Properties

Label 1620.3.t.b.269.3
Level $1620$
Weight $3$
Character 1620.269
Analytic conductor $44.142$
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] = [1620,3,Mod(269,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.269");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.t (of order \(6\), 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: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 269.3
Root \(-0.535233 + 0.309017i\) of defining polynomial
Character \(\chi\) \(=\) 1620.269
Dual form 1620.3.t.b.1349.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.75495 - 4.17256i) q^{5} +(6.92820 + 4.00000i) q^{7} +O(q^{10})\) \(q+(2.75495 - 4.17256i) q^{5} +(6.92820 + 4.00000i) q^{7} +(-7.74597 - 4.47214i) q^{11} +(10.3923 - 6.00000i) q^{13} -31.3050 q^{17} -6.00000 q^{19} +(-2.23607 - 3.87298i) q^{23} +(-9.82051 - 22.9904i) q^{25} +(-23.2379 - 13.4164i) q^{29} +(-17.0000 - 29.4449i) q^{31} +(35.7771 - 17.8885i) q^{35} -44.0000i q^{37} +(-15.4919 + 8.94427i) q^{41} +(24.2487 + 14.0000i) q^{43} +(-2.23607 + 3.87298i) q^{47} +(7.50000 + 12.9904i) q^{49} +40.2492 q^{53} +(-40.0000 + 20.0000i) q^{55} +(-85.2056 + 49.1935i) q^{59} +(-37.0000 + 64.0859i) q^{61} +(3.59492 - 59.8922i) q^{65} +(79.6743 - 46.0000i) q^{67} -53.6656i q^{71} +56.0000i q^{73} +(-35.7771 - 61.9677i) q^{77} +(39.0000 - 67.5500i) q^{79} +(51.4296 - 89.0786i) q^{83} +(-86.2436 + 130.622i) q^{85} +17.8885i q^{89} +96.0000 q^{91} +(-16.5297 + 25.0354i) q^{95} +(-27.7128 - 16.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{19} + 60 q^{25} - 136 q^{31} + 60 q^{49} - 320 q^{55} - 296 q^{61} + 312 q^{79} + 280 q^{85} + 768 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) 2.75495 4.17256i 0.550990 0.834512i
\(6\) 0 0
\(7\) 6.92820 + 4.00000i 0.989743 + 0.571429i 0.905198 0.424991i \(-0.139723\pi\)
0.0845458 + 0.996420i \(0.473056\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −7.74597 4.47214i −0.704179 0.406558i 0.104723 0.994501i \(-0.466604\pi\)
−0.808902 + 0.587944i \(0.799938\pi\)
\(12\) 0 0
\(13\) 10.3923 6.00000i 0.799408 0.461538i −0.0438561 0.999038i \(-0.513964\pi\)
0.843264 + 0.537499i \(0.180631\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −31.3050 −1.84147 −0.920734 0.390191i \(-0.872409\pi\)
−0.920734 + 0.390191i \(0.872409\pi\)
\(18\) 0 0
\(19\) −6.00000 −0.315789 −0.157895 0.987456i \(-0.550471\pi\)
−0.157895 + 0.987456i \(0.550471\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.23607 3.87298i −0.0972203 0.168391i 0.813313 0.581827i \(-0.197662\pi\)
−0.910533 + 0.413436i \(0.864328\pi\)
\(24\) 0 0
\(25\) −9.82051 22.9904i −0.392820 0.919615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −23.2379 13.4164i −0.801307 0.462635i 0.0426210 0.999091i \(-0.486429\pi\)
−0.843928 + 0.536457i \(0.819763\pi\)
\(30\) 0 0
\(31\) −17.0000 29.4449i −0.548387 0.949834i −0.998385 0.0568049i \(-0.981909\pi\)
0.449998 0.893029i \(-0.351425\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 35.7771 17.8885i 1.02220 0.511101i
\(36\) 0 0
\(37\) 44.0000i 1.18919i −0.804026 0.594595i \(-0.797313\pi\)
0.804026 0.594595i \(-0.202687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −15.4919 + 8.94427i −0.377852 + 0.218153i −0.676883 0.736090i \(-0.736670\pi\)
0.299031 + 0.954243i \(0.403337\pi\)
\(42\) 0 0
\(43\) 24.2487 + 14.0000i 0.563924 + 0.325581i 0.754719 0.656048i \(-0.227773\pi\)
−0.190795 + 0.981630i \(0.561107\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.23607 + 3.87298i −0.0475759 + 0.0824039i −0.888833 0.458232i \(-0.848483\pi\)
0.841257 + 0.540636i \(0.181816\pi\)
\(48\) 0 0
\(49\) 7.50000 + 12.9904i 0.153061 + 0.265110i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 40.2492 0.759419 0.379710 0.925106i \(-0.376024\pi\)
0.379710 + 0.925106i \(0.376024\pi\)
\(54\) 0 0
\(55\) −40.0000 + 20.0000i −0.727273 + 0.363636i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −85.2056 + 49.1935i −1.44416 + 0.833788i −0.998124 0.0612253i \(-0.980499\pi\)
−0.446039 + 0.895013i \(0.647166\pi\)
\(60\) 0 0
\(61\) −37.0000 + 64.0859i −0.606557 + 1.05059i 0.385246 + 0.922814i \(0.374117\pi\)
−0.991803 + 0.127774i \(0.959217\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.59492 59.8922i 0.0553064 0.921419i
\(66\) 0 0
\(67\) 79.6743 46.0000i 1.18917 0.686567i 0.231051 0.972942i \(-0.425783\pi\)
0.958118 + 0.286374i \(0.0924501\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 53.6656i 0.755854i −0.925835 0.377927i \(-0.876637\pi\)
0.925835 0.377927i \(-0.123363\pi\)
\(72\) 0 0
\(73\) 56.0000i 0.767123i 0.923515 + 0.383562i \(0.125303\pi\)
−0.923515 + 0.383562i \(0.874697\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −35.7771 61.9677i −0.464638 0.804776i
\(78\) 0 0
\(79\) 39.0000 67.5500i 0.493671 0.855063i −0.506303 0.862356i \(-0.668988\pi\)
0.999973 + 0.00729288i \(0.00232142\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 51.4296 89.0786i 0.619633 1.07324i −0.369919 0.929064i \(-0.620615\pi\)
0.989553 0.144172i \(-0.0460520\pi\)
\(84\) 0 0
\(85\) −86.2436 + 130.622i −1.01463 + 1.53673i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 17.8885i 0.200995i 0.994937 + 0.100497i \(0.0320434\pi\)
−0.994937 + 0.100497i \(0.967957\pi\)
\(90\) 0 0
\(91\) 96.0000 1.05495
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.5297 + 25.0354i −0.173997 + 0.263530i
\(96\) 0 0
\(97\) −27.7128 16.0000i −0.285699 0.164948i 0.350302 0.936637i \(-0.386079\pi\)
−0.636001 + 0.771689i \(0.719412\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −131.681 76.0263i −1.30378 0.752736i −0.322727 0.946492i \(-0.604600\pi\)
−0.981050 + 0.193756i \(0.937933\pi\)
\(102\) 0 0
\(103\) −90.0666 + 52.0000i −0.874433 + 0.504854i −0.868819 0.495130i \(-0.835120\pi\)
−0.00561444 + 0.999984i \(0.501787\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 147.580 1.37926 0.689628 0.724163i \(-0.257774\pi\)
0.689628 + 0.724163i \(0.257774\pi\)
\(108\) 0 0
\(109\) 74.0000 0.678899 0.339450 0.940624i \(-0.389759\pi\)
0.339450 + 0.940624i \(0.389759\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −20.1246 34.8569i −0.178094 0.308468i 0.763134 0.646241i \(-0.223660\pi\)
−0.941228 + 0.337773i \(0.890326\pi\)
\(114\) 0 0
\(115\) −22.3205 1.33975i −0.194091 0.0116500i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −216.887 125.220i −1.82258 1.05227i
\(120\) 0 0
\(121\) −20.5000 35.5070i −0.169421 0.293447i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −122.984 22.3607i −0.983870 0.178885i
\(126\) 0 0
\(127\) 16.0000i 0.125984i 0.998014 + 0.0629921i \(0.0200643\pi\)
−0.998014 + 0.0629921i \(0.979936\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −69.7137 + 40.2492i −0.532166 + 0.307246i −0.741898 0.670513i \(-0.766074\pi\)
0.209732 + 0.977759i \(0.432741\pi\)
\(132\) 0 0
\(133\) −41.5692 24.0000i −0.312551 0.180451i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 87.2067 151.046i 0.636545 1.10253i −0.349641 0.936884i \(-0.613696\pi\)
0.986186 0.165644i \(-0.0529703\pi\)
\(138\) 0 0
\(139\) 59.0000 + 102.191i 0.424460 + 0.735187i 0.996370 0.0851297i \(-0.0271305\pi\)
−0.571909 + 0.820317i \(0.693797\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −107.331 −0.750568
\(144\) 0 0
\(145\) −120.000 + 60.0000i −0.827586 + 0.413793i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −85.2056 + 49.1935i −0.571850 + 0.330158i −0.757888 0.652385i \(-0.773769\pi\)
0.186038 + 0.982543i \(0.440435\pi\)
\(150\) 0 0
\(151\) −17.0000 + 29.4449i −0.112583 + 0.194999i −0.916811 0.399322i \(-0.869246\pi\)
0.804228 + 0.594321i \(0.202579\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −169.695 10.1856i −1.09480 0.0657135i
\(156\) 0 0
\(157\) 79.6743 46.0000i 0.507480 0.292994i −0.224317 0.974516i \(-0.572015\pi\)
0.731797 + 0.681523i \(0.238682\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 35.7771i 0.222218i
\(162\) 0 0
\(163\) 68.0000i 0.417178i −0.978003 0.208589i \(-0.933113\pi\)
0.978003 0.208589i \(-0.0668871\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 33.5410 + 58.0948i 0.200844 + 0.347873i 0.948801 0.315875i \(-0.102298\pi\)
−0.747956 + 0.663748i \(0.768965\pi\)
\(168\) 0 0
\(169\) −12.5000 + 21.6506i −0.0739645 + 0.128110i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −38.0132 + 65.8407i −0.219729 + 0.380582i −0.954725 0.297489i \(-0.903851\pi\)
0.734996 + 0.678072i \(0.237184\pi\)
\(174\) 0 0
\(175\) 23.9230 198.564i 0.136703 1.13465i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 259.384i 1.44907i −0.689237 0.724536i \(-0.742054\pi\)
0.689237 0.724536i \(-0.257946\pi\)
\(180\) 0 0
\(181\) −166.000 −0.917127 −0.458564 0.888662i \(-0.651636\pi\)
−0.458564 + 0.888662i \(0.651636\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −183.593 121.218i −0.992393 0.655231i
\(186\) 0 0
\(187\) 242.487 + 140.000i 1.29672 + 0.748663i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −185.903 107.331i −0.973315 0.561944i −0.0730699 0.997327i \(-0.523280\pi\)
−0.900245 + 0.435383i \(0.856613\pi\)
\(192\) 0 0
\(193\) 27.7128 16.0000i 0.143590 0.0829016i −0.426484 0.904495i \(-0.640248\pi\)
0.570074 + 0.821594i \(0.306915\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.47214 0.0227012 0.0113506 0.999936i \(-0.496387\pi\)
0.0113506 + 0.999936i \(0.496387\pi\)
\(198\) 0 0
\(199\) 114.000 0.572864 0.286432 0.958101i \(-0.407531\pi\)
0.286432 + 0.958101i \(0.407531\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −107.331 185.903i −0.528725 0.915779i
\(204\) 0 0
\(205\) −5.35898 + 89.2820i −0.0261414 + 0.435522i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 46.4758 + 26.8328i 0.222372 + 0.128387i
\(210\) 0 0
\(211\) 3.00000 + 5.19615i 0.0142180 + 0.0246263i 0.873047 0.487636i \(-0.162141\pi\)
−0.858829 + 0.512263i \(0.828807\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 125.220 62.6099i 0.582418 0.291209i
\(216\) 0 0
\(217\) 272.000i 1.25346i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −325.331 + 187.830i −1.47208 + 0.849908i
\(222\) 0 0
\(223\) −235.559 136.000i −1.05632 0.609865i −0.131907 0.991262i \(-0.542110\pi\)
−0.924411 + 0.381397i \(0.875443\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 122.984 213.014i 0.541779 0.938388i −0.457023 0.889455i \(-0.651084\pi\)
0.998802 0.0489334i \(-0.0155822\pi\)
\(228\) 0 0
\(229\) −77.0000 133.368i −0.336245 0.582393i 0.647479 0.762084i \(-0.275824\pi\)
−0.983723 + 0.179691i \(0.942490\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 183.358 0.786942 0.393471 0.919337i \(-0.371274\pi\)
0.393471 + 0.919337i \(0.371274\pi\)
\(234\) 0 0
\(235\) 10.0000 + 20.0000i 0.0425532 + 0.0851064i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 154.919 89.4427i 0.648198 0.374237i −0.139567 0.990213i \(-0.544571\pi\)
0.787766 + 0.615975i \(0.211238\pi\)
\(240\) 0 0
\(241\) 103.000 178.401i 0.427386 0.740254i −0.569254 0.822162i \(-0.692768\pi\)
0.996640 + 0.0819076i \(0.0261012\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 74.8653 + 4.49364i 0.305572 + 0.0183414i
\(246\) 0 0
\(247\) −62.3538 + 36.0000i −0.252445 + 0.145749i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 26.8328i 0.106904i −0.998570 0.0534518i \(-0.982978\pi\)
0.998570 0.0534518i \(-0.0170224\pi\)
\(252\) 0 0
\(253\) 40.0000i 0.158103i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.1246 34.8569i −0.0783059 0.135630i 0.824213 0.566280i \(-0.191618\pi\)
−0.902519 + 0.430650i \(0.858284\pi\)
\(258\) 0 0
\(259\) 176.000 304.841i 0.679537 1.17699i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 105.095 182.030i 0.399602 0.692130i −0.594075 0.804410i \(-0.702482\pi\)
0.993677 + 0.112279i \(0.0358152\pi\)
\(264\) 0 0
\(265\) 110.885 167.942i 0.418432 0.633744i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 134.164i 0.498751i 0.968407 + 0.249376i \(0.0802254\pi\)
−0.968407 + 0.249376i \(0.919775\pi\)
\(270\) 0 0
\(271\) −398.000 −1.46863 −0.734317 0.678806i \(-0.762498\pi\)
−0.734317 + 0.678806i \(0.762498\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −26.7468 + 222.001i −0.0972610 + 0.807278i
\(276\) 0 0
\(277\) −252.879 146.000i −0.912922 0.527076i −0.0315519 0.999502i \(-0.510045\pi\)
−0.881370 + 0.472426i \(0.843378\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 46.4758 + 26.8328i 0.165394 + 0.0954904i 0.580412 0.814323i \(-0.302891\pi\)
−0.415018 + 0.909813i \(0.636225\pi\)
\(282\) 0 0
\(283\) 45.0333 26.0000i 0.159128 0.0918728i −0.418321 0.908299i \(-0.637381\pi\)
0.577450 + 0.816426i \(0.304048\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −143.108 −0.498635
\(288\) 0 0
\(289\) 691.000 2.39100
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 194.538 + 336.950i 0.663952 + 1.15000i 0.979568 + 0.201112i \(0.0644554\pi\)
−0.315616 + 0.948887i \(0.602211\pi\)
\(294\) 0 0
\(295\) −29.4744 + 491.051i −0.0999133 + 1.66458i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −46.4758 26.8328i −0.155437 0.0897419i
\(300\) 0 0
\(301\) 112.000 + 193.990i 0.372093 + 0.644484i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 165.469 + 330.938i 0.542521 + 1.08504i
\(306\) 0 0
\(307\) 492.000i 1.60261i 0.598259 + 0.801303i \(0.295859\pi\)
−0.598259 + 0.801303i \(0.704141\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 418.282 241.495i 1.34496 0.776512i 0.357428 0.933941i \(-0.383654\pi\)
0.987530 + 0.157428i \(0.0503203\pi\)
\(312\) 0 0
\(313\) 491.902 + 284.000i 1.57157 + 0.907348i 0.995977 + 0.0896145i \(0.0285635\pi\)
0.575597 + 0.817734i \(0.304770\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −306.341 + 530.599i −0.966376 + 1.67381i −0.260506 + 0.965472i \(0.583889\pi\)
−0.705871 + 0.708341i \(0.749444\pi\)
\(318\) 0 0
\(319\) 120.000 + 207.846i 0.376176 + 0.651555i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 187.830 0.581516
\(324\) 0 0
\(325\) −240.000 180.000i −0.738462 0.553846i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −30.9839 + 17.8885i −0.0941759 + 0.0543725i
\(330\) 0 0
\(331\) −101.000 + 174.937i −0.305136 + 0.528511i −0.977292 0.211899i \(-0.932035\pi\)
0.672156 + 0.740410i \(0.265369\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 27.5610 459.174i 0.0822717 1.37067i
\(336\) 0 0
\(337\) −318.697 + 184.000i −0.945689 + 0.545994i −0.891739 0.452550i \(-0.850515\pi\)
−0.0539502 + 0.998544i \(0.517181\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 304.105i 0.891804i
\(342\) 0 0
\(343\) 272.000i 0.793003i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −127.456 220.760i −0.367308 0.636196i 0.621836 0.783148i \(-0.286387\pi\)
−0.989144 + 0.146952i \(0.953054\pi\)
\(348\) 0 0
\(349\) 59.0000 102.191i 0.169054 0.292811i −0.769033 0.639209i \(-0.779262\pi\)
0.938088 + 0.346398i \(0.112595\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.6525 27.1109i 0.0443413 0.0768014i −0.843003 0.537909i \(-0.819214\pi\)
0.887344 + 0.461108i \(0.152548\pi\)
\(354\) 0 0
\(355\) −223.923 147.846i −0.630769 0.416468i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 53.6656i 0.149486i −0.997203 0.0747432i \(-0.976186\pi\)
0.997203 0.0747432i \(-0.0238137\pi\)
\(360\) 0 0
\(361\) −325.000 −0.900277
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 233.663 + 154.277i 0.640174 + 0.422677i
\(366\) 0 0
\(367\) −304.841 176.000i −0.830629 0.479564i 0.0234388 0.999725i \(-0.492539\pi\)
−0.854068 + 0.520161i \(0.825872\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 278.855 + 160.997i 0.751630 + 0.433954i
\(372\) 0 0
\(373\) 114.315 66.0000i 0.306475 0.176944i −0.338873 0.940832i \(-0.610046\pi\)
0.645348 + 0.763889i \(0.276712\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −321.994 −0.854095
\(378\) 0 0
\(379\) 394.000 1.03958 0.519789 0.854295i \(-0.326011\pi\)
0.519789 + 0.854295i \(0.326011\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −38.0132 65.8407i −0.0992511 0.171908i 0.812124 0.583485i \(-0.198311\pi\)
−0.911375 + 0.411577i \(0.864978\pi\)
\(384\) 0 0
\(385\) −357.128 21.4359i −0.927606 0.0556778i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 240.125 + 138.636i 0.617288 + 0.356391i 0.775812 0.630964i \(-0.217340\pi\)
−0.158524 + 0.987355i \(0.550674\pi\)
\(390\) 0 0
\(391\) 70.0000 + 121.244i 0.179028 + 0.310086i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −174.413 348.827i −0.441553 0.883105i
\(396\) 0 0
\(397\) 652.000i 1.64232i 0.570700 + 0.821159i \(0.306672\pi\)
−0.570700 + 0.821159i \(0.693328\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 154.919 89.4427i 0.386333 0.223049i −0.294237 0.955732i \(-0.595066\pi\)
0.680570 + 0.732683i \(0.261732\pi\)
\(402\) 0 0
\(403\) −353.338 204.000i −0.876770 0.506203i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −196.774 + 340.823i −0.483474 + 0.837402i
\(408\) 0 0
\(409\) 103.000 + 178.401i 0.251834 + 0.436189i 0.964031 0.265791i \(-0.0856331\pi\)
−0.712197 + 0.701980i \(0.752300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −787.096 −1.90580
\(414\) 0 0
\(415\) −230.000 460.000i −0.554217 1.10843i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 178.157 102.859i 0.425196 0.245487i −0.272102 0.962268i \(-0.587719\pi\)
0.697298 + 0.716781i \(0.254385\pi\)
\(420\) 0 0
\(421\) 19.0000 32.9090i 0.0451306 0.0781686i −0.842578 0.538575i \(-0.818963\pi\)
0.887708 + 0.460406i \(0.152296\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 307.431 + 719.713i 0.723366 + 1.69344i
\(426\) 0 0
\(427\) −512.687 + 296.000i −1.20067 + 0.693208i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 608.210i 1.41116i −0.708630 0.705581i \(-0.750686\pi\)
0.708630 0.705581i \(-0.249314\pi\)
\(432\) 0 0
\(433\) 272.000i 0.628176i 0.949394 + 0.314088i \(0.101699\pi\)
−0.949394 + 0.314088i \(0.898301\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.4164 + 23.2379i 0.0307012 + 0.0531760i
\(438\) 0 0
\(439\) 183.000 316.965i 0.416856 0.722017i −0.578765 0.815494i \(-0.696465\pi\)
0.995621 + 0.0934778i \(0.0297984\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −288.453 + 499.615i −0.651135 + 1.12780i 0.331713 + 0.943380i \(0.392373\pi\)
−0.982848 + 0.184418i \(0.940960\pi\)
\(444\) 0 0
\(445\) 74.6410 + 49.2820i 0.167733 + 0.110746i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 429.325i 0.956181i 0.878311 + 0.478090i \(0.158671\pi\)
−0.878311 + 0.478090i \(0.841329\pi\)
\(450\) 0 0
\(451\) 160.000 0.354767
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 264.475 400.566i 0.581264 0.880364i
\(456\) 0 0
\(457\) 90.0666 + 52.0000i 0.197082 + 0.113786i 0.595294 0.803508i \(-0.297036\pi\)
−0.398211 + 0.917294i \(0.630369\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 441.520 + 254.912i 0.957744 + 0.552954i 0.895478 0.445106i \(-0.146834\pi\)
0.0622662 + 0.998060i \(0.480167\pi\)
\(462\) 0 0
\(463\) 83.1384 48.0000i 0.179565 0.103672i −0.407523 0.913195i \(-0.633608\pi\)
0.587088 + 0.809523i \(0.300274\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 147.580 0.316018 0.158009 0.987438i \(-0.449492\pi\)
0.158009 + 0.987438i \(0.449492\pi\)
\(468\) 0 0
\(469\) 736.000 1.56930
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −125.220 216.887i −0.264735 0.458535i
\(474\) 0 0
\(475\) 58.9230 + 137.942i 0.124049 + 0.290405i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 495.742 + 286.217i 1.03495 + 0.597530i 0.918399 0.395655i \(-0.129482\pi\)
0.116552 + 0.993185i \(0.462816\pi\)
\(480\) 0 0
\(481\) −264.000 457.261i −0.548857 0.950647i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −143.108 + 71.5542i −0.295069 + 0.147534i
\(486\) 0 0
\(487\) 648.000i 1.33060i −0.746578 0.665298i \(-0.768305\pi\)
0.746578 0.665298i \(-0.231695\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −116.190 + 67.0820i −0.236638 + 0.136623i −0.613631 0.789593i \(-0.710292\pi\)
0.376992 + 0.926216i \(0.376958\pi\)
\(492\) 0 0
\(493\) 727.461 + 420.000i 1.47558 + 0.851927i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 214.663 371.806i 0.431917 0.748101i
\(498\) 0 0
\(499\) 243.000 + 420.888i 0.486974 + 0.843464i 0.999888 0.0149764i \(-0.00476733\pi\)
−0.512914 + 0.858440i \(0.671434\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 791.568 1.57369 0.786847 0.617148i \(-0.211712\pi\)
0.786847 + 0.617148i \(0.211712\pi\)
\(504\) 0 0
\(505\) −680.000 + 340.000i −1.34653 + 0.673267i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 23.2379 13.4164i 0.0456540 0.0263584i −0.476999 0.878904i \(-0.658276\pi\)
0.522653 + 0.852545i \(0.324942\pi\)
\(510\) 0 0
\(511\) −224.000 + 387.979i −0.438356 + 0.759255i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −31.1559 + 519.066i −0.0604970 + 1.00789i
\(516\) 0 0
\(517\) 34.6410 20.0000i 0.0670039 0.0386847i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 983.870i 1.88843i 0.329336 + 0.944213i \(0.393175\pi\)
−0.329336 + 0.944213i \(0.606825\pi\)
\(522\) 0 0
\(523\) 292.000i 0.558317i 0.960245 + 0.279159i \(0.0900556\pi\)
−0.960245 + 0.279159i \(0.909944\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 532.184 + 921.770i 1.00984 + 1.74909i
\(528\) 0 0
\(529\) 254.500 440.807i 0.481096 0.833283i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −107.331 + 185.903i −0.201372 + 0.348786i
\(534\) 0 0
\(535\) 406.577 615.788i 0.759957 1.15101i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 134.164i 0.248913i
\(540\) 0 0
\(541\) −86.0000 −0.158965 −0.0794824 0.996836i \(-0.525327\pi\)
−0.0794824 + 0.996836i \(0.525327\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 203.866 308.769i 0.374067 0.566549i
\(546\) 0 0
\(547\) 142.028 + 82.0000i 0.259649 + 0.149909i 0.624175 0.781285i \(-0.285435\pi\)
−0.364525 + 0.931193i \(0.618769\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 139.427 + 80.4984i 0.253044 + 0.146095i
\(552\) 0 0
\(553\) 540.400 312.000i 0.977215 0.564195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 362.243 0.650347 0.325173 0.945654i \(-0.394577\pi\)
0.325173 + 0.945654i \(0.394577\pi\)
\(558\) 0 0
\(559\) 336.000 0.601073
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 498.643 + 863.675i 0.885689 + 1.53406i 0.844921 + 0.534890i \(0.179647\pi\)
0.0407680 + 0.999169i \(0.487020\pi\)
\(564\) 0 0
\(565\) −200.885 12.0577i −0.355548 0.0213411i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 542.218 + 313.050i 0.952931 + 0.550175i 0.893990 0.448087i \(-0.147894\pi\)
0.0589407 + 0.998261i \(0.481228\pi\)
\(570\) 0 0
\(571\) −197.000 341.214i −0.345009 0.597573i 0.640347 0.768086i \(-0.278791\pi\)
−0.985355 + 0.170513i \(0.945457\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −67.0820 + 89.4427i −0.116664 + 0.155553i
\(576\) 0 0
\(577\) 608.000i 1.05373i −0.849950 0.526863i \(-0.823368\pi\)
0.849950 0.526863i \(-0.176632\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 712.629 411.437i 1.22656 0.708152i
\(582\) 0 0
\(583\) −311.769 180.000i −0.534767 0.308748i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 355.535 615.804i 0.605681 1.04907i −0.386262 0.922389i \(-0.626234\pi\)
0.991943 0.126681i \(-0.0404326\pi\)
\(588\) 0 0
\(589\) 102.000 + 176.669i 0.173175 + 0.299948i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −603.738 −1.01811 −0.509054 0.860734i \(-0.670005\pi\)
−0.509054 + 0.860734i \(0.670005\pi\)
\(594\) 0 0
\(595\) −1120.00 + 560.000i −1.88235 + 0.941176i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 46.4758 26.8328i 0.0775890 0.0447960i −0.460704 0.887554i \(-0.652403\pi\)
0.538293 + 0.842758i \(0.319070\pi\)
\(600\) 0 0
\(601\) −217.000 + 375.855i −0.361065 + 0.625383i −0.988136 0.153579i \(-0.950920\pi\)
0.627071 + 0.778962i \(0.284253\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −204.632 12.2826i −0.338234 0.0203019i
\(606\) 0 0
\(607\) 568.113 328.000i 0.935935 0.540362i 0.0472514 0.998883i \(-0.484954\pi\)
0.888684 + 0.458521i \(0.151620\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 53.6656i 0.0878325i
\(612\) 0 0
\(613\) 844.000i 1.37684i −0.725315 0.688418i \(-0.758306\pi\)
0.725315 0.688418i \(-0.241694\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 230.315 + 398.917i 0.373282 + 0.646543i 0.990068 0.140587i \(-0.0448991\pi\)
−0.616786 + 0.787131i \(0.711566\pi\)
\(618\) 0 0
\(619\) 523.000 905.863i 0.844911 1.46343i −0.0407872 0.999168i \(-0.512987\pi\)
0.885698 0.464261i \(-0.153680\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −71.5542 + 123.935i −0.114854 + 0.198933i
\(624\) 0 0
\(625\) −432.115 + 451.554i −0.691384 + 0.722487i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1377.42i 2.18985i
\(630\) 0 0
\(631\) −46.0000 −0.0729002 −0.0364501 0.999335i \(-0.511605\pi\)
−0.0364501 + 0.999335i \(0.511605\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 66.7610 + 44.0792i 0.105135 + 0.0694160i
\(636\) 0 0
\(637\) 155.885 + 90.0000i 0.244717 + 0.141287i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 991.484 + 572.433i 1.54678 + 0.893032i 0.998385 + 0.0568101i \(0.0180930\pi\)
0.548391 + 0.836222i \(0.315240\pi\)
\(642\) 0 0
\(643\) −696.284 + 402.000i −1.08287 + 0.625194i −0.931669 0.363309i \(-0.881647\pi\)
−0.151200 + 0.988503i \(0.548314\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 576.906 0.891662 0.445831 0.895117i \(-0.352908\pi\)
0.445831 + 0.895117i \(0.352908\pi\)
\(648\) 0 0
\(649\) 880.000 1.35593
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −538.892 933.389i −0.825256 1.42939i −0.901723 0.432314i \(-0.857697\pi\)
0.0764669 0.997072i \(-0.475636\pi\)
\(654\) 0 0
\(655\) −24.1154 + 401.769i −0.0368174 + 0.613388i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −704.883 406.964i −1.06963 0.617548i −0.141546 0.989932i \(-0.545207\pi\)
−0.928079 + 0.372383i \(0.878541\pi\)
\(660\) 0 0
\(661\) −541.000 937.039i −0.818457 1.41761i −0.906819 0.421520i \(-0.861497\pi\)
0.0883620 0.996088i \(-0.471837\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −214.663 + 107.331i −0.322801 + 0.161400i
\(666\) 0 0
\(667\) 120.000i 0.179910i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 573.202 330.938i 0.854250 0.493201i
\(672\) 0 0
\(673\) −914.523 528.000i −1.35887 0.784547i −0.369403 0.929269i \(-0.620438\pi\)
−0.989472 + 0.144723i \(0.953771\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −306.341 + 530.599i −0.452498 + 0.783750i −0.998541 0.0540077i \(-0.982800\pi\)
0.546042 + 0.837758i \(0.316134\pi\)
\(678\) 0 0
\(679\) −128.000 221.703i −0.188513 0.326513i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −317.522 −0.464893 −0.232446 0.972609i \(-0.574673\pi\)
−0.232446 + 0.972609i \(0.574673\pi\)
\(684\) 0 0
\(685\) −390.000 780.000i −0.569343 1.13869i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 418.282 241.495i 0.607086 0.350501i
\(690\) 0 0
\(691\) −461.000 + 798.475i −0.667149 + 1.15554i 0.311549 + 0.950230i \(0.399152\pi\)
−0.978698 + 0.205306i \(0.934181\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 588.940 + 35.3500i 0.847396 + 0.0508633i
\(696\) 0 0
\(697\) 484.974 280.000i 0.695802 0.401722i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 474.046i 0.676243i −0.941102 0.338122i \(-0.890209\pi\)
0.941102 0.338122i \(-0.109791\pi\)
\(702\) 0 0
\(703\) 264.000i 0.375533i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −608.210 1053.45i −0.860269 1.49003i
\(708\) 0 0
\(709\) 483.000 836.581i 0.681241 1.17994i −0.293361 0.956002i \(-0.594774\pi\)
0.974602 0.223943i \(-0.0718928\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −76.0263 + 131.681i −0.106629 + 0.184686i
\(714\) 0 0
\(715\) −295.692 + 447.846i −0.413556 + 0.626358i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1109.09i 1.54254i −0.636505 0.771272i \(-0.719621\pi\)
0.636505 0.771272i \(-0.280379\pi\)
\(720\) 0 0
\(721\) −832.000 −1.15395
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −80.2403 + 666.004i −0.110676 + 0.918626i
\(726\) 0 0
\(727\) 353.338 + 204.000i 0.486023 + 0.280605i 0.722923 0.690929i \(-0.242798\pi\)
−0.236900 + 0.971534i \(0.576132\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −759.105 438.269i −1.03845 0.599548i
\(732\) 0 0
\(733\) −142.028 + 82.0000i −0.193763 + 0.111869i −0.593743 0.804655i \(-0.702350\pi\)
0.399980 + 0.916524i \(0.369017\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −822.873 −1.11652
\(738\) 0 0
\(739\) 1082.00 1.46414 0.732070 0.681229i \(-0.238554\pi\)
0.732070 + 0.681229i \(0.238554\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 570.197 + 987.611i 0.767426 + 1.32922i 0.938955 + 0.344041i \(0.111796\pi\)
−0.171529 + 0.985179i \(0.554871\pi\)
\(744\) 0 0
\(745\) −29.4744 + 491.051i −0.0395630 + 0.659129i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1022.47 + 590.322i 1.36511 + 0.788147i
\(750\) 0 0
\(751\) 479.000 + 829.652i 0.637816 + 1.10473i 0.985911 + 0.167271i \(0.0534954\pi\)
−0.348095 + 0.937459i \(0.613171\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 76.0263 + 152.053i 0.100697 + 0.201394i
\(756\) 0 0
\(757\) 772.000i 1.01982i 0.860229 + 0.509908i \(0.170320\pi\)
−0.860229 + 0.509908i \(0.829680\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −975.992 + 563.489i −1.28251 + 0.740459i −0.977307 0.211828i \(-0.932058\pi\)
−0.305205 + 0.952287i \(0.598725\pi\)
\(762\) 0 0
\(763\) 512.687 + 296.000i 0.671936 + 0.387942i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −590.322 + 1022.47i −0.769651 + 1.33307i
\(768\) 0 0
\(769\) 663.000 + 1148.35i 0.862159 + 1.49330i 0.869841 + 0.493332i \(0.164221\pi\)
−0.00768240 + 0.999970i \(0.502445\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 147.580 0.190919 0.0954596 0.995433i \(-0.469568\pi\)
0.0954596 + 0.995433i \(0.469568\pi\)
\(774\) 0 0
\(775\) −510.000 + 680.000i −0.658065 + 0.877419i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 92.9516 53.6656i 0.119322 0.0688904i
\(780\) 0 0
\(781\) −240.000 + 415.692i −0.307298 + 0.532256i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 27.5610 459.174i 0.0351096 0.584935i
\(786\) 0 0
\(787\) 980.341 566.000i 1.24567 0.719187i 0.275426 0.961322i \(-0.411181\pi\)
0.970242 + 0.242136i \(0.0778479\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 321.994i 0.407072i
\(792\) 0 0
\(793\) 888.000i 1.11980i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −306.341 530.599i −0.384368 0.665745i 0.607313 0.794462i \(-0.292247\pi\)
−0.991681 + 0.128718i \(0.958914\pi\)
\(798\) 0 0
\(799\) 70.0000 121.244i 0.0876095 0.151744i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 250.440 433.774i 0.311880 0.540192i
\(804\) 0 0
\(805\) −149.282 98.5641i −0.185444 0.122440i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1234.31i 1.52572i −0.646562 0.762861i \(-0.723794\pi\)
0.646562 0.762861i \(-0.276206\pi\)
\(810\) 0 0
\(811\) 874.000 1.07768 0.538841 0.842408i \(-0.318862\pi\)
0.538841 + 0.842408i \(0.318862\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −283.734 187.337i −0.348140 0.229861i
\(816\) 0 0
\(817\) −145.492 84.0000i −0.178081 0.102815i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −224.633 129.692i −0.273609 0.157968i 0.356918 0.934136i \(-0.383828\pi\)
−0.630527 + 0.776168i \(0.717161\pi\)
\(822\) 0 0
\(823\) −1184.72 + 684.000i −1.43952 + 0.831106i −0.997816 0.0660564i \(-0.978958\pi\)
−0.441701 + 0.897162i \(0.645625\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −460.630 −0.556989 −0.278495 0.960438i \(-0.589835\pi\)
−0.278495 + 0.960438i \(0.589835\pi\)
\(828\) 0 0
\(829\) 1002.00 1.20869 0.604343 0.796725i \(-0.293436\pi\)
0.604343 + 0.796725i \(0.293436\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −234.787 406.663i −0.281857 0.488191i
\(834\) 0 0
\(835\) 334.808 + 20.0962i 0.400967 + 0.0240673i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −263.363 152.053i −0.313901 0.181231i 0.334770 0.942300i \(-0.391341\pi\)
−0.648671 + 0.761069i \(0.724675\pi\)
\(840\) 0 0
\(841\) −60.5000 104.789i −0.0719382 0.124601i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 55.9017 + 111.803i 0.0661559 + 0.132312i
\(846\) 0 0
\(847\) 328.000i 0.387249i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −170.411 + 98.3870i −0.200248 + 0.115613i
\(852\) 0 0
\(853\) −668.572 386.000i −0.783789 0.452521i 0.0539827 0.998542i \(-0.482808\pi\)
−0.837771 + 0.546021i \(0.816142\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −449.450 + 778.470i −0.524445 + 0.908366i 0.475150 + 0.879905i \(0.342394\pi\)
−0.999595 + 0.0284609i \(0.990939\pi\)
\(858\) 0 0
\(859\) 139.000 + 240.755i 0.161816 + 0.280274i 0.935520 0.353274i \(-0.114932\pi\)
−0.773704 + 0.633547i \(0.781598\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 576.906 0.668488 0.334244 0.942486i \(-0.391519\pi\)
0.334244 + 0.942486i \(0.391519\pi\)
\(864\) 0 0
\(865\) 170.000 + 340.000i 0.196532 + 0.393064i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −604.185 + 348.827i −0.695265 + 0.401412i
\(870\) 0 0
\(871\) 552.000 956.092i 0.633754 1.09769i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −762.614 646.854i −0.871558 0.739262i
\(876\) 0 0
\(877\) −543.864 + 314.000i −0.620141 + 0.358039i −0.776924 0.629594i \(-0.783221\pi\)
0.156783 + 0.987633i \(0.449888\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 536.656i 0.609145i 0.952489 + 0.304572i \(0.0985135\pi\)
−0.952489 + 0.304572i \(0.901486\pi\)
\(882\) 0 0
\(883\) 948.000i 1.07361i −0.843705 0.536806i \(-0.819631\pi\)
0.843705 0.536806i \(-0.180369\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 140.872 + 243.998i 0.158819 + 0.275082i 0.934443 0.356113i \(-0.115898\pi\)
−0.775624 + 0.631195i \(0.782565\pi\)
\(888\) 0 0
\(889\) −64.0000 + 110.851i −0.0719910 + 0.124692i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 13.4164 23.2379i 0.0150240 0.0260223i
\(894\) 0 0
\(895\) −1082.29 714.589i −1.20927 0.798424i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 912.316i 1.01481i
\(900\) 0 0
\(901\) −1260.00 −1.39845
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −457.322 + 692.645i −0.505328 + 0.765353i
\(906\) 0 0
\(907\) −446.869 258.000i −0.492689 0.284454i 0.233000 0.972477i \(-0.425146\pi\)
−0.725689 + 0.688022i \(0.758479\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −278.855 160.997i −0.306097 0.176725i 0.339081 0.940757i \(-0.389884\pi\)
−0.645179 + 0.764032i \(0.723217\pi\)
\(912\) 0 0
\(913\) −796.743 + 460.000i −0.872665 + 0.503834i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −643.988 −0.702277
\(918\) 0 0
\(919\) 1314.00 1.42982 0.714908 0.699219i \(-0.246469\pi\)
0.714908 + 0.699219i \(0.246469\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −321.994 557.710i −0.348856 0.604236i
\(924\) 0 0
\(925\) −1011.58 + 432.102i −1.09360 + 0.467138i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1301.32 751.319i −1.40078 0.808739i −0.406305 0.913737i \(-0.633183\pi\)
−0.994472 + 0.104998i \(0.966516\pi\)
\(930\) 0 0
\(931\) −45.0000 77.9423i −0.0483351 0.0837189i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1252.20 626.099i 1.33925 0.669625i
\(936\) 0 0
\(937\) 1288.00i 1.37460i −0.726374 0.687300i \(-0.758796\pi\)
0.726374 0.687300i \(-0.241204\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −317.585 + 183.358i −0.337497 + 0.194854i −0.659165 0.751999i \(-0.729090\pi\)
0.321668 + 0.946853i \(0.395757\pi\)
\(942\) 0 0
\(943\) 69.2820 + 40.0000i 0.0734698 + 0.0424178i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 838.525 1452.37i 0.885455 1.53365i 0.0402628 0.999189i \(-0.487180\pi\)
0.845192 0.534463i \(-0.179486\pi\)
\(948\) 0 0
\(949\) 336.000 + 581.969i 0.354057 + 0.613245i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 827.345 0.868148 0.434074 0.900877i \(-0.357076\pi\)
0.434074 + 0.900877i \(0.357076\pi\)
\(954\) 0 0
\(955\) −960.000 + 480.000i −1.00524 + 0.502618i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1208.37 697.653i 1.26003 0.727480i
\(960\) 0 0
\(961\) −97.5000 + 168.875i −0.101457 + 0.175728i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.58644 159.713i 0.00993414 0.165505i
\(966\) 0 0
\(967\) 464.190 268.000i 0.480031 0.277146i −0.240399 0.970674i \(-0.577278\pi\)
0.720429 + 0.693528i \(0.243945\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 134.164i 0.138171i −0.997611 0.0690855i \(-0.977992\pi\)
0.997611 0.0690855i \(-0.0220081\pi\)
\(972\) 0 0
\(973\) 944.000i 0.970195i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 87.2067 + 151.046i 0.0892596 + 0.154602i 0.907198 0.420703i \(-0.138217\pi\)
−0.817939 + 0.575305i \(0.804883\pi\)
\(978\) 0 0
\(979\) 80.0000 138.564i 0.0817160 0.141536i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 927.968 1607.29i 0.944016 1.63508i 0.186308 0.982491i \(-0.440348\pi\)
0.757709 0.652593i \(-0.226319\pi\)
\(984\) 0 0
\(985\) 12.3205 18.6603i 0.0125081 0.0189444i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 125.220i 0.126613i
\(990\) 0 0
\(991\) −1646.00 −1.66095 −0.830474 0.557057i \(-0.811930\pi\)
−0.830474 + 0.557057i \(0.811930\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 314.064 475.672i 0.315642 0.478062i
\(996\) 0 0
\(997\) 1008.05 + 582.000i 1.01109 + 0.583751i 0.911510 0.411279i \(-0.134918\pi\)
0.0995771 + 0.995030i \(0.468251\pi\)
\(998\) 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 1620.3.t.b.269.3 8
3.2 odd 2 inner 1620.3.t.b.269.2 8
5.4 even 2 inner 1620.3.t.b.269.4 8
9.2 odd 6 60.3.b.a.29.1 4
9.4 even 3 inner 1620.3.t.b.1349.1 8
9.5 odd 6 inner 1620.3.t.b.1349.4 8
9.7 even 3 60.3.b.a.29.3 yes 4
15.14 odd 2 inner 1620.3.t.b.269.1 8
36.7 odd 6 240.3.c.d.209.2 4
36.11 even 6 240.3.c.d.209.4 4
45.2 even 12 300.3.g.e.101.2 2
45.4 even 6 inner 1620.3.t.b.1349.2 8
45.7 odd 12 300.3.g.e.101.1 2
45.14 odd 6 inner 1620.3.t.b.1349.3 8
45.29 odd 6 60.3.b.a.29.4 yes 4
45.34 even 6 60.3.b.a.29.2 yes 4
45.38 even 12 300.3.g.h.101.1 2
45.43 odd 12 300.3.g.h.101.2 2
72.11 even 6 960.3.c.g.449.1 4
72.29 odd 6 960.3.c.h.449.4 4
72.43 odd 6 960.3.c.g.449.3 4
72.61 even 6 960.3.c.h.449.2 4
180.7 even 12 1200.3.l.q.401.2 2
180.43 even 12 1200.3.l.h.401.1 2
180.47 odd 12 1200.3.l.q.401.1 2
180.79 odd 6 240.3.c.d.209.3 4
180.83 odd 12 1200.3.l.h.401.2 2
180.119 even 6 240.3.c.d.209.1 4
360.29 odd 6 960.3.c.h.449.1 4
360.259 odd 6 960.3.c.g.449.2 4
360.299 even 6 960.3.c.g.449.4 4
360.349 even 6 960.3.c.h.449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.3.b.a.29.1 4 9.2 odd 6
60.3.b.a.29.2 yes 4 45.34 even 6
60.3.b.a.29.3 yes 4 9.7 even 3
60.3.b.a.29.4 yes 4 45.29 odd 6
240.3.c.d.209.1 4 180.119 even 6
240.3.c.d.209.2 4 36.7 odd 6
240.3.c.d.209.3 4 180.79 odd 6
240.3.c.d.209.4 4 36.11 even 6
300.3.g.e.101.1 2 45.7 odd 12
300.3.g.e.101.2 2 45.2 even 12
300.3.g.h.101.1 2 45.38 even 12
300.3.g.h.101.2 2 45.43 odd 12
960.3.c.g.449.1 4 72.11 even 6
960.3.c.g.449.2 4 360.259 odd 6
960.3.c.g.449.3 4 72.43 odd 6
960.3.c.g.449.4 4 360.299 even 6
960.3.c.h.449.1 4 360.29 odd 6
960.3.c.h.449.2 4 72.61 even 6
960.3.c.h.449.3 4 360.349 even 6
960.3.c.h.449.4 4 72.29 odd 6
1200.3.l.h.401.1 2 180.43 even 12
1200.3.l.h.401.2 2 180.83 odd 12
1200.3.l.q.401.1 2 180.47 odd 12
1200.3.l.q.401.2 2 180.7 even 12
1620.3.t.b.269.1 8 15.14 odd 2 inner
1620.3.t.b.269.2 8 3.2 odd 2 inner
1620.3.t.b.269.3 8 1.1 even 1 trivial
1620.3.t.b.269.4 8 5.4 even 2 inner
1620.3.t.b.1349.1 8 9.4 even 3 inner
1620.3.t.b.1349.2 8 45.4 even 6 inner
1620.3.t.b.1349.3 8 45.14 odd 6 inner
1620.3.t.b.1349.4 8 9.5 odd 6 inner