Properties

Label 138.4.a.d.1.2
Level $138$
Weight $4$
Character 138.1
Self dual yes
Analytic conductor $8.142$
Analytic rank $0$
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] = [138,4,Mod(1,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, 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("138.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.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: \(8.14226358079\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{277}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 69 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-7.82166\) of defining polynomial
Character \(\chi\) \(=\) 138.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +17.6433 q^{5} -6.00000 q^{6} +14.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +17.6433 q^{5} -6.00000 q^{6} +14.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -35.2866 q^{10} -17.6433 q^{11} +12.0000 q^{12} -9.28663 q^{13} -28.0000 q^{14} +52.9300 q^{15} +16.0000 q^{16} -35.2866 q^{17} -18.0000 q^{18} +55.6433 q^{19} +70.5733 q^{20} +42.0000 q^{21} +35.2866 q^{22} +23.0000 q^{23} -24.0000 q^{24} +186.287 q^{25} +18.5733 q^{26} +27.0000 q^{27} +56.0000 q^{28} -74.4332 q^{29} -105.860 q^{30} +187.287 q^{31} -32.0000 q^{32} -52.9300 q^{33} +70.5733 q^{34} +247.006 q^{35} +36.0000 q^{36} +234.930 q^{37} -111.287 q^{38} -27.8599 q^{39} -141.147 q^{40} -393.440 q^{41} -84.0000 q^{42} +209.503 q^{43} -70.5733 q^{44} +158.790 q^{45} -46.0000 q^{46} -64.2802 q^{47} +48.0000 q^{48} -147.000 q^{49} -372.573 q^{50} -105.860 q^{51} -37.1465 q^{52} -379.783 q^{53} -54.0000 q^{54} -311.287 q^{55} -112.000 q^{56} +166.930 q^{57} +148.866 q^{58} -396.000 q^{59} +211.720 q^{60} +594.803 q^{61} -374.573 q^{62} +126.000 q^{63} +64.0000 q^{64} -163.847 q^{65} +105.860 q^{66} -1065.81 q^{67} -141.147 q^{68} +69.0000 q^{69} -494.013 q^{70} +911.159 q^{71} -72.0000 q^{72} -941.593 q^{73} -469.860 q^{74} +558.860 q^{75} +222.573 q^{76} -247.006 q^{77} +55.7198 q^{78} -520.293 q^{79} +282.293 q^{80} +81.0000 q^{81} +786.879 q^{82} -418.382 q^{83} +168.000 q^{84} -622.573 q^{85} -419.006 q^{86} -223.300 q^{87} +141.147 q^{88} +599.159 q^{89} -317.580 q^{90} -130.013 q^{91} +92.0000 q^{92} +561.860 q^{93} +128.560 q^{94} +981.733 q^{95} -96.0000 q^{96} +40.7262 q^{97} +294.000 q^{98} -158.790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 2 q^{5} - 12 q^{6} + 28 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 2 q^{5} - 12 q^{6} + 28 q^{7} - 16 q^{8} + 18 q^{9} - 4 q^{10} - 2 q^{11} + 24 q^{12} + 48 q^{13} - 56 q^{14} + 6 q^{15} + 32 q^{16} - 4 q^{17} - 36 q^{18} + 78 q^{19} + 8 q^{20} + 84 q^{21} + 4 q^{22} + 46 q^{23} - 48 q^{24} + 306 q^{25} - 96 q^{26} + 54 q^{27} + 112 q^{28} + 184 q^{29} - 12 q^{30} + 308 q^{31} - 64 q^{32} - 6 q^{33} + 8 q^{34} + 28 q^{35} + 72 q^{36} + 370 q^{37} - 156 q^{38} + 144 q^{39} - 16 q^{40} + 12 q^{41} - 168 q^{42} + 186 q^{43} - 8 q^{44} + 18 q^{45} - 92 q^{46} - 528 q^{47} + 96 q^{48} - 294 q^{49} - 612 q^{50} - 12 q^{51} + 192 q^{52} - 926 q^{53} - 108 q^{54} - 556 q^{55} - 224 q^{56} + 234 q^{57} - 368 q^{58} - 792 q^{59} + 24 q^{60} - 42 q^{61} - 616 q^{62} + 252 q^{63} + 128 q^{64} - 1060 q^{65} + 12 q^{66} - 434 q^{67} - 16 q^{68} + 138 q^{69} - 56 q^{70} + 624 q^{71} - 144 q^{72} - 352 q^{73} - 740 q^{74} + 918 q^{75} + 312 q^{76} - 28 q^{77} - 288 q^{78} - 508 q^{79} + 32 q^{80} + 162 q^{81} - 24 q^{82} + 994 q^{83} + 336 q^{84} - 1112 q^{85} - 372 q^{86} + 552 q^{87} + 16 q^{88} - 36 q^{90} + 672 q^{91} + 184 q^{92} + 924 q^{93} + 1056 q^{94} + 632 q^{95} - 192 q^{96} - 784 q^{97} + 588 q^{98} - 18 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 17.6433 1.57807 0.789033 0.614351i \(-0.210582\pi\)
0.789033 + 0.614351i \(0.210582\pi\)
\(6\) −6.00000 −0.408248
\(7\) 14.0000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −35.2866 −1.11586
\(11\) −17.6433 −0.483605 −0.241803 0.970325i \(-0.577739\pi\)
−0.241803 + 0.970325i \(0.577739\pi\)
\(12\) 12.0000 0.288675
\(13\) −9.28663 −0.198127 −0.0990634 0.995081i \(-0.531585\pi\)
−0.0990634 + 0.995081i \(0.531585\pi\)
\(14\) −28.0000 −0.534522
\(15\) 52.9300 0.911097
\(16\) 16.0000 0.250000
\(17\) −35.2866 −0.503427 −0.251714 0.967802i \(-0.580994\pi\)
−0.251714 + 0.967802i \(0.580994\pi\)
\(18\) −18.0000 −0.235702
\(19\) 55.6433 0.671866 0.335933 0.941886i \(-0.390948\pi\)
0.335933 + 0.941886i \(0.390948\pi\)
\(20\) 70.5733 0.789033
\(21\) 42.0000 0.436436
\(22\) 35.2866 0.341961
\(23\) 23.0000 0.208514
\(24\) −24.0000 −0.204124
\(25\) 186.287 1.49029
\(26\) 18.5733 0.140097
\(27\) 27.0000 0.192450
\(28\) 56.0000 0.377964
\(29\) −74.4332 −0.476617 −0.238308 0.971190i \(-0.576593\pi\)
−0.238308 + 0.971190i \(0.576593\pi\)
\(30\) −105.860 −0.644243
\(31\) 187.287 1.08509 0.542543 0.840028i \(-0.317462\pi\)
0.542543 + 0.840028i \(0.317462\pi\)
\(32\) −32.0000 −0.176777
\(33\) −52.9300 −0.279210
\(34\) 70.5733 0.355977
\(35\) 247.006 1.19291
\(36\) 36.0000 0.166667
\(37\) 234.930 1.04384 0.521922 0.852993i \(-0.325215\pi\)
0.521922 + 0.852993i \(0.325215\pi\)
\(38\) −111.287 −0.475081
\(39\) −27.8599 −0.114389
\(40\) −141.147 −0.557931
\(41\) −393.440 −1.49866 −0.749329 0.662198i \(-0.769624\pi\)
−0.749329 + 0.662198i \(0.769624\pi\)
\(42\) −84.0000 −0.308607
\(43\) 209.503 0.742999 0.371499 0.928433i \(-0.378844\pi\)
0.371499 + 0.928433i \(0.378844\pi\)
\(44\) −70.5733 −0.241803
\(45\) 158.790 0.526022
\(46\) −46.0000 −0.147442
\(47\) −64.2802 −0.199494 −0.0997471 0.995013i \(-0.531803\pi\)
−0.0997471 + 0.995013i \(0.531803\pi\)
\(48\) 48.0000 0.144338
\(49\) −147.000 −0.428571
\(50\) −372.573 −1.05380
\(51\) −105.860 −0.290654
\(52\) −37.1465 −0.0990634
\(53\) −379.783 −0.984288 −0.492144 0.870514i \(-0.663787\pi\)
−0.492144 + 0.870514i \(0.663787\pi\)
\(54\) −54.0000 −0.136083
\(55\) −311.287 −0.763161
\(56\) −112.000 −0.267261
\(57\) 166.930 0.387902
\(58\) 148.866 0.337019
\(59\) −396.000 −0.873810 −0.436905 0.899508i \(-0.643925\pi\)
−0.436905 + 0.899508i \(0.643925\pi\)
\(60\) 211.720 0.455548
\(61\) 594.803 1.24847 0.624235 0.781237i \(-0.285411\pi\)
0.624235 + 0.781237i \(0.285411\pi\)
\(62\) −374.573 −0.767272
\(63\) 126.000 0.251976
\(64\) 64.0000 0.125000
\(65\) −163.847 −0.312657
\(66\) 105.860 0.197431
\(67\) −1065.81 −1.94342 −0.971711 0.236172i \(-0.924107\pi\)
−0.971711 + 0.236172i \(0.924107\pi\)
\(68\) −141.147 −0.251714
\(69\) 69.0000 0.120386
\(70\) −494.013 −0.843512
\(71\) 911.159 1.52302 0.761512 0.648151i \(-0.224457\pi\)
0.761512 + 0.648151i \(0.224457\pi\)
\(72\) −72.0000 −0.117851
\(73\) −941.593 −1.50966 −0.754829 0.655921i \(-0.772280\pi\)
−0.754829 + 0.655921i \(0.772280\pi\)
\(74\) −469.860 −0.738110
\(75\) 558.860 0.860421
\(76\) 222.573 0.335933
\(77\) −247.006 −0.365571
\(78\) 55.7198 0.0808849
\(79\) −520.293 −0.740982 −0.370491 0.928836i \(-0.620810\pi\)
−0.370491 + 0.928836i \(0.620810\pi\)
\(80\) 282.293 0.394517
\(81\) 81.0000 0.111111
\(82\) 786.879 1.05971
\(83\) −418.382 −0.553294 −0.276647 0.960972i \(-0.589223\pi\)
−0.276647 + 0.960972i \(0.589223\pi\)
\(84\) 168.000 0.218218
\(85\) −622.573 −0.794442
\(86\) −419.006 −0.525379
\(87\) −223.300 −0.275175
\(88\) 141.147 0.170980
\(89\) 599.159 0.713604 0.356802 0.934180i \(-0.383867\pi\)
0.356802 + 0.934180i \(0.383867\pi\)
\(90\) −317.580 −0.371954
\(91\) −130.013 −0.149770
\(92\) 92.0000 0.104257
\(93\) 561.860 0.626475
\(94\) 128.560 0.141064
\(95\) 981.733 1.06025
\(96\) −96.0000 −0.102062
\(97\) 40.7262 0.0426301 0.0213151 0.999773i \(-0.493215\pi\)
0.0213151 + 0.999773i \(0.493215\pi\)
\(98\) 294.000 0.303046
\(99\) −158.790 −0.161202
\(100\) 745.147 0.745147
\(101\) −1461.31 −1.43966 −0.719832 0.694149i \(-0.755781\pi\)
−0.719832 + 0.694149i \(0.755781\pi\)
\(102\) 211.720 0.205523
\(103\) 1573.03 1.50481 0.752405 0.658701i \(-0.228894\pi\)
0.752405 + 0.658701i \(0.228894\pi\)
\(104\) 74.2931 0.0700484
\(105\) 741.019 0.688725
\(106\) 759.567 0.695997
\(107\) 1698.82 1.53487 0.767433 0.641129i \(-0.221534\pi\)
0.767433 + 0.641129i \(0.221534\pi\)
\(108\) 108.000 0.0962250
\(109\) 389.376 0.342160 0.171080 0.985257i \(-0.445274\pi\)
0.171080 + 0.985257i \(0.445274\pi\)
\(110\) 622.573 0.539637
\(111\) 704.790 0.602664
\(112\) 224.000 0.188982
\(113\) 110.853 0.0922851 0.0461426 0.998935i \(-0.485307\pi\)
0.0461426 + 0.998935i \(0.485307\pi\)
\(114\) −333.860 −0.274288
\(115\) 405.796 0.329050
\(116\) −297.733 −0.238308
\(117\) −83.5797 −0.0660423
\(118\) 792.000 0.617877
\(119\) −494.013 −0.380555
\(120\) −423.440 −0.322121
\(121\) −1019.71 −0.766126
\(122\) −1189.61 −0.882802
\(123\) −1180.32 −0.865250
\(124\) 749.147 0.542543
\(125\) 1081.30 0.773715
\(126\) −252.000 −0.178174
\(127\) −437.427 −0.305633 −0.152816 0.988255i \(-0.548834\pi\)
−0.152816 + 0.988255i \(0.548834\pi\)
\(128\) −128.000 −0.0883883
\(129\) 628.510 0.428970
\(130\) 327.694 0.221082
\(131\) −2139.06 −1.42664 −0.713322 0.700836i \(-0.752810\pi\)
−0.713322 + 0.700836i \(0.752810\pi\)
\(132\) −211.720 −0.139605
\(133\) 779.006 0.507883
\(134\) 2131.62 1.37421
\(135\) 476.370 0.303699
\(136\) 282.293 0.177988
\(137\) −753.019 −0.469597 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(138\) −138.000 −0.0851257
\(139\) −615.159 −0.375375 −0.187688 0.982229i \(-0.560099\pi\)
−0.187688 + 0.982229i \(0.560099\pi\)
\(140\) 988.026 0.596453
\(141\) −192.841 −0.115178
\(142\) −1822.32 −1.07694
\(143\) 163.847 0.0958152
\(144\) 144.000 0.0833333
\(145\) −1313.25 −0.752133
\(146\) 1883.19 1.06749
\(147\) −441.000 −0.247436
\(148\) 939.720 0.521922
\(149\) −3004.42 −1.65189 −0.825946 0.563750i \(-0.809358\pi\)
−0.825946 + 0.563750i \(0.809358\pi\)
\(150\) −1117.72 −0.608410
\(151\) −1863.29 −1.00419 −0.502093 0.864814i \(-0.667437\pi\)
−0.502093 + 0.864814i \(0.667437\pi\)
\(152\) −445.147 −0.237540
\(153\) −317.580 −0.167809
\(154\) 494.013 0.258498
\(155\) 3304.36 1.71234
\(156\) −111.440 −0.0571943
\(157\) −1471.92 −0.748231 −0.374116 0.927382i \(-0.622054\pi\)
−0.374116 + 0.927382i \(0.622054\pi\)
\(158\) 1040.59 0.523953
\(159\) −1139.35 −0.568279
\(160\) −564.586 −0.278965
\(161\) 322.000 0.157622
\(162\) −162.000 −0.0785674
\(163\) 467.235 0.224520 0.112260 0.993679i \(-0.464191\pi\)
0.112260 + 0.993679i \(0.464191\pi\)
\(164\) −1573.76 −0.749329
\(165\) −933.860 −0.440611
\(166\) 836.765 0.391238
\(167\) 1858.57 0.861202 0.430601 0.902542i \(-0.358302\pi\)
0.430601 + 0.902542i \(0.358302\pi\)
\(168\) −336.000 −0.154303
\(169\) −2110.76 −0.960746
\(170\) 1245.15 0.561755
\(171\) 500.790 0.223955
\(172\) 838.013 0.371499
\(173\) 2224.78 0.977727 0.488863 0.872360i \(-0.337412\pi\)
0.488863 + 0.872360i \(0.337412\pi\)
\(174\) 446.599 0.194578
\(175\) 2608.01 1.12656
\(176\) −282.293 −0.120901
\(177\) −1188.00 −0.504495
\(178\) −1198.32 −0.504594
\(179\) 3839.29 1.60314 0.801569 0.597902i \(-0.203999\pi\)
0.801569 + 0.597902i \(0.203999\pi\)
\(180\) 635.159 0.263011
\(181\) −4215.26 −1.73104 −0.865519 0.500876i \(-0.833011\pi\)
−0.865519 + 0.500876i \(0.833011\pi\)
\(182\) 260.026 0.105903
\(183\) 1784.41 0.720805
\(184\) −184.000 −0.0737210
\(185\) 4144.94 1.64726
\(186\) −1123.72 −0.442985
\(187\) 622.573 0.243460
\(188\) −257.121 −0.0997471
\(189\) 378.000 0.145479
\(190\) −1963.47 −0.749709
\(191\) −3829.55 −1.45077 −0.725384 0.688344i \(-0.758338\pi\)
−0.725384 + 0.688344i \(0.758338\pi\)
\(192\) 192.000 0.0721688
\(193\) −262.077 −0.0977447 −0.0488724 0.998805i \(-0.515563\pi\)
−0.0488724 + 0.998805i \(0.515563\pi\)
\(194\) −81.4525 −0.0301441
\(195\) −491.541 −0.180513
\(196\) −588.000 −0.214286
\(197\) −5380.40 −1.94587 −0.972937 0.231069i \(-0.925778\pi\)
−0.972937 + 0.231069i \(0.925778\pi\)
\(198\) 317.580 0.113987
\(199\) 3908.04 1.39213 0.696064 0.717980i \(-0.254933\pi\)
0.696064 + 0.717980i \(0.254933\pi\)
\(200\) −1490.29 −0.526898
\(201\) −3197.43 −1.12204
\(202\) 2922.62 1.01800
\(203\) −1042.06 −0.360289
\(204\) −423.440 −0.145327
\(205\) −6941.58 −2.36498
\(206\) −3146.06 −1.06406
\(207\) 207.000 0.0695048
\(208\) −148.586 −0.0495317
\(209\) −981.733 −0.324918
\(210\) −1482.04 −0.487002
\(211\) 2113.91 0.689705 0.344852 0.938657i \(-0.387929\pi\)
0.344852 + 0.938657i \(0.387929\pi\)
\(212\) −1519.13 −0.492144
\(213\) 2733.48 0.879319
\(214\) −3397.63 −1.08531
\(215\) 3696.33 1.17250
\(216\) −216.000 −0.0680414
\(217\) 2622.01 0.820248
\(218\) −778.752 −0.241944
\(219\) −2824.78 −0.871602
\(220\) −1245.15 −0.381581
\(221\) 327.694 0.0997425
\(222\) −1409.58 −0.426148
\(223\) 4783.70 1.43650 0.718251 0.695784i \(-0.244943\pi\)
0.718251 + 0.695784i \(0.244943\pi\)
\(224\) −448.000 −0.133631
\(225\) 1676.58 0.496764
\(226\) −221.707 −0.0652554
\(227\) −663.630 −0.194038 −0.0970192 0.995283i \(-0.530931\pi\)
−0.0970192 + 0.995283i \(0.530931\pi\)
\(228\) 667.720 0.193951
\(229\) 2959.47 0.854004 0.427002 0.904251i \(-0.359570\pi\)
0.427002 + 0.904251i \(0.359570\pi\)
\(230\) −811.593 −0.232673
\(231\) −741.019 −0.211063
\(232\) 595.465 0.168510
\(233\) 2503.76 0.703977 0.351989 0.936004i \(-0.385506\pi\)
0.351989 + 0.936004i \(0.385506\pi\)
\(234\) 167.159 0.0466989
\(235\) −1134.12 −0.314815
\(236\) −1584.00 −0.436905
\(237\) −1560.88 −0.427806
\(238\) 988.026 0.269093
\(239\) 6400.36 1.73224 0.866119 0.499838i \(-0.166607\pi\)
0.866119 + 0.499838i \(0.166607\pi\)
\(240\) 846.879 0.227774
\(241\) −873.209 −0.233396 −0.116698 0.993167i \(-0.537231\pi\)
−0.116698 + 0.993167i \(0.537231\pi\)
\(242\) 2039.43 0.541733
\(243\) 243.000 0.0641500
\(244\) 2379.21 0.624235
\(245\) −2593.57 −0.676314
\(246\) 2360.64 0.611824
\(247\) −516.739 −0.133115
\(248\) −1498.29 −0.383636
\(249\) −1255.15 −0.319445
\(250\) −2162.60 −0.547099
\(251\) −1698.87 −0.427217 −0.213608 0.976919i \(-0.568522\pi\)
−0.213608 + 0.976919i \(0.568522\pi\)
\(252\) 504.000 0.125988
\(253\) −405.796 −0.100839
\(254\) 874.853 0.216115
\(255\) −1867.72 −0.458671
\(256\) 256.000 0.0625000
\(257\) 5625.01 1.36529 0.682643 0.730752i \(-0.260831\pi\)
0.682643 + 0.730752i \(0.260831\pi\)
\(258\) −1257.02 −0.303328
\(259\) 3289.02 0.789072
\(260\) −655.388 −0.156329
\(261\) −669.899 −0.158872
\(262\) 4278.12 1.00879
\(263\) −1360.69 −0.319025 −0.159513 0.987196i \(-0.550992\pi\)
−0.159513 + 0.987196i \(0.550992\pi\)
\(264\) 423.440 0.0987156
\(265\) −6700.64 −1.55327
\(266\) −1558.01 −0.359127
\(267\) 1797.48 0.412000
\(268\) −4263.24 −0.971711
\(269\) 1666.53 0.377734 0.188867 0.982003i \(-0.439519\pi\)
0.188867 + 0.982003i \(0.439519\pi\)
\(270\) −952.739 −0.214748
\(271\) −3321.20 −0.744459 −0.372230 0.928141i \(-0.621407\pi\)
−0.372230 + 0.928141i \(0.621407\pi\)
\(272\) −564.586 −0.125857
\(273\) −390.039 −0.0864696
\(274\) 1506.04 0.332055
\(275\) −3286.71 −0.720714
\(276\) 276.000 0.0601929
\(277\) −157.490 −0.0341611 −0.0170806 0.999854i \(-0.505437\pi\)
−0.0170806 + 0.999854i \(0.505437\pi\)
\(278\) 1230.32 0.265430
\(279\) 1685.58 0.361695
\(280\) −1976.05 −0.421756
\(281\) −150.802 −0.0320146 −0.0160073 0.999872i \(-0.505095\pi\)
−0.0160073 + 0.999872i \(0.505095\pi\)
\(282\) 385.681 0.0814432
\(283\) 6881.58 1.44547 0.722734 0.691126i \(-0.242885\pi\)
0.722734 + 0.691126i \(0.242885\pi\)
\(284\) 3644.64 0.761512
\(285\) 2945.20 0.612135
\(286\) −327.694 −0.0677516
\(287\) −5508.15 −1.13288
\(288\) −288.000 −0.0589256
\(289\) −3667.85 −0.746561
\(290\) 2626.50 0.531838
\(291\) 122.179 0.0246125
\(292\) −3766.37 −0.754829
\(293\) 5650.84 1.12671 0.563354 0.826215i \(-0.309511\pi\)
0.563354 + 0.826215i \(0.309511\pi\)
\(294\) 882.000 0.174964
\(295\) −6986.75 −1.37893
\(296\) −1879.44 −0.369055
\(297\) −476.370 −0.0930699
\(298\) 6008.84 1.16806
\(299\) −213.593 −0.0413123
\(300\) 2235.44 0.430211
\(301\) 2933.05 0.561654
\(302\) 3726.57 0.710067
\(303\) −4383.94 −0.831190
\(304\) 890.293 0.167966
\(305\) 10494.3 1.97017
\(306\) 635.159 0.118659
\(307\) 4071.26 0.756870 0.378435 0.925628i \(-0.376462\pi\)
0.378435 + 0.925628i \(0.376462\pi\)
\(308\) −988.026 −0.182786
\(309\) 4719.10 0.868803
\(310\) −6608.71 −1.21081
\(311\) −532.535 −0.0970973 −0.0485487 0.998821i \(-0.515460\pi\)
−0.0485487 + 0.998821i \(0.515460\pi\)
\(312\) 222.879 0.0404425
\(313\) 9759.63 1.76245 0.881225 0.472697i \(-0.156719\pi\)
0.881225 + 0.472697i \(0.156719\pi\)
\(314\) 2943.85 0.529079
\(315\) 2223.06 0.397635
\(316\) −2081.17 −0.370491
\(317\) 3997.16 0.708211 0.354106 0.935205i \(-0.384785\pi\)
0.354106 + 0.935205i \(0.384785\pi\)
\(318\) 2278.70 0.401834
\(319\) 1313.25 0.230495
\(320\) 1129.17 0.197258
\(321\) 5096.45 0.886156
\(322\) −644.000 −0.111456
\(323\) −1963.47 −0.338236
\(324\) 324.000 0.0555556
\(325\) −1729.98 −0.295267
\(326\) −934.470 −0.158759
\(327\) 1168.13 0.197546
\(328\) 3147.52 0.529855
\(329\) −899.923 −0.150803
\(330\) 1867.72 0.311559
\(331\) 6349.73 1.05442 0.527210 0.849735i \(-0.323238\pi\)
0.527210 + 0.849735i \(0.323238\pi\)
\(332\) −1673.53 −0.276647
\(333\) 2114.37 0.347948
\(334\) −3717.15 −0.608962
\(335\) −18804.4 −3.06685
\(336\) 672.000 0.109109
\(337\) 2530.74 0.409074 0.204537 0.978859i \(-0.434431\pi\)
0.204537 + 0.978859i \(0.434431\pi\)
\(338\) 4221.52 0.679350
\(339\) 332.560 0.0532808
\(340\) −2490.29 −0.397221
\(341\) −3304.36 −0.524754
\(342\) −1001.58 −0.158360
\(343\) −6860.00 −1.07990
\(344\) −1676.03 −0.262690
\(345\) 1217.39 0.189977
\(346\) −4449.56 −0.691357
\(347\) 4947.72 0.765440 0.382720 0.923864i \(-0.374987\pi\)
0.382720 + 0.923864i \(0.374987\pi\)
\(348\) −893.198 −0.137587
\(349\) 3314.15 0.508317 0.254158 0.967163i \(-0.418202\pi\)
0.254158 + 0.967163i \(0.418202\pi\)
\(350\) −5216.03 −0.796595
\(351\) −250.739 −0.0381295
\(352\) 564.586 0.0854902
\(353\) 4137.26 0.623808 0.311904 0.950114i \(-0.399033\pi\)
0.311904 + 0.950114i \(0.399033\pi\)
\(354\) 2376.00 0.356732
\(355\) 16075.9 2.40343
\(356\) 2396.64 0.356802
\(357\) −1482.04 −0.219714
\(358\) −7678.57 −1.13359
\(359\) −10388.3 −1.52723 −0.763615 0.645672i \(-0.776577\pi\)
−0.763615 + 0.645672i \(0.776577\pi\)
\(360\) −1270.32 −0.185977
\(361\) −3762.82 −0.548596
\(362\) 8430.52 1.22403
\(363\) −3059.14 −0.442323
\(364\) −520.052 −0.0748849
\(365\) −16612.8 −2.38234
\(366\) −3568.82 −0.509686
\(367\) −6576.09 −0.935338 −0.467669 0.883904i \(-0.654906\pi\)
−0.467669 + 0.883904i \(0.654906\pi\)
\(368\) 368.000 0.0521286
\(369\) −3540.96 −0.499552
\(370\) −8289.89 −1.16479
\(371\) −5316.97 −0.744052
\(372\) 2247.44 0.313237
\(373\) 6765.53 0.939158 0.469579 0.882891i \(-0.344406\pi\)
0.469579 + 0.882891i \(0.344406\pi\)
\(374\) −1245.15 −0.172152
\(375\) 3243.90 0.446705
\(376\) 514.242 0.0705319
\(377\) 691.234 0.0944306
\(378\) −756.000 −0.102869
\(379\) −155.159 −0.0210289 −0.0105145 0.999945i \(-0.503347\pi\)
−0.0105145 + 0.999945i \(0.503347\pi\)
\(380\) 3926.93 0.530124
\(381\) −1312.28 −0.176457
\(382\) 7659.11 1.02585
\(383\) −7842.88 −1.04635 −0.523175 0.852225i \(-0.675253\pi\)
−0.523175 + 0.852225i \(0.675253\pi\)
\(384\) −384.000 −0.0510310
\(385\) −4358.01 −0.576896
\(386\) 524.155 0.0691160
\(387\) 1885.53 0.247666
\(388\) 162.905 0.0213151
\(389\) 932.447 0.121535 0.0607673 0.998152i \(-0.480645\pi\)
0.0607673 + 0.998152i \(0.480645\pi\)
\(390\) 983.082 0.127642
\(391\) −811.593 −0.104972
\(392\) 1176.00 0.151523
\(393\) −6417.17 −0.823673
\(394\) 10760.8 1.37594
\(395\) −9179.70 −1.16932
\(396\) −635.159 −0.0806009
\(397\) 11841.7 1.49702 0.748512 0.663121i \(-0.230768\pi\)
0.748512 + 0.663121i \(0.230768\pi\)
\(398\) −7816.08 −0.984383
\(399\) 2337.02 0.293226
\(400\) 2980.59 0.372573
\(401\) −10096.7 −1.25737 −0.628687 0.777658i \(-0.716407\pi\)
−0.628687 + 0.777658i \(0.716407\pi\)
\(402\) 6394.85 0.793399
\(403\) −1739.26 −0.214985
\(404\) −5845.25 −0.719832
\(405\) 1429.11 0.175341
\(406\) 2084.13 0.254762
\(407\) −4144.94 −0.504809
\(408\) 846.879 0.102762
\(409\) −7527.86 −0.910095 −0.455047 0.890467i \(-0.650378\pi\)
−0.455047 + 0.890467i \(0.650378\pi\)
\(410\) 13883.2 1.67229
\(411\) −2259.06 −0.271122
\(412\) 6292.13 0.752405
\(413\) −5544.00 −0.660539
\(414\) −414.000 −0.0491473
\(415\) −7381.65 −0.873135
\(416\) 297.172 0.0350242
\(417\) −1845.48 −0.216723
\(418\) 1963.47 0.229752
\(419\) −6805.49 −0.793484 −0.396742 0.917930i \(-0.629859\pi\)
−0.396742 + 0.917930i \(0.629859\pi\)
\(420\) 2964.08 0.344362
\(421\) −15768.0 −1.82538 −0.912691 0.408651i \(-0.865999\pi\)
−0.912691 + 0.408651i \(0.865999\pi\)
\(422\) −4227.82 −0.487695
\(423\) −578.522 −0.0664981
\(424\) 3038.27 0.347998
\(425\) −6573.43 −0.750254
\(426\) −5466.96 −0.621772
\(427\) 8327.24 0.943755
\(428\) 6795.26 0.767433
\(429\) 491.541 0.0553189
\(430\) −7392.66 −0.829083
\(431\) −1074.27 −0.120059 −0.0600297 0.998197i \(-0.519120\pi\)
−0.0600297 + 0.998197i \(0.519120\pi\)
\(432\) 432.000 0.0481125
\(433\) 16863.2 1.87158 0.935788 0.352563i \(-0.114690\pi\)
0.935788 + 0.352563i \(0.114690\pi\)
\(434\) −5244.03 −0.580003
\(435\) −3939.74 −0.434244
\(436\) 1557.50 0.171080
\(437\) 1279.80 0.140094
\(438\) 5649.56 0.616316
\(439\) −1592.20 −0.173102 −0.0865509 0.996247i \(-0.527585\pi\)
−0.0865509 + 0.996247i \(0.527585\pi\)
\(440\) 2490.29 0.269818
\(441\) −1323.00 −0.142857
\(442\) −655.388 −0.0705286
\(443\) 7007.67 0.751567 0.375784 0.926707i \(-0.377374\pi\)
0.375784 + 0.926707i \(0.377374\pi\)
\(444\) 2819.16 0.301332
\(445\) 10571.2 1.12611
\(446\) −9567.39 −1.01576
\(447\) −9013.26 −0.953720
\(448\) 896.000 0.0944911
\(449\) 15939.5 1.67534 0.837672 0.546174i \(-0.183916\pi\)
0.837672 + 0.546174i \(0.183916\pi\)
\(450\) −3353.16 −0.351265
\(451\) 6941.58 0.724759
\(452\) 443.414 0.0461426
\(453\) −5589.86 −0.579767
\(454\) 1327.26 0.137206
\(455\) −2293.86 −0.236347
\(456\) −1335.44 −0.137144
\(457\) −7205.18 −0.737515 −0.368757 0.929526i \(-0.620217\pi\)
−0.368757 + 0.929526i \(0.620217\pi\)
\(458\) −5918.93 −0.603872
\(459\) −952.739 −0.0968846
\(460\) 1623.19 0.164525
\(461\) −12480.1 −1.26086 −0.630431 0.776246i \(-0.717122\pi\)
−0.630431 + 0.776246i \(0.717122\pi\)
\(462\) 1482.04 0.149244
\(463\) 62.0613 0.00622945 0.00311472 0.999995i \(-0.499009\pi\)
0.00311472 + 0.999995i \(0.499009\pi\)
\(464\) −1190.93 −0.119154
\(465\) 9913.07 0.988619
\(466\) −5007.52 −0.497787
\(467\) 7018.84 0.695489 0.347744 0.937589i \(-0.386948\pi\)
0.347744 + 0.937589i \(0.386948\pi\)
\(468\) −334.319 −0.0330211
\(469\) −14921.3 −1.46909
\(470\) 2268.23 0.222608
\(471\) −4415.77 −0.431991
\(472\) 3168.00 0.308939
\(473\) −3696.33 −0.359318
\(474\) 3121.76 0.302504
\(475\) 10365.6 1.00128
\(476\) −1976.05 −0.190278
\(477\) −3418.05 −0.328096
\(478\) −12800.7 −1.22488
\(479\) 18588.6 1.77315 0.886573 0.462589i \(-0.153080\pi\)
0.886573 + 0.462589i \(0.153080\pi\)
\(480\) −1693.76 −0.161061
\(481\) −2181.71 −0.206814
\(482\) 1746.42 0.165036
\(483\) 966.000 0.0910032
\(484\) −4078.85 −0.383063
\(485\) 718.546 0.0672732
\(486\) −486.000 −0.0453609
\(487\) −2248.82 −0.209248 −0.104624 0.994512i \(-0.533364\pi\)
−0.104624 + 0.994512i \(0.533364\pi\)
\(488\) −4758.42 −0.441401
\(489\) 1401.71 0.129626
\(490\) 5187.14 0.478226
\(491\) 228.382 0.0209913 0.0104956 0.999945i \(-0.496659\pi\)
0.0104956 + 0.999945i \(0.496659\pi\)
\(492\) −4721.28 −0.432625
\(493\) 2626.50 0.239942
\(494\) 1033.48 0.0941263
\(495\) −2801.58 −0.254387
\(496\) 2996.59 0.271272
\(497\) 12756.2 1.15130
\(498\) 2510.29 0.225882
\(499\) 13676.7 1.22696 0.613481 0.789710i \(-0.289769\pi\)
0.613481 + 0.789710i \(0.289769\pi\)
\(500\) 4325.20 0.386857
\(501\) 5575.72 0.497215
\(502\) 3397.73 0.302088
\(503\) −3414.19 −0.302646 −0.151323 0.988484i \(-0.548353\pi\)
−0.151323 + 0.988484i \(0.548353\pi\)
\(504\) −1008.00 −0.0890871
\(505\) −25782.4 −2.27188
\(506\) 811.593 0.0713037
\(507\) −6332.28 −0.554687
\(508\) −1749.71 −0.152816
\(509\) 19713.9 1.71671 0.858353 0.513060i \(-0.171488\pi\)
0.858353 + 0.513060i \(0.171488\pi\)
\(510\) 3735.44 0.324330
\(511\) −13182.3 −1.14119
\(512\) −512.000 −0.0441942
\(513\) 1502.37 0.129301
\(514\) −11250.0 −0.965403
\(515\) 27753.5 2.37469
\(516\) 2514.04 0.214485
\(517\) 1134.12 0.0964765
\(518\) −6578.04 −0.557958
\(519\) 6674.33 0.564491
\(520\) 1310.78 0.110541
\(521\) 16581.6 1.39434 0.697172 0.716904i \(-0.254441\pi\)
0.697172 + 0.716904i \(0.254441\pi\)
\(522\) 1339.80 0.112340
\(523\) −11603.8 −0.970170 −0.485085 0.874467i \(-0.661211\pi\)
−0.485085 + 0.874467i \(0.661211\pi\)
\(524\) −8556.23 −0.713322
\(525\) 7824.04 0.650417
\(526\) 2721.38 0.225585
\(527\) −6608.71 −0.546262
\(528\) −846.879 −0.0698024
\(529\) 529.000 0.0434783
\(530\) 13401.3 1.09833
\(531\) −3564.00 −0.291270
\(532\) 3116.03 0.253941
\(533\) 3653.73 0.296924
\(534\) −3594.96 −0.291328
\(535\) 29972.7 2.42212
\(536\) 8526.47 0.687104
\(537\) 11517.9 0.925572
\(538\) −3333.07 −0.267098
\(539\) 2593.57 0.207259
\(540\) 1905.48 0.151849
\(541\) 22175.8 1.76231 0.881156 0.472826i \(-0.156766\pi\)
0.881156 + 0.472826i \(0.156766\pi\)
\(542\) 6642.40 0.526412
\(543\) −12645.8 −0.999415
\(544\) 1129.17 0.0889942
\(545\) 6869.88 0.539951
\(546\) 780.077 0.0611433
\(547\) −7899.03 −0.617437 −0.308719 0.951153i \(-0.599900\pi\)
−0.308719 + 0.951153i \(0.599900\pi\)
\(548\) −3012.08 −0.234798
\(549\) 5353.22 0.416157
\(550\) 6573.43 0.509622
\(551\) −4141.71 −0.320223
\(552\) −552.000 −0.0425628
\(553\) −7284.10 −0.560129
\(554\) 314.979 0.0241556
\(555\) 12434.8 0.951044
\(556\) −2460.64 −0.187688
\(557\) 23535.7 1.79037 0.895187 0.445692i \(-0.147042\pi\)
0.895187 + 0.445692i \(0.147042\pi\)
\(558\) −3371.16 −0.255757
\(559\) −1945.58 −0.147208
\(560\) 3952.10 0.298226
\(561\) 1867.72 0.140562
\(562\) 301.604 0.0226377
\(563\) 5337.85 0.399580 0.199790 0.979839i \(-0.435974\pi\)
0.199790 + 0.979839i \(0.435974\pi\)
\(564\) −771.362 −0.0575890
\(565\) 1955.82 0.145632
\(566\) −13763.2 −1.02210
\(567\) 1134.00 0.0839921
\(568\) −7289.28 −0.538470
\(569\) −15639.6 −1.15228 −0.576140 0.817351i \(-0.695442\pi\)
−0.576140 + 0.817351i \(0.695442\pi\)
\(570\) −5890.40 −0.432845
\(571\) 7640.99 0.560010 0.280005 0.959999i \(-0.409664\pi\)
0.280005 + 0.959999i \(0.409664\pi\)
\(572\) 655.388 0.0479076
\(573\) −11488.7 −0.837602
\(574\) 11016.3 0.801066
\(575\) 4284.59 0.310748
\(576\) 576.000 0.0416667
\(577\) −7382.08 −0.532617 −0.266308 0.963888i \(-0.585804\pi\)
−0.266308 + 0.963888i \(0.585804\pi\)
\(578\) 7335.71 0.527898
\(579\) −786.232 −0.0564329
\(580\) −5252.99 −0.376067
\(581\) −5857.35 −0.418251
\(582\) −244.357 −0.0174037
\(583\) 6700.64 0.476007
\(584\) 7532.74 0.533745
\(585\) −1474.62 −0.104219
\(586\) −11301.7 −0.796703
\(587\) 8046.01 0.565749 0.282874 0.959157i \(-0.408712\pi\)
0.282874 + 0.959157i \(0.408712\pi\)
\(588\) −1764.00 −0.123718
\(589\) 10421.2 0.729032
\(590\) 13973.5 0.975051
\(591\) −16141.2 −1.12345
\(592\) 3758.88 0.260961
\(593\) −4264.63 −0.295324 −0.147662 0.989038i \(-0.547175\pi\)
−0.147662 + 0.989038i \(0.547175\pi\)
\(594\) 952.739 0.0658104
\(595\) −8716.03 −0.600542
\(596\) −12017.7 −0.825946
\(597\) 11724.1 0.803746
\(598\) 427.185 0.0292122
\(599\) −24148.6 −1.64722 −0.823612 0.567154i \(-0.808044\pi\)
−0.823612 + 0.567154i \(0.808044\pi\)
\(600\) −4470.88 −0.304205
\(601\) −6389.01 −0.433632 −0.216816 0.976212i \(-0.569567\pi\)
−0.216816 + 0.976212i \(0.569567\pi\)
\(602\) −5866.09 −0.397149
\(603\) −9592.28 −0.647808
\(604\) −7453.15 −0.502093
\(605\) −17991.1 −1.20900
\(606\) 8767.87 0.587740
\(607\) −1666.19 −0.111414 −0.0557072 0.998447i \(-0.517741\pi\)
−0.0557072 + 0.998447i \(0.517741\pi\)
\(608\) −1780.59 −0.118770
\(609\) −3126.19 −0.208013
\(610\) −20988.6 −1.39312
\(611\) 596.947 0.0395252
\(612\) −1270.32 −0.0839046
\(613\) 21040.3 1.38631 0.693155 0.720789i \(-0.256220\pi\)
0.693155 + 0.720789i \(0.256220\pi\)
\(614\) −8142.52 −0.535188
\(615\) −20824.7 −1.36542
\(616\) 1976.05 0.129249
\(617\) 13670.0 0.891950 0.445975 0.895045i \(-0.352857\pi\)
0.445975 + 0.895045i \(0.352857\pi\)
\(618\) −9438.19 −0.614336
\(619\) −22747.0 −1.47702 −0.738512 0.674241i \(-0.764471\pi\)
−0.738512 + 0.674241i \(0.764471\pi\)
\(620\) 13217.4 0.856169
\(621\) 621.000 0.0401286
\(622\) 1065.07 0.0686582
\(623\) 8388.23 0.539434
\(624\) −445.758 −0.0285971
\(625\) −4208.12 −0.269320
\(626\) −19519.3 −1.24624
\(627\) −2945.20 −0.187592
\(628\) −5887.69 −0.374116
\(629\) −8289.89 −0.525500
\(630\) −4446.12 −0.281171
\(631\) −4308.14 −0.271798 −0.135899 0.990723i \(-0.543392\pi\)
−0.135899 + 0.990723i \(0.543392\pi\)
\(632\) 4162.34 0.261977
\(633\) 6341.73 0.398201
\(634\) −7994.32 −0.500781
\(635\) −7717.66 −0.482309
\(636\) −4557.40 −0.284139
\(637\) 1365.14 0.0849115
\(638\) −2626.50 −0.162984
\(639\) 8200.43 0.507675
\(640\) −2258.34 −0.139483
\(641\) 897.151 0.0552813 0.0276407 0.999618i \(-0.491201\pi\)
0.0276407 + 0.999618i \(0.491201\pi\)
\(642\) −10192.9 −0.626607
\(643\) −19935.4 −1.22267 −0.611333 0.791374i \(-0.709366\pi\)
−0.611333 + 0.791374i \(0.709366\pi\)
\(644\) 1288.00 0.0788110
\(645\) 11089.0 0.676944
\(646\) 3926.93 0.239169
\(647\) 4670.58 0.283801 0.141901 0.989881i \(-0.454679\pi\)
0.141901 + 0.989881i \(0.454679\pi\)
\(648\) −648.000 −0.0392837
\(649\) 6986.75 0.422579
\(650\) 3459.95 0.208785
\(651\) 7866.04 0.473570
\(652\) 1868.94 0.112260
\(653\) 24406.1 1.46261 0.731304 0.682052i \(-0.238912\pi\)
0.731304 + 0.682052i \(0.238912\pi\)
\(654\) −2336.26 −0.139686
\(655\) −37740.1 −2.25134
\(656\) −6295.03 −0.374664
\(657\) −8474.33 −0.503220
\(658\) 1799.85 0.106634
\(659\) 6906.99 0.408283 0.204141 0.978941i \(-0.434560\pi\)
0.204141 + 0.978941i \(0.434560\pi\)
\(660\) −3735.44 −0.220306
\(661\) 2861.61 0.168387 0.0841934 0.996449i \(-0.473169\pi\)
0.0841934 + 0.996449i \(0.473169\pi\)
\(662\) −12699.5 −0.745587
\(663\) 983.082 0.0575863
\(664\) 3347.06 0.195619
\(665\) 13744.3 0.801473
\(666\) −4228.74 −0.246037
\(667\) −1711.96 −0.0993815
\(668\) 7434.29 0.430601
\(669\) 14351.1 0.829365
\(670\) 37608.8 2.16859
\(671\) −10494.3 −0.603767
\(672\) −1344.00 −0.0771517
\(673\) −28273.2 −1.61939 −0.809696 0.586849i \(-0.800368\pi\)
−0.809696 + 0.586849i \(0.800368\pi\)
\(674\) −5061.48 −0.289259
\(675\) 5029.74 0.286807
\(676\) −8443.03 −0.480373
\(677\) −22741.0 −1.29100 −0.645501 0.763759i \(-0.723352\pi\)
−0.645501 + 0.763759i \(0.723352\pi\)
\(678\) −665.121 −0.0376752
\(679\) 570.167 0.0322254
\(680\) 4980.59 0.280878
\(681\) −1990.89 −0.112028
\(682\) 6608.71 0.371057
\(683\) 31438.8 1.76131 0.880653 0.473763i \(-0.157105\pi\)
0.880653 + 0.473763i \(0.157105\pi\)
\(684\) 2003.16 0.111978
\(685\) −13285.8 −0.741055
\(686\) 13720.0 0.763604
\(687\) 8878.40 0.493060
\(688\) 3352.05 0.185750
\(689\) 3526.91 0.195014
\(690\) −2434.78 −0.134334
\(691\) 2549.94 0.140382 0.0701912 0.997534i \(-0.477639\pi\)
0.0701912 + 0.997534i \(0.477639\pi\)
\(692\) 8899.11 0.488863
\(693\) −2223.06 −0.121857
\(694\) −9895.44 −0.541248
\(695\) −10853.5 −0.592367
\(696\) 1786.40 0.0972890
\(697\) 13883.2 0.754465
\(698\) −6628.31 −0.359434
\(699\) 7511.28 0.406441
\(700\) 10432.1 0.563278
\(701\) −6358.28 −0.342580 −0.171290 0.985221i \(-0.554794\pi\)
−0.171290 + 0.985221i \(0.554794\pi\)
\(702\) 501.478 0.0269616
\(703\) 13072.3 0.701324
\(704\) −1129.17 −0.0604507
\(705\) −3402.35 −0.181759
\(706\) −8274.52 −0.441099
\(707\) −20458.4 −1.08828
\(708\) −4752.00 −0.252247
\(709\) −11553.3 −0.611978 −0.305989 0.952035i \(-0.598987\pi\)
−0.305989 + 0.952035i \(0.598987\pi\)
\(710\) −32151.7 −1.69948
\(711\) −4682.64 −0.246994
\(712\) −4793.28 −0.252297
\(713\) 4307.59 0.226256
\(714\) 2964.08 0.155361
\(715\) 2890.81 0.151203
\(716\) 15357.1 0.801569
\(717\) 19201.1 1.00011
\(718\) 20776.7 1.07991
\(719\) 17141.5 0.889110 0.444555 0.895751i \(-0.353362\pi\)
0.444555 + 0.895751i \(0.353362\pi\)
\(720\) 2540.64 0.131506
\(721\) 22022.5 1.13753
\(722\) 7525.64 0.387916
\(723\) −2619.63 −0.134751
\(724\) −16861.0 −0.865519
\(725\) −13865.9 −0.710299
\(726\) 6118.28 0.312770
\(727\) −2741.52 −0.139859 −0.0699294 0.997552i \(-0.522277\pi\)
−0.0699294 + 0.997552i \(0.522277\pi\)
\(728\) 1040.10 0.0529516
\(729\) 729.000 0.0370370
\(730\) 33225.6 1.68457
\(731\) −7392.66 −0.374046
\(732\) 7137.63 0.360402
\(733\) −9419.32 −0.474639 −0.237320 0.971432i \(-0.576269\pi\)
−0.237320 + 0.971432i \(0.576269\pi\)
\(734\) 13152.2 0.661384
\(735\) −7780.70 −0.390470
\(736\) −736.000 −0.0368605
\(737\) 18804.4 0.939850
\(738\) 7081.91 0.353237
\(739\) 14408.2 0.717203 0.358602 0.933491i \(-0.383254\pi\)
0.358602 + 0.933491i \(0.383254\pi\)
\(740\) 16579.8 0.823628
\(741\) −1550.22 −0.0768538
\(742\) 10633.9 0.526124
\(743\) −8023.72 −0.396180 −0.198090 0.980184i \(-0.563474\pi\)
−0.198090 + 0.980184i \(0.563474\pi\)
\(744\) −4494.88 −0.221492
\(745\) −53008.0 −2.60679
\(746\) −13531.1 −0.664085
\(747\) −3765.44 −0.184431
\(748\) 2490.29 0.121730
\(749\) 23783.4 1.16025
\(750\) −6487.80 −0.315868
\(751\) −16501.1 −0.801775 −0.400888 0.916127i \(-0.631298\pi\)
−0.400888 + 0.916127i \(0.631298\pi\)
\(752\) −1028.48 −0.0498736
\(753\) −5096.60 −0.246654
\(754\) −1382.47 −0.0667725
\(755\) −32874.6 −1.58467
\(756\) 1512.00 0.0727393
\(757\) −17112.3 −0.821607 −0.410804 0.911724i \(-0.634752\pi\)
−0.410804 + 0.911724i \(0.634752\pi\)
\(758\) 310.317 0.0148697
\(759\) −1217.39 −0.0582193
\(760\) −7853.86 −0.374855
\(761\) −21846.4 −1.04065 −0.520323 0.853970i \(-0.674188\pi\)
−0.520323 + 0.853970i \(0.674188\pi\)
\(762\) 2624.56 0.124774
\(763\) 5451.26 0.258649
\(764\) −15318.2 −0.725384
\(765\) −5603.16 −0.264814
\(766\) 15685.8 0.739882
\(767\) 3677.51 0.173125
\(768\) 768.000 0.0360844
\(769\) −15782.8 −0.740107 −0.370053 0.929010i \(-0.620661\pi\)
−0.370053 + 0.929010i \(0.620661\pi\)
\(770\) 8716.03 0.407927
\(771\) 16875.0 0.788248
\(772\) −1048.31 −0.0488724
\(773\) 26095.1 1.21420 0.607100 0.794625i \(-0.292333\pi\)
0.607100 + 0.794625i \(0.292333\pi\)
\(774\) −3771.06 −0.175126
\(775\) 34889.0 1.61710
\(776\) −325.810 −0.0150720
\(777\) 9867.06 0.455571
\(778\) −1864.89 −0.0859379
\(779\) −21892.3 −1.00690
\(780\) −1966.16 −0.0902564
\(781\) −16075.9 −0.736543
\(782\) 1623.19 0.0742263
\(783\) −2009.70 −0.0917250
\(784\) −2352.00 −0.107143
\(785\) −25969.6 −1.18076
\(786\) 12834.3 0.582425
\(787\) 8756.20 0.396601 0.198300 0.980141i \(-0.436458\pi\)
0.198300 + 0.980141i \(0.436458\pi\)
\(788\) −21521.6 −0.972937
\(789\) −4082.07 −0.184189
\(790\) 18359.4 0.826833
\(791\) 1551.95 0.0697610
\(792\) 1270.32 0.0569935
\(793\) −5523.72 −0.247355
\(794\) −23683.4 −1.05856
\(795\) −20101.9 −0.896782
\(796\) 15632.2 0.696064
\(797\) −17161.4 −0.762719 −0.381359 0.924427i \(-0.624544\pi\)
−0.381359 + 0.924427i \(0.624544\pi\)
\(798\) −4674.04 −0.207342
\(799\) 2268.23 0.100431
\(800\) −5961.17 −0.263449
\(801\) 5392.43 0.237868
\(802\) 20193.5 0.889098
\(803\) 16612.8 0.730079
\(804\) −12789.7 −0.561018
\(805\) 5681.15 0.248738
\(806\) 3478.52 0.152017
\(807\) 4999.60 0.218085
\(808\) 11690.5 0.508998
\(809\) 20060.1 0.871785 0.435893 0.899999i \(-0.356433\pi\)
0.435893 + 0.899999i \(0.356433\pi\)
\(810\) −2858.22 −0.123985
\(811\) 18329.3 0.793622 0.396811 0.917900i \(-0.370117\pi\)
0.396811 + 0.917900i \(0.370117\pi\)
\(812\) −4168.26 −0.180144
\(813\) −9963.59 −0.429814
\(814\) 8289.89 0.356954
\(815\) 8243.58 0.354307
\(816\) −1693.76 −0.0726635
\(817\) 11657.5 0.499195
\(818\) 15055.7 0.643534
\(819\) −1170.12 −0.0499233
\(820\) −27766.3 −1.18249
\(821\) 32619.7 1.38665 0.693323 0.720627i \(-0.256146\pi\)
0.693323 + 0.720627i \(0.256146\pi\)
\(822\) 4518.12 0.191712
\(823\) 21374.7 0.905316 0.452658 0.891684i \(-0.350476\pi\)
0.452658 + 0.891684i \(0.350476\pi\)
\(824\) −12584.3 −0.532031
\(825\) −9860.14 −0.416104
\(826\) 11088.0 0.467071
\(827\) 3536.45 0.148699 0.0743497 0.997232i \(-0.476312\pi\)
0.0743497 + 0.997232i \(0.476312\pi\)
\(828\) 828.000 0.0347524
\(829\) 1345.14 0.0563553 0.0281776 0.999603i \(-0.491030\pi\)
0.0281776 + 0.999603i \(0.491030\pi\)
\(830\) 14763.3 0.617400
\(831\) −472.469 −0.0197229
\(832\) −594.345 −0.0247659
\(833\) 5187.14 0.215755
\(834\) 3690.96 0.153246
\(835\) 32791.4 1.35903
\(836\) −3926.93 −0.162459
\(837\) 5056.74 0.208825
\(838\) 13611.0 0.561078
\(839\) 14522.5 0.597584 0.298792 0.954318i \(-0.403416\pi\)
0.298792 + 0.954318i \(0.403416\pi\)
\(840\) −5928.15 −0.243501
\(841\) −18848.7 −0.772836
\(842\) 31536.0 1.29074
\(843\) −452.406 −0.0184836
\(844\) 8455.65 0.344852
\(845\) −37240.8 −1.51612
\(846\) 1157.04 0.0470212
\(847\) −14276.0 −0.579137
\(848\) −6076.53 −0.246072
\(849\) 20644.7 0.834541
\(850\) 13146.9 0.530510
\(851\) 5403.39 0.217657
\(852\) 10933.9 0.439659
\(853\) −20073.4 −0.805746 −0.402873 0.915256i \(-0.631988\pi\)
−0.402873 + 0.915256i \(0.631988\pi\)
\(854\) −16654.5 −0.667335
\(855\) 8835.59 0.353416
\(856\) −13590.5 −0.542657
\(857\) −16038.0 −0.639262 −0.319631 0.947542i \(-0.603559\pi\)
−0.319631 + 0.947542i \(0.603559\pi\)
\(858\) −983.082 −0.0391164
\(859\) 42326.4 1.68121 0.840605 0.541648i \(-0.182200\pi\)
0.840605 + 0.541648i \(0.182200\pi\)
\(860\) 14785.3 0.586251
\(861\) −16524.5 −0.654068
\(862\) 2148.53 0.0848948
\(863\) −14075.7 −0.555205 −0.277602 0.960696i \(-0.589540\pi\)
−0.277602 + 0.960696i \(0.589540\pi\)
\(864\) −864.000 −0.0340207
\(865\) 39252.5 1.54292
\(866\) −33726.4 −1.32340
\(867\) −11003.6 −0.431027
\(868\) 10488.1 0.410124
\(869\) 9179.70 0.358343
\(870\) 7879.49 0.307057
\(871\) 9897.78 0.385044
\(872\) −3115.01 −0.120972
\(873\) 366.536 0.0142100
\(874\) −2559.59 −0.0990612
\(875\) 15138.2 0.584874
\(876\) −11299.1 −0.435801
\(877\) −18552.9 −0.714351 −0.357175 0.934037i \(-0.616260\pi\)
−0.357175 + 0.934037i \(0.616260\pi\)
\(878\) 3184.41 0.122401
\(879\) 16952.5 0.650506
\(880\) −4980.59 −0.190790
\(881\) −18290.7 −0.699465 −0.349732 0.936850i \(-0.613728\pi\)
−0.349732 + 0.936850i \(0.613728\pi\)
\(882\) 2646.00 0.101015
\(883\) −37336.9 −1.42298 −0.711488 0.702698i \(-0.751979\pi\)
−0.711488 + 0.702698i \(0.751979\pi\)
\(884\) 1310.78 0.0498712
\(885\) −20960.3 −0.796126
\(886\) −14015.3 −0.531438
\(887\) −7792.73 −0.294988 −0.147494 0.989063i \(-0.547121\pi\)
−0.147494 + 0.989063i \(0.547121\pi\)
\(888\) −5638.32 −0.213074
\(889\) −6123.97 −0.231037
\(890\) −21142.3 −0.796283
\(891\) −1429.11 −0.0537339
\(892\) 19134.8 0.718251
\(893\) −3576.76 −0.134033
\(894\) 18026.5 0.674382
\(895\) 67737.8 2.52986
\(896\) −1792.00 −0.0668153
\(897\) −640.778 −0.0238517
\(898\) −31878.9 −1.18465
\(899\) −13940.3 −0.517171
\(900\) 6706.32 0.248382
\(901\) 13401.3 0.495518
\(902\) −13883.2 −0.512482
\(903\) 8799.14 0.324271
\(904\) −886.828 −0.0326277
\(905\) −74371.2 −2.73169
\(906\) 11179.7 0.409957
\(907\) 12764.9 0.467313 0.233657 0.972319i \(-0.424931\pi\)
0.233657 + 0.972319i \(0.424931\pi\)
\(908\) −2654.52 −0.0970192
\(909\) −13151.8 −0.479888
\(910\) 4587.72 0.167122
\(911\) −20843.2 −0.758030 −0.379015 0.925390i \(-0.623737\pi\)
−0.379015 + 0.925390i \(0.623737\pi\)
\(912\) 2670.88 0.0969755
\(913\) 7381.65 0.267576
\(914\) 14410.4 0.521502
\(915\) 31482.9 1.13748
\(916\) 11837.9 0.427002
\(917\) −29946.8 −1.07844
\(918\) 1905.48 0.0685078
\(919\) −24938.9 −0.895167 −0.447583 0.894242i \(-0.647715\pi\)
−0.447583 + 0.894242i \(0.647715\pi\)
\(920\) −3246.37 −0.116337
\(921\) 12213.8 0.436979
\(922\) 24960.2 0.891563
\(923\) −8461.60 −0.301752
\(924\) −2964.08 −0.105531
\(925\) 43764.3 1.55563
\(926\) −124.123 −0.00440488
\(927\) 14157.3 0.501603
\(928\) 2381.86 0.0842548
\(929\) 43378.9 1.53199 0.765993 0.642849i \(-0.222248\pi\)
0.765993 + 0.642849i \(0.222248\pi\)
\(930\) −19826.1 −0.699059
\(931\) −8179.57 −0.287943
\(932\) 10015.0 0.351989
\(933\) −1597.60 −0.0560592
\(934\) −14037.7 −0.491785
\(935\) 10984.3 0.384196
\(936\) 668.638 0.0233495
\(937\) 22043.0 0.768530 0.384265 0.923223i \(-0.374455\pi\)
0.384265 + 0.923223i \(0.374455\pi\)
\(938\) 29842.7 1.03880
\(939\) 29278.9 1.01755
\(940\) −4536.46 −0.157408
\(941\) −11791.8 −0.408504 −0.204252 0.978918i \(-0.565476\pi\)
−0.204252 + 0.978918i \(0.565476\pi\)
\(942\) 8831.54 0.305464
\(943\) −9049.11 −0.312492
\(944\) −6336.00 −0.218453
\(945\) 6669.17 0.229575
\(946\) 7392.66 0.254076
\(947\) −34148.3 −1.17178 −0.585888 0.810392i \(-0.699254\pi\)
−0.585888 + 0.810392i \(0.699254\pi\)
\(948\) −6243.52 −0.213903
\(949\) 8744.23 0.299104
\(950\) −20731.2 −0.708010
\(951\) 11991.5 0.408886
\(952\) 3952.10 0.134547
\(953\) 24045.9 0.817338 0.408669 0.912683i \(-0.365993\pi\)
0.408669 + 0.912683i \(0.365993\pi\)
\(954\) 6836.10 0.231999
\(955\) −67566.0 −2.28941
\(956\) 25601.4 0.866119
\(957\) 3939.74 0.133076
\(958\) −37177.3 −1.25380
\(959\) −10542.3 −0.354982
\(960\) 3387.52 0.113887
\(961\) 5285.28 0.177412
\(962\) 4363.42 0.146239
\(963\) 15289.3 0.511622
\(964\) −3492.84 −0.116698
\(965\) −4623.91 −0.154248
\(966\) −1932.00 −0.0643489
\(967\) −14125.6 −0.469751 −0.234876 0.972025i \(-0.575468\pi\)
−0.234876 + 0.972025i \(0.575468\pi\)
\(968\) 8157.71 0.270866
\(969\) −5890.40 −0.195280
\(970\) −1437.09 −0.0475693
\(971\) 9607.55 0.317529 0.158765 0.987316i \(-0.449249\pi\)
0.158765 + 0.987316i \(0.449249\pi\)
\(972\) 972.000 0.0320750
\(973\) −8612.23 −0.283757
\(974\) 4497.64 0.147961
\(975\) −5189.93 −0.170473
\(976\) 9516.84 0.312118
\(977\) 35829.9 1.17328 0.586642 0.809846i \(-0.300450\pi\)
0.586642 + 0.809846i \(0.300450\pi\)
\(978\) −2803.41 −0.0916597
\(979\) −10571.2 −0.345103
\(980\) −10374.3 −0.338157
\(981\) 3504.38 0.114053
\(982\) −456.763 −0.0148431
\(983\) −32173.8 −1.04393 −0.521966 0.852966i \(-0.674801\pi\)
−0.521966 + 0.852966i \(0.674801\pi\)
\(984\) 9442.55 0.305912
\(985\) −94928.0 −3.07072
\(986\) −5252.99 −0.169665
\(987\) −2699.77 −0.0870664
\(988\) −2066.96 −0.0665573
\(989\) 4818.57 0.154926
\(990\) 5603.16 0.179879
\(991\) 7190.75 0.230496 0.115248 0.993337i \(-0.463234\pi\)
0.115248 + 0.993337i \(0.463234\pi\)
\(992\) −5993.17 −0.191818
\(993\) 19049.2 0.608770
\(994\) −25512.5 −0.814091
\(995\) 68950.8 2.19687
\(996\) −5020.59 −0.159722
\(997\) 4706.41 0.149502 0.0747510 0.997202i \(-0.476184\pi\)
0.0747510 + 0.997202i \(0.476184\pi\)
\(998\) −27353.4 −0.867593
\(999\) 6343.11 0.200888
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 138.4.a.d.1.2 2
3.2 odd 2 414.4.a.k.1.1 2
4.3 odd 2 1104.4.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.4.a.d.1.2 2 1.1 even 1 trivial
414.4.a.k.1.1 2 3.2 odd 2
1104.4.a.k.1.2 2 4.3 odd 2