Properties

Label 77.4.a.e.1.2
Level $77$
Weight $4$
Character 77.1
Self dual yes
Analytic conductor $4.543$
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] = [77,4,Mod(1,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, 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("77.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.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: \(4.54314707044\)
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: \( x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.18888\) of defining polynomial
Character \(\chi\) \(=\) 77.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.18888 q^{2} +6.48496 q^{3} -3.20880 q^{4} +7.60736 q^{5} -14.1948 q^{6} +7.00000 q^{7} +24.5347 q^{8} +15.0547 q^{9} -16.6516 q^{10} +11.0000 q^{11} -20.8090 q^{12} +0.174238 q^{13} -15.3222 q^{14} +49.3335 q^{15} -28.0332 q^{16} +128.863 q^{17} -32.9530 q^{18} +141.685 q^{19} -24.4105 q^{20} +45.3947 q^{21} -24.0777 q^{22} -133.369 q^{23} +159.107 q^{24} -67.1280 q^{25} -0.381386 q^{26} -77.4647 q^{27} -22.4616 q^{28} -177.002 q^{29} -107.985 q^{30} +48.2757 q^{31} -134.917 q^{32} +71.3346 q^{33} -282.065 q^{34} +53.2515 q^{35} -48.3076 q^{36} +161.625 q^{37} -310.131 q^{38} +1.12993 q^{39} +186.645 q^{40} -195.689 q^{41} -99.3636 q^{42} -488.447 q^{43} -35.2968 q^{44} +114.527 q^{45} +291.928 q^{46} -171.705 q^{47} -181.794 q^{48} +49.0000 q^{49} +146.935 q^{50} +835.671 q^{51} -0.559095 q^{52} -431.477 q^{53} +169.561 q^{54} +83.6810 q^{55} +171.743 q^{56} +918.821 q^{57} +387.437 q^{58} +194.176 q^{59} -158.301 q^{60} +585.008 q^{61} -105.670 q^{62} +105.383 q^{63} +519.582 q^{64} +1.32549 q^{65} -156.143 q^{66} +155.905 q^{67} -413.496 q^{68} -864.891 q^{69} -116.561 q^{70} -374.994 q^{71} +369.364 q^{72} +210.419 q^{73} -353.778 q^{74} -435.323 q^{75} -454.639 q^{76} +77.0000 q^{77} -2.47327 q^{78} -7.00618 q^{79} -213.258 q^{80} -908.833 q^{81} +428.339 q^{82} -93.6417 q^{83} -145.663 q^{84} +980.307 q^{85} +1069.15 q^{86} -1147.85 q^{87} +269.882 q^{88} -307.119 q^{89} -250.685 q^{90} +1.21967 q^{91} +427.954 q^{92} +313.066 q^{93} +375.842 q^{94} +1077.85 q^{95} -874.929 q^{96} +965.991 q^{97} -107.255 q^{98} +165.602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 2 q^{3} + 45 q^{4} - 24 q^{5} + 4 q^{6} + 35 q^{7} + 57 q^{8} + 63 q^{9} - 10 q^{10} + 55 q^{11} + 24 q^{12} - 50 q^{13} + 7 q^{14} - 146 q^{15} + 433 q^{16} + 222 q^{17} + 245 q^{18} + 160 q^{19}+ \cdots + 693 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18888 −0.773886 −0.386943 0.922104i \(-0.626469\pi\)
−0.386943 + 0.922104i \(0.626469\pi\)
\(3\) 6.48496 1.24803 0.624016 0.781412i \(-0.285500\pi\)
0.624016 + 0.781412i \(0.285500\pi\)
\(4\) −3.20880 −0.401100
\(5\) 7.60736 0.680423 0.340212 0.940349i \(-0.389501\pi\)
0.340212 + 0.940349i \(0.389501\pi\)
\(6\) −14.1948 −0.965834
\(7\) 7.00000 0.377964
\(8\) 24.5347 1.08429
\(9\) 15.0547 0.557582
\(10\) −16.6516 −0.526570
\(11\) 11.0000 0.301511
\(12\) −20.8090 −0.500586
\(13\) 0.174238 0.00371730 0.00185865 0.999998i \(-0.499408\pi\)
0.00185865 + 0.999998i \(0.499408\pi\)
\(14\) −15.3222 −0.292501
\(15\) 49.3335 0.849190
\(16\) −28.0332 −0.438018
\(17\) 128.863 1.83846 0.919231 0.393719i \(-0.128812\pi\)
0.919231 + 0.393719i \(0.128812\pi\)
\(18\) −32.9530 −0.431505
\(19\) 141.685 1.71078 0.855388 0.517988i \(-0.173319\pi\)
0.855388 + 0.517988i \(0.173319\pi\)
\(20\) −24.4105 −0.272918
\(21\) 45.3947 0.471712
\(22\) −24.0777 −0.233335
\(23\) −133.369 −1.20910 −0.604550 0.796567i \(-0.706647\pi\)
−0.604550 + 0.796567i \(0.706647\pi\)
\(24\) 159.107 1.35323
\(25\) −67.1280 −0.537024
\(26\) −0.381386 −0.00287677
\(27\) −77.4647 −0.552151
\(28\) −22.4616 −0.151602
\(29\) −177.002 −1.13340 −0.566698 0.823925i \(-0.691779\pi\)
−0.566698 + 0.823925i \(0.691779\pi\)
\(30\) −107.985 −0.657176
\(31\) 48.2757 0.279696 0.139848 0.990173i \(-0.455339\pi\)
0.139848 + 0.990173i \(0.455339\pi\)
\(32\) −134.917 −0.745316
\(33\) 71.3346 0.376296
\(34\) −282.065 −1.42276
\(35\) 53.2515 0.257176
\(36\) −48.3076 −0.223647
\(37\) 161.625 0.718136 0.359068 0.933311i \(-0.383095\pi\)
0.359068 + 0.933311i \(0.383095\pi\)
\(38\) −310.131 −1.32394
\(39\) 1.12993 0.00463931
\(40\) 186.645 0.737778
\(41\) −195.689 −0.745402 −0.372701 0.927952i \(-0.621568\pi\)
−0.372701 + 0.927952i \(0.621568\pi\)
\(42\) −99.3636 −0.365051
\(43\) −488.447 −1.73227 −0.866133 0.499814i \(-0.833402\pi\)
−0.866133 + 0.499814i \(0.833402\pi\)
\(44\) −35.2968 −0.120936
\(45\) 114.527 0.379392
\(46\) 291.928 0.935705
\(47\) −171.705 −0.532889 −0.266444 0.963850i \(-0.585849\pi\)
−0.266444 + 0.963850i \(0.585849\pi\)
\(48\) −181.794 −0.546660
\(49\) 49.0000 0.142857
\(50\) 146.935 0.415596
\(51\) 835.671 2.29446
\(52\) −0.559095 −0.00149101
\(53\) −431.477 −1.11826 −0.559131 0.829079i \(-0.688865\pi\)
−0.559131 + 0.829079i \(0.688865\pi\)
\(54\) 169.561 0.427302
\(55\) 83.6810 0.205155
\(56\) 171.743 0.409824
\(57\) 918.821 2.13510
\(58\) 387.437 0.877120
\(59\) 194.176 0.428468 0.214234 0.976782i \(-0.431275\pi\)
0.214234 + 0.976782i \(0.431275\pi\)
\(60\) −158.301 −0.340610
\(61\) 585.008 1.22791 0.613955 0.789341i \(-0.289577\pi\)
0.613955 + 0.789341i \(0.289577\pi\)
\(62\) −105.670 −0.216453
\(63\) 105.383 0.210746
\(64\) 519.582 1.01481
\(65\) 1.32549 0.00252934
\(66\) −156.143 −0.291210
\(67\) 155.905 0.284281 0.142140 0.989847i \(-0.454602\pi\)
0.142140 + 0.989847i \(0.454602\pi\)
\(68\) −413.496 −0.737408
\(69\) −864.891 −1.50899
\(70\) −116.561 −0.199025
\(71\) −374.994 −0.626811 −0.313405 0.949619i \(-0.601470\pi\)
−0.313405 + 0.949619i \(0.601470\pi\)
\(72\) 369.364 0.604582
\(73\) 210.419 0.337365 0.168683 0.985670i \(-0.446049\pi\)
0.168683 + 0.985670i \(0.446049\pi\)
\(74\) −353.778 −0.555755
\(75\) −435.323 −0.670223
\(76\) −454.639 −0.686193
\(77\) 77.0000 0.113961
\(78\) −2.47327 −0.00359029
\(79\) −7.00618 −0.00997793 −0.00498897 0.999988i \(-0.501588\pi\)
−0.00498897 + 0.999988i \(0.501588\pi\)
\(80\) −213.258 −0.298038
\(81\) −908.833 −1.24668
\(82\) 428.339 0.576856
\(83\) −93.6417 −0.123837 −0.0619187 0.998081i \(-0.519722\pi\)
−0.0619187 + 0.998081i \(0.519722\pi\)
\(84\) −145.663 −0.189204
\(85\) 980.307 1.25093
\(86\) 1069.15 1.34058
\(87\) −1147.85 −1.41451
\(88\) 269.882 0.326926
\(89\) −307.119 −0.365781 −0.182891 0.983133i \(-0.558545\pi\)
−0.182891 + 0.983133i \(0.558545\pi\)
\(90\) −250.685 −0.293606
\(91\) 1.21967 0.00140501
\(92\) 427.954 0.484970
\(93\) 313.066 0.349069
\(94\) 375.842 0.412395
\(95\) 1077.85 1.16405
\(96\) −874.929 −0.930178
\(97\) 965.991 1.01115 0.505575 0.862783i \(-0.331280\pi\)
0.505575 + 0.862783i \(0.331280\pi\)
\(98\) −107.255 −0.110555
\(99\) 165.602 0.168117
\(100\) 215.401 0.215401
\(101\) −577.986 −0.569424 −0.284712 0.958613i \(-0.591898\pi\)
−0.284712 + 0.958613i \(0.591898\pi\)
\(102\) −1829.18 −1.77565
\(103\) 133.232 0.127453 0.0637267 0.997967i \(-0.479701\pi\)
0.0637267 + 0.997967i \(0.479701\pi\)
\(104\) 4.27488 0.00403064
\(105\) 345.334 0.320963
\(106\) 944.451 0.865408
\(107\) 8.33627 0.00753175 0.00376587 0.999993i \(-0.498801\pi\)
0.00376587 + 0.999993i \(0.498801\pi\)
\(108\) 248.569 0.221468
\(109\) −317.052 −0.278606 −0.139303 0.990250i \(-0.544486\pi\)
−0.139303 + 0.990250i \(0.544486\pi\)
\(110\) −183.168 −0.158767
\(111\) 1048.13 0.896256
\(112\) −196.232 −0.165555
\(113\) −2170.00 −1.80652 −0.903260 0.429093i \(-0.858833\pi\)
−0.903260 + 0.429093i \(0.858833\pi\)
\(114\) −2011.19 −1.65232
\(115\) −1014.58 −0.822700
\(116\) 567.966 0.454606
\(117\) 2.62310 0.00207270
\(118\) −425.028 −0.331585
\(119\) 902.040 0.694873
\(120\) 1210.38 0.920769
\(121\) 121.000 0.0909091
\(122\) −1280.51 −0.950263
\(123\) −1269.03 −0.930285
\(124\) −154.907 −0.112186
\(125\) −1461.59 −1.04583
\(126\) −230.671 −0.163094
\(127\) 1301.24 0.909184 0.454592 0.890700i \(-0.349785\pi\)
0.454592 + 0.890700i \(0.349785\pi\)
\(128\) −57.9688 −0.0400294
\(129\) −3167.56 −2.16192
\(130\) −2.90134 −0.00195742
\(131\) 2522.22 1.68220 0.841098 0.540883i \(-0.181910\pi\)
0.841098 + 0.540883i \(0.181910\pi\)
\(132\) −228.899 −0.150932
\(133\) 991.794 0.646612
\(134\) −341.257 −0.220001
\(135\) −589.302 −0.375696
\(136\) 3161.62 1.99343
\(137\) 2574.70 1.60563 0.802815 0.596228i \(-0.203334\pi\)
0.802815 + 0.596228i \(0.203334\pi\)
\(138\) 1893.14 1.16779
\(139\) −2741.49 −1.67288 −0.836438 0.548061i \(-0.815366\pi\)
−0.836438 + 0.548061i \(0.815366\pi\)
\(140\) −170.874 −0.103153
\(141\) −1113.50 −0.665062
\(142\) 820.816 0.485080
\(143\) 1.91662 0.00112081
\(144\) −422.031 −0.244231
\(145\) −1346.52 −0.771189
\(146\) −460.582 −0.261082
\(147\) 317.763 0.178290
\(148\) −518.624 −0.288045
\(149\) −1021.98 −0.561907 −0.280954 0.959721i \(-0.590651\pi\)
−0.280954 + 0.959721i \(0.590651\pi\)
\(150\) 952.869 0.518676
\(151\) 1016.40 0.547769 0.273885 0.961763i \(-0.411691\pi\)
0.273885 + 0.961763i \(0.411691\pi\)
\(152\) 3476.20 1.85498
\(153\) 1940.00 1.02509
\(154\) −168.544 −0.0881925
\(155\) 367.251 0.190312
\(156\) −3.62571 −0.00186083
\(157\) −1580.49 −0.803422 −0.401711 0.915767i \(-0.631584\pi\)
−0.401711 + 0.915767i \(0.631584\pi\)
\(158\) 15.3357 0.00772178
\(159\) −2798.11 −1.39563
\(160\) −1026.36 −0.507130
\(161\) −933.581 −0.456997
\(162\) 1989.33 0.964792
\(163\) 768.358 0.369218 0.184609 0.982812i \(-0.440898\pi\)
0.184609 + 0.982812i \(0.440898\pi\)
\(164\) 627.927 0.298981
\(165\) 542.668 0.256040
\(166\) 204.970 0.0958361
\(167\) 3092.73 1.43307 0.716534 0.697552i \(-0.245727\pi\)
0.716534 + 0.697552i \(0.245727\pi\)
\(168\) 1113.75 0.511473
\(169\) −2196.97 −0.999986
\(170\) −2145.77 −0.968079
\(171\) 2133.03 0.953898
\(172\) 1567.33 0.694812
\(173\) −1327.46 −0.583380 −0.291690 0.956513i \(-0.594218\pi\)
−0.291690 + 0.956513i \(0.594218\pi\)
\(174\) 2512.51 1.09467
\(175\) −469.896 −0.202976
\(176\) −308.365 −0.132067
\(177\) 1259.23 0.534741
\(178\) 672.247 0.283073
\(179\) 3141.37 1.31171 0.655857 0.754885i \(-0.272307\pi\)
0.655857 + 0.754885i \(0.272307\pi\)
\(180\) −367.494 −0.152174
\(181\) −3683.36 −1.51261 −0.756303 0.654221i \(-0.772997\pi\)
−0.756303 + 0.654221i \(0.772997\pi\)
\(182\) −2.66970 −0.00108732
\(183\) 3793.75 1.53247
\(184\) −3272.16 −1.31102
\(185\) 1229.54 0.488636
\(186\) −685.264 −0.270140
\(187\) 1417.49 0.554317
\(188\) 550.968 0.213742
\(189\) −542.253 −0.208694
\(190\) −2359.28 −0.900843
\(191\) −2862.38 −1.08437 −0.542185 0.840259i \(-0.682403\pi\)
−0.542185 + 0.840259i \(0.682403\pi\)
\(192\) 3369.47 1.26651
\(193\) 2023.91 0.754839 0.377420 0.926042i \(-0.376811\pi\)
0.377420 + 0.926042i \(0.376811\pi\)
\(194\) −2114.44 −0.782515
\(195\) 8.59576 0.00315669
\(196\) −157.231 −0.0573001
\(197\) −4767.82 −1.72433 −0.862165 0.506628i \(-0.830892\pi\)
−0.862165 + 0.506628i \(0.830892\pi\)
\(198\) −362.483 −0.130104
\(199\) 3546.72 1.26342 0.631709 0.775205i \(-0.282354\pi\)
0.631709 + 0.775205i \(0.282354\pi\)
\(200\) −1646.97 −0.582291
\(201\) 1011.04 0.354791
\(202\) 1265.14 0.440669
\(203\) −1239.02 −0.428384
\(204\) −2681.50 −0.920308
\(205\) −1488.68 −0.507189
\(206\) −291.628 −0.0986343
\(207\) −2007.83 −0.674173
\(208\) −4.88444 −0.00162824
\(209\) 1558.53 0.515818
\(210\) −755.895 −0.248389
\(211\) 1453.49 0.474230 0.237115 0.971482i \(-0.423798\pi\)
0.237115 + 0.971482i \(0.423798\pi\)
\(212\) 1384.52 0.448536
\(213\) −2431.82 −0.782279
\(214\) −18.2471 −0.00582872
\(215\) −3715.79 −1.17867
\(216\) −1900.57 −0.598693
\(217\) 337.930 0.105715
\(218\) 693.989 0.215610
\(219\) 1364.56 0.421042
\(220\) −268.516 −0.0822879
\(221\) 22.4528 0.00683411
\(222\) −2294.24 −0.693600
\(223\) 6440.10 1.93391 0.966953 0.254955i \(-0.0820606\pi\)
0.966953 + 0.254955i \(0.0820606\pi\)
\(224\) −944.416 −0.281703
\(225\) −1010.59 −0.299435
\(226\) 4749.88 1.39804
\(227\) −1313.21 −0.383970 −0.191985 0.981398i \(-0.561492\pi\)
−0.191985 + 0.981398i \(0.561492\pi\)
\(228\) −2948.31 −0.856390
\(229\) −3780.00 −1.09078 −0.545392 0.838181i \(-0.683619\pi\)
−0.545392 + 0.838181i \(0.683619\pi\)
\(230\) 2220.80 0.636676
\(231\) 499.342 0.142226
\(232\) −4342.70 −1.22893
\(233\) 2322.65 0.653055 0.326527 0.945188i \(-0.394121\pi\)
0.326527 + 0.945188i \(0.394121\pi\)
\(234\) −5.74166 −0.00160403
\(235\) −1306.22 −0.362590
\(236\) −623.073 −0.171859
\(237\) −45.4348 −0.0124528
\(238\) −1974.46 −0.537753
\(239\) 2204.79 0.596718 0.298359 0.954454i \(-0.403561\pi\)
0.298359 + 0.954454i \(0.403561\pi\)
\(240\) −1382.97 −0.371960
\(241\) −4610.39 −1.23229 −0.616144 0.787634i \(-0.711306\pi\)
−0.616144 + 0.787634i \(0.711306\pi\)
\(242\) −264.855 −0.0703533
\(243\) −3802.20 −1.00375
\(244\) −1877.17 −0.492516
\(245\) 372.761 0.0972033
\(246\) 2777.76 0.719934
\(247\) 24.6869 0.00635946
\(248\) 1184.43 0.303272
\(249\) −607.263 −0.154553
\(250\) 3199.24 0.809351
\(251\) 4981.12 1.25261 0.626306 0.779577i \(-0.284566\pi\)
0.626306 + 0.779577i \(0.284566\pi\)
\(252\) −338.154 −0.0845304
\(253\) −1467.06 −0.364557
\(254\) −2848.26 −0.703605
\(255\) 6357.25 1.56120
\(256\) −4029.77 −0.983829
\(257\) 2726.08 0.661665 0.330833 0.943689i \(-0.392670\pi\)
0.330833 + 0.943689i \(0.392670\pi\)
\(258\) 6933.40 1.67308
\(259\) 1131.38 0.271430
\(260\) −4.25324 −0.00101452
\(261\) −2664.72 −0.631962
\(262\) −5520.84 −1.30183
\(263\) −4228.84 −0.991489 −0.495744 0.868468i \(-0.665105\pi\)
−0.495744 + 0.868468i \(0.665105\pi\)
\(264\) 1750.17 0.408014
\(265\) −3282.40 −0.760892
\(266\) −2170.92 −0.500404
\(267\) −1991.65 −0.456507
\(268\) −500.268 −0.114025
\(269\) −2396.04 −0.543081 −0.271541 0.962427i \(-0.587533\pi\)
−0.271541 + 0.962427i \(0.587533\pi\)
\(270\) 1289.91 0.290746
\(271\) −2461.68 −0.551795 −0.275897 0.961187i \(-0.588975\pi\)
−0.275897 + 0.961187i \(0.588975\pi\)
\(272\) −3612.43 −0.805279
\(273\) 7.90948 0.00175349
\(274\) −5635.71 −1.24258
\(275\) −738.408 −0.161919
\(276\) 2775.26 0.605258
\(277\) 5890.54 1.27772 0.638860 0.769323i \(-0.279406\pi\)
0.638860 + 0.769323i \(0.279406\pi\)
\(278\) 6000.79 1.29462
\(279\) 726.777 0.155953
\(280\) 1306.51 0.278854
\(281\) −1964.26 −0.417004 −0.208502 0.978022i \(-0.566859\pi\)
−0.208502 + 0.978022i \(0.566859\pi\)
\(282\) 2437.32 0.514682
\(283\) 8513.60 1.78827 0.894136 0.447796i \(-0.147791\pi\)
0.894136 + 0.447796i \(0.147791\pi\)
\(284\) 1203.28 0.251414
\(285\) 6989.80 1.45277
\(286\) −4.19524 −0.000867378 0
\(287\) −1369.82 −0.281735
\(288\) −2031.13 −0.415575
\(289\) 11692.6 2.37994
\(290\) 2947.37 0.596813
\(291\) 6264.42 1.26195
\(292\) −675.193 −0.135317
\(293\) 5219.58 1.04072 0.520360 0.853947i \(-0.325798\pi\)
0.520360 + 0.853947i \(0.325798\pi\)
\(294\) −695.545 −0.137976
\(295\) 1477.17 0.291539
\(296\) 3965.43 0.778669
\(297\) −852.111 −0.166480
\(298\) 2237.00 0.434852
\(299\) −23.2379 −0.00449459
\(300\) 1396.86 0.268827
\(301\) −3419.13 −0.654735
\(302\) −2224.77 −0.423911
\(303\) −3748.22 −0.710659
\(304\) −3971.87 −0.749350
\(305\) 4450.37 0.835499
\(306\) −4246.42 −0.793306
\(307\) 2887.29 0.536764 0.268382 0.963313i \(-0.413511\pi\)
0.268382 + 0.963313i \(0.413511\pi\)
\(308\) −247.078 −0.0457096
\(309\) 864.001 0.159066
\(310\) −803.868 −0.147279
\(311\) −5876.90 −1.07154 −0.535769 0.844364i \(-0.679978\pi\)
−0.535769 + 0.844364i \(0.679978\pi\)
\(312\) 27.7224 0.00503036
\(313\) −7683.14 −1.38747 −0.693733 0.720233i \(-0.744035\pi\)
−0.693733 + 0.720233i \(0.744035\pi\)
\(314\) 3459.51 0.621757
\(315\) 801.687 0.143397
\(316\) 22.4814 0.00400215
\(317\) −9642.33 −1.70841 −0.854207 0.519933i \(-0.825957\pi\)
−0.854207 + 0.519933i \(0.825957\pi\)
\(318\) 6124.73 1.08006
\(319\) −1947.03 −0.341732
\(320\) 3952.65 0.690499
\(321\) 54.0604 0.00939986
\(322\) 2043.50 0.353663
\(323\) 18257.9 3.14519
\(324\) 2916.27 0.500046
\(325\) −11.6962 −0.00199628
\(326\) −1681.84 −0.285732
\(327\) −2056.07 −0.347710
\(328\) −4801.17 −0.808233
\(329\) −1201.94 −0.201413
\(330\) −1187.84 −0.198146
\(331\) −2069.87 −0.343717 −0.171859 0.985122i \(-0.554977\pi\)
−0.171859 + 0.985122i \(0.554977\pi\)
\(332\) 300.478 0.0496713
\(333\) 2433.22 0.400420
\(334\) −6769.61 −1.10903
\(335\) 1186.03 0.193431
\(336\) −1272.56 −0.206618
\(337\) 7547.59 1.22001 0.610005 0.792397i \(-0.291167\pi\)
0.610005 + 0.792397i \(0.291167\pi\)
\(338\) 4808.90 0.773875
\(339\) −14072.4 −2.25459
\(340\) −3145.61 −0.501749
\(341\) 531.033 0.0843315
\(342\) −4668.94 −0.738208
\(343\) 343.000 0.0539949
\(344\) −11983.9 −1.87828
\(345\) −6579.54 −1.02675
\(346\) 2905.65 0.451469
\(347\) −6353.22 −0.982877 −0.491439 0.870912i \(-0.663529\pi\)
−0.491439 + 0.870912i \(0.663529\pi\)
\(348\) 3683.23 0.567362
\(349\) −1352.47 −0.207439 −0.103719 0.994607i \(-0.533074\pi\)
−0.103719 + 0.994607i \(0.533074\pi\)
\(350\) 1028.55 0.157080
\(351\) −13.4973 −0.00205251
\(352\) −1484.08 −0.224721
\(353\) 4203.80 0.633840 0.316920 0.948452i \(-0.397351\pi\)
0.316920 + 0.948452i \(0.397351\pi\)
\(354\) −2756.29 −0.413829
\(355\) −2852.71 −0.426497
\(356\) 985.484 0.146715
\(357\) 5849.70 0.867223
\(358\) −6876.08 −1.01512
\(359\) −11100.7 −1.63195 −0.815976 0.578086i \(-0.803800\pi\)
−0.815976 + 0.578086i \(0.803800\pi\)
\(360\) 2809.88 0.411372
\(361\) 13215.6 1.92675
\(362\) 8062.43 1.17059
\(363\) 784.680 0.113457
\(364\) −3.91367 −0.000563549 0
\(365\) 1600.73 0.229551
\(366\) −8304.07 −1.18596
\(367\) −4963.35 −0.705953 −0.352977 0.935632i \(-0.614830\pi\)
−0.352977 + 0.935632i \(0.614830\pi\)
\(368\) 3738.74 0.529607
\(369\) −2946.04 −0.415623
\(370\) −2691.32 −0.378149
\(371\) −3020.34 −0.422663
\(372\) −1004.57 −0.140012
\(373\) −8882.54 −1.23303 −0.616515 0.787343i \(-0.711456\pi\)
−0.616515 + 0.787343i \(0.711456\pi\)
\(374\) −3102.72 −0.428978
\(375\) −9478.34 −1.30522
\(376\) −4212.74 −0.577807
\(377\) −30.8405 −0.00421317
\(378\) 1186.93 0.161505
\(379\) −1218.83 −0.165190 −0.0825952 0.996583i \(-0.526321\pi\)
−0.0825952 + 0.996583i \(0.526321\pi\)
\(380\) −3458.60 −0.466901
\(381\) 8438.49 1.13469
\(382\) 6265.41 0.839179
\(383\) 112.614 0.0150243 0.00751214 0.999972i \(-0.497609\pi\)
0.00751214 + 0.999972i \(0.497609\pi\)
\(384\) −375.926 −0.0499580
\(385\) 585.767 0.0775414
\(386\) −4430.09 −0.584159
\(387\) −7353.43 −0.965880
\(388\) −3099.68 −0.405573
\(389\) 12656.2 1.64961 0.824804 0.565419i \(-0.191286\pi\)
0.824804 + 0.565419i \(0.191286\pi\)
\(390\) −18.8151 −0.00244292
\(391\) −17186.3 −2.22288
\(392\) 1202.20 0.154899
\(393\) 16356.5 2.09943
\(394\) 10436.2 1.33443
\(395\) −53.2985 −0.00678922
\(396\) −531.384 −0.0674320
\(397\) 3707.14 0.468655 0.234327 0.972158i \(-0.424711\pi\)
0.234327 + 0.972158i \(0.424711\pi\)
\(398\) −7763.35 −0.977742
\(399\) 6431.74 0.806992
\(400\) 1881.81 0.235226
\(401\) 5671.06 0.706232 0.353116 0.935579i \(-0.385122\pi\)
0.353116 + 0.935579i \(0.385122\pi\)
\(402\) −2213.04 −0.274568
\(403\) 8.41146 0.00103971
\(404\) 1854.64 0.228396
\(405\) −6913.82 −0.848273
\(406\) 2712.06 0.331520
\(407\) 1777.88 0.216526
\(408\) 20503.0 2.48786
\(409\) 260.465 0.0314894 0.0157447 0.999876i \(-0.494988\pi\)
0.0157447 + 0.999876i \(0.494988\pi\)
\(410\) 3258.53 0.392506
\(411\) 16696.8 2.00388
\(412\) −427.514 −0.0511216
\(413\) 1359.23 0.161946
\(414\) 4394.90 0.521733
\(415\) −712.366 −0.0842619
\(416\) −23.5076 −0.00277056
\(417\) −17778.4 −2.08780
\(418\) −3411.44 −0.399184
\(419\) 4153.96 0.484330 0.242165 0.970235i \(-0.422142\pi\)
0.242165 + 0.970235i \(0.422142\pi\)
\(420\) −1108.11 −0.128739
\(421\) 4240.11 0.490856 0.245428 0.969415i \(-0.421072\pi\)
0.245428 + 0.969415i \(0.421072\pi\)
\(422\) −3181.52 −0.367000
\(423\) −2584.97 −0.297129
\(424\) −10586.2 −1.21252
\(425\) −8650.31 −0.987298
\(426\) 5322.96 0.605395
\(427\) 4095.05 0.464107
\(428\) −26.7494 −0.00302099
\(429\) 12.4292 0.00139880
\(430\) 8133.42 0.912159
\(431\) 1012.17 0.113119 0.0565595 0.998399i \(-0.481987\pi\)
0.0565595 + 0.998399i \(0.481987\pi\)
\(432\) 2171.58 0.241852
\(433\) 1604.50 0.178077 0.0890386 0.996028i \(-0.471621\pi\)
0.0890386 + 0.996028i \(0.471621\pi\)
\(434\) −739.688 −0.0818114
\(435\) −8732.13 −0.962469
\(436\) 1017.36 0.111749
\(437\) −18896.3 −2.06850
\(438\) −2986.85 −0.325839
\(439\) 10671.6 1.16020 0.580099 0.814546i \(-0.303014\pi\)
0.580099 + 0.814546i \(0.303014\pi\)
\(440\) 2053.09 0.222448
\(441\) 737.681 0.0796546
\(442\) −49.1465 −0.00528882
\(443\) 2193.74 0.235277 0.117638 0.993056i \(-0.462468\pi\)
0.117638 + 0.993056i \(0.462468\pi\)
\(444\) −3363.25 −0.359489
\(445\) −2336.37 −0.248886
\(446\) −14096.6 −1.49662
\(447\) −6627.52 −0.701278
\(448\) 3637.07 0.383561
\(449\) −7070.36 −0.743143 −0.371571 0.928404i \(-0.621181\pi\)
−0.371571 + 0.928404i \(0.621181\pi\)
\(450\) 2212.07 0.231729
\(451\) −2152.58 −0.224747
\(452\) 6963.12 0.724596
\(453\) 6591.29 0.683633
\(454\) 2874.47 0.297149
\(455\) 9.27844 0.000956000 0
\(456\) 22543.0 2.31507
\(457\) 16245.8 1.66290 0.831451 0.555599i \(-0.187511\pi\)
0.831451 + 0.555599i \(0.187511\pi\)
\(458\) 8273.98 0.844143
\(459\) −9982.32 −1.01511
\(460\) 3255.60 0.329985
\(461\) −5235.18 −0.528908 −0.264454 0.964398i \(-0.585192\pi\)
−0.264454 + 0.964398i \(0.585192\pi\)
\(462\) −1093.00 −0.110067
\(463\) −11093.9 −1.11355 −0.556777 0.830662i \(-0.687962\pi\)
−0.556777 + 0.830662i \(0.687962\pi\)
\(464\) 4961.93 0.496448
\(465\) 2381.61 0.237515
\(466\) −5084.00 −0.505390
\(467\) −7813.14 −0.774195 −0.387097 0.922039i \(-0.626522\pi\)
−0.387097 + 0.922039i \(0.626522\pi\)
\(468\) −8.41702 −0.000831361 0
\(469\) 1091.33 0.107448
\(470\) 2859.17 0.280603
\(471\) −10249.4 −1.00270
\(472\) 4764.06 0.464584
\(473\) −5372.91 −0.522298
\(474\) 99.4513 0.00963703
\(475\) −9511.02 −0.918728
\(476\) −2894.47 −0.278714
\(477\) −6495.76 −0.623523
\(478\) −4826.01 −0.461792
\(479\) 5298.54 0.505420 0.252710 0.967542i \(-0.418678\pi\)
0.252710 + 0.967542i \(0.418678\pi\)
\(480\) −6655.90 −0.632915
\(481\) 28.1612 0.00266953
\(482\) 10091.6 0.953650
\(483\) −6054.23 −0.570346
\(484\) −388.265 −0.0364637
\(485\) 7348.65 0.688010
\(486\) 8322.56 0.776788
\(487\) 5580.89 0.519290 0.259645 0.965704i \(-0.416394\pi\)
0.259645 + 0.965704i \(0.416394\pi\)
\(488\) 14353.0 1.33141
\(489\) 4982.77 0.460795
\(490\) −815.929 −0.0752243
\(491\) 16718.8 1.53668 0.768341 0.640041i \(-0.221083\pi\)
0.768341 + 0.640041i \(0.221083\pi\)
\(492\) 4072.08 0.373138
\(493\) −22809.0 −2.08371
\(494\) −54.0366 −0.00492150
\(495\) 1259.79 0.114391
\(496\) −1353.32 −0.122512
\(497\) −2624.96 −0.236912
\(498\) 1329.23 0.119606
\(499\) 19594.4 1.75784 0.878922 0.476965i \(-0.158263\pi\)
0.878922 + 0.476965i \(0.158263\pi\)
\(500\) 4689.95 0.419482
\(501\) 20056.2 1.78851
\(502\) −10903.1 −0.969379
\(503\) 9008.04 0.798506 0.399253 0.916841i \(-0.369269\pi\)
0.399253 + 0.916841i \(0.369269\pi\)
\(504\) 2585.54 0.228511
\(505\) −4396.95 −0.387449
\(506\) 3211.21 0.282126
\(507\) −14247.3 −1.24801
\(508\) −4175.42 −0.364674
\(509\) −3379.88 −0.294324 −0.147162 0.989112i \(-0.547014\pi\)
−0.147162 + 0.989112i \(0.547014\pi\)
\(510\) −13915.3 −1.20819
\(511\) 1472.93 0.127512
\(512\) 9284.42 0.801401
\(513\) −10975.6 −0.944606
\(514\) −5967.05 −0.512053
\(515\) 1013.54 0.0867222
\(516\) 10164.1 0.867148
\(517\) −1888.76 −0.160672
\(518\) −2476.45 −0.210056
\(519\) −8608.51 −0.728076
\(520\) 32.5206 0.00274254
\(521\) −736.886 −0.0619646 −0.0309823 0.999520i \(-0.509864\pi\)
−0.0309823 + 0.999520i \(0.509864\pi\)
\(522\) 5832.75 0.489066
\(523\) 9541.00 0.797703 0.398852 0.917015i \(-0.369409\pi\)
0.398852 + 0.917015i \(0.369409\pi\)
\(524\) −8093.32 −0.674729
\(525\) −3047.26 −0.253320
\(526\) 9256.43 0.767299
\(527\) 6220.95 0.514210
\(528\) −1999.73 −0.164824
\(529\) 5620.20 0.461922
\(530\) 7184.78 0.588843
\(531\) 2923.27 0.238906
\(532\) −3182.47 −0.259356
\(533\) −34.0964 −0.00277088
\(534\) 4359.49 0.353284
\(535\) 63.4170 0.00512478
\(536\) 3825.08 0.308243
\(537\) 20371.7 1.63706
\(538\) 5244.64 0.420283
\(539\) 539.000 0.0430730
\(540\) 1890.95 0.150692
\(541\) 16721.0 1.32882 0.664411 0.747368i \(-0.268683\pi\)
0.664411 + 0.747368i \(0.268683\pi\)
\(542\) 5388.32 0.427026
\(543\) −23886.4 −1.88778
\(544\) −17385.7 −1.37023
\(545\) −2411.93 −0.189570
\(546\) −17.3129 −0.00135700
\(547\) 8958.59 0.700259 0.350129 0.936701i \(-0.386138\pi\)
0.350129 + 0.936701i \(0.386138\pi\)
\(548\) −8261.70 −0.644019
\(549\) 8807.13 0.684661
\(550\) 1616.29 0.125307
\(551\) −25078.5 −1.93899
\(552\) −21219.9 −1.63619
\(553\) −49.0432 −0.00377130
\(554\) −12893.7 −0.988809
\(555\) 7973.53 0.609834
\(556\) 8796.89 0.670992
\(557\) −14329.6 −1.09006 −0.545031 0.838416i \(-0.683482\pi\)
−0.545031 + 0.838416i \(0.683482\pi\)
\(558\) −1590.83 −0.120690
\(559\) −85.1059 −0.00643935
\(560\) −1492.81 −0.112648
\(561\) 9192.38 0.691805
\(562\) 4299.54 0.322713
\(563\) 15791.2 1.18209 0.591047 0.806637i \(-0.298715\pi\)
0.591047 + 0.806637i \(0.298715\pi\)
\(564\) 3573.01 0.266757
\(565\) −16508.0 −1.22920
\(566\) −18635.2 −1.38392
\(567\) −6361.83 −0.471202
\(568\) −9200.37 −0.679646
\(569\) −13996.4 −1.03121 −0.515605 0.856826i \(-0.672433\pi\)
−0.515605 + 0.856826i \(0.672433\pi\)
\(570\) −15299.8 −1.12428
\(571\) −6642.54 −0.486833 −0.243417 0.969922i \(-0.578268\pi\)
−0.243417 + 0.969922i \(0.578268\pi\)
\(572\) −6.15005 −0.000449557 0
\(573\) −18562.4 −1.35333
\(574\) 2998.38 0.218031
\(575\) 8952.78 0.649316
\(576\) 7822.16 0.565839
\(577\) −85.5448 −0.00617206 −0.00308603 0.999995i \(-0.500982\pi\)
−0.00308603 + 0.999995i \(0.500982\pi\)
\(578\) −25593.8 −1.84180
\(579\) 13125.0 0.942063
\(580\) 4320.72 0.309324
\(581\) −655.492 −0.0468062
\(582\) −13712.1 −0.976603
\(583\) −4746.25 −0.337169
\(584\) 5162.57 0.365802
\(585\) 19.9549 0.00141031
\(586\) −11425.0 −0.805399
\(587\) 11205.0 0.787869 0.393934 0.919139i \(-0.371114\pi\)
0.393934 + 0.919139i \(0.371114\pi\)
\(588\) −1019.64 −0.0715123
\(589\) 6839.93 0.478497
\(590\) −3233.35 −0.225618
\(591\) −30919.1 −2.15202
\(592\) −4530.87 −0.314556
\(593\) 11664.6 0.807773 0.403886 0.914809i \(-0.367659\pi\)
0.403886 + 0.914809i \(0.367659\pi\)
\(594\) 1865.17 0.128836
\(595\) 6862.15 0.472808
\(596\) 3279.34 0.225381
\(597\) 23000.3 1.57679
\(598\) 50.8649 0.00347830
\(599\) −20545.1 −1.40142 −0.700711 0.713445i \(-0.747134\pi\)
−0.700711 + 0.713445i \(0.747134\pi\)
\(600\) −10680.5 −0.726718
\(601\) 3885.01 0.263682 0.131841 0.991271i \(-0.457911\pi\)
0.131841 + 0.991271i \(0.457911\pi\)
\(602\) 7484.06 0.506690
\(603\) 2347.11 0.158510
\(604\) −3261.42 −0.219710
\(605\) 920.491 0.0618567
\(606\) 8204.40 0.549969
\(607\) 8439.80 0.564351 0.282175 0.959363i \(-0.408944\pi\)
0.282175 + 0.959363i \(0.408944\pi\)
\(608\) −19115.6 −1.27507
\(609\) −8034.97 −0.534636
\(610\) −9741.32 −0.646581
\(611\) −29.9175 −0.00198091
\(612\) −6225.06 −0.411165
\(613\) 17997.8 1.18584 0.592922 0.805260i \(-0.297974\pi\)
0.592922 + 0.805260i \(0.297974\pi\)
\(614\) −6319.94 −0.415394
\(615\) −9654.01 −0.632987
\(616\) 1889.17 0.123567
\(617\) −26336.6 −1.71843 −0.859214 0.511616i \(-0.829047\pi\)
−0.859214 + 0.511616i \(0.829047\pi\)
\(618\) −1891.20 −0.123099
\(619\) 27875.5 1.81003 0.905016 0.425376i \(-0.139858\pi\)
0.905016 + 0.425376i \(0.139858\pi\)
\(620\) −1178.44 −0.0763341
\(621\) 10331.4 0.667606
\(622\) 12863.8 0.829249
\(623\) −2149.83 −0.138252
\(624\) −31.6754 −0.00203210
\(625\) −2727.83 −0.174581
\(626\) 16817.5 1.07374
\(627\) 10107.0 0.643757
\(628\) 5071.50 0.322253
\(629\) 20827.5 1.32027
\(630\) −1754.80 −0.110973
\(631\) 8242.15 0.519992 0.259996 0.965610i \(-0.416279\pi\)
0.259996 + 0.965610i \(0.416279\pi\)
\(632\) −171.895 −0.0108190
\(633\) 9425.84 0.591854
\(634\) 21105.9 1.32212
\(635\) 9899.00 0.618630
\(636\) 8978.59 0.559786
\(637\) 8.53766 0.000531043 0
\(638\) 4261.80 0.264462
\(639\) −5645.43 −0.349499
\(640\) −440.990 −0.0272370
\(641\) −7001.97 −0.431453 −0.215726 0.976454i \(-0.569212\pi\)
−0.215726 + 0.976454i \(0.569212\pi\)
\(642\) −118.332 −0.00727442
\(643\) 2473.25 0.151688 0.0758442 0.997120i \(-0.475835\pi\)
0.0758442 + 0.997120i \(0.475835\pi\)
\(644\) 2995.68 0.183302
\(645\) −24096.8 −1.47102
\(646\) −39964.4 −2.43402
\(647\) 9153.21 0.556182 0.278091 0.960555i \(-0.410298\pi\)
0.278091 + 0.960555i \(0.410298\pi\)
\(648\) −22298.0 −1.35177
\(649\) 2135.94 0.129188
\(650\) 25.6017 0.00154489
\(651\) 2191.46 0.131936
\(652\) −2465.51 −0.148093
\(653\) 725.254 0.0434630 0.0217315 0.999764i \(-0.493082\pi\)
0.0217315 + 0.999764i \(0.493082\pi\)
\(654\) 4500.49 0.269088
\(655\) 19187.5 1.14461
\(656\) 5485.78 0.326499
\(657\) 3167.80 0.188109
\(658\) 2630.89 0.155871
\(659\) 14332.9 0.847242 0.423621 0.905840i \(-0.360759\pi\)
0.423621 + 0.905840i \(0.360759\pi\)
\(660\) −1741.32 −0.102698
\(661\) 26773.3 1.57543 0.787715 0.616040i \(-0.211264\pi\)
0.787715 + 0.616040i \(0.211264\pi\)
\(662\) 4530.70 0.265998
\(663\) 145.606 0.00852919
\(664\) −2297.47 −0.134276
\(665\) 7544.94 0.439970
\(666\) −5326.04 −0.309879
\(667\) 23606.6 1.37039
\(668\) −9923.95 −0.574804
\(669\) 41763.8 2.41358
\(670\) −2596.07 −0.149694
\(671\) 6435.08 0.370229
\(672\) −6124.50 −0.351574
\(673\) −25423.9 −1.45619 −0.728096 0.685475i \(-0.759595\pi\)
−0.728096 + 0.685475i \(0.759595\pi\)
\(674\) −16520.8 −0.944149
\(675\) 5200.05 0.296519
\(676\) 7049.64 0.401095
\(677\) 9287.48 0.527248 0.263624 0.964625i \(-0.415082\pi\)
0.263624 + 0.964625i \(0.415082\pi\)
\(678\) 30802.8 1.74480
\(679\) 6761.94 0.382179
\(680\) 24051.6 1.35638
\(681\) −8516.15 −0.479206
\(682\) −1162.37 −0.0652629
\(683\) −31888.8 −1.78652 −0.893258 0.449544i \(-0.851586\pi\)
−0.893258 + 0.449544i \(0.851586\pi\)
\(684\) −6844.46 −0.382609
\(685\) 19586.7 1.09251
\(686\) −750.786 −0.0417859
\(687\) −24513.2 −1.36133
\(688\) 13692.7 0.758763
\(689\) −75.1796 −0.00415692
\(690\) 14401.8 0.794591
\(691\) −18650.3 −1.02676 −0.513379 0.858162i \(-0.671606\pi\)
−0.513379 + 0.858162i \(0.671606\pi\)
\(692\) 4259.55 0.233994
\(693\) 1159.21 0.0635424
\(694\) 13906.4 0.760635
\(695\) −20855.5 −1.13826
\(696\) −28162.3 −1.53375
\(697\) −25217.0 −1.37039
\(698\) 2960.40 0.160534
\(699\) 15062.3 0.815033
\(700\) 1507.80 0.0814138
\(701\) 8566.01 0.461532 0.230766 0.973009i \(-0.425877\pi\)
0.230766 + 0.973009i \(0.425877\pi\)
\(702\) 29.5439 0.00158841
\(703\) 22899.8 1.22857
\(704\) 5715.40 0.305976
\(705\) −8470.81 −0.452524
\(706\) −9201.61 −0.490520
\(707\) −4045.90 −0.215222
\(708\) −4040.61 −0.214485
\(709\) 680.116 0.0360258 0.0180129 0.999838i \(-0.494266\pi\)
0.0180129 + 0.999838i \(0.494266\pi\)
\(710\) 6244.25 0.330060
\(711\) −105.476 −0.00556352
\(712\) −7535.08 −0.396614
\(713\) −6438.47 −0.338180
\(714\) −12804.3 −0.671132
\(715\) 14.5804 0.000762624 0
\(716\) −10080.0 −0.526129
\(717\) 14297.9 0.744723
\(718\) 24298.0 1.26294
\(719\) 12557.6 0.651346 0.325673 0.945482i \(-0.394409\pi\)
0.325673 + 0.945482i \(0.394409\pi\)
\(720\) −3210.55 −0.166181
\(721\) 932.621 0.0481728
\(722\) −28927.3 −1.49109
\(723\) −29898.2 −1.53793
\(724\) 11819.2 0.606707
\(725\) 11881.8 0.608661
\(726\) −1717.57 −0.0878031
\(727\) 22253.6 1.13527 0.567635 0.823281i \(-0.307859\pi\)
0.567635 + 0.823281i \(0.307859\pi\)
\(728\) 29.9242 0.00152344
\(729\) −118.633 −0.00602716
\(730\) −3503.81 −0.177646
\(731\) −62942.6 −3.18470
\(732\) −12173.4 −0.614675
\(733\) 6852.72 0.345308 0.172654 0.984982i \(-0.444766\pi\)
0.172654 + 0.984982i \(0.444766\pi\)
\(734\) 10864.2 0.546327
\(735\) 2417.34 0.121313
\(736\) 17993.6 0.901161
\(737\) 1714.95 0.0857139
\(738\) 6448.53 0.321645
\(739\) −9133.20 −0.454628 −0.227314 0.973821i \(-0.572994\pi\)
−0.227314 + 0.973821i \(0.572994\pi\)
\(740\) −3945.36 −0.195992
\(741\) 160.093 0.00793681
\(742\) 6611.16 0.327093
\(743\) −33458.7 −1.65206 −0.826031 0.563624i \(-0.809407\pi\)
−0.826031 + 0.563624i \(0.809407\pi\)
\(744\) 7680.99 0.378493
\(745\) −7774.60 −0.382335
\(746\) 19442.8 0.954225
\(747\) −1409.75 −0.0690496
\(748\) −4548.45 −0.222337
\(749\) 58.3539 0.00284673
\(750\) 20746.9 1.01010
\(751\) −37530.0 −1.82355 −0.911776 0.410688i \(-0.865289\pi\)
−0.911776 + 0.410688i \(0.865289\pi\)
\(752\) 4813.44 0.233415
\(753\) 32302.4 1.56330
\(754\) 67.5062 0.00326052
\(755\) 7732.10 0.372715
\(756\) 1739.98 0.0837071
\(757\) 32174.8 1.54480 0.772399 0.635138i \(-0.219057\pi\)
0.772399 + 0.635138i \(0.219057\pi\)
\(758\) 2667.88 0.127839
\(759\) −9513.80 −0.454979
\(760\) 26444.7 1.26217
\(761\) 4457.43 0.212328 0.106164 0.994349i \(-0.466143\pi\)
0.106164 + 0.994349i \(0.466143\pi\)
\(762\) −18470.8 −0.878121
\(763\) −2219.37 −0.105303
\(764\) 9184.82 0.434941
\(765\) 14758.2 0.697497
\(766\) −246.498 −0.0116271
\(767\) 33.8329 0.00159274
\(768\) −26132.9 −1.22785
\(769\) −38329.0 −1.79737 −0.898687 0.438591i \(-0.855478\pi\)
−0.898687 + 0.438591i \(0.855478\pi\)
\(770\) −1282.17 −0.0600082
\(771\) 17678.5 0.825779
\(772\) −6494.32 −0.302766
\(773\) −9529.23 −0.443393 −0.221696 0.975116i \(-0.571159\pi\)
−0.221696 + 0.975116i \(0.571159\pi\)
\(774\) 16095.8 0.747481
\(775\) −3240.65 −0.150203
\(776\) 23700.3 1.09638
\(777\) 7336.94 0.338753
\(778\) −27703.0 −1.27661
\(779\) −27726.1 −1.27521
\(780\) −27.5821 −0.00126615
\(781\) −4124.93 −0.188991
\(782\) 37618.7 1.72026
\(783\) 13711.4 0.625806
\(784\) −1373.62 −0.0625740
\(785\) −12023.4 −0.546667
\(786\) −35802.5 −1.62472
\(787\) 29705.1 1.34545 0.672726 0.739892i \(-0.265123\pi\)
0.672726 + 0.739892i \(0.265123\pi\)
\(788\) 15299.0 0.691629
\(789\) −27423.9 −1.23741
\(790\) 116.664 0.00525408
\(791\) −15190.0 −0.682801
\(792\) 4063.00 0.182288
\(793\) 101.931 0.00456451
\(794\) −8114.48 −0.362685
\(795\) −21286.2 −0.949617
\(796\) −11380.7 −0.506758
\(797\) 8596.99 0.382084 0.191042 0.981582i \(-0.438813\pi\)
0.191042 + 0.981582i \(0.438813\pi\)
\(798\) −14078.3 −0.624520
\(799\) −22126.4 −0.979695
\(800\) 9056.69 0.400253
\(801\) −4623.59 −0.203953
\(802\) −12413.3 −0.546543
\(803\) 2314.61 0.101719
\(804\) −3244.22 −0.142307
\(805\) −7102.09 −0.310951
\(806\) −18.4117 −0.000804620 0
\(807\) −15538.2 −0.677783
\(808\) −14180.7 −0.617422
\(809\) 16630.2 0.722727 0.361364 0.932425i \(-0.382311\pi\)
0.361364 + 0.932425i \(0.382311\pi\)
\(810\) 15133.5 0.656467
\(811\) 18584.2 0.804660 0.402330 0.915495i \(-0.368201\pi\)
0.402330 + 0.915495i \(0.368201\pi\)
\(812\) 3975.76 0.171825
\(813\) −15963.9 −0.688657
\(814\) −3891.56 −0.167567
\(815\) 5845.18 0.251224
\(816\) −23426.5 −1.00501
\(817\) −69205.5 −2.96352
\(818\) −570.127 −0.0243692
\(819\) 18.3617 0.000783407 0
\(820\) 4776.87 0.203434
\(821\) −4088.88 −0.173816 −0.0869079 0.996216i \(-0.527699\pi\)
−0.0869079 + 0.996216i \(0.527699\pi\)
\(822\) −36547.3 −1.55077
\(823\) −12830.0 −0.543410 −0.271705 0.962381i \(-0.587587\pi\)
−0.271705 + 0.962381i \(0.587587\pi\)
\(824\) 3268.80 0.138197
\(825\) −4788.55 −0.202080
\(826\) −2975.20 −0.125327
\(827\) −13768.7 −0.578942 −0.289471 0.957187i \(-0.593479\pi\)
−0.289471 + 0.957187i \(0.593479\pi\)
\(828\) 6442.73 0.270411
\(829\) −10907.8 −0.456987 −0.228494 0.973545i \(-0.573380\pi\)
−0.228494 + 0.973545i \(0.573380\pi\)
\(830\) 1559.28 0.0652091
\(831\) 38199.9 1.59463
\(832\) 90.5308 0.00377234
\(833\) 6314.28 0.262637
\(834\) 38914.9 1.61572
\(835\) 23527.5 0.975093
\(836\) −5001.03 −0.206895
\(837\) −3739.66 −0.154434
\(838\) −9092.53 −0.374817
\(839\) −27174.8 −1.11821 −0.559105 0.829097i \(-0.688855\pi\)
−0.559105 + 0.829097i \(0.688855\pi\)
\(840\) 8472.68 0.348018
\(841\) 6940.81 0.284588
\(842\) −9281.09 −0.379866
\(843\) −12738.2 −0.520434
\(844\) −4663.97 −0.190214
\(845\) −16713.1 −0.680414
\(846\) 5658.20 0.229944
\(847\) 847.000 0.0343604
\(848\) 12095.7 0.489819
\(849\) 55210.3 2.23182
\(850\) 18934.5 0.764056
\(851\) −21555.7 −0.868298
\(852\) 7803.23 0.313773
\(853\) −33975.1 −1.36376 −0.681878 0.731466i \(-0.738837\pi\)
−0.681878 + 0.731466i \(0.738837\pi\)
\(854\) −8963.58 −0.359166
\(855\) 16226.7 0.649054
\(856\) 204.528 0.00816662
\(857\) 20423.7 0.814073 0.407037 0.913412i \(-0.366562\pi\)
0.407037 + 0.913412i \(0.366562\pi\)
\(858\) −27.2060 −0.00108251
\(859\) −36704.2 −1.45789 −0.728947 0.684570i \(-0.759990\pi\)
−0.728947 + 0.684570i \(0.759990\pi\)
\(860\) 11923.2 0.472766
\(861\) −8883.24 −0.351615
\(862\) −2215.51 −0.0875413
\(863\) 26222.8 1.03434 0.517170 0.855883i \(-0.326985\pi\)
0.517170 + 0.855883i \(0.326985\pi\)
\(864\) 10451.3 0.411527
\(865\) −10098.5 −0.396945
\(866\) −3512.06 −0.137811
\(867\) 75826.4 2.97024
\(868\) −1084.35 −0.0424024
\(869\) −77.0680 −0.00300846
\(870\) 19113.6 0.744841
\(871\) 27.1645 0.00105676
\(872\) −7778.79 −0.302091
\(873\) 14542.7 0.563799
\(874\) 41361.8 1.60078
\(875\) −10231.1 −0.395285
\(876\) −4378.60 −0.168880
\(877\) 44950.8 1.73076 0.865382 0.501113i \(-0.167076\pi\)
0.865382 + 0.501113i \(0.167076\pi\)
\(878\) −23358.8 −0.897861
\(879\) 33848.8 1.29885
\(880\) −2345.84 −0.0898617
\(881\) 3896.87 0.149023 0.0745113 0.997220i \(-0.476260\pi\)
0.0745113 + 0.997220i \(0.476260\pi\)
\(882\) −1614.70 −0.0616436
\(883\) 6257.41 0.238481 0.119240 0.992865i \(-0.461954\pi\)
0.119240 + 0.992865i \(0.461954\pi\)
\(884\) −72.0466 −0.00274117
\(885\) 9579.38 0.363850
\(886\) −4801.83 −0.182077
\(887\) 40626.2 1.53787 0.768937 0.639324i \(-0.220786\pi\)
0.768937 + 0.639324i \(0.220786\pi\)
\(888\) 25715.7 0.971803
\(889\) 9108.68 0.343639
\(890\) 5114.02 0.192610
\(891\) −9997.16 −0.375889
\(892\) −20665.0 −0.775691
\(893\) −24328.0 −0.911653
\(894\) 14506.9 0.542709
\(895\) 23897.5 0.892521
\(896\) −405.782 −0.0151297
\(897\) −150.697 −0.00560938
\(898\) 15476.2 0.575108
\(899\) −8544.91 −0.317006
\(900\) 3242.80 0.120104
\(901\) −55601.4 −2.05588
\(902\) 4711.73 0.173929
\(903\) −22172.9 −0.817129
\(904\) −53240.5 −1.95880
\(905\) −28020.6 −1.02921
\(906\) −14427.5 −0.529054
\(907\) 42366.9 1.55101 0.775506 0.631340i \(-0.217495\pi\)
0.775506 + 0.631340i \(0.217495\pi\)
\(908\) 4213.85 0.154010
\(909\) −8701.42 −0.317501
\(910\) −20.3094 −0.000739835 0
\(911\) 26231.4 0.953992 0.476996 0.878906i \(-0.341726\pi\)
0.476996 + 0.878906i \(0.341726\pi\)
\(912\) −25757.4 −0.935213
\(913\) −1030.06 −0.0373384
\(914\) −35560.1 −1.28690
\(915\) 28860.5 1.04273
\(916\) 12129.3 0.437514
\(917\) 17655.6 0.635810
\(918\) 21850.1 0.785578
\(919\) 2542.24 0.0912522 0.0456261 0.998959i \(-0.485472\pi\)
0.0456261 + 0.998959i \(0.485472\pi\)
\(920\) −24892.5 −0.892047
\(921\) 18724.0 0.669898
\(922\) 11459.2 0.409314
\(923\) −65.3381 −0.00233004
\(924\) −1602.29 −0.0570471
\(925\) −10849.6 −0.385656
\(926\) 24283.2 0.861764
\(927\) 2005.76 0.0710657
\(928\) 23880.5 0.844739
\(929\) −20097.8 −0.709782 −0.354891 0.934908i \(-0.615482\pi\)
−0.354891 + 0.934908i \(0.615482\pi\)
\(930\) −5213.05 −0.183809
\(931\) 6942.56 0.244396
\(932\) −7452.92 −0.261941
\(933\) −38111.5 −1.33731
\(934\) 17102.0 0.599138
\(935\) 10783.4 0.377170
\(936\) 64.3571 0.00224741
\(937\) −202.788 −0.00707023 −0.00353511 0.999994i \(-0.501125\pi\)
−0.00353511 + 0.999994i \(0.501125\pi\)
\(938\) −2388.80 −0.0831525
\(939\) −49824.9 −1.73160
\(940\) 4191.41 0.145435
\(941\) 12419.1 0.430235 0.215118 0.976588i \(-0.430987\pi\)
0.215118 + 0.976588i \(0.430987\pi\)
\(942\) 22434.8 0.775972
\(943\) 26098.8 0.901265
\(944\) −5443.37 −0.187677
\(945\) −4125.11 −0.142000
\(946\) 11760.7 0.404199
\(947\) 369.282 0.0126717 0.00633583 0.999980i \(-0.497983\pi\)
0.00633583 + 0.999980i \(0.497983\pi\)
\(948\) 145.791 0.00499481
\(949\) 36.6629 0.00125409
\(950\) 20818.5 0.710990
\(951\) −62530.1 −2.13215
\(952\) 22131.3 0.753445
\(953\) 25994.1 0.883558 0.441779 0.897124i \(-0.354348\pi\)
0.441779 + 0.897124i \(0.354348\pi\)
\(954\) 14218.5 0.482536
\(955\) −21775.2 −0.737830
\(956\) −7074.72 −0.239344
\(957\) −12626.4 −0.426492
\(958\) −11597.9 −0.391138
\(959\) 18022.9 0.606871
\(960\) 25632.8 0.861764
\(961\) −27460.5 −0.921770
\(962\) −61.6416 −0.00206591
\(963\) 125.500 0.00419957
\(964\) 14793.8 0.494271
\(965\) 15396.6 0.513610
\(966\) 13252.0 0.441383
\(967\) −53830.0 −1.79013 −0.895065 0.445937i \(-0.852871\pi\)
−0.895065 + 0.445937i \(0.852871\pi\)
\(968\) 2968.70 0.0985720
\(969\) 118402. 3.92530
\(970\) −16085.3 −0.532441
\(971\) −1349.54 −0.0446022 −0.0223011 0.999751i \(-0.507099\pi\)
−0.0223011 + 0.999751i \(0.507099\pi\)
\(972\) 12200.5 0.402605
\(973\) −19190.4 −0.632288
\(974\) −12215.9 −0.401872
\(975\) −75.8497 −0.00249142
\(976\) −16399.6 −0.537847
\(977\) 6611.71 0.216507 0.108253 0.994123i \(-0.465474\pi\)
0.108253 + 0.994123i \(0.465474\pi\)
\(978\) −10906.7 −0.356603
\(979\) −3378.31 −0.110287
\(980\) −1196.12 −0.0389883
\(981\) −4773.13 −0.155346
\(982\) −36595.5 −1.18922
\(983\) 40595.9 1.31720 0.658601 0.752493i \(-0.271149\pi\)
0.658601 + 0.752493i \(0.271149\pi\)
\(984\) −31135.4 −1.00870
\(985\) −36270.5 −1.17327
\(986\) 49926.2 1.61255
\(987\) −7794.51 −0.251370
\(988\) −79.2153 −0.00255078
\(989\) 65143.5 2.09448
\(990\) −2757.54 −0.0885256
\(991\) −48877.7 −1.56675 −0.783376 0.621548i \(-0.786504\pi\)
−0.783376 + 0.621548i \(0.786504\pi\)
\(992\) −6513.19 −0.208462
\(993\) −13423.0 −0.428970
\(994\) 5745.71 0.183343
\(995\) 26981.2 0.859660
\(996\) 1948.59 0.0619913
\(997\) 13127.4 0.417000 0.208500 0.978022i \(-0.433142\pi\)
0.208500 + 0.978022i \(0.433142\pi\)
\(998\) −42889.7 −1.36037
\(999\) −12520.2 −0.396520
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 77.4.a.e.1.2 5
3.2 odd 2 693.4.a.o.1.4 5
4.3 odd 2 1232.4.a.y.1.2 5
5.4 even 2 1925.4.a.r.1.4 5
7.6 odd 2 539.4.a.h.1.2 5
11.10 odd 2 847.4.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.e.1.2 5 1.1 even 1 trivial
539.4.a.h.1.2 5 7.6 odd 2
693.4.a.o.1.4 5 3.2 odd 2
847.4.a.f.1.4 5 11.10 odd 2
1232.4.a.y.1.2 5 4.3 odd 2
1925.4.a.r.1.4 5 5.4 even 2