Properties

Label 1620.4.i.m.541.2
Level $1620$
Weight $4$
Character 1620.541
Analytic conductor $95.583$
Analytic rank $0$
Dimension $4$
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] = [1620,4,Mod(541,1620)]
 
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, 4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (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: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 11x^{2} + 10x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 541.2
Root \(1.85078 - 3.20565i\) of defining polynomial
Character \(\chi\) \(=\) 1620.541
Dual form 1620.4.i.m.1081.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.50000 - 4.33013i) q^{5} +(1.55234 - 2.68874i) q^{7} +O(q^{10})\) \(q+(-2.50000 - 4.33013i) q^{5} +(1.55234 - 2.68874i) q^{7} +(20.2617 - 35.0943i) q^{11} +(16.7094 + 28.9415i) q^{13} +70.5234 q^{17} +146.361 q^{19} +(-50.2617 - 87.0558i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-50.2617 + 87.0558i) q^{29} +(12.9477 + 22.4260i) q^{31} -15.5234 q^{35} -218.570 q^{37} +(46.5703 + 80.6621i) q^{41} +(-8.63360 + 14.9538i) q^{43} +(-206.309 + 357.337i) q^{47} +(166.680 + 288.699i) q^{49} +471.047 q^{53} -202.617 q^{55} +(-321.047 - 556.069i) q^{59} +(-245.170 + 424.646i) q^{61} +(83.5469 - 144.707i) q^{65} +(-218.798 - 378.969i) q^{67} +1113.14 q^{71} +222.570 q^{73} +(-62.9063 - 108.957i) q^{77} +(201.641 - 349.252i) q^{79} +(330.234 - 571.983i) q^{83} +(-176.309 - 305.375i) q^{85} +173.719 q^{89} +103.755 q^{91} +(-365.902 - 633.761i) q^{95} +(295.018 - 510.986i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{5} - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{5} - 13 q^{7} - 15 q^{11} - 10 q^{13} + 90 q^{17} + 86 q^{19} - 105 q^{23} - 50 q^{25} - 105 q^{29} + 71 q^{31} + 130 q^{35} - 298 q^{37} - 390 q^{41} - 169 q^{43} - 345 q^{47} + 417 q^{49} + 1500 q^{53} + 150 q^{55} - 900 q^{59} - q^{61} - 50 q^{65} + 335 q^{67} + 3300 q^{71} + 314 q^{73} - 1020 q^{77} - 346 q^{79} - 600 q^{83} - 225 q^{85} + 3000 q^{89} + 1606 q^{91} - 215 q^{95} + 623 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −2.50000 4.33013i −0.223607 0.387298i
\(6\) 0 0
\(7\) 1.55234 2.68874i 0.0838187 0.145178i −0.821069 0.570830i \(-0.806622\pi\)
0.904887 + 0.425651i \(0.139955\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 20.2617 35.0943i 0.555376 0.961940i −0.442498 0.896769i \(-0.645908\pi\)
0.997874 0.0651702i \(-0.0207590\pi\)
\(12\) 0 0
\(13\) 16.7094 + 28.9415i 0.356488 + 0.617456i 0.987371 0.158422i \(-0.0506407\pi\)
−0.630883 + 0.775878i \(0.717307\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 70.5234 1.00614 0.503072 0.864245i \(-0.332203\pi\)
0.503072 + 0.864245i \(0.332203\pi\)
\(18\) 0 0
\(19\) 146.361 1.76724 0.883618 0.468208i \(-0.155100\pi\)
0.883618 + 0.468208i \(0.155100\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −50.2617 87.0558i −0.455665 0.789235i 0.543061 0.839693i \(-0.317265\pi\)
−0.998726 + 0.0504583i \(0.983932\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −50.2617 + 87.0558i −0.321840 + 0.557444i −0.980868 0.194675i \(-0.937635\pi\)
0.659028 + 0.752119i \(0.270968\pi\)
\(30\) 0 0
\(31\) 12.9477 + 22.4260i 0.0750151 + 0.129930i 0.901093 0.433626i \(-0.142766\pi\)
−0.826078 + 0.563556i \(0.809433\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −15.5234 −0.0749697
\(36\) 0 0
\(37\) −218.570 −0.971155 −0.485578 0.874194i \(-0.661391\pi\)
−0.485578 + 0.874194i \(0.661391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 46.5703 + 80.6621i 0.177392 + 0.307251i 0.940986 0.338444i \(-0.109901\pi\)
−0.763595 + 0.645696i \(0.776567\pi\)
\(42\) 0 0
\(43\) −8.63360 + 14.9538i −0.0306189 + 0.0530334i −0.880929 0.473249i \(-0.843081\pi\)
0.850310 + 0.526282i \(0.176414\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −206.309 + 357.337i −0.640281 + 1.10900i 0.345089 + 0.938570i \(0.387849\pi\)
−0.985370 + 0.170429i \(0.945485\pi\)
\(48\) 0 0
\(49\) 166.680 + 288.699i 0.485949 + 0.841688i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 471.047 1.22082 0.610408 0.792087i \(-0.291005\pi\)
0.610408 + 0.792087i \(0.291005\pi\)
\(54\) 0 0
\(55\) −202.617 −0.496743
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −321.047 556.069i −0.708419 1.22702i −0.965443 0.260613i \(-0.916075\pi\)
0.257024 0.966405i \(-0.417258\pi\)
\(60\) 0 0
\(61\) −245.170 + 424.646i −0.514602 + 0.891317i 0.485254 + 0.874373i \(0.338727\pi\)
−0.999856 + 0.0169441i \(0.994606\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 83.5469 144.707i 0.159426 0.276135i
\(66\) 0 0
\(67\) −218.798 378.969i −0.398961 0.691021i 0.594637 0.803994i \(-0.297296\pi\)
−0.993598 + 0.112974i \(0.963962\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1113.14 1.86064 0.930320 0.366748i \(-0.119529\pi\)
0.930320 + 0.366748i \(0.119529\pi\)
\(72\) 0 0
\(73\) 222.570 0.356848 0.178424 0.983954i \(-0.442900\pi\)
0.178424 + 0.983954i \(0.442900\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −62.9063 108.957i −0.0931018 0.161257i
\(78\) 0 0
\(79\) 201.641 349.252i 0.287169 0.497391i −0.685964 0.727636i \(-0.740619\pi\)
0.973133 + 0.230244i \(0.0739526\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 330.234 571.983i 0.436722 0.756425i −0.560712 0.828011i \(-0.689473\pi\)
0.997434 + 0.0715859i \(0.0228060\pi\)
\(84\) 0 0
\(85\) −176.309 305.375i −0.224981 0.389678i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 173.719 0.206901 0.103450 0.994635i \(-0.467012\pi\)
0.103450 + 0.994635i \(0.467012\pi\)
\(90\) 0 0
\(91\) 103.755 0.119521
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −365.902 633.761i −0.395166 0.684448i
\(96\) 0 0
\(97\) 295.018 510.986i 0.308810 0.534874i −0.669293 0.742999i \(-0.733403\pi\)
0.978102 + 0.208125i \(0.0667361\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −482.590 + 835.870i −0.475440 + 0.823487i −0.999604 0.0281306i \(-0.991045\pi\)
0.524164 + 0.851617i \(0.324378\pi\)
\(102\) 0 0
\(103\) 320.996 + 555.981i 0.307075 + 0.531869i 0.977721 0.209908i \(-0.0673166\pi\)
−0.670647 + 0.741777i \(0.733983\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1145.81 1.03523 0.517616 0.855613i \(-0.326819\pi\)
0.517616 + 0.855613i \(0.326819\pi\)
\(108\) 0 0
\(109\) 165.978 0.145852 0.0729258 0.997337i \(-0.476766\pi\)
0.0729258 + 0.997337i \(0.476766\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −868.113 1503.62i −0.722701 1.25175i −0.959913 0.280297i \(-0.909567\pi\)
0.237212 0.971458i \(-0.423766\pi\)
\(114\) 0 0
\(115\) −251.309 + 435.279i −0.203780 + 0.352956i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 109.477 189.619i 0.0843336 0.146070i
\(120\) 0 0
\(121\) −155.574 269.463i −0.116885 0.202451i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 29.4218 0.0205572 0.0102786 0.999947i \(-0.496728\pi\)
0.0102786 + 0.999947i \(0.496728\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 29.4492 + 51.0075i 0.0196411 + 0.0340194i 0.875679 0.482894i \(-0.160414\pi\)
−0.856038 + 0.516913i \(0.827081\pi\)
\(132\) 0 0
\(133\) 227.202 393.526i 0.148127 0.256564i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1188.14 2057.92i 0.740947 1.28336i −0.211118 0.977461i \(-0.567710\pi\)
0.952065 0.305897i \(-0.0989562\pi\)
\(138\) 0 0
\(139\) −512.137 887.047i −0.312510 0.541283i 0.666395 0.745599i \(-0.267836\pi\)
−0.978905 + 0.204316i \(0.934503\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1354.24 0.791940
\(144\) 0 0
\(145\) 502.617 0.287863
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 590.496 + 1022.77i 0.324667 + 0.562339i 0.981445 0.191745i \(-0.0614146\pi\)
−0.656778 + 0.754084i \(0.728081\pi\)
\(150\) 0 0
\(151\) −648.756 + 1123.68i −0.349636 + 0.605587i −0.986185 0.165649i \(-0.947028\pi\)
0.636549 + 0.771236i \(0.280361\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 64.7383 112.130i 0.0335478 0.0581064i
\(156\) 0 0
\(157\) 1493.99 + 2587.66i 0.759447 + 1.31540i 0.943133 + 0.332415i \(0.107864\pi\)
−0.183687 + 0.982985i \(0.558803\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −312.094 −0.152773
\(162\) 0 0
\(163\) −1273.44 −0.611924 −0.305962 0.952044i \(-0.598978\pi\)
−0.305962 + 0.952044i \(0.598978\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 72.0937 + 124.870i 0.0334059 + 0.0578607i 0.882245 0.470791i \(-0.156031\pi\)
−0.848839 + 0.528651i \(0.822698\pi\)
\(168\) 0 0
\(169\) 540.094 935.470i 0.245832 0.425794i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1699.48 2943.58i 0.746872 1.29362i −0.202443 0.979294i \(-0.564888\pi\)
0.949315 0.314326i \(-0.101778\pi\)
\(174\) 0 0
\(175\) 38.8086 + 67.2184i 0.0167637 + 0.0290356i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3994.08 1.66777 0.833887 0.551936i \(-0.186111\pi\)
0.833887 + 0.551936i \(0.186111\pi\)
\(180\) 0 0
\(181\) 2244.20 0.921601 0.460800 0.887504i \(-0.347562\pi\)
0.460800 + 0.887504i \(0.347562\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 546.426 + 946.437i 0.217157 + 0.376127i
\(186\) 0 0
\(187\) 1428.93 2474.97i 0.558788 0.967850i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1846.80 3198.76i 0.699634 1.21180i −0.268959 0.963152i \(-0.586680\pi\)
0.968593 0.248650i \(-0.0799869\pi\)
\(192\) 0 0
\(193\) −2061.05 3569.84i −0.768691 1.33141i −0.938273 0.345896i \(-0.887575\pi\)
0.169582 0.985516i \(-0.445758\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1423.03 0.514654 0.257327 0.966324i \(-0.417158\pi\)
0.257327 + 0.966324i \(0.417158\pi\)
\(198\) 0 0
\(199\) −3962.59 −1.41156 −0.705781 0.708430i \(-0.749404\pi\)
−0.705781 + 0.708430i \(0.749404\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 156.047 + 270.281i 0.0539524 + 0.0934483i
\(204\) 0 0
\(205\) 232.851 403.311i 0.0793320 0.137407i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2965.52 5136.44i 0.981481 1.69998i
\(210\) 0 0
\(211\) −1792.90 3105.40i −0.584969 1.01320i −0.994879 0.101070i \(-0.967773\pi\)
0.409910 0.912126i \(-0.365560\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 86.3360 0.0273863
\(216\) 0 0
\(217\) 80.3968 0.0251507
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1178.40 + 2041.05i 0.358678 + 0.621249i
\(222\) 0 0
\(223\) −1391.28 + 2409.77i −0.417791 + 0.723634i −0.995717 0.0924545i \(-0.970529\pi\)
0.577926 + 0.816089i \(0.303862\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 600.867 1040.73i 0.175687 0.304299i −0.764712 0.644373i \(-0.777119\pi\)
0.940399 + 0.340074i \(0.110452\pi\)
\(228\) 0 0
\(229\) 440.427 + 762.841i 0.127093 + 0.220131i 0.922549 0.385880i \(-0.126102\pi\)
−0.795456 + 0.606011i \(0.792769\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3309.42 −0.930504 −0.465252 0.885178i \(-0.654036\pi\)
−0.465252 + 0.885178i \(0.654036\pi\)
\(234\) 0 0
\(235\) 2063.09 0.572685
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −60.2343 104.329i −0.0163022 0.0282363i 0.857759 0.514052i \(-0.171856\pi\)
−0.874061 + 0.485815i \(0.838523\pi\)
\(240\) 0 0
\(241\) −1277.17 + 2212.13i −0.341369 + 0.591268i −0.984687 0.174331i \(-0.944224\pi\)
0.643318 + 0.765599i \(0.277557\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 833.402 1443.50i 0.217323 0.376414i
\(246\) 0 0
\(247\) 2445.60 + 4235.90i 0.629999 + 1.09119i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1380.63 0.347190 0.173595 0.984817i \(-0.444462\pi\)
0.173595 + 0.984817i \(0.444462\pi\)
\(252\) 0 0
\(253\) −4073.55 −1.01226
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −170.965 296.119i −0.0414960 0.0718733i 0.844531 0.535506i \(-0.179879\pi\)
−0.886027 + 0.463633i \(0.846546\pi\)
\(258\) 0 0
\(259\) −339.296 + 587.678i −0.0814009 + 0.140991i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1061.63 1838.79i 0.248907 0.431120i −0.714316 0.699824i \(-0.753262\pi\)
0.963223 + 0.268704i \(0.0865952\pi\)
\(264\) 0 0
\(265\) −1177.62 2039.69i −0.272983 0.472820i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1939.95 −0.439705 −0.219852 0.975533i \(-0.570558\pi\)
−0.219852 + 0.975533i \(0.570558\pi\)
\(270\) 0 0
\(271\) 8278.84 1.85573 0.927866 0.372913i \(-0.121641\pi\)
0.927866 + 0.372913i \(0.121641\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 506.543 + 877.358i 0.111075 + 0.192388i
\(276\) 0 0
\(277\) −99.1935 + 171.808i −0.0215161 + 0.0372670i −0.876583 0.481251i \(-0.840183\pi\)
0.855067 + 0.518518i \(0.173516\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4438.90 7688.40i 0.942358 1.63221i 0.181400 0.983409i \(-0.441937\pi\)
0.760957 0.648802i \(-0.224730\pi\)
\(282\) 0 0
\(283\) 3795.25 + 6573.56i 0.797187 + 1.38077i 0.921441 + 0.388518i \(0.127013\pi\)
−0.124254 + 0.992250i \(0.539654\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 289.172 0.0594749
\(288\) 0 0
\(289\) 60.5544 0.0123253
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2526.98 + 4376.86i 0.503850 + 0.872693i 0.999990 + 0.00445108i \(0.00141683\pi\)
−0.496140 + 0.868242i \(0.665250\pi\)
\(294\) 0 0
\(295\) −1605.23 + 2780.35i −0.316815 + 0.548739i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1679.68 2909.30i 0.324878 0.562706i
\(300\) 0 0
\(301\) 26.8046 + 46.4270i 0.00513286 + 0.00889038i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2451.70 0.460274
\(306\) 0 0
\(307\) −7370.65 −1.37024 −0.685122 0.728428i \(-0.740251\pi\)
−0.685122 + 0.728428i \(0.740251\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4445.02 7698.99i −0.810462 1.40376i −0.912541 0.408985i \(-0.865883\pi\)
0.102079 0.994776i \(-0.467451\pi\)
\(312\) 0 0
\(313\) 2228.91 3860.59i 0.402510 0.697167i −0.591518 0.806291i \(-0.701471\pi\)
0.994028 + 0.109124i \(0.0348047\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5062.45 8768.43i 0.896958 1.55358i 0.0655954 0.997846i \(-0.479105\pi\)
0.831363 0.555730i \(-0.187561\pi\)
\(318\) 0 0
\(319\) 2036.78 + 3527.80i 0.357485 + 0.619182i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10321.9 1.77809
\(324\) 0 0
\(325\) −835.469 −0.142595
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 640.523 + 1109.42i 0.107335 + 0.185910i
\(330\) 0 0
\(331\) −763.012 + 1321.58i −0.126704 + 0.219457i −0.922398 0.386242i \(-0.873773\pi\)
0.795694 + 0.605699i \(0.207106\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1093.99 + 1894.84i −0.178421 + 0.309034i
\(336\) 0 0
\(337\) 1001.58 + 1734.79i 0.161898 + 0.280415i 0.935549 0.353196i \(-0.114905\pi\)
−0.773652 + 0.633611i \(0.781572\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1049.37 0.166646
\(342\) 0 0
\(343\) 2099.89 0.330564
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1325.41 + 2295.68i 0.205049 + 0.355155i 0.950148 0.311798i \(-0.100931\pi\)
−0.745099 + 0.666953i \(0.767598\pi\)
\(348\) 0 0
\(349\) 2381.29 4124.52i 0.365237 0.632609i −0.623577 0.781762i \(-0.714321\pi\)
0.988814 + 0.149153i \(0.0476546\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1187.19 + 2056.28i −0.179002 + 0.310041i −0.941539 0.336904i \(-0.890620\pi\)
0.762537 + 0.646945i \(0.223954\pi\)
\(354\) 0 0
\(355\) −2782.85 4820.04i −0.416052 0.720623i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3973.36 0.584139 0.292070 0.956397i \(-0.405656\pi\)
0.292070 + 0.956397i \(0.405656\pi\)
\(360\) 0 0
\(361\) 14562.5 2.12313
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −556.426 963.758i −0.0797936 0.138207i
\(366\) 0 0
\(367\) 1717.81 2975.33i 0.244329 0.423190i −0.717614 0.696441i \(-0.754766\pi\)
0.961943 + 0.273251i \(0.0880990\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 731.226 1266.52i 0.102327 0.177236i
\(372\) 0 0
\(373\) −4607.95 7981.20i −0.639653 1.10791i −0.985509 0.169624i \(-0.945745\pi\)
0.345856 0.938288i \(-0.387589\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3359.37 −0.458929
\(378\) 0 0
\(379\) 3064.74 0.415370 0.207685 0.978196i \(-0.433407\pi\)
0.207685 + 0.978196i \(0.433407\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 864.574 + 1497.49i 0.115346 + 0.199786i 0.917918 0.396770i \(-0.129869\pi\)
−0.802572 + 0.596556i \(0.796536\pi\)
\(384\) 0 0
\(385\) −314.531 + 544.784i −0.0416364 + 0.0721163i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2975.10 + 5153.02i −0.387772 + 0.671641i −0.992150 0.125057i \(-0.960089\pi\)
0.604377 + 0.796698i \(0.293422\pi\)
\(390\) 0 0
\(391\) −3544.63 6139.48i −0.458464 0.794084i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2016.41 −0.256852
\(396\) 0 0
\(397\) −13663.9 −1.72739 −0.863694 0.504016i \(-0.831855\pi\)
−0.863694 + 0.504016i \(0.831855\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −925.992 1603.87i −0.115316 0.199734i 0.802590 0.596531i \(-0.203455\pi\)
−0.917906 + 0.396798i \(0.870121\pi\)
\(402\) 0 0
\(403\) −432.694 + 749.449i −0.0534840 + 0.0926370i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4428.61 + 7670.58i −0.539356 + 0.934192i
\(408\) 0 0
\(409\) 6154.37 + 10659.7i 0.744044 + 1.28872i 0.950640 + 0.310295i \(0.100428\pi\)
−0.206597 + 0.978426i \(0.566239\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1993.50 −0.237515
\(414\) 0 0
\(415\) −3302.34 −0.390616
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2643.69 4579.01i −0.308241 0.533888i 0.669737 0.742598i \(-0.266407\pi\)
−0.977978 + 0.208710i \(0.933073\pi\)
\(420\) 0 0
\(421\) −7564.55 + 13102.2i −0.875709 + 1.51677i −0.0197041 + 0.999806i \(0.506272\pi\)
−0.856005 + 0.516967i \(0.827061\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −881.543 + 1526.88i −0.100614 + 0.174269i
\(426\) 0 0
\(427\) 761.174 + 1318.39i 0.0862665 + 0.149418i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15509.7 1.73335 0.866676 0.498872i \(-0.166252\pi\)
0.866676 + 0.498872i \(0.166252\pi\)
\(432\) 0 0
\(433\) −11171.2 −1.23984 −0.619922 0.784663i \(-0.712836\pi\)
−0.619922 + 0.784663i \(0.712836\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7356.35 12741.6i −0.805268 1.39476i
\(438\) 0 0
\(439\) 1667.77 2888.66i 0.181317 0.314051i −0.761012 0.648738i \(-0.775297\pi\)
0.942329 + 0.334687i \(0.108631\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2271.05 + 3933.57i −0.243568 + 0.421872i −0.961728 0.274006i \(-0.911651\pi\)
0.718160 + 0.695878i \(0.244985\pi\)
\(444\) 0 0
\(445\) −434.297 752.225i −0.0462644 0.0801323i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7617.80 −0.800682 −0.400341 0.916366i \(-0.631108\pi\)
−0.400341 + 0.916366i \(0.631108\pi\)
\(450\) 0 0
\(451\) 3774.38 0.394076
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −259.387 449.271i −0.0267258 0.0462905i
\(456\) 0 0
\(457\) 7384.84 12790.9i 0.755904 1.30926i −0.189020 0.981973i \(-0.560531\pi\)
0.944924 0.327291i \(-0.106136\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4492.15 + 7780.63i −0.453840 + 0.786074i −0.998621 0.0525047i \(-0.983280\pi\)
0.544781 + 0.838579i \(0.316613\pi\)
\(462\) 0 0
\(463\) −4982.11 8629.27i −0.500083 0.866169i −1.00000 9.58172e-5i \(-0.999970\pi\)
0.499917 0.866073i \(-0.333364\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15489.1 1.53480 0.767398 0.641171i \(-0.221551\pi\)
0.767398 + 0.641171i \(0.221551\pi\)
\(468\) 0 0
\(469\) −1358.60 −0.133761
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 349.863 + 605.981i 0.0340100 + 0.0589070i
\(474\) 0 0
\(475\) −1829.51 + 3168.81i −0.176724 + 0.306094i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1675.19 2901.52i 0.159795 0.276772i −0.775000 0.631961i \(-0.782250\pi\)
0.934795 + 0.355189i \(0.115583\pi\)
\(480\) 0 0
\(481\) −3652.17 6325.75i −0.346205 0.599645i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2950.18 −0.276208
\(486\) 0 0
\(487\) 5435.64 0.505775 0.252887 0.967496i \(-0.418620\pi\)
0.252887 + 0.967496i \(0.418620\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3967.29 + 6871.55i 0.364646 + 0.631586i 0.988719 0.149780i \(-0.0478567\pi\)
−0.624073 + 0.781366i \(0.714523\pi\)
\(492\) 0 0
\(493\) −3544.63 + 6139.48i −0.323818 + 0.560868i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1727.98 2992.94i 0.155956 0.270124i
\(498\) 0 0
\(499\) −3833.06 6639.05i −0.343870 0.595601i 0.641278 0.767309i \(-0.278405\pi\)
−0.985148 + 0.171708i \(0.945071\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 10069.6 0.892608 0.446304 0.894881i \(-0.352740\pi\)
0.446304 + 0.894881i \(0.352740\pi\)
\(504\) 0 0
\(505\) 4825.90 0.425247
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8845.75 15321.3i −0.770296 1.33419i −0.937401 0.348253i \(-0.886775\pi\)
0.167104 0.985939i \(-0.446558\pi\)
\(510\) 0 0
\(511\) 345.505 598.433i 0.0299105 0.0518065i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1604.98 2779.91i 0.137328 0.237859i
\(516\) 0 0
\(517\) 8360.33 + 14480.5i 0.711193 + 1.23182i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −20972.5 −1.76357 −0.881786 0.471649i \(-0.843659\pi\)
−0.881786 + 0.471649i \(0.843659\pi\)
\(522\) 0 0
\(523\) −18424.0 −1.54039 −0.770196 0.637807i \(-0.779842\pi\)
−0.770196 + 0.637807i \(0.779842\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 913.113 + 1581.56i 0.0754760 + 0.130728i
\(528\) 0 0
\(529\) 1031.02 1785.78i 0.0847390 0.146772i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1556.32 + 2695.63i −0.126476 + 0.219063i
\(534\) 0 0
\(535\) −2864.53 4961.51i −0.231485 0.400944i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13508.9 1.07954
\(540\) 0 0
\(541\) 13367.9 1.06235 0.531174 0.847263i \(-0.321751\pi\)
0.531174 + 0.847263i \(0.321751\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −414.945 718.706i −0.0326134 0.0564881i
\(546\) 0 0
\(547\) 7804.60 13518.0i 0.610056 1.05665i −0.381174 0.924503i \(-0.624480\pi\)
0.991230 0.132145i \(-0.0421864\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7356.35 + 12741.6i −0.568768 + 0.985135i
\(552\) 0 0
\(553\) −626.031 1084.32i −0.0481402 0.0833813i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1379.34 0.104927 0.0524637 0.998623i \(-0.483293\pi\)
0.0524637 + 0.998623i \(0.483293\pi\)
\(558\) 0 0
\(559\) −577.048 −0.0436611
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5237.88 9072.28i −0.392097 0.679131i 0.600629 0.799528i \(-0.294917\pi\)
−0.992726 + 0.120396i \(0.961583\pi\)
\(564\) 0 0
\(565\) −4340.57 + 7518.08i −0.323202 + 0.559802i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10557.5 + 18286.1i −0.777842 + 1.34726i 0.155341 + 0.987861i \(0.450352\pi\)
−0.933183 + 0.359401i \(0.882981\pi\)
\(570\) 0 0
\(571\) 8776.02 + 15200.5i 0.643196 + 1.11405i 0.984715 + 0.174173i \(0.0557253\pi\)
−0.341519 + 0.939875i \(0.610941\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2513.09 0.182266
\(576\) 0 0
\(577\) 22028.2 1.58933 0.794667 0.607046i \(-0.207646\pi\)
0.794667 + 0.607046i \(0.207646\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1025.27 1775.83i −0.0732109 0.126805i
\(582\) 0 0
\(583\) 9544.22 16531.1i 0.678012 1.17435i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11171.2 + 19349.1i −0.785494 + 1.36052i 0.143210 + 0.989692i \(0.454258\pi\)
−0.928704 + 0.370823i \(0.879076\pi\)
\(588\) 0 0
\(589\) 1895.03 + 3282.29i 0.132569 + 0.229617i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −13126.8 −0.909027 −0.454514 0.890740i \(-0.650187\pi\)
−0.454514 + 0.890740i \(0.650187\pi\)
\(594\) 0 0
\(595\) −1094.77 −0.0754303
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11874.1 + 20566.5i 0.809954 + 1.40288i 0.912895 + 0.408194i \(0.133841\pi\)
−0.102942 + 0.994687i \(0.532825\pi\)
\(600\) 0 0
\(601\) −2131.75 + 3692.30i −0.144685 + 0.250602i −0.929255 0.369438i \(-0.879550\pi\)
0.784570 + 0.620040i \(0.212884\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −777.871 + 1347.31i −0.0522727 + 0.0905389i
\(606\) 0 0
\(607\) 9277.26 + 16068.7i 0.620350 + 1.07448i 0.989420 + 0.145076i \(0.0463427\pi\)
−0.369071 + 0.929401i \(0.620324\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13789.1 −0.913010
\(612\) 0 0
\(613\) −3110.17 −0.204924 −0.102462 0.994737i \(-0.532672\pi\)
−0.102462 + 0.994737i \(0.532672\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12074.4 + 20913.5i 0.787841 + 1.36458i 0.927287 + 0.374351i \(0.122135\pi\)
−0.139446 + 0.990230i \(0.544532\pi\)
\(618\) 0 0
\(619\) −2029.62 + 3515.40i −0.131789 + 0.228265i −0.924366 0.381507i \(-0.875405\pi\)
0.792577 + 0.609771i \(0.208739\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 269.671 467.084i 0.0173421 0.0300375i
\(624\) 0 0
\(625\) −312.500 541.266i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −15414.3 −0.977122
\(630\) 0 0
\(631\) 22017.9 1.38909 0.694547 0.719447i \(-0.255605\pi\)
0.694547 + 0.719447i \(0.255605\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −73.5544 127.400i −0.00459672 0.00796176i
\(636\) 0 0
\(637\) −5570.25 + 9647.96i −0.346470 + 0.600104i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5434.38 + 9412.63i −0.334860 + 0.579994i −0.983458 0.181137i \(-0.942022\pi\)
0.648598 + 0.761131i \(0.275356\pi\)
\(642\) 0 0
\(643\) −11644.7 20169.2i −0.714188 1.23701i −0.963272 0.268528i \(-0.913463\pi\)
0.249084 0.968482i \(-0.419870\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −31678.8 −1.92492 −0.962460 0.271424i \(-0.912505\pi\)
−0.962460 + 0.271424i \(0.912505\pi\)
\(648\) 0 0
\(649\) −26019.8 −1.57376
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13544.1 23459.1i −0.811672 1.40586i −0.911693 0.410872i \(-0.865224\pi\)
0.100021 0.994985i \(-0.468109\pi\)
\(654\) 0 0
\(655\) 147.246 255.037i 0.00878377 0.0152139i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2340.94 4054.62i 0.138376 0.239675i −0.788506 0.615027i \(-0.789145\pi\)
0.926882 + 0.375352i \(0.122478\pi\)
\(660\) 0 0
\(661\) −1850.96 3205.95i −0.108917 0.188649i 0.806415 0.591350i \(-0.201405\pi\)
−0.915332 + 0.402701i \(0.868071\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2272.02 −0.132489
\(666\) 0 0
\(667\) 10105.0 0.586605
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9935.11 + 17208.1i 0.571596 + 0.990033i
\(672\) 0 0
\(673\) −14150.1 + 24508.7i −0.810469 + 1.40377i 0.102067 + 0.994778i \(0.467454\pi\)
−0.912536 + 0.408996i \(0.865879\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7197.66 + 12466.7i −0.408609 + 0.707732i −0.994734 0.102489i \(-0.967319\pi\)
0.586125 + 0.810221i \(0.300653\pi\)
\(678\) 0 0
\(679\) −915.938 1586.45i −0.0517680 0.0896648i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −31758.3 −1.77921 −0.889604 0.456733i \(-0.849019\pi\)
−0.889604 + 0.456733i \(0.849019\pi\)
\(684\) 0 0
\(685\) −11881.4 −0.662723
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7870.90 + 13632.8i 0.435207 + 0.753800i
\(690\) 0 0
\(691\) −11613.2 + 20114.6i −0.639343 + 1.10737i 0.346234 + 0.938148i \(0.387460\pi\)
−0.985577 + 0.169227i \(0.945873\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2560.68 + 4435.23i −0.139759 + 0.242069i
\(696\) 0 0
\(697\) 3284.30 + 5688.57i 0.178482 + 0.309139i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14493.9 0.780921 0.390461 0.920620i \(-0.372316\pi\)
0.390461 + 0.920620i \(0.372316\pi\)
\(702\) 0 0
\(703\) −31990.2 −1.71626
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1498.29 + 2595.11i 0.0797015 + 0.138047i
\(708\) 0 0
\(709\) 8502.48 14726.7i 0.450377 0.780076i −0.548032 0.836457i \(-0.684623\pi\)
0.998409 + 0.0563811i \(0.0179562\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1301.54 2254.34i 0.0683635 0.118409i
\(714\) 0 0
\(715\) −3385.61 5864.04i −0.177083 0.306717i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5275.78 −0.273649 −0.136824 0.990595i \(-0.543690\pi\)
−0.136824 + 0.990595i \(0.543690\pi\)
\(720\) 0 0
\(721\) 1993.18 0.102954
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1256.54 2176.40i −0.0643680 0.111489i
\(726\) 0 0
\(727\) 2382.68 4126.92i 0.121552 0.210535i −0.798828 0.601560i \(-0.794546\pi\)
0.920380 + 0.391025i \(0.127879\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −608.871 + 1054.60i −0.0308070 + 0.0533593i
\(732\) 0 0
\(733\) −11709.3 20281.1i −0.590031 1.02196i −0.994228 0.107292i \(-0.965782\pi\)
0.404197 0.914672i \(-0.367551\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −17732.9 −0.886294
\(738\) 0 0
\(739\) 11307.4 0.562855 0.281427 0.959583i \(-0.409192\pi\)
0.281427 + 0.959583i \(0.409192\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9039.50 + 15656.9i 0.446335 + 0.773076i 0.998144 0.0608950i \(-0.0193955\pi\)
−0.551809 + 0.833971i \(0.686062\pi\)
\(744\) 0 0
\(745\) 2952.48 5113.85i 0.145195 0.251486i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1778.69 3080.79i 0.0867718 0.150293i
\(750\) 0 0
\(751\) 13250.0 + 22949.6i 0.643805 + 1.11510i 0.984576 + 0.174957i \(0.0559786\pi\)
−0.340771 + 0.940146i \(0.610688\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6487.56 0.312724
\(756\) 0 0
\(757\) −9631.16 −0.462418 −0.231209 0.972904i \(-0.574268\pi\)
−0.231209 + 0.972904i \(0.574268\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4856.30 8411.35i −0.231328 0.400672i 0.726871 0.686774i \(-0.240974\pi\)
−0.958199 + 0.286102i \(0.907640\pi\)
\(762\) 0 0
\(763\) 257.655 446.271i 0.0122251 0.0211745i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10729.0 18583.1i 0.505086 0.874835i
\(768\) 0 0
\(769\) 1482.08 + 2567.04i 0.0694995 + 0.120377i 0.898681 0.438603i \(-0.144526\pi\)
−0.829182 + 0.558979i \(0.811193\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −30022.1 −1.39692 −0.698461 0.715648i \(-0.746131\pi\)
−0.698461 + 0.715648i \(0.746131\pi\)
\(774\) 0 0
\(775\) −647.383 −0.0300060
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6816.07 + 11805.8i 0.313493 + 0.542986i
\(780\) 0 0
\(781\) 22554.1 39064.9i 1.03336 1.78982i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7469.93 12938.3i 0.339635 0.588265i
\(786\) 0 0
\(787\) 45.3685 + 78.5805i 0.00205491 + 0.00355920i 0.867051 0.498219i \(-0.166013\pi\)
−0.864996 + 0.501779i \(0.832679\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5390.44 −0.242303
\(792\) 0 0
\(793\) −16386.5 −0.733798
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7210.69 12489.3i −0.320471 0.555073i 0.660114 0.751165i \(-0.270508\pi\)
−0.980585 + 0.196093i \(0.937175\pi\)
\(798\) 0 0
\(799\) −14549.6 + 25200.6i −0.644215 + 1.11581i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4509.66 7810.95i 0.198185 0.343266i
\(804\) 0 0
\(805\) 780.234 + 1351.41i 0.0341611 + 0.0591687i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −28357.7 −1.23239 −0.616195 0.787594i \(-0.711327\pi\)
−0.616195 + 0.787594i \(0.711327\pi\)
\(810\) 0 0
\(811\) 28791.5 1.24662 0.623308 0.781977i \(-0.285788\pi\)
0.623308 + 0.781977i \(0.285788\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3183.60 + 5514.16i 0.136830 + 0.236997i
\(816\) 0 0
\(817\) −1263.62 + 2188.66i −0.0541108 + 0.0937226i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3512.84 + 6084.41i −0.149329 + 0.258645i −0.930980 0.365071i \(-0.881045\pi\)
0.781651 + 0.623716i \(0.214378\pi\)
\(822\) 0 0
\(823\) 2386.74 + 4133.95i 0.101089 + 0.175092i 0.912134 0.409893i \(-0.134434\pi\)
−0.811044 + 0.584985i \(0.801101\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −13428.7 −0.564645 −0.282323 0.959319i \(-0.591105\pi\)
−0.282323 + 0.959319i \(0.591105\pi\)
\(828\) 0 0
\(829\) 32961.5 1.38094 0.690470 0.723361i \(-0.257404\pi\)
0.690470 + 0.723361i \(0.257404\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 11754.9 + 20360.0i 0.488934 + 0.846859i
\(834\) 0 0
\(835\) 360.469 624.350i 0.0149396 0.0258761i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16466.0 + 28520.0i −0.677558 + 1.17356i 0.298157 + 0.954517i \(0.403628\pi\)
−0.975714 + 0.219047i \(0.929705\pi\)
\(840\) 0 0
\(841\) 7142.02 + 12370.3i 0.292838 + 0.507210i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5400.94 −0.219879
\(846\) 0 0
\(847\) −966.019 −0.0391887
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10985.7 + 19027.8i 0.442521 + 0.766469i
\(852\) 0 0
\(853\) 267.138 462.697i 0.0107229 0.0185726i −0.860614 0.509258i \(-0.829920\pi\)
0.871337 + 0.490685i \(0.163253\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15515.5 + 26873.7i −0.618437 + 1.07116i 0.371334 + 0.928499i \(0.378900\pi\)
−0.989771 + 0.142665i \(0.954433\pi\)
\(858\) 0 0
\(859\) 6495.49 + 11250.5i 0.258001 + 0.446871i 0.965706 0.259637i \(-0.0836028\pi\)
−0.707705 + 0.706508i \(0.750269\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16721.3 0.659558 0.329779 0.944058i \(-0.393026\pi\)
0.329779 + 0.944058i \(0.393026\pi\)
\(864\) 0 0
\(865\) −16994.8 −0.668022
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8171.17 14152.9i −0.318973 0.552478i
\(870\) 0 0
\(871\) 7311.94 12664.7i 0.284450 0.492681i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 194.043 336.092i 0.00749697 0.0129851i
\(876\) 0 0
\(877\) −12192.7 21118.4i −0.469462 0.813133i 0.529928 0.848043i \(-0.322219\pi\)
−0.999390 + 0.0349099i \(0.988886\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −50589.8 −1.93464 −0.967318 0.253565i \(-0.918397\pi\)
−0.967318 + 0.253565i \(0.918397\pi\)
\(882\) 0 0
\(883\) −37437.1 −1.42679 −0.713396 0.700761i \(-0.752844\pi\)
−0.713396 + 0.700761i \(0.752844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19565.6 + 33888.6i 0.740640 + 1.28283i 0.952204 + 0.305462i \(0.0988108\pi\)
−0.211565 + 0.977364i \(0.567856\pi\)
\(888\) 0 0
\(889\) 45.6727 79.1074i 0.00172307 0.00298445i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −30195.5 + 52300.2i −1.13153 + 1.95986i
\(894\) 0 0
\(895\) −9985.19 17294.9i −0.372925 0.645926i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2603.09 −0.0965715
\(900\) 0 0
\(901\) 33219.8 1.22832
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5610.49 9717.65i −0.206076 0.356934i
\(906\) 0 0
\(907\) −7809.06 + 13525.7i −0.285883 + 0.495163i −0.972823 0.231551i \(-0.925620\pi\)
0.686940 + 0.726714i \(0.258953\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4316.53 7476.45i 0.156985 0.271905i −0.776795 0.629753i \(-0.783156\pi\)
0.933780 + 0.357848i \(0.116489\pi\)
\(912\) 0 0
\(913\) −13382.2 23178.7i −0.485090 0.840200i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 182.861 0.00658517
\(918\) 0 0
\(919\) 49659.5 1.78250 0.891249 0.453514i \(-0.149830\pi\)
0.891249 + 0.453514i \(0.149830\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18599.9 + 32215.9i 0.663296 + 1.14886i
\(924\) 0 0
\(925\) 2732.13 4732.19i 0.0971155 0.168209i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 710.648 1230.88i 0.0250975 0.0434702i −0.853204 0.521578i \(-0.825344\pi\)
0.878301 + 0.478107i \(0.158677\pi\)
\(930\) 0 0
\(931\) 24395.5 + 42254.3i 0.858787 + 1.48746i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −14289.3 −0.499795
\(936\) 0 0
\(937\) 14023.3 0.488922 0.244461 0.969659i \(-0.421389\pi\)
0.244461 + 0.969659i \(0.421389\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −6742.00 11677.5i −0.233563 0.404543i 0.725291 0.688442i \(-0.241705\pi\)
−0.958854 + 0.283899i \(0.908372\pi\)
\(942\) 0 0
\(943\) 4681.41 8108.43i 0.161662 0.280007i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13539.1 23450.4i 0.464584 0.804682i −0.534599 0.845106i \(-0.679537\pi\)
0.999183 + 0.0404235i \(0.0128707\pi\)
\(948\) 0 0
\(949\) 3719.01 + 6441.51i 0.127212 + 0.220338i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 31779.6 1.08021 0.540107 0.841597i \(-0.318384\pi\)
0.540107 + 0.841597i \(0.318384\pi\)
\(954\) 0 0
\(955\) −18468.0 −0.625772
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3688.80 6389.20i −0.124210 0.215139i
\(960\) 0 0
\(961\) 14560.2 25219.0i 0.488745 0.846532i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −10305.2 + 17849.2i −0.343769 + 0.595425i
\(966\) 0 0
\(967\) −8567.00 14838.5i −0.284898 0.493457i 0.687687 0.726008i \(-0.258626\pi\)
−0.972584 + 0.232550i \(0.925293\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3605.56 0.119164 0.0595818 0.998223i \(-0.481023\pi\)
0.0595818 + 0.998223i \(0.481023\pi\)
\(972\) 0 0
\(973\) −3180.05 −0.104777
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −21533.5 37297.1i −0.705136 1.22133i −0.966642 0.256130i \(-0.917552\pi\)
0.261506 0.965202i \(-0.415781\pi\)
\(978\) 0 0
\(979\) 3519.84 6096.54i 0.114908 0.199026i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6802.30 11781.9i 0.220712 0.382284i −0.734313 0.678812i \(-0.762495\pi\)
0.955024 + 0.296528i \(0.0958286\pi\)
\(984\) 0 0
\(985\) −3557.58 6161.90i −0.115080 0.199324i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1735.76 0.0558078
\(990\) 0 0
\(991\) −17228.8 −0.552260 −0.276130 0.961120i \(-0.589052\pi\)
−0.276130 + 0.961120i \(0.589052\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9906.48 + 17158.5i 0.315635 + 0.546696i
\(996\) 0 0
\(997\) −24359.0 + 42191.0i −0.773778 + 1.34022i 0.161701 + 0.986840i \(0.448302\pi\)
−0.935479 + 0.353383i \(0.885031\pi\)
\(998\) 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 1620.4.i.m.541.2 4
3.2 odd 2 1620.4.i.p.541.2 4
9.2 odd 6 540.4.a.g.1.1 2
9.4 even 3 inner 1620.4.i.m.1081.2 4
9.5 odd 6 1620.4.i.p.1081.2 4
9.7 even 3 540.4.a.j.1.1 yes 2
36.7 odd 6 2160.4.a.z.1.2 2
36.11 even 6 2160.4.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
540.4.a.g.1.1 2 9.2 odd 6
540.4.a.j.1.1 yes 2 9.7 even 3
1620.4.i.m.541.2 4 1.1 even 1 trivial
1620.4.i.m.1081.2 4 9.4 even 3 inner
1620.4.i.p.541.2 4 3.2 odd 2
1620.4.i.p.1081.2 4 9.5 odd 6
2160.4.a.u.1.2 2 36.11 even 6
2160.4.a.z.1.2 2 36.7 odd 6