Properties

Label 1764.4.k.k.361.1
Level $1764$
Weight $4$
Character 1764.361
Analytic conductor $104.079$
Analytic rank $0$
Dimension $2$
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] = [1764,4,Mod(361,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), 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: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1764.361
Dual form 1764.4.k.k.1549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.00000 + 5.19615i) q^{5} +O(q^{10})\) \(q+(3.00000 + 5.19615i) q^{5} +(-6.00000 + 10.3923i) q^{11} -82.0000 q^{13} +(-15.0000 + 25.9808i) q^{17} +(-34.0000 - 58.8897i) q^{19} +(108.000 + 187.061i) q^{23} +(44.5000 - 77.0763i) q^{25} -246.000 q^{29} +(56.0000 - 96.9948i) q^{31} +(-55.0000 - 95.2628i) q^{37} +246.000 q^{41} -172.000 q^{43} +(96.0000 + 166.277i) q^{47} +(279.000 - 483.242i) q^{53} -72.0000 q^{55} +(270.000 - 467.654i) q^{59} +(-55.0000 - 95.2628i) q^{61} +(-246.000 - 426.084i) q^{65} +(-70.0000 + 121.244i) q^{67} +840.000 q^{71} +(275.000 - 476.314i) q^{73} +(104.000 + 180.133i) q^{79} -516.000 q^{83} -180.000 q^{85} +(-699.000 - 1210.70i) q^{89} +(204.000 - 353.338i) q^{95} +1586.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} - 12 q^{11} - 164 q^{13} - 30 q^{17} - 68 q^{19} + 216 q^{23} + 89 q^{25} - 492 q^{29} + 112 q^{31} - 110 q^{37} + 492 q^{41} - 344 q^{43} + 192 q^{47} + 558 q^{53} - 144 q^{55} + 540 q^{59} - 110 q^{61} - 492 q^{65} - 140 q^{67} + 1680 q^{71} + 550 q^{73} + 208 q^{79} - 1032 q^{83} - 360 q^{85} - 1398 q^{89} + 408 q^{95} + 3172 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) 3.00000 + 5.19615i 0.268328 + 0.464758i 0.968430 0.249285i \(-0.0801955\pi\)
−0.700102 + 0.714043i \(0.746862\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 + 10.3923i −0.164461 + 0.284854i −0.936464 0.350765i \(-0.885922\pi\)
0.772003 + 0.635619i \(0.219255\pi\)
\(12\) 0 0
\(13\) −82.0000 −1.74944 −0.874720 0.484629i \(-0.838954\pi\)
−0.874720 + 0.484629i \(0.838954\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −15.0000 + 25.9808i −0.214002 + 0.370662i −0.952963 0.303085i \(-0.901983\pi\)
0.738961 + 0.673748i \(0.235317\pi\)
\(18\) 0 0
\(19\) −34.0000 58.8897i −0.410533 0.711065i 0.584415 0.811455i \(-0.301324\pi\)
−0.994948 + 0.100390i \(0.967991\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 108.000 + 187.061i 0.979111 + 1.69587i 0.665641 + 0.746272i \(0.268158\pi\)
0.313470 + 0.949598i \(0.398508\pi\)
\(24\) 0 0
\(25\) 44.5000 77.0763i 0.356000 0.616610i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −246.000 −1.57521 −0.787604 0.616181i \(-0.788679\pi\)
−0.787604 + 0.616181i \(0.788679\pi\)
\(30\) 0 0
\(31\) 56.0000 96.9948i 0.324448 0.561961i −0.656952 0.753932i \(-0.728155\pi\)
0.981401 + 0.191971i \(0.0614880\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −55.0000 95.2628i −0.244377 0.423273i 0.717579 0.696477i \(-0.245250\pi\)
−0.961956 + 0.273204i \(0.911917\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 246.000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −172.000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 96.0000 + 166.277i 0.297937 + 0.516042i 0.975664 0.219272i \(-0.0703681\pi\)
−0.677727 + 0.735314i \(0.737035\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 279.000 483.242i 0.723087 1.25242i −0.236670 0.971590i \(-0.576056\pi\)
0.959757 0.280833i \(-0.0906107\pi\)
\(54\) 0 0
\(55\) −72.0000 −0.176518
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 270.000 467.654i 0.595780 1.03192i −0.397657 0.917534i \(-0.630176\pi\)
0.993436 0.114386i \(-0.0364902\pi\)
\(60\) 0 0
\(61\) −55.0000 95.2628i −0.115443 0.199953i 0.802514 0.596634i \(-0.203495\pi\)
−0.917957 + 0.396680i \(0.870162\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −246.000 426.084i −0.469424 0.813066i
\(66\) 0 0
\(67\) −70.0000 + 121.244i −0.127640 + 0.221078i −0.922762 0.385371i \(-0.874073\pi\)
0.795122 + 0.606450i \(0.207407\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 840.000 1.40408 0.702040 0.712138i \(-0.252273\pi\)
0.702040 + 0.712138i \(0.252273\pi\)
\(72\) 0 0
\(73\) 275.000 476.314i 0.440908 0.763676i −0.556849 0.830614i \(-0.687990\pi\)
0.997757 + 0.0669381i \(0.0213230\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 104.000 + 180.133i 0.148113 + 0.256539i 0.930530 0.366216i \(-0.119347\pi\)
−0.782417 + 0.622755i \(0.786013\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −516.000 −0.682390 −0.341195 0.939993i \(-0.610832\pi\)
−0.341195 + 0.939993i \(0.610832\pi\)
\(84\) 0 0
\(85\) −180.000 −0.229691
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −699.000 1210.70i −0.832515 1.44196i −0.896038 0.443978i \(-0.853567\pi\)
0.0635224 0.997980i \(-0.479767\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 204.000 353.338i 0.220315 0.381597i
\(96\) 0 0
\(97\) 1586.00 1.66014 0.830072 0.557657i \(-0.188299\pi\)
0.830072 + 0.557657i \(0.188299\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −621.000 + 1075.60i −0.611800 + 1.05967i 0.379137 + 0.925341i \(0.376221\pi\)
−0.990937 + 0.134328i \(0.957112\pi\)
\(102\) 0 0
\(103\) −340.000 588.897i −0.325254 0.563357i 0.656309 0.754492i \(-0.272117\pi\)
−0.981564 + 0.191135i \(0.938783\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 498.000 + 862.561i 0.449939 + 0.779317i 0.998382 0.0568710i \(-0.0181124\pi\)
−0.548442 + 0.836188i \(0.684779\pi\)
\(108\) 0 0
\(109\) −691.000 + 1196.85i −0.607209 + 1.05172i 0.384489 + 0.923130i \(0.374378\pi\)
−0.991698 + 0.128588i \(0.958956\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 750.000 0.624372 0.312186 0.950021i \(-0.398939\pi\)
0.312186 + 0.950021i \(0.398939\pi\)
\(114\) 0 0
\(115\) −648.000 + 1122.37i −0.525446 + 0.910099i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 593.500 + 1027.97i 0.445905 + 0.772331i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1284.00 0.918756
\(126\) 0 0
\(127\) 176.000 0.122972 0.0614861 0.998108i \(-0.480416\pi\)
0.0614861 + 0.998108i \(0.480416\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −774.000 1340.61i −0.516219 0.894118i −0.999823 0.0188305i \(-0.994006\pi\)
0.483604 0.875287i \(-0.339328\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 189.000 327.358i 0.117864 0.204146i −0.801057 0.598588i \(-0.795729\pi\)
0.918921 + 0.394442i \(0.129062\pi\)
\(138\) 0 0
\(139\) −2500.00 −1.52552 −0.762760 0.646682i \(-0.776156\pi\)
−0.762760 + 0.646682i \(0.776156\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 492.000 852.169i 0.287714 0.498335i
\(144\) 0 0
\(145\) −738.000 1278.25i −0.422673 0.732091i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 423.000 + 732.657i 0.232574 + 0.402830i 0.958565 0.284874i \(-0.0919519\pi\)
−0.725991 + 0.687704i \(0.758619\pi\)
\(150\) 0 0
\(151\) 1268.00 2196.24i 0.683367 1.18363i −0.290580 0.956851i \(-0.593848\pi\)
0.973947 0.226775i \(-0.0728183\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 672.000 0.348234
\(156\) 0 0
\(157\) 593.000 1027.11i 0.301443 0.522115i −0.675020 0.737799i \(-0.735865\pi\)
0.976463 + 0.215685i \(0.0691984\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1054.00 1825.58i −0.506476 0.877243i −0.999972 0.00749450i \(-0.997614\pi\)
0.493496 0.869748i \(-0.335719\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1944.00 0.900786 0.450393 0.892830i \(-0.351284\pi\)
0.450393 + 0.892830i \(0.351284\pi\)
\(168\) 0 0
\(169\) 4527.00 2.06054
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −681.000 1179.53i −0.299280 0.518368i 0.676691 0.736267i \(-0.263413\pi\)
−0.975971 + 0.217898i \(0.930080\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 798.000 1382.18i 0.333214 0.577144i −0.649926 0.759997i \(-0.725200\pi\)
0.983140 + 0.182854i \(0.0585335\pi\)
\(180\) 0 0
\(181\) −1690.00 −0.694015 −0.347007 0.937862i \(-0.612802\pi\)
−0.347007 + 0.937862i \(0.612802\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 330.000 571.577i 0.131146 0.227152i
\(186\) 0 0
\(187\) −180.000 311.769i −0.0703899 0.121919i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1776.00 + 3076.12i 0.672811 + 1.16534i 0.977104 + 0.212764i \(0.0682465\pi\)
−0.304293 + 0.952579i \(0.598420\pi\)
\(192\) 0 0
\(193\) 1343.00 2326.14i 0.500887 0.867562i −0.499112 0.866537i \(-0.666340\pi\)
0.999999 0.00102491i \(-0.000326238\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1410.00 0.509941 0.254970 0.966949i \(-0.417934\pi\)
0.254970 + 0.966949i \(0.417934\pi\)
\(198\) 0 0
\(199\) 1484.00 2570.36i 0.528633 0.915619i −0.470810 0.882235i \(-0.656038\pi\)
0.999443 0.0333844i \(-0.0106286\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 738.000 + 1278.25i 0.251435 + 0.435498i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 816.000 0.270067
\(210\) 0 0
\(211\) −1348.00 −0.439811 −0.219906 0.975521i \(-0.570575\pi\)
−0.219906 + 0.975521i \(0.570575\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −516.000 893.738i −0.163679 0.283500i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1230.00 2130.42i 0.374384 0.648451i
\(222\) 0 0
\(223\) 3872.00 1.16273 0.581364 0.813644i \(-0.302519\pi\)
0.581364 + 0.813644i \(0.302519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2682.00 4645.36i 0.784188 1.35825i −0.145296 0.989388i \(-0.546413\pi\)
0.929483 0.368865i \(-0.120253\pi\)
\(228\) 0 0
\(229\) 437.000 + 756.906i 0.126104 + 0.218418i 0.922164 0.386799i \(-0.126419\pi\)
−0.796060 + 0.605218i \(0.793086\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 189.000 + 327.358i 0.0531408 + 0.0920425i 0.891372 0.453272i \(-0.149743\pi\)
−0.838231 + 0.545315i \(0.816410\pi\)
\(234\) 0 0
\(235\) −576.000 + 997.661i −0.159890 + 0.276937i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1920.00 −0.519642 −0.259821 0.965657i \(-0.583664\pi\)
−0.259821 + 0.965657i \(0.583664\pi\)
\(240\) 0 0
\(241\) −2161.00 + 3742.96i −0.577603 + 1.00044i 0.418151 + 0.908378i \(0.362678\pi\)
−0.995754 + 0.0920596i \(0.970655\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2788.00 + 4828.96i 0.718203 + 1.24396i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5292.00 −1.33079 −0.665395 0.746492i \(-0.731737\pi\)
−0.665395 + 0.746492i \(0.731737\pi\)
\(252\) 0 0
\(253\) −2592.00 −0.644101
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2559.00 4432.32i −0.621113 1.07580i −0.989279 0.146039i \(-0.953347\pi\)
0.368166 0.929760i \(-0.379986\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1884.00 3263.18i 0.441720 0.765082i −0.556097 0.831117i \(-0.687702\pi\)
0.997817 + 0.0660355i \(0.0210351\pi\)
\(264\) 0 0
\(265\) 3348.00 0.776098
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1959.00 3393.09i 0.444024 0.769071i −0.553960 0.832543i \(-0.686884\pi\)
0.997984 + 0.0634719i \(0.0202173\pi\)
\(270\) 0 0
\(271\) −2440.00 4226.20i −0.546935 0.947320i −0.998482 0.0550723i \(-0.982461\pi\)
0.451547 0.892247i \(-0.350872\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 534.000 + 924.915i 0.117096 + 0.202816i
\(276\) 0 0
\(277\) 1769.00 3064.00i 0.383714 0.664613i −0.607875 0.794032i \(-0.707978\pi\)
0.991590 + 0.129419i \(0.0413114\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5430.00 1.15276 0.576382 0.817180i \(-0.304464\pi\)
0.576382 + 0.817180i \(0.304464\pi\)
\(282\) 0 0
\(283\) 3218.00 5573.74i 0.675937 1.17076i −0.300257 0.953858i \(-0.597072\pi\)
0.976194 0.216899i \(-0.0695943\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2006.50 + 3475.36i 0.408406 + 0.707380i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1350.00 −0.269174 −0.134587 0.990902i \(-0.542971\pi\)
−0.134587 + 0.990902i \(0.542971\pi\)
\(294\) 0 0
\(295\) 3240.00 0.639458
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8856.00 15339.0i −1.71290 2.96682i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 330.000 571.577i 0.0619533 0.107306i
\(306\) 0 0
\(307\) 3332.00 0.619437 0.309719 0.950828i \(-0.399765\pi\)
0.309719 + 0.950828i \(0.399765\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2364.00 + 4094.57i −0.431029 + 0.746565i −0.996962 0.0778865i \(-0.975183\pi\)
0.565933 + 0.824451i \(0.308516\pi\)
\(312\) 0 0
\(313\) −2557.00 4428.85i −0.461758 0.799788i 0.537291 0.843397i \(-0.319448\pi\)
−0.999049 + 0.0436091i \(0.986114\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3603.00 + 6240.58i 0.638374 + 1.10570i 0.985790 + 0.167985i \(0.0537261\pi\)
−0.347415 + 0.937711i \(0.612941\pi\)
\(318\) 0 0
\(319\) 1476.00 2556.51i 0.259060 0.448705i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2040.00 0.351420
\(324\) 0 0
\(325\) −3649.00 + 6320.25i −0.622800 + 1.07872i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3130.00 5421.32i −0.519759 0.900250i −0.999736 0.0229685i \(-0.992688\pi\)
0.479977 0.877281i \(-0.340645\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −840.000 −0.136997
\(336\) 0 0
\(337\) −5326.00 −0.860907 −0.430454 0.902613i \(-0.641646\pi\)
−0.430454 + 0.902613i \(0.641646\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 672.000 + 1163.94i 0.106718 + 0.184841i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000 31.1769i 0.00278470 0.00482324i −0.864630 0.502410i \(-0.832447\pi\)
0.867414 + 0.497586i \(0.165780\pi\)
\(348\) 0 0
\(349\) 3134.00 0.480685 0.240343 0.970688i \(-0.422740\pi\)
0.240343 + 0.970688i \(0.422740\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 609.000 1054.82i 0.0918238 0.159043i −0.816455 0.577409i \(-0.804064\pi\)
0.908279 + 0.418366i \(0.137397\pi\)
\(354\) 0 0
\(355\) 2520.00 + 4364.77i 0.376754 + 0.652557i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5004.00 8667.18i −0.735657 1.27420i −0.954434 0.298421i \(-0.903540\pi\)
0.218777 0.975775i \(-0.429793\pi\)
\(360\) 0 0
\(361\) 1117.50 1935.57i 0.162925 0.282194i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3300.00 0.473233
\(366\) 0 0
\(367\) 536.000 928.379i 0.0762370 0.132046i −0.825387 0.564568i \(-0.809043\pi\)
0.901624 + 0.432522i \(0.142376\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 137.000 + 237.291i 0.0190177 + 0.0329396i 0.875378 0.483440i \(-0.160613\pi\)
−0.856360 + 0.516379i \(0.827279\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20172.0 2.75573
\(378\) 0 0
\(379\) 7652.00 1.03709 0.518545 0.855051i \(-0.326474\pi\)
0.518545 + 0.855051i \(0.326474\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1080.00 + 1870.61i 0.144087 + 0.249566i 0.929032 0.369999i \(-0.120642\pi\)
−0.784945 + 0.619566i \(0.787309\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −537.000 + 930.111i −0.0699922 + 0.121230i −0.898898 0.438159i \(-0.855631\pi\)
0.828905 + 0.559389i \(0.188964\pi\)
\(390\) 0 0
\(391\) −6480.00 −0.838127
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −624.000 + 1080.80i −0.0794857 + 0.137673i
\(396\) 0 0
\(397\) −3463.00 5998.09i −0.437791 0.758276i 0.559728 0.828676i \(-0.310906\pi\)
−0.997519 + 0.0704004i \(0.977572\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 969.000 + 1678.36i 0.120672 + 0.209010i 0.920033 0.391841i \(-0.128162\pi\)
−0.799361 + 0.600851i \(0.794828\pi\)
\(402\) 0 0
\(403\) −4592.00 + 7953.58i −0.567603 + 0.983116i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1320.00 0.160762
\(408\) 0 0
\(409\) 4787.00 8291.33i 0.578733 1.00240i −0.416892 0.908956i \(-0.636881\pi\)
0.995625 0.0934393i \(-0.0297861\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1548.00 2681.21i −0.183104 0.317146i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5052.00 0.589037 0.294518 0.955646i \(-0.404841\pi\)
0.294518 + 0.955646i \(0.404841\pi\)
\(420\) 0 0
\(421\) 3422.00 0.396147 0.198074 0.980187i \(-0.436531\pi\)
0.198074 + 0.980187i \(0.436531\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1335.00 + 2312.29i 0.152369 + 0.263912i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1104.00 1912.18i 0.123382 0.213705i −0.797717 0.603032i \(-0.793959\pi\)
0.921099 + 0.389327i \(0.127292\pi\)
\(432\) 0 0
\(433\) −6814.00 −0.756259 −0.378129 0.925753i \(-0.623433\pi\)
−0.378129 + 0.925753i \(0.623433\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7344.00 12720.2i 0.803916 1.39242i
\(438\) 0 0
\(439\) −6292.00 10898.1i −0.684056 1.18482i −0.973732 0.227696i \(-0.926881\pi\)
0.289676 0.957125i \(-0.406452\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3498.00 + 6058.71i 0.375158 + 0.649793i 0.990351 0.138584i \(-0.0442551\pi\)
−0.615193 + 0.788377i \(0.710922\pi\)
\(444\) 0 0
\(445\) 4194.00 7264.22i 0.446775 0.773836i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9474.00 −0.995781 −0.497891 0.867240i \(-0.665892\pi\)
−0.497891 + 0.867240i \(0.665892\pi\)
\(450\) 0 0
\(451\) −1476.00 + 2556.51i −0.154107 + 0.266921i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2893.00 5010.82i −0.296124 0.512902i 0.679121 0.734026i \(-0.262361\pi\)
−0.975246 + 0.221123i \(0.929028\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3438.00 −0.347340 −0.173670 0.984804i \(-0.555563\pi\)
−0.173670 + 0.984804i \(0.555563\pi\)
\(462\) 0 0
\(463\) 9392.00 0.942728 0.471364 0.881939i \(-0.343762\pi\)
0.471364 + 0.881939i \(0.343762\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2478.00 4292.02i −0.245542 0.425291i 0.716742 0.697339i \(-0.245633\pi\)
−0.962284 + 0.272047i \(0.912299\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1032.00 1787.48i 0.100320 0.173760i
\(474\) 0 0
\(475\) −6052.00 −0.584600
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10296.0 + 17833.2i −0.982122 + 1.70108i −0.328034 + 0.944666i \(0.606386\pi\)
−0.654088 + 0.756418i \(0.726947\pi\)
\(480\) 0 0
\(481\) 4510.00 + 7811.55i 0.427522 + 0.740491i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4758.00 + 8241.10i 0.445463 + 0.771565i
\(486\) 0 0
\(487\) 6716.00 11632.5i 0.624910 1.08238i −0.363649 0.931536i \(-0.618469\pi\)
0.988558 0.150839i \(-0.0481975\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14172.0 1.30259 0.651297 0.758823i \(-0.274225\pi\)
0.651297 + 0.758823i \(0.274225\pi\)
\(492\) 0 0
\(493\) 3690.00 6391.27i 0.337098 0.583871i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2978.00 + 5158.05i 0.267162 + 0.462737i 0.968128 0.250457i \(-0.0805809\pi\)
−0.700966 + 0.713195i \(0.747248\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16968.0 −1.50411 −0.752053 0.659102i \(-0.770936\pi\)
−0.752053 + 0.659102i \(0.770936\pi\)
\(504\) 0 0
\(505\) −7452.00 −0.656653
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2607.00 + 4515.46i 0.227020 + 0.393210i 0.956924 0.290340i \(-0.0937683\pi\)
−0.729903 + 0.683550i \(0.760435\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2040.00 3533.38i 0.174550 0.302329i
\(516\) 0 0
\(517\) −2304.00 −0.195996
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −699.000 + 1210.70i −0.0587788 + 0.101808i −0.893917 0.448232i \(-0.852054\pi\)
0.835139 + 0.550039i \(0.185387\pi\)
\(522\) 0 0
\(523\) 9290.00 + 16090.8i 0.776718 + 1.34531i 0.933824 + 0.357733i \(0.116450\pi\)
−0.157106 + 0.987582i \(0.550217\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1680.00 + 2909.85i 0.138865 + 0.240522i
\(528\) 0 0
\(529\) −17244.5 + 29868.4i −1.41732 + 2.45487i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −20172.0 −1.63930
\(534\) 0 0
\(535\) −2988.00 + 5175.37i −0.241463 + 0.418226i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9485.00 + 16428.5i 0.753774 + 1.30558i 0.945981 + 0.324221i \(0.105102\pi\)
−0.192207 + 0.981354i \(0.561564\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8292.00 −0.651725
\(546\) 0 0
\(547\) −16036.0 −1.25347 −0.626737 0.779231i \(-0.715610\pi\)
−0.626737 + 0.779231i \(0.715610\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8364.00 + 14486.9i 0.646676 + 1.12008i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4155.00 7196.67i 0.316074 0.547456i −0.663592 0.748095i \(-0.730969\pi\)
0.979665 + 0.200640i \(0.0643020\pi\)
\(558\) 0 0
\(559\) 14104.0 1.06715
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3546.00 6141.85i 0.265446 0.459766i −0.702234 0.711946i \(-0.747814\pi\)
0.967680 + 0.252180i \(0.0811475\pi\)
\(564\) 0 0
\(565\) 2250.00 + 3897.11i 0.167537 + 0.290182i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3579.00 6199.01i −0.263690 0.456724i 0.703530 0.710666i \(-0.251606\pi\)
−0.967220 + 0.253942i \(0.918273\pi\)
\(570\) 0 0
\(571\) −3250.00 + 5629.17i −0.238193 + 0.412563i −0.960196 0.279328i \(-0.909888\pi\)
0.722003 + 0.691890i \(0.243222\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 19224.0 1.39425
\(576\) 0 0
\(577\) −10897.0 + 18874.2i −0.786218 + 1.36177i 0.142050 + 0.989859i \(0.454631\pi\)
−0.928268 + 0.371911i \(0.878703\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3348.00 + 5798.91i 0.237839 + 0.411949i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9756.00 −0.685985 −0.342993 0.939338i \(-0.611441\pi\)
−0.342993 + 0.939338i \(0.611441\pi\)
\(588\) 0 0
\(589\) −7616.00 −0.532787
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2793.00 + 4837.62i 0.193414 + 0.335004i 0.946380 0.323057i \(-0.104710\pi\)
−0.752965 + 0.658060i \(0.771377\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 + 20.7846i −0.000818542 + 0.00141776i −0.866434 0.499291i \(-0.833594\pi\)
0.865616 + 0.500709i \(0.166927\pi\)
\(600\) 0 0
\(601\) 4298.00 0.291712 0.145856 0.989306i \(-0.453406\pi\)
0.145856 + 0.989306i \(0.453406\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3561.00 + 6167.83i −0.239298 + 0.414476i
\(606\) 0 0
\(607\) −4240.00 7343.90i −0.283519 0.491070i 0.688730 0.725018i \(-0.258169\pi\)
−0.972249 + 0.233948i \(0.924835\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7872.00 13634.7i −0.521223 0.902784i
\(612\) 0 0
\(613\) 953.000 1650.64i 0.0627917 0.108758i −0.832921 0.553393i \(-0.813333\pi\)
0.895712 + 0.444634i \(0.146666\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7482.00 −0.488191 −0.244096 0.969751i \(-0.578491\pi\)
−0.244096 + 0.969751i \(0.578491\pi\)
\(618\) 0 0
\(619\) 3674.00 6363.55i 0.238563 0.413203i −0.721739 0.692165i \(-0.756657\pi\)
0.960302 + 0.278962i \(0.0899903\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1710.50 2962.67i −0.109472 0.189611i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3300.00 0.209189
\(630\) 0 0
\(631\) 4520.00 0.285164 0.142582 0.989783i \(-0.454460\pi\)
0.142582 + 0.989783i \(0.454460\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 528.000 + 914.523i 0.0329969 + 0.0571523i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9903.00 + 17152.5i −0.610211 + 1.05692i 0.380994 + 0.924577i \(0.375582\pi\)
−0.991205 + 0.132338i \(0.957751\pi\)
\(642\) 0 0
\(643\) −5020.00 −0.307884 −0.153942 0.988080i \(-0.549197\pi\)
−0.153942 + 0.988080i \(0.549197\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14196.0 24588.2i 0.862600 1.49407i −0.00681018 0.999977i \(-0.502168\pi\)
0.869410 0.494091i \(-0.164499\pi\)
\(648\) 0 0
\(649\) 3240.00 + 5611.84i 0.195965 + 0.339421i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8781.00 15209.1i −0.526228 0.911454i −0.999533 0.0305554i \(-0.990272\pi\)
0.473305 0.880899i \(-0.343061\pi\)
\(654\) 0 0
\(655\) 4644.00 8043.64i 0.277032 0.479834i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4716.00 −0.278770 −0.139385 0.990238i \(-0.544513\pi\)
−0.139385 + 0.990238i \(0.544513\pi\)
\(660\) 0 0
\(661\) 11381.0 19712.5i 0.669697 1.15995i −0.308292 0.951292i \(-0.599757\pi\)
0.977989 0.208657i \(-0.0669093\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −26568.0 46017.1i −1.54230 2.67135i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1320.00 0.0759434
\(672\) 0 0
\(673\) 4802.00 0.275042 0.137521 0.990499i \(-0.456086\pi\)
0.137521 + 0.990499i \(0.456086\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10779.0 + 18669.8i 0.611921 + 1.05988i 0.990916 + 0.134480i \(0.0429364\pi\)
−0.378995 + 0.925399i \(0.623730\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1890.00 3273.58i 0.105884 0.183397i −0.808215 0.588888i \(-0.799566\pi\)
0.914099 + 0.405491i \(0.132899\pi\)
\(684\) 0 0
\(685\) 2268.00 0.126505
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −22878.0 + 39625.9i −1.26500 + 2.19104i
\(690\) 0 0
\(691\) 2750.00 + 4763.14i 0.151396 + 0.262226i 0.931741 0.363123i \(-0.118290\pi\)
−0.780345 + 0.625350i \(0.784956\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7500.00 12990.4i −0.409340 0.708997i
\(696\) 0 0
\(697\) −3690.00 + 6391.27i −0.200529 + 0.347326i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10230.0 −0.551187 −0.275593 0.961274i \(-0.588874\pi\)
−0.275593 + 0.961274i \(0.588874\pi\)
\(702\) 0 0
\(703\) −3740.00 + 6477.87i −0.200650 + 0.347536i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −5095.00 8824.80i −0.269883 0.467450i 0.698949 0.715172i \(-0.253652\pi\)
−0.968831 + 0.247721i \(0.920318\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24192.0 1.27068
\(714\) 0 0
\(715\) 5904.00 0.308807
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4704.00 + 8147.57i 0.243991 + 0.422605i 0.961847 0.273586i \(-0.0882099\pi\)
−0.717856 + 0.696191i \(0.754877\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −10947.0 + 18960.8i −0.560774 + 0.971290i
\(726\) 0 0
\(727\) −33064.0 −1.68676 −0.843381 0.537316i \(-0.819438\pi\)
−0.843381 + 0.537316i \(0.819438\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2580.00 4468.69i 0.130540 0.226102i
\(732\) 0 0
\(733\) 3161.00 + 5475.01i 0.159283 + 0.275886i 0.934610 0.355674i \(-0.115749\pi\)
−0.775328 + 0.631559i \(0.782415\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −840.000 1454.92i −0.0419834 0.0727175i
\(738\) 0 0
\(739\) 10370.0 17961.4i 0.516193 0.894072i −0.483630 0.875272i \(-0.660682\pi\)
0.999823 0.0188001i \(-0.00598461\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32040.0 −1.58201 −0.791005 0.611810i \(-0.790442\pi\)
−0.791005 + 0.611810i \(0.790442\pi\)
\(744\) 0 0
\(745\) −2538.00 + 4395.94i −0.124812 + 0.216181i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 6416.00 + 11112.8i 0.311749 + 0.539964i 0.978741 0.205100i \(-0.0657520\pi\)
−0.666992 + 0.745064i \(0.732419\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15216.0 0.733466
\(756\) 0 0
\(757\) −19906.0 −0.955741 −0.477870 0.878430i \(-0.658591\pi\)
−0.477870 + 0.878430i \(0.658591\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5421.00 + 9389.45i 0.258227 + 0.447263i 0.965767 0.259411i \(-0.0835283\pi\)
−0.707540 + 0.706674i \(0.750195\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22140.0 + 38347.6i −1.04228 + 1.80528i
\(768\) 0 0
\(769\) 28274.0 1.32586 0.662930 0.748681i \(-0.269313\pi\)
0.662930 + 0.748681i \(0.269313\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16173.0 + 28012.5i −0.752526 + 1.30341i 0.194069 + 0.980988i \(0.437831\pi\)
−0.946595 + 0.322425i \(0.895502\pi\)
\(774\) 0 0
\(775\) −4984.00 8632.54i −0.231007 0.400116i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8364.00 14486.9i −0.384687 0.666298i
\(780\) 0 0
\(781\) −5040.00 + 8729.54i −0.230916 + 0.399958i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7116.00 0.323543
\(786\) 0 0
\(787\) −15058.0 + 26081.2i −0.682033 + 1.18132i 0.292327 + 0.956318i \(0.405570\pi\)
−0.974360 + 0.224997i \(0.927763\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4510.00 + 7811.55i 0.201961 + 0.349806i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6594.00 0.293063 0.146532 0.989206i \(-0.453189\pi\)
0.146532 + 0.989206i \(0.453189\pi\)
\(798\) 0 0
\(799\) −5760.00 −0.255036
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3300.00 + 5715.77i 0.145024 + 0.251189i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21507.0 + 37251.2i −0.934667 + 1.61889i −0.159441 + 0.987207i \(0.550969\pi\)
−0.775226 + 0.631684i \(0.782364\pi\)
\(810\) 0 0
\(811\) −14164.0 −0.613274 −0.306637 0.951827i \(-0.599204\pi\)
−0.306637 + 0.951827i \(0.599204\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6324.00 10953.5i 0.271804 0.470778i
\(816\) 0 0
\(817\) 5848.00 + 10129.0i 0.250423 + 0.433745i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17415.0 + 30163.7i 0.740302 + 1.28224i 0.952358 + 0.304983i \(0.0986507\pi\)
−0.212056 + 0.977257i \(0.568016\pi\)
\(822\) 0 0
\(823\) −15508.0 + 26860.6i −0.656835 + 1.13767i 0.324596 + 0.945853i \(0.394772\pi\)
−0.981430 + 0.191818i \(0.938562\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9876.00 −0.415263 −0.207631 0.978207i \(-0.566575\pi\)
−0.207631 + 0.978207i \(0.566575\pi\)
\(828\) 0 0
\(829\) 1577.00 2731.44i 0.0660693 0.114435i −0.831099 0.556125i \(-0.812287\pi\)
0.897168 + 0.441690i \(0.145621\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 5832.00 + 10101.3i 0.241706 + 0.418647i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −36936.0 −1.51987 −0.759936 0.649998i \(-0.774770\pi\)
−0.759936 + 0.649998i \(0.774770\pi\)
\(840\) 0 0
\(841\) 36127.0 1.48128
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 13581.0 + 23523.0i 0.552900 + 0.957651i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11880.0 20576.8i 0.478544 0.828863i
\(852\) 0 0
\(853\) 9638.00 0.386869 0.193434 0.981113i \(-0.438037\pi\)
0.193434 + 0.981113i \(0.438037\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5133.00 8890.62i 0.204597 0.354373i −0.745407 0.666610i \(-0.767745\pi\)
0.950004 + 0.312237i \(0.101078\pi\)
\(858\) 0 0
\(859\) 2042.00 + 3536.85i 0.0811084 + 0.140484i 0.903726 0.428111i \(-0.140821\pi\)
−0.822618 + 0.568595i \(0.807487\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −96.0000 166.277i −0.00378665 0.00655867i 0.864126 0.503276i \(-0.167872\pi\)
−0.867913 + 0.496717i \(0.834539\pi\)
\(864\) 0 0
\(865\) 4086.00 7077.16i 0.160611 0.278186i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2496.00 −0.0974350
\(870\) 0 0
\(871\) 5740.00 9941.97i 0.223298 0.386763i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −9955.00 17242.6i −0.383303 0.663900i 0.608229 0.793761i \(-0.291880\pi\)
−0.991532 + 0.129862i \(0.958547\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −14802.0 −0.566052 −0.283026 0.959112i \(-0.591338\pi\)
−0.283026 + 0.959112i \(0.591338\pi\)
\(882\) 0 0
\(883\) −32548.0 −1.24046 −0.620231 0.784419i \(-0.712961\pi\)
−0.620231 + 0.784419i \(0.712961\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 732.000 + 1267.86i 0.0277093 + 0.0479939i 0.879548 0.475811i \(-0.157845\pi\)
−0.851838 + 0.523805i \(0.824512\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6528.00 11306.8i 0.244626 0.423705i
\(894\) 0 0
\(895\) 9576.00 0.357643
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −13776.0 + 23860.7i −0.511074 + 0.885206i
\(900\) 0 0
\(901\) 8370.00 + 14497.3i 0.309484 + 0.536042i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5070.00 8781.50i −0.186224 0.322549i
\(906\) 0 0
\(907\) 24782.0 42923.7i 0.907247 1.57140i 0.0893742 0.995998i \(-0.471513\pi\)
0.817873 0.575399i \(-0.195153\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8448.00 −0.307239 −0.153619 0.988130i \(-0.549093\pi\)
−0.153619 + 0.988130i \(0.549093\pi\)
\(912\) 0 0
\(913\) 3096.00 5362.43i 0.112226 0.194382i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −7300.00 12644.0i −0.262029 0.453848i 0.704752 0.709454i \(-0.251058\pi\)
−0.966781 + 0.255606i \(0.917725\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −68880.0 −2.45635
\(924\) 0 0
\(925\) −9790.00 −0.347993
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −10551.0 18274.9i −0.372623 0.645403i 0.617345 0.786693i \(-0.288208\pi\)
−0.989968 + 0.141290i \(0.954875\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1080.00 1870.61i 0.0377752 0.0654285i
\(936\) 0 0
\(937\) −20806.0 −0.725403 −0.362701 0.931905i \(-0.618146\pi\)
−0.362701 + 0.931905i \(0.618146\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12255.0 21226.3i 0.424550 0.735342i −0.571828 0.820373i \(-0.693766\pi\)
0.996378 + 0.0850311i \(0.0270990\pi\)
\(942\) 0 0
\(943\) 26568.0 + 46017.1i 0.917469 + 1.58910i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22074.0 38233.3i −0.757454 1.31195i −0.944145 0.329530i \(-0.893110\pi\)
0.186692 0.982419i \(-0.440223\pi\)
\(948\) 0 0
\(949\) −22550.0 + 39057.7i −0.771342 + 1.33600i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −27114.0 −0.921625 −0.460812 0.887498i \(-0.652442\pi\)
−0.460812 + 0.887498i \(0.652442\pi\)
\(954\) 0 0
\(955\) −10656.0 + 18456.7i −0.361068 + 0.625388i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8623.50 + 14936.3i 0.289467 + 0.501371i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16116.0 0.537609
\(966\) 0 0
\(967\) −10264.0 −0.341332 −0.170666 0.985329i \(-0.554592\pi\)
−0.170666 + 0.985329i \(0.554592\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 25734.0 + 44572.6i 0.850508 + 1.47312i 0.880750 + 0.473581i \(0.157039\pi\)
−0.0302424 + 0.999543i \(0.509628\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −11895.0 + 20602.7i −0.389514 + 0.674657i −0.992384 0.123182i \(-0.960690\pi\)
0.602871 + 0.797839i \(0.294024\pi\)
\(978\) 0 0
\(979\) 16776.0 0.547664
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13212.0 22883.9i 0.428685 0.742504i −0.568072 0.822979i \(-0.692310\pi\)
0.996757 + 0.0804749i \(0.0256437\pi\)
\(984\) 0 0
\(985\) 4230.00 + 7326.57i 0.136831 + 0.236999i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −18576.0 32174.6i −0.597252 1.03447i
\(990\) 0 0
\(991\) −19744.0 + 34197.6i −0.632885 + 1.09619i 0.354074 + 0.935217i \(0.384796\pi\)
−0.986959 + 0.160971i \(0.948537\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17808.0 0.567388
\(996\) 0 0
\(997\) −15427.0 + 26720.3i −0.490048 + 0.848788i −0.999934 0.0114536i \(-0.996354\pi\)
0.509886 + 0.860242i \(0.329687\pi\)
\(998\) 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 1764.4.k.k.361.1 2
3.2 odd 2 196.4.e.c.165.1 2
7.2 even 3 inner 1764.4.k.k.1549.1 2
7.3 odd 6 1764.4.a.k.1.1 1
7.4 even 3 252.4.a.c.1.1 1
7.5 odd 6 1764.4.k.e.1549.1 2
7.6 odd 2 1764.4.k.e.361.1 2
21.2 odd 6 196.4.e.c.177.1 2
21.5 even 6 196.4.e.d.177.1 2
21.11 odd 6 28.4.a.b.1.1 1
21.17 even 6 196.4.a.b.1.1 1
21.20 even 2 196.4.e.d.165.1 2
28.11 odd 6 1008.4.a.f.1.1 1
84.11 even 6 112.4.a.c.1.1 1
84.59 odd 6 784.4.a.n.1.1 1
105.32 even 12 700.4.e.f.449.1 2
105.53 even 12 700.4.e.f.449.2 2
105.74 odd 6 700.4.a.e.1.1 1
168.11 even 6 448.4.a.m.1.1 1
168.53 odd 6 448.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.4.a.b.1.1 1 21.11 odd 6
112.4.a.c.1.1 1 84.11 even 6
196.4.a.b.1.1 1 21.17 even 6
196.4.e.c.165.1 2 3.2 odd 2
196.4.e.c.177.1 2 21.2 odd 6
196.4.e.d.165.1 2 21.20 even 2
196.4.e.d.177.1 2 21.5 even 6
252.4.a.c.1.1 1 7.4 even 3
448.4.a.d.1.1 1 168.53 odd 6
448.4.a.m.1.1 1 168.11 even 6
700.4.a.e.1.1 1 105.74 odd 6
700.4.e.f.449.1 2 105.32 even 12
700.4.e.f.449.2 2 105.53 even 12
784.4.a.n.1.1 1 84.59 odd 6
1008.4.a.f.1.1 1 28.11 odd 6
1764.4.a.k.1.1 1 7.3 odd 6
1764.4.k.e.361.1 2 7.6 odd 2
1764.4.k.e.1549.1 2 7.5 odd 6
1764.4.k.k.361.1 2 1.1 even 1 trivial
1764.4.k.k.1549.1 2 7.2 even 3 inner