Properties

Label 1764.4.k.bc.1549.4
Level $1764$
Weight $4$
Character 1764.1549
Analytic conductor $104.079$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.7027860559216896.42
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 88x^{6} + 278x^{5} + 3869x^{4} - 8206x^{3} - 71054x^{2} + 75204x + 945876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1549.4
Root \(-3.39728 - 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1549
Dual form 1764.4.k.bc.361.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.29150 - 9.16515i) q^{5} +O(q^{10})\) \(q+(5.29150 - 9.16515i) q^{5} +(27.6225 + 47.8435i) q^{11} -83.5225 q^{13} +(47.6235 + 82.4864i) q^{17} +(41.7612 - 72.3326i) q^{19} +(82.8674 - 143.530i) q^{23} +(6.50000 + 11.2583i) q^{25} +110.490 q^{29} +(-41.7612 - 72.3326i) q^{31} +(-39.0000 + 67.5500i) q^{37} -412.737 q^{41} +148.000 q^{43} +(-232.826 + 403.267i) q^{47} +(-55.2449 - 95.6870i) q^{53} +584.657 q^{55} +(275.158 + 476.588i) q^{59} +(-292.329 + 506.328i) q^{61} +(-441.959 + 765.496i) q^{65} +(130.000 + 225.167i) q^{67} -718.184 q^{71} +(334.090 + 578.661i) q^{73} +(-332.000 + 575.041i) q^{79} -126.996 q^{83} +1008.00 q^{85} +(439.195 - 760.708i) q^{89} +(-441.959 - 765.496i) q^{95} -1169.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 52 q^{25} - 312 q^{37} + 1184 q^{43} + 1040 q^{67} - 2656 q^{79} + 8064 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 5.29150 9.16515i 0.473286 0.819756i −0.526246 0.850332i \(-0.676401\pi\)
0.999532 + 0.0305763i \(0.00973425\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 27.6225 + 47.8435i 0.757135 + 1.31140i 0.944306 + 0.329069i \(0.106735\pi\)
−0.187171 + 0.982327i \(0.559932\pi\)
\(12\) 0 0
\(13\) −83.5225 −1.78192 −0.890960 0.454082i \(-0.849967\pi\)
−0.890960 + 0.454082i \(0.849967\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 47.6235 + 82.4864i 0.679435 + 1.17682i 0.975151 + 0.221541i \(0.0711085\pi\)
−0.295716 + 0.955276i \(0.595558\pi\)
\(18\) 0 0
\(19\) 41.7612 72.3326i 0.504246 0.873380i −0.495742 0.868470i \(-0.665104\pi\)
0.999988 0.00491032i \(-0.00156301\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 82.8674 143.530i 0.751263 1.30122i −0.195948 0.980614i \(-0.562779\pi\)
0.947211 0.320611i \(-0.103888\pi\)
\(24\) 0 0
\(25\) 6.50000 + 11.2583i 0.0520000 + 0.0900666i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 110.490 0.707498 0.353749 0.935340i \(-0.384907\pi\)
0.353749 + 0.935340i \(0.384907\pi\)
\(30\) 0 0
\(31\) −41.7612 72.3326i −0.241953 0.419075i 0.719318 0.694681i \(-0.244455\pi\)
−0.961270 + 0.275607i \(0.911121\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −39.0000 + 67.5500i −0.173285 + 0.300139i −0.939567 0.342366i \(-0.888772\pi\)
0.766281 + 0.642505i \(0.222105\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −412.737 −1.57216 −0.786082 0.618122i \(-0.787894\pi\)
−0.786082 + 0.618122i \(0.787894\pi\)
\(42\) 0 0
\(43\) 148.000 0.524879 0.262439 0.964948i \(-0.415473\pi\)
0.262439 + 0.964948i \(0.415473\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −232.826 + 403.267i −0.722578 + 1.25154i 0.237385 + 0.971416i \(0.423710\pi\)
−0.959963 + 0.280127i \(0.909624\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −55.2449 95.6870i −0.143179 0.247993i 0.785513 0.618845i \(-0.212399\pi\)
−0.928692 + 0.370852i \(0.879066\pi\)
\(54\) 0 0
\(55\) 584.657 1.43337
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 275.158 + 476.588i 0.607162 + 1.05163i 0.991706 + 0.128528i \(0.0410253\pi\)
−0.384544 + 0.923107i \(0.625641\pi\)
\(60\) 0 0
\(61\) −292.329 + 506.328i −0.613588 + 1.06276i 0.377043 + 0.926196i \(0.376941\pi\)
−0.990631 + 0.136569i \(0.956392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −441.959 + 765.496i −0.843358 + 1.46074i
\(66\) 0 0
\(67\) 130.000 + 225.167i 0.237045 + 0.410574i 0.959865 0.280462i \(-0.0904877\pi\)
−0.722820 + 0.691036i \(0.757154\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −718.184 −1.20046 −0.600231 0.799827i \(-0.704925\pi\)
−0.600231 + 0.799827i \(0.704925\pi\)
\(72\) 0 0
\(73\) 334.090 + 578.661i 0.535647 + 0.927768i 0.999132 + 0.0416633i \(0.0132657\pi\)
−0.463484 + 0.886105i \(0.653401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −332.000 + 575.041i −0.472822 + 0.818951i −0.999516 0.0311034i \(-0.990098\pi\)
0.526694 + 0.850055i \(0.323431\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −126.996 −0.167947 −0.0839737 0.996468i \(-0.526761\pi\)
−0.0839737 + 0.996468i \(0.526761\pi\)
\(84\) 0 0
\(85\) 1008.00 1.28627
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 439.195 760.708i 0.523085 0.906009i −0.476554 0.879145i \(-0.658114\pi\)
0.999639 0.0268644i \(-0.00855223\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −441.959 765.496i −0.477306 0.826718i
\(96\) 0 0
\(97\) −1169.31 −1.22398 −0.611989 0.790866i \(-0.709630\pi\)
−0.611989 + 0.790866i \(0.709630\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 227.535 + 394.102i 0.224164 + 0.388263i 0.956068 0.293144i \(-0.0947016\pi\)
−0.731904 + 0.681407i \(0.761368\pi\)
\(102\) 0 0
\(103\) 542.896 940.323i 0.519351 0.899542i −0.480396 0.877052i \(-0.659507\pi\)
0.999747 0.0224903i \(-0.00715950\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 469.582 813.339i 0.424263 0.734846i −0.572088 0.820192i \(-0.693866\pi\)
0.996351 + 0.0853466i \(0.0271997\pi\)
\(108\) 0 0
\(109\) 325.000 + 562.917i 0.285590 + 0.494657i 0.972752 0.231847i \(-0.0744770\pi\)
−0.687162 + 0.726504i \(0.741144\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1546.86 1.28775 0.643877 0.765129i \(-0.277325\pi\)
0.643877 + 0.765129i \(0.277325\pi\)
\(114\) 0 0
\(115\) −876.986 1518.98i −0.711125 1.23170i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −860.500 + 1490.43i −0.646506 + 1.11978i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1460.45 1.04502
\(126\) 0 0
\(127\) 2288.00 1.59864 0.799320 0.600906i \(-0.205193\pi\)
0.799320 + 0.600906i \(0.205193\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −444.486 + 769.873i −0.296450 + 0.513466i −0.975321 0.220791i \(-0.929136\pi\)
0.678871 + 0.734257i \(0.262469\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1436.37 + 2487.86i 0.895746 + 1.55148i 0.832879 + 0.553455i \(0.186691\pi\)
0.0628668 + 0.998022i \(0.479976\pi\)
\(138\) 0 0
\(139\) −501.135 −0.305796 −0.152898 0.988242i \(-0.548861\pi\)
−0.152898 + 0.988242i \(0.548861\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2307.10 3996.01i −1.34915 2.33680i
\(144\) 0 0
\(145\) 584.657 1012.66i 0.334849 0.579976i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 939.163 1626.68i 0.516371 0.894381i −0.483448 0.875373i \(-0.660616\pi\)
0.999819 0.0190078i \(-0.00605074\pi\)
\(150\) 0 0
\(151\) 988.000 + 1711.27i 0.532466 + 0.922257i 0.999281 + 0.0379029i \(0.0120677\pi\)
−0.466816 + 0.884355i \(0.654599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −883.919 −0.458052
\(156\) 0 0
\(157\) 292.329 + 506.328i 0.148601 + 0.257385i 0.930711 0.365756i \(-0.119190\pi\)
−0.782110 + 0.623141i \(0.785856\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1674.00 + 2899.45i −0.804404 + 1.39327i 0.112289 + 0.993676i \(0.464182\pi\)
−0.916693 + 0.399592i \(0.869152\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −84.6640 −0.0392305 −0.0196153 0.999808i \(-0.506244\pi\)
−0.0196153 + 0.999808i \(0.506244\pi\)
\(168\) 0 0
\(169\) 4779.00 2.17524
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 756.685 1310.62i 0.332542 0.575979i −0.650468 0.759534i \(-0.725427\pi\)
0.983009 + 0.183555i \(0.0587605\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 82.8674 + 143.530i 0.0346022 + 0.0599328i 0.882808 0.469734i \(-0.155650\pi\)
−0.848206 + 0.529667i \(0.822317\pi\)
\(180\) 0 0
\(181\) 3591.47 1.47487 0.737435 0.675418i \(-0.236037\pi\)
0.737435 + 0.675418i \(0.236037\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 412.737 + 714.882i 0.164027 + 0.284104i
\(186\) 0 0
\(187\) −2630.96 + 4556.95i −1.02885 + 1.78202i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1077.28 + 1865.90i −0.408110 + 0.706867i −0.994678 0.103033i \(-0.967145\pi\)
0.586568 + 0.809900i \(0.300479\pi\)
\(192\) 0 0
\(193\) 689.000 + 1193.38i 0.256970 + 0.445086i 0.965429 0.260667i \(-0.0839423\pi\)
−0.708458 + 0.705753i \(0.750609\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −994.408 −0.359638 −0.179819 0.983700i \(-0.557551\pi\)
−0.179819 + 0.983700i \(0.557551\pi\)
\(198\) 0 0
\(199\) 2338.63 + 4050.62i 0.833070 + 1.44292i 0.895592 + 0.444876i \(0.146752\pi\)
−0.0625216 + 0.998044i \(0.519914\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2184.00 + 3782.80i −0.744084 + 1.28879i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4614.19 1.52713
\(210\) 0 0
\(211\) 5108.00 1.66658 0.833292 0.552833i \(-0.186453\pi\)
0.833292 + 0.552833i \(0.186453\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 783.142 1356.44i 0.248418 0.430273i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3977.63 6889.46i −1.21070 2.09699i
\(222\) 0 0
\(223\) −1169.31 −0.351135 −0.175567 0.984467i \(-0.556176\pi\)
−0.175567 + 0.984467i \(0.556176\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 677.312 + 1173.14i 0.198039 + 0.343013i 0.947892 0.318590i \(-0.103209\pi\)
−0.749854 + 0.661604i \(0.769876\pi\)
\(228\) 0 0
\(229\) 542.896 940.323i 0.156662 0.271346i −0.777001 0.629499i \(-0.783260\pi\)
0.933663 + 0.358153i \(0.116593\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1878.33 + 3253.36i −0.528126 + 0.914741i 0.471337 + 0.881953i \(0.343772\pi\)
−0.999462 + 0.0327872i \(0.989562\pi\)
\(234\) 0 0
\(235\) 2464.00 + 4267.77i 0.683973 + 1.18468i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3148.96 0.852256 0.426128 0.904663i \(-0.359877\pi\)
0.426128 + 0.904663i \(0.359877\pi\)
\(240\) 0 0
\(241\) −2923.29 5063.28i −0.781350 1.35334i −0.931155 0.364623i \(-0.881198\pi\)
0.149805 0.988716i \(-0.452135\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3488.00 + 6041.39i −0.898527 + 1.55629i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6011.15 −1.51163 −0.755817 0.654783i \(-0.772760\pi\)
−0.755817 + 0.654783i \(0.772760\pi\)
\(252\) 0 0
\(253\) 9156.00 2.27523
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 502.693 870.689i 0.122012 0.211331i −0.798549 0.601930i \(-0.794399\pi\)
0.920561 + 0.390599i \(0.127732\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1850.70 + 3205.51i 0.433914 + 0.751561i 0.997206 0.0746968i \(-0.0237989\pi\)
−0.563293 + 0.826258i \(0.690466\pi\)
\(264\) 0 0
\(265\) −1169.31 −0.271058
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2100.73 + 3638.57i 0.476147 + 0.824711i 0.999627 0.0273274i \(-0.00869966\pi\)
−0.523479 + 0.852038i \(0.675366\pi\)
\(270\) 0 0
\(271\) 3800.27 6582.26i 0.851845 1.47544i −0.0276959 0.999616i \(-0.508817\pi\)
0.879541 0.475823i \(-0.157850\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −359.092 + 621.965i −0.0787420 + 0.136385i
\(276\) 0 0
\(277\) 529.000 + 916.255i 0.114746 + 0.198745i 0.917678 0.397325i \(-0.130061\pi\)
−0.802932 + 0.596070i \(0.796728\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8618.21 −1.82961 −0.914803 0.403901i \(-0.867654\pi\)
−0.914803 + 0.403901i \(0.867654\pi\)
\(282\) 0 0
\(283\) −542.896 940.323i −0.114035 0.197514i 0.803359 0.595495i \(-0.203044\pi\)
−0.917393 + 0.397981i \(0.869711\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2079.50 + 3601.80i −0.423265 + 0.733116i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6000.56 −1.19644 −0.598220 0.801332i \(-0.704125\pi\)
−0.598220 + 0.801332i \(0.704125\pi\)
\(294\) 0 0
\(295\) 5824.00 1.14945
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6921.29 + 11988.0i −1.33869 + 2.31868i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3093.71 + 5358.47i 0.580805 + 1.00598i
\(306\) 0 0
\(307\) −6932.36 −1.28877 −0.644383 0.764703i \(-0.722886\pi\)
−0.644383 + 0.764703i \(0.722886\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −275.158 476.588i −0.0501697 0.0868965i 0.839850 0.542819i \(-0.182643\pi\)
−0.890020 + 0.455922i \(0.849310\pi\)
\(312\) 0 0
\(313\) 250.567 433.995i 0.0452489 0.0783734i −0.842514 0.538675i \(-0.818925\pi\)
0.887763 + 0.460301i \(0.152259\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4806.31 8324.77i 0.851574 1.47497i −0.0282125 0.999602i \(-0.508981\pi\)
0.879787 0.475368i \(-0.157685\pi\)
\(318\) 0 0
\(319\) 3052.00 + 5286.22i 0.535671 + 0.927810i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7955.27 1.37041
\(324\) 0 0
\(325\) −542.896 940.323i −0.0926598 0.160492i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4162.00 + 7208.80i −0.691131 + 1.19707i 0.280337 + 0.959902i \(0.409554\pi\)
−0.971468 + 0.237172i \(0.923780\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2751.58 0.448761
\(336\) 0 0
\(337\) 9794.00 1.58313 0.791563 0.611088i \(-0.209268\pi\)
0.791563 + 0.611088i \(0.209268\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2307.10 3996.01i 0.366382 0.634592i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1077.28 + 1865.90i 0.166660 + 0.288664i 0.937244 0.348675i \(-0.113368\pi\)
−0.770583 + 0.637339i \(0.780035\pi\)
\(348\) 0 0
\(349\) 1753.97 0.269020 0.134510 0.990912i \(-0.457054\pi\)
0.134510 + 0.990912i \(0.457054\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3095.53 + 5361.61i 0.466738 + 0.808413i 0.999278 0.0379913i \(-0.0120959\pi\)
−0.532540 + 0.846405i \(0.678763\pi\)
\(354\) 0 0
\(355\) −3800.27 + 6582.26i −0.568162 + 0.984085i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1463.99 2535.71i 0.215227 0.372784i −0.738116 0.674674i \(-0.764284\pi\)
0.953343 + 0.301890i \(0.0976176\pi\)
\(360\) 0 0
\(361\) −58.5000 101.325i −0.00852894 0.0147726i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7071.35 1.01406
\(366\) 0 0
\(367\) 4343.17 + 7522.59i 0.617743 + 1.06996i 0.989897 + 0.141790i \(0.0452859\pi\)
−0.372154 + 0.928171i \(0.621381\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2327.00 + 4030.48i −0.323023 + 0.559492i −0.981110 0.193450i \(-0.938032\pi\)
0.658087 + 0.752942i \(0.271366\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9228.38 −1.26070
\(378\) 0 0
\(379\) 2388.00 0.323650 0.161825 0.986819i \(-0.448262\pi\)
0.161825 + 0.986819i \(0.448262\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4127.37 7148.82i 0.550650 0.953753i −0.447578 0.894245i \(-0.647713\pi\)
0.998228 0.0595085i \(-0.0189533\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2154.55 3731.79i −0.280823 0.486399i 0.690765 0.723080i \(-0.257274\pi\)
−0.971588 + 0.236680i \(0.923941\pi\)
\(390\) 0 0
\(391\) 15785.7 2.04174
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3513.56 + 6085.66i 0.447560 + 0.775197i
\(396\) 0 0
\(397\) −5220.15 + 9041.57i −0.659929 + 1.14303i 0.320704 + 0.947179i \(0.396081\pi\)
−0.980633 + 0.195852i \(0.937253\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −331.469 + 574.122i −0.0412788 + 0.0714970i −0.885927 0.463825i \(-0.846477\pi\)
0.844648 + 0.535322i \(0.179810\pi\)
\(402\) 0 0
\(403\) 3488.00 + 6041.39i 0.431141 + 0.746757i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4309.10 −0.524802
\(408\) 0 0
\(409\) 3758.51 + 6509.93i 0.454392 + 0.787030i 0.998653 0.0518858i \(-0.0165232\pi\)
−0.544261 + 0.838916i \(0.683190\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −672.000 + 1163.94i −0.0794872 + 0.137676i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13419.3 1.56461 0.782307 0.622893i \(-0.214043\pi\)
0.782307 + 0.622893i \(0.214043\pi\)
\(420\) 0 0
\(421\) 8238.00 0.953671 0.476836 0.878993i \(-0.341784\pi\)
0.476836 + 0.878993i \(0.341784\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −619.106 + 1072.32i −0.0706613 + 0.122389i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4170.99 7224.37i −0.466147 0.807391i 0.533105 0.846049i \(-0.321025\pi\)
−0.999253 + 0.0386580i \(0.987692\pi\)
\(432\) 0 0
\(433\) −1837.49 −0.203936 −0.101968 0.994788i \(-0.532514\pi\)
−0.101968 + 0.994788i \(0.532514\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6921.29 11988.0i −0.757643 1.31228i
\(438\) 0 0
\(439\) 5261.91 9113.90i 0.572067 0.990849i −0.424286 0.905528i \(-0.639475\pi\)
0.996353 0.0853213i \(-0.0271917\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6877.99 + 11913.0i −0.737660 + 1.27766i 0.215887 + 0.976418i \(0.430736\pi\)
−0.953547 + 0.301246i \(0.902598\pi\)
\(444\) 0 0
\(445\) −4648.00 8050.57i −0.495138 0.857604i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9281.14 −0.975511 −0.487755 0.872980i \(-0.662184\pi\)
−0.487755 + 0.872980i \(0.662184\pi\)
\(450\) 0 0
\(451\) −11400.8 19746.8i −1.19034 2.06173i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1445.00 2502.81i 0.147909 0.256185i −0.782546 0.622593i \(-0.786079\pi\)
0.930454 + 0.366408i \(0.119413\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14191.8 1.43379 0.716896 0.697180i \(-0.245562\pi\)
0.716896 + 0.697180i \(0.245562\pi\)
\(462\) 0 0
\(463\) −8632.00 −0.866443 −0.433221 0.901288i \(-0.642623\pi\)
−0.433221 + 0.901288i \(0.642623\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2201.27 3812.70i 0.218121 0.377796i −0.736113 0.676859i \(-0.763341\pi\)
0.954233 + 0.299063i \(0.0966740\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4088.12 + 7080.84i 0.397404 + 0.688324i
\(474\) 0 0
\(475\) 1085.79 0.104883
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4635.36 8028.67i −0.442160 0.765844i 0.555689 0.831390i \(-0.312454\pi\)
−0.997850 + 0.0655459i \(0.979121\pi\)
\(480\) 0 0
\(481\) 3257.38 5641.94i 0.308781 0.534824i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6187.43 + 10716.9i −0.579292 + 1.00336i
\(486\) 0 0
\(487\) 6268.00 + 10856.5i 0.583224 + 1.01017i 0.995094 + 0.0989314i \(0.0315424\pi\)
−0.411870 + 0.911243i \(0.635124\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −19391.0 −1.78228 −0.891142 0.453724i \(-0.850095\pi\)
−0.891142 + 0.453724i \(0.850095\pi\)
\(492\) 0 0
\(493\) 5261.91 + 9113.90i 0.480699 + 0.832595i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3378.00 + 5850.87i −0.303046 + 0.524891i −0.976824 0.214042i \(-0.931337\pi\)
0.673778 + 0.738934i \(0.264670\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16721.1 −1.48222 −0.741112 0.671381i \(-0.765701\pi\)
−0.741112 + 0.671381i \(0.765701\pi\)
\(504\) 0 0
\(505\) 4816.00 0.424375
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5497.87 + 9522.59i −0.478760 + 0.829237i −0.999703 0.0243545i \(-0.992247\pi\)
0.520943 + 0.853591i \(0.325580\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5745.47 9951.45i −0.491603 0.851482i
\(516\) 0 0
\(517\) −25724.9 −2.18836
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10387.2 17991.2i −0.873459 1.51288i −0.858395 0.512990i \(-0.828538\pi\)
−0.0150646 0.999887i \(-0.504795\pi\)
\(522\) 0 0
\(523\) −835.225 + 1446.65i −0.0698314 + 0.120952i −0.898827 0.438304i \(-0.855579\pi\)
0.828996 + 0.559255i \(0.188913\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3977.63 6889.46i 0.328783 0.569468i
\(528\) 0 0
\(529\) −7650.50 13251.1i −0.628791 1.08910i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 34472.8 2.80147
\(534\) 0 0
\(535\) −4969.59 8607.58i −0.401596 0.695585i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9685.00 16774.9i 0.769669 1.33310i −0.168074 0.985774i \(-0.553755\pi\)
0.937743 0.347331i \(-0.112912\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6878.95 0.540664
\(546\) 0 0
\(547\) −20396.0 −1.59428 −0.797139 0.603796i \(-0.793654\pi\)
−0.797139 + 0.603796i \(0.793654\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4614.19 7992.01i 0.356753 0.617915i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2154.55 3731.79i −0.163898 0.283880i 0.772365 0.635179i \(-0.219074\pi\)
−0.936263 + 0.351299i \(0.885740\pi\)
\(558\) 0 0
\(559\) −12361.3 −0.935292
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.1660 36.6606i −0.00158444 0.00274433i 0.865232 0.501372i \(-0.167171\pi\)
−0.866817 + 0.498627i \(0.833838\pi\)
\(564\) 0 0
\(565\) 8185.20 14177.2i 0.609476 1.05564i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3314.69 5741.22i 0.244217 0.422995i −0.717694 0.696358i \(-0.754803\pi\)
0.961911 + 0.273363i \(0.0881359\pi\)
\(570\) 0 0
\(571\) −654.000 1132.76i −0.0479318 0.0830203i 0.841064 0.540935i \(-0.181930\pi\)
−0.888996 + 0.457915i \(0.848596\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2154.55 0.156263
\(576\) 0 0
\(577\) 4343.17 + 7522.59i 0.313359 + 0.542755i 0.979087 0.203440i \(-0.0652122\pi\)
−0.665728 + 0.746195i \(0.731879\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3052.00 5286.22i 0.216811 0.375528i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25737.9 1.80974 0.904868 0.425691i \(-0.139969\pi\)
0.904868 + 0.425691i \(0.139969\pi\)
\(588\) 0 0
\(589\) −6976.00 −0.488015
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 111.122 192.468i 0.00769514 0.0133284i −0.862152 0.506649i \(-0.830884\pi\)
0.869847 + 0.493321i \(0.164217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11187.1 + 19376.6i 0.763092 + 1.32171i 0.941249 + 0.337713i \(0.109653\pi\)
−0.178157 + 0.984002i \(0.557014\pi\)
\(600\) 0 0
\(601\) 21715.8 1.47389 0.736945 0.675953i \(-0.236268\pi\)
0.736945 + 0.675953i \(0.236268\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9106.68 + 15773.2i 0.611965 + 1.05996i
\(606\) 0 0
\(607\) −6765.32 + 11717.9i −0.452382 + 0.783548i −0.998533 0.0541379i \(-0.982759\pi\)
0.546152 + 0.837686i \(0.316092\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19446.2 33681.8i 1.28758 2.23015i
\(612\) 0 0
\(613\) −6385.00 11059.1i −0.420698 0.728670i 0.575310 0.817935i \(-0.304881\pi\)
−0.996008 + 0.0892655i \(0.971548\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11049.0 −0.720932 −0.360466 0.932772i \(-0.617382\pi\)
−0.360466 + 0.932772i \(0.617382\pi\)
\(618\) 0 0
\(619\) 9103.95 + 15768.5i 0.591145 + 1.02389i 0.994079 + 0.108663i \(0.0346571\pi\)
−0.402934 + 0.915229i \(0.632010\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6915.50 11978.0i 0.442592 0.766592i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −7429.27 −0.470945
\(630\) 0 0
\(631\) −23952.0 −1.51112 −0.755558 0.655082i \(-0.772634\pi\)
−0.755558 + 0.655082i \(0.772634\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12107.0 20969.9i 0.756614 1.31049i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3867.14 + 6698.09i 0.238289 + 0.412728i 0.960223 0.279233i \(-0.0900803\pi\)
−0.721935 + 0.691961i \(0.756747\pi\)
\(642\) 0 0
\(643\) 16286.9 0.998899 0.499449 0.866343i \(-0.333536\pi\)
0.499449 + 0.866343i \(0.333536\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6603.80 + 11438.1i 0.401270 + 0.695021i 0.993880 0.110469i \(-0.0352354\pi\)
−0.592609 + 0.805490i \(0.701902\pi\)
\(648\) 0 0
\(649\) −15201.1 + 26329.1i −0.919407 + 1.59246i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9336.39 + 16171.1i −0.559512 + 0.969103i 0.438025 + 0.898963i \(0.355678\pi\)
−0.997537 + 0.0701402i \(0.977655\pi\)
\(654\) 0 0
\(655\) 4704.00 + 8147.57i 0.280611 + 0.486033i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6353.16 0.375545 0.187772 0.982213i \(-0.439873\pi\)
0.187772 + 0.982213i \(0.439873\pi\)
\(660\) 0 0
\(661\) −9980.93 17287.5i −0.587312 1.01725i −0.994583 0.103947i \(-0.966853\pi\)
0.407271 0.913307i \(-0.366481\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9156.00 15858.7i 0.531517 0.920614i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −32299.3 −1.85827
\(672\) 0 0
\(673\) −12046.0 −0.689954 −0.344977 0.938611i \(-0.612113\pi\)
−0.344977 + 0.938611i \(0.612113\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13826.7 + 23948.5i −0.784938 + 1.35955i 0.144098 + 0.989563i \(0.453972\pi\)
−0.929036 + 0.369989i \(0.879361\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3950.01 + 6841.62i 0.221293 + 0.383290i 0.955201 0.295959i \(-0.0956390\pi\)
−0.733908 + 0.679249i \(0.762306\pi\)
\(684\) 0 0
\(685\) 30402.2 1.69578
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4614.19 + 7992.01i 0.255133 + 0.441903i
\(690\) 0 0
\(691\) 3842.03 6654.60i 0.211516 0.366357i −0.740673 0.671866i \(-0.765493\pi\)
0.952189 + 0.305509i \(0.0988265\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2651.76 + 4592.98i −0.144729 + 0.250678i
\(696\) 0 0
\(697\) −19656.0 34045.2i −1.06818 1.85015i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12043.4 −0.648891 −0.324445 0.945904i \(-0.605178\pi\)
−0.324445 + 0.945904i \(0.605178\pi\)
\(702\) 0 0
\(703\) 3257.38 + 5641.94i 0.174757 + 0.302688i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12769.0 22116.6i 0.676375 1.17152i −0.299690 0.954037i \(-0.596883\pi\)
0.976065 0.217479i \(-0.0697834\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −13842.6 −0.727080
\(714\) 0 0
\(715\) −48832.0 −2.55414
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2476.42 4289.29i 0.128449 0.222481i −0.794627 0.607098i \(-0.792333\pi\)
0.923076 + 0.384618i \(0.125667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 718.184 + 1243.93i 0.0367899 + 0.0637220i
\(726\) 0 0
\(727\) −14115.3 −0.720093 −0.360046 0.932934i \(-0.617239\pi\)
−0.360046 + 0.932934i \(0.617239\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7048.28 + 12208.0i 0.356621 + 0.617686i
\(732\) 0 0
\(733\) 6472.99 11211.5i 0.326174 0.564949i −0.655575 0.755130i \(-0.727574\pi\)
0.981749 + 0.190180i \(0.0609072\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7181.84 + 12439.3i −0.358950 + 0.621720i
\(738\) 0 0
\(739\) 7270.00 + 12592.0i 0.361883 + 0.626799i 0.988271 0.152712i \(-0.0488008\pi\)
−0.626388 + 0.779511i \(0.715467\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1160.14 −0.0572833 −0.0286417 0.999590i \(-0.509118\pi\)
−0.0286417 + 0.999590i \(0.509118\pi\)
\(744\) 0 0
\(745\) −9939.17 17215.2i −0.488783 0.846596i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 3384.00 5861.26i 0.164426 0.284794i −0.772025 0.635592i \(-0.780756\pi\)
0.936451 + 0.350798i \(0.114089\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 20912.0 1.00803
\(756\) 0 0
\(757\) −2354.00 −0.113022 −0.0565110 0.998402i \(-0.517998\pi\)
−0.0565110 + 0.998402i \(0.517998\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −16535.9 + 28641.1i −0.787684 + 1.36431i 0.139698 + 0.990194i \(0.455387\pi\)
−0.927382 + 0.374115i \(0.877947\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22981.9 39805.8i −1.08191 1.87393i
\(768\) 0 0
\(769\) −22885.2 −1.07316 −0.536580 0.843850i \(-0.680284\pi\)
−0.536580 + 0.843850i \(0.680284\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −12175.7 21089.0i −0.566535 0.981267i −0.996905 0.0786143i \(-0.974950\pi\)
0.430371 0.902652i \(-0.358383\pi\)
\(774\) 0 0
\(775\) 542.896 940.323i 0.0251631 0.0435838i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17236.4 + 29854.3i −0.792758 + 1.37310i
\(780\) 0 0
\(781\) −19838.0 34360.4i −0.908911 1.57428i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6187.43 0.281323
\(786\) 0 0
\(787\) 18124.4 + 31392.3i 0.820920 + 1.42188i 0.904998 + 0.425416i \(0.139872\pi\)
−0.0840778 + 0.996459i \(0.526794\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 24416.0 42289.8i 1.09336 1.89376i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −33431.7 −1.48584 −0.742918 0.669382i \(-0.766559\pi\)
−0.742918 + 0.669382i \(0.766559\pi\)
\(798\) 0 0
\(799\) −44352.0 −1.96378
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −18456.8 + 31968.0i −0.811115 + 1.40489i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −17899.4 31002.6i −0.777883 1.34733i −0.933160 0.359462i \(-0.882960\pi\)
0.155276 0.987871i \(-0.450373\pi\)
\(810\) 0 0
\(811\) −30402.2 −1.31636 −0.658178 0.752862i \(-0.728673\pi\)
−0.658178 + 0.752862i \(0.728673\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 17716.0 + 30684.9i 0.761427 + 1.31883i
\(816\) 0 0
\(817\) 6180.66 10705.2i 0.264668 0.458419i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16518.2 28610.4i 0.702180 1.21621i −0.265519 0.964106i \(-0.585543\pi\)
0.967699 0.252107i \(-0.0811233\pi\)
\(822\) 0 0
\(823\) 5200.00 + 9006.66i 0.220244 + 0.381473i 0.954882 0.296986i \(-0.0959814\pi\)
−0.734638 + 0.678459i \(0.762648\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10109.8 0.425094 0.212547 0.977151i \(-0.431824\pi\)
0.212547 + 0.977151i \(0.431824\pi\)
\(828\) 0 0
\(829\) −7057.65 12224.2i −0.295684 0.512140i 0.679460 0.733713i \(-0.262214\pi\)
−0.975144 + 0.221573i \(0.928881\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −448.000 + 775.959i −0.0185673 + 0.0321595i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21843.3 −0.898826 −0.449413 0.893324i \(-0.648367\pi\)
−0.449413 + 0.893324i \(0.648367\pi\)
\(840\) 0 0
\(841\) −12181.0 −0.499446
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25288.1 43800.3i 1.02951 1.78317i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6463.65 + 11195.4i 0.260366 + 0.450967i
\(852\) 0 0
\(853\) −25641.4 −1.02924 −0.514622 0.857417i \(-0.672068\pi\)
−0.514622 + 0.857417i \(0.672068\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −365.114 632.395i −0.0145531 0.0252068i 0.858657 0.512550i \(-0.171299\pi\)
−0.873210 + 0.487344i \(0.837966\pi\)
\(858\) 0 0
\(859\) 11400.8 19746.8i 0.452841 0.784344i −0.545720 0.837968i \(-0.683744\pi\)
0.998561 + 0.0536234i \(0.0170770\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −16159.1 + 27988.4i −0.637385 + 1.10398i 0.348619 + 0.937264i \(0.386651\pi\)
−0.986004 + 0.166719i \(0.946683\pi\)
\(864\) 0 0
\(865\) −8008.00 13870.3i −0.314775 0.545206i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −36682.6 −1.43196
\(870\) 0 0
\(871\) −10857.9 18806.5i −0.422396 0.731611i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7083.00 12268.1i 0.272721 0.472366i −0.696837 0.717230i \(-0.745410\pi\)
0.969558 + 0.244864i \(0.0787432\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19673.8 0.752358 0.376179 0.926547i \(-0.377238\pi\)
0.376179 + 0.926547i \(0.377238\pi\)
\(882\) 0 0
\(883\) −44876.0 −1.71030 −0.855152 0.518378i \(-0.826536\pi\)
−0.855152 + 0.518378i \(0.826536\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7873.76 13637.7i 0.298055 0.516247i −0.677636 0.735398i \(-0.736995\pi\)
0.975691 + 0.219151i \(0.0703288\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 19446.2 + 33681.8i 0.728715 + 1.26217i
\(894\) 0 0
\(895\) 1753.97 0.0655070
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4614.19 7992.01i −0.171181 0.296494i
\(900\) 0 0
\(901\) 5261.91 9113.90i 0.194561 0.336990i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19004.2 32916.3i 0.698036 1.20903i
\(906\) 0 0
\(907\) 178.000 + 308.305i 0.00651642 + 0.0112868i 0.869265 0.494346i \(-0.164592\pi\)
−0.862749 + 0.505633i \(0.831259\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4916.80 −0.178815 −0.0894077 0.995995i \(-0.528497\pi\)
−0.0894077 + 0.995995i \(0.528497\pi\)
\(912\) 0 0
\(913\) −3507.94 6075.94i −0.127159 0.220245i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 21528.0 37287.6i 0.772735 1.33842i −0.163324 0.986572i \(-0.552222\pi\)
0.936059 0.351843i \(-0.114445\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 59984.5 2.13913
\(924\) 0 0
\(925\) −1014.00 −0.0360434
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21605.2 37421.3i 0.763018 1.32159i −0.178270 0.983982i \(-0.557050\pi\)
0.941288 0.337604i \(-0.109617\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 27843.4 + 48226.2i 0.973880 + 1.68681i
\(936\) 0 0
\(937\) 52118.0 1.81710 0.908549 0.417778i \(-0.137191\pi\)
0.908549 + 0.417778i \(0.137191\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2333.55 + 4041.83i 0.0808413 + 0.140021i 0.903612 0.428353i \(-0.140906\pi\)
−0.822770 + 0.568374i \(0.807573\pi\)
\(942\) 0 0
\(943\) −34202.4 + 59240.4i −1.18111 + 2.04574i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 359.092 621.965i 0.0123220 0.0213423i −0.859799 0.510633i \(-0.829411\pi\)
0.872121 + 0.489291i \(0.162744\pi\)
\(948\) 0 0
\(949\) −27904.0 48331.1i −0.954481 1.65321i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36461.6 1.23936 0.619679 0.784855i \(-0.287263\pi\)
0.619679 + 0.784855i \(0.287263\pi\)
\(954\) 0 0
\(955\) 11400.8 + 19746.8i 0.386305 + 0.669101i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 11407.5 19758.4i 0.382918 0.663233i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 14583.4 0.486483
\(966\) 0 0
\(967\) −39992.0 −1.32994 −0.664972 0.746868i \(-0.731557\pi\)
−0.664972 + 0.746868i \(0.731557\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19091.7 + 33067.9i −0.630982 + 1.09289i 0.356370 + 0.934345i \(0.384014\pi\)
−0.987351 + 0.158547i \(0.949319\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5524.49 9568.70i −0.180905 0.313337i 0.761284 0.648419i \(-0.224569\pi\)
−0.942189 + 0.335082i \(0.891236\pi\)
\(978\) 0 0
\(979\) 48526.5 1.58418
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −23155.6 40106.7i −0.751322 1.30133i −0.947182 0.320696i \(-0.896083\pi\)
0.195860 0.980632i \(-0.437250\pi\)
\(984\) 0 0
\(985\) −5261.91 + 9113.90i −0.170212 + 0.294815i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12264.4 21242.5i 0.394322 0.682985i
\(990\) 0 0
\(991\) 25532.0 + 44222.7i 0.818416 + 1.41754i 0.906849 + 0.421457i \(0.138481\pi\)
−0.0884321 + 0.996082i \(0.528186\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 49499.4 1.57712
\(996\) 0 0
\(997\) 7057.65 + 12224.2i 0.224191 + 0.388309i 0.956076 0.293118i \(-0.0946929\pi\)
−0.731886 + 0.681427i \(0.761360\pi\)
\(998\) 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.bc.1549.4 8
3.2 odd 2 inner 1764.4.k.bc.1549.1 8
7.2 even 3 1764.4.a.bb.1.1 4
7.3 odd 6 inner 1764.4.k.bc.361.2 8
7.4 even 3 inner 1764.4.k.bc.361.4 8
7.5 odd 6 1764.4.a.bb.1.3 yes 4
7.6 odd 2 inner 1764.4.k.bc.1549.2 8
21.2 odd 6 1764.4.a.bb.1.4 yes 4
21.5 even 6 1764.4.a.bb.1.2 yes 4
21.11 odd 6 inner 1764.4.k.bc.361.1 8
21.17 even 6 inner 1764.4.k.bc.361.3 8
21.20 even 2 inner 1764.4.k.bc.1549.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.4.a.bb.1.1 4 7.2 even 3
1764.4.a.bb.1.2 yes 4 21.5 even 6
1764.4.a.bb.1.3 yes 4 7.5 odd 6
1764.4.a.bb.1.4 yes 4 21.2 odd 6
1764.4.k.bc.361.1 8 21.11 odd 6 inner
1764.4.k.bc.361.2 8 7.3 odd 6 inner
1764.4.k.bc.361.3 8 21.17 even 6 inner
1764.4.k.bc.361.4 8 7.4 even 3 inner
1764.4.k.bc.1549.1 8 3.2 odd 2 inner
1764.4.k.bc.1549.2 8 7.6 odd 2 inner
1764.4.k.bc.1549.3 8 21.20 even 2 inner
1764.4.k.bc.1549.4 8 1.1 even 1 trivial