Properties

Label 338.4.a.k.1.2
Level $338$
Weight $4$
Character 338.1
Self dual yes
Analytic conductor $19.943$
Analytic rank $1$
Dimension $3$
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] = [338,4,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(19.9426455819\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.66487 q^{3} +4.00000 q^{4} +8.53079 q^{5} -7.32975 q^{6} -4.20105 q^{7} +8.00000 q^{8} -13.5687 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.66487 q^{3} +4.00000 q^{4} +8.53079 q^{5} -7.32975 q^{6} -4.20105 q^{7} +8.00000 q^{8} -13.5687 q^{9} +17.0616 q^{10} -65.3726 q^{11} -14.6595 q^{12} -8.40209 q^{14} -31.2643 q^{15} +16.0000 q^{16} -26.9019 q^{17} -27.1374 q^{18} -13.3730 q^{19} +34.1232 q^{20} +15.3963 q^{21} -130.745 q^{22} +159.929 q^{23} -29.3190 q^{24} -52.2255 q^{25} +148.679 q^{27} -16.8042 q^{28} -301.288 q^{29} -62.5286 q^{30} -73.0232 q^{31} +32.0000 q^{32} +239.582 q^{33} -53.8038 q^{34} -35.8383 q^{35} -54.2748 q^{36} -118.781 q^{37} -26.7459 q^{38} +68.2464 q^{40} -432.901 q^{41} +30.7926 q^{42} -356.508 q^{43} -261.490 q^{44} -115.752 q^{45} +319.858 q^{46} +588.614 q^{47} -58.6380 q^{48} -325.351 q^{49} -104.451 q^{50} +98.5921 q^{51} -269.462 q^{53} +297.358 q^{54} -557.680 q^{55} -33.6084 q^{56} +49.0102 q^{57} -602.576 q^{58} +230.340 q^{59} -125.057 q^{60} -380.816 q^{61} -146.046 q^{62} +57.0027 q^{63} +64.0000 q^{64} +479.164 q^{66} +435.848 q^{67} -107.608 q^{68} -586.120 q^{69} -71.6765 q^{70} +65.9622 q^{71} -108.550 q^{72} +885.517 q^{73} -237.562 q^{74} +191.400 q^{75} -53.4918 q^{76} +274.633 q^{77} -385.463 q^{79} +136.493 q^{80} -178.536 q^{81} -865.803 q^{82} -254.207 q^{83} +61.5852 q^{84} -229.495 q^{85} -713.016 q^{86} +1104.18 q^{87} -522.980 q^{88} -372.612 q^{89} -231.504 q^{90} +639.716 q^{92} +267.621 q^{93} +1177.23 q^{94} -114.082 q^{95} -117.276 q^{96} +1313.88 q^{97} -650.702 q^{98} +887.020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} - 12 q^{3} + 12 q^{4} - 12 q^{5} - 24 q^{6} + 27 q^{7} + 24 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{2} - 12 q^{3} + 12 q^{4} - 12 q^{5} - 24 q^{6} + 27 q^{7} + 24 q^{8} + 9 q^{9} - 24 q^{10} - 82 q^{11} - 48 q^{12} + 54 q^{14} + 90 q^{15} + 48 q^{16} - 90 q^{17} + 18 q^{18} - 130 q^{19} - 48 q^{20} - 234 q^{21} - 164 q^{22} + 19 q^{23} - 96 q^{24} - 61 q^{25} - 69 q^{27} + 108 q^{28} - 101 q^{29} + 180 q^{30} - 519 q^{31} + 96 q^{32} + 146 q^{33} - 180 q^{34} - 458 q^{35} + 36 q^{36} - 84 q^{37} - 260 q^{38} - 96 q^{40} - 187 q^{41} - 468 q^{42} - 1205 q^{43} - 328 q^{44} - 645 q^{45} + 38 q^{46} + 536 q^{47} - 192 q^{48} - 184 q^{49} - 122 q^{50} - 207 q^{51} - 1095 q^{53} - 138 q^{54} - 526 q^{55} + 216 q^{56} + 1409 q^{57} - 202 q^{58} + 1413 q^{59} + 360 q^{60} - 1108 q^{61} - 1038 q^{62} + 1404 q^{63} + 192 q^{64} + 292 q^{66} + 1605 q^{67} - 360 q^{68} - 314 q^{69} - 916 q^{70} + 909 q^{71} + 72 q^{72} - 287 q^{73} - 168 q^{74} - 505 q^{75} - 520 q^{76} + 480 q^{77} - 1961 q^{79} - 192 q^{80} + 915 q^{81} - 374 q^{82} - 191 q^{83} - 936 q^{84} - 67 q^{85} - 2410 q^{86} + 1636 q^{87} - 656 q^{88} - 1091 q^{89} - 1290 q^{90} + 76 q^{92} + 1614 q^{93} + 1072 q^{94} + 1829 q^{95} - 384 q^{96} + 947 q^{97} - 368 q^{98} + 2057 q^{99}+O(q^{100}) \) 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
\(3\) −3.66487 −0.705305 −0.352653 0.935754i \(-0.614720\pi\)
−0.352653 + 0.935754i \(0.614720\pi\)
\(4\) 4.00000 0.500000
\(5\) 8.53079 0.763017 0.381509 0.924365i \(-0.375405\pi\)
0.381509 + 0.924365i \(0.375405\pi\)
\(6\) −7.32975 −0.498726
\(7\) −4.20105 −0.226835 −0.113418 0.993547i \(-0.536180\pi\)
−0.113418 + 0.993547i \(0.536180\pi\)
\(8\) 8.00000 0.353553
\(9\) −13.5687 −0.502544
\(10\) 17.0616 0.539535
\(11\) −65.3726 −1.79187 −0.895935 0.444186i \(-0.853493\pi\)
−0.895935 + 0.444186i \(0.853493\pi\)
\(12\) −14.6595 −0.352653
\(13\) 0 0
\(14\) −8.40209 −0.160397
\(15\) −31.2643 −0.538160
\(16\) 16.0000 0.250000
\(17\) −26.9019 −0.383804 −0.191902 0.981414i \(-0.561466\pi\)
−0.191902 + 0.981414i \(0.561466\pi\)
\(18\) −27.1374 −0.355352
\(19\) −13.3730 −0.161472 −0.0807360 0.996736i \(-0.525727\pi\)
−0.0807360 + 0.996736i \(0.525727\pi\)
\(20\) 34.1232 0.381509
\(21\) 15.3963 0.159988
\(22\) −130.745 −1.26704
\(23\) 159.929 1.44989 0.724946 0.688806i \(-0.241865\pi\)
0.724946 + 0.688806i \(0.241865\pi\)
\(24\) −29.3190 −0.249363
\(25\) −52.2255 −0.417804
\(26\) 0 0
\(27\) 148.679 1.05975
\(28\) −16.8042 −0.113418
\(29\) −301.288 −1.92923 −0.964616 0.263658i \(-0.915071\pi\)
−0.964616 + 0.263658i \(0.915071\pi\)
\(30\) −62.5286 −0.380537
\(31\) −73.0232 −0.423076 −0.211538 0.977370i \(-0.567847\pi\)
−0.211538 + 0.977370i \(0.567847\pi\)
\(32\) 32.0000 0.176777
\(33\) 239.582 1.26382
\(34\) −53.8038 −0.271390
\(35\) −35.8383 −0.173079
\(36\) −54.2748 −0.251272
\(37\) −118.781 −0.527769 −0.263885 0.964554i \(-0.585004\pi\)
−0.263885 + 0.964554i \(0.585004\pi\)
\(38\) −26.7459 −0.114178
\(39\) 0 0
\(40\) 68.2464 0.269767
\(41\) −432.901 −1.64897 −0.824486 0.565883i \(-0.808535\pi\)
−0.824486 + 0.565883i \(0.808535\pi\)
\(42\) 30.7926 0.113129
\(43\) −356.508 −1.26435 −0.632174 0.774827i \(-0.717837\pi\)
−0.632174 + 0.774827i \(0.717837\pi\)
\(44\) −261.490 −0.895935
\(45\) −115.752 −0.383450
\(46\) 319.858 1.02523
\(47\) 588.614 1.82677 0.913385 0.407096i \(-0.133459\pi\)
0.913385 + 0.407096i \(0.133459\pi\)
\(48\) −58.6380 −0.176326
\(49\) −325.351 −0.948546
\(50\) −104.451 −0.295432
\(51\) 98.5921 0.270699
\(52\) 0 0
\(53\) −269.462 −0.698366 −0.349183 0.937054i \(-0.613541\pi\)
−0.349183 + 0.937054i \(0.613541\pi\)
\(54\) 297.358 0.749358
\(55\) −557.680 −1.36723
\(56\) −33.6084 −0.0801983
\(57\) 49.0102 0.113887
\(58\) −602.576 −1.36417
\(59\) 230.340 0.508265 0.254133 0.967169i \(-0.418210\pi\)
0.254133 + 0.967169i \(0.418210\pi\)
\(60\) −125.057 −0.269080
\(61\) −380.816 −0.799319 −0.399659 0.916664i \(-0.630872\pi\)
−0.399659 + 0.916664i \(0.630872\pi\)
\(62\) −146.046 −0.299160
\(63\) 57.0027 0.113995
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 479.164 0.893652
\(67\) 435.848 0.794736 0.397368 0.917659i \(-0.369924\pi\)
0.397368 + 0.917659i \(0.369924\pi\)
\(68\) −107.608 −0.191902
\(69\) −586.120 −1.02262
\(70\) −71.6765 −0.122385
\(71\) 65.9622 0.110257 0.0551287 0.998479i \(-0.482443\pi\)
0.0551287 + 0.998479i \(0.482443\pi\)
\(72\) −108.550 −0.177676
\(73\) 885.517 1.41975 0.709876 0.704326i \(-0.248751\pi\)
0.709876 + 0.704326i \(0.248751\pi\)
\(74\) −237.562 −0.373189
\(75\) 191.400 0.294680
\(76\) −53.4918 −0.0807360
\(77\) 274.633 0.406459
\(78\) 0 0
\(79\) −385.463 −0.548962 −0.274481 0.961592i \(-0.588506\pi\)
−0.274481 + 0.961592i \(0.588506\pi\)
\(80\) 136.493 0.190754
\(81\) −178.536 −0.244905
\(82\) −865.803 −1.16600
\(83\) −254.207 −0.336179 −0.168089 0.985772i \(-0.553760\pi\)
−0.168089 + 0.985772i \(0.553760\pi\)
\(84\) 61.5852 0.0799940
\(85\) −229.495 −0.292849
\(86\) −713.016 −0.894029
\(87\) 1104.18 1.36070
\(88\) −522.980 −0.633522
\(89\) −372.612 −0.443785 −0.221892 0.975071i \(-0.571223\pi\)
−0.221892 + 0.975071i \(0.571223\pi\)
\(90\) −231.504 −0.271140
\(91\) 0 0
\(92\) 639.716 0.724946
\(93\) 267.621 0.298398
\(94\) 1177.23 1.29172
\(95\) −114.082 −0.123206
\(96\) −117.276 −0.124682
\(97\) 1313.88 1.37531 0.687653 0.726039i \(-0.258641\pi\)
0.687653 + 0.726039i \(0.258641\pi\)
\(98\) −650.702 −0.670723
\(99\) 887.020 0.900494
\(100\) −208.902 −0.208902
\(101\) 1463.72 1.44204 0.721019 0.692915i \(-0.243674\pi\)
0.721019 + 0.692915i \(0.243674\pi\)
\(102\) 197.184 0.191413
\(103\) 210.886 0.201740 0.100870 0.994900i \(-0.467837\pi\)
0.100870 + 0.994900i \(0.467837\pi\)
\(104\) 0 0
\(105\) 131.343 0.122074
\(106\) −538.924 −0.493820
\(107\) 391.061 0.353321 0.176660 0.984272i \(-0.443471\pi\)
0.176660 + 0.984272i \(0.443471\pi\)
\(108\) 594.717 0.529876
\(109\) 1331.40 1.16996 0.584978 0.811049i \(-0.301103\pi\)
0.584978 + 0.811049i \(0.301103\pi\)
\(110\) −1115.36 −0.966776
\(111\) 435.317 0.372238
\(112\) −67.2167 −0.0567088
\(113\) −711.912 −0.592665 −0.296332 0.955085i \(-0.595764\pi\)
−0.296332 + 0.955085i \(0.595764\pi\)
\(114\) 98.0204 0.0805303
\(115\) 1364.32 1.10629
\(116\) −1205.15 −0.964616
\(117\) 0 0
\(118\) 460.679 0.359398
\(119\) 113.016 0.0870603
\(120\) −250.114 −0.190268
\(121\) 2942.57 2.21080
\(122\) −761.631 −0.565204
\(123\) 1586.53 1.16303
\(124\) −292.093 −0.211538
\(125\) −1511.87 −1.08181
\(126\) 114.005 0.0806064
\(127\) −1171.70 −0.818677 −0.409338 0.912383i \(-0.634240\pi\)
−0.409338 + 0.912383i \(0.634240\pi\)
\(128\) 128.000 0.0883883
\(129\) 1306.56 0.891751
\(130\) 0 0
\(131\) −1128.61 −0.752726 −0.376363 0.926472i \(-0.622825\pi\)
−0.376363 + 0.926472i \(0.622825\pi\)
\(132\) 958.329 0.631908
\(133\) 56.1804 0.0366275
\(134\) 871.696 0.561963
\(135\) 1268.35 0.808610
\(136\) −215.215 −0.135695
\(137\) −1747.93 −1.09004 −0.545022 0.838422i \(-0.683479\pi\)
−0.545022 + 0.838422i \(0.683479\pi\)
\(138\) −1172.24 −0.723099
\(139\) 1676.54 1.02304 0.511518 0.859273i \(-0.329083\pi\)
0.511518 + 0.859273i \(0.329083\pi\)
\(140\) −143.353 −0.0865396
\(141\) −2157.20 −1.28843
\(142\) 131.924 0.0779637
\(143\) 0 0
\(144\) −217.099 −0.125636
\(145\) −2570.22 −1.47204
\(146\) 1771.03 1.00392
\(147\) 1192.37 0.669014
\(148\) −475.124 −0.263885
\(149\) −896.961 −0.493167 −0.246584 0.969122i \(-0.579308\pi\)
−0.246584 + 0.969122i \(0.579308\pi\)
\(150\) 382.800 0.208370
\(151\) −2078.28 −1.12005 −0.560027 0.828475i \(-0.689209\pi\)
−0.560027 + 0.828475i \(0.689209\pi\)
\(152\) −106.984 −0.0570890
\(153\) 365.024 0.192879
\(154\) 549.266 0.287410
\(155\) −622.946 −0.322814
\(156\) 0 0
\(157\) −3494.75 −1.77650 −0.888252 0.459357i \(-0.848080\pi\)
−0.888252 + 0.459357i \(0.848080\pi\)
\(158\) −770.927 −0.388175
\(159\) 987.544 0.492562
\(160\) 272.985 0.134884
\(161\) −671.869 −0.328887
\(162\) −357.071 −0.173174
\(163\) −451.254 −0.216840 −0.108420 0.994105i \(-0.534579\pi\)
−0.108420 + 0.994105i \(0.534579\pi\)
\(164\) −1731.61 −0.824486
\(165\) 2043.83 0.964313
\(166\) −508.414 −0.237714
\(167\) 3203.35 1.48433 0.742164 0.670218i \(-0.233800\pi\)
0.742164 + 0.670218i \(0.233800\pi\)
\(168\) 123.170 0.0565643
\(169\) 0 0
\(170\) −458.989 −0.207076
\(171\) 181.454 0.0811468
\(172\) −1426.03 −0.632174
\(173\) 1899.38 0.834725 0.417362 0.908740i \(-0.362955\pi\)
0.417362 + 0.908740i \(0.362955\pi\)
\(174\) 2208.36 0.962159
\(175\) 219.402 0.0947727
\(176\) −1045.96 −0.447967
\(177\) −844.165 −0.358482
\(178\) −745.225 −0.313803
\(179\) 2650.89 1.10691 0.553455 0.832879i \(-0.313309\pi\)
0.553455 + 0.832879i \(0.313309\pi\)
\(180\) −463.007 −0.191725
\(181\) −2289.94 −0.940387 −0.470194 0.882563i \(-0.655816\pi\)
−0.470194 + 0.882563i \(0.655816\pi\)
\(182\) 0 0
\(183\) 1395.64 0.563764
\(184\) 1279.43 0.512614
\(185\) −1013.30 −0.402697
\(186\) 535.242 0.210999
\(187\) 1758.65 0.687727
\(188\) 2354.46 0.913385
\(189\) −624.608 −0.240389
\(190\) −228.164 −0.0871198
\(191\) −338.313 −0.128165 −0.0640825 0.997945i \(-0.520412\pi\)
−0.0640825 + 0.997945i \(0.520412\pi\)
\(192\) −234.552 −0.0881632
\(193\) 2684.05 1.00105 0.500524 0.865723i \(-0.333141\pi\)
0.500524 + 0.865723i \(0.333141\pi\)
\(194\) 2627.77 0.972489
\(195\) 0 0
\(196\) −1301.40 −0.474273
\(197\) 1897.24 0.686156 0.343078 0.939307i \(-0.388530\pi\)
0.343078 + 0.939307i \(0.388530\pi\)
\(198\) 1774.04 0.636745
\(199\) −2415.60 −0.860491 −0.430245 0.902712i \(-0.641573\pi\)
−0.430245 + 0.902712i \(0.641573\pi\)
\(200\) −417.804 −0.147716
\(201\) −1597.33 −0.560532
\(202\) 2927.44 1.01967
\(203\) 1265.72 0.437618
\(204\) 394.368 0.135350
\(205\) −3692.99 −1.25819
\(206\) 421.772 0.142652
\(207\) −2170.03 −0.728635
\(208\) 0 0
\(209\) 874.225 0.289337
\(210\) 262.685 0.0863191
\(211\) 2668.42 0.870623 0.435312 0.900280i \(-0.356638\pi\)
0.435312 + 0.900280i \(0.356638\pi\)
\(212\) −1077.85 −0.349183
\(213\) −241.743 −0.0777651
\(214\) 782.122 0.249835
\(215\) −3041.30 −0.964719
\(216\) 1189.43 0.374679
\(217\) 306.774 0.0959685
\(218\) 2662.81 0.827284
\(219\) −3245.31 −1.00136
\(220\) −2230.72 −0.683614
\(221\) 0 0
\(222\) 870.634 0.263212
\(223\) −286.358 −0.0859907 −0.0429953 0.999075i \(-0.513690\pi\)
−0.0429953 + 0.999075i \(0.513690\pi\)
\(224\) −134.433 −0.0400992
\(225\) 708.632 0.209965
\(226\) −1423.82 −0.419077
\(227\) 5201.24 1.52079 0.760393 0.649463i \(-0.225006\pi\)
0.760393 + 0.649463i \(0.225006\pi\)
\(228\) 196.041 0.0569435
\(229\) 890.458 0.256957 0.128478 0.991712i \(-0.458991\pi\)
0.128478 + 0.991712i \(0.458991\pi\)
\(230\) 2728.64 0.782267
\(231\) −1006.50 −0.286678
\(232\) −2410.30 −0.682087
\(233\) 4753.11 1.33642 0.668212 0.743971i \(-0.267060\pi\)
0.668212 + 0.743971i \(0.267060\pi\)
\(234\) 0 0
\(235\) 5021.35 1.39386
\(236\) 921.358 0.254133
\(237\) 1412.67 0.387186
\(238\) 226.032 0.0615609
\(239\) −2292.62 −0.620491 −0.310245 0.950656i \(-0.600411\pi\)
−0.310245 + 0.950656i \(0.600411\pi\)
\(240\) −500.229 −0.134540
\(241\) 1975.21 0.527943 0.263972 0.964530i \(-0.414967\pi\)
0.263972 + 0.964530i \(0.414967\pi\)
\(242\) 5885.14 1.56327
\(243\) −3360.03 −0.887020
\(244\) −1523.26 −0.399659
\(245\) −2775.50 −0.723757
\(246\) 3173.06 0.822385
\(247\) 0 0
\(248\) −584.186 −0.149580
\(249\) 931.637 0.237109
\(250\) −3023.75 −0.764955
\(251\) −7465.74 −1.87742 −0.938711 0.344705i \(-0.887979\pi\)
−0.938711 + 0.344705i \(0.887979\pi\)
\(252\) 228.011 0.0569974
\(253\) −10455.0 −2.59802
\(254\) −2343.41 −0.578892
\(255\) 841.069 0.206548
\(256\) 256.000 0.0625000
\(257\) −554.966 −0.134700 −0.0673499 0.997729i \(-0.521454\pi\)
−0.0673499 + 0.997729i \(0.521454\pi\)
\(258\) 2613.11 0.630563
\(259\) 499.004 0.119717
\(260\) 0 0
\(261\) 4088.08 0.969525
\(262\) −2257.22 −0.532258
\(263\) −1993.07 −0.467292 −0.233646 0.972322i \(-0.575066\pi\)
−0.233646 + 0.972322i \(0.575066\pi\)
\(264\) 1916.66 0.446826
\(265\) −2298.72 −0.532866
\(266\) 112.361 0.0258996
\(267\) 1365.58 0.313004
\(268\) 1743.39 0.397368
\(269\) −5207.76 −1.18038 −0.590191 0.807263i \(-0.700948\pi\)
−0.590191 + 0.807263i \(0.700948\pi\)
\(270\) 2536.70 0.571773
\(271\) 4084.90 0.915646 0.457823 0.889043i \(-0.348629\pi\)
0.457823 + 0.889043i \(0.348629\pi\)
\(272\) −430.430 −0.0959510
\(273\) 0 0
\(274\) −3495.87 −0.770778
\(275\) 3414.12 0.748651
\(276\) −2344.48 −0.511308
\(277\) 437.993 0.0950051 0.0475026 0.998871i \(-0.484874\pi\)
0.0475026 + 0.998871i \(0.484874\pi\)
\(278\) 3353.07 0.723395
\(279\) 990.829 0.212614
\(280\) −286.706 −0.0611927
\(281\) 5462.34 1.15963 0.579815 0.814748i \(-0.303125\pi\)
0.579815 + 0.814748i \(0.303125\pi\)
\(282\) −4314.40 −0.911058
\(283\) −796.676 −0.167341 −0.0836704 0.996493i \(-0.526664\pi\)
−0.0836704 + 0.996493i \(0.526664\pi\)
\(284\) 263.849 0.0551287
\(285\) 418.096 0.0868978
\(286\) 0 0
\(287\) 1818.64 0.374045
\(288\) −434.198 −0.0888381
\(289\) −4189.29 −0.852694
\(290\) −5140.45 −1.04089
\(291\) −4815.22 −0.970011
\(292\) 3542.07 0.709876
\(293\) 1697.32 0.338426 0.169213 0.985580i \(-0.445877\pi\)
0.169213 + 0.985580i \(0.445877\pi\)
\(294\) 2384.74 0.473065
\(295\) 1964.98 0.387815
\(296\) −950.247 −0.186595
\(297\) −9719.54 −1.89894
\(298\) −1793.92 −0.348722
\(299\) 0 0
\(300\) 765.600 0.147340
\(301\) 1497.71 0.286798
\(302\) −4156.56 −0.791997
\(303\) −5364.36 −1.01708
\(304\) −213.967 −0.0403680
\(305\) −3248.66 −0.609894
\(306\) 730.047 0.136386
\(307\) −9385.86 −1.74488 −0.872442 0.488718i \(-0.837465\pi\)
−0.872442 + 0.488718i \(0.837465\pi\)
\(308\) 1098.53 0.203230
\(309\) −772.870 −0.142288
\(310\) −1245.89 −0.228264
\(311\) −3282.65 −0.598527 −0.299264 0.954170i \(-0.596741\pi\)
−0.299264 + 0.954170i \(0.596741\pi\)
\(312\) 0 0
\(313\) −5924.67 −1.06991 −0.534955 0.844880i \(-0.679672\pi\)
−0.534955 + 0.844880i \(0.679672\pi\)
\(314\) −6989.49 −1.25618
\(315\) 486.278 0.0869800
\(316\) −1541.85 −0.274481
\(317\) −6467.88 −1.14597 −0.572985 0.819566i \(-0.694215\pi\)
−0.572985 + 0.819566i \(0.694215\pi\)
\(318\) 1975.09 0.348294
\(319\) 19696.0 3.45693
\(320\) 545.971 0.0953772
\(321\) −1433.19 −0.249199
\(322\) −1343.74 −0.232558
\(323\) 359.758 0.0619736
\(324\) −714.143 −0.122452
\(325\) 0 0
\(326\) −902.508 −0.153329
\(327\) −4879.42 −0.825177
\(328\) −3463.21 −0.582999
\(329\) −2472.80 −0.414376
\(330\) 4087.65 0.681872
\(331\) −3509.24 −0.582735 −0.291368 0.956611i \(-0.594110\pi\)
−0.291368 + 0.956611i \(0.594110\pi\)
\(332\) −1016.83 −0.168089
\(333\) 1611.70 0.265227
\(334\) 6406.70 1.04958
\(335\) 3718.13 0.606398
\(336\) 246.341 0.0399970
\(337\) −1834.82 −0.296584 −0.148292 0.988944i \(-0.547378\pi\)
−0.148292 + 0.988944i \(0.547378\pi\)
\(338\) 0 0
\(339\) 2609.07 0.418010
\(340\) −917.978 −0.146425
\(341\) 4773.71 0.758097
\(342\) 362.907 0.0573795
\(343\) 2807.77 0.441999
\(344\) −2852.06 −0.447014
\(345\) −5000.07 −0.780274
\(346\) 3798.76 0.590239
\(347\) −7259.30 −1.12305 −0.561527 0.827459i \(-0.689786\pi\)
−0.561527 + 0.827459i \(0.689786\pi\)
\(348\) 4416.73 0.680349
\(349\) 1795.24 0.275349 0.137674 0.990478i \(-0.456037\pi\)
0.137674 + 0.990478i \(0.456037\pi\)
\(350\) 438.804 0.0670144
\(351\) 0 0
\(352\) −2091.92 −0.316761
\(353\) 4552.38 0.686398 0.343199 0.939263i \(-0.388489\pi\)
0.343199 + 0.939263i \(0.388489\pi\)
\(354\) −1688.33 −0.253485
\(355\) 562.710 0.0841283
\(356\) −1490.45 −0.221892
\(357\) −414.190 −0.0614041
\(358\) 5301.78 0.782703
\(359\) −10165.4 −1.49445 −0.747227 0.664569i \(-0.768615\pi\)
−0.747227 + 0.664569i \(0.768615\pi\)
\(360\) −926.014 −0.135570
\(361\) −6680.16 −0.973927
\(362\) −4579.88 −0.664954
\(363\) −10784.2 −1.55929
\(364\) 0 0
\(365\) 7554.16 1.08330
\(366\) 2791.28 0.398641
\(367\) −10401.6 −1.47945 −0.739723 0.672911i \(-0.765044\pi\)
−0.739723 + 0.672911i \(0.765044\pi\)
\(368\) 2558.87 0.362473
\(369\) 5873.91 0.828681
\(370\) −2026.59 −0.284750
\(371\) 1132.02 0.158414
\(372\) 1070.48 0.149199
\(373\) −7953.41 −1.10405 −0.552027 0.833826i \(-0.686145\pi\)
−0.552027 + 0.833826i \(0.686145\pi\)
\(374\) 3517.29 0.486296
\(375\) 5540.83 0.763006
\(376\) 4708.91 0.645861
\(377\) 0 0
\(378\) −1249.22 −0.169981
\(379\) −9014.81 −1.22179 −0.610897 0.791710i \(-0.709191\pi\)
−0.610897 + 0.791710i \(0.709191\pi\)
\(380\) −456.328 −0.0616030
\(381\) 4294.15 0.577417
\(382\) −676.627 −0.0906263
\(383\) 2496.24 0.333034 0.166517 0.986039i \(-0.446748\pi\)
0.166517 + 0.986039i \(0.446748\pi\)
\(384\) −469.104 −0.0623408
\(385\) 2342.84 0.310135
\(386\) 5368.10 0.707848
\(387\) 4837.35 0.635391
\(388\) 5255.54 0.687653
\(389\) 10896.9 1.42029 0.710145 0.704056i \(-0.248629\pi\)
0.710145 + 0.704056i \(0.248629\pi\)
\(390\) 0 0
\(391\) −4302.40 −0.556475
\(392\) −2602.81 −0.335362
\(393\) 4136.21 0.530902
\(394\) 3794.48 0.485186
\(395\) −3288.31 −0.418868
\(396\) 3548.08 0.450247
\(397\) −10887.1 −1.37635 −0.688174 0.725546i \(-0.741587\pi\)
−0.688174 + 0.725546i \(0.741587\pi\)
\(398\) −4831.21 −0.608459
\(399\) −205.894 −0.0258336
\(400\) −835.609 −0.104451
\(401\) −9968.76 −1.24144 −0.620718 0.784034i \(-0.713159\pi\)
−0.620718 + 0.784034i \(0.713159\pi\)
\(402\) −3194.66 −0.396356
\(403\) 0 0
\(404\) 5854.89 0.721019
\(405\) −1523.05 −0.186867
\(406\) 2531.45 0.309442
\(407\) 7765.01 0.945693
\(408\) 788.737 0.0957066
\(409\) −4036.32 −0.487979 −0.243989 0.969778i \(-0.578456\pi\)
−0.243989 + 0.969778i \(0.578456\pi\)
\(410\) −7385.98 −0.889677
\(411\) 6405.96 0.768814
\(412\) 843.543 0.100870
\(413\) −967.667 −0.115292
\(414\) −4340.06 −0.515223
\(415\) −2168.59 −0.256510
\(416\) 0 0
\(417\) −6144.29 −0.721552
\(418\) 1748.45 0.204592
\(419\) 15194.9 1.77164 0.885821 0.464028i \(-0.153596\pi\)
0.885821 + 0.464028i \(0.153596\pi\)
\(420\) 525.371 0.0610368
\(421\) −10154.8 −1.17556 −0.587782 0.809019i \(-0.699999\pi\)
−0.587782 + 0.809019i \(0.699999\pi\)
\(422\) 5336.84 0.615624
\(423\) −7986.73 −0.918033
\(424\) −2155.69 −0.246910
\(425\) 1404.97 0.160355
\(426\) −483.486 −0.0549882
\(427\) 1599.82 0.181314
\(428\) 1564.24 0.176660
\(429\) 0 0
\(430\) −6082.59 −0.682159
\(431\) 6616.84 0.739494 0.369747 0.929133i \(-0.379444\pi\)
0.369747 + 0.929133i \(0.379444\pi\)
\(432\) 2378.87 0.264938
\(433\) 5697.60 0.632354 0.316177 0.948700i \(-0.397601\pi\)
0.316177 + 0.948700i \(0.397601\pi\)
\(434\) 613.548 0.0678600
\(435\) 9419.55 1.03824
\(436\) 5325.61 0.584978
\(437\) −2138.73 −0.234117
\(438\) −6490.62 −0.708068
\(439\) 436.007 0.0474020 0.0237010 0.999719i \(-0.492455\pi\)
0.0237010 + 0.999719i \(0.492455\pi\)
\(440\) −4461.44 −0.483388
\(441\) 4414.59 0.476686
\(442\) 0 0
\(443\) 6609.58 0.708873 0.354436 0.935080i \(-0.384673\pi\)
0.354436 + 0.935080i \(0.384673\pi\)
\(444\) 1741.27 0.186119
\(445\) −3178.68 −0.338616
\(446\) −572.715 −0.0608046
\(447\) 3287.25 0.347833
\(448\) −268.867 −0.0283544
\(449\) −8015.31 −0.842463 −0.421231 0.906953i \(-0.638402\pi\)
−0.421231 + 0.906953i \(0.638402\pi\)
\(450\) 1417.26 0.148468
\(451\) 28299.9 2.95474
\(452\) −2847.65 −0.296332
\(453\) 7616.64 0.789980
\(454\) 10402.5 1.07536
\(455\) 0 0
\(456\) 392.082 0.0402652
\(457\) 4507.03 0.461335 0.230668 0.973033i \(-0.425909\pi\)
0.230668 + 0.973033i \(0.425909\pi\)
\(458\) 1780.92 0.181696
\(459\) −3999.75 −0.406737
\(460\) 5457.29 0.553146
\(461\) −2103.55 −0.212521 −0.106260 0.994338i \(-0.533888\pi\)
−0.106260 + 0.994338i \(0.533888\pi\)
\(462\) −2012.99 −0.202712
\(463\) −5468.28 −0.548883 −0.274441 0.961604i \(-0.588493\pi\)
−0.274441 + 0.961604i \(0.588493\pi\)
\(464\) −4820.60 −0.482308
\(465\) 2283.02 0.227683
\(466\) 9506.22 0.944994
\(467\) −6043.79 −0.598872 −0.299436 0.954116i \(-0.596799\pi\)
−0.299436 + 0.954116i \(0.596799\pi\)
\(468\) 0 0
\(469\) −1831.02 −0.180274
\(470\) 10042.7 0.985606
\(471\) 12807.8 1.25298
\(472\) 1842.72 0.179699
\(473\) 23305.8 2.26555
\(474\) 2825.35 0.273782
\(475\) 698.410 0.0674637
\(476\) 452.065 0.0435301
\(477\) 3656.25 0.350960
\(478\) −4585.24 −0.438753
\(479\) 8481.30 0.809020 0.404510 0.914534i \(-0.367442\pi\)
0.404510 + 0.914534i \(0.367442\pi\)
\(480\) −1000.46 −0.0951342
\(481\) 0 0
\(482\) 3950.42 0.373312
\(483\) 2462.32 0.231965
\(484\) 11770.3 1.10540
\(485\) 11208.5 1.04938
\(486\) −6720.05 −0.627218
\(487\) 6623.36 0.616289 0.308145 0.951340i \(-0.400292\pi\)
0.308145 + 0.951340i \(0.400292\pi\)
\(488\) −3046.53 −0.282602
\(489\) 1653.79 0.152939
\(490\) −5551.01 −0.511774
\(491\) 16379.4 1.50548 0.752739 0.658319i \(-0.228732\pi\)
0.752739 + 0.658319i \(0.228732\pi\)
\(492\) 6346.12 0.581514
\(493\) 8105.21 0.740447
\(494\) 0 0
\(495\) 7566.99 0.687093
\(496\) −1168.37 −0.105769
\(497\) −277.110 −0.0250102
\(498\) 1863.27 0.167661
\(499\) 7915.70 0.710131 0.355065 0.934841i \(-0.384459\pi\)
0.355065 + 0.934841i \(0.384459\pi\)
\(500\) −6047.50 −0.540905
\(501\) −11739.9 −1.04690
\(502\) −14931.5 −1.32754
\(503\) −479.511 −0.0425056 −0.0212528 0.999774i \(-0.506765\pi\)
−0.0212528 + 0.999774i \(0.506765\pi\)
\(504\) 456.022 0.0403032
\(505\) 12486.7 1.10030
\(506\) −20909.9 −1.83708
\(507\) 0 0
\(508\) −4686.82 −0.409338
\(509\) −1192.33 −0.103829 −0.0519145 0.998652i \(-0.516532\pi\)
−0.0519145 + 0.998652i \(0.516532\pi\)
\(510\) 1682.14 0.146052
\(511\) −3720.10 −0.322050
\(512\) 512.000 0.0441942
\(513\) −1988.28 −0.171120
\(514\) −1109.93 −0.0952471
\(515\) 1799.02 0.153931
\(516\) 5226.23 0.445876
\(517\) −38479.2 −3.27333
\(518\) 998.008 0.0846524
\(519\) −6961.00 −0.588736
\(520\) 0 0
\(521\) −23238.5 −1.95412 −0.977062 0.212955i \(-0.931691\pi\)
−0.977062 + 0.212955i \(0.931691\pi\)
\(522\) 8176.16 0.685557
\(523\) −11026.3 −0.921884 −0.460942 0.887430i \(-0.652488\pi\)
−0.460942 + 0.887430i \(0.652488\pi\)
\(524\) −4514.44 −0.376363
\(525\) −804.080 −0.0668437
\(526\) −3986.14 −0.330425
\(527\) 1964.46 0.162378
\(528\) 3833.32 0.315954
\(529\) 13410.3 1.10219
\(530\) −4597.45 −0.376793
\(531\) −3125.41 −0.255426
\(532\) 224.722 0.0183138
\(533\) 0 0
\(534\) 2731.16 0.221327
\(535\) 3336.06 0.269590
\(536\) 3486.79 0.280982
\(537\) −9715.18 −0.780709
\(538\) −10415.5 −0.834657
\(539\) 21269.0 1.69967
\(540\) 5073.41 0.404305
\(541\) −8987.70 −0.714254 −0.357127 0.934056i \(-0.616244\pi\)
−0.357127 + 0.934056i \(0.616244\pi\)
\(542\) 8169.80 0.647459
\(543\) 8392.35 0.663260
\(544\) −860.861 −0.0678476
\(545\) 11357.9 0.892697
\(546\) 0 0
\(547\) −10734.8 −0.839101 −0.419550 0.907732i \(-0.637812\pi\)
−0.419550 + 0.907732i \(0.637812\pi\)
\(548\) −6991.74 −0.545022
\(549\) 5167.17 0.401693
\(550\) 6828.23 0.529376
\(551\) 4029.11 0.311517
\(552\) −4688.96 −0.361550
\(553\) 1619.35 0.124524
\(554\) 875.985 0.0671788
\(555\) 3713.60 0.284024
\(556\) 6706.14 0.511518
\(557\) −13228.3 −1.00628 −0.503141 0.864204i \(-0.667822\pi\)
−0.503141 + 0.864204i \(0.667822\pi\)
\(558\) 1981.66 0.150341
\(559\) 0 0
\(560\) −573.412 −0.0432698
\(561\) −6445.22 −0.485058
\(562\) 10924.7 0.819983
\(563\) 10734.0 0.803522 0.401761 0.915745i \(-0.368398\pi\)
0.401761 + 0.915745i \(0.368398\pi\)
\(564\) −8628.79 −0.644216
\(565\) −6073.18 −0.452213
\(566\) −1593.35 −0.118328
\(567\) 750.037 0.0555531
\(568\) 527.698 0.0389819
\(569\) 5313.45 0.391479 0.195739 0.980656i \(-0.437289\pi\)
0.195739 + 0.980656i \(0.437289\pi\)
\(570\) 836.192 0.0614460
\(571\) 4629.37 0.339287 0.169644 0.985505i \(-0.445738\pi\)
0.169644 + 0.985505i \(0.445738\pi\)
\(572\) 0 0
\(573\) 1239.88 0.0903954
\(574\) 3637.28 0.264490
\(575\) −8352.38 −0.605771
\(576\) −868.397 −0.0628180
\(577\) 504.750 0.0364177 0.0182088 0.999834i \(-0.494204\pi\)
0.0182088 + 0.999834i \(0.494204\pi\)
\(578\) −8378.58 −0.602946
\(579\) −9836.71 −0.706045
\(580\) −10280.9 −0.736019
\(581\) 1067.94 0.0762572
\(582\) −9630.44 −0.685901
\(583\) 17615.4 1.25138
\(584\) 7084.14 0.501958
\(585\) 0 0
\(586\) 3394.65 0.239303
\(587\) −5532.93 −0.389044 −0.194522 0.980898i \(-0.562316\pi\)
−0.194522 + 0.980898i \(0.562316\pi\)
\(588\) 4769.49 0.334507
\(589\) 976.536 0.0683149
\(590\) 3929.96 0.274227
\(591\) −6953.15 −0.483950
\(592\) −1900.49 −0.131942
\(593\) 18079.8 1.25202 0.626009 0.779816i \(-0.284687\pi\)
0.626009 + 0.779816i \(0.284687\pi\)
\(594\) −19439.1 −1.34275
\(595\) 964.117 0.0664285
\(596\) −3587.84 −0.246584
\(597\) 8852.88 0.606909
\(598\) 0 0
\(599\) −1837.55 −0.125342 −0.0626712 0.998034i \(-0.519962\pi\)
−0.0626712 + 0.998034i \(0.519962\pi\)
\(600\) 1531.20 0.104185
\(601\) −23487.3 −1.59412 −0.797060 0.603901i \(-0.793612\pi\)
−0.797060 + 0.603901i \(0.793612\pi\)
\(602\) 2995.41 0.202797
\(603\) −5913.89 −0.399390
\(604\) −8313.12 −0.560027
\(605\) 25102.5 1.68688
\(606\) −10728.7 −0.719182
\(607\) −16310.8 −1.09067 −0.545335 0.838218i \(-0.683598\pi\)
−0.545335 + 0.838218i \(0.683598\pi\)
\(608\) −427.935 −0.0285445
\(609\) −4638.72 −0.308654
\(610\) −6497.32 −0.431260
\(611\) 0 0
\(612\) 1460.09 0.0964393
\(613\) −8453.11 −0.556962 −0.278481 0.960442i \(-0.589831\pi\)
−0.278481 + 0.960442i \(0.589831\pi\)
\(614\) −18771.7 −1.23382
\(615\) 13534.4 0.887411
\(616\) 2197.06 0.143705
\(617\) 13410.3 0.875008 0.437504 0.899216i \(-0.355863\pi\)
0.437504 + 0.899216i \(0.355863\pi\)
\(618\) −1545.74 −0.100613
\(619\) 890.135 0.0577990 0.0288995 0.999582i \(-0.490800\pi\)
0.0288995 + 0.999582i \(0.490800\pi\)
\(620\) −2491.78 −0.161407
\(621\) 23778.1 1.53653
\(622\) −6565.30 −0.423223
\(623\) 1565.36 0.100666
\(624\) 0 0
\(625\) −6369.30 −0.407635
\(626\) −11849.3 −0.756541
\(627\) −3203.92 −0.204071
\(628\) −13979.0 −0.888252
\(629\) 3195.43 0.202560
\(630\) 972.557 0.0615041
\(631\) 10582.4 0.667639 0.333819 0.942637i \(-0.391662\pi\)
0.333819 + 0.942637i \(0.391662\pi\)
\(632\) −3083.71 −0.194087
\(633\) −9779.42 −0.614055
\(634\) −12935.8 −0.810323
\(635\) −9995.57 −0.624665
\(636\) 3950.17 0.246281
\(637\) 0 0
\(638\) 39391.9 2.44442
\(639\) −895.021 −0.0554092
\(640\) 1091.94 0.0674419
\(641\) 26884.2 1.65657 0.828286 0.560305i \(-0.189316\pi\)
0.828286 + 0.560305i \(0.189316\pi\)
\(642\) −2866.38 −0.176210
\(643\) −5691.12 −0.349045 −0.174523 0.984653i \(-0.555838\pi\)
−0.174523 + 0.984653i \(0.555838\pi\)
\(644\) −2687.48 −0.164443
\(645\) 11146.0 0.680422
\(646\) 719.516 0.0438220
\(647\) 1809.79 0.109969 0.0549847 0.998487i \(-0.482489\pi\)
0.0549847 + 0.998487i \(0.482489\pi\)
\(648\) −1428.29 −0.0865870
\(649\) −15057.9 −0.910745
\(650\) 0 0
\(651\) −1124.29 −0.0676871
\(652\) −1805.02 −0.108420
\(653\) 9459.16 0.566869 0.283435 0.958992i \(-0.408526\pi\)
0.283435 + 0.958992i \(0.408526\pi\)
\(654\) −9758.85 −0.583488
\(655\) −9627.94 −0.574343
\(656\) −6926.42 −0.412243
\(657\) −12015.3 −0.713488
\(658\) −4945.59 −0.293008
\(659\) 1015.32 0.0600171 0.0300085 0.999550i \(-0.490447\pi\)
0.0300085 + 0.999550i \(0.490447\pi\)
\(660\) 8175.31 0.482157
\(661\) −23648.4 −1.39156 −0.695778 0.718257i \(-0.744940\pi\)
−0.695778 + 0.718257i \(0.744940\pi\)
\(662\) −7018.49 −0.412056
\(663\) 0 0
\(664\) −2033.66 −0.118857
\(665\) 479.264 0.0279474
\(666\) 3223.40 0.187544
\(667\) −48184.7 −2.79718
\(668\) 12813.4 0.742164
\(669\) 1049.46 0.0606497
\(670\) 7436.26 0.428788
\(671\) 24894.9 1.43228
\(672\) 492.682 0.0282822
\(673\) 17747.0 1.01649 0.508243 0.861214i \(-0.330295\pi\)
0.508243 + 0.861214i \(0.330295\pi\)
\(674\) −3669.63 −0.209717
\(675\) −7764.85 −0.442769
\(676\) 0 0
\(677\) 10754.2 0.610511 0.305256 0.952270i \(-0.401258\pi\)
0.305256 + 0.952270i \(0.401258\pi\)
\(678\) 5218.14 0.295577
\(679\) −5519.69 −0.311968
\(680\) −1835.96 −0.103538
\(681\) −19061.9 −1.07262
\(682\) 9547.42 0.536055
\(683\) 23981.7 1.34354 0.671768 0.740762i \(-0.265535\pi\)
0.671768 + 0.740762i \(0.265535\pi\)
\(684\) 725.815 0.0405734
\(685\) −14911.3 −0.831723
\(686\) 5615.55 0.312540
\(687\) −3263.42 −0.181233
\(688\) −5704.13 −0.316087
\(689\) 0 0
\(690\) −10000.1 −0.551737
\(691\) −29001.4 −1.59662 −0.798312 0.602244i \(-0.794273\pi\)
−0.798312 + 0.602244i \(0.794273\pi\)
\(692\) 7597.53 0.417362
\(693\) −3726.41 −0.204264
\(694\) −14518.6 −0.794119
\(695\) 14302.2 0.780594
\(696\) 8833.46 0.481079
\(697\) 11645.9 0.632882
\(698\) 3590.47 0.194701
\(699\) −17419.6 −0.942587
\(700\) 877.608 0.0473864
\(701\) −31031.5 −1.67196 −0.835979 0.548761i \(-0.815100\pi\)
−0.835979 + 0.548761i \(0.815100\pi\)
\(702\) 0 0
\(703\) 1588.45 0.0852199
\(704\) −4183.84 −0.223984
\(705\) −18402.6 −0.983096
\(706\) 9104.75 0.485357
\(707\) −6149.16 −0.327105
\(708\) −3376.66 −0.179241
\(709\) 4382.59 0.232146 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(710\) 1125.42 0.0594877
\(711\) 5230.23 0.275878
\(712\) −2980.90 −0.156902
\(713\) −11678.5 −0.613414
\(714\) −828.380 −0.0434192
\(715\) 0 0
\(716\) 10603.6 0.553455
\(717\) 8402.17 0.437636
\(718\) −20330.8 −1.05674
\(719\) −28764.6 −1.49199 −0.745994 0.665953i \(-0.768025\pi\)
−0.745994 + 0.665953i \(0.768025\pi\)
\(720\) −1852.03 −0.0958625
\(721\) −885.941 −0.0457617
\(722\) −13360.3 −0.688670
\(723\) −7238.89 −0.372361
\(724\) −9159.77 −0.470194
\(725\) 15734.9 0.806042
\(726\) −21568.3 −1.10258
\(727\) 25408.5 1.29621 0.648107 0.761549i \(-0.275561\pi\)
0.648107 + 0.761549i \(0.275561\pi\)
\(728\) 0 0
\(729\) 17134.5 0.870525
\(730\) 15108.3 0.766006
\(731\) 9590.74 0.485262
\(732\) 5582.57 0.281882
\(733\) 22341.8 1.12580 0.562902 0.826524i \(-0.309685\pi\)
0.562902 + 0.826524i \(0.309685\pi\)
\(734\) −20803.1 −1.04613
\(735\) 10171.9 0.510470
\(736\) 5117.73 0.256307
\(737\) −28492.5 −1.42406
\(738\) 11747.8 0.585966
\(739\) 8210.90 0.408718 0.204359 0.978896i \(-0.434489\pi\)
0.204359 + 0.978896i \(0.434489\pi\)
\(740\) −4053.18 −0.201349
\(741\) 0 0
\(742\) 2264.04 0.112016
\(743\) 32752.3 1.61718 0.808590 0.588373i \(-0.200231\pi\)
0.808590 + 0.588373i \(0.200231\pi\)
\(744\) 2140.97 0.105500
\(745\) −7651.79 −0.376295
\(746\) −15906.8 −0.780684
\(747\) 3449.26 0.168945
\(748\) 7034.58 0.343863
\(749\) −1642.87 −0.0801455
\(750\) 11081.7 0.539527
\(751\) 8907.41 0.432804 0.216402 0.976304i \(-0.430568\pi\)
0.216402 + 0.976304i \(0.430568\pi\)
\(752\) 9417.83 0.456693
\(753\) 27361.0 1.32416
\(754\) 0 0
\(755\) −17729.4 −0.854620
\(756\) −2498.43 −0.120195
\(757\) −10744.6 −0.515879 −0.257940 0.966161i \(-0.583044\pi\)
−0.257940 + 0.966161i \(0.583044\pi\)
\(758\) −18029.6 −0.863938
\(759\) 38316.2 1.83240
\(760\) −912.656 −0.0435599
\(761\) 16997.8 0.809683 0.404841 0.914387i \(-0.367327\pi\)
0.404841 + 0.914387i \(0.367327\pi\)
\(762\) 8588.30 0.408296
\(763\) −5593.28 −0.265387
\(764\) −1353.25 −0.0640825
\(765\) 3113.94 0.147170
\(766\) 4992.48 0.235490
\(767\) 0 0
\(768\) −938.208 −0.0440816
\(769\) −19032.4 −0.892492 −0.446246 0.894910i \(-0.647239\pi\)
−0.446246 + 0.894910i \(0.647239\pi\)
\(770\) 4685.68 0.219299
\(771\) 2033.88 0.0950045
\(772\) 10736.2 0.500524
\(773\) −23750.6 −1.10511 −0.552555 0.833477i \(-0.686347\pi\)
−0.552555 + 0.833477i \(0.686347\pi\)
\(774\) 9674.69 0.449289
\(775\) 3813.68 0.176763
\(776\) 10511.1 0.486244
\(777\) −1828.79 −0.0844368
\(778\) 21793.7 1.00430
\(779\) 5789.17 0.266263
\(780\) 0 0
\(781\) −4312.12 −0.197567
\(782\) −8604.79 −0.393487
\(783\) −44795.2 −2.04451
\(784\) −5205.62 −0.237136
\(785\) −29813.0 −1.35550
\(786\) 8272.42 0.375404
\(787\) −17825.9 −0.807403 −0.403702 0.914891i \(-0.632277\pi\)
−0.403702 + 0.914891i \(0.632277\pi\)
\(788\) 7588.96 0.343078
\(789\) 7304.34 0.329584
\(790\) −6576.62 −0.296184
\(791\) 2990.78 0.134437
\(792\) 7096.16 0.318373
\(793\) 0 0
\(794\) −21774.3 −0.973225
\(795\) 8424.53 0.375833
\(796\) −9662.42 −0.430245
\(797\) −26461.2 −1.17604 −0.588020 0.808846i \(-0.700092\pi\)
−0.588020 + 0.808846i \(0.700092\pi\)
\(798\) −411.788 −0.0182671
\(799\) −15834.8 −0.701122
\(800\) −1671.22 −0.0738581
\(801\) 5055.87 0.223021
\(802\) −19937.5 −0.877828
\(803\) −57888.5 −2.54401
\(804\) −6389.32 −0.280266
\(805\) −5731.58 −0.250946
\(806\) 0 0
\(807\) 19085.8 0.832530
\(808\) 11709.8 0.509837
\(809\) 21873.3 0.950588 0.475294 0.879827i \(-0.342342\pi\)
0.475294 + 0.879827i \(0.342342\pi\)
\(810\) −3046.10 −0.132135
\(811\) −39113.8 −1.69355 −0.846776 0.531949i \(-0.821460\pi\)
−0.846776 + 0.531949i \(0.821460\pi\)
\(812\) 5062.90 0.218809
\(813\) −14970.6 −0.645810
\(814\) 15530.0 0.668706
\(815\) −3849.56 −0.165453
\(816\) 1577.47 0.0676748
\(817\) 4767.57 0.204157
\(818\) −8072.65 −0.345053
\(819\) 0 0
\(820\) −14772.0 −0.629097
\(821\) 26296.8 1.11786 0.558931 0.829214i \(-0.311212\pi\)
0.558931 + 0.829214i \(0.311212\pi\)
\(822\) 12811.9 0.543634
\(823\) 24681.1 1.04536 0.522678 0.852530i \(-0.324933\pi\)
0.522678 + 0.852530i \(0.324933\pi\)
\(824\) 1687.09 0.0713258
\(825\) −12512.3 −0.528027
\(826\) −1935.33 −0.0815241
\(827\) 3081.68 0.129578 0.0647888 0.997899i \(-0.479363\pi\)
0.0647888 + 0.997899i \(0.479363\pi\)
\(828\) −8680.12 −0.364317
\(829\) −8498.53 −0.356051 −0.178026 0.984026i \(-0.556971\pi\)
−0.178026 + 0.984026i \(0.556971\pi\)
\(830\) −4337.18 −0.181380
\(831\) −1605.19 −0.0670076
\(832\) 0 0
\(833\) 8752.57 0.364056
\(834\) −12288.6 −0.510215
\(835\) 27327.1 1.13257
\(836\) 3496.90 0.144668
\(837\) −10857.0 −0.448356
\(838\) 30389.7 1.25274
\(839\) −18680.1 −0.768664 −0.384332 0.923195i \(-0.625568\pi\)
−0.384332 + 0.923195i \(0.625568\pi\)
\(840\) 1050.74 0.0431596
\(841\) 66385.3 2.72194
\(842\) −20309.5 −0.831249
\(843\) −20018.8 −0.817894
\(844\) 10673.7 0.435312
\(845\) 0 0
\(846\) −15973.5 −0.649147
\(847\) −12361.9 −0.501486
\(848\) −4311.39 −0.174592
\(849\) 2919.72 0.118026
\(850\) 2809.93 0.113388
\(851\) −18996.5 −0.765208
\(852\) −966.973 −0.0388826
\(853\) −24129.9 −0.968574 −0.484287 0.874909i \(-0.660921\pi\)
−0.484287 + 0.874909i \(0.660921\pi\)
\(854\) 3199.65 0.128208
\(855\) 1547.94 0.0619164
\(856\) 3128.49 0.124918
\(857\) −7739.30 −0.308482 −0.154241 0.988033i \(-0.549293\pi\)
−0.154241 + 0.988033i \(0.549293\pi\)
\(858\) 0 0
\(859\) 4302.59 0.170899 0.0854496 0.996342i \(-0.472767\pi\)
0.0854496 + 0.996342i \(0.472767\pi\)
\(860\) −12165.2 −0.482360
\(861\) −6665.08 −0.263816
\(862\) 13233.7 0.522901
\(863\) 16253.2 0.641096 0.320548 0.947232i \(-0.396133\pi\)
0.320548 + 0.947232i \(0.396133\pi\)
\(864\) 4757.73 0.187340
\(865\) 16203.2 0.636910
\(866\) 11395.2 0.447142
\(867\) 15353.2 0.601410
\(868\) 1227.10 0.0479842
\(869\) 25198.7 0.983669
\(870\) 18839.1 0.734144
\(871\) 0 0
\(872\) 10651.2 0.413642
\(873\) −17827.7 −0.691152
\(874\) −4277.45 −0.165546
\(875\) 6351.46 0.245392
\(876\) −12981.2 −0.500680
\(877\) 9441.33 0.363525 0.181762 0.983342i \(-0.441820\pi\)
0.181762 + 0.983342i \(0.441820\pi\)
\(878\) 872.014 0.0335183
\(879\) −6220.48 −0.238693
\(880\) −8922.88 −0.341807
\(881\) −11872.4 −0.454021 −0.227011 0.973892i \(-0.572895\pi\)
−0.227011 + 0.973892i \(0.572895\pi\)
\(882\) 8829.18 0.337068
\(883\) −493.891 −0.0188231 −0.00941153 0.999956i \(-0.502996\pi\)
−0.00941153 + 0.999956i \(0.502996\pi\)
\(884\) 0 0
\(885\) −7201.40 −0.273528
\(886\) 13219.2 0.501249
\(887\) 20543.6 0.777664 0.388832 0.921309i \(-0.372879\pi\)
0.388832 + 0.921309i \(0.372879\pi\)
\(888\) 3482.54 0.131606
\(889\) 4922.38 0.185705
\(890\) −6357.36 −0.239437
\(891\) 11671.3 0.438838
\(892\) −1145.43 −0.0429953
\(893\) −7871.52 −0.294972
\(894\) 6574.50 0.245955
\(895\) 22614.2 0.844592
\(896\) −537.734 −0.0200496
\(897\) 0 0
\(898\) −16030.6 −0.595711
\(899\) 22001.0 0.816212
\(900\) 2834.53 0.104983
\(901\) 7249.03 0.268036
\(902\) 56599.7 2.08932
\(903\) −5488.90 −0.202280
\(904\) −5695.30 −0.209539
\(905\) −19535.0 −0.717532
\(906\) 15233.3 0.558600
\(907\) −5386.61 −0.197199 −0.0985994 0.995127i \(-0.531436\pi\)
−0.0985994 + 0.995127i \(0.531436\pi\)
\(908\) 20805.0 0.760393
\(909\) −19860.8 −0.724688
\(910\) 0 0
\(911\) 31793.5 1.15627 0.578137 0.815940i \(-0.303780\pi\)
0.578137 + 0.815940i \(0.303780\pi\)
\(912\) 784.164 0.0284718
\(913\) 16618.2 0.602389
\(914\) 9014.07 0.326213
\(915\) 11905.9 0.430162
\(916\) 3561.83 0.128478
\(917\) 4741.34 0.170745
\(918\) −7999.51 −0.287607
\(919\) −42991.4 −1.54315 −0.771575 0.636139i \(-0.780531\pi\)
−0.771575 + 0.636139i \(0.780531\pi\)
\(920\) 10914.6 0.391134
\(921\) 34398.0 1.23068
\(922\) −4207.10 −0.150275
\(923\) 0 0
\(924\) −4025.98 −0.143339
\(925\) 6203.40 0.220504
\(926\) −10936.6 −0.388119
\(927\) −2861.45 −0.101383
\(928\) −9641.21 −0.341043
\(929\) 40676.7 1.43655 0.718277 0.695757i \(-0.244931\pi\)
0.718277 + 0.695757i \(0.244931\pi\)
\(930\) 4566.04 0.160996
\(931\) 4350.91 0.153164
\(932\) 19012.4 0.668212
\(933\) 12030.5 0.422145
\(934\) −12087.6 −0.423467
\(935\) 15002.6 0.524748
\(936\) 0 0
\(937\) 21008.3 0.732456 0.366228 0.930525i \(-0.380649\pi\)
0.366228 + 0.930525i \(0.380649\pi\)
\(938\) −3662.04 −0.127473
\(939\) 21713.2 0.754614
\(940\) 20085.4 0.696929
\(941\) −4525.50 −0.156777 −0.0783885 0.996923i \(-0.524977\pi\)
−0.0783885 + 0.996923i \(0.524977\pi\)
\(942\) 25615.6 0.885989
\(943\) −69233.5 −2.39083
\(944\) 3685.43 0.127066
\(945\) −5328.40 −0.183421
\(946\) 46611.7 1.60198
\(947\) −27401.6 −0.940267 −0.470134 0.882595i \(-0.655794\pi\)
−0.470134 + 0.882595i \(0.655794\pi\)
\(948\) 5650.70 0.193593
\(949\) 0 0
\(950\) 1396.82 0.0477040
\(951\) 23704.0 0.808259
\(952\) 904.129 0.0307805
\(953\) 26226.2 0.891449 0.445725 0.895170i \(-0.352946\pi\)
0.445725 + 0.895170i \(0.352946\pi\)
\(954\) 7312.49 0.248166
\(955\) −2886.08 −0.0977921
\(956\) −9170.48 −0.310245
\(957\) −72183.2 −2.43819
\(958\) 16962.6 0.572063
\(959\) 7343.15 0.247260
\(960\) −2000.91 −0.0672700
\(961\) −24458.6 −0.821007
\(962\) 0 0
\(963\) −5306.19 −0.177559
\(964\) 7900.83 0.263972
\(965\) 22897.1 0.763817
\(966\) 4924.63 0.164024
\(967\) 20843.6 0.693158 0.346579 0.938021i \(-0.387343\pi\)
0.346579 + 0.938021i \(0.387343\pi\)
\(968\) 23540.6 0.781635
\(969\) −1318.47 −0.0437103
\(970\) 22417.0 0.742026
\(971\) 32577.3 1.07668 0.538339 0.842729i \(-0.319052\pi\)
0.538339 + 0.842729i \(0.319052\pi\)
\(972\) −13440.1 −0.443510
\(973\) −7043.20 −0.232060
\(974\) 13246.7 0.435782
\(975\) 0 0
\(976\) −6093.05 −0.199830
\(977\) −43864.1 −1.43637 −0.718186 0.695851i \(-0.755027\pi\)
−0.718186 + 0.695851i \(0.755027\pi\)
\(978\) 3307.58 0.108144
\(979\) 24358.6 0.795204
\(980\) −11102.0 −0.361879
\(981\) −18065.4 −0.587955
\(982\) 32758.7 1.06453
\(983\) −14758.8 −0.478875 −0.239437 0.970912i \(-0.576963\pi\)
−0.239437 + 0.970912i \(0.576963\pi\)
\(984\) 12692.2 0.411193
\(985\) 16185.0 0.523549
\(986\) 16210.4 0.523575
\(987\) 9062.49 0.292262
\(988\) 0 0
\(989\) −57016.0 −1.83317
\(990\) 15134.0 0.485848
\(991\) 48622.4 1.55857 0.779284 0.626671i \(-0.215583\pi\)
0.779284 + 0.626671i \(0.215583\pi\)
\(992\) −2336.74 −0.0747900
\(993\) 12860.9 0.411006
\(994\) −554.220 −0.0176849
\(995\) −20607.0 −0.656569
\(996\) 3726.55 0.118554
\(997\) −18812.3 −0.597586 −0.298793 0.954318i \(-0.596584\pi\)
−0.298793 + 0.954318i \(0.596584\pi\)
\(998\) 15831.4 0.502138
\(999\) −17660.2 −0.559305
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 338.4.a.k.1.2 yes 3
13.2 odd 12 338.4.e.h.147.5 12
13.3 even 3 338.4.c.k.191.2 6
13.4 even 6 338.4.c.l.315.2 6
13.5 odd 4 338.4.b.f.337.2 6
13.6 odd 12 338.4.e.h.23.2 12
13.7 odd 12 338.4.e.h.23.5 12
13.8 odd 4 338.4.b.f.337.5 6
13.9 even 3 338.4.c.k.315.2 6
13.10 even 6 338.4.c.l.191.2 6
13.11 odd 12 338.4.e.h.147.2 12
13.12 even 2 338.4.a.j.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
338.4.a.j.1.2 3 13.12 even 2
338.4.a.k.1.2 yes 3 1.1 even 1 trivial
338.4.b.f.337.2 6 13.5 odd 4
338.4.b.f.337.5 6 13.8 odd 4
338.4.c.k.191.2 6 13.3 even 3
338.4.c.k.315.2 6 13.9 even 3
338.4.c.l.191.2 6 13.10 even 6
338.4.c.l.315.2 6 13.4 even 6
338.4.e.h.23.2 12 13.6 odd 12
338.4.e.h.23.5 12 13.7 odd 12
338.4.e.h.147.2 12 13.11 odd 12
338.4.e.h.147.5 12 13.2 odd 12