Properties

Label 322.4.a.h.1.3
Level $322$
Weight $4$
Character 322.1
Self dual yes
Analytic conductor $18.999$
Analytic rank $0$
Dimension $5$
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] = [322,4,Mod(1,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, 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("322.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 322.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: \(18.9986150218\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 92x^{3} - 28x^{2} + 1593x - 1782 \) 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 1.3
Root \(1.26012\) of defining polynomial
Character \(\chi\) \(=\) 322.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -0.260120 q^{3} +4.00000 q^{4} +15.2484 q^{5} -0.520240 q^{6} -7.00000 q^{7} +8.00000 q^{8} -26.9323 q^{9} +30.4969 q^{10} +63.1379 q^{11} -1.04048 q^{12} +40.8864 q^{13} -14.0000 q^{14} -3.96643 q^{15} +16.0000 q^{16} -57.0717 q^{17} -53.8647 q^{18} +34.4579 q^{19} +60.9938 q^{20} +1.82084 q^{21} +126.276 q^{22} +23.0000 q^{23} -2.08096 q^{24} +107.515 q^{25} +81.7727 q^{26} +14.0289 q^{27} -28.0000 q^{28} +180.951 q^{29} -7.93285 q^{30} -5.92805 q^{31} +32.0000 q^{32} -16.4234 q^{33} -114.143 q^{34} -106.739 q^{35} -107.729 q^{36} -161.864 q^{37} +68.9158 q^{38} -10.6354 q^{39} +121.988 q^{40} -104.432 q^{41} +3.64168 q^{42} +16.5725 q^{43} +252.552 q^{44} -410.676 q^{45} +46.0000 q^{46} +333.754 q^{47} -4.16192 q^{48} +49.0000 q^{49} +215.030 q^{50} +14.8455 q^{51} +163.545 q^{52} +296.103 q^{53} +28.0578 q^{54} +962.756 q^{55} -56.0000 q^{56} -8.96319 q^{57} +361.902 q^{58} -805.644 q^{59} -15.8657 q^{60} +715.603 q^{61} -11.8561 q^{62} +188.526 q^{63} +64.0000 q^{64} +623.454 q^{65} -32.8469 q^{66} +301.232 q^{67} -228.287 q^{68} -5.98276 q^{69} -213.478 q^{70} -871.046 q^{71} -215.459 q^{72} +894.771 q^{73} -323.728 q^{74} -27.9668 q^{75} +137.832 q^{76} -441.966 q^{77} -21.2707 q^{78} -1246.19 q^{79} +243.975 q^{80} +723.524 q^{81} -208.863 q^{82} -1223.93 q^{83} +7.28336 q^{84} -870.255 q^{85} +33.1451 q^{86} -47.0690 q^{87} +505.104 q^{88} -86.3579 q^{89} -821.353 q^{90} -286.205 q^{91} +92.0000 q^{92} +1.54200 q^{93} +667.507 q^{94} +525.429 q^{95} -8.32384 q^{96} +343.939 q^{97} +98.0000 q^{98} -1700.45 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{2} + 5 q^{3} + 20 q^{4} + 22 q^{5} + 10 q^{6} - 35 q^{7} + 40 q^{8} + 54 q^{9} + 44 q^{10} + 42 q^{11} + 20 q^{12} + 107 q^{13} - 70 q^{14} + 122 q^{15} + 80 q^{16} + 218 q^{17} + 108 q^{18}+ \cdots - 2616 q^{99}+O(q^{100}) \) 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
\(3\) −0.260120 −0.0500601 −0.0250301 0.999687i \(-0.507968\pi\)
−0.0250301 + 0.999687i \(0.507968\pi\)
\(4\) 4.00000 0.500000
\(5\) 15.2484 1.36386 0.681931 0.731416i \(-0.261140\pi\)
0.681931 + 0.731416i \(0.261140\pi\)
\(6\) −0.520240 −0.0353978
\(7\) −7.00000 −0.377964
\(8\) 8.00000 0.353553
\(9\) −26.9323 −0.997494
\(10\) 30.4969 0.964397
\(11\) 63.1379 1.73062 0.865309 0.501238i \(-0.167122\pi\)
0.865309 + 0.501238i \(0.167122\pi\)
\(12\) −1.04048 −0.0250301
\(13\) 40.8864 0.872295 0.436148 0.899875i \(-0.356343\pi\)
0.436148 + 0.899875i \(0.356343\pi\)
\(14\) −14.0000 −0.267261
\(15\) −3.96643 −0.0682751
\(16\) 16.0000 0.250000
\(17\) −57.0717 −0.814231 −0.407115 0.913377i \(-0.633465\pi\)
−0.407115 + 0.913377i \(0.633465\pi\)
\(18\) −53.8647 −0.705335
\(19\) 34.4579 0.416062 0.208031 0.978122i \(-0.433294\pi\)
0.208031 + 0.978122i \(0.433294\pi\)
\(20\) 60.9938 0.681931
\(21\) 1.82084 0.0189209
\(22\) 126.276 1.22373
\(23\) 23.0000 0.208514
\(24\) −2.08096 −0.0176989
\(25\) 107.515 0.860121
\(26\) 81.7727 0.616806
\(27\) 14.0289 0.0999948
\(28\) −28.0000 −0.188982
\(29\) 180.951 1.15868 0.579340 0.815086i \(-0.303310\pi\)
0.579340 + 0.815086i \(0.303310\pi\)
\(30\) −7.93285 −0.0482778
\(31\) −5.92805 −0.0343455 −0.0171727 0.999853i \(-0.505467\pi\)
−0.0171727 + 0.999853i \(0.505467\pi\)
\(32\) 32.0000 0.176777
\(33\) −16.4234 −0.0866350
\(34\) −114.143 −0.575748
\(35\) −106.739 −0.515492
\(36\) −107.729 −0.498747
\(37\) −161.864 −0.719196 −0.359598 0.933107i \(-0.617086\pi\)
−0.359598 + 0.933107i \(0.617086\pi\)
\(38\) 68.9158 0.294200
\(39\) −10.6354 −0.0436672
\(40\) 121.988 0.482198
\(41\) −104.432 −0.397792 −0.198896 0.980021i \(-0.563736\pi\)
−0.198896 + 0.980021i \(0.563736\pi\)
\(42\) 3.64168 0.0133791
\(43\) 16.5725 0.0587742 0.0293871 0.999568i \(-0.490644\pi\)
0.0293871 + 0.999568i \(0.490644\pi\)
\(44\) 252.552 0.865309
\(45\) −410.676 −1.36044
\(46\) 46.0000 0.147442
\(47\) 333.754 1.03581 0.517904 0.855439i \(-0.326713\pi\)
0.517904 + 0.855439i \(0.326713\pi\)
\(48\) −4.16192 −0.0125150
\(49\) 49.0000 0.142857
\(50\) 215.030 0.608198
\(51\) 14.8455 0.0407605
\(52\) 163.545 0.436148
\(53\) 296.103 0.767412 0.383706 0.923455i \(-0.374648\pi\)
0.383706 + 0.923455i \(0.374648\pi\)
\(54\) 28.0578 0.0707070
\(55\) 962.756 2.36033
\(56\) −56.0000 −0.133631
\(57\) −8.96319 −0.0208281
\(58\) 361.902 0.819311
\(59\) −805.644 −1.77773 −0.888864 0.458171i \(-0.848505\pi\)
−0.888864 + 0.458171i \(0.848505\pi\)
\(60\) −15.8657 −0.0341376
\(61\) 715.603 1.50202 0.751012 0.660288i \(-0.229566\pi\)
0.751012 + 0.660288i \(0.229566\pi\)
\(62\) −11.8561 −0.0242859
\(63\) 188.526 0.377017
\(64\) 64.0000 0.125000
\(65\) 623.454 1.18969
\(66\) −32.8469 −0.0612602
\(67\) 301.232 0.549275 0.274637 0.961548i \(-0.411442\pi\)
0.274637 + 0.961548i \(0.411442\pi\)
\(68\) −228.287 −0.407115
\(69\) −5.98276 −0.0104383
\(70\) −213.478 −0.364508
\(71\) −871.046 −1.45597 −0.727987 0.685591i \(-0.759544\pi\)
−0.727987 + 0.685591i \(0.759544\pi\)
\(72\) −215.459 −0.352667
\(73\) 894.771 1.43459 0.717295 0.696770i \(-0.245380\pi\)
0.717295 + 0.696770i \(0.245380\pi\)
\(74\) −323.728 −0.508548
\(75\) −27.9668 −0.0430578
\(76\) 137.832 0.208031
\(77\) −441.966 −0.654112
\(78\) −21.2707 −0.0308774
\(79\) −1246.19 −1.77477 −0.887386 0.461028i \(-0.847481\pi\)
−0.887386 + 0.461028i \(0.847481\pi\)
\(80\) 243.975 0.340966
\(81\) 723.524 0.992488
\(82\) −208.863 −0.281281
\(83\) −1223.93 −1.61860 −0.809300 0.587395i \(-0.800154\pi\)
−0.809300 + 0.587395i \(0.800154\pi\)
\(84\) 7.28336 0.00946047
\(85\) −870.255 −1.11050
\(86\) 33.1451 0.0415596
\(87\) −47.0690 −0.0580037
\(88\) 505.104 0.611866
\(89\) −86.3579 −0.102853 −0.0514265 0.998677i \(-0.516377\pi\)
−0.0514265 + 0.998677i \(0.516377\pi\)
\(90\) −821.353 −0.961980
\(91\) −286.205 −0.329697
\(92\) 92.0000 0.104257
\(93\) 1.54200 0.00171934
\(94\) 667.507 0.732426
\(95\) 525.429 0.567452
\(96\) −8.32384 −0.00884946
\(97\) 343.939 0.360018 0.180009 0.983665i \(-0.442387\pi\)
0.180009 + 0.983665i \(0.442387\pi\)
\(98\) 98.0000 0.101015
\(99\) −1700.45 −1.72628
\(100\) 430.061 0.430061
\(101\) 458.505 0.451713 0.225856 0.974161i \(-0.427482\pi\)
0.225856 + 0.974161i \(0.427482\pi\)
\(102\) 29.6910 0.0288220
\(103\) −574.886 −0.549954 −0.274977 0.961451i \(-0.588670\pi\)
−0.274977 + 0.961451i \(0.588670\pi\)
\(104\) 327.091 0.308403
\(105\) 27.7650 0.0258056
\(106\) 592.205 0.542642
\(107\) −1451.76 −1.31165 −0.655825 0.754913i \(-0.727679\pi\)
−0.655825 + 0.754913i \(0.727679\pi\)
\(108\) 56.1155 0.0499974
\(109\) −1812.89 −1.59306 −0.796530 0.604598i \(-0.793334\pi\)
−0.796530 + 0.604598i \(0.793334\pi\)
\(110\) 1925.51 1.66900
\(111\) 42.1040 0.0360030
\(112\) −112.000 −0.0944911
\(113\) −1693.79 −1.41007 −0.705035 0.709173i \(-0.749069\pi\)
−0.705035 + 0.709173i \(0.749069\pi\)
\(114\) −17.9264 −0.0147277
\(115\) 350.714 0.284385
\(116\) 723.804 0.579340
\(117\) −1101.17 −0.870109
\(118\) −1611.29 −1.25704
\(119\) 399.502 0.307750
\(120\) −31.7314 −0.0241389
\(121\) 2655.40 1.99504
\(122\) 1431.21 1.06209
\(123\) 27.1647 0.0199135
\(124\) −23.7122 −0.0171727
\(125\) −266.616 −0.190775
\(126\) 377.053 0.266591
\(127\) 251.950 0.176039 0.0880194 0.996119i \(-0.471946\pi\)
0.0880194 + 0.996119i \(0.471946\pi\)
\(128\) 128.000 0.0883883
\(129\) −4.31085 −0.00294224
\(130\) 1246.91 0.841238
\(131\) −1844.44 −1.23015 −0.615074 0.788469i \(-0.710874\pi\)
−0.615074 + 0.788469i \(0.710874\pi\)
\(132\) −65.6938 −0.0433175
\(133\) −241.205 −0.157257
\(134\) 602.465 0.388396
\(135\) 213.919 0.136379
\(136\) −456.574 −0.287874
\(137\) −1169.93 −0.729592 −0.364796 0.931087i \(-0.618861\pi\)
−0.364796 + 0.931087i \(0.618861\pi\)
\(138\) −11.9655 −0.00738096
\(139\) −1618.44 −0.987585 −0.493792 0.869580i \(-0.664390\pi\)
−0.493792 + 0.869580i \(0.664390\pi\)
\(140\) −426.957 −0.257746
\(141\) −86.8160 −0.0518526
\(142\) −1742.09 −1.02953
\(143\) 2581.48 1.50961
\(144\) −430.917 −0.249373
\(145\) 2759.22 1.58028
\(146\) 1789.54 1.01441
\(147\) −12.7459 −0.00715144
\(148\) −647.455 −0.359598
\(149\) −1745.63 −0.959785 −0.479892 0.877327i \(-0.659324\pi\)
−0.479892 + 0.877327i \(0.659324\pi\)
\(150\) −55.9337 −0.0304464
\(151\) −2520.41 −1.35833 −0.679167 0.733984i \(-0.737659\pi\)
−0.679167 + 0.733984i \(0.737659\pi\)
\(152\) 275.663 0.147100
\(153\) 1537.07 0.812190
\(154\) −883.931 −0.462527
\(155\) −90.3936 −0.0468425
\(156\) −42.5414 −0.0218336
\(157\) 1122.56 0.570635 0.285318 0.958433i \(-0.407901\pi\)
0.285318 + 0.958433i \(0.407901\pi\)
\(158\) −2492.37 −1.25495
\(159\) −77.0222 −0.0384167
\(160\) 487.950 0.241099
\(161\) −161.000 −0.0788110
\(162\) 1447.05 0.701795
\(163\) 149.438 0.0718091 0.0359045 0.999355i \(-0.488569\pi\)
0.0359045 + 0.999355i \(0.488569\pi\)
\(164\) −417.726 −0.198896
\(165\) −250.432 −0.118158
\(166\) −2447.86 −1.14452
\(167\) 1950.78 0.903927 0.451963 0.892037i \(-0.350724\pi\)
0.451963 + 0.892037i \(0.350724\pi\)
\(168\) 14.5667 0.00668956
\(169\) −525.306 −0.239101
\(170\) −1740.51 −0.785241
\(171\) −928.032 −0.415020
\(172\) 66.2902 0.0293871
\(173\) 2850.79 1.25284 0.626420 0.779486i \(-0.284520\pi\)
0.626420 + 0.779486i \(0.284520\pi\)
\(174\) −94.1379 −0.0410148
\(175\) −752.606 −0.325095
\(176\) 1010.21 0.432655
\(177\) 209.564 0.0889933
\(178\) −172.716 −0.0727280
\(179\) 1847.62 0.771493 0.385747 0.922605i \(-0.373944\pi\)
0.385747 + 0.922605i \(0.373944\pi\)
\(180\) −1642.71 −0.680222
\(181\) 3260.42 1.33892 0.669461 0.742847i \(-0.266525\pi\)
0.669461 + 0.742847i \(0.266525\pi\)
\(182\) −572.409 −0.233131
\(183\) −186.143 −0.0751915
\(184\) 184.000 0.0737210
\(185\) −2468.17 −0.980884
\(186\) 3.08401 0.00121576
\(187\) −3603.39 −1.40912
\(188\) 1335.01 0.517904
\(189\) −98.2021 −0.0377945
\(190\) 1050.86 0.401249
\(191\) 1616.13 0.612247 0.306123 0.951992i \(-0.400968\pi\)
0.306123 + 0.951992i \(0.400968\pi\)
\(192\) −16.6477 −0.00625751
\(193\) −4746.87 −1.77040 −0.885200 0.465211i \(-0.845979\pi\)
−0.885200 + 0.465211i \(0.845979\pi\)
\(194\) 687.878 0.254571
\(195\) −162.173 −0.0595560
\(196\) 196.000 0.0714286
\(197\) −1471.77 −0.532280 −0.266140 0.963934i \(-0.585748\pi\)
−0.266140 + 0.963934i \(0.585748\pi\)
\(198\) −3400.90 −1.22067
\(199\) 3296.74 1.17437 0.587184 0.809453i \(-0.300236\pi\)
0.587184 + 0.809453i \(0.300236\pi\)
\(200\) 860.121 0.304099
\(201\) −78.3566 −0.0274967
\(202\) 917.011 0.319409
\(203\) −1266.66 −0.437940
\(204\) 59.3820 0.0203802
\(205\) −1592.42 −0.542533
\(206\) −1149.77 −0.388876
\(207\) −619.444 −0.207992
\(208\) 654.182 0.218074
\(209\) 2175.60 0.720045
\(210\) 55.5300 0.0182473
\(211\) −2436.09 −0.794821 −0.397410 0.917641i \(-0.630091\pi\)
−0.397410 + 0.917641i \(0.630091\pi\)
\(212\) 1184.41 0.383706
\(213\) 226.577 0.0728862
\(214\) −2903.51 −0.927477
\(215\) 252.706 0.0801599
\(216\) 112.231 0.0353535
\(217\) 41.4964 0.0129814
\(218\) −3625.79 −1.12646
\(219\) −232.748 −0.0718157
\(220\) 3851.02 1.18016
\(221\) −2333.45 −0.710250
\(222\) 84.2080 0.0254580
\(223\) 1464.12 0.439662 0.219831 0.975538i \(-0.429449\pi\)
0.219831 + 0.975538i \(0.429449\pi\)
\(224\) −224.000 −0.0668153
\(225\) −2895.64 −0.857966
\(226\) −3387.57 −0.997070
\(227\) −3419.39 −0.999794 −0.499897 0.866085i \(-0.666629\pi\)
−0.499897 + 0.866085i \(0.666629\pi\)
\(228\) −35.8527 −0.0104141
\(229\) −3437.40 −0.991922 −0.495961 0.868345i \(-0.665184\pi\)
−0.495961 + 0.868345i \(0.665184\pi\)
\(230\) 701.429 0.201091
\(231\) 114.964 0.0327449
\(232\) 1447.61 0.409656
\(233\) 3049.48 0.857415 0.428708 0.903443i \(-0.358969\pi\)
0.428708 + 0.903443i \(0.358969\pi\)
\(234\) −2202.33 −0.615260
\(235\) 5089.22 1.41270
\(236\) −3222.58 −0.888864
\(237\) 324.158 0.0888453
\(238\) 799.004 0.217612
\(239\) 1961.10 0.530767 0.265383 0.964143i \(-0.414501\pi\)
0.265383 + 0.964143i \(0.414501\pi\)
\(240\) −63.4628 −0.0170688
\(241\) 6121.70 1.63624 0.818119 0.575049i \(-0.195017\pi\)
0.818119 + 0.575049i \(0.195017\pi\)
\(242\) 5310.80 1.41071
\(243\) −566.983 −0.149679
\(244\) 2862.41 0.751012
\(245\) 747.174 0.194838
\(246\) 54.3295 0.0140810
\(247\) 1408.86 0.362929
\(248\) −47.4244 −0.0121430
\(249\) 318.369 0.0810273
\(250\) −533.233 −0.134898
\(251\) 4112.95 1.03429 0.517145 0.855898i \(-0.326995\pi\)
0.517145 + 0.855898i \(0.326995\pi\)
\(252\) 754.105 0.188509
\(253\) 1452.17 0.360859
\(254\) 503.899 0.124478
\(255\) 226.371 0.0555917
\(256\) 256.000 0.0625000
\(257\) −2135.52 −0.518328 −0.259164 0.965833i \(-0.583447\pi\)
−0.259164 + 0.965833i \(0.583447\pi\)
\(258\) −8.62170 −0.00208048
\(259\) 1133.05 0.271830
\(260\) 2493.81 0.594845
\(261\) −4873.43 −1.15578
\(262\) −3688.88 −0.869846
\(263\) 7005.80 1.64257 0.821286 0.570517i \(-0.193257\pi\)
0.821286 + 0.570517i \(0.193257\pi\)
\(264\) −131.388 −0.0306301
\(265\) 4515.10 1.04664
\(266\) −482.410 −0.111197
\(267\) 22.4634 0.00514883
\(268\) 1204.93 0.274637
\(269\) 6566.34 1.48831 0.744157 0.668004i \(-0.232851\pi\)
0.744157 + 0.668004i \(0.232851\pi\)
\(270\) 427.837 0.0964346
\(271\) −5759.43 −1.29100 −0.645499 0.763761i \(-0.723351\pi\)
−0.645499 + 0.763761i \(0.723351\pi\)
\(272\) −913.147 −0.203558
\(273\) 74.4475 0.0165046
\(274\) −2339.87 −0.515900
\(275\) 6788.29 1.48854
\(276\) −23.9310 −0.00521913
\(277\) 356.886 0.0774123 0.0387061 0.999251i \(-0.487676\pi\)
0.0387061 + 0.999251i \(0.487676\pi\)
\(278\) −3236.88 −0.698328
\(279\) 159.656 0.0342594
\(280\) −853.913 −0.182254
\(281\) −925.745 −0.196531 −0.0982657 0.995160i \(-0.531329\pi\)
−0.0982657 + 0.995160i \(0.531329\pi\)
\(282\) −173.632 −0.0366654
\(283\) 6928.18 1.45526 0.727629 0.685971i \(-0.240622\pi\)
0.727629 + 0.685971i \(0.240622\pi\)
\(284\) −3484.19 −0.727987
\(285\) −136.675 −0.0284067
\(286\) 5162.96 1.06746
\(287\) 731.021 0.150351
\(288\) −861.835 −0.176334
\(289\) −1655.82 −0.337028
\(290\) 5518.44 1.11743
\(291\) −89.4655 −0.0180225
\(292\) 3579.08 0.717295
\(293\) 148.190 0.0295472 0.0147736 0.999891i \(-0.495297\pi\)
0.0147736 + 0.999891i \(0.495297\pi\)
\(294\) −25.4918 −0.00505684
\(295\) −12284.8 −2.42458
\(296\) −1294.91 −0.254274
\(297\) 885.755 0.173053
\(298\) −3491.27 −0.678670
\(299\) 940.386 0.181886
\(300\) −111.867 −0.0215289
\(301\) −116.008 −0.0222145
\(302\) −5040.83 −0.960487
\(303\) −119.266 −0.0226128
\(304\) 551.326 0.104016
\(305\) 10911.8 2.04856
\(306\) 3074.15 0.574305
\(307\) −10096.8 −1.87705 −0.938527 0.345207i \(-0.887809\pi\)
−0.938527 + 0.345207i \(0.887809\pi\)
\(308\) −1767.86 −0.327056
\(309\) 149.539 0.0275308
\(310\) −180.787 −0.0331227
\(311\) −1063.81 −0.193965 −0.0969826 0.995286i \(-0.530919\pi\)
−0.0969826 + 0.995286i \(0.530919\pi\)
\(312\) −85.0829 −0.0154387
\(313\) 1953.88 0.352844 0.176422 0.984315i \(-0.443548\pi\)
0.176422 + 0.984315i \(0.443548\pi\)
\(314\) 2245.11 0.403500
\(315\) 2874.73 0.514200
\(316\) −4984.75 −0.887386
\(317\) −2939.51 −0.520819 −0.260409 0.965498i \(-0.583858\pi\)
−0.260409 + 0.965498i \(0.583858\pi\)
\(318\) −154.044 −0.0271647
\(319\) 11424.9 2.00523
\(320\) 975.901 0.170483
\(321\) 377.631 0.0656614
\(322\) −322.000 −0.0557278
\(323\) −1966.57 −0.338771
\(324\) 2894.10 0.496244
\(325\) 4395.90 0.750280
\(326\) 298.876 0.0507767
\(327\) 471.570 0.0797488
\(328\) −835.452 −0.140641
\(329\) −2336.27 −0.391498
\(330\) −500.864 −0.0835505
\(331\) −8808.46 −1.46271 −0.731354 0.681998i \(-0.761111\pi\)
−0.731354 + 0.681998i \(0.761111\pi\)
\(332\) −4895.72 −0.809300
\(333\) 4359.37 0.717393
\(334\) 3901.56 0.639173
\(335\) 4593.33 0.749135
\(336\) 29.1334 0.00473024
\(337\) 3966.73 0.641191 0.320596 0.947216i \(-0.396117\pi\)
0.320596 + 0.947216i \(0.396117\pi\)
\(338\) −1050.61 −0.169070
\(339\) 440.587 0.0705883
\(340\) −3481.02 −0.555250
\(341\) −374.285 −0.0594389
\(342\) −1856.06 −0.293463
\(343\) −343.000 −0.0539949
\(344\) 132.580 0.0207798
\(345\) −91.2278 −0.0142363
\(346\) 5701.57 0.885891
\(347\) 5747.24 0.889129 0.444565 0.895747i \(-0.353358\pi\)
0.444565 + 0.895747i \(0.353358\pi\)
\(348\) −188.276 −0.0290018
\(349\) 8367.21 1.28334 0.641671 0.766980i \(-0.278241\pi\)
0.641671 + 0.766980i \(0.278241\pi\)
\(350\) −1505.21 −0.229877
\(351\) 573.590 0.0872249
\(352\) 2020.41 0.305933
\(353\) 4056.10 0.611571 0.305785 0.952100i \(-0.401081\pi\)
0.305785 + 0.952100i \(0.401081\pi\)
\(354\) 419.128 0.0629278
\(355\) −13282.1 −1.98575
\(356\) −345.431 −0.0514265
\(357\) −103.918 −0.0154060
\(358\) 3695.23 0.545528
\(359\) −2580.13 −0.379315 −0.189658 0.981850i \(-0.560738\pi\)
−0.189658 + 0.981850i \(0.560738\pi\)
\(360\) −3285.41 −0.480990
\(361\) −5671.65 −0.826892
\(362\) 6520.83 0.946761
\(363\) −690.722 −0.0998720
\(364\) −1144.82 −0.164848
\(365\) 13643.9 1.95658
\(366\) −372.285 −0.0531684
\(367\) 938.253 0.133451 0.0667254 0.997771i \(-0.478745\pi\)
0.0667254 + 0.997771i \(0.478745\pi\)
\(368\) 368.000 0.0521286
\(369\) 2812.59 0.396795
\(370\) −4936.34 −0.693590
\(371\) −2072.72 −0.290054
\(372\) 6.16802 0.000859669 0
\(373\) 10951.6 1.52025 0.760125 0.649777i \(-0.225138\pi\)
0.760125 + 0.649777i \(0.225138\pi\)
\(374\) −7206.78 −0.996401
\(375\) 69.3522 0.00955023
\(376\) 2670.03 0.366213
\(377\) 7398.42 1.01071
\(378\) −196.404 −0.0267247
\(379\) 9520.76 1.29037 0.645183 0.764028i \(-0.276781\pi\)
0.645183 + 0.764028i \(0.276781\pi\)
\(380\) 2101.72 0.283726
\(381\) −65.5371 −0.00881252
\(382\) 3232.26 0.432924
\(383\) −3099.25 −0.413484 −0.206742 0.978396i \(-0.566286\pi\)
−0.206742 + 0.978396i \(0.566286\pi\)
\(384\) −33.2954 −0.00442473
\(385\) −6739.29 −0.892119
\(386\) −9493.74 −1.25186
\(387\) −446.337 −0.0586269
\(388\) 1375.76 0.180009
\(389\) 4368.35 0.569368 0.284684 0.958621i \(-0.408111\pi\)
0.284684 + 0.958621i \(0.408111\pi\)
\(390\) −324.345 −0.0421125
\(391\) −1312.65 −0.169779
\(392\) 392.000 0.0505076
\(393\) 479.775 0.0615813
\(394\) −2943.54 −0.376379
\(395\) −19002.4 −2.42054
\(396\) −6801.81 −0.863141
\(397\) 6306.06 0.797210 0.398605 0.917123i \(-0.369495\pi\)
0.398605 + 0.917123i \(0.369495\pi\)
\(398\) 6593.47 0.830404
\(399\) 62.7423 0.00787229
\(400\) 1720.24 0.215030
\(401\) −83.0874 −0.0103471 −0.00517355 0.999987i \(-0.501647\pi\)
−0.00517355 + 0.999987i \(0.501647\pi\)
\(402\) −156.713 −0.0194431
\(403\) −242.376 −0.0299594
\(404\) 1834.02 0.225856
\(405\) 11032.6 1.35362
\(406\) −2533.31 −0.309671
\(407\) −10219.7 −1.24465
\(408\) 118.764 0.0144110
\(409\) 1594.33 0.192750 0.0963749 0.995345i \(-0.469275\pi\)
0.0963749 + 0.995345i \(0.469275\pi\)
\(410\) −3184.84 −0.383629
\(411\) 304.323 0.0365235
\(412\) −2299.55 −0.274977
\(413\) 5639.51 0.671918
\(414\) −1238.89 −0.147072
\(415\) −18663.0 −2.20755
\(416\) 1308.36 0.154201
\(417\) 420.988 0.0494386
\(418\) 4351.20 0.509149
\(419\) −3260.67 −0.380177 −0.190089 0.981767i \(-0.560878\pi\)
−0.190089 + 0.981767i \(0.560878\pi\)
\(420\) 111.060 0.0129028
\(421\) −7104.66 −0.822471 −0.411235 0.911529i \(-0.634903\pi\)
−0.411235 + 0.911529i \(0.634903\pi\)
\(422\) −4872.18 −0.562023
\(423\) −8988.76 −1.03321
\(424\) 2368.82 0.271321
\(425\) −6136.08 −0.700337
\(426\) 453.153 0.0515384
\(427\) −5009.22 −0.567712
\(428\) −5807.02 −0.655825
\(429\) −671.495 −0.0755712
\(430\) 505.411 0.0566816
\(431\) 10760.4 1.20258 0.601290 0.799031i \(-0.294654\pi\)
0.601290 + 0.799031i \(0.294654\pi\)
\(432\) 224.462 0.0249987
\(433\) −11537.4 −1.28049 −0.640245 0.768171i \(-0.721167\pi\)
−0.640245 + 0.768171i \(0.721167\pi\)
\(434\) 82.9927 0.00917921
\(435\) −717.729 −0.0791091
\(436\) −7251.57 −0.796530
\(437\) 792.532 0.0867550
\(438\) −465.496 −0.0507814
\(439\) −10785.2 −1.17255 −0.586277 0.810111i \(-0.699407\pi\)
−0.586277 + 0.810111i \(0.699407\pi\)
\(440\) 7702.05 0.834501
\(441\) −1319.68 −0.142499
\(442\) −4666.91 −0.502222
\(443\) 2483.46 0.266349 0.133175 0.991093i \(-0.457483\pi\)
0.133175 + 0.991093i \(0.457483\pi\)
\(444\) 168.416 0.0180015
\(445\) −1316.82 −0.140277
\(446\) 2928.24 0.310888
\(447\) 454.074 0.0480469
\(448\) −448.000 −0.0472456
\(449\) −3010.79 −0.316454 −0.158227 0.987403i \(-0.550578\pi\)
−0.158227 + 0.987403i \(0.550578\pi\)
\(450\) −5791.27 −0.606674
\(451\) −6593.59 −0.688426
\(452\) −6775.14 −0.705035
\(453\) 655.610 0.0679983
\(454\) −6838.79 −0.706961
\(455\) −4364.17 −0.449661
\(456\) −71.7055 −0.00736385
\(457\) −12906.5 −1.32110 −0.660549 0.750783i \(-0.729677\pi\)
−0.660549 + 0.750783i \(0.729677\pi\)
\(458\) −6874.81 −0.701395
\(459\) −800.652 −0.0814188
\(460\) 1402.86 0.142193
\(461\) 1928.01 0.194786 0.0973931 0.995246i \(-0.468950\pi\)
0.0973931 + 0.995246i \(0.468950\pi\)
\(462\) 229.928 0.0231542
\(463\) 2880.38 0.289121 0.144560 0.989496i \(-0.453823\pi\)
0.144560 + 0.989496i \(0.453823\pi\)
\(464\) 2895.22 0.289670
\(465\) 23.5132 0.00234494
\(466\) 6098.95 0.606284
\(467\) 6861.47 0.679895 0.339947 0.940444i \(-0.389591\pi\)
0.339947 + 0.940444i \(0.389591\pi\)
\(468\) −4404.66 −0.435055
\(469\) −2108.63 −0.207606
\(470\) 10178.4 0.998929
\(471\) −291.999 −0.0285661
\(472\) −6445.16 −0.628522
\(473\) 1046.36 0.101716
\(474\) 648.316 0.0628231
\(475\) 3704.75 0.357864
\(476\) 1598.01 0.153875
\(477\) −7974.73 −0.765488
\(478\) 3922.21 0.375309
\(479\) −4580.05 −0.436885 −0.218443 0.975850i \(-0.570098\pi\)
−0.218443 + 0.975850i \(0.570098\pi\)
\(480\) −126.926 −0.0120695
\(481\) −6618.02 −0.627351
\(482\) 12243.4 1.15700
\(483\) 41.8793 0.00394529
\(484\) 10621.6 0.997521
\(485\) 5244.54 0.491015
\(486\) −1133.97 −0.105839
\(487\) 8844.29 0.822943 0.411471 0.911423i \(-0.365015\pi\)
0.411471 + 0.911423i \(0.365015\pi\)
\(488\) 5724.82 0.531046
\(489\) −38.8718 −0.00359477
\(490\) 1494.35 0.137771
\(491\) −4687.22 −0.430817 −0.215408 0.976524i \(-0.569108\pi\)
−0.215408 + 0.976524i \(0.569108\pi\)
\(492\) 108.659 0.00995675
\(493\) −10327.2 −0.943434
\(494\) 2817.72 0.256630
\(495\) −25929.3 −2.35441
\(496\) −94.8488 −0.00858637
\(497\) 6097.32 0.550307
\(498\) 636.738 0.0572950
\(499\) −10245.1 −0.919107 −0.459553 0.888150i \(-0.651991\pi\)
−0.459553 + 0.888150i \(0.651991\pi\)
\(500\) −1066.47 −0.0953876
\(501\) −507.436 −0.0452507
\(502\) 8225.89 0.731354
\(503\) −10020.9 −0.888286 −0.444143 0.895956i \(-0.646492\pi\)
−0.444143 + 0.895956i \(0.646492\pi\)
\(504\) 1508.21 0.133296
\(505\) 6991.50 0.616074
\(506\) 2904.35 0.255166
\(507\) 136.643 0.0119694
\(508\) 1007.80 0.0880194
\(509\) −15666.6 −1.36427 −0.682133 0.731228i \(-0.738947\pi\)
−0.682133 + 0.731228i \(0.738947\pi\)
\(510\) 452.741 0.0393093
\(511\) −6263.40 −0.542224
\(512\) 512.000 0.0441942
\(513\) 483.406 0.0416041
\(514\) −4271.05 −0.366513
\(515\) −8766.13 −0.750062
\(516\) −17.2434 −0.00147112
\(517\) 21072.5 1.79259
\(518\) 2266.09 0.192213
\(519\) −741.546 −0.0627173
\(520\) 4987.63 0.420619
\(521\) 12571.8 1.05716 0.528581 0.848883i \(-0.322724\pi\)
0.528581 + 0.848883i \(0.322724\pi\)
\(522\) −9746.86 −0.817258
\(523\) 12266.4 1.02557 0.512785 0.858517i \(-0.328614\pi\)
0.512785 + 0.858517i \(0.328614\pi\)
\(524\) −7377.76 −0.615074
\(525\) 195.768 0.0162743
\(526\) 14011.6 1.16147
\(527\) 338.324 0.0279651
\(528\) −262.775 −0.0216587
\(529\) 529.000 0.0434783
\(530\) 9030.21 0.740089
\(531\) 21697.9 1.77327
\(532\) −964.821 −0.0786284
\(533\) −4269.83 −0.346992
\(534\) 44.9268 0.00364077
\(535\) −22137.0 −1.78891
\(536\) 2409.86 0.194198
\(537\) −480.602 −0.0386210
\(538\) 13132.7 1.05240
\(539\) 3093.76 0.247231
\(540\) 855.675 0.0681896
\(541\) −17141.8 −1.36226 −0.681131 0.732161i \(-0.738512\pi\)
−0.681131 + 0.732161i \(0.738512\pi\)
\(542\) −11518.9 −0.912874
\(543\) −848.100 −0.0670266
\(544\) −1826.29 −0.143937
\(545\) −27643.8 −2.17272
\(546\) 148.895 0.0116705
\(547\) −13619.5 −1.06459 −0.532294 0.846560i \(-0.678670\pi\)
−0.532294 + 0.846560i \(0.678670\pi\)
\(548\) −4679.73 −0.364796
\(549\) −19272.8 −1.49826
\(550\) 13576.6 1.05256
\(551\) 6235.19 0.482083
\(552\) −47.8621 −0.00369048
\(553\) 8723.31 0.670801
\(554\) 713.772 0.0547387
\(555\) 642.021 0.0491032
\(556\) −6473.76 −0.493792
\(557\) −14755.1 −1.12243 −0.561214 0.827671i \(-0.689666\pi\)
−0.561214 + 0.827671i \(0.689666\pi\)
\(558\) 319.313 0.0242251
\(559\) 677.591 0.0512684
\(560\) −1707.83 −0.128873
\(561\) 937.314 0.0705409
\(562\) −1851.49 −0.138969
\(563\) 21018.7 1.57341 0.786706 0.617327i \(-0.211785\pi\)
0.786706 + 0.617327i \(0.211785\pi\)
\(564\) −347.264 −0.0259263
\(565\) −25827.6 −1.92314
\(566\) 13856.4 1.02902
\(567\) −5064.67 −0.375125
\(568\) −6968.37 −0.514765
\(569\) 13750.9 1.01313 0.506563 0.862203i \(-0.330916\pi\)
0.506563 + 0.862203i \(0.330916\pi\)
\(570\) −273.349 −0.0200866
\(571\) 10636.3 0.779535 0.389768 0.920913i \(-0.372555\pi\)
0.389768 + 0.920913i \(0.372555\pi\)
\(572\) 10325.9 0.754805
\(573\) −420.388 −0.0306491
\(574\) 1462.04 0.106314
\(575\) 2472.85 0.179348
\(576\) −1723.67 −0.124687
\(577\) 7290.64 0.526019 0.263010 0.964793i \(-0.415285\pi\)
0.263010 + 0.964793i \(0.415285\pi\)
\(578\) −3311.64 −0.238315
\(579\) 1234.76 0.0886264
\(580\) 11036.9 0.790141
\(581\) 8567.52 0.611774
\(582\) −178.931 −0.0127439
\(583\) 18695.3 1.32810
\(584\) 7158.17 0.507204
\(585\) −16791.1 −1.18671
\(586\) 296.379 0.0208930
\(587\) −6315.96 −0.444102 −0.222051 0.975035i \(-0.571275\pi\)
−0.222051 + 0.975035i \(0.571275\pi\)
\(588\) −50.9835 −0.00357572
\(589\) −204.268 −0.0142899
\(590\) −24569.7 −1.71444
\(591\) 382.836 0.0266460
\(592\) −2589.82 −0.179799
\(593\) 19795.1 1.37081 0.685403 0.728164i \(-0.259626\pi\)
0.685403 + 0.728164i \(0.259626\pi\)
\(594\) 1771.51 0.122367
\(595\) 6091.79 0.419729
\(596\) −6982.54 −0.479892
\(597\) −857.547 −0.0587890
\(598\) 1880.77 0.128613
\(599\) 27201.1 1.85544 0.927720 0.373277i \(-0.121766\pi\)
0.927720 + 0.373277i \(0.121766\pi\)
\(600\) −223.735 −0.0152232
\(601\) 11695.1 0.793763 0.396882 0.917870i \(-0.370092\pi\)
0.396882 + 0.917870i \(0.370092\pi\)
\(602\) −232.016 −0.0157081
\(603\) −8112.89 −0.547898
\(604\) −10081.7 −0.679167
\(605\) 40490.7 2.72096
\(606\) −238.533 −0.0159897
\(607\) −4523.78 −0.302495 −0.151247 0.988496i \(-0.548329\pi\)
−0.151247 + 0.988496i \(0.548329\pi\)
\(608\) 1102.65 0.0735501
\(609\) 329.483 0.0219233
\(610\) 21823.7 1.44855
\(611\) 13646.0 0.903530
\(612\) 6148.30 0.406095
\(613\) 27979.4 1.84352 0.921758 0.387765i \(-0.126753\pi\)
0.921758 + 0.387765i \(0.126753\pi\)
\(614\) −20193.6 −1.32728
\(615\) 414.220 0.0271593
\(616\) −3535.72 −0.231264
\(617\) 27419.8 1.78911 0.894553 0.446961i \(-0.147494\pi\)
0.894553 + 0.446961i \(0.147494\pi\)
\(618\) 299.079 0.0194672
\(619\) 4941.83 0.320887 0.160443 0.987045i \(-0.448708\pi\)
0.160443 + 0.987045i \(0.448708\pi\)
\(620\) −361.574 −0.0234213
\(621\) 322.664 0.0208504
\(622\) −2127.62 −0.137154
\(623\) 604.505 0.0388748
\(624\) −170.166 −0.0109168
\(625\) −17504.9 −1.12031
\(626\) 3907.77 0.249498
\(627\) −565.917 −0.0360455
\(628\) 4490.22 0.285318
\(629\) 9237.84 0.585591
\(630\) 5749.47 0.363594
\(631\) −13077.3 −0.825037 −0.412519 0.910949i \(-0.635351\pi\)
−0.412519 + 0.910949i \(0.635351\pi\)
\(632\) −9969.49 −0.627477
\(633\) 633.675 0.0397888
\(634\) −5879.03 −0.368274
\(635\) 3841.84 0.240093
\(636\) −308.089 −0.0192084
\(637\) 2003.43 0.124614
\(638\) 22849.7 1.41792
\(639\) 23459.3 1.45233
\(640\) 1951.80 0.120550
\(641\) −1272.72 −0.0784235 −0.0392117 0.999231i \(-0.512485\pi\)
−0.0392117 + 0.999231i \(0.512485\pi\)
\(642\) 755.262 0.0464296
\(643\) 113.036 0.00693269 0.00346634 0.999994i \(-0.498897\pi\)
0.00346634 + 0.999994i \(0.498897\pi\)
\(644\) −644.000 −0.0394055
\(645\) −65.7338 −0.00401281
\(646\) −3933.14 −0.239547
\(647\) −1829.78 −0.111184 −0.0555919 0.998454i \(-0.517705\pi\)
−0.0555919 + 0.998454i \(0.517705\pi\)
\(648\) 5788.19 0.350898
\(649\) −50866.7 −3.07657
\(650\) 8791.81 0.530528
\(651\) −10.7940 −0.000649849 0
\(652\) 597.751 0.0359045
\(653\) 1936.86 0.116072 0.0580361 0.998314i \(-0.481516\pi\)
0.0580361 + 0.998314i \(0.481516\pi\)
\(654\) 943.139 0.0563909
\(655\) −28124.8 −1.67775
\(656\) −1670.90 −0.0994480
\(657\) −24098.3 −1.43099
\(658\) −4672.55 −0.276831
\(659\) −22889.3 −1.35302 −0.676509 0.736434i \(-0.736508\pi\)
−0.676509 + 0.736434i \(0.736508\pi\)
\(660\) −1001.73 −0.0590791
\(661\) 17818.0 1.04847 0.524237 0.851573i \(-0.324351\pi\)
0.524237 + 0.851573i \(0.324351\pi\)
\(662\) −17616.9 −1.03429
\(663\) 606.978 0.0355552
\(664\) −9791.45 −0.572262
\(665\) −3678.01 −0.214477
\(666\) 8718.74 0.507274
\(667\) 4161.87 0.241602
\(668\) 7803.11 0.451963
\(669\) −380.847 −0.0220095
\(670\) 9186.65 0.529718
\(671\) 45181.7 2.59943
\(672\) 58.2669 0.00334478
\(673\) 20993.6 1.20244 0.601222 0.799082i \(-0.294681\pi\)
0.601222 + 0.799082i \(0.294681\pi\)
\(674\) 7933.46 0.453391
\(675\) 1508.32 0.0860076
\(676\) −2101.22 −0.119551
\(677\) 5364.75 0.304555 0.152278 0.988338i \(-0.451339\pi\)
0.152278 + 0.988338i \(0.451339\pi\)
\(678\) 881.175 0.0499134
\(679\) −2407.57 −0.136074
\(680\) −6962.04 −0.392621
\(681\) 889.453 0.0500498
\(682\) −748.570 −0.0420297
\(683\) −31720.5 −1.77709 −0.888543 0.458793i \(-0.848282\pi\)
−0.888543 + 0.458793i \(0.848282\pi\)
\(684\) −3712.13 −0.207510
\(685\) −17839.7 −0.995063
\(686\) −686.000 −0.0381802
\(687\) 894.138 0.0496557
\(688\) 265.161 0.0146935
\(689\) 12106.6 0.669409
\(690\) −182.456 −0.0100666
\(691\) −12501.1 −0.688228 −0.344114 0.938928i \(-0.611821\pi\)
−0.344114 + 0.938928i \(0.611821\pi\)
\(692\) 11403.1 0.626420
\(693\) 11903.2 0.652473
\(694\) 11494.5 0.628709
\(695\) −24678.7 −1.34693
\(696\) −376.552 −0.0205074
\(697\) 5960.09 0.323894
\(698\) 16734.4 0.907460
\(699\) −793.230 −0.0429223
\(700\) −3010.43 −0.162548
\(701\) −4917.13 −0.264932 −0.132466 0.991188i \(-0.542290\pi\)
−0.132466 + 0.991188i \(0.542290\pi\)
\(702\) 1147.18 0.0616773
\(703\) −5577.48 −0.299230
\(704\) 4040.83 0.216327
\(705\) −1323.81 −0.0707199
\(706\) 8112.20 0.432446
\(707\) −3209.54 −0.170731
\(708\) 838.257 0.0444966
\(709\) 2043.21 0.108229 0.0541145 0.998535i \(-0.482766\pi\)
0.0541145 + 0.998535i \(0.482766\pi\)
\(710\) −26564.2 −1.40414
\(711\) 33562.7 1.77032
\(712\) −690.863 −0.0363640
\(713\) −136.345 −0.00716153
\(714\) −207.837 −0.0108937
\(715\) 39363.6 2.05890
\(716\) 7390.46 0.385747
\(717\) −510.122 −0.0265703
\(718\) −5160.27 −0.268216
\(719\) 24690.8 1.28069 0.640343 0.768089i \(-0.278792\pi\)
0.640343 + 0.768089i \(0.278792\pi\)
\(720\) −6570.82 −0.340111
\(721\) 4024.21 0.207863
\(722\) −11343.3 −0.584701
\(723\) −1592.38 −0.0819103
\(724\) 13041.7 0.669461
\(725\) 19455.0 0.996606
\(726\) −1381.44 −0.0706202
\(727\) −23243.0 −1.18574 −0.592872 0.805297i \(-0.702006\pi\)
−0.592872 + 0.805297i \(0.702006\pi\)
\(728\) −2289.64 −0.116565
\(729\) −19387.7 −0.984995
\(730\) 27287.7 1.38351
\(731\) −945.823 −0.0478557
\(732\) −744.570 −0.0375958
\(733\) −35315.4 −1.77954 −0.889772 0.456406i \(-0.849137\pi\)
−0.889772 + 0.456406i \(0.849137\pi\)
\(734\) 1876.51 0.0943639
\(735\) −194.355 −0.00975359
\(736\) 736.000 0.0368605
\(737\) 19019.2 0.950585
\(738\) 5625.17 0.280576
\(739\) 10783.2 0.536763 0.268382 0.963313i \(-0.413511\pi\)
0.268382 + 0.963313i \(0.413511\pi\)
\(740\) −9872.69 −0.490442
\(741\) −366.472 −0.0181683
\(742\) −4145.44 −0.205099
\(743\) −4972.73 −0.245534 −0.122767 0.992436i \(-0.539177\pi\)
−0.122767 + 0.992436i \(0.539177\pi\)
\(744\) 12.3360 0.000607878 0
\(745\) −26618.2 −1.30901
\(746\) 21903.2 1.07498
\(747\) 32963.3 1.61454
\(748\) −14413.6 −0.704562
\(749\) 10162.3 0.495757
\(750\) 138.704 0.00675303
\(751\) −29604.7 −1.43847 −0.719234 0.694768i \(-0.755507\pi\)
−0.719234 + 0.694768i \(0.755507\pi\)
\(752\) 5340.06 0.258952
\(753\) −1069.86 −0.0517767
\(754\) 14796.8 0.714681
\(755\) −38432.4 −1.85258
\(756\) −392.809 −0.0188972
\(757\) −9958.05 −0.478113 −0.239056 0.971006i \(-0.576838\pi\)
−0.239056 + 0.971006i \(0.576838\pi\)
\(758\) 19041.5 0.912426
\(759\) −377.739 −0.0180646
\(760\) 4203.44 0.200625
\(761\) 28654.4 1.36494 0.682472 0.730911i \(-0.260905\pi\)
0.682472 + 0.730911i \(0.260905\pi\)
\(762\) −131.074 −0.00623139
\(763\) 12690.2 0.602120
\(764\) 6464.52 0.306123
\(765\) 23438.0 1.10772
\(766\) −6198.50 −0.292377
\(767\) −32939.9 −1.55070
\(768\) −66.5907 −0.00312876
\(769\) 1580.21 0.0741011 0.0370506 0.999313i \(-0.488204\pi\)
0.0370506 + 0.999313i \(0.488204\pi\)
\(770\) −13478.6 −0.630824
\(771\) 555.493 0.0259476
\(772\) −18987.5 −0.885200
\(773\) 36297.7 1.68892 0.844461 0.535617i \(-0.179921\pi\)
0.844461 + 0.535617i \(0.179921\pi\)
\(774\) −892.675 −0.0414555
\(775\) −637.356 −0.0295413
\(776\) 2751.51 0.127286
\(777\) −294.728 −0.0136079
\(778\) 8736.69 0.402604
\(779\) −3598.49 −0.165506
\(780\) −648.691 −0.0297780
\(781\) −54996.1 −2.51974
\(782\) −2625.30 −0.120052
\(783\) 2538.54 0.115862
\(784\) 784.000 0.0357143
\(785\) 17117.2 0.778268
\(786\) 959.551 0.0435446
\(787\) 3840.61 0.173956 0.0869778 0.996210i \(-0.472279\pi\)
0.0869778 + 0.996210i \(0.472279\pi\)
\(788\) −5887.07 −0.266140
\(789\) −1822.35 −0.0822273
\(790\) −38004.8 −1.71158
\(791\) 11856.5 0.532956
\(792\) −13603.6 −0.610333
\(793\) 29258.4 1.31021
\(794\) 12612.1 0.563712
\(795\) −1174.47 −0.0523951
\(796\) 13186.9 0.587184
\(797\) −24193.7 −1.07526 −0.537632 0.843179i \(-0.680681\pi\)
−0.537632 + 0.843179i \(0.680681\pi\)
\(798\) 125.485 0.00556655
\(799\) −19047.9 −0.843386
\(800\) 3440.49 0.152049
\(801\) 2325.82 0.102595
\(802\) −166.175 −0.00731650
\(803\) 56494.0 2.48273
\(804\) −313.426 −0.0137484
\(805\) −2455.00 −0.107487
\(806\) −484.753 −0.0211845
\(807\) −1708.04 −0.0745052
\(808\) 3668.04 0.159705
\(809\) −16476.8 −0.716063 −0.358031 0.933710i \(-0.616552\pi\)
−0.358031 + 0.933710i \(0.616552\pi\)
\(810\) 22065.2 0.957152
\(811\) 2891.58 0.125200 0.0625999 0.998039i \(-0.480061\pi\)
0.0625999 + 0.998039i \(0.480061\pi\)
\(812\) −5066.63 −0.218970
\(813\) 1498.14 0.0646275
\(814\) −20439.5 −0.880103
\(815\) 2278.70 0.0979377
\(816\) 237.528 0.0101901
\(817\) 571.055 0.0244537
\(818\) 3188.66 0.136295
\(819\) 7708.16 0.328870
\(820\) −6369.68 −0.271267
\(821\) 36315.1 1.54373 0.771867 0.635785i \(-0.219323\pi\)
0.771867 + 0.635785i \(0.219323\pi\)
\(822\) 608.646 0.0258260
\(823\) −18744.4 −0.793913 −0.396956 0.917838i \(-0.629934\pi\)
−0.396956 + 0.917838i \(0.629934\pi\)
\(824\) −4599.09 −0.194438
\(825\) −1765.77 −0.0745166
\(826\) 11279.0 0.475118
\(827\) 25294.4 1.06357 0.531784 0.846880i \(-0.321522\pi\)
0.531784 + 0.846880i \(0.321522\pi\)
\(828\) −2477.78 −0.103996
\(829\) −39438.4 −1.65230 −0.826148 0.563453i \(-0.809473\pi\)
−0.826148 + 0.563453i \(0.809473\pi\)
\(830\) −37326.1 −1.56097
\(831\) −92.8332 −0.00387527
\(832\) 2616.73 0.109037
\(833\) −2796.51 −0.116319
\(834\) 841.977 0.0349584
\(835\) 29746.3 1.23283
\(836\) 8702.40 0.360023
\(837\) −83.1639 −0.00343437
\(838\) −6521.34 −0.268826
\(839\) −30165.0 −1.24125 −0.620627 0.784106i \(-0.713122\pi\)
−0.620627 + 0.784106i \(0.713122\pi\)
\(840\) 222.120 0.00912365
\(841\) 8354.24 0.342541
\(842\) −14209.3 −0.581574
\(843\) 240.805 0.00983838
\(844\) −9744.35 −0.397410
\(845\) −8010.10 −0.326101
\(846\) −17977.5 −0.730591
\(847\) −18587.8 −0.754055
\(848\) 4737.64 0.191853
\(849\) −1802.16 −0.0728503
\(850\) −12272.2 −0.495213
\(851\) −3722.87 −0.149963
\(852\) 906.306 0.0364431
\(853\) 8977.27 0.360347 0.180173 0.983635i \(-0.442334\pi\)
0.180173 + 0.983635i \(0.442334\pi\)
\(854\) −10018.4 −0.401433
\(855\) −14151.0 −0.566030
\(856\) −11614.0 −0.463738
\(857\) −40593.6 −1.61803 −0.809015 0.587789i \(-0.799999\pi\)
−0.809015 + 0.587789i \(0.799999\pi\)
\(858\) −1342.99 −0.0534369
\(859\) −14117.7 −0.560755 −0.280377 0.959890i \(-0.590460\pi\)
−0.280377 + 0.959890i \(0.590460\pi\)
\(860\) 1010.82 0.0400799
\(861\) −190.153 −0.00752660
\(862\) 21520.9 0.850352
\(863\) −7080.02 −0.279266 −0.139633 0.990203i \(-0.544592\pi\)
−0.139633 + 0.990203i \(0.544592\pi\)
\(864\) 448.924 0.0176767
\(865\) 43470.1 1.70870
\(866\) −23074.8 −0.905443
\(867\) 430.712 0.0168717
\(868\) 165.985 0.00649068
\(869\) −78681.7 −3.07145
\(870\) −1435.46 −0.0559386
\(871\) 12316.3 0.479129
\(872\) −14503.1 −0.563232
\(873\) −9263.09 −0.359116
\(874\) 1585.06 0.0613450
\(875\) 1866.31 0.0721062
\(876\) −930.991 −0.0359079
\(877\) −29775.4 −1.14646 −0.573229 0.819395i \(-0.694309\pi\)
−0.573229 + 0.819395i \(0.694309\pi\)
\(878\) −21570.5 −0.829120
\(879\) −38.5471 −0.00147914
\(880\) 15404.1 0.590082
\(881\) 32974.9 1.26101 0.630506 0.776184i \(-0.282847\pi\)
0.630506 + 0.776184i \(0.282847\pi\)
\(882\) −2639.37 −0.100762
\(883\) −8286.80 −0.315825 −0.157912 0.987453i \(-0.550476\pi\)
−0.157912 + 0.987453i \(0.550476\pi\)
\(884\) −9333.82 −0.355125
\(885\) 3195.53 0.121375
\(886\) 4966.91 0.188337
\(887\) −13354.7 −0.505533 −0.252767 0.967527i \(-0.581341\pi\)
−0.252767 + 0.967527i \(0.581341\pi\)
\(888\) 336.832 0.0127290
\(889\) −1763.65 −0.0665364
\(890\) −2633.65 −0.0991911
\(891\) 45681.8 1.71762
\(892\) 5856.48 0.219831
\(893\) 11500.4 0.430960
\(894\) 908.149 0.0339743
\(895\) 28173.3 1.05221
\(896\) −896.000 −0.0334077
\(897\) −244.613 −0.00910524
\(898\) −6021.57 −0.223767
\(899\) −1072.69 −0.0397954
\(900\) −11582.5 −0.428983
\(901\) −16899.1 −0.624850
\(902\) −13187.2 −0.486791
\(903\) 30.1759 0.00111206
\(904\) −13550.3 −0.498535
\(905\) 49716.3 1.82611
\(906\) 1311.22 0.0480821
\(907\) 53901.6 1.97329 0.986645 0.162885i \(-0.0520801\pi\)
0.986645 + 0.162885i \(0.0520801\pi\)
\(908\) −13677.6 −0.499897
\(909\) −12348.6 −0.450581
\(910\) −8728.35 −0.317958
\(911\) −14861.5 −0.540488 −0.270244 0.962792i \(-0.587104\pi\)
−0.270244 + 0.962792i \(0.587104\pi\)
\(912\) −143.411 −0.00520703
\(913\) −77276.5 −2.80118
\(914\) −25813.1 −0.934158
\(915\) −2838.38 −0.102551
\(916\) −13749.6 −0.495961
\(917\) 12911.1 0.464952
\(918\) −1601.30 −0.0575718
\(919\) 39976.2 1.43492 0.717460 0.696599i \(-0.245304\pi\)
0.717460 + 0.696599i \(0.245304\pi\)
\(920\) 2805.71 0.100545
\(921\) 2626.38 0.0939655
\(922\) 3856.02 0.137735
\(923\) −35613.9 −1.27004
\(924\) 459.856 0.0163725
\(925\) −17402.8 −0.618596
\(926\) 5760.77 0.204439
\(927\) 15483.0 0.548576
\(928\) 5790.43 0.204828
\(929\) −7480.52 −0.264185 −0.132093 0.991237i \(-0.542170\pi\)
−0.132093 + 0.991237i \(0.542170\pi\)
\(930\) 47.0264 0.00165812
\(931\) 1688.44 0.0594375
\(932\) 12197.9 0.428708
\(933\) 276.718 0.00970992
\(934\) 13722.9 0.480758
\(935\) −54946.1 −1.92185
\(936\) −8809.32 −0.307630
\(937\) −18057.0 −0.629558 −0.314779 0.949165i \(-0.601930\pi\)
−0.314779 + 0.949165i \(0.601930\pi\)
\(938\) −4217.25 −0.146800
\(939\) −508.244 −0.0176634
\(940\) 20356.9 0.706350
\(941\) −18921.5 −0.655496 −0.327748 0.944765i \(-0.606290\pi\)
−0.327748 + 0.944765i \(0.606290\pi\)
\(942\) −583.999 −0.0201993
\(943\) −2401.93 −0.0829453
\(944\) −12890.3 −0.444432
\(945\) −1497.43 −0.0515465
\(946\) 2092.71 0.0719238
\(947\) −10147.2 −0.348193 −0.174097 0.984729i \(-0.555701\pi\)
−0.174097 + 0.984729i \(0.555701\pi\)
\(948\) 1296.63 0.0444226
\(949\) 36583.9 1.25139
\(950\) 7409.49 0.253048
\(951\) 764.626 0.0260722
\(952\) 3196.02 0.108806
\(953\) 6284.28 0.213607 0.106804 0.994280i \(-0.465938\pi\)
0.106804 + 0.994280i \(0.465938\pi\)
\(954\) −15949.5 −0.541282
\(955\) 24643.5 0.835020
\(956\) 7844.42 0.265383
\(957\) −2971.84 −0.100382
\(958\) −9160.11 −0.308925
\(959\) 8189.53 0.275760
\(960\) −253.851 −0.00853439
\(961\) −29755.9 −0.998820
\(962\) −13236.0 −0.443604
\(963\) 39099.2 1.30836
\(964\) 24486.8 0.818119
\(965\) −72382.4 −2.41458
\(966\) 83.7586 0.00278974
\(967\) −10750.9 −0.357524 −0.178762 0.983892i \(-0.557209\pi\)
−0.178762 + 0.983892i \(0.557209\pi\)
\(968\) 21243.2 0.705354
\(969\) 511.544 0.0169589
\(970\) 10489.1 0.347200
\(971\) 21337.3 0.705198 0.352599 0.935774i \(-0.385298\pi\)
0.352599 + 0.935774i \(0.385298\pi\)
\(972\) −2267.93 −0.0748394
\(973\) 11329.1 0.373272
\(974\) 17688.6 0.581908
\(975\) −1143.46 −0.0375591
\(976\) 11449.6 0.375506
\(977\) −713.051 −0.0233496 −0.0116748 0.999932i \(-0.503716\pi\)
−0.0116748 + 0.999932i \(0.503716\pi\)
\(978\) −77.7435 −0.00254189
\(979\) −5452.46 −0.177999
\(980\) 2988.70 0.0974188
\(981\) 48825.4 1.58907
\(982\) −9374.43 −0.304634
\(983\) 31306.7 1.01580 0.507898 0.861417i \(-0.330423\pi\)
0.507898 + 0.861417i \(0.330423\pi\)
\(984\) 217.318 0.00704049
\(985\) −22442.2 −0.725956
\(986\) −20654.4 −0.667108
\(987\) 607.712 0.0195985
\(988\) 5635.43 0.181465
\(989\) 381.168 0.0122553
\(990\) −51858.5 −1.66482
\(991\) 13538.1 0.433956 0.216978 0.976176i \(-0.430380\pi\)
0.216978 + 0.976176i \(0.430380\pi\)
\(992\) −189.698 −0.00607148
\(993\) 2291.26 0.0732234
\(994\) 12194.6 0.389125
\(995\) 50270.1 1.60168
\(996\) 1273.48 0.0405137
\(997\) 44983.5 1.42893 0.714464 0.699672i \(-0.246671\pi\)
0.714464 + 0.699672i \(0.246671\pi\)
\(998\) −20490.2 −0.649907
\(999\) −2270.77 −0.0719158
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 322.4.a.h.1.3 5
7.6 odd 2 2254.4.a.l.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.4.a.h.1.3 5 1.1 even 1 trivial
2254.4.a.l.1.3 5 7.6 odd 2