Properties

Label 384.5.h.e.65.3
Level $384$
Weight $5$
Character 384.65
Analytic conductor $39.694$
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] = [384,5,Mod(65,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.3
Root \(-1.58114 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.65
Dual form 384.5.h.e.65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.00000 + 8.48528i) q^{3} -37.9473 q^{5} -37.9473 q^{7} +(-63.0000 - 50.9117i) q^{9} +O(q^{10})\) \(q+(-3.00000 + 8.48528i) q^{3} -37.9473 q^{5} -37.9473 q^{7} +(-63.0000 - 50.9117i) q^{9} +134.000 q^{11} -321.994i q^{13} +(113.842 - 321.994i) q^{15} +237.588i q^{17} -492.146i q^{19} +(113.842 - 321.994i) q^{21} +643.988i q^{23} +815.000 q^{25} +(621.000 - 381.838i) q^{27} -796.894 q^{29} -1100.47 q^{31} +(-402.000 + 1137.03i) q^{33} +1440.00 q^{35} +1609.97i q^{37} +(2732.21 + 965.981i) q^{39} +1289.76i q^{41} +424.264i q^{43} +(2390.68 + 1931.96i) q^{45} -2575.95i q^{47} -961.000 q^{49} +(-2016.00 - 712.764i) q^{51} +1631.74 q^{53} -5084.94 q^{55} +(4176.00 + 1476.44i) q^{57} -1382.00 q^{59} -321.994i q^{61} +(2390.68 + 1931.96i) q^{63} +12218.8i q^{65} -186.676i q^{67} +(-5464.42 - 1931.96i) q^{69} -1931.96i q^{71} +4894.00 q^{73} +(-2445.00 + 6915.50i) q^{75} -5084.94 q^{77} +1479.95 q^{79} +(1377.00 + 6414.87i) q^{81} -5914.00 q^{83} -9015.83i q^{85} +(2390.68 - 6761.87i) q^{87} +9062.28i q^{89} +12218.8i q^{91} +(3301.42 - 9337.82i) q^{93} +18675.6i q^{95} +5858.00 q^{97} +(-8442.00 - 6822.17i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 252 q^{9} + 536 q^{11} + 3260 q^{25} + 2484 q^{27} - 1608 q^{33} + 5760 q^{35} - 3844 q^{49} - 8064 q^{51} + 16704 q^{57} - 5528 q^{59} + 19576 q^{73} - 9780 q^{75} + 5508 q^{81} - 23656 q^{83} + 23432 q^{97} - 33768 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −3.00000 + 8.48528i −0.333333 + 0.942809i
\(4\) 0 0
\(5\) −37.9473 −1.51789 −0.758947 0.651153i \(-0.774286\pi\)
−0.758947 + 0.651153i \(0.774286\pi\)
\(6\) 0 0
\(7\) −37.9473 −0.774435 −0.387218 0.921988i \(-0.626564\pi\)
−0.387218 + 0.921988i \(0.626564\pi\)
\(8\) 0 0
\(9\) −63.0000 50.9117i −0.777778 0.628539i
\(10\) 0 0
\(11\) 134.000 1.10744 0.553719 0.832704i \(-0.313208\pi\)
0.553719 + 0.832704i \(0.313208\pi\)
\(12\) 0 0
\(13\) 321.994i 1.90529i −0.304087 0.952644i \(-0.598351\pi\)
0.304087 0.952644i \(-0.401649\pi\)
\(14\) 0 0
\(15\) 113.842 321.994i 0.505964 1.43108i
\(16\) 0 0
\(17\) 237.588i 0.822103i 0.911612 + 0.411052i \(0.134838\pi\)
−0.911612 + 0.411052i \(0.865162\pi\)
\(18\) 0 0
\(19\) 492.146i 1.36329i −0.731685 0.681643i \(-0.761266\pi\)
0.731685 0.681643i \(-0.238734\pi\)
\(20\) 0 0
\(21\) 113.842 321.994i 0.258145 0.730145i
\(22\) 0 0
\(23\) 643.988i 1.21737i 0.793413 + 0.608684i \(0.208302\pi\)
−0.793413 + 0.608684i \(0.791698\pi\)
\(24\) 0 0
\(25\) 815.000 1.30400
\(26\) 0 0
\(27\) 621.000 381.838i 0.851852 0.523783i
\(28\) 0 0
\(29\) −796.894 −0.947555 −0.473778 0.880645i \(-0.657110\pi\)
−0.473778 + 0.880645i \(0.657110\pi\)
\(30\) 0 0
\(31\) −1100.47 −1.14513 −0.572566 0.819858i \(-0.694052\pi\)
−0.572566 + 0.819858i \(0.694052\pi\)
\(32\) 0 0
\(33\) −402.000 + 1137.03i −0.369146 + 1.04410i
\(34\) 0 0
\(35\) 1440.00 1.17551
\(36\) 0 0
\(37\) 1609.97i 1.17602i 0.808854 + 0.588009i \(0.200088\pi\)
−0.808854 + 0.588009i \(0.799912\pi\)
\(38\) 0 0
\(39\) 2732.21 + 965.981i 1.79632 + 0.635096i
\(40\) 0 0
\(41\) 1289.76i 0.767259i 0.923487 + 0.383630i \(0.125326\pi\)
−0.923487 + 0.383630i \(0.874674\pi\)
\(42\) 0 0
\(43\) 424.264i 0.229456i 0.993397 + 0.114728i \(0.0365996\pi\)
−0.993397 + 0.114728i \(0.963400\pi\)
\(44\) 0 0
\(45\) 2390.68 + 1931.96i 1.18058 + 0.954056i
\(46\) 0 0
\(47\) 2575.95i 1.16612i −0.812431 0.583058i \(-0.801856\pi\)
0.812431 0.583058i \(-0.198144\pi\)
\(48\) 0 0
\(49\) −961.000 −0.400250
\(50\) 0 0
\(51\) −2016.00 712.764i −0.775087 0.274034i
\(52\) 0 0
\(53\) 1631.74 0.580895 0.290448 0.956891i \(-0.406196\pi\)
0.290448 + 0.956891i \(0.406196\pi\)
\(54\) 0 0
\(55\) −5084.94 −1.68097
\(56\) 0 0
\(57\) 4176.00 + 1476.44i 1.28532 + 0.454429i
\(58\) 0 0
\(59\) −1382.00 −0.397012 −0.198506 0.980100i \(-0.563609\pi\)
−0.198506 + 0.980100i \(0.563609\pi\)
\(60\) 0 0
\(61\) 321.994i 0.0865342i −0.999064 0.0432671i \(-0.986223\pi\)
0.999064 0.0432671i \(-0.0137766\pi\)
\(62\) 0 0
\(63\) 2390.68 + 1931.96i 0.602339 + 0.486763i
\(64\) 0 0
\(65\) 12218.8i 2.89202i
\(66\) 0 0
\(67\) 186.676i 0.0415853i −0.999784 0.0207926i \(-0.993381\pi\)
0.999784 0.0207926i \(-0.00661898\pi\)
\(68\) 0 0
\(69\) −5464.42 1931.96i −1.14775 0.405789i
\(70\) 0 0
\(71\) 1931.96i 0.383250i −0.981468 0.191625i \(-0.938624\pi\)
0.981468 0.191625i \(-0.0613757\pi\)
\(72\) 0 0
\(73\) 4894.00 0.918371 0.459186 0.888340i \(-0.348141\pi\)
0.459186 + 0.888340i \(0.348141\pi\)
\(74\) 0 0
\(75\) −2445.00 + 6915.50i −0.434667 + 1.22942i
\(76\) 0 0
\(77\) −5084.94 −0.857639
\(78\) 0 0
\(79\) 1479.95 0.237133 0.118566 0.992946i \(-0.462170\pi\)
0.118566 + 0.992946i \(0.462170\pi\)
\(80\) 0 0
\(81\) 1377.00 + 6414.87i 0.209877 + 0.977728i
\(82\) 0 0
\(83\) −5914.00 −0.858470 −0.429235 0.903193i \(-0.641217\pi\)
−0.429235 + 0.903193i \(0.641217\pi\)
\(84\) 0 0
\(85\) 9015.83i 1.24787i
\(86\) 0 0
\(87\) 2390.68 6761.87i 0.315852 0.893364i
\(88\) 0 0
\(89\) 9062.28i 1.14408i 0.820225 + 0.572041i \(0.193848\pi\)
−0.820225 + 0.572041i \(0.806152\pi\)
\(90\) 0 0
\(91\) 12218.8i 1.47552i
\(92\) 0 0
\(93\) 3301.42 9337.82i 0.381711 1.07964i
\(94\) 0 0
\(95\) 18675.6i 2.06932i
\(96\) 0 0
\(97\) 5858.00 0.622595 0.311298 0.950312i \(-0.399236\pi\)
0.311298 + 0.950312i \(0.399236\pi\)
\(98\) 0 0
\(99\) −8442.00 6822.17i −0.861341 0.696068i
\(100\) 0 0
\(101\) 14837.4 1.45451 0.727253 0.686370i \(-0.240797\pi\)
0.727253 + 0.686370i \(0.240797\pi\)
\(102\) 0 0
\(103\) 2846.05 0.268267 0.134134 0.990963i \(-0.457175\pi\)
0.134134 + 0.990963i \(0.457175\pi\)
\(104\) 0 0
\(105\) −4320.00 + 12218.8i −0.391837 + 1.10828i
\(106\) 0 0
\(107\) 21274.0 1.85815 0.929077 0.369887i \(-0.120604\pi\)
0.929077 + 0.369887i \(0.120604\pi\)
\(108\) 0 0
\(109\) 7405.86i 0.623336i 0.950191 + 0.311668i \(0.100888\pi\)
−0.950191 + 0.311668i \(0.899112\pi\)
\(110\) 0 0
\(111\) −13661.0 4829.91i −1.10876 0.392006i
\(112\) 0 0
\(113\) 3461.99i 0.271125i 0.990769 + 0.135562i \(0.0432842\pi\)
−0.990769 + 0.135562i \(0.956716\pi\)
\(114\) 0 0
\(115\) 24437.6i 1.84783i
\(116\) 0 0
\(117\) −16393.2 + 20285.6i −1.19755 + 1.48189i
\(118\) 0 0
\(119\) 9015.83i 0.636666i
\(120\) 0 0
\(121\) 3315.00 0.226419
\(122\) 0 0
\(123\) −10944.0 3869.29i −0.723379 0.255753i
\(124\) 0 0
\(125\) −7209.99 −0.461440
\(126\) 0 0
\(127\) 21364.3 1.32459 0.662296 0.749242i \(-0.269582\pi\)
0.662296 + 0.749242i \(0.269582\pi\)
\(128\) 0 0
\(129\) −3600.00 1272.79i −0.216333 0.0764853i
\(130\) 0 0
\(131\) 17338.0 1.01031 0.505157 0.863027i \(-0.331435\pi\)
0.505157 + 0.863027i \(0.331435\pi\)
\(132\) 0 0
\(133\) 18675.6i 1.05578i
\(134\) 0 0
\(135\) −23565.3 + 14489.7i −1.29302 + 0.795046i
\(136\) 0 0
\(137\) 27560.2i 1.46839i 0.678939 + 0.734195i \(0.262440\pi\)
−0.678939 + 0.734195i \(0.737560\pi\)
\(138\) 0 0
\(139\) 19668.9i 1.01801i 0.860765 + 0.509003i \(0.169986\pi\)
−0.860765 + 0.509003i \(0.830014\pi\)
\(140\) 0 0
\(141\) 21857.7 + 7727.85i 1.09942 + 0.388705i
\(142\) 0 0
\(143\) 43147.2i 2.10999i
\(144\) 0 0
\(145\) 30240.0 1.43829
\(146\) 0 0
\(147\) 2883.00 8154.36i 0.133417 0.377359i
\(148\) 0 0
\(149\) 9372.99 0.422188 0.211094 0.977466i \(-0.432297\pi\)
0.211094 + 0.977466i \(0.432297\pi\)
\(150\) 0 0
\(151\) 23489.4 1.03019 0.515096 0.857133i \(-0.327756\pi\)
0.515096 + 0.857133i \(0.327756\pi\)
\(152\) 0 0
\(153\) 12096.0 14968.0i 0.516724 0.639414i
\(154\) 0 0
\(155\) 41760.0 1.73819
\(156\) 0 0
\(157\) 4185.92i 0.169821i −0.996389 0.0849105i \(-0.972940\pi\)
0.996389 0.0849105i \(-0.0270604\pi\)
\(158\) 0 0
\(159\) −4895.21 + 13845.7i −0.193632 + 0.547673i
\(160\) 0 0
\(161\) 24437.6i 0.942773i
\(162\) 0 0
\(163\) 44955.0i 1.69201i −0.533175 0.846005i \(-0.679001\pi\)
0.533175 0.846005i \(-0.320999\pi\)
\(164\) 0 0
\(165\) 15254.8 43147.2i 0.560324 1.58484i
\(166\) 0 0
\(167\) 40571.2i 1.45474i 0.686246 + 0.727370i \(0.259257\pi\)
−0.686246 + 0.727370i \(0.740743\pi\)
\(168\) 0 0
\(169\) −75119.0 −2.63012
\(170\) 0 0
\(171\) −25056.0 + 31005.2i −0.856879 + 1.06033i
\(172\) 0 0
\(173\) −23261.7 −0.777230 −0.388615 0.921400i \(-0.627046\pi\)
−0.388615 + 0.921400i \(0.627046\pi\)
\(174\) 0 0
\(175\) −30927.1 −1.00986
\(176\) 0 0
\(177\) 4146.00 11726.7i 0.132337 0.374307i
\(178\) 0 0
\(179\) 6586.00 0.205549 0.102775 0.994705i \(-0.467228\pi\)
0.102775 + 0.994705i \(0.467228\pi\)
\(180\) 0 0
\(181\) 30589.4i 0.933714i −0.884333 0.466857i \(-0.845386\pi\)
0.884333 0.466857i \(-0.154614\pi\)
\(182\) 0 0
\(183\) 2732.21 + 965.981i 0.0815852 + 0.0288447i
\(184\) 0 0
\(185\) 61094.0i 1.78507i
\(186\) 0 0
\(187\) 31836.8i 0.910429i
\(188\) 0 0
\(189\) −23565.3 + 14489.7i −0.659704 + 0.405636i
\(190\) 0 0
\(191\) 11591.8i 0.317748i 0.987299 + 0.158874i \(0.0507864\pi\)
−0.987299 + 0.158874i \(0.949214\pi\)
\(192\) 0 0
\(193\) −24386.0 −0.654675 −0.327338 0.944907i \(-0.606151\pi\)
−0.327338 + 0.944907i \(0.606151\pi\)
\(194\) 0 0
\(195\) −103680. 36656.4i −2.72663 0.964008i
\(196\) 0 0
\(197\) −64244.8 −1.65541 −0.827705 0.561163i \(-0.810354\pi\)
−0.827705 + 0.561163i \(0.810354\pi\)
\(198\) 0 0
\(199\) 34114.7 0.861459 0.430730 0.902481i \(-0.358256\pi\)
0.430730 + 0.902481i \(0.358256\pi\)
\(200\) 0 0
\(201\) 1584.00 + 560.029i 0.0392070 + 0.0138618i
\(202\) 0 0
\(203\) 30240.0 0.733820
\(204\) 0 0
\(205\) 48943.1i 1.16462i
\(206\) 0 0
\(207\) 32786.5 40571.2i 0.765164 0.946842i
\(208\) 0 0
\(209\) 65947.6i 1.50975i
\(210\) 0 0
\(211\) 17666.4i 0.396810i 0.980120 + 0.198405i \(0.0635760\pi\)
−0.980120 + 0.198405i \(0.936424\pi\)
\(212\) 0 0
\(213\) 16393.2 + 5795.89i 0.361331 + 0.127750i
\(214\) 0 0
\(215\) 16099.7i 0.348290i
\(216\) 0 0
\(217\) 41760.0 0.886831
\(218\) 0 0
\(219\) −14682.0 + 41527.0i −0.306124 + 0.865849i
\(220\) 0 0
\(221\) 76501.8 1.56634
\(222\) 0 0
\(223\) −43146.1 −0.867625 −0.433812 0.901003i \(-0.642832\pi\)
−0.433812 + 0.901003i \(0.642832\pi\)
\(224\) 0 0
\(225\) −51345.0 41493.0i −1.01422 0.819615i
\(226\) 0 0
\(227\) 32678.0 0.634167 0.317084 0.948398i \(-0.397296\pi\)
0.317084 + 0.948398i \(0.397296\pi\)
\(228\) 0 0
\(229\) 55704.9i 1.06224i 0.847297 + 0.531120i \(0.178229\pi\)
−0.847297 + 0.531120i \(0.821771\pi\)
\(230\) 0 0
\(231\) 15254.8 43147.2i 0.285880 0.808590i
\(232\) 0 0
\(233\) 47687.3i 0.878397i −0.898390 0.439198i \(-0.855263\pi\)
0.898390 0.439198i \(-0.144737\pi\)
\(234\) 0 0
\(235\) 97750.4i 1.77004i
\(236\) 0 0
\(237\) −4439.84 + 12557.8i −0.0790443 + 0.223571i
\(238\) 0 0
\(239\) 74702.6i 1.30780i 0.756583 + 0.653898i \(0.226867\pi\)
−0.756583 + 0.653898i \(0.773133\pi\)
\(240\) 0 0
\(241\) 766.000 0.0131885 0.00659424 0.999978i \(-0.497901\pi\)
0.00659424 + 0.999978i \(0.497901\pi\)
\(242\) 0 0
\(243\) −58563.0 7560.39i −0.991770 0.128036i
\(244\) 0 0
\(245\) 36467.4 0.607537
\(246\) 0 0
\(247\) −158468. −2.59745
\(248\) 0 0
\(249\) 17742.0 50182.0i 0.286157 0.809373i
\(250\) 0 0
\(251\) 2054.00 0.0326027 0.0163013 0.999867i \(-0.494811\pi\)
0.0163013 + 0.999867i \(0.494811\pi\)
\(252\) 0 0
\(253\) 86294.3i 1.34816i
\(254\) 0 0
\(255\) 76501.8 + 27047.5i 1.17650 + 0.415955i
\(256\) 0 0
\(257\) 18667.6i 0.282633i 0.989964 + 0.141316i \(0.0451335\pi\)
−0.989964 + 0.141316i \(0.954866\pi\)
\(258\) 0 0
\(259\) 61094.0i 0.910750i
\(260\) 0 0
\(261\) 50204.3 + 40571.2i 0.736987 + 0.595576i
\(262\) 0 0
\(263\) 92090.2i 1.33138i −0.746228 0.665690i \(-0.768137\pi\)
0.746228 0.665690i \(-0.231863\pi\)
\(264\) 0 0
\(265\) −61920.0 −0.881737
\(266\) 0 0
\(267\) −76896.0 27186.8i −1.07865 0.381361i
\(268\) 0 0
\(269\) −7627.41 −0.105408 −0.0527039 0.998610i \(-0.516784\pi\)
−0.0527039 + 0.998610i \(0.516784\pi\)
\(270\) 0 0
\(271\) 53999.1 0.735271 0.367636 0.929970i \(-0.380167\pi\)
0.367636 + 0.929970i \(0.380167\pi\)
\(272\) 0 0
\(273\) −103680. 36656.4i −1.39114 0.491841i
\(274\) 0 0
\(275\) 109210. 1.44410
\(276\) 0 0
\(277\) 23505.5i 0.306345i 0.988199 + 0.153173i \(0.0489490\pi\)
−0.988199 + 0.153173i \(0.951051\pi\)
\(278\) 0 0
\(279\) 69329.8 + 56026.9i 0.890659 + 0.719761i
\(280\) 0 0
\(281\) 118624.i 1.50231i −0.660123 0.751157i \(-0.729496\pi\)
0.660123 0.751157i \(-0.270504\pi\)
\(282\) 0 0
\(283\) 134763.i 1.68267i −0.540515 0.841334i \(-0.681771\pi\)
0.540515 0.841334i \(-0.318229\pi\)
\(284\) 0 0
\(285\) −158468. 56026.9i −1.95098 0.689774i
\(286\) 0 0
\(287\) 48943.1i 0.594193i
\(288\) 0 0
\(289\) 27073.0 0.324146
\(290\) 0 0
\(291\) −17574.0 + 49706.8i −0.207532 + 0.586989i
\(292\) 0 0
\(293\) −34494.1 −0.401800 −0.200900 0.979612i \(-0.564387\pi\)
−0.200900 + 0.979612i \(0.564387\pi\)
\(294\) 0 0
\(295\) 52443.2 0.602622
\(296\) 0 0
\(297\) 83214.0 51166.2i 0.943373 0.580057i
\(298\) 0 0
\(299\) 207360. 2.31944
\(300\) 0 0
\(301\) 16099.7i 0.177699i
\(302\) 0 0
\(303\) −44512.2 + 125900.i −0.484835 + 1.37132i
\(304\) 0 0
\(305\) 12218.8i 0.131350i
\(306\) 0 0
\(307\) 23402.4i 0.248304i −0.992263 0.124152i \(-0.960379\pi\)
0.992263 0.124152i \(-0.0396210\pi\)
\(308\) 0 0
\(309\) −8538.15 + 24149.5i −0.0894225 + 0.252925i
\(310\) 0 0
\(311\) 19963.6i 0.206404i 0.994660 + 0.103202i \(0.0329088\pi\)
−0.994660 + 0.103202i \(0.967091\pi\)
\(312\) 0 0
\(313\) −27458.0 −0.280272 −0.140136 0.990132i \(-0.544754\pi\)
−0.140136 + 0.990132i \(0.544754\pi\)
\(314\) 0 0
\(315\) −90720.0 73312.8i −0.914286 0.738854i
\(316\) 0 0
\(317\) 168752. 1.67931 0.839653 0.543123i \(-0.182758\pi\)
0.839653 + 0.543123i \(0.182758\pi\)
\(318\) 0 0
\(319\) −106784. −1.04936
\(320\) 0 0
\(321\) −63822.0 + 180516.i −0.619385 + 1.75188i
\(322\) 0 0
\(323\) 116928. 1.12076
\(324\) 0 0
\(325\) 262425.i 2.48450i
\(326\) 0 0
\(327\) −62840.8 22217.6i −0.587687 0.207779i
\(328\) 0 0
\(329\) 97750.4i 0.903081i
\(330\) 0 0
\(331\) 79065.9i 0.721661i 0.932631 + 0.360830i \(0.117507\pi\)
−0.932631 + 0.360830i \(0.882493\pi\)
\(332\) 0 0
\(333\) 81966.2 101428.i 0.739174 0.914681i
\(334\) 0 0
\(335\) 7083.86i 0.0631220i
\(336\) 0 0
\(337\) 63454.0 0.558726 0.279363 0.960186i \(-0.409877\pi\)
0.279363 + 0.960186i \(0.409877\pi\)
\(338\) 0 0
\(339\) −29376.0 10386.0i −0.255619 0.0903750i
\(340\) 0 0
\(341\) −147463. −1.26816
\(342\) 0 0
\(343\) 127579. 1.08440
\(344\) 0 0
\(345\) 207360. + 73312.8i 1.74216 + 0.615945i
\(346\) 0 0
\(347\) 28166.0 0.233919 0.116960 0.993137i \(-0.462685\pi\)
0.116960 + 0.993137i \(0.462685\pi\)
\(348\) 0 0
\(349\) 67296.7i 0.552514i −0.961084 0.276257i \(-0.910906\pi\)
0.961084 0.276257i \(-0.0890940\pi\)
\(350\) 0 0
\(351\) −122949. 199958.i −0.997957 1.62302i
\(352\) 0 0
\(353\) 27831.7i 0.223352i 0.993745 + 0.111676i \(0.0356220\pi\)
−0.993745 + 0.111676i \(0.964378\pi\)
\(354\) 0 0
\(355\) 73312.8i 0.581732i
\(356\) 0 0
\(357\) 76501.8 + 27047.5i 0.600254 + 0.212222i
\(358\) 0 0
\(359\) 147473.i 1.14426i 0.820164 + 0.572129i \(0.193882\pi\)
−0.820164 + 0.572129i \(0.806118\pi\)
\(360\) 0 0
\(361\) −111887. −0.858549
\(362\) 0 0
\(363\) −9945.00 + 28128.7i −0.0754730 + 0.213470i
\(364\) 0 0
\(365\) −185714. −1.39399
\(366\) 0 0
\(367\) 216565. 1.60789 0.803946 0.594702i \(-0.202730\pi\)
0.803946 + 0.594702i \(0.202730\pi\)
\(368\) 0 0
\(369\) 65664.0 81255.1i 0.482253 0.596757i
\(370\) 0 0
\(371\) −61920.0 −0.449866
\(372\) 0 0
\(373\) 163895.i 1.17801i 0.808131 + 0.589003i \(0.200479\pi\)
−0.808131 + 0.589003i \(0.799521\pi\)
\(374\) 0 0
\(375\) 21630.0 61178.8i 0.153813 0.435049i
\(376\) 0 0
\(377\) 256595.i 1.80537i
\(378\) 0 0
\(379\) 259836.i 1.80893i −0.426550 0.904464i \(-0.640271\pi\)
0.426550 0.904464i \(-0.359729\pi\)
\(380\) 0 0
\(381\) −64093.0 + 181283.i −0.441531 + 1.24884i
\(382\) 0 0
\(383\) 222820.i 1.51899i −0.650511 0.759497i \(-0.725445\pi\)
0.650511 0.759497i \(-0.274555\pi\)
\(384\) 0 0
\(385\) 192960. 1.30180
\(386\) 0 0
\(387\) 21600.0 26728.6i 0.144222 0.178466i
\(388\) 0 0
\(389\) 59008.1 0.389953 0.194977 0.980808i \(-0.437537\pi\)
0.194977 + 0.980808i \(0.437537\pi\)
\(390\) 0 0
\(391\) −153004. −1.00080
\(392\) 0 0
\(393\) −52014.0 + 147118.i −0.336771 + 0.952533i
\(394\) 0 0
\(395\) −56160.0 −0.359942
\(396\) 0 0
\(397\) 236665.i 1.50160i 0.660530 + 0.750799i \(0.270331\pi\)
−0.660530 + 0.750799i \(0.729669\pi\)
\(398\) 0 0
\(399\) −158468. 56026.9i −0.995396 0.351926i
\(400\) 0 0
\(401\) 193906.i 1.20587i 0.797789 + 0.602937i \(0.206003\pi\)
−0.797789 + 0.602937i \(0.793997\pi\)
\(402\) 0 0
\(403\) 354345.i 2.18181i
\(404\) 0 0
\(405\) −52253.5 243427.i −0.318570 1.48409i
\(406\) 0 0
\(407\) 215736.i 1.30237i
\(408\) 0 0
\(409\) −125086. −0.747760 −0.373880 0.927477i \(-0.621973\pi\)
−0.373880 + 0.927477i \(0.621973\pi\)
\(410\) 0 0
\(411\) −233856. 82680.6i −1.38441 0.489463i
\(412\) 0 0
\(413\) 52443.2 0.307460
\(414\) 0 0
\(415\) 224421. 1.30307
\(416\) 0 0
\(417\) −166896. 59006.6i −0.959785 0.339335i
\(418\) 0 0
\(419\) −72346.0 −0.412085 −0.206042 0.978543i \(-0.566058\pi\)
−0.206042 + 0.978543i \(0.566058\pi\)
\(420\) 0 0
\(421\) 254053.i 1.43338i 0.697394 + 0.716688i \(0.254343\pi\)
−0.697394 + 0.716688i \(0.745657\pi\)
\(422\) 0 0
\(423\) −131146. + 162285.i −0.732950 + 0.906979i
\(424\) 0 0
\(425\) 193634.i 1.07202i
\(426\) 0 0
\(427\) 12218.8i 0.0670151i
\(428\) 0 0
\(429\) 366116. + 129442.i 1.98932 + 0.703330i
\(430\) 0 0
\(431\) 280779.i 1.51150i 0.654858 + 0.755752i \(0.272729\pi\)
−0.654858 + 0.755752i \(0.727271\pi\)
\(432\) 0 0
\(433\) 303266. 1.61751 0.808757 0.588143i \(-0.200141\pi\)
0.808757 + 0.588143i \(0.200141\pi\)
\(434\) 0 0
\(435\) −90720.0 + 256595.i −0.479429 + 1.35603i
\(436\) 0 0
\(437\) 316936. 1.65962
\(438\) 0 0
\(439\) 235387. 1.22139 0.610694 0.791866i \(-0.290890\pi\)
0.610694 + 0.791866i \(0.290890\pi\)
\(440\) 0 0
\(441\) 60543.0 + 48926.1i 0.311305 + 0.251573i
\(442\) 0 0
\(443\) −151354. −0.771235 −0.385617 0.922659i \(-0.626011\pi\)
−0.385617 + 0.922659i \(0.626011\pi\)
\(444\) 0 0
\(445\) 343889.i 1.73660i
\(446\) 0 0
\(447\) −28119.0 + 79532.5i −0.140729 + 0.398042i
\(448\) 0 0
\(449\) 86312.3i 0.428134i 0.976819 + 0.214067i \(0.0686711\pi\)
−0.976819 + 0.214067i \(0.931329\pi\)
\(450\) 0 0
\(451\) 172828.i 0.849692i
\(452\) 0 0
\(453\) −70468.2 + 199314.i −0.343397 + 0.971274i
\(454\) 0 0
\(455\) 463671.i 2.23969i
\(456\) 0 0
\(457\) 30626.0 0.146642 0.0733209 0.997308i \(-0.476640\pi\)
0.0733209 + 0.997308i \(0.476640\pi\)
\(458\) 0 0
\(459\) 90720.0 + 147542.i 0.430604 + 0.700310i
\(460\) 0 0
\(461\) −326233. −1.53506 −0.767532 0.641011i \(-0.778515\pi\)
−0.767532 + 0.641011i \(0.778515\pi\)
\(462\) 0 0
\(463\) −305286. −1.42412 −0.712058 0.702121i \(-0.752237\pi\)
−0.712058 + 0.702121i \(0.752237\pi\)
\(464\) 0 0
\(465\) −125280. + 354345.i −0.579396 + 1.63878i
\(466\) 0 0
\(467\) 299558. 1.37356 0.686779 0.726866i \(-0.259024\pi\)
0.686779 + 0.726866i \(0.259024\pi\)
\(468\) 0 0
\(469\) 7083.86i 0.0322051i
\(470\) 0 0
\(471\) 35518.7 + 12557.8i 0.160109 + 0.0566070i
\(472\) 0 0
\(473\) 56851.4i 0.254108i
\(474\) 0 0
\(475\) 401099.i 1.77773i
\(476\) 0 0
\(477\) −102799. 83074.4i −0.451808 0.365116i
\(478\) 0 0
\(479\) 43791.2i 0.190860i 0.995436 + 0.0954301i \(0.0304226\pi\)
−0.995436 + 0.0954301i \(0.969577\pi\)
\(480\) 0 0
\(481\) 518400. 2.24065
\(482\) 0 0
\(483\) 207360. + 73312.8i 0.888855 + 0.314258i
\(484\) 0 0
\(485\) −222295. −0.945033
\(486\) 0 0
\(487\) 152662. 0.643685 0.321842 0.946793i \(-0.395698\pi\)
0.321842 + 0.946793i \(0.395698\pi\)
\(488\) 0 0
\(489\) 381456. + 134865.i 1.59524 + 0.564003i
\(490\) 0 0
\(491\) −367142. −1.52290 −0.761449 0.648224i \(-0.775512\pi\)
−0.761449 + 0.648224i \(0.775512\pi\)
\(492\) 0 0
\(493\) 189332.i 0.778988i
\(494\) 0 0
\(495\) 320351. + 258883.i 1.30742 + 1.05656i
\(496\) 0 0
\(497\) 73312.8i 0.296802i
\(498\) 0 0
\(499\) 60160.6i 0.241608i 0.992676 + 0.120804i \(0.0385473\pi\)
−0.992676 + 0.120804i \(0.961453\pi\)
\(500\) 0 0
\(501\) −344258. 121714.i −1.37154 0.484913i
\(502\) 0 0
\(503\) 263391.i 1.04103i −0.853851 0.520517i \(-0.825739\pi\)
0.853851 0.520517i \(-0.174261\pi\)
\(504\) 0 0
\(505\) −563040. −2.20778
\(506\) 0 0
\(507\) 225357. 637406.i 0.876708 2.47971i
\(508\) 0 0
\(509\) −15975.8 −0.0616634 −0.0308317 0.999525i \(-0.509816\pi\)
−0.0308317 + 0.999525i \(0.509816\pi\)
\(510\) 0 0
\(511\) −185714. −0.711219
\(512\) 0 0
\(513\) −187920. 305623.i −0.714066 1.16132i
\(514\) 0 0
\(515\) −108000. −0.407201
\(516\) 0 0
\(517\) 345177.i 1.29140i
\(518\) 0 0
\(519\) 69785.1 197382.i 0.259077 0.732779i
\(520\) 0 0
\(521\) 210707.i 0.776252i −0.921606 0.388126i \(-0.873123\pi\)
0.921606 0.388126i \(-0.126877\pi\)
\(522\) 0 0
\(523\) 92812.0i 0.339313i 0.985503 + 0.169657i \(0.0542659\pi\)
−0.985503 + 0.169657i \(0.945734\pi\)
\(524\) 0 0
\(525\) 92781.2 262425.i 0.336621 0.952109i
\(526\) 0 0
\(527\) 261459.i 0.941418i
\(528\) 0 0
\(529\) −134879. −0.481984
\(530\) 0 0
\(531\) 87066.0 + 70360.0i 0.308787 + 0.249538i
\(532\) 0 0
\(533\) 415296. 1.46185
\(534\) 0 0
\(535\) −807292. −2.82048
\(536\) 0 0
\(537\) −19758.0 + 55884.1i −0.0685164 + 0.193794i
\(538\) 0 0
\(539\) −128774. −0.443252
\(540\) 0 0
\(541\) 249545.i 0.852618i 0.904578 + 0.426309i \(0.140186\pi\)
−0.904578 + 0.426309i \(0.859814\pi\)
\(542\) 0 0
\(543\) 259560. + 91768.2i 0.880314 + 0.311238i
\(544\) 0 0
\(545\) 281033.i 0.946158i
\(546\) 0 0
\(547\) 258309.i 0.863306i −0.902040 0.431653i \(-0.857930\pi\)
0.902040 0.431653i \(-0.142070\pi\)
\(548\) 0 0
\(549\) −16393.2 + 20285.6i −0.0543902 + 0.0673044i
\(550\) 0 0
\(551\) 392188.i 1.29179i
\(552\) 0 0
\(553\) −56160.0 −0.183644
\(554\) 0 0
\(555\) 518400. + 183282.i 1.68298 + 0.595023i
\(556\) 0 0
\(557\) 151448. 0.488149 0.244075 0.969756i \(-0.421516\pi\)
0.244075 + 0.969756i \(0.421516\pi\)
\(558\) 0 0
\(559\) 136610. 0.437180
\(560\) 0 0
\(561\) −270144. 95510.3i −0.858360 0.303476i
\(562\) 0 0
\(563\) 238118. 0.751234 0.375617 0.926775i \(-0.377431\pi\)
0.375617 + 0.926775i \(0.377431\pi\)
\(564\) 0 0
\(565\) 131373.i 0.411539i
\(566\) 0 0
\(567\) −52253.5 243427.i −0.162536 0.757187i
\(568\) 0 0
\(569\) 163868.i 0.506138i 0.967448 + 0.253069i \(0.0814400\pi\)
−0.967448 + 0.253069i \(0.918560\pi\)
\(570\) 0 0
\(571\) 207126.i 0.635275i −0.948212 0.317638i \(-0.897110\pi\)
0.948212 0.317638i \(-0.102890\pi\)
\(572\) 0 0
\(573\) −98359.5 34775.3i −0.299576 0.105916i
\(574\) 0 0
\(575\) 524850.i 1.58745i
\(576\) 0 0
\(577\) −306146. −0.919553 −0.459777 0.888035i \(-0.652071\pi\)
−0.459777 + 0.888035i \(0.652071\pi\)
\(578\) 0 0
\(579\) 73158.0 206922.i 0.218225 0.617234i
\(580\) 0 0
\(581\) 224421. 0.664830
\(582\) 0 0
\(583\) 218653. 0.643306
\(584\) 0 0
\(585\) 622080. 769785.i 1.81775 2.24935i
\(586\) 0 0
\(587\) 460954. 1.33777 0.668885 0.743366i \(-0.266772\pi\)
0.668885 + 0.743366i \(0.266772\pi\)
\(588\) 0 0
\(589\) 541594.i 1.56114i
\(590\) 0 0
\(591\) 192734. 545135.i 0.551804 1.56074i
\(592\) 0 0
\(593\) 254966.i 0.725057i −0.931973 0.362529i \(-0.881913\pi\)
0.931973 0.362529i \(-0.118087\pi\)
\(594\) 0 0
\(595\) 342127.i 0.966391i
\(596\) 0 0
\(597\) −102344. + 289472.i −0.287153 + 0.812192i
\(598\) 0 0
\(599\) 245359.i 0.683831i 0.939731 + 0.341916i \(0.111076\pi\)
−0.939731 + 0.341916i \(0.888924\pi\)
\(600\) 0 0
\(601\) 74206.0 0.205442 0.102721 0.994710i \(-0.467245\pi\)
0.102721 + 0.994710i \(0.467245\pi\)
\(602\) 0 0
\(603\) −9504.00 + 11760.6i −0.0261380 + 0.0323441i
\(604\) 0 0
\(605\) −125795. −0.343680
\(606\) 0 0
\(607\) −299518. −0.812917 −0.406458 0.913669i \(-0.633236\pi\)
−0.406458 + 0.913669i \(0.633236\pi\)
\(608\) 0 0
\(609\) −90720.0 + 256595.i −0.244607 + 0.691852i
\(610\) 0 0
\(611\) −829440. −2.22179
\(612\) 0 0
\(613\) 206398.i 0.549268i 0.961549 + 0.274634i \(0.0885567\pi\)
−0.961549 + 0.274634i \(0.911443\pi\)
\(614\) 0 0
\(615\) 415296. + 146829.i 1.09801 + 0.388206i
\(616\) 0 0
\(617\) 716667.i 1.88255i −0.337638 0.941276i \(-0.609628\pi\)
0.337638 0.941276i \(-0.390372\pi\)
\(618\) 0 0
\(619\) 153329.i 0.400169i 0.979779 + 0.200084i \(0.0641216\pi\)
−0.979779 + 0.200084i \(0.935878\pi\)
\(620\) 0 0
\(621\) 245899. + 399916.i 0.637636 + 1.03702i
\(622\) 0 0
\(623\) 343889.i 0.886018i
\(624\) 0 0
\(625\) −235775. −0.603584
\(626\) 0 0
\(627\) 559584. + 197843.i 1.42341 + 0.503252i
\(628\) 0 0
\(629\) −382509. −0.966809
\(630\) 0 0
\(631\) 448955. 1.12757 0.563786 0.825921i \(-0.309344\pi\)
0.563786 + 0.825921i \(0.309344\pi\)
\(632\) 0 0
\(633\) −149904. 52999.1i −0.374116 0.132270i
\(634\) 0 0
\(635\) −810720. −2.01059
\(636\) 0 0
\(637\) 309436.i 0.762592i
\(638\) 0 0
\(639\) −98359.5 + 121714.i −0.240888 + 0.298083i
\(640\) 0 0
\(641\) 73482.5i 0.178841i 0.995994 + 0.0894207i \(0.0285016\pi\)
−0.995994 + 0.0894207i \(0.971498\pi\)
\(642\) 0 0
\(643\) 260753.i 0.630677i −0.948979 0.315338i \(-0.897882\pi\)
0.948979 0.315338i \(-0.102118\pi\)
\(644\) 0 0
\(645\) 136610. + 48299.1i 0.328371 + 0.116097i
\(646\) 0 0
\(647\) 226040.i 0.539978i 0.962863 + 0.269989i \(0.0870201\pi\)
−0.962863 + 0.269989i \(0.912980\pi\)
\(648\) 0 0
\(649\) −185188. −0.439667
\(650\) 0 0
\(651\) −125280. + 354345.i −0.295610 + 0.836113i
\(652\) 0 0
\(653\) −144390. −0.338618 −0.169309 0.985563i \(-0.554154\pi\)
−0.169309 + 0.985563i \(0.554154\pi\)
\(654\) 0 0
\(655\) −657931. −1.53355
\(656\) 0 0
\(657\) −308322. 249162.i −0.714289 0.577232i
\(658\) 0 0
\(659\) −383174. −0.882318 −0.441159 0.897429i \(-0.645433\pi\)
−0.441159 + 0.897429i \(0.645433\pi\)
\(660\) 0 0
\(661\) 57636.9i 0.131916i −0.997822 0.0659580i \(-0.978990\pi\)
0.997822 0.0659580i \(-0.0210103\pi\)
\(662\) 0 0
\(663\) −229505. + 649139.i −0.522115 + 1.47676i
\(664\) 0 0
\(665\) 708691.i 1.60256i
\(666\) 0 0
\(667\) 513190.i 1.15352i
\(668\) 0 0
\(669\) 129438. 366107.i 0.289208 0.818005i
\(670\) 0 0
\(671\) 43147.2i 0.0958313i
\(672\) 0 0
\(673\) 679778. 1.50085 0.750424 0.660956i \(-0.229849\pi\)
0.750424 + 0.660956i \(0.229849\pi\)
\(674\) 0 0
\(675\) 506115. 311198.i 1.11081 0.683013i
\(676\) 0 0
\(677\) −6716.68 −0.0146547 −0.00732735 0.999973i \(-0.502332\pi\)
−0.00732735 + 0.999973i \(0.502332\pi\)
\(678\) 0 0
\(679\) −222295. −0.482160
\(680\) 0 0
\(681\) −98034.0 + 277282.i −0.211389 + 0.597899i
\(682\) 0 0
\(683\) −552634. −1.18467 −0.592333 0.805693i \(-0.701793\pi\)
−0.592333 + 0.805693i \(0.701793\pi\)
\(684\) 0 0
\(685\) 1.04584e6i 2.22886i
\(686\) 0 0
\(687\) −472672. 167115.i −1.00149 0.354080i
\(688\) 0 0
\(689\) 525409.i 1.10677i
\(690\) 0 0
\(691\) 327719.i 0.686349i 0.939272 + 0.343174i \(0.111502\pi\)
−0.939272 + 0.343174i \(0.888498\pi\)
\(692\) 0 0
\(693\) 320351. + 258883.i 0.667053 + 0.539060i
\(694\) 0 0
\(695\) 746382.i 1.54522i
\(696\) 0 0
\(697\) −306432. −0.630766
\(698\) 0 0
\(699\) 404640. + 143062.i 0.828160 + 0.292799i
\(700\) 0 0
\(701\) 838750. 1.70685 0.853427 0.521212i \(-0.174520\pi\)
0.853427 + 0.521212i \(0.174520\pi\)
\(702\) 0 0
\(703\) 792340. 1.60325
\(704\) 0 0
\(705\) −829440. 293251.i −1.66881 0.590013i
\(706\) 0 0
\(707\) −563040. −1.12642
\(708\) 0 0
\(709\) 3541.93i 0.00704608i −0.999994 0.00352304i \(-0.998879\pi\)
0.999994 0.00352304i \(-0.00112142\pi\)
\(710\) 0 0
\(711\) −93236.6 75346.5i −0.184437 0.149047i
\(712\) 0 0
\(713\) 708691.i 1.39405i
\(714\) 0 0
\(715\) 1.63732e6i 3.20274i
\(716\) 0 0
\(717\) −633872. 224108.i −1.23300 0.435932i
\(718\) 0 0
\(719\) 1.02909e6i 1.99066i −0.0965451 0.995329i \(-0.530779\pi\)
0.0965451 0.995329i \(-0.469221\pi\)
\(720\) 0 0
\(721\) −108000. −0.207756
\(722\) 0 0
\(723\) −2298.00 + 6499.73i −0.00439616 + 0.0124342i
\(724\) 0 0
\(725\) −649469. −1.23561
\(726\) 0 0
\(727\) 821901. 1.55507 0.777537 0.628838i \(-0.216469\pi\)
0.777537 + 0.628838i \(0.216469\pi\)
\(728\) 0 0
\(729\) 239841. 474242.i 0.451303 0.892371i
\(730\) 0 0
\(731\) −100800. −0.188637
\(732\) 0 0
\(733\) 118816.i 0.221139i −0.993868 0.110570i \(-0.964732\pi\)
0.993868 0.110570i \(-0.0352675\pi\)
\(734\) 0 0
\(735\) −109402. + 309436.i −0.202512 + 0.572791i
\(736\) 0 0
\(737\) 25014.6i 0.0460531i
\(738\) 0 0
\(739\) 349169.i 0.639363i −0.947525 0.319681i \(-0.896424\pi\)
0.947525 0.319681i \(-0.103576\pi\)
\(740\) 0 0
\(741\) 475404. 1.34465e6i 0.865818 2.44890i
\(742\) 0 0
\(743\) 532578.i 0.964729i 0.875971 + 0.482365i \(0.160222\pi\)
−0.875971 + 0.482365i \(0.839778\pi\)
\(744\) 0 0
\(745\) −355680. −0.640836
\(746\) 0 0
\(747\) 372582. + 301092.i 0.667699 + 0.539582i
\(748\) 0 0
\(749\) −807292. −1.43902
\(750\) 0 0
\(751\) 753444. 1.33589 0.667946 0.744210i \(-0.267174\pi\)
0.667946 + 0.744210i \(0.267174\pi\)
\(752\) 0 0
\(753\) −6162.00 + 17428.8i −0.0108676 + 0.0307381i
\(754\) 0 0
\(755\) −891360. −1.56372
\(756\) 0 0
\(757\) 87904.3i 0.153398i 0.997054 + 0.0766988i \(0.0244380\pi\)
−0.997054 + 0.0766988i \(0.975562\pi\)
\(758\) 0 0
\(759\) −732232. 258883.i −1.27106 0.449386i
\(760\) 0 0
\(761\) 1.06243e6i 1.83455i 0.398257 + 0.917274i \(0.369615\pi\)
−0.398257 + 0.917274i \(0.630385\pi\)
\(762\) 0 0
\(763\) 281033.i 0.482734i
\(764\) 0 0
\(765\) −459011. + 567997.i −0.784332 + 0.970562i
\(766\) 0 0
\(767\) 444995.i 0.756423i
\(768\) 0 0
\(769\) 259006. 0.437983 0.218991 0.975727i \(-0.429723\pi\)
0.218991 + 0.975727i \(0.429723\pi\)
\(770\) 0 0
\(771\) −158400. 56002.9i −0.266469 0.0942110i
\(772\) 0 0
\(773\) −597784. −1.00043 −0.500214 0.865902i \(-0.666745\pi\)
−0.500214 + 0.865902i \(0.666745\pi\)
\(774\) 0 0
\(775\) −896885. −1.49325
\(776\) 0 0
\(777\) 518400. + 183282.i 0.858663 + 0.303583i
\(778\) 0 0
\(779\) 634752. 1.04599
\(780\) 0 0
\(781\) 258883.i 0.424426i
\(782\) 0 0
\(783\) −494871. + 304284.i −0.807177 + 0.496313i
\(784\) 0 0
\(785\) 158844.i 0.257770i
\(786\) 0 0
\(787\) 916156.i 1.47918i −0.673060 0.739588i \(-0.735020\pi\)
0.673060 0.739588i \(-0.264980\pi\)
\(788\) 0 0
\(789\) 781411. + 276271.i 1.25524 + 0.443793i
\(790\) 0 0
\(791\) 131373.i 0.209969i
\(792\) 0 0
\(793\) −103680. −0.164873
\(794\) 0 0
\(795\) 185760. 525409.i 0.293912 0.831310i
\(796\) 0 0
\(797\) −33735.2 −0.0531088 −0.0265544 0.999647i \(-0.508454\pi\)
−0.0265544 + 0.999647i \(0.508454\pi\)
\(798\) 0 0
\(799\) 612015. 0.958668
\(800\) 0 0
\(801\) 461376. 570924.i 0.719101 0.889842i
\(802\) 0 0
\(803\) 655796. 1.01704
\(804\) 0 0
\(805\) 927342.i 1.43103i
\(806\) 0 0
\(807\) 22882.2 64720.8i 0.0351359 0.0993794i
\(808\) 0 0
\(809\) 914035.i 1.39658i 0.715815 + 0.698290i \(0.246055\pi\)
−0.715815 + 0.698290i \(0.753945\pi\)
\(810\) 0 0
\(811\) 1.20567e6i 1.83311i 0.399911 + 0.916554i \(0.369041\pi\)
−0.399911 + 0.916554i \(0.630959\pi\)
\(812\) 0 0
\(813\) −161997. + 458197.i −0.245090 + 0.693220i
\(814\) 0 0
\(815\) 1.70592e6i 2.56829i
\(816\) 0 0
\(817\) 208800. 0.312814
\(818\) 0 0
\(819\) 622080. 769785.i 0.927424 1.14763i
\(820\) 0 0
\(821\) 447741. 0.664263 0.332132 0.943233i \(-0.392232\pi\)
0.332132 + 0.943233i \(0.392232\pi\)
\(822\) 0 0
\(823\) 327220. 0.483103 0.241552 0.970388i \(-0.422344\pi\)
0.241552 + 0.970388i \(0.422344\pi\)
\(824\) 0 0
\(825\) −327630. + 926678.i −0.481366 + 1.36151i
\(826\) 0 0
\(827\) −44966.0 −0.0657466 −0.0328733 0.999460i \(-0.510466\pi\)
−0.0328733 + 0.999460i \(0.510466\pi\)
\(828\) 0 0
\(829\) 6761.87i 0.00983915i −0.999988 0.00491958i \(-0.998434\pi\)
0.999988 0.00491958i \(-0.00156596\pi\)
\(830\) 0 0
\(831\) −199451. 70516.6i −0.288825 0.102115i
\(832\) 0 0
\(833\) 228322.i 0.329047i
\(834\) 0 0
\(835\) 1.53957e6i 2.20814i
\(836\) 0 0
\(837\) −683394. + 420202.i −0.975484 + 0.599801i
\(838\) 0 0
\(839\) 455299.i 0.646804i −0.946262 0.323402i \(-0.895173\pi\)
0.946262 0.323402i \(-0.104827\pi\)
\(840\) 0 0
\(841\) −72241.0 −0.102139
\(842\) 0 0
\(843\) 1.00656e6 + 355873.i 1.41640 + 0.500771i
\(844\) 0 0
\(845\) 2.85057e6 3.99225
\(846\) 0 0
\(847\) −125795. −0.175347
\(848\) 0 0
\(849\) 1.14350e6 + 404290.i 1.58644 + 0.560890i
\(850\) 0 0
\(851\) −1.03680e6 −1.43165
\(852\) 0 0
\(853\) 629498.i 0.865160i −0.901596 0.432580i \(-0.857603\pi\)
0.901596 0.432580i \(-0.142397\pi\)
\(854\) 0 0
\(855\) 950808. 1.17657e6i 1.30065 1.60947i
\(856\) 0 0
\(857\) 118522.i 0.161376i −0.996739 0.0806880i \(-0.974288\pi\)
0.996739 0.0806880i \(-0.0257117\pi\)
\(858\) 0 0
\(859\) 667944.i 0.905220i −0.891709 0.452610i \(-0.850493\pi\)
0.891709 0.452610i \(-0.149507\pi\)
\(860\) 0 0
\(861\) 415296. + 146829.i 0.560210 + 0.198064i
\(862\) 0 0
\(863\) 233124.i 0.313015i −0.987677 0.156507i \(-0.949977\pi\)
0.987677 0.156507i \(-0.0500235\pi\)
\(864\) 0 0
\(865\) 882720. 1.17975
\(866\) 0 0
\(867\) −81219.0 + 229722.i −0.108049 + 0.305608i
\(868\) 0 0
\(869\) 198313. 0.262610
\(870\) 0 0
\(871\) −60108.6 −0.0792319
\(872\) 0 0
\(873\) −369054. 298241.i −0.484241 0.391326i
\(874\) 0 0
\(875\) 273600. 0.357355
\(876\) 0 0
\(877\) 930884.i 1.21031i 0.796108 + 0.605155i \(0.206889\pi\)
−0.796108 + 0.605155i \(0.793111\pi\)
\(878\) 0 0
\(879\) 103482. 292692.i 0.133933 0.378821i
\(880\) 0 0
\(881\) 862444.i 1.11117i 0.831461 + 0.555583i \(0.187505\pi\)
−0.831461 + 0.555583i \(0.812495\pi\)
\(882\) 0 0
\(883\) 402219.i 0.515871i −0.966162 0.257936i \(-0.916958\pi\)
0.966162 0.257936i \(-0.0830423\pi\)
\(884\) 0 0
\(885\) −157330. + 444995.i −0.200874 + 0.568158i
\(886\) 0 0
\(887\) 465603.i 0.591791i 0.955220 + 0.295896i \(0.0956181\pi\)
−0.955220 + 0.295896i \(0.904382\pi\)
\(888\) 0 0
\(889\) −810720. −1.02581
\(890\) 0 0
\(891\) 184518. + 859593.i 0.232425 + 1.08277i
\(892\) 0 0
\(893\) −1.26774e6 −1.58975
\(894\) 0 0
\(895\) −249921. −0.312002
\(896\) 0 0
\(897\) −622080. + 1.75951e6i −0.773146 + 2.18679i
\(898\) 0 0
\(899\) 876960. 1.08508
\(900\) 0 0
\(901\) 387681.i 0.477556i
\(902\) 0 0
\(903\) 136610. + 48299.1i 0.167536 + 0.0592329i
\(904\) 0 0
\(905\) 1.16079e6i 1.41728i
\(906\) 0 0
\(907\) 1.04785e6i 1.27375i 0.770968 + 0.636874i \(0.219773\pi\)
−0.770968 + 0.636874i \(0.780227\pi\)
\(908\) 0 0
\(909\) −934757. 755397.i −1.13128 0.914214i
\(910\) 0 0
\(911\) 56670.9i 0.0682847i −0.999417 0.0341424i \(-0.989130\pi\)
0.999417 0.0341424i \(-0.0108700\pi\)
\(912\) 0 0
\(913\) −792476. −0.950702
\(914\) 0 0
\(915\) −103680. 36656.4i −0.123838 0.0437832i
\(916\) 0 0
\(917\) −657931. −0.782423
\(918\) 0 0
\(919\) −887702. −1.05108 −0.525540 0.850769i \(-0.676137\pi\)
−0.525540 + 0.850769i \(0.676137\pi\)
\(920\) 0 0
\(921\) 198576. + 70207.2i 0.234103 + 0.0827680i
\(922\) 0 0
\(923\) −622080. −0.730202
\(924\) 0 0
\(925\) 1.31212e6i 1.53353i
\(926\) 0 0
\(927\) −179301. 144897.i −0.208652 0.168617i
\(928\) 0 0
\(929\) 255068.i 0.295545i −0.989021 0.147773i \(-0.952790\pi\)
0.989021 0.147773i \(-0.0472104\pi\)
\(930\) 0 0
\(931\) 472953.i 0.545655i
\(932\) 0 0
\(933\) −169397. 59890.8i −0.194600 0.0688014i
\(934\) 0 0
\(935\) 1.20812e6i 1.38193i
\(936\) 0 0
\(937\) 952706. 1.08512 0.542562 0.840015i \(-0.317454\pi\)
0.542562 + 0.840015i \(0.317454\pi\)
\(938\) 0 0
\(939\) 82374.0 232989.i 0.0934241 0.264243i
\(940\) 0 0
\(941\) −691438. −0.780862 −0.390431 0.920632i \(-0.627674\pi\)
−0.390431 + 0.920632i \(0.627674\pi\)
\(942\) 0 0
\(943\) −830591. −0.934037
\(944\) 0 0
\(945\) 894240. 549846.i 1.00136 0.615712i
\(946\) 0 0
\(947\) 1.24172e6 1.38460 0.692300 0.721610i \(-0.256597\pi\)
0.692300 + 0.721610i \(0.256597\pi\)
\(948\) 0 0
\(949\) 1.57584e6i 1.74976i
\(950\) 0 0
\(951\) −506255. + 1.43191e6i −0.559769 + 1.58326i
\(952\) 0 0
\(953\) 101552.i 0.111816i 0.998436 + 0.0559078i \(0.0178053\pi\)
−0.998436 + 0.0559078i \(0.982195\pi\)
\(954\) 0 0
\(955\) 439877.i 0.482308i
\(956\) 0 0
\(957\) 320351. 906091.i 0.349786 0.989345i
\(958\) 0 0
\(959\) 1.04584e6i 1.13717i
\(960\) 0 0
\(961\) 287519. 0.311329
\(962\) 0 0
\(963\) −1.34026e6 1.08310e6i −1.44523 1.16792i
\(964\) 0 0
\(965\) 925384. 0.993727
\(966\) 0 0
\(967\) 675121. 0.721986 0.360993 0.932569i \(-0.382438\pi\)
0.360993 + 0.932569i \(0.382438\pi\)
\(968\) 0 0
\(969\) −350784. + 992167.i −0.373587 + 1.05666i
\(970\) 0 0
\(971\) 125894. 0.133526 0.0667631 0.997769i \(-0.478733\pi\)
0.0667631 + 0.997769i \(0.478733\pi\)
\(972\) 0 0
\(973\) 746382.i 0.788379i
\(974\) 0 0
\(975\) 2.22675e6 + 787275.i 2.34241 + 0.828165i
\(976\) 0 0
\(977\) 1.77319e6i 1.85766i 0.370512 + 0.928828i \(0.379182\pi\)
−0.370512 + 0.928828i \(0.620818\pi\)
\(978\) 0 0
\(979\) 1.21435e6i 1.26700i
\(980\) 0 0
\(981\) 377045. 466569.i 0.391791 0.484817i
\(982\) 0 0
\(983\) 839116.i 0.868390i 0.900819 + 0.434195i \(0.142967\pi\)
−0.900819 + 0.434195i \(0.857033\pi\)
\(984\) 0 0
\(985\) 2.43792e6 2.51274
\(986\) 0 0
\(987\) −829440. 293251.i −0.851433 0.301027i
\(988\) 0 0
\(989\) −273221. −0.279332
\(990\) 0 0
\(991\) −1.48909e6 −1.51626 −0.758131 0.652103i \(-0.773887\pi\)
−0.758131 + 0.652103i \(0.773887\pi\)
\(992\) 0 0
\(993\) −670896. 237198.i −0.680388 0.240554i
\(994\) 0 0
\(995\) −1.29456e6 −1.30760
\(996\) 0 0
\(997\) 1.59290e6i 1.60250i −0.598327 0.801252i \(-0.704168\pi\)
0.598327 0.801252i \(-0.295832\pi\)
\(998\) 0 0
\(999\) 614747. + 999791.i 0.615978 + 1.00179i
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 384.5.h.e.65.3 yes 4
3.2 odd 2 384.5.h.f.65.4 yes 4
4.3 odd 2 384.5.h.f.65.1 yes 4
8.3 odd 2 inner 384.5.h.e.65.4 yes 4
8.5 even 2 384.5.h.f.65.2 yes 4
12.11 even 2 inner 384.5.h.e.65.2 yes 4
24.5 odd 2 inner 384.5.h.e.65.1 4
24.11 even 2 384.5.h.f.65.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.h.e.65.1 4 24.5 odd 2 inner
384.5.h.e.65.2 yes 4 12.11 even 2 inner
384.5.h.e.65.3 yes 4 1.1 even 1 trivial
384.5.h.e.65.4 yes 4 8.3 odd 2 inner
384.5.h.f.65.1 yes 4 4.3 odd 2
384.5.h.f.65.2 yes 4 8.5 even 2
384.5.h.f.65.3 yes 4 24.11 even 2
384.5.h.f.65.4 yes 4 3.2 odd 2