.0649481i −0.738835 0.673887i \(-0.764624\pi\)
0.673887 + 0.738835i \(0.264624\pi\)
\(108\) 0 0
\(109\) −1.17937e6 + 1.17937e6i −0.910694 + 0.910694i −0.996327 0.0856332i \(-0.972709\pi\)
0.0856332 + 0.996327i \(0.472709\pi\)
\(110\) 0 0
\(111\) 265090.i 0.193832i
\(112\) 0 0
\(113\) 2.00498e6 1.38955 0.694777 0.719226i \(-0.255503\pi\)
0.694777 + 0.719226i \(0.255503\pi\)
\(114\) 0 0
\(115\) 1.12712e6 + 1.12712e6i 0.741099 + 0.741099i
\(116\) 0 0
\(117\) −52136.8 52136.8i −0.0325527 0.0325527i
\(118\) 0 0
\(119\) 2.05225e6 1.21784
\(120\) 0 0
\(121\) 806839.i 0.455439i
\(122\) 0 0
\(123\) −61712.7 + 61712.7i −0.0331634 + 0.0331634i
\(124\) 0 0
\(125\) 1.39607e6 1.39607e6i 0.714786 0.714786i
\(126\) 0 0
\(127\) 3.68727e6i 1.80009i 0.435800 + 0.900043i \(0.356465\pi\)
−0.435800 + 0.900043i \(0.643535\pi\)
\(128\) 0 0
\(129\) 270353. 0.125940
\(130\) 0 0
\(131\) 1.70561e6 + 1.70561e6i 0.758691 + 0.758691i 0.976084 0.217393i \(-0.0697554\pi\)
−0.217393 + 0.976084i \(0.569755\pi\)
\(132\) 0 0
\(133\) 3.58731e6 + 3.58731e6i 1.52481 + 1.52481i
\(134\) 0 0
\(135\) 298837. 0.121460
\(136\) 0 0
\(137\) 2.48475e6i 0.966318i 0.875533 + 0.483159i \(0.160511\pi\)
−0.875533 + 0.483159i \(0.839489\pi\)
\(138\) 0 0
\(139\) 3.44106e6 3.44106e6i 1.28129 1.28129i 0.341359 0.939933i \(-0.389113\pi\)
0.939933 0.341359i \(-0.110887\pi\)
\(140\) 0 0
\(141\) 1.39184e6 1.39184e6i 0.496516 0.496516i
\(142\) 0 0
\(143\) 298026.i 0.101917i
\(144\) 0 0
\(145\) −2.87785e6 −0.943983
\(146\) 0 0
\(147\) −1.42319e6 1.42319e6i −0.448034 0.448034i
\(148\) 0 0
\(149\) 2.43736e6 + 2.43736e6i 0.736819 + 0.736819i 0.971961 0.235142i \(-0.0755555\pi\)
−0.235142 + 0.971961i \(0.575556\pi\)
\(150\) 0 0
\(151\) 6.62460e6 1.92411 0.962053 0.272863i \(-0.0879706\pi\)
0.962053 + 0.272863i \(0.0879706\pi\)
\(152\) 0 0
\(153\) 1.00391e6i 0.280299i
\(154\) 0 0
\(155\) 1.98278e6 1.98278e6i 0.532449 0.532449i
\(156\) 0 0
\(157\) −3.45318e6 + 3.45318e6i −0.892320 + 0.892320i −0.994741 0.102421i \(-0.967341\pi\)
0.102421 + 0.994741i \(0.467341\pi\)
\(158\) 0 0
\(159\) 1.52351e6i 0.379012i
\(160\) 0 0
\(161\) −1.00369e7 −2.40505
\(162\) 0 0
\(163\) −1.39390e6 1.39390e6i −0.321860 0.321860i 0.527620 0.849480i \(-0.323084\pi\)
−0.849480 + 0.527620i \(0.823084\pi\)
\(164\) 0 0
\(165\) −854110. 854110.i −0.190135 0.190135i
\(166\) 0 0
\(167\) −6.08424e6 −1.30634 −0.653171 0.757211i \(-0.726562\pi\)
−0.653171 + 0.757211i \(0.726562\pi\)
\(168\) 0 0
\(169\) 4.73474e6i 0.980926i
\(170\) 0 0
\(171\) 1.75483e6 1.75483e6i 0.350951 0.350951i
\(172\) 0 0
\(173\) −1.58288e6 + 1.58288e6i −0.305710 + 0.305710i −0.843243 0.537533i \(-0.819356\pi\)
0.537533 + 0.843243i \(0.319356\pi\)
\(174\) 0 0
\(175\) 4.67012e6i 0.871393i
\(176\) 0 0
\(177\) 2.23170e6 0.402454
\(178\) 0 0
\(179\) 5.83249e6 + 5.83249e6i 1.01694 + 1.01694i 0.999854 + 0.0170847i \(0.00543851\pi\)
0.0170847 + 0.999854i \(0.494561\pi\)
\(180\) 0 0
\(181\) −1.51655e6 1.51655e6i −0.255754 0.255754i 0.567571 0.823325i \(-0.307883\pi\)
−0.823325 + 0.567571i \(0.807883\pi\)
\(182\) 0 0
\(183\) 2.72718e6 0.445000
\(184\) 0 0
\(185\) 1.34157e6i 0.211885i
\(186\) 0 0
\(187\) −2.86930e6 + 2.86930e6i −0.438784 + 0.438784i
\(188\) 0 0
\(189\) −1.33056e6 + 1.33056e6i −0.197083 + 0.197083i
\(190\) 0 0
\(191\) 8.72584e6i 1.25230i −0.779704 0.626148i \(-0.784631\pi\)
0.779704 0.626148i \(-0.215369\pi\)
\(192\) 0 0
\(193\) 3.56415e6 0.495774 0.247887 0.968789i \(-0.420264\pi\)
0.247887 + 0.968789i \(0.420264\pi\)
\(194\) 0 0
\(195\) 263855. + 263855.i 0.0355846 + 0.0355846i
\(196\) 0 0
\(197\) 5.20680e6 + 5.20680e6i 0.681039 + 0.681039i 0.960234 0.279195i \(-0.0900677\pi\)
−0.279195 + 0.960234i \(0.590068\pi\)
\(198\) 0 0
\(199\) −6.78464e6 −0.860929 −0.430464 0.902608i \(-0.641650\pi\)
−0.430464 + 0.902608i \(0.641650\pi\)
\(200\) 0 0
\(201\) 8.69393e6i 1.07060i
\(202\) 0 0
\(203\) 1.28135e7 1.28135e7i 1.53173 1.53173i
\(204\) 0 0
\(205\) 312317. 312317.i 0.0362522 0.0362522i
\(206\) 0 0
\(207\) 4.90983e6i 0.553549i
\(208\) 0 0
\(209\) −1.00310e7 −1.09877
\(210\) 0 0
\(211\) 6.46771e6 + 6.46771e6i 0.688499 + 0.688499i 0.961900 0.273401i \(-0.0881487\pi\)
−0.273401 + 0.961900i \(0.588149\pi\)
\(212\) 0 0
\(213\) −2.91262e6 2.91262e6i −0.301401 0.301401i
\(214\) 0 0
\(215\) −1.36821e6 −0.137670
\(216\) 0 0
\(217\) 1.76565e7i 1.72793i
\(218\) 0 0
\(219\) −2.54606e6 + 2.54606e6i −0.242402 + 0.242402i
\(220\) 0 0
\(221\) 886396. 886396.i 0.0821204 0.0821204i
\(222\) 0 0
\(223\) 9.39293e6i 0.847006i −0.905895 0.423503i \(-0.860800\pi\)
0.905895 0.423503i \(-0.139200\pi\)
\(224\) 0 0
\(225\) 2.28452e6 0.200561
\(226\) 0 0
\(227\) 1.85693e6 + 1.85693e6i 0.158751 + 0.158751i 0.782013 0.623262i \(-0.214193\pi\)
−0.623262 + 0.782013i \(0.714193\pi\)
\(228\) 0 0
\(229\) 2.24949e6 + 2.24949e6i 0.187317 + 0.187317i 0.794535 0.607218i \(-0.207715\pi\)
−0.607218 + 0.794535i \(0.707715\pi\)
\(230\) 0 0
\(231\) 7.60580e6 0.617034
\(232\) 0 0
\(233\) 1.06680e7i 0.843363i −0.906744 0.421681i \(-0.861440\pi\)
0.906744 0.421681i \(-0.138560\pi\)
\(234\) 0 0
\(235\) −7.04387e6 + 7.04387e6i −0.542760 + 0.542760i
\(236\) 0 0
\(237\) −3.06945e6 + 3.06945e6i −0.230577 + 0.230577i
\(238\) 0 0
\(239\) 3.18359e6i 0.233198i 0.993179 + 0.116599i \(0.0371992\pi\)
−0.993179 + 0.116599i \(0.962801\pi\)
\(240\) 0 0
\(241\) 2.52000e7 1.80032 0.900158 0.435564i \(-0.143451\pi\)
0.900158 + 0.435564i \(0.143451\pi\)
\(242\) 0 0
\(243\) 650880. + 650880.i 0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 7.20252e6 + 7.20252e6i 0.489763 + 0.489763i
\(246\) 0 0
\(247\) 3.09882e6 0.205639
\(248\) 0 0
\(249\) 1.33389e7i 0.864018i
\(250\) 0 0
\(251\) −903606. + 903606.i −0.0571423 + 0.0571423i −0.735100 0.677958i \(-0.762865\pi\)
0.677958 + 0.735100i \(0.262865\pi\)
\(252\) 0 0
\(253\) 1.40329e7 1.40329e7i 0.866533 0.866533i
\(254\) 0 0
\(255\) 5.08063e6i 0.306406i
\(256\) 0 0
\(257\) −909145. −0.0535591 −0.0267796 0.999641i \(-0.508525\pi\)
−0.0267796 + 0.999641i \(0.508525\pi\)
\(258\) 0 0
\(259\) −5.97332e6 5.97332e6i −0.343809 0.343809i
\(260\) 0 0
\(261\) −6.26809e6 6.26809e6i −0.352544 0.352544i
\(262\) 0 0
\(263\) 1.10225e7 0.605915 0.302957 0.953004i \(-0.402026\pi\)
0.302957 + 0.953004i \(0.402026\pi\)
\(264\) 0 0
\(265\) 7.71021e6i 0.414313i
\(266\) 0 0
\(267\) 5.80292e6 5.80292e6i 0.304869 0.304869i
\(268\) 0 0
\(269\) −1.78576e7 + 1.78576e7i −0.917416 + 0.917416i −0.996841 0.0794249i \(-0.974692\pi\)
0.0794249 + 0.996841i \(0.474692\pi\)
\(270\) 0 0
\(271\) 2.40422e6i 0.120800i −0.998174 0.0604000i \(-0.980762\pi\)
0.998174 0.0604000i \(-0.0192376\pi\)
\(272\) 0 0
\(273\) −2.34962e6 −0.115481
\(274\) 0 0
\(275\) −6.52941e6 6.52941e6i −0.313961 0.313961i
\(276\) 0 0
\(277\) −6.70939e6 6.70939e6i −0.315678 0.315678i 0.531427 0.847104i \(-0.321656\pi\)
−0.847104 + 0.531427i \(0.821656\pi\)
\(278\) 0 0
\(279\) 8.63715e6 0.397702
\(280\) 0 0
\(281\) 1.26165e7i 0.568616i −0.958733 0.284308i \(-0.908236\pi\)
0.958733 0.284308i \(-0.0917638\pi\)
\(282\) 0 0
\(283\) −1.79614e7 + 1.79614e7i −0.792464 + 0.792464i −0.981894 0.189430i \(-0.939336\pi\)
0.189430 + 0.981894i \(0.439336\pi\)
\(284\) 0 0
\(285\) −8.88089e6 + 8.88089e6i −0.383638 + 0.383638i
\(286\) 0 0
\(287\) 2.78117e6i 0.117647i
\(288\) 0 0
\(289\) −7.06970e6 −0.292892
\(290\) 0 0
\(291\) −9.26173e6 9.26173e6i −0.375849 0.375849i
\(292\) 0 0
\(293\) −1.10111e7 1.10111e7i −0.437752 0.437752i 0.453503 0.891255i \(-0.350174\pi\)
−0.891255 + 0.453503i \(0.850174\pi\)
\(294\) 0 0
\(295\) −1.12942e7 −0.439938
\(296\) 0 0
\(297\) 3.72058e6i 0.142017i
\(298\) 0 0
\(299\) −4.33510e6 + 4.33510e6i −0.162175 + 0.162175i
\(300\) 0 0
\(301\) 6.09192e6 6.09192e6i 0.223385 0.223385i
\(302\) 0 0
\(303\) 4.91037e6i 0.176517i
\(304\) 0 0
\(305\) −1.38018e7 −0.486446
\(306\) 0 0
\(307\) −3.55938e7 3.55938e7i −1.23015 1.23015i −0.963904 0.266250i \(-0.914215\pi\)
−0.266250 0.963904i \(-0.585785\pi\)
\(308\) 0 0
\(309\) −7.27962e6 7.27962e6i −0.246737 0.246737i
\(310\) 0 0
\(311\) −1.92583e7 −0.640230 −0.320115 0.947379i \(-0.603722\pi\)
−0.320115 + 0.947379i \(0.603722\pi\)
\(312\) 0 0
\(313\) 4.68811e7i 1.52885i −0.644714 0.764424i \(-0.723023\pi\)
0.644714 0.764424i \(-0.276977\pi\)
\(314\) 0 0
\(315\) 6.73374e6 6.73374e6i 0.215439 0.215439i
\(316\) 0 0
\(317\) 3.62609e7 3.62609e7i 1.13831 1.13831i 0.149556 0.988753i \(-0.452216\pi\)
0.988753 0.149556i \(-0.0477844\pi\)
\(318\) 0 0
\(319\) 3.58298e7i 1.10376i
\(320\) 0 0
\(321\) 1.75403e6 0.0530299
\(322\) 0 0
\(323\) 2.98345e7 + 2.98345e7i 0.885342 + 0.885342i
\(324\) 0 0
\(325\) 2.01709e6 + 2.01709e6i 0.0587592 + 0.0587592i
\(326\) 0 0
\(327\) 2.59998e7 0.743578
\(328\) 0 0
\(329\) 6.27253e7i 1.76139i
\(330\) 0 0
\(331\) −4.94385e7 + 4.94385e7i −1.36327 + 1.36327i −0.493550 + 0.869717i \(0.664301\pi\)
−0.869717 + 0.493550i \(0.835699\pi\)
\(332\) 0 0
\(333\) −2.92201e6 + 2.92201e6i −0.0791314 + 0.0791314i
\(334\) 0 0
\(335\) 4.39985e7i 1.17032i
\(336\) 0 0
\(337\) 6.97311e7 1.82195 0.910976 0.412460i \(-0.135330\pi\)
0.910976 + 0.412460i \(0.135330\pi\)
\(338\) 0 0
\(339\) −2.21003e7 2.21003e7i −0.567283 0.567283i
\(340\) 0 0
\(341\) −2.46860e7 2.46860e7i −0.622568 0.622568i
\(342\) 0 0
\(343\) −5.69554e6 −0.141141
\(344\) 0 0
\(345\) 2.48478e7i 0.605105i
\(346\) 0 0
\(347\) −2.21474e7 + 2.21474e7i −0.530070 + 0.530070i −0.920593 0.390523i \(-0.872294\pi\)
0.390523 + 0.920593i \(0.372294\pi\)
\(348\) 0 0
\(349\) −3.60324e7 + 3.60324e7i −0.847651 + 0.847651i −0.989840 0.142188i \(-0.954586\pi\)
0.142188 + 0.989840i \(0.454586\pi\)
\(350\) 0 0
\(351\) 1.14938e6i 0.0265792i
\(352\) 0 0
\(353\) 2.11146e7 0.480020 0.240010 0.970770i \(-0.422849\pi\)
0.240010 + 0.970770i \(0.422849\pi\)
\(354\) 0 0
\(355\) 1.47403e7 + 1.47403e7i 0.329473 + 0.329473i
\(356\) 0 0
\(357\) −2.26213e7 2.26213e7i −0.497180 0.497180i
\(358\) 0 0
\(359\) −4.86766e7 −1.05205 −0.526025 0.850469i \(-0.676318\pi\)
−0.526025 + 0.850469i \(0.676318\pi\)
\(360\) 0 0
\(361\) 5.72549e7i 1.21700i
\(362\) 0 0
\(363\) 8.89354e6 8.89354e6i 0.185932 0.185932i
\(364\) 0 0
\(365\) 1.28851e7 1.28851e7i 0.264979 0.264979i
\(366\) 0 0
\(367\) 7.99759e6i 0.161793i 0.996722 + 0.0808967i \(0.0257784\pi\)
−0.996722 + 0.0808967i \(0.974222\pi\)
\(368\) 0 0
\(369\) 1.36048e6 0.0270778
\(370\) 0 0
\(371\) −3.43295e7 3.43295e7i −0.672272 0.672272i
\(372\) 0 0
\(373\) 2.69021e7 + 2.69021e7i 0.518394 + 0.518394i 0.917085 0.398692i \(-0.130536\pi\)
−0.398692 + 0.917085i \(0.630536\pi\)
\(374\) 0 0
\(375\) −3.07768e7 −0.583620
\(376\) 0 0
\(377\) 1.10687e7i 0.206573i
\(378\) 0 0
\(379\) −3.06914e7 + 3.06914e7i −0.563765 + 0.563765i −0.930375 0.366610i \(-0.880519\pi\)
0.366610 + 0.930375i \(0.380519\pi\)
\(380\) 0 0
\(381\) 4.06437e7 4.06437e7i 0.734882 0.734882i
\(382\) 0 0
\(383\) 4.04427e7i 0.719853i 0.932981 + 0.359926i \(0.117198\pi\)
−0.932981 + 0.359926i \(0.882802\pi\)
\(384\) 0 0
\(385\) −3.84916e7 −0.674503
\(386\) 0 0
\(387\) −2.98003e6 2.98003e6i −0.0514147 0.0514147i
\(388\) 0 0
\(389\) 1.80046e7 + 1.80046e7i 0.305869 + 0.305869i 0.843305 0.537436i \(-0.180607\pi\)
−0.537436 + 0.843305i \(0.680607\pi\)
\(390\) 0 0
\(391\) −8.34738e7 −1.39643
\(392\) 0 0
\(393\) 3.76008e7i 0.619468i
\(394\) 0 0
\(395\) 1.55339e7 1.55339e7i 0.252052 0.252052i
\(396\) 0 0
\(397\) 2.46768e7 2.46768e7i 0.394383 0.394383i −0.481863 0.876246i \(-0.660040\pi\)
0.876246 + 0.481863i \(0.160040\pi\)
\(398\) 0 0
\(399\) 7.90838e7i 1.24500i
\(400\) 0 0
\(401\) −6.93037e7 −1.07479 −0.537394 0.843331i \(-0.680591\pi\)
−0.537394 + 0.843331i \(0.680591\pi\)
\(402\) 0 0
\(403\) 7.62609e6 + 7.62609e6i 0.116516 + 0.116516i
\(404\) 0 0
\(405\) −3.29399e6 3.29399e6i −0.0495858 0.0495858i
\(406\) 0 0
\(407\) 1.67029e7 0.247747
\(408\) 0 0
\(409\) 7.56422e7i 1.10559i 0.833317 + 0.552795i \(0.186439\pi\)
−0.833317 + 0.552795i \(0.813561\pi\)
\(410\) 0 0
\(411\) 2.73886e7 2.73886e7i 0.394498 0.394498i
\(412\) 0 0
\(413\) 5.02873e7 5.02873e7i 0.713852 0.713852i
\(414\) 0 0
\(415\) 6.75060e7i 0.944491i
\(416\) 0 0
\(417\) −7.58596e7 −1.04617
\(418\) 0 0
\(419\) −2.91448e7 2.91448e7i −0.396204 0.396204i 0.480688 0.876892i \(-0.340387\pi\)
−0.876892 + 0.480688i \(0.840387\pi\)
\(420\) 0 0
\(421\) 2.16659e6 + 2.16659e6i 0.0290355 + 0.0290355i 0.721476 0.692440i \(-0.243464\pi\)
−0.692440 + 0.721476i \(0.743464\pi\)
\(422\) 0 0
\(423\) −3.06837e7 −0.405403
\(424\) 0 0
\(425\) 3.88398e7i 0.505954i
\(426\) 0 0
\(427\) 6.14519e7 6.14519e7i 0.789318 0.789318i
\(428\) 0 0
\(429\) 3.28505e6 3.28505e6i 0.0416074 0.0416074i
\(430\) 0 0
\(431\) 1.18801e8i 1.48385i −0.670485 0.741923i \(-0.733914\pi\)
0.670485 0.741923i \(-0.266086\pi\)
\(432\) 0 0
\(433\) −2.58626e7 −0.318572 −0.159286 0.987232i \(-0.550919\pi\)
−0.159286 + 0.987232i \(0.550919\pi\)
\(434\) 0 0
\(435\) 3.17217e7 + 3.17217e7i 0.385379 + 0.385379i
\(436\) 0 0
\(437\) −1.45911e8 1.45911e8i −1.74842 1.74842i
\(438\) 0 0
\(439\) −1.54103e8 −1.82145 −0.910725 0.413013i \(-0.864477\pi\)
−0.910725 + 0.413013i \(0.864477\pi\)
\(440\) 0 0
\(441\) 3.13748e7i 0.365818i
\(442\) 0 0
\(443\) 8.89562e7 8.89562e7i 1.02321 1.02321i 0.0234864 0.999724i \(-0.492523\pi\)
0.999724 0.0234864i \(-0.00747663\pi\)
\(444\) 0 0
\(445\) −2.93676e7 + 2.93676e7i −0.333264 + 0.333264i
\(446\) 0 0
\(447\) 5.37326e7i 0.601610i
\(448\) 0 0
\(449\) 1.15494e8 1.27591 0.637956 0.770073i \(-0.279780\pi\)
0.637956 + 0.770073i \(0.279780\pi\)
\(450\) 0 0
\(451\) −3.88842e6 3.88842e6i −0.0423880 0.0423880i
\(452\) 0 0
\(453\) −7.30210e7 7.30210e7i −0.785513 0.785513i
\(454\) 0 0
\(455\) 1.18910e7 0.126236
\(456\) 0 0
\(457\) 1.19475e8i 1.25178i −0.779913 0.625888i \(-0.784737\pi\)
0.779913 0.625888i \(-0.215263\pi\)
\(458\) 0 0
\(459\) −1.10658e7 + 1.10658e7i −0.114432 + 0.114432i
\(460\) 0 0
\(461\) −4.11301e7 + 4.11301e7i −0.419814 + 0.419814i −0.885140 0.465326i \(-0.845937\pi\)
0.465326 + 0.885140i \(0.345937\pi\)
\(462\) 0 0
\(463\) 7.63639e7i 0.769388i 0.923044 + 0.384694i \(0.125693\pi\)
−0.923044 + 0.384694i \(0.874307\pi\)
\(464\) 0 0
\(465\) −4.37111e7 −0.434743
\(466\) 0 0
\(467\) −5.02857e7 5.02857e7i −0.493734 0.493734i 0.415746 0.909481i \(-0.363520\pi\)
−0.909481 + 0.415746i \(0.863520\pi\)
\(468\) 0 0
\(469\) 1.95902e8 + 1.95902e8i 1.89898 + 1.89898i
\(470\) 0 0
\(471\) 7.61269e7 0.728577
\(472\) 0 0
\(473\) 1.70345e7i 0.160971i
\(474\) 0 0
\(475\) −6.78917e7 + 6.78917e7i −0.633484 + 0.633484i
\(476\) 0 0
\(477\) −1.67932e7 + 1.67932e7i −0.154731 + 0.154731i
\(478\) 0 0
\(479\) 7.03911e7i 0.640489i 0.947335 + 0.320244i \(0.103765\pi\)
−0.947335 + 0.320244i \(0.896235\pi\)
\(480\) 0 0
\(481\) −5.15993e6 −0.0463669
\(482\) 0 0
\(483\) 1.10634e8 + 1.10634e8i 0.981856 + 0.981856i
\(484\) 0 0
\(485\) 4.68720e7 + 4.68720e7i 0.410854 + 0.410854i
\(486\) 0 0
\(487\) −6.50586e7 −0.563272 −0.281636 0.959521i \(-0.590877\pi\)
−0.281636 + 0.959521i \(0.590877\pi\)
\(488\) 0 0
\(489\) 3.07290e7i 0.262798i
\(490\) 0 0
\(491\) −7.38935e7 + 7.38935e7i −0.624254 + 0.624254i −0.946616 0.322362i \(-0.895523\pi\)
0.322362 + 0.946616i \(0.395523\pi\)
\(492\) 0 0
\(493\) 1.06566e8 1.06566e8i 0.889360 0.889360i
\(494\) 0 0
\(495\) 1.88292e7i 0.155245i
\(496\) 0 0
\(497\) −1.31261e8 −1.06922
\(498\) 0 0
\(499\) −1.88982e7 1.88982e7i −0.152097 0.152097i 0.626957 0.779054i \(-0.284300\pi\)
−0.779054 + 0.626957i \(0.784300\pi\)
\(500\) 0 0
\(501\) 6.70647e7 + 6.70647e7i 0.533312 + 0.533312i
\(502\) 0 0
\(503\) 1.97908e8 1.55511 0.777554 0.628817i \(-0.216460\pi\)
0.777554 + 0.628817i \(0.216460\pi\)
\(504\) 0 0
\(505\) 2.48505e7i 0.192958i
\(506\) 0 0
\(507\) 5.21897e7 5.21897e7i 0.400461 0.400461i
\(508\) 0 0
\(509\) −3.61874e7 + 3.61874e7i −0.274412 + 0.274412i −0.830874 0.556461i \(-0.812159\pi\)
0.556461 + 0.830874i \(0.312159\pi\)
\(510\) 0 0
\(511\) 1.14741e8i 0.859919i
\(512\) 0 0
\(513\) −3.86859e7 −0.286550
\(514\) 0 0
\(515\) 3.68409e7 + 3.68409e7i 0.269717 + 0.269717i
\(516\) 0 0
\(517\) 8.76977e7 + 8.76977e7i 0.634624 + 0.634624i
\(518\) 0 0
\(519\) 3.48952e7 0.249611
\(520\) 0 0
\(521\) 1.38334e8i 0.978175i −0.872235 0.489087i \(-0.837330\pi\)
0.872235 0.489087i \(-0.162670\pi\)
\(522\) 0 0
\(523\) −8.07473e7 + 8.07473e7i −0.564446 + 0.564446i −0.930567 0.366121i \(-0.880686\pi\)
0.366121 + 0.930567i \(0.380686\pi\)
\(524\) 0 0
\(525\) 5.14774e7 5.14774e7i 0.355745 0.355745i
\(526\) 0 0
\(527\) 1.46843e8i 1.00328i
\(528\) 0 0
\(529\) 2.60209e8 1.75774
\(530\) 0 0
\(531\) −2.45994e7 2.45994e7i −0.164301 0.164301i
\(532\) 0 0
\(533\) 1.20123e6 + 1.20123e6i 0.00793311 + 0.00793311i
\(534\) 0 0
\(535\) −8.87681e6 −0.0579690
\(536\) 0 0
\(537\) 1.28580e8i 0.830327i
\(538\) 0 0
\(539\) 8.96729e7 8.96729e7i 0.572658 0.572658i
\(540\) 0 0
\(541\) 4.12188e7 4.12188e7i 0.260318 0.260318i −0.564865 0.825183i \(-0.691072\pi\)
0.825183 + 0.564865i \(0.191072\pi\)
\(542\) 0 0
\(543\) 3.34330e7i 0.208822i
\(544\) 0 0
\(545\) −1.31580e8 −0.812834
\(546\) 0 0
\(547\) 3.64952e7 + 3.64952e7i 0.222984 + 0.222984i 0.809754 0.586770i \(-0.199601\pi\)
−0.586770 + 0.809754i \(0.699601\pi\)
\(548\) 0 0
\(549\) −3.00609e7 3.00609e7i −0.181670 0.181670i
\(550\) 0 0
\(551\) 3.72552e8 2.22706
\(552\) 0 0
\(553\) 1.38329e8i 0.817970i
\(554\) 0 0
\(555\) 1.47878e7 1.47878e7i 0.0865016 0.0865016i
\(556\) 0 0
\(557\) −1.55783e8 + 1.55783e8i −0.901477 + 0.901477i −0.995564 0.0940866i \(-0.970007\pi\)
0.0940866 + 0.995564i \(0.470007\pi\)
\(558\) 0 0
\(559\) 5.26238e6i 0.0301264i
\(560\) 0 0
\(561\) 6.32549e7 0.358266
\(562\) 0 0
\(563\) 5.52051e7 + 5.52051e7i 0.309353 + 0.309353i 0.844658 0.535306i \(-0.179804\pi\)
−0.535306 + 0.844658i \(0.679804\pi\)
\(564\) 0 0
\(565\) 1.11846e8 + 1.11846e8i 0.620118 + 0.620118i
\(566\) 0 0
\(567\) 2.93328e7 0.160918
\(568\) 0 0
\(569\) 3.17482e8i 1.72338i 0.507433 + 0.861691i \(0.330595\pi\)
−0.507433 + 0.861691i \(0.669405\pi\)
\(570\) 0 0
\(571\) −5.83480e7 + 5.83480e7i −0.313413 + 0.313413i −0.846230 0.532817i \(-0.821133\pi\)
0.532817 + 0.846230i \(0.321133\pi\)
\(572\) 0 0
\(573\) −9.61823e7 + 9.61823e7i −0.511248 + 0.511248i
\(574\) 0 0
\(575\) 1.89954e8i 0.999182i
\(576\) 0 0
\(577\) 1.26597e8 0.659015 0.329507 0.944153i \(-0.393117\pi\)
0.329507 + 0.944153i \(0.393117\pi\)
\(578\) 0 0
\(579\) −3.92865e7 3.92865e7i −0.202399 0.202399i
\(580\) 0 0
\(581\) −3.00568e8 3.00568e8i −1.53255 1.53255i
\(582\) 0 0
\(583\) 9.59937e7 0.484437
\(584\) 0 0
\(585\) 5.81680e6i 0.0290547i
\(586\) 0 0
\(587\) 1.49945e8 1.49945e8i 0.741342 0.741342i −0.231494 0.972836i \(-0.574361\pi\)
0.972836 + 0.231494i \(0.0743614\pi\)
\(588\) 0 0
\(589\) −2.56680e8 + 2.56680e8i −1.25617 + 1.25617i
\(590\) 0 0
\(591\) 1.14786e8i 0.556066i
\(592\) 0 0
\(593\) 2.26564e8 1.08649 0.543245 0.839574i \(-0.317195\pi\)
0.543245 + 0.839574i \(0.317195\pi\)
\(594\) 0 0
\(595\) 1.14483e8 + 1.14483e8i 0.543486 + 0.543486i
\(596\) 0 0
\(597\) 7.47850e7 + 7.47850e7i 0.351473 + 0.351473i
\(598\) 0 0
\(599\) 1.65861e7 0.0771727 0.0385864 0.999255i \(-0.487715\pi\)
0.0385864 + 0.999255i \(0.487715\pi\)
\(600\) 0 0
\(601\) 3.72359e8i 1.71529i 0.514238 + 0.857647i \(0.328075\pi\)
−0.514238 + 0.857647i \(0.671925\pi\)
\(602\) 0 0
\(603\) 9.58307e7 9.58307e7i 0.437072 0.437072i
\(604\) 0 0
\(605\) −4.50087e7 + 4.50087e7i −0.203250 + 0.203250i
\(606\) 0 0
\(607\) 2.68278e8i 1.19955i −0.800168 0.599776i \(-0.795256\pi\)
0.800168 0.599776i \(-0.204744\pi\)
\(608\) 0 0
\(609\) −2.82480e8 −1.25065
\(610\) 0 0
\(611\) −2.70920e7 2.70920e7i −0.118773 0.118773i
\(612\) 0 0
\(613\) 1.58322e7 + 1.58322e7i 0.0687322 + 0.0687322i 0.740637 0.671905i \(-0.234524\pi\)
−0.671905 + 0.740637i \(0.734524\pi\)
\(614\) 0 0
\(615\) −6.88516e6 −0.0295998
\(616\) 0 0
\(617\) 1.28338e7i 0.0546387i −0.999627 0.0273193i \(-0.991303\pi\)
0.999627 0.0273193i \(-0.00869710\pi\)
\(618\) 0 0
\(619\) 1.76902e8 1.76902e8i 0.745864 0.745864i −0.227835 0.973700i \(-0.573165\pi\)
0.973700 + 0.227835i \(0.0731648\pi\)
\(620\) 0 0
\(621\) 5.41196e7 5.41196e7i 0.225985 0.225985i
\(622\) 0 0
\(623\) 2.61516e8i 1.08152i
\(624\) 0 0
\(625\) 8.86098e6 0.0362946
\(626\) 0 0
\(627\) 1.10569e8 + 1.10569e8i 0.448570 + 0.448570i
\(628\) 0 0
\(629\) −4.96781e7 4.96781e7i −0.199624 0.199624i
\(630\) 0 0
\(631\) 2.96825e8 1.18144 0.590721 0.806876i \(-0.298843\pi\)
0.590721 + 0.806876i \(0.298843\pi\)
\(632\) 0 0
\(633\) 1.42583e8i 0.562157i
\(634\) 0 0
\(635\) −2.05690e8 + 2.05690e8i −0.803328 + 0.803328i
\(636\) 0 0
\(637\) −2.77021e7 + 2.77021e7i −0.107175 + 0.107175i
\(638\) 0 0
\(639\) 6.42099e7i 0.246093i
\(640\) 0 0
\(641\) 4.80696e8 1.82514 0.912571 0.408918i \(-0.134094\pi\)
0.912571 + 0.408918i \(0.134094\pi\)
\(642\) 0 0
\(643\) 1.10391e8 + 1.10391e8i 0.415242 + 0.415242i 0.883560 0.468318i \(-0.155140\pi\)
−0.468318 + 0.883560i \(0.655140\pi\)
\(644\) 0 0
\(645\) 1.50814e7 + 1.50814e7i 0.0562034 + 0.0562034i
\(646\) 0 0
\(647\) 1.38174e6 0.00510170 0.00255085 0.999997i \(-0.499188\pi\)
0.00255085 + 0.999997i \(0.499188\pi\)
\(648\) 0 0
\(649\) 1.40616e8i 0.514399i
\(650\) 0 0
\(651\) 1.94622e8 1.94622e8i 0.705423 0.705423i
\(652\) 0 0
\(653\) −9.70802e7 + 9.70802e7i −0.348651 + 0.348651i −0.859607 0.510956i \(-0.829292\pi\)
0.510956 + 0.859607i \(0.329292\pi\)
\(654\) 0 0
\(655\) 1.90291e8i 0.677165i
\(656\) 0 0
\(657\) 5.61289e7 0.197920
\(658\) 0 0
\(659\) −1.68632e8 1.68632e8i −0.589229 0.589229i 0.348194 0.937423i \(-0.386795\pi\)
−0.937423 + 0.348194i \(0.886795\pi\)
\(660\) 0 0
\(661\) 3.28246e7 + 3.28246e7i 0.113657 + 0.113657i 0.761648 0.647991i \(-0.224391\pi\)
−0.647991 + 0.761648i \(0.724391\pi\)
\(662\) 0 0
\(663\) −1.95410e7 −0.0670510
\(664\) 0 0
\(665\) 4.00229e8i 1.36095i
\(666\) 0 0
\(667\) −5.21182e8 + 5.21182e8i −1.75635 + 1.75635i
\(668\) 0 0
\(669\) −1.03535e8 + 1.03535e8i −0.345789 + 0.345789i
\(670\) 0 0
\(671\) 1.71835e8i 0.568779i
\(672\) 0 0
\(673\) −2.71251e8 −0.889870 −0.444935 0.895563i \(-0.646773\pi\)
−0.444935 + 0.895563i \(0.646773\pi\)
\(674\) 0 0
\(675\) −2.51815e7 2.51815e7i −0.0818787 0.0818787i
\(676\) 0 0
\(677\) 2.72402e8 + 2.72402e8i 0.877897 + 0.877897i 0.993317 0.115420i \(-0.0368214\pi\)
−0.115420 + 0.993317i \(0.536821\pi\)
\(678\) 0 0
\(679\) −4.17392e8 −1.33332
\(680\) 0 0
\(681\) 4.09367e7i 0.129620i
\(682\) 0 0
\(683\) 1.80546e8 1.80546e8i 0.566664 0.566664i −0.364528 0.931192i \(-0.618770\pi\)
0.931192 + 0.364528i \(0.118770\pi\)
\(684\) 0 0
\(685\) −1.38609e8 + 1.38609e8i −0.431241 + 0.431241i
\(686\) 0 0
\(687\) 4.95908e7i 0.152944i
\(688\) 0 0
\(689\) −2.96548e7 −0.0906645
\(690\) 0 0
\(691\) 2.41248e8 + 2.41248e8i 0.731189 + 0.731189i 0.970855 0.239666i \(-0.0770380\pi\)
−0.239666 + 0.970855i \(0.577038\pi\)
\(692\) 0 0
\(693\) −8.38365e7 8.38365e7i −0.251903 0.251903i
\(694\) 0 0
\(695\) 3.83912e8 1.14361
\(696\) 0 0
\(697\) 2.31300e7i 0.0683090i
\(698\) 0 0
\(699\) −1.17590e8 + 1.17590e8i −0.344301 + 0.344301i
\(700\) 0 0
\(701\) 7.64461e7 7.64461e7i 0.221922 0.221922i −0.587385 0.809308i \(-0.699843\pi\)
0.809308 + 0.587385i \(0.199843\pi\)
\(702\) 0 0
\(703\) 1.73674e8i 0.499883i
\(704\) 0 0
\(705\) 1.55285e8 0.443162
\(706\) 0 0
\(707\) −1.10646e8 1.10646e8i −0.313097 0.313097i
\(708\) 0 0
\(709\) −9.09088e7 9.09088e7i −0.255074 0.255074i 0.567973 0.823047i \(-0.307728\pi\)
−0.823047 + 0.567973i \(0.807728\pi\)
\(710\) 0 0
\(711\) 6.76673e7 0.188265
\(712\) 0 0
\(713\) 7.18165e8i 1.98133i
\(714\) 0 0
\(715\) −1.66251e7 + 1.66251e7i −0.0454826 + 0.0454826i
\(716\) 0 0
\(717\) 3.50918e7 3.50918e7i 0.0952025 0.0952025i
\(718\) 0 0
\(719\) 4.32498e8i 1.16358i 0.813338 + 0.581791i \(0.197648\pi\)
−0.813338 + 0.581791i \(0.802352\pi\)
\(720\) 0 0
\(721\) −3.28066e8 −0.875297
\(722\) 0 0
\(723\) −2.77772e8 2.77772e8i −0.734976 0.734976i
\(724\) 0 0
\(725\) 2.42503e8 + 2.42503e8i 0.636359 + 0.636359i
\(726\) 0 0
\(727\) 4.23063e8 1.10104 0.550518 0.834823i \(-0.314430\pi\)
0.550518 + 0.834823i \(0.314430\pi\)
\(728\) 0 0
\(729\) 1.43489e7i 0.0370370i
\(730\) 0 0
\(731\) 5.06645e7 5.06645e7i 0.129703 0.129703i
\(732\) 0 0
\(733\) 2.01686e8 2.01686e8i 0.512111 0.512111i −0.403062 0.915173i \(-0.632054\pi\)
0.915173 + 0.403062i \(0.132054\pi\)
\(734\) 0 0
\(735\) 1.58783e8i 0.399890i
\(736\) 0 0
\(737\) −5.47790e8 −1.36840
\(738\) 0 0
\(739\) −2.56114e8 2.56114e8i −0.634601 0.634601i 0.314617 0.949219i \(-0.398124\pi\)
−0.949219 + 0.314617i \(0.898124\pi\)
\(740\) 0 0
\(741\) −3.41574e7 3.41574e7i −0.0839519 0.0839519i
\(742\) 0 0
\(743\) −4.27129e8 −1.04134 −0.520670 0.853758i \(-0.674318\pi\)
−0.520670 + 0.853758i \(0.674318\pi\)
\(744\) 0 0
\(745\) 2.71931e8i 0.657643i
\(746\) 0 0
\(747\) −1.47031e8 + 1.47031e8i −0.352734 + 0.352734i
\(748\) 0 0
\(749\) 3.95238e7 3.95238e7i 0.0940617 0.0940617i
\(750\) 0 0
\(751\) 5.54989e8i 1.31028i 0.755507 + 0.655140i \(0.227391\pi\)
−0.755507 + 0.655140i \(0.772609\pi\)
\(752\) 0 0
\(753\) 1.99204e7 0.0466565
\(754\) 0 0
\(755\) 3.69547e8 + 3.69547e8i 0.858674 + 0.858674i
\(756\) 0 0
\(757\) 1.31247e8 + 1.31247e8i 0.302553 + 0.302553i 0.842012 0.539459i \(-0.181371\pi\)
−0.539459 + 0.842012i \(0.681371\pi\)
\(758\) 0 0
\(759\) −3.09360e8 −0.707521
\(760\) 0 0
\(761\) 4.29874e8i 0.975411i −0.873008 0.487705i \(-0.837834\pi\)
0.873008 0.487705i \(-0.162166\pi\)
\(762\) 0 0
\(763\) 5.85858e8 5.85858e8i 1.31892 1.31892i
\(764\) 0 0
\(765\) 5.60022e7 5.60022e7i 0.125090 0.125090i
\(766\) 0 0
\(767\) 4.34396e7i 0.0962720i
\(768\) 0 0
\(769\) 1.99225e8 0.438091 0.219045 0.975715i \(-0.429706\pi\)
0.219045 + 0.975715i \(0.429706\pi\)
\(770\) 0 0
\(771\) 1.00212e7 + 1.00212e7i 0.0218654 + 0.0218654i
\(772\) 0 0
\(773\) −1.29385e8 1.29385e8i −0.280121 0.280121i 0.553036 0.833157i \(-0.313469\pi\)
−0.833157 + 0.553036i \(0.813469\pi\)
\(774\) 0 0
\(775\) −3.34158e8 −0.717871
\(776\) 0 0
\(777\) 1.31684e8i 0.280718i
\(778\) 0 0
\(779\) −4.04311e7 + 4.04311e7i −0.0855270 + 0.0855270i
\(780\) 0 0
\(781\) 1.83519e8 1.83519e8i 0.385238 0.385238i
\(782\) 0 0
\(783\) 1.38182e8i 0.287851i
\(784\) 0 0
\(785\) −3.85265e8 −0.796435
\(786\) 0 0
\(787\) −2.68473e7 2.68473e7i −0.0550778 0.0550778i 0.679031 0.734109i \(-0.262400\pi\)
−0.734109 + 0.679031i \(0.762400\pi\)
\(788\) 0 0
\(789\) −1.21497e8 1.21497e8i −0.247364 0.247364i
\(790\) 0 0
\(791\) −9.95981e8 −2.01243
\(792\) 0 0
\(793\) 5.30840e7i 0.106450i
\(794\) 0 0
\(795\) 8.49873e7 8.49873e7i 0.169143 0.169143i
\(796\) 0 0
\(797\) 5.83661e8 5.83661e8i 1.15288 1.15288i 0.166913 0.985972i \(-0.446620\pi\)
0.985972 0.166913i \(-0.0533798\pi\)
\(798\) 0 0
\(799\) 5.21665e8i 1.02271i
\(800\) 0 0
\(801\) −1.27928e8 −0.248924
\(802\) 0 0
\(803\) −1.60423e8 1.60423e8i −0.309827 0.309827i
\(804\) 0 0
\(805\) −5.59900e8 5.59900e8i −1.07330 1.07330i
\(806\) 0 0
\(807\) 3.93678e8 0.749067
\(808\) 0 0
\(809\) 2.19319e8i 0.414219i −0.978318 0.207110i \(-0.933594\pi\)
0.978318 0.207110i \(-0.0664057\pi\)
\(810\) 0 0
\(811\) 5.12226e6 5.12226e6i 0.00960282 0.00960282i −0.702289 0.711892i \(-0.747839\pi\)
0.711892 + 0.702289i \(0.247839\pi\)
\(812\) 0 0
\(813\) −2.65010e7 + 2.65010e7i −0.0493164 + 0.0493164i
\(814\) 0 0
\(815\) 1.55514e8i 0.287274i
\(816\) 0 0
\(817\) 1.77122e8 0.324793
\(818\) 0 0
\(819\) 2.58991e7 + 2.58991e7i 0.0471447 + 0.0471447i
\(820\) 0 0
\(821\) 6.18920e8 + 6.18920e8i 1.11842 + 1.11842i 0.991973 + 0.126447i \(0.0403574\pi\)
0.126447 + 0.991973i \(0.459643\pi\)
\(822\) 0 0
\(823\) −4.08281e8 −0.732420 −0.366210 0.930532i \(-0.619345\pi\)
−0.366210 + 0.930532i \(0.619345\pi\)
\(824\) 0 0
\(825\) 1.43943e8i 0.256348i
\(826\) 0 0
\(827\) 2.35915e8 2.35915e8i 0.417099 0.417099i −0.467103 0.884203i \(-0.654702\pi\)
0.884203 + 0.467103i \(0.154702\pi\)
\(828\) 0 0
\(829\) 1.79611e8 1.79611e8i 0.315260 0.315260i −0.531683 0.846943i \(-0.678440\pi\)
0.846943 + 0.531683i \(0.178440\pi\)
\(830\) 0 0
\(831\) 1.47911e8i 0.257750i
\(832\) 0 0
\(833\) −5.33415e8 −0.922847
\(834\) 0 0
\(835\) −3.39403e8 3.39403e8i −0.582983 0.582983i
\(836\) 0 0
\(837\) −9.52047e7 9.52047e7i −0.162361 0.162361i
\(838\) 0 0
\(839\) −1.16608e7 −0.0197444 −0.00987221 0.999951i \(-0.503142\pi\)
−0.00987221 + 0.999951i \(0.503142\pi\)
\(840\) 0 0
\(841\) 7.35898e8i 1.23717i
\(842\) 0 0
\(843\) −1.39068e8 + 1.39068e8i −0.232136 + 0.232136i
\(844\) 0 0
\(845\) −2.64123e8 + 2.64123e8i −0.437760 + 0.437760i
\(846\) 0 0
\(847\) 4.00800e8i 0.659594i
\(848\) 0 0
\(849\) 3.95965e8 0.647044
\(850\) 0 0
\(851\) 2.42961e8 + 2.42961e8i 0.394228 + 0.394228i
\(852\) 0 0
\(853\) −9.74736e7 9.74736e7i −0.157051 0.157051i 0.624208 0.781258i \(-0.285422\pi\)
−0.781258 + 0.624208i \(0.785422\pi\)
\(854\) 0 0
\(855\) 1.95783e8 0.313239
\(856\) 0 0
\(857\) 9.03691e8i 1.43575i 0.696174 + 0.717873i \(0.254884\pi\)
−0.696174 + 0.717873i \(0.745116\pi\)
\(858\) 0 0
\(859\) 1.52064e8 1.52064e8i 0.239909 0.239909i −0.576903 0.816812i \(-0.695739\pi\)
0.816812 + 0.576903i \(0.195739\pi\)
\(860\) 0 0
\(861\) 3.06560e7 3.06560e7i 0.0480293 0.0480293i
\(862\) 0 0
\(863\) 3.80994e8i 0.592770i 0.955069 + 0.296385i \(0.0957811\pi\)
−0.955069 + 0.296385i \(0.904219\pi\)
\(864\) 0 0
\(865\) −1.76598e8 −0.272859
\(866\) 0 0
\(867\) 7.79272e7 + 7.79272e7i 0.119573 + 0.119573i
\(868\) 0 0
\(869\) −1.93401e8 1.93401e8i −0.294713 0.294713i
\(870\) 0 0
\(871\) 1.69226e8 0.256101
\(872\) 0 0
\(873\) 2.04179e8i 0.306879i
\(874\) 0 0
\(875\) −6.93500e8 + 6.93500e8i −1.03520 + 1.03520i
\(876\) 0 0
\(877\) 5.62865e8 5.62865e8i 0.834460 0.834460i −0.153663 0.988123i \(-0.549107\pi\)
0.988123 + 0.153663i \(0.0491071\pi\)
\(878\) 0 0
\(879\) 2.42744e8i 0.357423i
\(880\) 0 0
\(881\) 1.19313e9 1.74486 0.872428 0.488743i \(-0.162544\pi\)
0.872428 + 0.488743i \(0.162544\pi\)
\(882\) 0 0
\(883\) −3.93429e8 3.93429e8i −0.571458 0.571458i 0.361078 0.932536i \(-0.382409\pi\)
−0.932536 + 0.361078i \(0.882409\pi\)
\(884\) 0 0
\(885\) 1.24493e8 + 1.24493e8i 0.179604 + 0.179604i
\(886\) 0 0
\(887\) −6.77947e8 −0.971460 −0.485730 0.874109i \(-0.661446\pi\)
−0.485730 + 0.874109i \(0.661446\pi\)
\(888\) 0 0
\(889\) 1.83166e9i 2.60699i
\(890\) 0 0
\(891\) −4.10108e7 + 4.10108e7i −0.0579783 + 0.0579783i
\(892\) 0 0
\(893\) 9.11866e8 9.11866e8i 1.28049 1.28049i
\(894\) 0 0
\(895\) 6.50719e8i 0.907662i
\(896\) 0 0
\(897\) 9.55689e7 0.132416
\(898\) 0 0
\(899\) 9.16838e8 + 9.16838e8i 1.26187 + 1.26187i
\(900\) 0 0
\(901\) −2.85507e8 2.85507e8i −0.390339 0.390339i
\(902\) 0 0
\(903\) −1.34299e8 −0.182393
\(904\) 0 0
\(905\) 1.69199e8i 0.228271i
\(906\) 0 0
\(907\) −4.68493e8 + 4.68493e8i −0.627887 + 0.627887i −0.947536 0.319649i \(-0.896435\pi\)
0.319649 + 0.947536i \(0.396435\pi\)
\(908\) 0 0
\(909\) −5.41256e7 + 5.41256e7i −0.0720628 + 0.0720628i
\(910\) 0 0
\(911\) 3.97237e8i 0.525406i −0.964877 0.262703i \(-0.915386\pi\)
0.964877 0.262703i \(-0.0846139\pi\)
\(912\) 0 0
\(913\) 8.40464e8 1.10435
\(914\) 0 0
\(915\) 1.52133e8 + 1.52133e8i 0.198591 + 0.198591i
\(916\) 0 0
\(917\) −8.47265e8 8.47265e8i −1.09878 1.09878i
\(918\) 0 0
\(919\) −5.37068e8 −0.691963 −0.345982 0.938241i \(-0.612454\pi\)
−0.345982 + 0.938241i \(0.612454\pi\)
\(920\) 0 0
\(921\) 7.84680e8i 1.00442i
\(922\) 0 0
\(923\) −5.66936e7 + 5.66936e7i −0.0720989 + 0.0720989i
\(924\) 0 0
\(925\) 1.13048e8 1.13048e8i 0.142836 0.142836i
\(926\) 0 0
\(927\) 1.60482e8i 0.201460i
\(928\) 0 0
\(929\) −1.33360e9 −1.66333 −0.831667 0.555275i \(-0.812613\pi\)
−0.831667 + 0.555275i \(0.812613\pi\)
\(930\) 0 0
\(931\) −9.32403e8 9.32403e8i −1.15546 1.15546i
\(932\) 0 0
\(933\) 2.12278e8 + 2.12278e8i 0.261373 + 0.261373i
\(934\) 0 0
\(935\) −3.20122e8 −0.391634
\(936\) 0 0
\(937\) 6.89471e8i 0.838102i −0.907963 0.419051i \(-0.862363\pi\)
0.907963 0.419051i \(-0.137637\pi\)
\(938\) 0 0
\(939\) −5.16756e8 + 5.16756e8i −0.624150 + 0.624150i
\(940\) 0 0
\(941\) −5.75999e8 + 5.75999e8i −0.691278 + 0.691278i −0.962513 0.271235i \(-0.912568\pi\)
0.271235 + 0.962513i \(0.412568\pi\)
\(942\) 0 0
\(943\) 1.13122e8i 0.134900i
\(944\) 0 0
\(945\) −1.48448e8 −0.175905
\(946\) 0 0
\(947\) −3.25772e8 3.25772e8i −0.383587 0.383587i 0.488806 0.872393i \(-0.337433\pi\)
−0.872393 + 0.488806i \(0.837433\pi\)
\(948\) 0 0
\(949\) 4.95585e7 + 4.95585e7i 0.0579855 + 0.0579855i
\(950\) 0 0
\(951\) −7.99385e8 −0.929426
\(952\) 0 0
\(953\) 7.99405e7i 0.0923609i −0.998933 0.0461804i \(-0.985295\pi\)
0.998933 0.0461804i \(-0.0147049\pi\)
\(954\) 0 0
\(955\) 4.86762e8 4.86762e8i 0.558864 0.558864i
\(956\) 0 0
\(957\) 3.94942e8 3.94942e8i 0.450606 0.450606i
\(958\) 0 0
\(959\) 1.23430e9i 1.39948i
\(960\) 0 0
\(961\) −3.75859e8 −0.423501
\(962\) 0 0
\(963\) −1.93341e7 1.93341e7i −0.0216494 0.0216494i
\(964\) 0 0
\(965\) 1.98822e8 + 1.98822e8i 0.221250 + 0.221250i
\(966\) 0 0
\(967\) 1.19083e9 1.31696 0.658479 0.752599i \(-0.271200\pi\)
0.658479 + 0.752599i \(0.271200\pi\)
\(968\) 0 0
\(969\) 6.57713e8i 0.722878i
\(970\) 0 0
\(971\) −7.36800e8 + 7.36800e8i −0.804808 + 0.804808i −0.983843 0.179035i \(-0.942702\pi\)
0.179035 + 0.983843i \(0.442702\pi\)
\(972\) 0 0
\(973\) −1.70936e9 + 1.70936e9i −1.85564 + 1.85564i
\(974\) 0 0
\(975\) 4.44676e7i 0.0479767i
\(976\) 0 0
\(977\) 7.27724e8 0.780338 0.390169 0.920743i \(-0.372417\pi\)
0.390169 + 0.920743i \(0.372417\pi\)
\(978\) 0 0
\(979\) 3.65632e8 + 3.65632e8i 0.389670 + 0.389670i
\(980\) 0 0
\(981\) −2.86588e8 2.86588e8i −0.303565 0.303565i
\(982\) 0 0
\(983\) −4.20246e8 −0.442428 −0.221214 0.975225i \(-0.571002\pi\)
−0.221214 + 0.975225i \(0.571002\pi\)
\(984\) 0 0
\(985\) 5.80912e8i 0.607857i
\(986\) 0 0
\(987\) −6.91402e8 + 6.91402e8i −0.719083 + 0.719083i
\(988\) 0 0
\(989\) −2.47785e8 + 2.47785e8i −0.256145 + 0.256145i
\(990\) 0 0
\(991\) 2.01934e8i 0.207486i 0.994604 + 0.103743i \(0.0330820\pi\)
−0.994604 + 0.103743i \(0.966918\pi\)
\(992\) 0 0
\(993\) 1.08989e9 1.11310
\(994\) 0 0
\(995\) −3.78474e8 3.78474e8i −0.384208 0.384208i
\(996\) 0 0
\(997\) −1.25907e9 1.25907e9i −1.27047 1.27047i −0.945842 0.324626i \(-0.894761\pi\)
−0.324626 0.945842i \(-0.605239\pi\)
\(998\) 0 0
\(999\) 6.44169e7 0.0646105
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 384.7.l.a.31.6 48
4.3 odd 2 384.7.l.b.31.19 48
8.3 odd 2 48.7.l.a.43.3 yes 48
8.5 even 2 192.7.l.a.79.19 48
16.3 odd 4 inner 384.7.l.a.223.6 48
16.5 even 4 48.7.l.a.19.3 48
16.11 odd 4 192.7.l.a.175.19 48
16.13 even 4 384.7.l.b.223.19 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.7.l.a.19.3 48 16.5 even 4
48.7.l.a.43.3 yes 48 8.3 odd 2
192.7.l.a.79.19 48 8.5 even 2
192.7.l.a.175.19 48 16.11 odd 4
384.7.l.a.31.6 48 1.1 even 1 trivial
384.7.l.a.223.6 48 16.3 odd 4 inner
384.7.l.b.31.19 48 4.3 odd 2
384.7.l.b.223.19 48 16.13 even 4