Properties

Label 363.4.a.i.1.2
Level $363$
Weight $4$
Character 363.1
Self dual yes
Analytic conductor $21.418$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

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: yes
Analytic conductor: \(21.4176933321\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.42443\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.42443 q^{2} -3.00000 q^{3} +11.5756 q^{4} +2.84886 q^{5} -13.2733 q^{6} -31.6977 q^{7} +15.8199 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.42443 q^{2} -3.00000 q^{3} +11.5756 q^{4} +2.84886 q^{5} -13.2733 q^{6} -31.6977 q^{7} +15.8199 q^{8} +9.00000 q^{9} +12.6046 q^{10} -34.7267 q^{12} -5.15114 q^{13} -140.244 q^{14} -8.54657 q^{15} -22.6107 q^{16} -121.942 q^{17} +39.8199 q^{18} -34.8489 q^{19} +32.9772 q^{20} +95.0931 q^{21} +116.244 q^{23} -47.4596 q^{24} -116.884 q^{25} -22.7909 q^{26} -27.0000 q^{27} -366.919 q^{28} +69.4534 q^{29} -37.8137 q^{30} +140.605 q^{31} -226.598 q^{32} -539.524 q^{34} -90.3023 q^{35} +104.180 q^{36} -420.070 q^{37} -154.186 q^{38} +15.4534 q^{39} +45.0685 q^{40} +322.058 q^{41} +420.733 q^{42} -321.035 q^{43} +25.6397 q^{45} +514.315 q^{46} -231.408 q^{47} +67.8322 q^{48} +661.745 q^{49} -517.145 q^{50} +365.826 q^{51} -59.6274 q^{52} +4.91916 q^{53} -119.460 q^{54} -501.453 q^{56} +104.547 q^{57} +307.292 q^{58} +406.443 q^{59} -98.9315 q^{60} +556.431 q^{61} +622.095 q^{62} -285.279 q^{63} -821.683 q^{64} -14.6749 q^{65} +84.7452 q^{67} -1411.55 q^{68} -348.733 q^{69} -399.536 q^{70} +49.0808 q^{71} +142.379 q^{72} -785.884 q^{73} -1858.57 q^{74} +350.652 q^{75} -403.395 q^{76} +68.3726 q^{78} +383.118 q^{79} -64.4147 q^{80} +81.0000 q^{81} +1424.92 q^{82} +930.211 q^{83} +1100.76 q^{84} -347.395 q^{85} -1420.40 q^{86} -208.360 q^{87} -732.559 q^{89} +113.441 q^{90} +163.279 q^{91} +1345.59 q^{92} -421.814 q^{93} -1023.85 q^{94} -99.2794 q^{95} +679.795 q^{96} -1171.49 q^{97} +2927.84 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} + 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} + 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9} + 104 q^{10} - 99 q^{12} - 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} - 106 q^{17} - 9 q^{18} - 50 q^{19} - 328 q^{20} + 72 q^{21} + 134 q^{23} + 171 q^{24} + 42 q^{25} + 112 q^{26} - 54 q^{27} - 202 q^{28} + 198 q^{29} - 312 q^{30} + 360 q^{31} - 857 q^{32} - 626 q^{34} - 220 q^{35} + 297 q^{36} - 328 q^{37} - 72 q^{38} + 90 q^{39} + 1272 q^{40} + 782 q^{41} + 546 q^{42} - 386 q^{43} - 126 q^{45} + 418 q^{46} + 266 q^{47} - 603 q^{48} + 378 q^{49} - 1379 q^{50} + 318 q^{51} - 592 q^{52} - 522 q^{53} + 27 q^{54} - 1062 q^{56} + 150 q^{57} - 390 q^{58} - 172 q^{59} + 984 q^{60} + 778 q^{61} - 568 q^{62} - 216 q^{63} + 809 q^{64} + 404 q^{65} - 776 q^{67} - 1070 q^{68} - 402 q^{69} + 304 q^{70} + 630 q^{71} - 513 q^{72} - 1296 q^{73} - 2358 q^{74} - 126 q^{75} - 728 q^{76} - 336 q^{78} - 652 q^{79} - 3832 q^{80} + 162 q^{81} - 1070 q^{82} + 324 q^{83} + 606 q^{84} - 616 q^{85} - 1068 q^{86} - 594 q^{87} - 756 q^{89} + 936 q^{90} - 28 q^{91} + 1726 q^{92} - 1080 q^{93} - 3722 q^{94} + 156 q^{95} + 2571 q^{96} - 452 q^{97} + 4467 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.42443 1.56427 0.782136 0.623108i \(-0.214130\pi\)
0.782136 + 0.623108i \(0.214130\pi\)
\(3\) −3.00000 −0.577350
\(4\) 11.5756 1.44695
\(5\) 2.84886 0.254810 0.127405 0.991851i \(-0.459335\pi\)
0.127405 + 0.991851i \(0.459335\pi\)
\(6\) −13.2733 −0.903133
\(7\) −31.6977 −1.71152 −0.855758 0.517377i \(-0.826909\pi\)
−0.855758 + 0.517377i \(0.826909\pi\)
\(8\) 15.8199 0.699146
\(9\) 9.00000 0.333333
\(10\) 12.6046 0.398591
\(11\) 0 0
\(12\) −34.7267 −0.835395
\(13\) −5.15114 −0.109898 −0.0549488 0.998489i \(-0.517500\pi\)
−0.0549488 + 0.998489i \(0.517500\pi\)
\(14\) −140.244 −2.67728
\(15\) −8.54657 −0.147114
\(16\) −22.6107 −0.353293
\(17\) −121.942 −1.73972 −0.869861 0.493297i \(-0.835792\pi\)
−0.869861 + 0.493297i \(0.835792\pi\)
\(18\) 39.8199 0.521424
\(19\) −34.8489 −0.420783 −0.210391 0.977617i \(-0.567474\pi\)
−0.210391 + 0.977617i \(0.567474\pi\)
\(20\) 32.9772 0.368696
\(21\) 95.0931 0.988144
\(22\) 0 0
\(23\) 116.244 1.05385 0.526926 0.849911i \(-0.323344\pi\)
0.526926 + 0.849911i \(0.323344\pi\)
\(24\) −47.4596 −0.403652
\(25\) −116.884 −0.935072
\(26\) −22.7909 −0.171910
\(27\) −27.0000 −0.192450
\(28\) −366.919 −2.47647
\(29\) 69.4534 0.444730 0.222365 0.974963i \(-0.428622\pi\)
0.222365 + 0.974963i \(0.428622\pi\)
\(30\) −37.8137 −0.230127
\(31\) 140.605 0.814623 0.407312 0.913289i \(-0.366466\pi\)
0.407312 + 0.913289i \(0.366466\pi\)
\(32\) −226.598 −1.25179
\(33\) 0 0
\(34\) −539.524 −2.72140
\(35\) −90.3023 −0.436111
\(36\) 104.180 0.482315
\(37\) −420.070 −1.86646 −0.933232 0.359276i \(-0.883024\pi\)
−0.933232 + 0.359276i \(0.883024\pi\)
\(38\) −154.186 −0.658219
\(39\) 15.4534 0.0634495
\(40\) 45.0685 0.178149
\(41\) 322.058 1.22676 0.613378 0.789789i \(-0.289810\pi\)
0.613378 + 0.789789i \(0.289810\pi\)
\(42\) 420.733 1.54573
\(43\) −321.035 −1.13854 −0.569272 0.822149i \(-0.692775\pi\)
−0.569272 + 0.822149i \(0.692775\pi\)
\(44\) 0 0
\(45\) 25.6397 0.0849365
\(46\) 514.315 1.64851
\(47\) −231.408 −0.718176 −0.359088 0.933304i \(-0.616912\pi\)
−0.359088 + 0.933304i \(0.616912\pi\)
\(48\) 67.8322 0.203974
\(49\) 661.745 1.92929
\(50\) −517.145 −1.46271
\(51\) 365.826 1.00443
\(52\) −59.6274 −0.159016
\(53\) 4.91916 0.0127490 0.00637452 0.999980i \(-0.497971\pi\)
0.00637452 + 0.999980i \(0.497971\pi\)
\(54\) −119.460 −0.301044
\(55\) 0 0
\(56\) −501.453 −1.19660
\(57\) 104.547 0.242939
\(58\) 307.292 0.695679
\(59\) 406.443 0.896854 0.448427 0.893820i \(-0.351984\pi\)
0.448427 + 0.893820i \(0.351984\pi\)
\(60\) −98.9315 −0.212867
\(61\) 556.431 1.16793 0.583964 0.811779i \(-0.301501\pi\)
0.583964 + 0.811779i \(0.301501\pi\)
\(62\) 622.095 1.27429
\(63\) −285.279 −0.570505
\(64\) −821.683 −1.60485
\(65\) −14.6749 −0.0280030
\(66\) 0 0
\(67\) 84.7452 0.154526 0.0772632 0.997011i \(-0.475382\pi\)
0.0772632 + 0.997011i \(0.475382\pi\)
\(68\) −1411.55 −2.51728
\(69\) −348.733 −0.608442
\(70\) −399.536 −0.682196
\(71\) 49.0808 0.0820398 0.0410199 0.999158i \(-0.486939\pi\)
0.0410199 + 0.999158i \(0.486939\pi\)
\(72\) 142.379 0.233049
\(73\) −785.884 −1.26001 −0.630005 0.776591i \(-0.716947\pi\)
−0.630005 + 0.776591i \(0.716947\pi\)
\(74\) −1858.57 −2.91966
\(75\) 350.652 0.539864
\(76\) −403.395 −0.608850
\(77\) 0 0
\(78\) 68.3726 0.0992522
\(79\) 383.118 0.545622 0.272811 0.962068i \(-0.412047\pi\)
0.272811 + 0.962068i \(0.412047\pi\)
\(80\) −64.4147 −0.0900223
\(81\) 81.0000 0.111111
\(82\) 1424.92 1.91898
\(83\) 930.211 1.23017 0.615084 0.788462i \(-0.289122\pi\)
0.615084 + 0.788462i \(0.289122\pi\)
\(84\) 1100.76 1.42979
\(85\) −347.395 −0.443298
\(86\) −1420.40 −1.78099
\(87\) −208.360 −0.256765
\(88\) 0 0
\(89\) −732.559 −0.872484 −0.436242 0.899829i \(-0.643691\pi\)
−0.436242 + 0.899829i \(0.643691\pi\)
\(90\) 113.441 0.132864
\(91\) 163.279 0.188092
\(92\) 1345.59 1.52487
\(93\) −421.814 −0.470323
\(94\) −1023.85 −1.12342
\(95\) −99.2794 −0.107220
\(96\) 679.795 0.722722
\(97\) −1171.49 −1.22626 −0.613128 0.789984i \(-0.710089\pi\)
−0.613128 + 0.789984i \(0.710089\pi\)
\(98\) 2927.84 3.01793
\(99\) 0 0
\(100\) −1353.00 −1.35300
\(101\) 1221.27 1.20318 0.601589 0.798806i \(-0.294535\pi\)
0.601589 + 0.798806i \(0.294535\pi\)
\(102\) 1618.57 1.57120
\(103\) 516.745 0.494334 0.247167 0.968973i \(-0.420500\pi\)
0.247167 + 0.968973i \(0.420500\pi\)
\(104\) −81.4903 −0.0768345
\(105\) 270.907 0.251789
\(106\) 21.7645 0.0199430
\(107\) 152.025 0.137353 0.0686765 0.997639i \(-0.478122\pi\)
0.0686765 + 0.997639i \(0.478122\pi\)
\(108\) −312.540 −0.278465
\(109\) −2170.32 −1.90714 −0.953572 0.301164i \(-0.902625\pi\)
−0.953572 + 0.301164i \(0.902625\pi\)
\(110\) 0 0
\(111\) 1260.21 1.07760
\(112\) 716.708 0.604666
\(113\) −646.397 −0.538123 −0.269062 0.963123i \(-0.586714\pi\)
−0.269062 + 0.963123i \(0.586714\pi\)
\(114\) 462.559 0.380023
\(115\) 331.163 0.268532
\(116\) 803.963 0.643501
\(117\) −46.3603 −0.0366326
\(118\) 1798.28 1.40292
\(119\) 3865.28 2.97756
\(120\) −135.206 −0.102854
\(121\) 0 0
\(122\) 2461.89 1.82696
\(123\) −966.174 −0.708268
\(124\) 1627.58 1.17872
\(125\) −689.093 −0.493075
\(126\) −1262.20 −0.892425
\(127\) 993.304 0.694027 0.347014 0.937860i \(-0.387196\pi\)
0.347014 + 0.937860i \(0.387196\pi\)
\(128\) −1822.69 −1.25863
\(129\) 963.105 0.657339
\(130\) −64.9279 −0.0438043
\(131\) −385.814 −0.257318 −0.128659 0.991689i \(-0.541067\pi\)
−0.128659 + 0.991689i \(0.541067\pi\)
\(132\) 0 0
\(133\) 1104.63 0.720177
\(134\) 374.949 0.241721
\(135\) −76.9192 −0.0490381
\(136\) −1929.11 −1.21632
\(137\) 884.840 0.551803 0.275901 0.961186i \(-0.411024\pi\)
0.275901 + 0.961186i \(0.411024\pi\)
\(138\) −1542.94 −0.951769
\(139\) 1091.94 0.666312 0.333156 0.942872i \(-0.391886\pi\)
0.333156 + 0.942872i \(0.391886\pi\)
\(140\) −1045.30 −0.631029
\(141\) 694.223 0.414639
\(142\) 217.155 0.128333
\(143\) 0 0
\(144\) −203.497 −0.117764
\(145\) 197.863 0.113322
\(146\) −3477.09 −1.97100
\(147\) −1985.24 −1.11387
\(148\) −4862.55 −2.70067
\(149\) −297.014 −0.163304 −0.0816522 0.996661i \(-0.526020\pi\)
−0.0816522 + 0.996661i \(0.526020\pi\)
\(150\) 1551.43 0.844494
\(151\) 1887.86 1.01743 0.508716 0.860935i \(-0.330120\pi\)
0.508716 + 0.860935i \(0.330120\pi\)
\(152\) −551.304 −0.294189
\(153\) −1097.48 −0.579907
\(154\) 0 0
\(155\) 400.562 0.207574
\(156\) 178.882 0.0918080
\(157\) −56.5343 −0.0287384 −0.0143692 0.999897i \(-0.504574\pi\)
−0.0143692 + 0.999897i \(0.504574\pi\)
\(158\) 1695.08 0.853501
\(159\) −14.7575 −0.00736066
\(160\) −645.547 −0.318968
\(161\) −3684.68 −1.80369
\(162\) 358.379 0.173808
\(163\) −49.2338 −0.0236582 −0.0118291 0.999930i \(-0.503765\pi\)
−0.0118291 + 0.999930i \(0.503765\pi\)
\(164\) 3728.01 1.77505
\(165\) 0 0
\(166\) 4115.65 1.92432
\(167\) −2068.75 −0.958589 −0.479294 0.877654i \(-0.659107\pi\)
−0.479294 + 0.877654i \(0.659107\pi\)
\(168\) 1504.36 0.690857
\(169\) −2170.47 −0.987923
\(170\) −1537.03 −0.693438
\(171\) −313.640 −0.140261
\(172\) −3716.17 −1.64741
\(173\) 604.012 0.265446 0.132723 0.991153i \(-0.457628\pi\)
0.132723 + 0.991153i \(0.457628\pi\)
\(174\) −921.875 −0.401650
\(175\) 3704.96 1.60039
\(176\) 0 0
\(177\) −1219.33 −0.517799
\(178\) −3241.15 −1.36480
\(179\) −2132.02 −0.890251 −0.445126 0.895468i \(-0.646841\pi\)
−0.445126 + 0.895468i \(0.646841\pi\)
\(180\) 296.794 0.122899
\(181\) −589.371 −0.242031 −0.121015 0.992651i \(-0.538615\pi\)
−0.121015 + 0.992651i \(0.538615\pi\)
\(182\) 722.418 0.294226
\(183\) −1669.29 −0.674304
\(184\) 1838.97 0.736796
\(185\) −1196.72 −0.475593
\(186\) −1866.28 −0.735713
\(187\) 0 0
\(188\) −2678.68 −1.03916
\(189\) 855.838 0.329381
\(190\) −439.255 −0.167720
\(191\) −2160.90 −0.818624 −0.409312 0.912395i \(-0.634231\pi\)
−0.409312 + 0.912395i \(0.634231\pi\)
\(192\) 2465.05 0.926560
\(193\) 1490.91 0.556052 0.278026 0.960574i \(-0.410320\pi\)
0.278026 + 0.960574i \(0.410320\pi\)
\(194\) −5183.18 −1.91820
\(195\) 44.0246 0.0161675
\(196\) 7660.08 2.79157
\(197\) 230.529 0.0833732 0.0416866 0.999131i \(-0.486727\pi\)
0.0416866 + 0.999131i \(0.486727\pi\)
\(198\) 0 0
\(199\) 22.4007 0.00797963 0.00398982 0.999992i \(-0.498730\pi\)
0.00398982 + 0.999992i \(0.498730\pi\)
\(200\) −1849.09 −0.653752
\(201\) −254.236 −0.0892159
\(202\) 5403.43 1.88210
\(203\) −2201.51 −0.761163
\(204\) 4234.65 1.45336
\(205\) 917.497 0.312589
\(206\) 2286.30 0.773273
\(207\) 1046.20 0.351284
\(208\) 116.471 0.0388260
\(209\) 0 0
\(210\) 1198.61 0.393866
\(211\) 1051.64 0.343117 0.171558 0.985174i \(-0.445120\pi\)
0.171558 + 0.985174i \(0.445120\pi\)
\(212\) 56.9421 0.0184472
\(213\) −147.243 −0.0473657
\(214\) 672.622 0.214857
\(215\) −914.583 −0.290112
\(216\) −427.136 −0.134551
\(217\) −4456.84 −1.39424
\(218\) −9602.42 −2.98329
\(219\) 2357.65 0.727467
\(220\) 0 0
\(221\) 628.141 0.191191
\(222\) 5575.71 1.68566
\(223\) 3861.80 1.15966 0.579832 0.814736i \(-0.303118\pi\)
0.579832 + 0.814736i \(0.303118\pi\)
\(224\) 7182.65 2.14246
\(225\) −1051.96 −0.311691
\(226\) −2859.94 −0.841771
\(227\) 872.721 0.255174 0.127587 0.991827i \(-0.459277\pi\)
0.127587 + 0.991827i \(0.459277\pi\)
\(228\) 1210.19 0.351520
\(229\) 1841.72 0.531459 0.265730 0.964048i \(-0.414387\pi\)
0.265730 + 0.964048i \(0.414387\pi\)
\(230\) 1465.21 0.420057
\(231\) 0 0
\(232\) 1098.74 0.310931
\(233\) −3932.14 −1.10559 −0.552796 0.833317i \(-0.686439\pi\)
−0.552796 + 0.833317i \(0.686439\pi\)
\(234\) −205.118 −0.0573033
\(235\) −659.248 −0.182998
\(236\) 4704.81 1.29770
\(237\) −1149.35 −0.315015
\(238\) 17101.7 4.65772
\(239\) −4772.10 −1.29155 −0.645777 0.763526i \(-0.723466\pi\)
−0.645777 + 0.763526i \(0.723466\pi\)
\(240\) 193.244 0.0519744
\(241\) −3988.84 −1.06616 −0.533078 0.846066i \(-0.678965\pi\)
−0.533078 + 0.846066i \(0.678965\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 6441.00 1.68993
\(245\) 1885.22 0.491601
\(246\) −4274.77 −1.10792
\(247\) 179.511 0.0462431
\(248\) 2224.34 0.569540
\(249\) −2790.63 −0.710238
\(250\) −3048.84 −0.771303
\(251\) −5474.22 −1.37661 −0.688306 0.725421i \(-0.741645\pi\)
−0.688306 + 0.725421i \(0.741645\pi\)
\(252\) −3302.27 −0.825491
\(253\) 0 0
\(254\) 4394.80 1.08565
\(255\) 1042.19 0.255938
\(256\) −1490.90 −0.363989
\(257\) −6434.01 −1.56164 −0.780822 0.624754i \(-0.785199\pi\)
−0.780822 + 0.624754i \(0.785199\pi\)
\(258\) 4261.19 1.02826
\(259\) 13315.3 3.19448
\(260\) −169.870 −0.0405188
\(261\) 625.081 0.148243
\(262\) −1707.01 −0.402516
\(263\) −7589.00 −1.77931 −0.889654 0.456636i \(-0.849054\pi\)
−0.889654 + 0.456636i \(0.849054\pi\)
\(264\) 0 0
\(265\) 14.0140 0.00324858
\(266\) 4887.35 1.12655
\(267\) 2197.68 0.503729
\(268\) 980.974 0.223591
\(269\) 478.178 0.108383 0.0541914 0.998531i \(-0.482742\pi\)
0.0541914 + 0.998531i \(0.482742\pi\)
\(270\) −340.323 −0.0767090
\(271\) 122.323 0.0274192 0.0137096 0.999906i \(-0.495636\pi\)
0.0137096 + 0.999906i \(0.495636\pi\)
\(272\) 2757.20 0.614631
\(273\) −489.838 −0.108595
\(274\) 3914.91 0.863170
\(275\) 0 0
\(276\) −4036.78 −0.880383
\(277\) −8199.41 −1.77854 −0.889269 0.457385i \(-0.848786\pi\)
−0.889269 + 0.457385i \(0.848786\pi\)
\(278\) 4831.22 1.04229
\(279\) 1265.44 0.271541
\(280\) −1428.57 −0.304905
\(281\) −6943.79 −1.47413 −0.737067 0.675820i \(-0.763790\pi\)
−0.737067 + 0.675820i \(0.763790\pi\)
\(282\) 3071.54 0.648609
\(283\) −1035.14 −0.217429 −0.108715 0.994073i \(-0.534673\pi\)
−0.108715 + 0.994073i \(0.534673\pi\)
\(284\) 568.139 0.118707
\(285\) 297.838 0.0619032
\(286\) 0 0
\(287\) −10208.5 −2.09961
\(288\) −2039.39 −0.417264
\(289\) 9956.85 2.02663
\(290\) 875.430 0.177266
\(291\) 3514.47 0.707979
\(292\) −9097.06 −1.82317
\(293\) 6144.81 1.22520 0.612600 0.790393i \(-0.290124\pi\)
0.612600 + 0.790393i \(0.290124\pi\)
\(294\) −8783.53 −1.74240
\(295\) 1157.90 0.228527
\(296\) −6645.45 −1.30493
\(297\) 0 0
\(298\) −1314.12 −0.255452
\(299\) −598.791 −0.115816
\(300\) 4059.00 0.781154
\(301\) 10176.1 1.94864
\(302\) 8352.72 1.59154
\(303\) −3663.81 −0.694655
\(304\) 787.958 0.148659
\(305\) 1585.19 0.297599
\(306\) −4855.71 −0.907133
\(307\) 2186.09 0.406406 0.203203 0.979137i \(-0.434865\pi\)
0.203203 + 0.979137i \(0.434865\pi\)
\(308\) 0 0
\(309\) −1550.24 −0.285404
\(310\) 1772.26 0.324702
\(311\) −7484.83 −1.36471 −0.682357 0.731019i \(-0.739045\pi\)
−0.682357 + 0.731019i \(0.739045\pi\)
\(312\) 244.471 0.0443604
\(313\) −6833.33 −1.23400 −0.617001 0.786962i \(-0.711653\pi\)
−0.617001 + 0.786962i \(0.711653\pi\)
\(314\) −250.132 −0.0449546
\(315\) −812.721 −0.145370
\(316\) 4434.81 0.789485
\(317\) 924.265 0.163760 0.0818800 0.996642i \(-0.473908\pi\)
0.0818800 + 0.996642i \(0.473908\pi\)
\(318\) −65.2934 −0.0115141
\(319\) 0 0
\(320\) −2340.86 −0.408931
\(321\) −456.074 −0.0793008
\(322\) −16302.6 −2.82145
\(323\) 4249.54 0.732046
\(324\) 937.621 0.160772
\(325\) 602.086 0.102762
\(326\) −217.831 −0.0370078
\(327\) 6510.95 1.10109
\(328\) 5094.91 0.857681
\(329\) 7335.10 1.22917
\(330\) 0 0
\(331\) −9820.46 −1.63076 −0.815380 0.578927i \(-0.803472\pi\)
−0.815380 + 0.578927i \(0.803472\pi\)
\(332\) 10767.7 1.77999
\(333\) −3780.63 −0.622154
\(334\) −9153.02 −1.49949
\(335\) 241.427 0.0393748
\(336\) −2150.12 −0.349104
\(337\) −600.808 −0.0971161 −0.0485580 0.998820i \(-0.515463\pi\)
−0.0485580 + 0.998820i \(0.515463\pi\)
\(338\) −9603.07 −1.54538
\(339\) 1939.19 0.310686
\(340\) −4021.30 −0.641428
\(341\) 0 0
\(342\) −1387.68 −0.219406
\(343\) −10103.5 −1.59049
\(344\) −5078.73 −0.796008
\(345\) −993.490 −0.155037
\(346\) 2672.41 0.415230
\(347\) 3143.41 0.486303 0.243152 0.969988i \(-0.421819\pi\)
0.243152 + 0.969988i \(0.421819\pi\)
\(348\) −2411.89 −0.371525
\(349\) −720.663 −0.110533 −0.0552667 0.998472i \(-0.517601\pi\)
−0.0552667 + 0.998472i \(0.517601\pi\)
\(350\) 16392.3 2.50345
\(351\) 139.081 0.0211498
\(352\) 0 0
\(353\) 1207.12 0.182007 0.0910034 0.995851i \(-0.470993\pi\)
0.0910034 + 0.995851i \(0.470993\pi\)
\(354\) −5394.83 −0.809978
\(355\) 139.824 0.0209045
\(356\) −8479.79 −1.26244
\(357\) −11595.8 −1.71910
\(358\) −9432.99 −1.39260
\(359\) −8748.31 −1.28612 −0.643062 0.765814i \(-0.722336\pi\)
−0.643062 + 0.765814i \(0.722336\pi\)
\(360\) 405.617 0.0593830
\(361\) −5644.56 −0.822942
\(362\) −2607.63 −0.378602
\(363\) 0 0
\(364\) 1890.05 0.272158
\(365\) −2238.87 −0.321063
\(366\) −7385.66 −1.05479
\(367\) −6730.45 −0.957293 −0.478647 0.878008i \(-0.658872\pi\)
−0.478647 + 0.878008i \(0.658872\pi\)
\(368\) −2628.37 −0.372318
\(369\) 2898.52 0.408919
\(370\) −5294.81 −0.743956
\(371\) −155.926 −0.0218202
\(372\) −4882.73 −0.680532
\(373\) 227.394 0.0315657 0.0157828 0.999875i \(-0.494976\pi\)
0.0157828 + 0.999875i \(0.494976\pi\)
\(374\) 0 0
\(375\) 2067.28 0.284677
\(376\) −3660.84 −0.502110
\(377\) −357.764 −0.0488748
\(378\) 3786.60 0.515242
\(379\) 11356.2 1.53913 0.769565 0.638568i \(-0.220473\pi\)
0.769565 + 0.638568i \(0.220473\pi\)
\(380\) −1149.22 −0.155141
\(381\) −2979.91 −0.400697
\(382\) −9560.74 −1.28055
\(383\) 10753.6 1.43468 0.717338 0.696725i \(-0.245360\pi\)
0.717338 + 0.696725i \(0.245360\pi\)
\(384\) 5468.07 0.726670
\(385\) 0 0
\(386\) 6596.43 0.869817
\(387\) −2889.32 −0.379515
\(388\) −13560.7 −1.77433
\(389\) −11727.1 −1.52850 −0.764252 0.644918i \(-0.776891\pi\)
−0.764252 + 0.644918i \(0.776891\pi\)
\(390\) 194.784 0.0252904
\(391\) −14175.1 −1.83341
\(392\) 10468.7 1.34885
\(393\) 1157.44 0.148563
\(394\) 1019.96 0.130418
\(395\) 1091.45 0.139030
\(396\) 0 0
\(397\) −359.905 −0.0454990 −0.0227495 0.999741i \(-0.507242\pi\)
−0.0227495 + 0.999741i \(0.507242\pi\)
\(398\) 99.1105 0.0124823
\(399\) −3313.89 −0.415794
\(400\) 2642.83 0.330354
\(401\) −4066.71 −0.506438 −0.253219 0.967409i \(-0.581489\pi\)
−0.253219 + 0.967409i \(0.581489\pi\)
\(402\) −1124.85 −0.139558
\(403\) −724.274 −0.0895252
\(404\) 14136.9 1.74093
\(405\) 230.757 0.0283122
\(406\) −9740.45 −1.19067
\(407\) 0 0
\(408\) 5787.32 0.702242
\(409\) 13488.8 1.63076 0.815379 0.578927i \(-0.196528\pi\)
0.815379 + 0.578927i \(0.196528\pi\)
\(410\) 4059.40 0.488975
\(411\) −2654.52 −0.318584
\(412\) 5981.62 0.715275
\(413\) −12883.3 −1.53498
\(414\) 4628.83 0.549504
\(415\) 2650.04 0.313459
\(416\) 1167.24 0.137569
\(417\) −3275.83 −0.384695
\(418\) 0 0
\(419\) −7040.12 −0.820841 −0.410420 0.911896i \(-0.634618\pi\)
−0.410420 + 0.911896i \(0.634618\pi\)
\(420\) 3135.90 0.364325
\(421\) 9171.74 1.06177 0.530883 0.847445i \(-0.321860\pi\)
0.530883 + 0.847445i \(0.321860\pi\)
\(422\) 4652.89 0.536728
\(423\) −2082.67 −0.239392
\(424\) 77.8204 0.00891343
\(425\) 14253.1 1.62677
\(426\) −651.464 −0.0740928
\(427\) −17637.6 −1.99893
\(428\) 1759.77 0.198742
\(429\) 0 0
\(430\) −4046.51 −0.453814
\(431\) −992.995 −0.110976 −0.0554882 0.998459i \(-0.517672\pi\)
−0.0554882 + 0.998459i \(0.517672\pi\)
\(432\) 610.490 0.0679912
\(433\) 3790.21 0.420660 0.210330 0.977630i \(-0.432546\pi\)
0.210330 + 0.977630i \(0.432546\pi\)
\(434\) −19719.0 −2.18097
\(435\) −593.589 −0.0654262
\(436\) −25122.7 −2.75954
\(437\) −4050.98 −0.443443
\(438\) 10431.3 1.13796
\(439\) 5136.97 0.558483 0.279242 0.960221i \(-0.409917\pi\)
0.279242 + 0.960221i \(0.409917\pi\)
\(440\) 0 0
\(441\) 5955.71 0.643095
\(442\) 2779.16 0.299075
\(443\) 10676.8 1.14508 0.572541 0.819876i \(-0.305958\pi\)
0.572541 + 0.819876i \(0.305958\pi\)
\(444\) 14587.7 1.55923
\(445\) −2086.96 −0.222317
\(446\) 17086.2 1.81403
\(447\) 891.042 0.0942838
\(448\) 26045.5 2.74672
\(449\) 10529.9 1.10676 0.553379 0.832929i \(-0.313338\pi\)
0.553379 + 0.832929i \(0.313338\pi\)
\(450\) −4654.30 −0.487569
\(451\) 0 0
\(452\) −7482.42 −0.778636
\(453\) −5663.59 −0.587414
\(454\) 3861.29 0.399162
\(455\) 465.160 0.0479275
\(456\) 1653.91 0.169850
\(457\) 14072.5 1.44045 0.720225 0.693741i \(-0.244039\pi\)
0.720225 + 0.693741i \(0.244039\pi\)
\(458\) 8148.55 0.831347
\(459\) 3292.43 0.334810
\(460\) 3833.41 0.388551
\(461\) 30.8173 0.00311346 0.00155673 0.999999i \(-0.499504\pi\)
0.00155673 + 0.999999i \(0.499504\pi\)
\(462\) 0 0
\(463\) 17591.3 1.76573 0.882867 0.469622i \(-0.155610\pi\)
0.882867 + 0.469622i \(0.155610\pi\)
\(464\) −1570.39 −0.157120
\(465\) −1201.69 −0.119843
\(466\) −17397.5 −1.72945
\(467\) 13273.1 1.31522 0.657609 0.753360i \(-0.271568\pi\)
0.657609 + 0.753360i \(0.271568\pi\)
\(468\) −536.647 −0.0530053
\(469\) −2686.23 −0.264474
\(470\) −2916.79 −0.286259
\(471\) 169.603 0.0165921
\(472\) 6429.87 0.627031
\(473\) 0 0
\(474\) −5085.23 −0.492769
\(475\) 4073.27 0.393462
\(476\) 44742.9 4.30837
\(477\) 44.2724 0.00424968
\(478\) −21113.8 −2.02034
\(479\) −2496.68 −0.238155 −0.119077 0.992885i \(-0.537994\pi\)
−0.119077 + 0.992885i \(0.537994\pi\)
\(480\) 1936.64 0.184156
\(481\) 2163.84 0.205120
\(482\) −17648.3 −1.66776
\(483\) 11054.0 1.04136
\(484\) 0 0
\(485\) −3337.41 −0.312462
\(486\) −1075.14 −0.100348
\(487\) −3464.42 −0.322357 −0.161178 0.986925i \(-0.551529\pi\)
−0.161178 + 0.986925i \(0.551529\pi\)
\(488\) 8802.65 0.816552
\(489\) 147.701 0.0136591
\(490\) 8341.01 0.768997
\(491\) 16224.6 1.49125 0.745625 0.666366i \(-0.232151\pi\)
0.745625 + 0.666366i \(0.232151\pi\)
\(492\) −11184.0 −1.02483
\(493\) −8469.29 −0.773707
\(494\) 794.236 0.0723367
\(495\) 0 0
\(496\) −3179.17 −0.287800
\(497\) −1555.75 −0.140412
\(498\) −12347.0 −1.11100
\(499\) 9993.81 0.896562 0.448281 0.893893i \(-0.352036\pi\)
0.448281 + 0.893893i \(0.352036\pi\)
\(500\) −7976.65 −0.713453
\(501\) 6206.24 0.553441
\(502\) −24220.3 −2.15340
\(503\) 15334.8 1.35933 0.679667 0.733520i \(-0.262124\pi\)
0.679667 + 0.733520i \(0.262124\pi\)
\(504\) −4513.08 −0.398866
\(505\) 3479.23 0.306581
\(506\) 0 0
\(507\) 6511.40 0.570377
\(508\) 11498.1 1.00422
\(509\) −7291.23 −0.634927 −0.317464 0.948270i \(-0.602831\pi\)
−0.317464 + 0.948270i \(0.602831\pi\)
\(510\) 4611.08 0.400357
\(511\) 24910.7 2.15653
\(512\) 7985.14 0.689251
\(513\) 940.919 0.0809797
\(514\) −28466.8 −2.44283
\(515\) 1472.13 0.125961
\(516\) 11148.5 0.951134
\(517\) 0 0
\(518\) 58912.5 4.99704
\(519\) −1812.04 −0.153255
\(520\) −232.154 −0.0195782
\(521\) 16794.3 1.41223 0.706114 0.708098i \(-0.250447\pi\)
0.706114 + 0.708098i \(0.250447\pi\)
\(522\) 2765.63 0.231893
\(523\) 21009.4 1.75655 0.878275 0.478157i \(-0.158695\pi\)
0.878275 + 0.478157i \(0.158695\pi\)
\(524\) −4466.01 −0.372326
\(525\) −11114.9 −0.923986
\(526\) −33577.0 −2.78332
\(527\) −17145.6 −1.41722
\(528\) 0 0
\(529\) 1345.73 0.110605
\(530\) 62.0039 0.00508166
\(531\) 3657.99 0.298951
\(532\) 12786.7 1.04206
\(533\) −1658.97 −0.134818
\(534\) 9723.46 0.787969
\(535\) 433.097 0.0349989
\(536\) 1340.66 0.108036
\(537\) 6396.07 0.513987
\(538\) 2115.66 0.169540
\(539\) 0 0
\(540\) −890.383 −0.0709555
\(541\) 16802.8 1.33532 0.667662 0.744464i \(-0.267295\pi\)
0.667662 + 0.744464i \(0.267295\pi\)
\(542\) 541.211 0.0428911
\(543\) 1768.11 0.139737
\(544\) 27631.9 2.17777
\(545\) −6182.93 −0.485959
\(546\) −2167.25 −0.169872
\(547\) −16784.5 −1.31198 −0.655990 0.754770i \(-0.727749\pi\)
−0.655990 + 0.754770i \(0.727749\pi\)
\(548\) 10242.5 0.798429
\(549\) 5007.88 0.389309
\(550\) 0 0
\(551\) −2420.37 −0.187135
\(552\) −5516.91 −0.425390
\(553\) −12144.0 −0.933840
\(554\) −36277.7 −2.78212
\(555\) 3590.16 0.274584
\(556\) 12639.9 0.964117
\(557\) −18127.0 −1.37893 −0.689467 0.724317i \(-0.742155\pi\)
−0.689467 + 0.724317i \(0.742155\pi\)
\(558\) 5598.85 0.424764
\(559\) 1653.70 0.125123
\(560\) 2041.80 0.154075
\(561\) 0 0
\(562\) −30722.3 −2.30595
\(563\) −2090.88 −0.156518 −0.0782592 0.996933i \(-0.524936\pi\)
−0.0782592 + 0.996933i \(0.524936\pi\)
\(564\) 8036.03 0.599961
\(565\) −1841.49 −0.137119
\(566\) −4579.89 −0.340119
\(567\) −2567.51 −0.190168
\(568\) 776.452 0.0573578
\(569\) −6249.23 −0.460424 −0.230212 0.973140i \(-0.573942\pi\)
−0.230212 + 0.973140i \(0.573942\pi\)
\(570\) 1317.76 0.0968335
\(571\) −6048.79 −0.443317 −0.221659 0.975124i \(-0.571147\pi\)
−0.221659 + 0.975124i \(0.571147\pi\)
\(572\) 0 0
\(573\) 6482.69 0.472633
\(574\) −45166.8 −3.28437
\(575\) −13587.1 −0.985428
\(576\) −7395.15 −0.534950
\(577\) −15729.1 −1.13486 −0.567429 0.823423i \(-0.692062\pi\)
−0.567429 + 0.823423i \(0.692062\pi\)
\(578\) 44053.4 3.17021
\(579\) −4472.73 −0.321037
\(580\) 2290.38 0.163970
\(581\) −29485.6 −2.10545
\(582\) 15549.5 1.10747
\(583\) 0 0
\(584\) −12432.6 −0.880931
\(585\) −132.074 −0.00933433
\(586\) 27187.3 1.91655
\(587\) 15620.5 1.09835 0.549173 0.835709i \(-0.314943\pi\)
0.549173 + 0.835709i \(0.314943\pi\)
\(588\) −22980.2 −1.61172
\(589\) −4899.91 −0.342780
\(590\) 5123.04 0.357478
\(591\) −691.587 −0.0481355
\(592\) 9498.09 0.659407
\(593\) 493.541 0.0341776 0.0170888 0.999854i \(-0.494560\pi\)
0.0170888 + 0.999854i \(0.494560\pi\)
\(594\) 0 0
\(595\) 11011.6 0.758711
\(596\) −3438.11 −0.236293
\(597\) −67.2022 −0.00460704
\(598\) −2649.31 −0.181168
\(599\) −12455.1 −0.849585 −0.424793 0.905291i \(-0.639653\pi\)
−0.424793 + 0.905291i \(0.639653\pi\)
\(600\) 5547.27 0.377444
\(601\) −12454.8 −0.845329 −0.422664 0.906286i \(-0.638905\pi\)
−0.422664 + 0.906286i \(0.638905\pi\)
\(602\) 45023.3 3.04820
\(603\) 762.707 0.0515088
\(604\) 21853.1 1.47217
\(605\) 0 0
\(606\) −16210.3 −1.08663
\(607\) 4243.19 0.283733 0.141867 0.989886i \(-0.454690\pi\)
0.141867 + 0.989886i \(0.454690\pi\)
\(608\) 7896.70 0.526732
\(609\) 6604.54 0.439458
\(610\) 7013.57 0.465526
\(611\) 1192.01 0.0789259
\(612\) −12703.9 −0.839095
\(613\) −5733.14 −0.377748 −0.188874 0.982001i \(-0.560484\pi\)
−0.188874 + 0.982001i \(0.560484\pi\)
\(614\) 9672.18 0.635729
\(615\) −2752.49 −0.180473
\(616\) 0 0
\(617\) 15642.1 1.02063 0.510314 0.859988i \(-0.329529\pi\)
0.510314 + 0.859988i \(0.329529\pi\)
\(618\) −6858.91 −0.446449
\(619\) −7467.40 −0.484879 −0.242440 0.970167i \(-0.577948\pi\)
−0.242440 + 0.970167i \(0.577948\pi\)
\(620\) 4636.74 0.300348
\(621\) −3138.60 −0.202814
\(622\) −33116.1 −2.13478
\(623\) 23220.4 1.49327
\(624\) −349.413 −0.0224162
\(625\) 12647.4 0.809432
\(626\) −30233.6 −1.93031
\(627\) 0 0
\(628\) −654.416 −0.0415829
\(629\) 51224.2 3.24713
\(630\) −3595.82 −0.227399
\(631\) −1486.38 −0.0937745 −0.0468872 0.998900i \(-0.514930\pi\)
−0.0468872 + 0.998900i \(0.514930\pi\)
\(632\) 6060.87 0.381469
\(633\) −3154.91 −0.198099
\(634\) 4089.35 0.256165
\(635\) 2829.78 0.176845
\(636\) −170.826 −0.0106505
\(637\) −3408.74 −0.212024
\(638\) 0 0
\(639\) 441.728 0.0273466
\(640\) −5192.58 −0.320711
\(641\) 12386.0 0.763211 0.381606 0.924325i \(-0.375371\pi\)
0.381606 + 0.924325i \(0.375371\pi\)
\(642\) −2017.87 −0.124048
\(643\) −14458.1 −0.886737 −0.443369 0.896339i \(-0.646217\pi\)
−0.443369 + 0.896339i \(0.646217\pi\)
\(644\) −42652.3 −2.60984
\(645\) 2743.75 0.167496
\(646\) 18801.8 1.14512
\(647\) 15792.8 0.959625 0.479813 0.877371i \(-0.340705\pi\)
0.479813 + 0.877371i \(0.340705\pi\)
\(648\) 1281.41 0.0776828
\(649\) 0 0
\(650\) 2663.89 0.160748
\(651\) 13370.5 0.804965
\(652\) −569.909 −0.0342321
\(653\) −3179.93 −0.190567 −0.0952837 0.995450i \(-0.530376\pi\)
−0.0952837 + 0.995450i \(0.530376\pi\)
\(654\) 28807.3 1.72240
\(655\) −1099.13 −0.0655672
\(656\) −7281.96 −0.433404
\(657\) −7072.96 −0.420003
\(658\) 32453.6 1.92276
\(659\) −11593.5 −0.685308 −0.342654 0.939462i \(-0.611326\pi\)
−0.342654 + 0.939462i \(0.611326\pi\)
\(660\) 0 0
\(661\) 3233.88 0.190293 0.0951464 0.995463i \(-0.469668\pi\)
0.0951464 + 0.995463i \(0.469668\pi\)
\(662\) −43449.9 −2.55095
\(663\) −1884.42 −0.110384
\(664\) 14715.8 0.860066
\(665\) 3146.93 0.183508
\(666\) −16727.1 −0.973219
\(667\) 8073.56 0.468680
\(668\) −23946.9 −1.38703
\(669\) −11585.4 −0.669532
\(670\) 1068.18 0.0615929
\(671\) 0 0
\(672\) −21548.0 −1.23695
\(673\) 5495.72 0.314776 0.157388 0.987537i \(-0.449693\pi\)
0.157388 + 0.987537i \(0.449693\pi\)
\(674\) −2658.23 −0.151916
\(675\) 3155.87 0.179955
\(676\) −25124.4 −1.42947
\(677\) −33836.7 −1.92090 −0.960451 0.278448i \(-0.910180\pi\)
−0.960451 + 0.278448i \(0.910180\pi\)
\(678\) 8579.82 0.485997
\(679\) 37133.6 2.09876
\(680\) −5495.75 −0.309930
\(681\) −2618.16 −0.147325
\(682\) 0 0
\(683\) −21080.3 −1.18099 −0.590493 0.807043i \(-0.701067\pi\)
−0.590493 + 0.807043i \(0.701067\pi\)
\(684\) −3630.56 −0.202950
\(685\) 2520.78 0.140605
\(686\) −44702.2 −2.48796
\(687\) −5525.16 −0.306838
\(688\) 7258.84 0.402239
\(689\) −25.3393 −0.00140109
\(690\) −4395.63 −0.242520
\(691\) 11811.3 0.650253 0.325127 0.945671i \(-0.394593\pi\)
0.325127 + 0.945671i \(0.394593\pi\)
\(692\) 6991.79 0.384087
\(693\) 0 0
\(694\) 13907.8 0.760711
\(695\) 3110.79 0.169783
\(696\) −3296.23 −0.179516
\(697\) −39272.4 −2.13422
\(698\) −3188.52 −0.172904
\(699\) 11796.4 0.638313
\(700\) 42887.0 2.31568
\(701\) −4244.99 −0.228718 −0.114359 0.993440i \(-0.536481\pi\)
−0.114359 + 0.993440i \(0.536481\pi\)
\(702\) 615.353 0.0330841
\(703\) 14639.0 0.785376
\(704\) 0 0
\(705\) 1977.74 0.105654
\(706\) 5340.81 0.284708
\(707\) −38711.5 −2.05926
\(708\) −14114.4 −0.749227
\(709\) −898.822 −0.0476107 −0.0238053 0.999717i \(-0.507578\pi\)
−0.0238053 + 0.999717i \(0.507578\pi\)
\(710\) 618.643 0.0327004
\(711\) 3448.06 0.181874
\(712\) −11589.0 −0.609993
\(713\) 16344.5 0.858493
\(714\) −51305.0 −2.68913
\(715\) 0 0
\(716\) −24679.4 −1.28815
\(717\) 14316.3 0.745679
\(718\) −38706.3 −2.01185
\(719\) 10741.8 0.557165 0.278582 0.960412i \(-0.410135\pi\)
0.278582 + 0.960412i \(0.410135\pi\)
\(720\) −579.733 −0.0300074
\(721\) −16379.6 −0.846061
\(722\) −24973.9 −1.28730
\(723\) 11966.5 0.615546
\(724\) −6822.30 −0.350206
\(725\) −8117.99 −0.415855
\(726\) 0 0
\(727\) 16794.2 0.856758 0.428379 0.903599i \(-0.359085\pi\)
0.428379 + 0.903599i \(0.359085\pi\)
\(728\) 2583.06 0.131503
\(729\) 729.000 0.0370370
\(730\) −9905.73 −0.502229
\(731\) 39147.7 1.98075
\(732\) −19323.0 −0.975681
\(733\) −8659.40 −0.436347 −0.218173 0.975910i \(-0.570010\pi\)
−0.218173 + 0.975910i \(0.570010\pi\)
\(734\) −29778.4 −1.49747
\(735\) −5655.65 −0.283826
\(736\) −26340.8 −1.31920
\(737\) 0 0
\(738\) 12824.3 0.639660
\(739\) −16705.7 −0.831567 −0.415783 0.909464i \(-0.636493\pi\)
−0.415783 + 0.909464i \(0.636493\pi\)
\(740\) −13852.7 −0.688157
\(741\) −538.534 −0.0266984
\(742\) −689.884 −0.0341327
\(743\) −1292.12 −0.0637996 −0.0318998 0.999491i \(-0.510156\pi\)
−0.0318998 + 0.999491i \(0.510156\pi\)
\(744\) −6673.03 −0.328824
\(745\) −846.151 −0.0416115
\(746\) 1006.09 0.0493773
\(747\) 8371.90 0.410056
\(748\) 0 0
\(749\) −4818.83 −0.235082
\(750\) 9146.53 0.445312
\(751\) −14980.4 −0.727886 −0.363943 0.931421i \(-0.618570\pi\)
−0.363943 + 0.931421i \(0.618570\pi\)
\(752\) 5232.30 0.253726
\(753\) 16422.7 0.794787
\(754\) −1582.90 −0.0764535
\(755\) 5378.25 0.259251
\(756\) 9906.82 0.476597
\(757\) 3003.41 0.144202 0.0721010 0.997397i \(-0.477030\pi\)
0.0721010 + 0.997397i \(0.477030\pi\)
\(758\) 50244.8 2.40762
\(759\) 0 0
\(760\) −1570.59 −0.0749621
\(761\) 20375.0 0.970555 0.485277 0.874360i \(-0.338719\pi\)
0.485277 + 0.874360i \(0.338719\pi\)
\(762\) −13184.4 −0.626799
\(763\) 68794.1 3.26411
\(764\) −25013.6 −1.18450
\(765\) −3126.56 −0.147766
\(766\) 47578.4 2.24422
\(767\) −2093.65 −0.0985621
\(768\) 4472.70 0.210149
\(769\) 12372.4 0.580184 0.290092 0.956999i \(-0.406314\pi\)
0.290092 + 0.956999i \(0.406314\pi\)
\(770\) 0 0
\(771\) 19302.0 0.901615
\(772\) 17258.1 0.804578
\(773\) 21023.6 0.978225 0.489113 0.872221i \(-0.337321\pi\)
0.489113 + 0.872221i \(0.337321\pi\)
\(774\) −12783.6 −0.593664
\(775\) −16434.4 −0.761732
\(776\) −18532.8 −0.857331
\(777\) −39945.8 −1.84433
\(778\) −51885.7 −2.39099
\(779\) −11223.4 −0.516198
\(780\) 509.610 0.0233935
\(781\) 0 0
\(782\) −62716.6 −2.86795
\(783\) −1875.24 −0.0855884
\(784\) −14962.5 −0.681602
\(785\) −161.058 −0.00732281
\(786\) 5121.02 0.232393
\(787\) −30286.2 −1.37177 −0.685886 0.727709i \(-0.740585\pi\)
−0.685886 + 0.727709i \(0.740585\pi\)
\(788\) 2668.51 0.120637
\(789\) 22767.0 1.02728
\(790\) 4829.03 0.217480
\(791\) 20489.3 0.921007
\(792\) 0 0
\(793\) −2866.25 −0.128353
\(794\) −1592.37 −0.0711729
\(795\) −42.0420 −0.00187557
\(796\) 259.301 0.0115461
\(797\) 32337.8 1.43722 0.718610 0.695413i \(-0.244779\pi\)
0.718610 + 0.695413i \(0.244779\pi\)
\(798\) −14662.1 −0.650415
\(799\) 28218.3 1.24943
\(800\) 26485.7 1.17052
\(801\) −6593.03 −0.290828
\(802\) −17992.9 −0.792207
\(803\) 0 0
\(804\) −2942.92 −0.129091
\(805\) −10497.1 −0.459596
\(806\) −3204.50 −0.140042
\(807\) −1434.53 −0.0625749
\(808\) 19320.3 0.841197
\(809\) 891.707 0.0387525 0.0193762 0.999812i \(-0.493832\pi\)
0.0193762 + 0.999812i \(0.493832\pi\)
\(810\) 1020.97 0.0442879
\(811\) 10114.9 0.437957 0.218978 0.975730i \(-0.429728\pi\)
0.218978 + 0.975730i \(0.429728\pi\)
\(812\) −25483.8 −1.10136
\(813\) −366.970 −0.0158305
\(814\) 0 0
\(815\) −140.260 −0.00602833
\(816\) −8271.59 −0.354857
\(817\) 11187.7 0.479080
\(818\) 59680.4 2.55095
\(819\) 1469.51 0.0626972
\(820\) 10620.6 0.452300
\(821\) −10833.5 −0.460525 −0.230262 0.973129i \(-0.573958\pi\)
−0.230262 + 0.973129i \(0.573958\pi\)
\(822\) −11744.7 −0.498351
\(823\) 31958.5 1.35359 0.676794 0.736173i \(-0.263369\pi\)
0.676794 + 0.736173i \(0.263369\pi\)
\(824\) 8174.84 0.345612
\(825\) 0 0
\(826\) −57001.3 −2.40112
\(827\) −34847.3 −1.46525 −0.732624 0.680634i \(-0.761704\pi\)
−0.732624 + 0.680634i \(0.761704\pi\)
\(828\) 12110.3 0.508289
\(829\) 6537.91 0.273910 0.136955 0.990577i \(-0.456268\pi\)
0.136955 + 0.990577i \(0.456268\pi\)
\(830\) 11724.9 0.490334
\(831\) 24598.2 1.02684
\(832\) 4232.60 0.176369
\(833\) −80694.5 −3.35642
\(834\) −14493.7 −0.601768
\(835\) −5893.56 −0.244258
\(836\) 0 0
\(837\) −3796.32 −0.156774
\(838\) −31148.5 −1.28402
\(839\) 2710.34 0.111527 0.0557635 0.998444i \(-0.482241\pi\)
0.0557635 + 0.998444i \(0.482241\pi\)
\(840\) 4285.71 0.176037
\(841\) −19565.2 −0.802215
\(842\) 40579.7 1.66089
\(843\) 20831.4 0.851092
\(844\) 12173.3 0.496471
\(845\) −6183.35 −0.251732
\(846\) −9214.62 −0.374474
\(847\) 0 0
\(848\) −111.226 −0.00450414
\(849\) 3105.41 0.125533
\(850\) 63061.7 2.54470
\(851\) −48830.8 −1.96698
\(852\) −1704.42 −0.0685356
\(853\) −9759.32 −0.391738 −0.195869 0.980630i \(-0.562753\pi\)
−0.195869 + 0.980630i \(0.562753\pi\)
\(854\) −78036.2 −3.12687
\(855\) −893.515 −0.0357398
\(856\) 2405.01 0.0960298
\(857\) 13649.8 0.544072 0.272036 0.962287i \(-0.412303\pi\)
0.272036 + 0.962287i \(0.412303\pi\)
\(858\) 0 0
\(859\) 7796.42 0.309674 0.154837 0.987940i \(-0.450515\pi\)
0.154837 + 0.987940i \(0.450515\pi\)
\(860\) −10586.8 −0.419776
\(861\) 30625.5 1.21221
\(862\) −4393.43 −0.173597
\(863\) 7183.57 0.283350 0.141675 0.989913i \(-0.454751\pi\)
0.141675 + 0.989913i \(0.454751\pi\)
\(864\) 6118.16 0.240907
\(865\) 1720.75 0.0676383
\(866\) 16769.5 0.658026
\(867\) −29870.6 −1.17008
\(868\) −51590.5 −2.01739
\(869\) 0 0
\(870\) −2626.29 −0.102344
\(871\) −436.534 −0.0169821
\(872\) −34334.1 −1.33337
\(873\) −10543.4 −0.408752
\(874\) −17923.3 −0.693666
\(875\) 21842.7 0.843905
\(876\) 27291.2 1.05261
\(877\) −17063.1 −0.656991 −0.328495 0.944506i \(-0.606542\pi\)
−0.328495 + 0.944506i \(0.606542\pi\)
\(878\) 22728.2 0.873620
\(879\) −18434.4 −0.707369
\(880\) 0 0
\(881\) −32174.9 −1.23042 −0.615210 0.788363i \(-0.710929\pi\)
−0.615210 + 0.788363i \(0.710929\pi\)
\(882\) 26350.6 1.00598
\(883\) 2843.68 0.108378 0.0541889 0.998531i \(-0.482743\pi\)
0.0541889 + 0.998531i \(0.482743\pi\)
\(884\) 7271.09 0.276644
\(885\) −3473.69 −0.131940
\(886\) 47238.9 1.79122
\(887\) 31417.8 1.18930 0.594649 0.803985i \(-0.297291\pi\)
0.594649 + 0.803985i \(0.297291\pi\)
\(888\) 19936.4 0.753401
\(889\) −31485.5 −1.18784
\(890\) −9233.59 −0.347765
\(891\) 0 0
\(892\) 44702.5 1.67797
\(893\) 8064.30 0.302196
\(894\) 3942.35 0.147485
\(895\) −6073.83 −0.226845
\(896\) 57775.1 2.15416
\(897\) 1796.37 0.0668664
\(898\) 46588.6 1.73127
\(899\) 9765.47 0.362288
\(900\) −12177.0 −0.451000
\(901\) −599.852 −0.0221798
\(902\) 0 0
\(903\) −30528.2 −1.12505
\(904\) −10225.9 −0.376227
\(905\) −1679.03 −0.0616718
\(906\) −25058.1 −0.918875
\(907\) 12253.1 0.448573 0.224287 0.974523i \(-0.427995\pi\)
0.224287 + 0.974523i \(0.427995\pi\)
\(908\) 10102.2 0.369223
\(909\) 10991.4 0.401059
\(910\) 2058.07 0.0749717
\(911\) −48422.4 −1.76104 −0.880518 0.474012i \(-0.842805\pi\)
−0.880518 + 0.474012i \(0.842805\pi\)
\(912\) −2363.87 −0.0858286
\(913\) 0 0
\(914\) 62262.9 2.25326
\(915\) −4755.57 −0.171819
\(916\) 21318.9 0.768993
\(917\) 12229.4 0.440404
\(918\) 14567.1 0.523733
\(919\) −5546.18 −0.199077 −0.0995385 0.995034i \(-0.531737\pi\)
−0.0995385 + 0.995034i \(0.531737\pi\)
\(920\) 5238.96 0.187743
\(921\) −6558.26 −0.234638
\(922\) 136.349 0.00487030
\(923\) −252.822 −0.00901598
\(924\) 0 0
\(925\) 49099.5 1.74528
\(926\) 77831.3 2.76209
\(927\) 4650.71 0.164778
\(928\) −15738.0 −0.556709
\(929\) −35684.5 −1.26025 −0.630125 0.776494i \(-0.716996\pi\)
−0.630125 + 0.776494i \(0.716996\pi\)
\(930\) −5316.78 −0.187467
\(931\) −23061.1 −0.811811
\(932\) −45516.7 −1.59973
\(933\) 22454.5 0.787918
\(934\) 58726.0 2.05736
\(935\) 0 0
\(936\) −733.413 −0.0256115
\(937\) 48903.6 1.70503 0.852514 0.522705i \(-0.175077\pi\)
0.852514 + 0.522705i \(0.175077\pi\)
\(938\) −11885.0 −0.413710
\(939\) 20500.0 0.712451
\(940\) −7631.17 −0.264789
\(941\) 23741.9 0.822490 0.411245 0.911525i \(-0.365094\pi\)
0.411245 + 0.911525i \(0.365094\pi\)
\(942\) 750.396 0.0259546
\(943\) 37437.4 1.29282
\(944\) −9189.97 −0.316852
\(945\) 2438.16 0.0839295
\(946\) 0 0
\(947\) 37612.4 1.29064 0.645321 0.763911i \(-0.276724\pi\)
0.645321 + 0.763911i \(0.276724\pi\)
\(948\) −13304.4 −0.455810
\(949\) 4048.20 0.138472
\(950\) 18021.9 0.615482
\(951\) −2772.80 −0.0945469
\(952\) 61148.2 2.08175
\(953\) 48294.3 1.64156 0.820779 0.571246i \(-0.193540\pi\)
0.820779 + 0.571246i \(0.193540\pi\)
\(954\) 195.880 0.00664765
\(955\) −6156.09 −0.208593
\(956\) −55239.8 −1.86881
\(957\) 0 0
\(958\) −11046.4 −0.372539
\(959\) −28047.4 −0.944419
\(960\) 7022.57 0.236096
\(961\) −10021.4 −0.336389
\(962\) 9573.76 0.320863
\(963\) 1368.22 0.0457843
\(964\) −46173.1 −1.54267
\(965\) 4247.39 0.141687
\(966\) 48907.8 1.62897
\(967\) −1840.92 −0.0612204 −0.0306102 0.999531i \(-0.509745\pi\)
−0.0306102 + 0.999531i \(0.509745\pi\)
\(968\) 0 0
\(969\) −12748.6 −0.422647
\(970\) −14766.1 −0.488775
\(971\) 31461.8 1.03981 0.519906 0.854223i \(-0.325967\pi\)
0.519906 + 0.854223i \(0.325967\pi\)
\(972\) −2812.86 −0.0928217
\(973\) −34612.1 −1.14040
\(974\) −15328.1 −0.504254
\(975\) −1806.26 −0.0593298
\(976\) −12581.3 −0.412620
\(977\) −7040.11 −0.230535 −0.115268 0.993334i \(-0.536773\pi\)
−0.115268 + 0.993334i \(0.536773\pi\)
\(978\) 653.494 0.0213665
\(979\) 0 0
\(980\) 21822.5 0.711320
\(981\) −19532.9 −0.635715
\(982\) 71784.4 2.33272
\(983\) −24610.9 −0.798541 −0.399270 0.916833i \(-0.630737\pi\)
−0.399270 + 0.916833i \(0.630737\pi\)
\(984\) −15284.7 −0.495183
\(985\) 656.744 0.0212443
\(986\) −37471.8 −1.21029
\(987\) −22005.3 −0.709662
\(988\) 2077.95 0.0669112
\(989\) −37318.5 −1.19986
\(990\) 0 0
\(991\) −40003.3 −1.28229 −0.641144 0.767421i \(-0.721540\pi\)
−0.641144 + 0.767421i \(0.721540\pi\)
\(992\) −31860.8 −1.01974
\(993\) 29461.4 0.941519
\(994\) −6883.31 −0.219643
\(995\) 63.8165 0.00203329
\(996\) −32303.2 −1.02768
\(997\) 7342.61 0.233242 0.116621 0.993176i \(-0.462794\pi\)
0.116621 + 0.993176i \(0.462794\pi\)
\(998\) 44216.9 1.40247
\(999\) 11341.9 0.359201
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 363.4.a.i.1.2 2
3.2 odd 2 1089.4.a.u.1.1 2
11.10 odd 2 33.4.a.c.1.1 2
33.32 even 2 99.4.a.f.1.2 2
44.43 even 2 528.4.a.p.1.2 2
55.32 even 4 825.4.c.h.199.2 4
55.43 even 4 825.4.c.h.199.3 4
55.54 odd 2 825.4.a.l.1.2 2
77.76 even 2 1617.4.a.k.1.1 2
88.21 odd 2 2112.4.a.bn.1.1 2
88.43 even 2 2112.4.a.bg.1.1 2
132.131 odd 2 1584.4.a.bj.1.1 2
165.164 even 2 2475.4.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 11.10 odd 2
99.4.a.f.1.2 2 33.32 even 2
363.4.a.i.1.2 2 1.1 even 1 trivial
528.4.a.p.1.2 2 44.43 even 2
825.4.a.l.1.2 2 55.54 odd 2
825.4.c.h.199.2 4 55.32 even 4
825.4.c.h.199.3 4 55.43 even 4
1089.4.a.u.1.1 2 3.2 odd 2
1584.4.a.bj.1.1 2 132.131 odd 2
1617.4.a.k.1.1 2 77.76 even 2
2112.4.a.bg.1.1 2 88.43 even 2
2112.4.a.bn.1.1 2 88.21 odd 2
2475.4.a.p.1.1 2 165.164 even 2