Properties

Label 252.4.x.a.209.15
Level $252$
Weight $4$
Character 252.209
Analytic conductor $14.868$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,4,Mod(41,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.41");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.x (of order \(6\), degree \(2\), 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: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 209.15
Character \(\chi\) \(=\) 252.209
Dual form 252.4.x.a.41.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.494604 - 5.17256i) q^{3} +(-2.99997 - 5.19610i) q^{5} +(16.2235 - 8.93305i) q^{7} +(-26.5107 - 5.11673i) q^{9} +O(q^{10})\) \(q+(0.494604 - 5.17256i) q^{3} +(-2.99997 - 5.19610i) q^{5} +(16.2235 - 8.93305i) q^{7} +(-26.5107 - 5.11673i) q^{9} +(-39.3810 - 22.7366i) q^{11} +(22.7627 - 13.1421i) q^{13} +(-28.3609 + 12.9475i) q^{15} +19.7249 q^{17} +27.9162i q^{19} +(-38.1826 - 88.3351i) q^{21} +(-60.3466 + 34.8411i) q^{23} +(44.5004 - 77.0769i) q^{25} +(-39.5789 + 134.598i) q^{27} +(-119.031 - 68.7227i) q^{29} +(-138.202 + 79.7908i) q^{31} +(-137.085 + 192.455i) q^{33} +(-95.0869 - 57.4998i) q^{35} -287.829 q^{37} +(-56.7196 - 124.242i) q^{39} +(20.4288 + 35.3837i) q^{41} +(55.4158 - 95.9830i) q^{43} +(52.9443 + 153.102i) q^{45} +(-109.666 + 189.948i) q^{47} +(183.401 - 289.850i) q^{49} +(9.75601 - 102.028i) q^{51} +209.770i q^{53} +272.837i q^{55} +(144.398 + 13.8074i) q^{57} +(-413.880 - 716.860i) q^{59} +(594.209 + 343.066i) q^{61} +(-475.804 + 153.811i) q^{63} +(-136.575 - 78.8515i) q^{65} +(-171.449 - 296.958i) q^{67} +(150.370 + 329.379i) q^{69} -387.476i q^{71} -220.721i q^{73} +(-376.675 - 268.303i) q^{75} +(-842.004 - 17.0742i) q^{77} +(242.301 - 419.677i) q^{79} +(676.638 + 271.297i) q^{81} +(354.323 - 613.706i) q^{83} +(-59.1741 - 102.493i) q^{85} +(-414.346 + 581.706i) q^{87} +140.929 q^{89} +(251.891 - 416.550i) q^{91} +(344.368 + 754.321i) q^{93} +(145.055 - 83.7476i) q^{95} +(1304.31 + 753.041i) q^{97} +(927.683 + 804.267i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 6 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 6 q^{7} + 60 q^{9} - 12 q^{11} + 192 q^{15} - 72 q^{21} - 408 q^{23} - 600 q^{25} - 84 q^{29} + 336 q^{37} + 36 q^{39} + 84 q^{43} + 318 q^{49} - 1812 q^{51} - 852 q^{57} - 564 q^{63} + 2964 q^{65} - 588 q^{67} + 2400 q^{77} + 204 q^{79} + 1980 q^{81} - 360 q^{85} - 1080 q^{91} + 2496 q^{93} + 300 q^{95} - 4968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.494604 5.17256i 0.0951865 0.995459i
\(4\) 0 0
\(5\) −2.99997 5.19610i −0.268325 0.464753i 0.700104 0.714041i \(-0.253137\pi\)
−0.968430 + 0.249288i \(0.919804\pi\)
\(6\) 0 0
\(7\) 16.2235 8.93305i 0.875984 0.482340i
\(8\) 0 0
\(9\) −26.5107 5.11673i −0.981879 0.189509i
\(10\) 0 0
\(11\) −39.3810 22.7366i −1.07944 0.623214i −0.148694 0.988883i \(-0.547507\pi\)
−0.930745 + 0.365669i \(0.880840\pi\)
\(12\) 0 0
\(13\) 22.7627 13.1421i 0.485634 0.280381i −0.237127 0.971479i \(-0.576206\pi\)
0.722761 + 0.691098i \(0.242873\pi\)
\(14\) 0 0
\(15\) −28.3609 + 12.9475i −0.488184 + 0.222869i
\(16\) 0 0
\(17\) 19.7249 0.281411 0.140706 0.990051i \(-0.455063\pi\)
0.140706 + 0.990051i \(0.455063\pi\)
\(18\) 0 0
\(19\) 27.9162i 0.337074i 0.985695 + 0.168537i \(0.0539043\pi\)
−0.985695 + 0.168537i \(0.946096\pi\)
\(20\) 0 0
\(21\) −38.1826 88.3351i −0.396768 0.917919i
\(22\) 0 0
\(23\) −60.3466 + 34.8411i −0.547093 + 0.315864i −0.747949 0.663757i \(-0.768961\pi\)
0.200856 + 0.979621i \(0.435628\pi\)
\(24\) 0 0
\(25\) 44.5004 77.0769i 0.356003 0.616615i
\(26\) 0 0
\(27\) −39.5789 + 134.598i −0.282110 + 0.959382i
\(28\) 0 0
\(29\) −119.031 68.7227i −0.762191 0.440051i 0.0678907 0.997693i \(-0.478373\pi\)
−0.830082 + 0.557641i \(0.811706\pi\)
\(30\) 0 0
\(31\) −138.202 + 79.7908i −0.800702 + 0.462286i −0.843717 0.536789i \(-0.819637\pi\)
0.0430146 + 0.999074i \(0.486304\pi\)
\(32\) 0 0
\(33\) −137.085 + 192.455i −0.723133 + 1.01522i
\(34\) 0 0
\(35\) −95.0869 57.4998i −0.459218 0.277693i
\(36\) 0 0
\(37\) −287.829 −1.27889 −0.639443 0.768839i \(-0.720835\pi\)
−0.639443 + 0.768839i \(0.720835\pi\)
\(38\) 0 0
\(39\) −56.7196 124.242i −0.232882 0.510117i
\(40\) 0 0
\(41\) 20.4288 + 35.3837i 0.0778157 + 0.134781i 0.902307 0.431093i \(-0.141872\pi\)
−0.824492 + 0.565874i \(0.808539\pi\)
\(42\) 0 0
\(43\) 55.4158 95.9830i 0.196531 0.340402i −0.750870 0.660450i \(-0.770366\pi\)
0.947401 + 0.320048i \(0.103699\pi\)
\(44\) 0 0
\(45\) 52.9443 + 153.102i 0.175388 + 0.507181i
\(46\) 0 0
\(47\) −109.666 + 189.948i −0.340351 + 0.589504i −0.984498 0.175397i \(-0.943879\pi\)
0.644147 + 0.764902i \(0.277212\pi\)
\(48\) 0 0
\(49\) 183.401 289.850i 0.534697 0.845044i
\(50\) 0 0
\(51\) 9.75601 102.028i 0.0267866 0.280133i
\(52\) 0 0
\(53\) 209.770i 0.543664i 0.962345 + 0.271832i \(0.0876295\pi\)
−0.962345 + 0.271832i \(0.912371\pi\)
\(54\) 0 0
\(55\) 272.837i 0.668897i
\(56\) 0 0
\(57\) 144.398 + 13.8074i 0.335544 + 0.0320849i
\(58\) 0 0
\(59\) −413.880 716.860i −0.913263 1.58182i −0.809425 0.587224i \(-0.800221\pi\)
−0.103839 0.994594i \(-0.533113\pi\)
\(60\) 0 0
\(61\) 594.209 + 343.066i 1.24722 + 0.720085i 0.970555 0.240881i \(-0.0774363\pi\)
0.276668 + 0.960965i \(0.410770\pi\)
\(62\) 0 0
\(63\) −475.804 + 153.811i −0.951518 + 0.307593i
\(64\) 0 0
\(65\) −136.575 78.8515i −0.260616 0.150467i
\(66\) 0 0
\(67\) −171.449 296.958i −0.312624 0.541480i 0.666306 0.745679i \(-0.267875\pi\)
−0.978930 + 0.204198i \(0.934541\pi\)
\(68\) 0 0
\(69\) 150.370 + 329.379i 0.262354 + 0.574675i
\(70\) 0 0
\(71\) 387.476i 0.647675i −0.946113 0.323838i \(-0.895027\pi\)
0.946113 0.323838i \(-0.104973\pi\)
\(72\) 0 0
\(73\) 220.721i 0.353882i −0.984221 0.176941i \(-0.943380\pi\)
0.984221 0.176941i \(-0.0566202\pi\)
\(74\) 0 0
\(75\) −376.675 268.303i −0.579929 0.413080i
\(76\) 0 0
\(77\) −842.004 17.0742i −1.24617 0.0252699i
\(78\) 0 0
\(79\) 242.301 419.677i 0.345076 0.597688i −0.640292 0.768132i \(-0.721187\pi\)
0.985367 + 0.170443i \(0.0545200\pi\)
\(80\) 0 0
\(81\) 676.638 + 271.297i 0.928173 + 0.372149i
\(82\) 0 0
\(83\) 354.323 613.706i 0.468579 0.811602i −0.530776 0.847512i \(-0.678100\pi\)
0.999355 + 0.0359099i \(0.0114329\pi\)
\(84\) 0 0
\(85\) −59.1741 102.493i −0.0755098 0.130787i
\(86\) 0 0
\(87\) −414.346 + 581.706i −0.510604 + 0.716844i
\(88\) 0 0
\(89\) 140.929 0.167848 0.0839240 0.996472i \(-0.473255\pi\)
0.0839240 + 0.996472i \(0.473255\pi\)
\(90\) 0 0
\(91\) 251.891 416.550i 0.290169 0.479850i
\(92\) 0 0
\(93\) 344.368 + 754.321i 0.383970 + 0.841070i
\(94\) 0 0
\(95\) 145.055 83.7476i 0.156656 0.0904455i
\(96\) 0 0
\(97\) 1304.31 + 753.041i 1.36528 + 0.788245i 0.990321 0.138796i \(-0.0443232\pi\)
0.374960 + 0.927041i \(0.377657\pi\)
\(98\) 0 0
\(99\) 927.683 + 804.267i 0.941774 + 0.816484i
\(100\) 0 0
\(101\) 923.516 1599.58i 0.909834 1.57588i 0.0955414 0.995425i \(-0.469542\pi\)
0.814293 0.580454i \(-0.197125\pi\)
\(102\) 0 0
\(103\) 696.481 402.114i 0.666275 0.384674i −0.128389 0.991724i \(-0.540981\pi\)
0.794664 + 0.607050i \(0.207647\pi\)
\(104\) 0 0
\(105\) −344.451 + 463.403i −0.320143 + 0.430700i
\(106\) 0 0
\(107\) 282.997i 0.255685i 0.991794 + 0.127843i \(0.0408053\pi\)
−0.991794 + 0.127843i \(0.959195\pi\)
\(108\) 0 0
\(109\) 1826.43 1.60496 0.802478 0.596681i \(-0.203514\pi\)
0.802478 + 0.596681i \(0.203514\pi\)
\(110\) 0 0
\(111\) −142.361 + 1488.81i −0.121733 + 1.27308i
\(112\) 0 0
\(113\) −815.811 + 471.008i −0.679159 + 0.392113i −0.799538 0.600615i \(-0.794922\pi\)
0.120379 + 0.992728i \(0.461589\pi\)
\(114\) 0 0
\(115\) 362.076 + 209.044i 0.293598 + 0.169509i
\(116\) 0 0
\(117\) −670.701 + 231.935i −0.529968 + 0.183268i
\(118\) 0 0
\(119\) 320.006 176.204i 0.246512 0.135736i
\(120\) 0 0
\(121\) 368.410 + 638.105i 0.276792 + 0.479418i
\(122\) 0 0
\(123\) 193.129 88.1683i 0.141576 0.0646331i
\(124\) 0 0
\(125\) −1283.99 −0.918749
\(126\) 0 0
\(127\) 320.551 0.223971 0.111985 0.993710i \(-0.464279\pi\)
0.111985 + 0.993710i \(0.464279\pi\)
\(128\) 0 0
\(129\) −469.069 334.115i −0.320149 0.228040i
\(130\) 0 0
\(131\) −306.907 531.578i −0.204691 0.354536i 0.745343 0.666681i \(-0.232286\pi\)
−0.950034 + 0.312145i \(0.898952\pi\)
\(132\) 0 0
\(133\) 249.377 + 452.897i 0.162584 + 0.295272i
\(134\) 0 0
\(135\) 818.118 198.133i 0.521573 0.126315i
\(136\) 0 0
\(137\) 1515.86 + 875.182i 0.945318 + 0.545780i 0.891623 0.452778i \(-0.149567\pi\)
0.0536948 + 0.998557i \(0.482900\pi\)
\(138\) 0 0
\(139\) 1120.72 647.048i 0.683872 0.394834i −0.117440 0.993080i \(-0.537469\pi\)
0.801312 + 0.598246i \(0.204135\pi\)
\(140\) 0 0
\(141\) 928.274 + 661.204i 0.554431 + 0.394918i
\(142\) 0 0
\(143\) −1195.23 −0.698950
\(144\) 0 0
\(145\) 824.664i 0.472308i
\(146\) 0 0
\(147\) −1408.56 1092.01i −0.790311 0.612706i
\(148\) 0 0
\(149\) −2499.23 + 1442.93i −1.37413 + 0.793352i −0.991445 0.130528i \(-0.958333\pi\)
−0.382681 + 0.923880i \(0.624999\pi\)
\(150\) 0 0
\(151\) 1540.63 2668.45i 0.830297 1.43812i −0.0675056 0.997719i \(-0.521504\pi\)
0.897803 0.440398i \(-0.145163\pi\)
\(152\) 0 0
\(153\) −522.922 100.927i −0.276312 0.0533298i
\(154\) 0 0
\(155\) 829.202 + 478.740i 0.429697 + 0.248086i
\(156\) 0 0
\(157\) 1090.13 629.388i 0.554153 0.319941i −0.196642 0.980475i \(-0.563004\pi\)
0.750795 + 0.660535i \(0.229670\pi\)
\(158\) 0 0
\(159\) 1085.05 + 103.753i 0.541195 + 0.0517495i
\(160\) 0 0
\(161\) −667.793 + 1104.32i −0.326891 + 0.540576i
\(162\) 0 0
\(163\) −2906.94 −1.39687 −0.698434 0.715675i \(-0.746119\pi\)
−0.698434 + 0.715675i \(0.746119\pi\)
\(164\) 0 0
\(165\) 1411.27 + 134.946i 0.665860 + 0.0636700i
\(166\) 0 0
\(167\) 286.775 + 496.709i 0.132882 + 0.230159i 0.924786 0.380487i \(-0.124244\pi\)
−0.791904 + 0.610645i \(0.790910\pi\)
\(168\) 0 0
\(169\) −753.072 + 1304.36i −0.342773 + 0.593700i
\(170\) 0 0
\(171\) 142.840 740.078i 0.0638785 0.330966i
\(172\) 0 0
\(173\) 1931.13 3344.82i 0.848679 1.46995i −0.0337092 0.999432i \(-0.510732\pi\)
0.882388 0.470523i \(-0.155935\pi\)
\(174\) 0 0
\(175\) 33.4177 1647.98i 0.0144351 0.711860i
\(176\) 0 0
\(177\) −3912.71 + 1786.25i −1.66157 + 0.758549i
\(178\) 0 0
\(179\) 2248.26i 0.938787i −0.882989 0.469394i \(-0.844473\pi\)
0.882989 0.469394i \(-0.155527\pi\)
\(180\) 0 0
\(181\) 166.407i 0.0683368i 0.999416 + 0.0341684i \(0.0108783\pi\)
−0.999416 + 0.0341684i \(0.989122\pi\)
\(182\) 0 0
\(183\) 2068.43 2903.90i 0.835534 1.17302i
\(184\) 0 0
\(185\) 863.478 + 1495.59i 0.343157 + 0.594366i
\(186\) 0 0
\(187\) −776.787 448.478i −0.303766 0.175379i
\(188\) 0 0
\(189\) 560.261 + 2537.20i 0.215624 + 0.976476i
\(190\) 0 0
\(191\) −1758.88 1015.49i −0.666324 0.384703i 0.128358 0.991728i \(-0.459029\pi\)
−0.794682 + 0.607025i \(0.792363\pi\)
\(192\) 0 0
\(193\) 251.118 + 434.949i 0.0936573 + 0.162219i 0.909047 0.416693i \(-0.136811\pi\)
−0.815390 + 0.578912i \(0.803478\pi\)
\(194\) 0 0
\(195\) −475.415 + 667.441i −0.174591 + 0.245110i
\(196\) 0 0
\(197\) 3949.11i 1.42823i −0.700026 0.714117i \(-0.746828\pi\)
0.700026 0.714117i \(-0.253172\pi\)
\(198\) 0 0
\(199\) 629.709i 0.224316i 0.993690 + 0.112158i \(0.0357763\pi\)
−0.993690 + 0.112158i \(0.964224\pi\)
\(200\) 0 0
\(201\) −1620.83 + 739.952i −0.568779 + 0.259663i
\(202\) 0 0
\(203\) −2545.00 51.6076i −0.879922 0.0178431i
\(204\) 0 0
\(205\) 122.572 212.300i 0.0417599 0.0723302i
\(206\) 0 0
\(207\) 1778.10 614.886i 0.597038 0.206462i
\(208\) 0 0
\(209\) 634.720 1099.37i 0.210069 0.363851i
\(210\) 0 0
\(211\) −1494.95 2589.33i −0.487757 0.844820i 0.512144 0.858900i \(-0.328851\pi\)
−0.999901 + 0.0140797i \(0.995518\pi\)
\(212\) 0 0
\(213\) −2004.24 191.647i −0.644735 0.0616500i
\(214\) 0 0
\(215\) −664.983 −0.210937
\(216\) 0 0
\(217\) −1529.33 + 2529.05i −0.478424 + 0.791165i
\(218\) 0 0
\(219\) −1141.69 109.169i −0.352275 0.0336848i
\(220\) 0 0
\(221\) 448.992 259.226i 0.136663 0.0789023i
\(222\) 0 0
\(223\) −2829.72 1633.74i −0.849740 0.490598i 0.0108230 0.999941i \(-0.496555\pi\)
−0.860563 + 0.509344i \(0.829888\pi\)
\(224\) 0 0
\(225\) −1574.12 + 1815.67i −0.466406 + 0.537976i
\(226\) 0 0
\(227\) −2836.86 + 4913.59i −0.829468 + 1.43668i 0.0689889 + 0.997617i \(0.478023\pi\)
−0.898457 + 0.439063i \(0.855311\pi\)
\(228\) 0 0
\(229\) −4528.48 + 2614.52i −1.30677 + 0.754465i −0.981556 0.191175i \(-0.938770\pi\)
−0.325215 + 0.945640i \(0.605437\pi\)
\(230\) 0 0
\(231\) −504.775 + 4346.87i −0.143774 + 1.23811i
\(232\) 0 0
\(233\) 3433.84i 0.965487i −0.875762 0.482743i \(-0.839640\pi\)
0.875762 0.482743i \(-0.160360\pi\)
\(234\) 0 0
\(235\) 1315.98 0.365299
\(236\) 0 0
\(237\) −2050.96 1460.89i −0.562128 0.400401i
\(238\) 0 0
\(239\) 469.677 271.168i 0.127117 0.0733908i −0.435093 0.900385i \(-0.643285\pi\)
0.562210 + 0.826995i \(0.309951\pi\)
\(240\) 0 0
\(241\) 3110.33 + 1795.75i 0.831343 + 0.479976i 0.854312 0.519760i \(-0.173979\pi\)
−0.0229693 + 0.999736i \(0.507312\pi\)
\(242\) 0 0
\(243\) 1737.97 3365.77i 0.458809 0.888535i
\(244\) 0 0
\(245\) −2056.29 83.4292i −0.536210 0.0217555i
\(246\) 0 0
\(247\) 366.876 + 635.448i 0.0945092 + 0.163695i
\(248\) 0 0
\(249\) −2999.18 2136.30i −0.763315 0.543705i
\(250\) 0 0
\(251\) −4334.39 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(252\) 0 0
\(253\) 3168.68 0.787404
\(254\) 0 0
\(255\) −559.416 + 255.388i −0.137380 + 0.0627178i
\(256\) 0 0
\(257\) 3282.88 + 5686.12i 0.796812 + 1.38012i 0.921682 + 0.387946i \(0.126815\pi\)
−0.124870 + 0.992173i \(0.539852\pi\)
\(258\) 0 0
\(259\) −4669.58 + 2571.19i −1.12028 + 0.616857i
\(260\) 0 0
\(261\) 2803.97 + 2430.94i 0.664986 + 0.576519i
\(262\) 0 0
\(263\) 6274.79 + 3622.75i 1.47118 + 0.849386i 0.999476 0.0323711i \(-0.0103058\pi\)
0.471704 + 0.881757i \(0.343639\pi\)
\(264\) 0 0
\(265\) 1089.99 629.305i 0.252669 0.145879i
\(266\) 0 0
\(267\) 69.7041 728.965i 0.0159769 0.167086i
\(268\) 0 0
\(269\) 3455.87 0.783302 0.391651 0.920114i \(-0.371904\pi\)
0.391651 + 0.920114i \(0.371904\pi\)
\(270\) 0 0
\(271\) 2471.30i 0.553951i 0.960877 + 0.276975i \(0.0893321\pi\)
−0.960877 + 0.276975i \(0.910668\pi\)
\(272\) 0 0
\(273\) −2030.04 1508.95i −0.450051 0.334527i
\(274\) 0 0
\(275\) −3504.94 + 2023.58i −0.768567 + 0.443732i
\(276\) 0 0
\(277\) 2256.85 3908.97i 0.489533 0.847897i −0.510394 0.859941i \(-0.670500\pi\)
0.999927 + 0.0120440i \(0.00383381\pi\)
\(278\) 0 0
\(279\) 4072.10 1408.17i 0.873800 0.302169i
\(280\) 0 0
\(281\) 2090.48 + 1206.94i 0.443799 + 0.256228i 0.705208 0.709001i \(-0.250854\pi\)
−0.261409 + 0.965228i \(0.584187\pi\)
\(282\) 0 0
\(283\) −3893.62 + 2247.98i −0.817851 + 0.472187i −0.849675 0.527307i \(-0.823202\pi\)
0.0318238 + 0.999493i \(0.489868\pi\)
\(284\) 0 0
\(285\) −361.445 791.728i −0.0751233 0.164554i
\(286\) 0 0
\(287\) 647.511 + 391.555i 0.133175 + 0.0805322i
\(288\) 0 0
\(289\) −4523.93 −0.920808
\(290\) 0 0
\(291\) 4540.27 6374.14i 0.914622 1.28405i
\(292\) 0 0
\(293\) −1569.64 2718.69i −0.312966 0.542073i 0.666037 0.745919i \(-0.267989\pi\)
−0.979003 + 0.203845i \(0.934656\pi\)
\(294\) 0 0
\(295\) −2483.25 + 4301.12i −0.490103 + 0.848884i
\(296\) 0 0
\(297\) 4618.96 4400.70i 0.902421 0.859779i
\(298\) 0 0
\(299\) −915.768 + 1586.16i −0.177125 + 0.306789i
\(300\) 0 0
\(301\) 41.6147 2052.21i 0.00796888 0.392981i
\(302\) 0 0
\(303\) −7817.13 5568.10i −1.48212 1.05571i
\(304\) 0 0
\(305\) 4116.75i 0.772868i
\(306\) 0 0
\(307\) 10294.0i 1.91371i −0.290560 0.956857i \(-0.593842\pi\)
0.290560 0.956857i \(-0.406158\pi\)
\(308\) 0 0
\(309\) −1735.47 3801.48i −0.319507 0.699865i
\(310\) 0 0
\(311\) 3940.59 + 6825.30i 0.718489 + 1.24446i 0.961598 + 0.274461i \(0.0884994\pi\)
−0.243109 + 0.969999i \(0.578167\pi\)
\(312\) 0 0
\(313\) −7096.84 4097.36i −1.28159 0.739925i −0.304449 0.952529i \(-0.598472\pi\)
−0.977139 + 0.212604i \(0.931806\pi\)
\(314\) 0 0
\(315\) 2226.61 + 2010.90i 0.398271 + 0.359686i
\(316\) 0 0
\(317\) 7590.80 + 4382.55i 1.34493 + 0.776494i 0.987526 0.157457i \(-0.0503295\pi\)
0.357401 + 0.933951i \(0.383663\pi\)
\(318\) 0 0
\(319\) 3125.05 + 5412.74i 0.548493 + 0.950017i
\(320\) 0 0
\(321\) 1463.82 + 139.971i 0.254524 + 0.0243378i
\(322\) 0 0
\(323\) 550.644i 0.0948564i
\(324\) 0 0
\(325\) 2339.31i 0.399266i
\(326\) 0 0
\(327\) 903.359 9447.32i 0.152770 1.59767i
\(328\) 0 0
\(329\) −82.3544 + 4061.26i −0.0138004 + 0.680561i
\(330\) 0 0
\(331\) −3431.76 + 5943.98i −0.569869 + 0.987041i 0.426710 + 0.904389i \(0.359673\pi\)
−0.996578 + 0.0826527i \(0.973661\pi\)
\(332\) 0 0
\(333\) 7630.55 + 1472.74i 1.25571 + 0.242360i
\(334\) 0 0
\(335\) −1028.68 + 1781.73i −0.167770 + 0.290586i
\(336\) 0 0
\(337\) 2323.81 + 4024.96i 0.375627 + 0.650605i 0.990421 0.138083i \(-0.0440941\pi\)
−0.614794 + 0.788688i \(0.710761\pi\)
\(338\) 0 0
\(339\) 2032.82 + 4452.79i 0.325686 + 0.713399i
\(340\) 0 0
\(341\) 7256.70 1.15241
\(342\) 0 0
\(343\) 386.154 6340.70i 0.0607882 0.998151i
\(344\) 0 0
\(345\) 1260.38 1769.46i 0.196686 0.276130i
\(346\) 0 0
\(347\) 8295.04 4789.14i 1.28329 0.740907i 0.305840 0.952083i \(-0.401063\pi\)
0.977448 + 0.211176i \(0.0677294\pi\)
\(348\) 0 0
\(349\) 755.168 + 435.997i 0.115826 + 0.0668721i 0.556794 0.830651i \(-0.312031\pi\)
−0.440968 + 0.897523i \(0.645365\pi\)
\(350\) 0 0
\(351\) 867.966 + 3583.96i 0.131990 + 0.545007i
\(352\) 0 0
\(353\) 4148.63 7185.64i 0.625523 1.08344i −0.362917 0.931821i \(-0.618219\pi\)
0.988440 0.151615i \(-0.0484475\pi\)
\(354\) 0 0
\(355\) −2013.36 + 1162.42i −0.301009 + 0.173788i
\(356\) 0 0
\(357\) −753.147 1742.40i −0.111655 0.258313i
\(358\) 0 0
\(359\) 976.696i 0.143588i 0.997419 + 0.0717939i \(0.0228724\pi\)
−0.997419 + 0.0717939i \(0.977128\pi\)
\(360\) 0 0
\(361\) 6079.69 0.886381
\(362\) 0 0
\(363\) 3482.85 1590.01i 0.503588 0.229901i
\(364\) 0 0
\(365\) −1146.89 + 662.155i −0.164468 + 0.0949556i
\(366\) 0 0
\(367\) −6001.51 3464.97i −0.853614 0.492834i 0.00825490 0.999966i \(-0.497372\pi\)
−0.861868 + 0.507132i \(0.830706\pi\)
\(368\) 0 0
\(369\) −360.534 1042.58i −0.0508635 0.147085i
\(370\) 0 0
\(371\) 1873.89 + 3403.20i 0.262231 + 0.476241i
\(372\) 0 0
\(373\) 2259.68 + 3913.89i 0.313678 + 0.543307i 0.979156 0.203111i \(-0.0651053\pi\)
−0.665477 + 0.746418i \(0.731772\pi\)
\(374\) 0 0
\(375\) −635.067 + 6641.52i −0.0874525 + 0.914578i
\(376\) 0 0
\(377\) −3612.63 −0.493528
\(378\) 0 0
\(379\) 2781.32 0.376958 0.188479 0.982077i \(-0.439644\pi\)
0.188479 + 0.982077i \(0.439644\pi\)
\(380\) 0 0
\(381\) 158.546 1658.07i 0.0213190 0.222954i
\(382\) 0 0
\(383\) 5449.44 + 9438.71i 0.727033 + 1.25926i 0.958132 + 0.286327i \(0.0924345\pi\)
−0.231099 + 0.972930i \(0.574232\pi\)
\(384\) 0 0
\(385\) 2437.27 + 4426.36i 0.322635 + 0.585943i
\(386\) 0 0
\(387\) −1960.23 + 2261.03i −0.257479 + 0.296989i
\(388\) 0 0
\(389\) 1922.09 + 1109.72i 0.250524 + 0.144640i 0.620004 0.784598i \(-0.287131\pi\)
−0.369480 + 0.929239i \(0.620464\pi\)
\(390\) 0 0
\(391\) −1190.33 + 687.237i −0.153958 + 0.0888877i
\(392\) 0 0
\(393\) −2901.41 + 1324.57i −0.372410 + 0.170015i
\(394\) 0 0
\(395\) −2907.58 −0.370370
\(396\) 0 0
\(397\) 10873.7i 1.37464i 0.726354 + 0.687321i \(0.241214\pi\)
−0.726354 + 0.687321i \(0.758786\pi\)
\(398\) 0 0
\(399\) 2465.98 1065.91i 0.309407 0.133740i
\(400\) 0 0
\(401\) −5301.00 + 3060.53i −0.660147 + 0.381136i −0.792333 0.610089i \(-0.791134\pi\)
0.132186 + 0.991225i \(0.457800\pi\)
\(402\) 0 0
\(403\) −2097.23 + 3632.51i −0.259232 + 0.449003i
\(404\) 0 0
\(405\) −620.209 4329.76i −0.0760949 0.531228i
\(406\) 0 0
\(407\) 11335.0 + 6544.26i 1.38048 + 0.797020i
\(408\) 0 0
\(409\) −12155.0 + 7017.69i −1.46950 + 0.848416i −0.999415 0.0342045i \(-0.989110\pi\)
−0.470085 + 0.882621i \(0.655777\pi\)
\(410\) 0 0
\(411\) 5276.68 7408.00i 0.633283 0.889075i
\(412\) 0 0
\(413\) −13118.3 7932.75i −1.56298 0.945145i
\(414\) 0 0
\(415\) −4251.83 −0.502926
\(416\) 0 0
\(417\) −2792.58 6117.03i −0.327946 0.718350i
\(418\) 0 0
\(419\) 4144.18 + 7177.94i 0.483190 + 0.836910i 0.999814 0.0193028i \(-0.00614466\pi\)
−0.516624 + 0.856213i \(0.672811\pi\)
\(420\) 0 0
\(421\) 7767.30 13453.4i 0.899181 1.55743i 0.0706368 0.997502i \(-0.477497\pi\)
0.828544 0.559924i \(-0.189170\pi\)
\(422\) 0 0
\(423\) 3879.25 4474.52i 0.445899 0.514323i
\(424\) 0 0
\(425\) 877.765 1520.33i 0.100183 0.173522i
\(426\) 0 0
\(427\) 12704.7 + 257.627i 1.43987 + 0.0291978i
\(428\) 0 0
\(429\) −591.163 + 6182.37i −0.0665306 + 0.695776i
\(430\) 0 0
\(431\) 3167.89i 0.354041i −0.984207 0.177021i \(-0.943354\pi\)
0.984207 0.177021i \(-0.0566459\pi\)
\(432\) 0 0
\(433\) 2187.70i 0.242804i 0.992603 + 0.121402i \(0.0387391\pi\)
−0.992603 + 0.121402i \(0.961261\pi\)
\(434\) 0 0
\(435\) 4265.62 + 407.882i 0.470163 + 0.0449573i
\(436\) 0 0
\(437\) −972.630 1684.65i −0.106470 0.184411i
\(438\) 0 0
\(439\) −1703.79 983.684i −0.185234 0.106945i 0.404516 0.914531i \(-0.367440\pi\)
−0.589749 + 0.807586i \(0.700773\pi\)
\(440\) 0 0
\(441\) −6345.18 + 6745.72i −0.685151 + 0.728401i
\(442\) 0 0
\(443\) 2229.21 + 1287.04i 0.239081 + 0.138034i 0.614754 0.788719i \(-0.289255\pi\)
−0.375673 + 0.926752i \(0.622588\pi\)
\(444\) 0 0
\(445\) −422.783 732.282i −0.0450379 0.0780079i
\(446\) 0 0
\(447\) 6227.51 + 13641.1i 0.658952 + 1.44340i
\(448\) 0 0
\(449\) 9168.11i 0.963630i −0.876273 0.481815i \(-0.839978\pi\)
0.876273 0.481815i \(-0.160022\pi\)
\(450\) 0 0
\(451\) 1857.93i 0.193983i
\(452\) 0 0
\(453\) −13040.7 9288.84i −1.35255 0.963416i
\(454\) 0 0
\(455\) −2920.10 59.2139i −0.300871 0.00610108i
\(456\) 0 0
\(457\) 449.418 778.415i 0.0460019 0.0796777i −0.842108 0.539310i \(-0.818685\pi\)
0.888110 + 0.459632i \(0.152019\pi\)
\(458\) 0 0
\(459\) −780.690 + 2654.92i −0.0793889 + 0.269981i
\(460\) 0 0
\(461\) 750.955 1300.69i 0.0758686 0.131408i −0.825595 0.564263i \(-0.809160\pi\)
0.901464 + 0.432855i \(0.142494\pi\)
\(462\) 0 0
\(463\) 515.882 + 893.534i 0.0517820 + 0.0896891i 0.890755 0.454485i \(-0.150177\pi\)
−0.838973 + 0.544174i \(0.816843\pi\)
\(464\) 0 0
\(465\) 2886.44 4052.31i 0.287861 0.404132i
\(466\) 0 0
\(467\) 8619.68 0.854114 0.427057 0.904225i \(-0.359550\pi\)
0.427057 + 0.904225i \(0.359550\pi\)
\(468\) 0 0
\(469\) −5434.23 3286.12i −0.535031 0.323537i
\(470\) 0 0
\(471\) −2716.37 5950.07i −0.265740 0.582091i
\(472\) 0 0
\(473\) −4364.66 + 2519.94i −0.424286 + 0.244962i
\(474\) 0 0
\(475\) 2151.69 + 1242.28i 0.207845 + 0.119999i
\(476\) 0 0
\(477\) 1073.34 5561.17i 0.103029 0.533812i
\(478\) 0 0
\(479\) 2037.51 3529.07i 0.194355 0.336634i −0.752334 0.658782i \(-0.771072\pi\)
0.946689 + 0.322149i \(0.104405\pi\)
\(480\) 0 0
\(481\) −6551.77 + 3782.66i −0.621070 + 0.358575i
\(482\) 0 0
\(483\) 5381.88 + 4000.40i 0.507006 + 0.376862i
\(484\) 0 0
\(485\) 9036.40i 0.846025i
\(486\) 0 0
\(487\) −14817.8 −1.37876 −0.689381 0.724399i \(-0.742117\pi\)
−0.689381 + 0.724399i \(0.742117\pi\)
\(488\) 0 0
\(489\) −1437.78 + 15036.3i −0.132963 + 1.39052i
\(490\) 0 0
\(491\) −6217.36 + 3589.59i −0.571457 + 0.329931i −0.757731 0.652567i \(-0.773692\pi\)
0.186274 + 0.982498i \(0.440359\pi\)
\(492\) 0 0
\(493\) −2347.88 1355.55i −0.214489 0.123835i
\(494\) 0 0
\(495\) 1396.03 7233.11i 0.126762 0.656776i
\(496\) 0 0
\(497\) −3461.34 6286.20i −0.312399 0.567353i
\(498\) 0 0
\(499\) −5380.59 9319.46i −0.482702 0.836065i 0.517100 0.855925i \(-0.327011\pi\)
−0.999803 + 0.0198598i \(0.993678\pi\)
\(500\) 0 0
\(501\) 2711.09 1237.69i 0.241762 0.110371i
\(502\) 0 0
\(503\) −11285.3 −1.00037 −0.500186 0.865918i \(-0.666735\pi\)
−0.500186 + 0.865918i \(0.666735\pi\)
\(504\) 0 0
\(505\) −11082.1 −0.976527
\(506\) 0 0
\(507\) 6374.41 + 4540.45i 0.558377 + 0.397729i
\(508\) 0 0
\(509\) 5404.52 + 9360.91i 0.470631 + 0.815157i 0.999436 0.0335864i \(-0.0106929\pi\)
−0.528805 + 0.848744i \(0.677360\pi\)
\(510\) 0 0
\(511\) −1971.71 3580.85i −0.170691 0.309995i
\(512\) 0 0
\(513\) −3757.45 1104.89i −0.323383 0.0950919i
\(514\) 0 0
\(515\) −4178.84 2412.66i −0.357557 0.206436i
\(516\) 0 0
\(517\) 8637.54 4986.89i 0.734775 0.424223i
\(518\) 0 0
\(519\) −16346.1 11643.3i −1.38250 0.984745i
\(520\) 0 0
\(521\) 3085.93 0.259495 0.129747 0.991547i \(-0.458583\pi\)
0.129747 + 0.991547i \(0.458583\pi\)
\(522\) 0 0
\(523\) 2155.94i 0.180254i 0.995930 + 0.0901268i \(0.0287272\pi\)
−0.995930 + 0.0901268i \(0.971273\pi\)
\(524\) 0 0
\(525\) −8507.74 987.951i −0.707253 0.0821290i
\(526\) 0 0
\(527\) −2726.01 + 1573.87i −0.225327 + 0.130092i
\(528\) 0 0
\(529\) −3655.69 + 6331.85i −0.300460 + 0.520412i
\(530\) 0 0
\(531\) 7304.27 + 21122.2i 0.596946 + 1.72623i
\(532\) 0 0
\(533\) 930.030 + 536.953i 0.0755799 + 0.0436361i
\(534\) 0 0
\(535\) 1470.48 848.982i 0.118831 0.0686069i
\(536\) 0 0
\(537\) −11629.3 1112.00i −0.934525 0.0893599i
\(538\) 0 0
\(539\) −13812.7 + 7244.67i −1.10382 + 0.578942i
\(540\) 0 0
\(541\) 19410.4 1.54255 0.771273 0.636504i \(-0.219620\pi\)
0.771273 + 0.636504i \(0.219620\pi\)
\(542\) 0 0
\(543\) 860.752 + 82.3057i 0.0680265 + 0.00650475i
\(544\) 0 0
\(545\) −5479.23 9490.31i −0.430651 0.745909i
\(546\) 0 0
\(547\) −2279.16 + 3947.62i −0.178153 + 0.308570i −0.941248 0.337716i \(-0.890346\pi\)
0.763095 + 0.646286i \(0.223679\pi\)
\(548\) 0 0
\(549\) −13997.5 12135.4i −1.08816 0.943396i
\(550\) 0 0
\(551\) 1918.48 3322.90i 0.148330 0.256915i
\(552\) 0 0
\(553\) 181.957 8973.10i 0.0139920 0.690009i
\(554\) 0 0
\(555\) 8163.09 3726.67i 0.624331 0.285024i
\(556\) 0 0
\(557\) 4710.38i 0.358322i −0.983820 0.179161i \(-0.942662\pi\)
0.983820 0.179161i \(-0.0573383\pi\)
\(558\) 0 0
\(559\) 2913.11i 0.220414i
\(560\) 0 0
\(561\) −2703.98 + 3796.16i −0.203498 + 0.285693i
\(562\) 0 0
\(563\) −2316.26 4011.87i −0.173390 0.300320i 0.766213 0.642587i \(-0.222139\pi\)
−0.939603 + 0.342267i \(0.888805\pi\)
\(564\) 0 0
\(565\) 4894.81 + 2826.02i 0.364471 + 0.210428i
\(566\) 0 0
\(567\) 13400.9 1643.07i 0.992567 0.121698i
\(568\) 0 0
\(569\) −6667.01 3849.20i −0.491205 0.283597i 0.233869 0.972268i \(-0.424861\pi\)
−0.725074 + 0.688671i \(0.758195\pi\)
\(570\) 0 0
\(571\) −8556.34 14820.0i −0.627096 1.08616i −0.988132 0.153610i \(-0.950910\pi\)
0.361036 0.932552i \(-0.382423\pi\)
\(572\) 0 0
\(573\) −6122.62 + 8595.64i −0.446381 + 0.626680i
\(574\) 0 0
\(575\) 6201.77i 0.449794i
\(576\) 0 0
\(577\) 2389.01i 0.172367i 0.996279 + 0.0861834i \(0.0274671\pi\)
−0.996279 + 0.0861834i \(0.972533\pi\)
\(578\) 0 0
\(579\) 2374.00 1083.79i 0.170398 0.0777910i
\(580\) 0 0
\(581\) 266.080 13121.6i 0.0189998 0.936965i
\(582\) 0 0
\(583\) 4769.48 8260.97i 0.338819 0.586852i
\(584\) 0 0
\(585\) 3217.24 + 2789.23i 0.227379 + 0.197129i
\(586\) 0 0
\(587\) −7877.22 + 13643.7i −0.553880 + 0.959348i 0.444110 + 0.895972i \(0.353520\pi\)
−0.997990 + 0.0633757i \(0.979813\pi\)
\(588\) 0 0
\(589\) −2227.45 3858.06i −0.155824 0.269896i
\(590\) 0 0
\(591\) −20427.0 1953.24i −1.42175 0.135949i
\(592\) 0 0
\(593\) 14439.8 0.999953 0.499976 0.866039i \(-0.333342\pi\)
0.499976 + 0.866039i \(0.333342\pi\)
\(594\) 0 0
\(595\) −1875.58 1134.18i −0.129229 0.0781458i
\(596\) 0 0
\(597\) 3257.21 + 311.457i 0.223298 + 0.0213519i
\(598\) 0 0
\(599\) −13069.6 + 7545.75i −0.891504 + 0.514710i −0.874434 0.485145i \(-0.838767\pi\)
−0.0170695 + 0.999854i \(0.505434\pi\)
\(600\) 0 0
\(601\) 21885.4 + 12635.5i 1.48540 + 0.857595i 0.999862 0.0166219i \(-0.00529115\pi\)
0.485536 + 0.874217i \(0.338624\pi\)
\(602\) 0 0
\(603\) 3025.78 + 8749.83i 0.204343 + 0.590913i
\(604\) 0 0
\(605\) 2210.44 3828.59i 0.148541 0.257280i
\(606\) 0 0
\(607\) −16922.0 + 9769.93i −1.13154 + 0.653293i −0.944321 0.329025i \(-0.893280\pi\)
−0.187216 + 0.982319i \(0.559947\pi\)
\(608\) 0 0
\(609\) −1525.71 + 13138.6i −0.101519 + 0.874228i
\(610\) 0 0
\(611\) 5764.97i 0.381711i
\(612\) 0 0
\(613\) 22968.2 1.51334 0.756671 0.653796i \(-0.226825\pi\)
0.756671 + 0.653796i \(0.226825\pi\)
\(614\) 0 0
\(615\) −1037.51 739.013i −0.0680268 0.0484551i
\(616\) 0 0
\(617\) −24539.5 + 14167.9i −1.60117 + 0.924437i −0.609920 + 0.792463i \(0.708798\pi\)
−0.991253 + 0.131974i \(0.957868\pi\)
\(618\) 0 0
\(619\) 17347.3 + 10015.5i 1.12641 + 0.650332i 0.943029 0.332710i \(-0.107963\pi\)
0.183379 + 0.983042i \(0.441297\pi\)
\(620\) 0 0
\(621\) −2301.08 9501.48i −0.148694 0.613979i
\(622\) 0 0
\(623\) 2286.36 1258.93i 0.147032 0.0809597i
\(624\) 0 0
\(625\) −1710.61 2962.87i −0.109479 0.189624i
\(626\) 0 0
\(627\) −5372.61 3826.88i −0.342203 0.243749i
\(628\) 0 0
\(629\) −5677.39 −0.359893
\(630\) 0 0
\(631\) 7219.42 0.455469 0.227734 0.973723i \(-0.426868\pi\)
0.227734 + 0.973723i \(0.426868\pi\)
\(632\) 0 0
\(633\) −14132.9 + 6452.03i −0.887412 + 0.405127i
\(634\) 0 0
\(635\) −961.643 1665.61i −0.0600970 0.104091i
\(636\) 0 0
\(637\) 365.481 9008.04i 0.0227329 0.560301i
\(638\) 0 0
\(639\) −1982.61 + 10272.3i −0.122740 + 0.635939i
\(640\) 0 0
\(641\) −14946.4 8629.29i −0.920976 0.531726i −0.0370299 0.999314i \(-0.511790\pi\)
−0.883946 + 0.467588i \(0.845123\pi\)
\(642\) 0 0
\(643\) 21951.3 12673.6i 1.34631 0.777291i 0.358584 0.933498i \(-0.383260\pi\)
0.987724 + 0.156206i \(0.0499265\pi\)
\(644\) 0 0
\(645\) −328.903 + 3439.66i −0.0200784 + 0.209979i
\(646\) 0 0
\(647\) −6799.46 −0.413160 −0.206580 0.978430i \(-0.566233\pi\)
−0.206580 + 0.978430i \(0.566233\pi\)
\(648\) 0 0
\(649\) 37640.9i 2.27663i
\(650\) 0 0
\(651\) 12325.2 + 9161.45i 0.742033 + 0.551560i
\(652\) 0 0
\(653\) 19452.8 11231.1i 1.16577 0.673058i 0.213090 0.977033i \(-0.431647\pi\)
0.952680 + 0.303975i \(0.0983139\pi\)
\(654\) 0 0
\(655\) −1841.42 + 3189.43i −0.109848 + 0.190262i
\(656\) 0 0
\(657\) −1129.37 + 5851.47i −0.0670637 + 0.347470i
\(658\) 0 0
\(659\) 3236.46 + 1868.57i 0.191312 + 0.110454i 0.592597 0.805499i \(-0.298103\pi\)
−0.401285 + 0.915953i \(0.631436\pi\)
\(660\) 0 0
\(661\) 10205.8 5892.33i 0.600545 0.346725i −0.168711 0.985666i \(-0.553960\pi\)
0.769256 + 0.638941i \(0.220627\pi\)
\(662\) 0 0
\(663\) −1118.79 2450.65i −0.0655356 0.143553i
\(664\) 0 0
\(665\) 1605.17 2654.46i 0.0936030 0.154790i
\(666\) 0 0
\(667\) 9577.50 0.555986
\(668\) 0 0
\(669\) −9850.20 + 13828.8i −0.569254 + 0.799184i
\(670\) 0 0
\(671\) −15600.4 27020.6i −0.897534 1.55457i
\(672\) 0 0
\(673\) 5712.35 9894.08i 0.327184 0.566699i −0.654768 0.755830i \(-0.727234\pi\)
0.981952 + 0.189131i \(0.0605670\pi\)
\(674\) 0 0
\(675\) 8613.09 + 9040.26i 0.491138 + 0.515496i
\(676\) 0 0
\(677\) 15013.5 26004.1i 0.852312 1.47625i −0.0268044 0.999641i \(-0.508533\pi\)
0.879116 0.476607i \(-0.158134\pi\)
\(678\) 0 0
\(679\) 27887.3 + 565.500i 1.57617 + 0.0319615i
\(680\) 0 0
\(681\) 24012.7 + 17104.1i 1.35120 + 0.962454i
\(682\) 0 0
\(683\) 18482.4i 1.03545i 0.855548 + 0.517723i \(0.173220\pi\)
−0.855548 + 0.517723i \(0.826780\pi\)
\(684\) 0 0
\(685\) 10502.1i 0.585786i
\(686\) 0 0
\(687\) 11284.0 + 24717.0i 0.626652 + 1.37265i
\(688\) 0 0
\(689\) 2756.82 + 4774.94i 0.152433 + 0.264022i
\(690\) 0 0
\(691\) −16408.4 9473.38i −0.903334 0.521540i −0.0250537 0.999686i \(-0.507976\pi\)
−0.878280 + 0.478146i \(0.841309\pi\)
\(692\) 0 0
\(693\) 22234.8 + 4760.96i 1.21880 + 0.260972i
\(694\) 0 0
\(695\) −6724.25 3882.25i −0.367001 0.211888i
\(696\) 0 0
\(697\) 402.956 + 697.941i 0.0218982 + 0.0379288i
\(698\) 0 0
\(699\) −17761.7 1698.39i −0.961103 0.0919013i
\(700\) 0 0
\(701\) 9045.09i 0.487344i −0.969858 0.243672i \(-0.921648\pi\)
0.969858 0.243672i \(-0.0783521\pi\)
\(702\) 0 0
\(703\) 8035.08i 0.431079i
\(704\) 0 0
\(705\) 650.890 6807.00i 0.0347715 0.363640i
\(706\) 0 0
\(707\) 693.518 34200.5i 0.0368917 1.81929i
\(708\) 0 0
\(709\) −1866.89 + 3233.55i −0.0988893 + 0.171281i −0.911225 0.411909i \(-0.864862\pi\)
0.812336 + 0.583190i \(0.198196\pi\)
\(710\) 0 0
\(711\) −8570.95 + 9886.16i −0.452089 + 0.521463i
\(712\) 0 0
\(713\) 5560.00 9630.20i 0.292039 0.505826i
\(714\) 0 0
\(715\) 3585.64 + 6210.51i 0.187546 + 0.324839i
\(716\) 0 0
\(717\) −1170.33 2563.55i −0.0609578 0.133525i
\(718\) 0 0
\(719\) 35410.2 1.83669 0.918343 0.395785i \(-0.129527\pi\)
0.918343 + 0.395785i \(0.129527\pi\)
\(720\) 0 0
\(721\) 7707.23 12745.4i 0.398103 0.658339i
\(722\) 0 0
\(723\) 10827.0 15200.2i 0.556929 0.781881i
\(724\) 0 0
\(725\) −10593.9 + 6116.37i −0.542685 + 0.313319i
\(726\) 0 0
\(727\) −2328.92 1344.60i −0.118810 0.0685951i 0.439417 0.898283i \(-0.355185\pi\)
−0.558227 + 0.829688i \(0.688518\pi\)
\(728\) 0 0
\(729\) −16550.0 10654.5i −0.840828 0.541302i
\(730\) 0 0
\(731\) 1093.07 1893.25i 0.0553060 0.0957928i
\(732\) 0 0
\(733\) −10520.4 + 6073.96i −0.530123 + 0.306066i −0.741066 0.671432i \(-0.765680\pi\)
0.210944 + 0.977498i \(0.432346\pi\)
\(734\) 0 0
\(735\) −1448.59 + 10595.0i −0.0726966 + 0.531704i
\(736\) 0 0
\(737\) 15592.7i 0.779326i
\(738\) 0 0
\(739\) −33517.3 −1.66841 −0.834204 0.551456i \(-0.814072\pi\)
−0.834204 + 0.551456i \(0.814072\pi\)
\(740\) 0 0
\(741\) 3468.35 1583.39i 0.171947 0.0784985i
\(742\) 0 0
\(743\) 21381.5 12344.6i 1.05573 0.609529i 0.131485 0.991318i \(-0.458025\pi\)
0.924249 + 0.381790i \(0.124692\pi\)
\(744\) 0 0
\(745\) 14995.2 + 8657.49i 0.737426 + 0.425753i
\(746\) 0 0
\(747\) −12533.5 + 14456.8i −0.613893 + 0.708095i
\(748\) 0 0
\(749\) 2528.03 + 4591.19i 0.123327 + 0.223976i
\(750\) 0 0
\(751\) −4794.74 8304.74i −0.232973 0.403521i 0.725709 0.688002i \(-0.241512\pi\)
−0.958682 + 0.284481i \(0.908179\pi\)
\(752\) 0 0
\(753\) −2143.80 + 22419.9i −0.103751 + 1.08503i
\(754\) 0 0
\(755\) −18487.4 −0.891159
\(756\) 0 0
\(757\) −13621.8 −0.654022 −0.327011 0.945021i \(-0.606041\pi\)
−0.327011 + 0.945021i \(0.606041\pi\)
\(758\) 0 0
\(759\) 1567.24 16390.2i 0.0749502 0.783829i
\(760\) 0 0
\(761\) −839.906 1454.76i −0.0400086 0.0692970i 0.845328 0.534248i \(-0.179405\pi\)
−0.885336 + 0.464951i \(0.846072\pi\)
\(762\) 0 0
\(763\) 29631.0 16315.6i 1.40592 0.774134i
\(764\) 0 0
\(765\) 1044.32 + 3019.93i 0.0493562 + 0.142727i
\(766\) 0 0
\(767\) −18842.0 10878.5i −0.887023 0.512123i
\(768\) 0 0
\(769\) 30695.1 17721.8i 1.43939 0.831034i 0.441586 0.897219i \(-0.354416\pi\)
0.997807 + 0.0661849i \(0.0210827\pi\)
\(770\) 0 0
\(771\) 31035.5 14168.5i 1.44970 0.661825i
\(772\) 0 0
\(773\) 13901.4 0.646829 0.323415 0.946257i \(-0.395169\pi\)
0.323415 + 0.946257i \(0.395169\pi\)
\(774\) 0 0
\(775\) 14202.9i 0.658300i
\(776\) 0 0
\(777\) 10990.0 + 25425.4i 0.507420 + 1.17391i
\(778\) 0 0
\(779\) −987.778 + 570.294i −0.0454311 + 0.0262297i
\(780\) 0 0
\(781\) −8809.91 + 15259.2i −0.403641 + 0.699126i
\(782\) 0 0
\(783\) 13961.0 13301.3i 0.637199 0.607090i
\(784\) 0 0
\(785\) −6540.73 3776.29i −0.297387 0.171696i
\(786\) 0 0
\(787\) −3906.63 + 2255.49i −0.176946 + 0.102160i −0.585857 0.810415i \(-0.699242\pi\)
0.408911 + 0.912574i \(0.365909\pi\)
\(788\) 0 0
\(789\) 21842.4 30664.9i 0.985566 1.38365i
\(790\) 0 0
\(791\) −9027.72 + 14929.1i −0.405801 + 0.671070i
\(792\) 0 0
\(793\) 18034.4 0.807592
\(794\) 0 0
\(795\) −2716.00 5949.28i −0.121166 0.265408i
\(796\) 0 0
\(797\) −15620.6 27055.6i −0.694240 1.20246i −0.970436 0.241358i \(-0.922407\pi\)
0.276196 0.961101i \(-0.410926\pi\)
\(798\) 0 0
\(799\) −2163.16 + 3746.70i −0.0957785 + 0.165893i
\(800\) 0 0
\(801\) −3736.14 721.097i −0.164806 0.0318086i
\(802\) 0 0
\(803\) −5018.45 + 8692.21i −0.220544 + 0.381994i
\(804\) 0 0
\(805\) 7741.52 + 156.983i 0.338948 + 0.00687319i
\(806\) 0 0
\(807\) 1709.29 17875.7i 0.0745598 0.779745i
\(808\) 0 0
\(809\) 42276.5i 1.83728i 0.395091 + 0.918642i \(0.370713\pi\)
−0.395091 + 0.918642i \(0.629287\pi\)
\(810\) 0 0
\(811\) 29402.1i 1.27305i −0.771254 0.636527i \(-0.780370\pi\)
0.771254 0.636527i \(-0.219630\pi\)
\(812\) 0 0
\(813\) 12782.9 + 1222.31i 0.551436 + 0.0527286i
\(814\) 0 0
\(815\) 8720.74 + 15104.8i 0.374815 + 0.649198i
\(816\) 0 0
\(817\) 2679.48 + 1547.00i 0.114741 + 0.0662455i
\(818\) 0 0
\(819\) −8809.20 + 9754.19i −0.375846 + 0.416165i
\(820\) 0 0
\(821\) 36104.3 + 20844.8i 1.53477 + 0.886103i 0.999132 + 0.0416579i \(0.0132640\pi\)
0.535643 + 0.844445i \(0.320069\pi\)
\(822\) 0 0
\(823\) 4162.09 + 7208.95i 0.176284 + 0.305332i 0.940605 0.339504i \(-0.110259\pi\)
−0.764321 + 0.644836i \(0.776926\pi\)
\(824\) 0 0
\(825\) 8733.52 + 19130.4i 0.368560 + 0.807315i
\(826\) 0 0
\(827\) 7260.76i 0.305298i 0.988280 + 0.152649i \(0.0487804\pi\)
−0.988280 + 0.152649i \(0.951220\pi\)
\(828\) 0 0
\(829\) 26509.9i 1.11065i 0.831634 + 0.555324i \(0.187406\pi\)
−0.831634 + 0.555324i \(0.812594\pi\)
\(830\) 0 0
\(831\) −19103.1 13607.1i −0.797450 0.568019i
\(832\) 0 0
\(833\) 3617.57 5717.26i 0.150470 0.237805i
\(834\) 0 0
\(835\) 1720.63 2980.22i 0.0713113 0.123515i
\(836\) 0 0
\(837\) −5269.78 21759.7i −0.217623 0.898594i
\(838\) 0 0
\(839\) −4438.40 + 7687.53i −0.182635 + 0.316332i −0.942777 0.333424i \(-0.891796\pi\)
0.760142 + 0.649757i \(0.225129\pi\)
\(840\) 0 0
\(841\) −2748.88 4761.19i −0.112710 0.195219i
\(842\) 0 0
\(843\) 7276.92 10216.2i 0.297308 0.417395i
\(844\) 0 0
\(845\) 9036.78 0.367899
\(846\) 0 0
\(847\) 11677.1 + 7061.25i 0.473708 + 0.286455i
\(848\) 0 0
\(849\) 9702.03 + 21251.9i 0.392194 + 0.859083i
\(850\) 0 0
\(851\) 17369.5 10028.3i 0.699669 0.403954i
\(852\) 0 0
\(853\) −13168.8 7603.02i −0.528596 0.305185i 0.211849 0.977302i \(-0.432052\pi\)
−0.740444 + 0.672118i \(0.765385\pi\)
\(854\) 0 0
\(855\) −4274.03 + 1478.00i −0.170958 + 0.0591189i
\(856\) 0 0
\(857\) −11826.9 + 20484.8i −0.471410 + 0.816506i −0.999465 0.0327038i \(-0.989588\pi\)
0.528055 + 0.849210i \(0.322922\pi\)
\(858\) 0 0
\(859\) −25545.4 + 14748.7i −1.01467 + 0.585818i −0.912555 0.408954i \(-0.865893\pi\)
−0.102112 + 0.994773i \(0.532560\pi\)
\(860\) 0 0
\(861\) 2345.60 3155.62i 0.0928431 0.124905i
\(862\) 0 0
\(863\) 36862.0i 1.45399i −0.686640 0.726997i \(-0.740915\pi\)
0.686640 0.726997i \(-0.259085\pi\)
\(864\) 0 0
\(865\) −23173.4 −0.910888
\(866\) 0 0
\(867\) −2237.55 + 23400.3i −0.0876485 + 0.916627i
\(868\) 0 0
\(869\) −19084.1 + 11018.2i −0.744976 + 0.430112i
\(870\) 0 0
\(871\) −7805.27 4506.38i −0.303641 0.175307i
\(872\) 0 0
\(873\) −30725.0 26637.5i −1.19116 1.03269i
\(874\) 0 0
\(875\) −20830.8 + 11470.0i −0.804810 + 0.443149i
\(876\) 0 0
\(877\) 4353.93 + 7541.22i 0.167642 + 0.290364i 0.937590 0.347742i \(-0.113052\pi\)
−0.769949 + 0.638106i \(0.779718\pi\)
\(878\) 0 0
\(879\) −14838.9 + 6774.36i −0.569402 + 0.259947i
\(880\) 0 0
\(881\) −18950.2 −0.724685 −0.362342 0.932045i \(-0.618023\pi\)
−0.362342 + 0.932045i \(0.618023\pi\)
\(882\) 0 0
\(883\) 5510.89 0.210030 0.105015 0.994471i \(-0.466511\pi\)
0.105015 + 0.994471i \(0.466511\pi\)
\(884\) 0 0
\(885\) 21019.6 + 14972.1i 0.798378 + 0.568680i
\(886\) 0 0
\(887\) 12652.8 + 21915.3i 0.478962 + 0.829586i 0.999709 0.0241247i \(-0.00767987\pi\)
−0.520747 + 0.853711i \(0.674347\pi\)
\(888\) 0 0
\(889\) 5200.44 2863.50i 0.196195 0.108030i
\(890\) 0 0
\(891\) −20478.3 26068.4i −0.769977 0.980163i
\(892\) 0 0
\(893\) −5302.61 3061.46i −0.198707 0.114723i
\(894\) 0 0
\(895\) −11682.2 + 6744.71i −0.436304 + 0.251900i
\(896\) 0 0
\(897\) 7751.55 + 5521.38i 0.288536 + 0.205522i
\(898\) 0 0
\(899\) 21933.8 0.813717
\(900\) 0 0
\(901\) 4137.70i 0.152993i
\(902\) 0 0
\(903\) −10594.6 1230.28i −0.390438 0.0453392i
\(904\) 0 0
\(905\) 864.669 499.217i 0.0317598 0.0183365i
\(906\) 0 0
\(907\) 9256.41 16032.6i 0.338869 0.586938i −0.645351 0.763886i \(-0.723289\pi\)
0.984220 + 0.176948i \(0.0566224\pi\)
\(908\) 0 0
\(909\) −32667.7 + 37680.6i −1.19199 + 1.37490i
\(910\) 0 0
\(911\) −27338.9 15784.1i −0.994266 0.574040i −0.0877196 0.996145i \(-0.527958\pi\)
−0.906547 + 0.422105i \(0.861291\pi\)
\(912\) 0 0
\(913\) −27907.2 + 16112.2i −1.01160 + 0.584050i
\(914\) 0 0
\(915\) −21294.2 2036.16i −0.769359 0.0735666i
\(916\) 0 0
\(917\) −9727.70 5882.42i −0.350313 0.211837i
\(918\) 0 0
\(919\) 2352.27 0.0844332 0.0422166 0.999108i \(-0.486558\pi\)
0.0422166 + 0.999108i \(0.486558\pi\)
\(920\) 0 0
\(921\) −53246.4 5091.45i −1.90502 0.182160i
\(922\) 0 0
\(923\) −5092.23 8820.01i −0.181596 0.314533i
\(924\) 0 0
\(925\) −12808.5 + 22185.0i −0.455287 + 0.788580i
\(926\) 0 0
\(927\) −20521.7 + 7096.62i −0.727100 + 0.251439i
\(928\) 0 0
\(929\) −4728.40 + 8189.83i −0.166990 + 0.289235i −0.937360 0.348362i \(-0.886738\pi\)
0.770370 + 0.637597i \(0.220071\pi\)
\(930\) 0 0
\(931\) 8091.50 + 5119.86i 0.284842 + 0.180233i
\(932\) 0 0
\(933\) 37253.3 17007.1i 1.30720 0.596771i
\(934\) 0 0
\(935\) 5381.68i 0.188235i
\(936\) 0 0
\(937\) 34791.9i 1.21302i −0.795075 0.606511i \(-0.792569\pi\)
0.795075 0.606511i \(-0.207431\pi\)
\(938\) 0 0
\(939\) −24704.0 + 34682.2i −0.858555 + 1.20534i
\(940\) 0 0
\(941\) 5515.38 + 9552.92i 0.191069 + 0.330942i 0.945605 0.325317i \(-0.105471\pi\)
−0.754535 + 0.656259i \(0.772138\pi\)
\(942\) 0 0
\(943\) −2465.62 1423.52i −0.0851448 0.0491584i
\(944\) 0 0
\(945\) 11502.8 10522.7i 0.395963 0.362225i
\(946\) 0 0
\(947\) −26402.6 15243.5i −0.905985 0.523071i −0.0268478 0.999640i \(-0.508547\pi\)
−0.879137 + 0.476569i \(0.841880\pi\)
\(948\) 0 0
\(949\) −2900.72 5024.20i −0.0992218 0.171857i
\(950\) 0 0
\(951\) 26423.4 37096.2i 0.900987 1.26491i
\(952\) 0 0
\(953\) 21378.9i 0.726685i −0.931656 0.363343i \(-0.881635\pi\)
0.931656 0.363343i \(-0.118365\pi\)
\(954\) 0 0
\(955\) 12185.7i 0.412902i
\(956\) 0 0
\(957\) 29543.4 13487.3i 0.997912 0.455573i
\(958\) 0 0
\(959\) 32410.5 + 657.221i 1.09134 + 0.0221301i
\(960\) 0 0
\(961\) −2162.36 + 3745.31i −0.0725842 + 0.125720i
\(962\) 0 0
\(963\) 1448.02 7502.45i 0.0484546 0.251052i
\(964\) 0 0
\(965\) 1506.69 2609.67i 0.0502613 0.0870550i
\(966\) 0 0
\(967\) 1999.31 + 3462.91i 0.0664877 + 0.115160i 0.897353 0.441314i \(-0.145487\pi\)
−0.830865 + 0.556474i \(0.812154\pi\)
\(968\) 0 0
\(969\) 2848.24 + 272.350i 0.0944257 + 0.00902905i
\(970\) 0 0
\(971\) 48223.6 1.59379 0.796895 0.604118i \(-0.206475\pi\)
0.796895 + 0.604118i \(0.206475\pi\)
\(972\) 0 0
\(973\) 12401.8 20508.8i 0.408618 0.675727i
\(974\) 0 0
\(975\) −12100.2 1157.03i −0.397453 0.0380047i
\(976\) 0 0
\(977\) −3157.95 + 1823.24i −0.103410 + 0.0597038i −0.550813 0.834629i \(-0.685682\pi\)
0.447403 + 0.894332i \(0.352349\pi\)
\(978\) 0 0
\(979\) −5549.94 3204.26i −0.181182 0.104605i
\(980\) 0 0
\(981\) −48420.0 9345.36i −1.57587 0.304153i
\(982\) 0 0
\(983\) 10473.4 18140.5i 0.339828 0.588600i −0.644572 0.764544i \(-0.722964\pi\)
0.984400 + 0.175944i \(0.0562978\pi\)
\(984\) 0 0
\(985\) −20519.9 + 11847.2i −0.663776 + 0.383232i
\(986\) 0 0
\(987\) 20966.4 + 2434.70i 0.676157 + 0.0785180i
\(988\) 0 0
\(989\) 7722.99i 0.248308i
\(990\) 0 0
\(991\) 6732.72 0.215814 0.107907 0.994161i \(-0.465585\pi\)
0.107907 + 0.994161i \(0.465585\pi\)
\(992\) 0 0
\(993\) 29048.2 + 20690.9i 0.928316 + 0.661234i
\(994\) 0 0
\(995\) 3272.03 1889.11i 0.104252 0.0601897i
\(996\) 0 0
\(997\) −35961.0 20762.1i −1.14232 0.659520i −0.195318 0.980740i \(-0.562574\pi\)
−0.947005 + 0.321220i \(0.895907\pi\)
\(998\) 0 0
\(999\) 11392.0 38741.1i 0.360786 1.22694i
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 252.4.x.a.209.15 yes 48
3.2 odd 2 756.4.x.a.629.16 48
7.6 odd 2 inner 252.4.x.a.209.10 yes 48
9.2 odd 6 2268.4.f.a.1133.17 48
9.4 even 3 756.4.x.a.125.9 48
9.5 odd 6 inner 252.4.x.a.41.10 48
9.7 even 3 2268.4.f.a.1133.32 48
21.20 even 2 756.4.x.a.629.9 48
63.13 odd 6 756.4.x.a.125.16 48
63.20 even 6 2268.4.f.a.1133.31 48
63.34 odd 6 2268.4.f.a.1133.18 48
63.41 even 6 inner 252.4.x.a.41.15 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.x.a.41.10 48 9.5 odd 6 inner
252.4.x.a.41.15 yes 48 63.41 even 6 inner
252.4.x.a.209.10 yes 48 7.6 odd 2 inner
252.4.x.a.209.15 yes 48 1.1 even 1 trivial
756.4.x.a.125.9 48 9.4 even 3
756.4.x.a.125.16 48 63.13 odd 6
756.4.x.a.629.9 48 21.20 even 2
756.4.x.a.629.16 48 3.2 odd 2
2268.4.f.a.1133.17 48 9.2 odd 6
2268.4.f.a.1133.18 48 63.34 odd 6
2268.4.f.a.1133.31 48 63.20 even 6
2268.4.f.a.1133.32 48 9.7 even 3