Properties

Label 6561.2.a.d.1.5
Level $6561$
Weight $2$
Character 6561.1
Self dual yes
Analytic conductor $52.390$
Analytic rank $0$
Dimension $72$
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] = [6561,2,Mod(1,6561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6561, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6561.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6561 = 3^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6561.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: \(52.3898487662\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: no (minimal twist has level 81)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 6561.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.27431 q^{2} +3.17247 q^{4} +0.586876 q^{5} +3.80961 q^{7} -2.66656 q^{8} +O(q^{10})\) \(q-2.27431 q^{2} +3.17247 q^{4} +0.586876 q^{5} +3.80961 q^{7} -2.66656 q^{8} -1.33474 q^{10} +4.98958 q^{11} -2.48589 q^{13} -8.66421 q^{14} -0.280361 q^{16} +2.85693 q^{17} +0.417601 q^{19} +1.86185 q^{20} -11.3478 q^{22} +7.35642 q^{23} -4.65558 q^{25} +5.65367 q^{26} +12.0859 q^{28} +5.28268 q^{29} +2.93826 q^{31} +5.97075 q^{32} -6.49755 q^{34} +2.23577 q^{35} -6.29234 q^{37} -0.949753 q^{38} -1.56494 q^{40} +3.65040 q^{41} -6.01707 q^{43} +15.8293 q^{44} -16.7308 q^{46} -6.58406 q^{47} +7.51309 q^{49} +10.5882 q^{50} -7.88641 q^{52} +2.47763 q^{53} +2.92826 q^{55} -10.1586 q^{56} -12.0144 q^{58} -4.63396 q^{59} +11.3950 q^{61} -6.68250 q^{62} -13.0186 q^{64} -1.45891 q^{65} +5.94721 q^{67} +9.06355 q^{68} -5.08482 q^{70} +2.82531 q^{71} -13.7368 q^{73} +14.3107 q^{74} +1.32483 q^{76} +19.0083 q^{77} +12.5673 q^{79} -0.164537 q^{80} -8.30214 q^{82} -2.09817 q^{83} +1.67667 q^{85} +13.6847 q^{86} -13.3050 q^{88} -0.345463 q^{89} -9.47025 q^{91} +23.3380 q^{92} +14.9742 q^{94} +0.245080 q^{95} +10.9705 q^{97} -17.0871 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.27431 −1.60818 −0.804089 0.594509i \(-0.797346\pi\)
−0.804089 + 0.594509i \(0.797346\pi\)
\(3\) 0 0
\(4\) 3.17247 1.58624
\(5\) 0.586876 0.262459 0.131230 0.991352i \(-0.458108\pi\)
0.131230 + 0.991352i \(0.458108\pi\)
\(6\) 0 0
\(7\) 3.80961 1.43990 0.719948 0.694028i \(-0.244166\pi\)
0.719948 + 0.694028i \(0.244166\pi\)
\(8\) −2.66656 −0.942773
\(9\) 0 0
\(10\) −1.33474 −0.422081
\(11\) 4.98958 1.50441 0.752207 0.658927i \(-0.228989\pi\)
0.752207 + 0.658927i \(0.228989\pi\)
\(12\) 0 0
\(13\) −2.48589 −0.689461 −0.344731 0.938702i \(-0.612030\pi\)
−0.344731 + 0.938702i \(0.612030\pi\)
\(14\) −8.66421 −2.31561
\(15\) 0 0
\(16\) −0.280361 −0.0700902
\(17\) 2.85693 0.692908 0.346454 0.938067i \(-0.387386\pi\)
0.346454 + 0.938067i \(0.387386\pi\)
\(18\) 0 0
\(19\) 0.417601 0.0958042 0.0479021 0.998852i \(-0.484746\pi\)
0.0479021 + 0.998852i \(0.484746\pi\)
\(20\) 1.86185 0.416322
\(21\) 0 0
\(22\) −11.3478 −2.41936
\(23\) 7.35642 1.53392 0.766960 0.641695i \(-0.221769\pi\)
0.766960 + 0.641695i \(0.221769\pi\)
\(24\) 0 0
\(25\) −4.65558 −0.931115
\(26\) 5.65367 1.10878
\(27\) 0 0
\(28\) 12.0859 2.28401
\(29\) 5.28268 0.980968 0.490484 0.871450i \(-0.336820\pi\)
0.490484 + 0.871450i \(0.336820\pi\)
\(30\) 0 0
\(31\) 2.93826 0.527727 0.263863 0.964560i \(-0.415003\pi\)
0.263863 + 0.964560i \(0.415003\pi\)
\(32\) 5.97075 1.05549
\(33\) 0 0
\(34\) −6.49755 −1.11432
\(35\) 2.23577 0.377914
\(36\) 0 0
\(37\) −6.29234 −1.03445 −0.517227 0.855848i \(-0.673036\pi\)
−0.517227 + 0.855848i \(0.673036\pi\)
\(38\) −0.949753 −0.154070
\(39\) 0 0
\(40\) −1.56494 −0.247439
\(41\) 3.65040 0.570097 0.285049 0.958513i \(-0.407990\pi\)
0.285049 + 0.958513i \(0.407990\pi\)
\(42\) 0 0
\(43\) −6.01707 −0.917595 −0.458797 0.888541i \(-0.651720\pi\)
−0.458797 + 0.888541i \(0.651720\pi\)
\(44\) 15.8293 2.38636
\(45\) 0 0
\(46\) −16.7308 −2.46682
\(47\) −6.58406 −0.960385 −0.480192 0.877163i \(-0.659433\pi\)
−0.480192 + 0.877163i \(0.659433\pi\)
\(48\) 0 0
\(49\) 7.51309 1.07330
\(50\) 10.5882 1.49740
\(51\) 0 0
\(52\) −7.88641 −1.09365
\(53\) 2.47763 0.340329 0.170165 0.985416i \(-0.445570\pi\)
0.170165 + 0.985416i \(0.445570\pi\)
\(54\) 0 0
\(55\) 2.92826 0.394847
\(56\) −10.1586 −1.35749
\(57\) 0 0
\(58\) −12.0144 −1.57757
\(59\) −4.63396 −0.603290 −0.301645 0.953420i \(-0.597536\pi\)
−0.301645 + 0.953420i \(0.597536\pi\)
\(60\) 0 0
\(61\) 11.3950 1.45899 0.729493 0.683988i \(-0.239756\pi\)
0.729493 + 0.683988i \(0.239756\pi\)
\(62\) −6.68250 −0.848678
\(63\) 0 0
\(64\) −13.0186 −1.62733
\(65\) −1.45891 −0.180955
\(66\) 0 0
\(67\) 5.94721 0.726568 0.363284 0.931679i \(-0.381656\pi\)
0.363284 + 0.931679i \(0.381656\pi\)
\(68\) 9.06355 1.09912
\(69\) 0 0
\(70\) −5.08482 −0.607752
\(71\) 2.82531 0.335302 0.167651 0.985846i \(-0.446382\pi\)
0.167651 + 0.985846i \(0.446382\pi\)
\(72\) 0 0
\(73\) −13.7368 −1.60778 −0.803888 0.594781i \(-0.797239\pi\)
−0.803888 + 0.594781i \(0.797239\pi\)
\(74\) 14.3107 1.66359
\(75\) 0 0
\(76\) 1.32483 0.151968
\(77\) 19.0083 2.16620
\(78\) 0 0
\(79\) 12.5673 1.41393 0.706964 0.707250i \(-0.250064\pi\)
0.706964 + 0.707250i \(0.250064\pi\)
\(80\) −0.164537 −0.0183958
\(81\) 0 0
\(82\) −8.30214 −0.916818
\(83\) −2.09817 −0.230305 −0.115152 0.993348i \(-0.536736\pi\)
−0.115152 + 0.993348i \(0.536736\pi\)
\(84\) 0 0
\(85\) 1.67667 0.181860
\(86\) 13.6847 1.47566
\(87\) 0 0
\(88\) −13.3050 −1.41832
\(89\) −0.345463 −0.0366190 −0.0183095 0.999832i \(-0.505828\pi\)
−0.0183095 + 0.999832i \(0.505828\pi\)
\(90\) 0 0
\(91\) −9.47025 −0.992752
\(92\) 23.3380 2.43316
\(93\) 0 0
\(94\) 14.9742 1.54447
\(95\) 0.245080 0.0251447
\(96\) 0 0
\(97\) 10.9705 1.11389 0.556944 0.830550i \(-0.311974\pi\)
0.556944 + 0.830550i \(0.311974\pi\)
\(98\) −17.0871 −1.72606
\(99\) 0 0
\(100\) −14.7697 −1.47697
\(101\) 1.77372 0.176492 0.0882458 0.996099i \(-0.471874\pi\)
0.0882458 + 0.996099i \(0.471874\pi\)
\(102\) 0 0
\(103\) 17.5976 1.73395 0.866974 0.498354i \(-0.166062\pi\)
0.866974 + 0.498354i \(0.166062\pi\)
\(104\) 6.62878 0.650005
\(105\) 0 0
\(106\) −5.63490 −0.547310
\(107\) −4.53980 −0.438879 −0.219439 0.975626i \(-0.570423\pi\)
−0.219439 + 0.975626i \(0.570423\pi\)
\(108\) 0 0
\(109\) 3.90189 0.373733 0.186866 0.982385i \(-0.440167\pi\)
0.186866 + 0.982385i \(0.440167\pi\)
\(110\) −6.65977 −0.634984
\(111\) 0 0
\(112\) −1.06806 −0.100923
\(113\) 13.0376 1.22647 0.613235 0.789900i \(-0.289868\pi\)
0.613235 + 0.789900i \(0.289868\pi\)
\(114\) 0 0
\(115\) 4.31731 0.402591
\(116\) 16.7591 1.55605
\(117\) 0 0
\(118\) 10.5390 0.970198
\(119\) 10.8838 0.997716
\(120\) 0 0
\(121\) 13.8959 1.26326
\(122\) −25.9158 −2.34631
\(123\) 0 0
\(124\) 9.32154 0.837099
\(125\) −5.66663 −0.506839
\(126\) 0 0
\(127\) −8.19348 −0.727054 −0.363527 0.931584i \(-0.618427\pi\)
−0.363527 + 0.931584i \(0.618427\pi\)
\(128\) 17.6668 1.56154
\(129\) 0 0
\(130\) 3.31801 0.291009
\(131\) 1.73569 0.151648 0.0758238 0.997121i \(-0.475841\pi\)
0.0758238 + 0.997121i \(0.475841\pi\)
\(132\) 0 0
\(133\) 1.59089 0.137948
\(134\) −13.5258 −1.16845
\(135\) 0 0
\(136\) −7.61820 −0.653255
\(137\) 2.74802 0.234780 0.117390 0.993086i \(-0.462547\pi\)
0.117390 + 0.993086i \(0.462547\pi\)
\(138\) 0 0
\(139\) 8.65405 0.734027 0.367013 0.930216i \(-0.380380\pi\)
0.367013 + 0.930216i \(0.380380\pi\)
\(140\) 7.09291 0.599460
\(141\) 0 0
\(142\) −6.42561 −0.539225
\(143\) −12.4035 −1.03724
\(144\) 0 0
\(145\) 3.10028 0.257464
\(146\) 31.2418 2.58559
\(147\) 0 0
\(148\) −19.9623 −1.64089
\(149\) −5.73323 −0.469685 −0.234842 0.972033i \(-0.575457\pi\)
−0.234842 + 0.972033i \(0.575457\pi\)
\(150\) 0 0
\(151\) 1.77462 0.144417 0.0722083 0.997390i \(-0.476995\pi\)
0.0722083 + 0.997390i \(0.476995\pi\)
\(152\) −1.11356 −0.0903216
\(153\) 0 0
\(154\) −43.2307 −3.48363
\(155\) 1.72439 0.138507
\(156\) 0 0
\(157\) 9.50907 0.758906 0.379453 0.925211i \(-0.376112\pi\)
0.379453 + 0.925211i \(0.376112\pi\)
\(158\) −28.5818 −2.27385
\(159\) 0 0
\(160\) 3.50409 0.277023
\(161\) 28.0251 2.20868
\(162\) 0 0
\(163\) 7.38623 0.578534 0.289267 0.957248i \(-0.406588\pi\)
0.289267 + 0.957248i \(0.406588\pi\)
\(164\) 11.5808 0.904309
\(165\) 0 0
\(166\) 4.77189 0.370371
\(167\) 4.55339 0.352352 0.176176 0.984359i \(-0.443627\pi\)
0.176176 + 0.984359i \(0.443627\pi\)
\(168\) 0 0
\(169\) −6.82036 −0.524643
\(170\) −3.81326 −0.292463
\(171\) 0 0
\(172\) −19.0890 −1.45552
\(173\) −4.65432 −0.353861 −0.176931 0.984223i \(-0.556617\pi\)
−0.176931 + 0.984223i \(0.556617\pi\)
\(174\) 0 0
\(175\) −17.7359 −1.34071
\(176\) −1.39888 −0.105445
\(177\) 0 0
\(178\) 0.785688 0.0588898
\(179\) 3.47274 0.259564 0.129782 0.991543i \(-0.458572\pi\)
0.129782 + 0.991543i \(0.458572\pi\)
\(180\) 0 0
\(181\) −15.2925 −1.13668 −0.568341 0.822793i \(-0.692415\pi\)
−0.568341 + 0.822793i \(0.692415\pi\)
\(182\) 21.5383 1.59652
\(183\) 0 0
\(184\) −19.6164 −1.44614
\(185\) −3.69283 −0.271502
\(186\) 0 0
\(187\) 14.2549 1.04242
\(188\) −20.8878 −1.52340
\(189\) 0 0
\(190\) −0.557388 −0.0404371
\(191\) −22.3873 −1.61989 −0.809945 0.586506i \(-0.800503\pi\)
−0.809945 + 0.586506i \(0.800503\pi\)
\(192\) 0 0
\(193\) −6.47290 −0.465930 −0.232965 0.972485i \(-0.574843\pi\)
−0.232965 + 0.972485i \(0.574843\pi\)
\(194\) −24.9504 −1.79133
\(195\) 0 0
\(196\) 23.8351 1.70251
\(197\) −5.53590 −0.394416 −0.197208 0.980362i \(-0.563187\pi\)
−0.197208 + 0.980362i \(0.563187\pi\)
\(198\) 0 0
\(199\) −16.8019 −1.19106 −0.595529 0.803334i \(-0.703057\pi\)
−0.595529 + 0.803334i \(0.703057\pi\)
\(200\) 12.4144 0.877830
\(201\) 0 0
\(202\) −4.03398 −0.283830
\(203\) 20.1249 1.41249
\(204\) 0 0
\(205\) 2.14234 0.149627
\(206\) −40.0225 −2.78850
\(207\) 0 0
\(208\) 0.696946 0.0483245
\(209\) 2.08365 0.144129
\(210\) 0 0
\(211\) −7.65013 −0.526656 −0.263328 0.964706i \(-0.584820\pi\)
−0.263328 + 0.964706i \(0.584820\pi\)
\(212\) 7.86022 0.539842
\(213\) 0 0
\(214\) 10.3249 0.705795
\(215\) −3.53128 −0.240831
\(216\) 0 0
\(217\) 11.1936 0.759871
\(218\) −8.87409 −0.601029
\(219\) 0 0
\(220\) 9.28984 0.626321
\(221\) −7.10202 −0.477734
\(222\) 0 0
\(223\) −8.45766 −0.566367 −0.283183 0.959066i \(-0.591390\pi\)
−0.283183 + 0.959066i \(0.591390\pi\)
\(224\) 22.7462 1.51980
\(225\) 0 0
\(226\) −29.6514 −1.97238
\(227\) −26.7392 −1.77475 −0.887373 0.461052i \(-0.847472\pi\)
−0.887373 + 0.461052i \(0.847472\pi\)
\(228\) 0 0
\(229\) −3.07865 −0.203443 −0.101722 0.994813i \(-0.532435\pi\)
−0.101722 + 0.994813i \(0.532435\pi\)
\(230\) −9.81889 −0.647438
\(231\) 0 0
\(232\) −14.0866 −0.924830
\(233\) 7.10490 0.465457 0.232729 0.972542i \(-0.425235\pi\)
0.232729 + 0.972542i \(0.425235\pi\)
\(234\) 0 0
\(235\) −3.86403 −0.252062
\(236\) −14.7011 −0.956961
\(237\) 0 0
\(238\) −24.7531 −1.60450
\(239\) −4.58602 −0.296645 −0.148323 0.988939i \(-0.547387\pi\)
−0.148323 + 0.988939i \(0.547387\pi\)
\(240\) 0 0
\(241\) 6.15729 0.396626 0.198313 0.980139i \(-0.436454\pi\)
0.198313 + 0.980139i \(0.436454\pi\)
\(242\) −31.6035 −2.03155
\(243\) 0 0
\(244\) 36.1505 2.31430
\(245\) 4.40926 0.281697
\(246\) 0 0
\(247\) −1.03811 −0.0660533
\(248\) −7.83505 −0.497526
\(249\) 0 0
\(250\) 12.8877 0.815087
\(251\) 29.3779 1.85431 0.927157 0.374673i \(-0.122245\pi\)
0.927157 + 0.374673i \(0.122245\pi\)
\(252\) 0 0
\(253\) 36.7054 2.30765
\(254\) 18.6345 1.16923
\(255\) 0 0
\(256\) −14.1425 −0.883908
\(257\) 18.9070 1.17938 0.589692 0.807628i \(-0.299249\pi\)
0.589692 + 0.807628i \(0.299249\pi\)
\(258\) 0 0
\(259\) −23.9713 −1.48951
\(260\) −4.62835 −0.287038
\(261\) 0 0
\(262\) −3.94748 −0.243876
\(263\) 14.6501 0.903362 0.451681 0.892180i \(-0.350825\pi\)
0.451681 + 0.892180i \(0.350825\pi\)
\(264\) 0 0
\(265\) 1.45406 0.0893224
\(266\) −3.61818 −0.221845
\(267\) 0 0
\(268\) 18.8674 1.15251
\(269\) −5.89036 −0.359142 −0.179571 0.983745i \(-0.557471\pi\)
−0.179571 + 0.983745i \(0.557471\pi\)
\(270\) 0 0
\(271\) −23.1929 −1.40887 −0.704435 0.709768i \(-0.748800\pi\)
−0.704435 + 0.709768i \(0.748800\pi\)
\(272\) −0.800973 −0.0485661
\(273\) 0 0
\(274\) −6.24985 −0.377567
\(275\) −23.2293 −1.40078
\(276\) 0 0
\(277\) −17.4388 −1.04780 −0.523899 0.851780i \(-0.675523\pi\)
−0.523899 + 0.851780i \(0.675523\pi\)
\(278\) −19.6820 −1.18045
\(279\) 0 0
\(280\) −5.96182 −0.356287
\(281\) 28.5863 1.70532 0.852659 0.522467i \(-0.174988\pi\)
0.852659 + 0.522467i \(0.174988\pi\)
\(282\) 0 0
\(283\) 17.8097 1.05868 0.529340 0.848410i \(-0.322440\pi\)
0.529340 + 0.848410i \(0.322440\pi\)
\(284\) 8.96321 0.531868
\(285\) 0 0
\(286\) 28.2094 1.66806
\(287\) 13.9066 0.820881
\(288\) 0 0
\(289\) −8.83792 −0.519878
\(290\) −7.05098 −0.414048
\(291\) 0 0
\(292\) −43.5798 −2.55031
\(293\) −16.6353 −0.971846 −0.485923 0.874002i \(-0.661516\pi\)
−0.485923 + 0.874002i \(0.661516\pi\)
\(294\) 0 0
\(295\) −2.71956 −0.158339
\(296\) 16.7789 0.975255
\(297\) 0 0
\(298\) 13.0391 0.755337
\(299\) −18.2872 −1.05758
\(300\) 0 0
\(301\) −22.9227 −1.32124
\(302\) −4.03604 −0.232248
\(303\) 0 0
\(304\) −0.117079 −0.00671494
\(305\) 6.68748 0.382924
\(306\) 0 0
\(307\) −14.1106 −0.805335 −0.402668 0.915346i \(-0.631917\pi\)
−0.402668 + 0.915346i \(0.631917\pi\)
\(308\) 60.3034 3.43610
\(309\) 0 0
\(310\) −3.92180 −0.222743
\(311\) 23.6481 1.34096 0.670480 0.741928i \(-0.266088\pi\)
0.670480 + 0.741928i \(0.266088\pi\)
\(312\) 0 0
\(313\) 8.18392 0.462583 0.231291 0.972885i \(-0.425705\pi\)
0.231291 + 0.972885i \(0.425705\pi\)
\(314\) −21.6265 −1.22046
\(315\) 0 0
\(316\) 39.8693 2.24282
\(317\) 4.09853 0.230196 0.115098 0.993354i \(-0.463282\pi\)
0.115098 + 0.993354i \(0.463282\pi\)
\(318\) 0 0
\(319\) 26.3583 1.47578
\(320\) −7.64031 −0.427106
\(321\) 0 0
\(322\) −63.7376 −3.55196
\(323\) 1.19306 0.0663836
\(324\) 0 0
\(325\) 11.5732 0.641968
\(326\) −16.7985 −0.930385
\(327\) 0 0
\(328\) −9.73404 −0.537472
\(329\) −25.0827 −1.38285
\(330\) 0 0
\(331\) −24.7961 −1.36292 −0.681458 0.731857i \(-0.738654\pi\)
−0.681458 + 0.731857i \(0.738654\pi\)
\(332\) −6.65640 −0.365318
\(333\) 0 0
\(334\) −10.3558 −0.566645
\(335\) 3.49028 0.190694
\(336\) 0 0
\(337\) 15.1613 0.825890 0.412945 0.910756i \(-0.364500\pi\)
0.412945 + 0.910756i \(0.364500\pi\)
\(338\) 15.5116 0.843719
\(339\) 0 0
\(340\) 5.31918 0.288473
\(341\) 14.6607 0.793919
\(342\) 0 0
\(343\) 1.95468 0.105543
\(344\) 16.0449 0.865083
\(345\) 0 0
\(346\) 10.5853 0.569072
\(347\) −25.1222 −1.34863 −0.674316 0.738443i \(-0.735561\pi\)
−0.674316 + 0.738443i \(0.735561\pi\)
\(348\) 0 0
\(349\) −17.3457 −0.928492 −0.464246 0.885706i \(-0.653675\pi\)
−0.464246 + 0.885706i \(0.653675\pi\)
\(350\) 40.3369 2.15610
\(351\) 0 0
\(352\) 29.7915 1.58789
\(353\) 21.7225 1.15617 0.578086 0.815976i \(-0.303800\pi\)
0.578086 + 0.815976i \(0.303800\pi\)
\(354\) 0 0
\(355\) 1.65811 0.0880031
\(356\) −1.09597 −0.0580863
\(357\) 0 0
\(358\) −7.89807 −0.417426
\(359\) −27.5292 −1.45293 −0.726467 0.687201i \(-0.758839\pi\)
−0.726467 + 0.687201i \(0.758839\pi\)
\(360\) 0 0
\(361\) −18.8256 −0.990822
\(362\) 34.7798 1.82799
\(363\) 0 0
\(364\) −30.0441 −1.57474
\(365\) −8.06183 −0.421975
\(366\) 0 0
\(367\) −24.3548 −1.27131 −0.635656 0.771972i \(-0.719270\pi\)
−0.635656 + 0.771972i \(0.719270\pi\)
\(368\) −2.06245 −0.107513
\(369\) 0 0
\(370\) 8.39862 0.436624
\(371\) 9.43880 0.490038
\(372\) 0 0
\(373\) 16.6726 0.863275 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(374\) −32.4200 −1.67640
\(375\) 0 0
\(376\) 17.5568 0.905424
\(377\) −13.1321 −0.676340
\(378\) 0 0
\(379\) −2.06537 −0.106091 −0.0530454 0.998592i \(-0.516893\pi\)
−0.0530454 + 0.998592i \(0.516893\pi\)
\(380\) 0.777510 0.0398854
\(381\) 0 0
\(382\) 50.9156 2.60507
\(383\) 14.4104 0.736339 0.368170 0.929759i \(-0.379985\pi\)
0.368170 + 0.929759i \(0.379985\pi\)
\(384\) 0 0
\(385\) 11.1555 0.568538
\(386\) 14.7214 0.749298
\(387\) 0 0
\(388\) 34.8037 1.76689
\(389\) 12.3423 0.625779 0.312890 0.949790i \(-0.398703\pi\)
0.312890 + 0.949790i \(0.398703\pi\)
\(390\) 0 0
\(391\) 21.0168 1.06287
\(392\) −20.0341 −1.01188
\(393\) 0 0
\(394\) 12.5903 0.634291
\(395\) 7.37543 0.371098
\(396\) 0 0
\(397\) 19.1683 0.962028 0.481014 0.876713i \(-0.340269\pi\)
0.481014 + 0.876713i \(0.340269\pi\)
\(398\) 38.2127 1.91543
\(399\) 0 0
\(400\) 1.30524 0.0652621
\(401\) −10.0840 −0.503569 −0.251784 0.967783i \(-0.581017\pi\)
−0.251784 + 0.967783i \(0.581017\pi\)
\(402\) 0 0
\(403\) −7.30418 −0.363847
\(404\) 5.62708 0.279957
\(405\) 0 0
\(406\) −45.7702 −2.27154
\(407\) −31.3961 −1.55625
\(408\) 0 0
\(409\) −3.46034 −0.171103 −0.0855514 0.996334i \(-0.527265\pi\)
−0.0855514 + 0.996334i \(0.527265\pi\)
\(410\) −4.87233 −0.240627
\(411\) 0 0
\(412\) 55.8281 2.75045
\(413\) −17.6536 −0.868675
\(414\) 0 0
\(415\) −1.23137 −0.0604455
\(416\) −14.8426 −0.727720
\(417\) 0 0
\(418\) −4.73886 −0.231785
\(419\) −5.82990 −0.284809 −0.142405 0.989809i \(-0.545483\pi\)
−0.142405 + 0.989809i \(0.545483\pi\)
\(420\) 0 0
\(421\) 38.5555 1.87908 0.939541 0.342436i \(-0.111252\pi\)
0.939541 + 0.342436i \(0.111252\pi\)
\(422\) 17.3987 0.846957
\(423\) 0 0
\(424\) −6.60677 −0.320853
\(425\) −13.3007 −0.645178
\(426\) 0 0
\(427\) 43.4106 2.10079
\(428\) −14.4024 −0.696165
\(429\) 0 0
\(430\) 8.03121 0.387299
\(431\) −0.202222 −0.00974071 −0.00487036 0.999988i \(-0.501550\pi\)
−0.00487036 + 0.999988i \(0.501550\pi\)
\(432\) 0 0
\(433\) −8.06122 −0.387398 −0.193699 0.981061i \(-0.562048\pi\)
−0.193699 + 0.981061i \(0.562048\pi\)
\(434\) −25.4577 −1.22201
\(435\) 0 0
\(436\) 12.3786 0.592829
\(437\) 3.07205 0.146956
\(438\) 0 0
\(439\) −2.18593 −0.104329 −0.0521645 0.998639i \(-0.516612\pi\)
−0.0521645 + 0.998639i \(0.516612\pi\)
\(440\) −7.80840 −0.372251
\(441\) 0 0
\(442\) 16.1522 0.768281
\(443\) −32.3583 −1.53739 −0.768695 0.639616i \(-0.779093\pi\)
−0.768695 + 0.639616i \(0.779093\pi\)
\(444\) 0 0
\(445\) −0.202744 −0.00961098
\(446\) 19.2353 0.910818
\(447\) 0 0
\(448\) −49.5958 −2.34318
\(449\) 38.8633 1.83407 0.917036 0.398805i \(-0.130575\pi\)
0.917036 + 0.398805i \(0.130575\pi\)
\(450\) 0 0
\(451\) 18.2140 0.857662
\(452\) 41.3613 1.94547
\(453\) 0 0
\(454\) 60.8132 2.85411
\(455\) −5.55787 −0.260557
\(456\) 0 0
\(457\) −29.5254 −1.38114 −0.690571 0.723265i \(-0.742641\pi\)
−0.690571 + 0.723265i \(0.742641\pi\)
\(458\) 7.00181 0.327173
\(459\) 0 0
\(460\) 13.6965 0.638605
\(461\) 21.5243 1.00249 0.501244 0.865306i \(-0.332876\pi\)
0.501244 + 0.865306i \(0.332876\pi\)
\(462\) 0 0
\(463\) −34.9400 −1.62380 −0.811901 0.583795i \(-0.801567\pi\)
−0.811901 + 0.583795i \(0.801567\pi\)
\(464\) −1.48106 −0.0687563
\(465\) 0 0
\(466\) −16.1587 −0.748538
\(467\) −25.1115 −1.16202 −0.581011 0.813896i \(-0.697343\pi\)
−0.581011 + 0.813896i \(0.697343\pi\)
\(468\) 0 0
\(469\) 22.6565 1.04618
\(470\) 8.78800 0.405360
\(471\) 0 0
\(472\) 12.3568 0.568766
\(473\) −30.0226 −1.38044
\(474\) 0 0
\(475\) −1.94417 −0.0892048
\(476\) 34.5285 1.58261
\(477\) 0 0
\(478\) 10.4300 0.477058
\(479\) −9.81192 −0.448318 −0.224159 0.974553i \(-0.571964\pi\)
−0.224159 + 0.974553i \(0.571964\pi\)
\(480\) 0 0
\(481\) 15.6421 0.713216
\(482\) −14.0036 −0.637845
\(483\) 0 0
\(484\) 44.0843 2.00383
\(485\) 6.43835 0.292350
\(486\) 0 0
\(487\) −17.0318 −0.771785 −0.385892 0.922544i \(-0.626106\pi\)
−0.385892 + 0.922544i \(0.626106\pi\)
\(488\) −30.3856 −1.37549
\(489\) 0 0
\(490\) −10.0280 −0.453019
\(491\) −8.40937 −0.379510 −0.189755 0.981831i \(-0.560769\pi\)
−0.189755 + 0.981831i \(0.560769\pi\)
\(492\) 0 0
\(493\) 15.0923 0.679721
\(494\) 2.36098 0.106225
\(495\) 0 0
\(496\) −0.823772 −0.0369885
\(497\) 10.7633 0.482800
\(498\) 0 0
\(499\) 24.1628 1.08168 0.540838 0.841127i \(-0.318107\pi\)
0.540838 + 0.841127i \(0.318107\pi\)
\(500\) −17.9772 −0.803966
\(501\) 0 0
\(502\) −66.8143 −2.98207
\(503\) −3.67529 −0.163873 −0.0819366 0.996638i \(-0.526111\pi\)
−0.0819366 + 0.996638i \(0.526111\pi\)
\(504\) 0 0
\(505\) 1.04095 0.0463218
\(506\) −83.4794 −3.71111
\(507\) 0 0
\(508\) −25.9936 −1.15328
\(509\) −29.3717 −1.30188 −0.650939 0.759130i \(-0.725625\pi\)
−0.650939 + 0.759130i \(0.725625\pi\)
\(510\) 0 0
\(511\) −52.3320 −2.31503
\(512\) −3.16917 −0.140059
\(513\) 0 0
\(514\) −43.0002 −1.89666
\(515\) 10.3276 0.455090
\(516\) 0 0
\(517\) −32.8517 −1.44482
\(518\) 54.5182 2.39539
\(519\) 0 0
\(520\) 3.89027 0.170600
\(521\) 34.6944 1.51999 0.759996 0.649928i \(-0.225201\pi\)
0.759996 + 0.649928i \(0.225201\pi\)
\(522\) 0 0
\(523\) 28.8835 1.26299 0.631493 0.775381i \(-0.282442\pi\)
0.631493 + 0.775381i \(0.282442\pi\)
\(524\) 5.50642 0.240549
\(525\) 0 0
\(526\) −33.3187 −1.45277
\(527\) 8.39441 0.365666
\(528\) 0 0
\(529\) 31.1169 1.35291
\(530\) −3.30699 −0.143646
\(531\) 0 0
\(532\) 5.04707 0.218818
\(533\) −9.07450 −0.393060
\(534\) 0 0
\(535\) −2.66430 −0.115188
\(536\) −15.8586 −0.684988
\(537\) 0 0
\(538\) 13.3965 0.577564
\(539\) 37.4871 1.61469
\(540\) 0 0
\(541\) −8.57548 −0.368689 −0.184344 0.982862i \(-0.559016\pi\)
−0.184344 + 0.982862i \(0.559016\pi\)
\(542\) 52.7479 2.26571
\(543\) 0 0
\(544\) 17.0581 0.731358
\(545\) 2.28992 0.0980896
\(546\) 0 0
\(547\) −25.1821 −1.07671 −0.538355 0.842718i \(-0.680954\pi\)
−0.538355 + 0.842718i \(0.680954\pi\)
\(548\) 8.71803 0.372416
\(549\) 0 0
\(550\) 52.8307 2.25271
\(551\) 2.20605 0.0939809
\(552\) 0 0
\(553\) 47.8763 2.03591
\(554\) 39.6613 1.68505
\(555\) 0 0
\(556\) 27.4547 1.16434
\(557\) −16.3019 −0.690734 −0.345367 0.938468i \(-0.612246\pi\)
−0.345367 + 0.938468i \(0.612246\pi\)
\(558\) 0 0
\(559\) 14.9578 0.632646
\(560\) −0.626822 −0.0264881
\(561\) 0 0
\(562\) −65.0141 −2.74246
\(563\) 43.7482 1.84377 0.921884 0.387467i \(-0.126650\pi\)
0.921884 + 0.387467i \(0.126650\pi\)
\(564\) 0 0
\(565\) 7.65143 0.321898
\(566\) −40.5048 −1.70254
\(567\) 0 0
\(568\) −7.53386 −0.316114
\(569\) 34.2317 1.43507 0.717533 0.696524i \(-0.245271\pi\)
0.717533 + 0.696524i \(0.245271\pi\)
\(570\) 0 0
\(571\) −32.3804 −1.35508 −0.677540 0.735486i \(-0.736954\pi\)
−0.677540 + 0.735486i \(0.736954\pi\)
\(572\) −39.3499 −1.64530
\(573\) 0 0
\(574\) −31.6279 −1.32012
\(575\) −34.2484 −1.42826
\(576\) 0 0
\(577\) −45.8114 −1.90715 −0.953576 0.301151i \(-0.902629\pi\)
−0.953576 + 0.301151i \(0.902629\pi\)
\(578\) 20.1002 0.836056
\(579\) 0 0
\(580\) 9.83555 0.408399
\(581\) −7.99322 −0.331614
\(582\) 0 0
\(583\) 12.3623 0.511996
\(584\) 36.6302 1.51577
\(585\) 0 0
\(586\) 37.8338 1.56290
\(587\) 43.0407 1.77648 0.888240 0.459380i \(-0.151928\pi\)
0.888240 + 0.459380i \(0.151928\pi\)
\(588\) 0 0
\(589\) 1.22702 0.0505584
\(590\) 6.18512 0.254637
\(591\) 0 0
\(592\) 1.76413 0.0725051
\(593\) 27.0758 1.11187 0.555935 0.831225i \(-0.312360\pi\)
0.555935 + 0.831225i \(0.312360\pi\)
\(594\) 0 0
\(595\) 6.38744 0.261860
\(596\) −18.1885 −0.745031
\(597\) 0 0
\(598\) 41.5908 1.70077
\(599\) 2.02845 0.0828803 0.0414401 0.999141i \(-0.486805\pi\)
0.0414401 + 0.999141i \(0.486805\pi\)
\(600\) 0 0
\(601\) 7.84512 0.320009 0.160005 0.987116i \(-0.448849\pi\)
0.160005 + 0.987116i \(0.448849\pi\)
\(602\) 52.1332 2.12479
\(603\) 0 0
\(604\) 5.62994 0.229079
\(605\) 8.15516 0.331554
\(606\) 0 0
\(607\) −20.5896 −0.835704 −0.417852 0.908515i \(-0.637217\pi\)
−0.417852 + 0.908515i \(0.637217\pi\)
\(608\) 2.49339 0.101120
\(609\) 0 0
\(610\) −15.2094 −0.615810
\(611\) 16.3672 0.662148
\(612\) 0 0
\(613\) −1.44225 −0.0582521 −0.0291261 0.999576i \(-0.509272\pi\)
−0.0291261 + 0.999576i \(0.509272\pi\)
\(614\) 32.0919 1.29512
\(615\) 0 0
\(616\) −50.6869 −2.04223
\(617\) −3.39364 −0.136623 −0.0683114 0.997664i \(-0.521761\pi\)
−0.0683114 + 0.997664i \(0.521761\pi\)
\(618\) 0 0
\(619\) 30.9382 1.24351 0.621755 0.783212i \(-0.286420\pi\)
0.621755 + 0.783212i \(0.286420\pi\)
\(620\) 5.47059 0.219704
\(621\) 0 0
\(622\) −53.7830 −2.15650
\(623\) −1.31608 −0.0527275
\(624\) 0 0
\(625\) 19.9523 0.798091
\(626\) −18.6128 −0.743915
\(627\) 0 0
\(628\) 30.1673 1.20381
\(629\) −17.9768 −0.716782
\(630\) 0 0
\(631\) 33.5776 1.33670 0.668352 0.743845i \(-0.267000\pi\)
0.668352 + 0.743845i \(0.267000\pi\)
\(632\) −33.5114 −1.33301
\(633\) 0 0
\(634\) −9.32132 −0.370197
\(635\) −4.80856 −0.190822
\(636\) 0 0
\(637\) −18.6767 −0.739998
\(638\) −59.9469 −2.37332
\(639\) 0 0
\(640\) 10.3682 0.409840
\(641\) 16.1196 0.636687 0.318343 0.947975i \(-0.396874\pi\)
0.318343 + 0.947975i \(0.396874\pi\)
\(642\) 0 0
\(643\) 7.50957 0.296149 0.148074 0.988976i \(-0.452693\pi\)
0.148074 + 0.988976i \(0.452693\pi\)
\(644\) 88.9087 3.50350
\(645\) 0 0
\(646\) −2.71338 −0.106757
\(647\) −18.2475 −0.717384 −0.358692 0.933456i \(-0.616777\pi\)
−0.358692 + 0.933456i \(0.616777\pi\)
\(648\) 0 0
\(649\) −23.1215 −0.907598
\(650\) −26.3211 −1.03240
\(651\) 0 0
\(652\) 23.4326 0.917691
\(653\) 8.51042 0.333038 0.166519 0.986038i \(-0.446747\pi\)
0.166519 + 0.986038i \(0.446747\pi\)
\(654\) 0 0
\(655\) 1.01863 0.0398013
\(656\) −1.02343 −0.0399582
\(657\) 0 0
\(658\) 57.0457 2.22387
\(659\) 13.1603 0.512653 0.256326 0.966590i \(-0.417488\pi\)
0.256326 + 0.966590i \(0.417488\pi\)
\(660\) 0 0
\(661\) −11.6091 −0.451542 −0.225771 0.974180i \(-0.572490\pi\)
−0.225771 + 0.974180i \(0.572490\pi\)
\(662\) 56.3939 2.19181
\(663\) 0 0
\(664\) 5.59492 0.217125
\(665\) 0.933659 0.0362057
\(666\) 0 0
\(667\) 38.8616 1.50473
\(668\) 14.4455 0.558914
\(669\) 0 0
\(670\) −7.93797 −0.306670
\(671\) 56.8565 2.19492
\(672\) 0 0
\(673\) 40.1759 1.54867 0.774333 0.632778i \(-0.218086\pi\)
0.774333 + 0.632778i \(0.218086\pi\)
\(674\) −34.4815 −1.32818
\(675\) 0 0
\(676\) −21.6374 −0.832208
\(677\) 13.6989 0.526491 0.263245 0.964729i \(-0.415207\pi\)
0.263245 + 0.964729i \(0.415207\pi\)
\(678\) 0 0
\(679\) 41.7934 1.60388
\(680\) −4.47094 −0.171453
\(681\) 0 0
\(682\) −33.3428 −1.27676
\(683\) −19.7916 −0.757304 −0.378652 0.925539i \(-0.623612\pi\)
−0.378652 + 0.925539i \(0.623612\pi\)
\(684\) 0 0
\(685\) 1.61275 0.0616200
\(686\) −4.44553 −0.169731
\(687\) 0 0
\(688\) 1.68695 0.0643144
\(689\) −6.15912 −0.234644
\(690\) 0 0
\(691\) 9.39957 0.357577 0.178788 0.983888i \(-0.442782\pi\)
0.178788 + 0.983888i \(0.442782\pi\)
\(692\) −14.7657 −0.561307
\(693\) 0 0
\(694\) 57.1357 2.16884
\(695\) 5.07886 0.192652
\(696\) 0 0
\(697\) 10.4290 0.395025
\(698\) 39.4494 1.49318
\(699\) 0 0
\(700\) −56.2667 −2.12668
\(701\) 19.3019 0.729024 0.364512 0.931199i \(-0.381236\pi\)
0.364512 + 0.931199i \(0.381236\pi\)
\(702\) 0 0
\(703\) −2.62769 −0.0991051
\(704\) −64.9573 −2.44817
\(705\) 0 0
\(706\) −49.4036 −1.85933
\(707\) 6.75717 0.254129
\(708\) 0 0
\(709\) 9.94291 0.373414 0.186707 0.982416i \(-0.440219\pi\)
0.186707 + 0.982416i \(0.440219\pi\)
\(710\) −3.77104 −0.141525
\(711\) 0 0
\(712\) 0.921198 0.0345234
\(713\) 21.6151 0.809490
\(714\) 0 0
\(715\) −7.27934 −0.272232
\(716\) 11.0172 0.411731
\(717\) 0 0
\(718\) 62.6098 2.33658
\(719\) 38.2949 1.42816 0.714080 0.700064i \(-0.246845\pi\)
0.714080 + 0.700064i \(0.246845\pi\)
\(720\) 0 0
\(721\) 67.0401 2.49670
\(722\) 42.8152 1.59342
\(723\) 0 0
\(724\) −48.5150 −1.80305
\(725\) −24.5939 −0.913394
\(726\) 0 0
\(727\) 23.4738 0.870596 0.435298 0.900286i \(-0.356643\pi\)
0.435298 + 0.900286i \(0.356643\pi\)
\(728\) 25.2530 0.935940
\(729\) 0 0
\(730\) 18.3351 0.678612
\(731\) −17.1904 −0.635809
\(732\) 0 0
\(733\) −20.7297 −0.765670 −0.382835 0.923817i \(-0.625052\pi\)
−0.382835 + 0.923817i \(0.625052\pi\)
\(734\) 55.3904 2.04450
\(735\) 0 0
\(736\) 43.9234 1.61904
\(737\) 29.6741 1.09306
\(738\) 0 0
\(739\) 18.7841 0.690984 0.345492 0.938422i \(-0.387712\pi\)
0.345492 + 0.938422i \(0.387712\pi\)
\(740\) −11.7154 −0.430666
\(741\) 0 0
\(742\) −21.4667 −0.788069
\(743\) −31.0779 −1.14014 −0.570068 0.821597i \(-0.693083\pi\)
−0.570068 + 0.821597i \(0.693083\pi\)
\(744\) 0 0
\(745\) −3.36470 −0.123273
\(746\) −37.9187 −1.38830
\(747\) 0 0
\(748\) 45.2233 1.65353
\(749\) −17.2948 −0.631939
\(750\) 0 0
\(751\) −24.1552 −0.881435 −0.440718 0.897646i \(-0.645276\pi\)
−0.440718 + 0.897646i \(0.645276\pi\)
\(752\) 1.84591 0.0673136
\(753\) 0 0
\(754\) 29.8665 1.08767
\(755\) 1.04148 0.0379035
\(756\) 0 0
\(757\) −0.915266 −0.0332659 −0.0166330 0.999862i \(-0.505295\pi\)
−0.0166330 + 0.999862i \(0.505295\pi\)
\(758\) 4.69728 0.170613
\(759\) 0 0
\(760\) −0.653522 −0.0237057
\(761\) 20.3847 0.738946 0.369473 0.929241i \(-0.379538\pi\)
0.369473 + 0.929241i \(0.379538\pi\)
\(762\) 0 0
\(763\) 14.8646 0.538136
\(764\) −71.0232 −2.56953
\(765\) 0 0
\(766\) −32.7738 −1.18416
\(767\) 11.5195 0.415945
\(768\) 0 0
\(769\) 22.5892 0.814589 0.407294 0.913297i \(-0.366472\pi\)
0.407294 + 0.913297i \(0.366472\pi\)
\(770\) −25.3711 −0.914311
\(771\) 0 0
\(772\) −20.5351 −0.739075
\(773\) 1.67594 0.0602793 0.0301396 0.999546i \(-0.490405\pi\)
0.0301396 + 0.999546i \(0.490405\pi\)
\(774\) 0 0
\(775\) −13.6793 −0.491374
\(776\) −29.2536 −1.05014
\(777\) 0 0
\(778\) −28.0702 −1.00636
\(779\) 1.52441 0.0546177
\(780\) 0 0
\(781\) 14.0971 0.504433
\(782\) −47.7987 −1.70928
\(783\) 0 0
\(784\) −2.10638 −0.0752277
\(785\) 5.58065 0.199182
\(786\) 0 0
\(787\) −31.3700 −1.11822 −0.559109 0.829094i \(-0.688857\pi\)
−0.559109 + 0.829094i \(0.688857\pi\)
\(788\) −17.5625 −0.625637
\(789\) 0 0
\(790\) −16.7740 −0.596792
\(791\) 49.6679 1.76599
\(792\) 0 0
\(793\) −28.3268 −1.00591
\(794\) −43.5946 −1.54711
\(795\) 0 0
\(796\) −53.3037 −1.88930
\(797\) −47.2210 −1.67265 −0.836326 0.548232i \(-0.815301\pi\)
−0.836326 + 0.548232i \(0.815301\pi\)
\(798\) 0 0
\(799\) −18.8102 −0.665459
\(800\) −27.7973 −0.982783
\(801\) 0 0
\(802\) 22.9340 0.809828
\(803\) −68.5410 −2.41876
\(804\) 0 0
\(805\) 16.4472 0.579689
\(806\) 16.6119 0.585131
\(807\) 0 0
\(808\) −4.72973 −0.166391
\(809\) −15.1675 −0.533260 −0.266630 0.963799i \(-0.585910\pi\)
−0.266630 + 0.963799i \(0.585910\pi\)
\(810\) 0 0
\(811\) 35.5509 1.24836 0.624181 0.781280i \(-0.285433\pi\)
0.624181 + 0.781280i \(0.285433\pi\)
\(812\) 63.8457 2.24055
\(813\) 0 0
\(814\) 71.4044 2.50272
\(815\) 4.33480 0.151841
\(816\) 0 0
\(817\) −2.51273 −0.0879095
\(818\) 7.86987 0.275164
\(819\) 0 0
\(820\) 6.79650 0.237344
\(821\) 12.0520 0.420616 0.210308 0.977635i \(-0.432553\pi\)
0.210308 + 0.977635i \(0.432553\pi\)
\(822\) 0 0
\(823\) 36.6728 1.27833 0.639167 0.769068i \(-0.279279\pi\)
0.639167 + 0.769068i \(0.279279\pi\)
\(824\) −46.9252 −1.63472
\(825\) 0 0
\(826\) 40.1496 1.39698
\(827\) 46.7729 1.62645 0.813227 0.581946i \(-0.197709\pi\)
0.813227 + 0.581946i \(0.197709\pi\)
\(828\) 0 0
\(829\) 0.600579 0.0208590 0.0104295 0.999946i \(-0.496680\pi\)
0.0104295 + 0.999946i \(0.496680\pi\)
\(830\) 2.80051 0.0972072
\(831\) 0 0
\(832\) 32.3628 1.12198
\(833\) 21.4644 0.743698
\(834\) 0 0
\(835\) 2.67228 0.0924780
\(836\) 6.61033 0.228623
\(837\) 0 0
\(838\) 13.2590 0.458024
\(839\) 43.3323 1.49600 0.747998 0.663701i \(-0.231015\pi\)
0.747998 + 0.663701i \(0.231015\pi\)
\(840\) 0 0
\(841\) −1.09334 −0.0377015
\(842\) −87.6871 −3.02190
\(843\) 0 0
\(844\) −24.2698 −0.835402
\(845\) −4.00271 −0.137697
\(846\) 0 0
\(847\) 52.9378 1.81896
\(848\) −0.694631 −0.0238537
\(849\) 0 0
\(850\) 30.2498 1.03756
\(851\) −46.2891 −1.58677
\(852\) 0 0
\(853\) 12.7838 0.437708 0.218854 0.975758i \(-0.429768\pi\)
0.218854 + 0.975758i \(0.429768\pi\)
\(854\) −98.7291 −3.37844
\(855\) 0 0
\(856\) 12.1057 0.413763
\(857\) −19.3404 −0.660655 −0.330327 0.943866i \(-0.607159\pi\)
−0.330327 + 0.943866i \(0.607159\pi\)
\(858\) 0 0
\(859\) 26.4521 0.902535 0.451268 0.892389i \(-0.350972\pi\)
0.451268 + 0.892389i \(0.350972\pi\)
\(860\) −11.2029 −0.382015
\(861\) 0 0
\(862\) 0.459916 0.0156648
\(863\) −3.26295 −0.111072 −0.0555360 0.998457i \(-0.517687\pi\)
−0.0555360 + 0.998457i \(0.517687\pi\)
\(864\) 0 0
\(865\) −2.73151 −0.0928741
\(866\) 18.3337 0.623004
\(867\) 0 0
\(868\) 35.5114 1.20534
\(869\) 62.7053 2.12713
\(870\) 0 0
\(871\) −14.7841 −0.500940
\(872\) −10.4046 −0.352345
\(873\) 0 0
\(874\) −6.98678 −0.236331
\(875\) −21.5876 −0.729795
\(876\) 0 0
\(877\) −40.1459 −1.35563 −0.677815 0.735233i \(-0.737073\pi\)
−0.677815 + 0.735233i \(0.737073\pi\)
\(878\) 4.97149 0.167780
\(879\) 0 0
\(880\) −0.820971 −0.0276749
\(881\) 45.4262 1.53045 0.765223 0.643765i \(-0.222629\pi\)
0.765223 + 0.643765i \(0.222629\pi\)
\(882\) 0 0
\(883\) −29.8695 −1.00519 −0.502594 0.864523i \(-0.667621\pi\)
−0.502594 + 0.864523i \(0.667621\pi\)
\(884\) −22.5310 −0.757798
\(885\) 0 0
\(886\) 73.5927 2.47240
\(887\) −27.8682 −0.935722 −0.467861 0.883802i \(-0.654975\pi\)
−0.467861 + 0.883802i \(0.654975\pi\)
\(888\) 0 0
\(889\) −31.2139 −1.04688
\(890\) 0.461102 0.0154562
\(891\) 0 0
\(892\) −26.8317 −0.898391
\(893\) −2.74951 −0.0920089
\(894\) 0 0
\(895\) 2.03807 0.0681250
\(896\) 67.3035 2.24845
\(897\) 0 0
\(898\) −88.3870 −2.94951
\(899\) 15.5219 0.517683
\(900\) 0 0
\(901\) 7.07843 0.235817
\(902\) −41.4242 −1.37927
\(903\) 0 0
\(904\) −34.7655 −1.15628
\(905\) −8.97480 −0.298333
\(906\) 0 0
\(907\) −27.5554 −0.914963 −0.457481 0.889219i \(-0.651248\pi\)
−0.457481 + 0.889219i \(0.651248\pi\)
\(908\) −84.8295 −2.81517
\(909\) 0 0
\(910\) 12.6403 0.419022
\(911\) 6.14871 0.203716 0.101858 0.994799i \(-0.467521\pi\)
0.101858 + 0.994799i \(0.467521\pi\)
\(912\) 0 0
\(913\) −10.4690 −0.346473
\(914\) 67.1499 2.22112
\(915\) 0 0
\(916\) −9.76695 −0.322709
\(917\) 6.61228 0.218357
\(918\) 0 0
\(919\) 16.7617 0.552917 0.276459 0.961026i \(-0.410839\pi\)
0.276459 + 0.961026i \(0.410839\pi\)
\(920\) −11.5124 −0.379552
\(921\) 0 0
\(922\) −48.9529 −1.61218
\(923\) −7.02339 −0.231178
\(924\) 0 0
\(925\) 29.2945 0.963196
\(926\) 79.4644 2.61136
\(927\) 0 0
\(928\) 31.5416 1.03540
\(929\) −54.8425 −1.79933 −0.899663 0.436586i \(-0.856188\pi\)
−0.899663 + 0.436586i \(0.856188\pi\)
\(930\) 0 0
\(931\) 3.13747 0.102827
\(932\) 22.5401 0.738325
\(933\) 0 0
\(934\) 57.1113 1.86874
\(935\) 8.36586 0.273593
\(936\) 0 0
\(937\) 15.1436 0.494719 0.247359 0.968924i \(-0.420437\pi\)
0.247359 + 0.968924i \(0.420437\pi\)
\(938\) −51.5279 −1.68245
\(939\) 0 0
\(940\) −12.2585 −0.399829
\(941\) 20.8740 0.680473 0.340237 0.940340i \(-0.389493\pi\)
0.340237 + 0.940340i \(0.389493\pi\)
\(942\) 0 0
\(943\) 26.8539 0.874484
\(944\) 1.29918 0.0422847
\(945\) 0 0
\(946\) 68.2807 2.22000
\(947\) −50.6227 −1.64502 −0.822509 0.568752i \(-0.807426\pi\)
−0.822509 + 0.568752i \(0.807426\pi\)
\(948\) 0 0
\(949\) 34.1483 1.10850
\(950\) 4.42165 0.143457
\(951\) 0 0
\(952\) −29.0223 −0.940619
\(953\) −54.4936 −1.76522 −0.882610 0.470107i \(-0.844215\pi\)
−0.882610 + 0.470107i \(0.844215\pi\)
\(954\) 0 0
\(955\) −13.1386 −0.425155
\(956\) −14.5490 −0.470549
\(957\) 0 0
\(958\) 22.3153 0.720976
\(959\) 10.4689 0.338058
\(960\) 0 0
\(961\) −22.3666 −0.721505
\(962\) −35.5748 −1.14698
\(963\) 0 0
\(964\) 19.5338 0.629143
\(965\) −3.79879 −0.122288
\(966\) 0 0
\(967\) 41.1758 1.32412 0.662062 0.749449i \(-0.269682\pi\)
0.662062 + 0.749449i \(0.269682\pi\)
\(968\) −37.0542 −1.19097
\(969\) 0 0
\(970\) −14.6428 −0.470151
\(971\) −9.59297 −0.307853 −0.153927 0.988082i \(-0.549192\pi\)
−0.153927 + 0.988082i \(0.549192\pi\)
\(972\) 0 0
\(973\) 32.9685 1.05692
\(974\) 38.7355 1.24117
\(975\) 0 0
\(976\) −3.19473 −0.102261
\(977\) 15.0831 0.482551 0.241276 0.970457i \(-0.422434\pi\)
0.241276 + 0.970457i \(0.422434\pi\)
\(978\) 0 0
\(979\) −1.72371 −0.0550901
\(980\) 13.9882 0.446838
\(981\) 0 0
\(982\) 19.1255 0.610319
\(983\) −56.0198 −1.78675 −0.893377 0.449307i \(-0.851671\pi\)
−0.893377 + 0.449307i \(0.851671\pi\)
\(984\) 0 0
\(985\) −3.24889 −0.103518
\(986\) −34.3244 −1.09311
\(987\) 0 0
\(988\) −3.29337 −0.104776
\(989\) −44.2641 −1.40752
\(990\) 0 0
\(991\) 19.1880 0.609527 0.304763 0.952428i \(-0.401423\pi\)
0.304763 + 0.952428i \(0.401423\pi\)
\(992\) 17.5436 0.557010
\(993\) 0 0
\(994\) −24.4790 −0.776428
\(995\) −9.86065 −0.312604
\(996\) 0 0
\(997\) 32.1559 1.01839 0.509195 0.860651i \(-0.329943\pi\)
0.509195 + 0.860651i \(0.329943\pi\)
\(998\) −54.9537 −1.73953
\(999\) 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 6561.2.a.d.1.5 72
3.2 odd 2 6561.2.a.c.1.68 72
81.2 odd 54 81.2.g.a.4.1 144
81.13 even 27 729.2.g.b.541.1 144
81.14 odd 54 729.2.g.d.55.8 144
81.25 even 27 729.2.g.b.190.1 144
81.29 odd 54 729.2.g.d.676.8 144
81.40 even 27 243.2.g.a.19.8 144
81.41 odd 54 81.2.g.a.61.1 yes 144
81.52 even 27 729.2.g.a.676.1 144
81.56 odd 54 729.2.g.c.190.8 144
81.67 even 27 729.2.g.a.55.1 144
81.68 odd 54 729.2.g.c.541.8 144
81.79 even 27 243.2.g.a.64.8 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.g.a.4.1 144 81.2 odd 54
81.2.g.a.61.1 yes 144 81.41 odd 54
243.2.g.a.19.8 144 81.40 even 27
243.2.g.a.64.8 144 81.79 even 27
729.2.g.a.55.1 144 81.67 even 27
729.2.g.a.676.1 144 81.52 even 27
729.2.g.b.190.1 144 81.25 even 27
729.2.g.b.541.1 144 81.13 even 27
729.2.g.c.190.8 144 81.56 odd 54
729.2.g.c.541.8 144 81.68 odd 54
729.2.g.d.55.8 144 81.14 odd 54
729.2.g.d.676.8 144 81.29 odd 54
6561.2.a.c.1.68 72 3.2 odd 2
6561.2.a.d.1.5 72 1.1 even 1 trivial