Properties

Label 231.4.a.k.1.4
Level $231$
Weight $4$
Character 231.1
Self dual yes
Analytic conductor $13.629$
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] = [231,4,Mod(1,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("231.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 231.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: \(13.6294412113\)
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: \( x^{5} - x^{4} - 30x^{3} + 11x^{2} + 185x + 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.59998\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.59998 q^{2} -3.00000 q^{3} -5.44007 q^{4} -17.2762 q^{5} -4.79994 q^{6} +7.00000 q^{7} -21.5038 q^{8} +9.00000 q^{9} -27.6416 q^{10} +11.0000 q^{11} +16.3202 q^{12} +46.1062 q^{13} +11.1999 q^{14} +51.8287 q^{15} +9.11483 q^{16} +19.8752 q^{17} +14.3998 q^{18} +76.5707 q^{19} +93.9838 q^{20} -21.0000 q^{21} +17.5998 q^{22} -163.205 q^{23} +64.5115 q^{24} +173.468 q^{25} +73.7690 q^{26} -27.0000 q^{27} -38.0805 q^{28} +158.858 q^{29} +82.9248 q^{30} +170.768 q^{31} +186.614 q^{32} -33.0000 q^{33} +31.7999 q^{34} -120.934 q^{35} -48.9606 q^{36} -245.971 q^{37} +122.511 q^{38} -138.319 q^{39} +371.505 q^{40} -3.33673 q^{41} -33.5996 q^{42} -122.798 q^{43} -59.8407 q^{44} -155.486 q^{45} -261.125 q^{46} +390.972 q^{47} -27.3445 q^{48} +49.0000 q^{49} +277.545 q^{50} -59.6255 q^{51} -250.821 q^{52} -410.957 q^{53} -43.1995 q^{54} -190.038 q^{55} -150.527 q^{56} -229.712 q^{57} +254.169 q^{58} +408.774 q^{59} -281.951 q^{60} -21.9747 q^{61} +273.225 q^{62} +63.0000 q^{63} +225.660 q^{64} -796.541 q^{65} -52.7993 q^{66} +618.424 q^{67} -108.122 q^{68} +489.616 q^{69} -193.491 q^{70} +929.041 q^{71} -193.534 q^{72} +868.090 q^{73} -393.549 q^{74} -520.404 q^{75} -416.549 q^{76} +77.0000 q^{77} -221.307 q^{78} +152.702 q^{79} -157.470 q^{80} +81.0000 q^{81} -5.33871 q^{82} -100.924 q^{83} +114.241 q^{84} -343.368 q^{85} -196.475 q^{86} -476.573 q^{87} -236.542 q^{88} -1063.23 q^{89} -248.774 q^{90} +322.743 q^{91} +887.848 q^{92} -512.303 q^{93} +625.547 q^{94} -1322.85 q^{95} -559.843 q^{96} +1415.50 q^{97} +78.3990 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} - 15 q^{3} + 21 q^{4} + 21 q^{5} + 3 q^{6} + 35 q^{7} - 42 q^{8} + 45 q^{9} - 23 q^{10} + 55 q^{11} - 63 q^{12} + 101 q^{13} - 7 q^{14} - 63 q^{15} - 7 q^{16} - 20 q^{17} - 9 q^{18} + 237 q^{19}+ \cdots + 495 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.59998 0.565678 0.282839 0.959167i \(-0.408724\pi\)
0.282839 + 0.959167i \(0.408724\pi\)
\(3\) −3.00000 −0.577350
\(4\) −5.44007 −0.680008
\(5\) −17.2762 −1.54523 −0.772616 0.634873i \(-0.781052\pi\)
−0.772616 + 0.634873i \(0.781052\pi\)
\(6\) −4.79994 −0.326594
\(7\) 7.00000 0.377964
\(8\) −21.5038 −0.950344
\(9\) 9.00000 0.333333
\(10\) −27.6416 −0.874104
\(11\) 11.0000 0.301511
\(12\) 16.3202 0.392603
\(13\) 46.1062 0.983658 0.491829 0.870692i \(-0.336328\pi\)
0.491829 + 0.870692i \(0.336328\pi\)
\(14\) 11.1999 0.213806
\(15\) 51.8287 0.892140
\(16\) 9.11483 0.142419
\(17\) 19.8752 0.283555 0.141778 0.989899i \(-0.454718\pi\)
0.141778 + 0.989899i \(0.454718\pi\)
\(18\) 14.3998 0.188559
\(19\) 76.5707 0.924553 0.462277 0.886736i \(-0.347033\pi\)
0.462277 + 0.886736i \(0.347033\pi\)
\(20\) 93.9838 1.05077
\(21\) −21.0000 −0.218218
\(22\) 17.5998 0.170558
\(23\) −163.205 −1.47960 −0.739798 0.672829i \(-0.765079\pi\)
−0.739798 + 0.672829i \(0.765079\pi\)
\(24\) 64.5115 0.548681
\(25\) 173.468 1.38774
\(26\) 73.7690 0.556434
\(27\) −27.0000 −0.192450
\(28\) −38.0805 −0.257019
\(29\) 158.858 1.01721 0.508605 0.861000i \(-0.330161\pi\)
0.508605 + 0.861000i \(0.330161\pi\)
\(30\) 82.9248 0.504664
\(31\) 170.768 0.989380 0.494690 0.869069i \(-0.335282\pi\)
0.494690 + 0.869069i \(0.335282\pi\)
\(32\) 186.614 1.03091
\(33\) −33.0000 −0.174078
\(34\) 31.7999 0.160401
\(35\) −120.934 −0.584043
\(36\) −48.9606 −0.226669
\(37\) −245.971 −1.09290 −0.546452 0.837490i \(-0.684022\pi\)
−0.546452 + 0.837490i \(0.684022\pi\)
\(38\) 122.511 0.523000
\(39\) −138.319 −0.567915
\(40\) 371.505 1.46850
\(41\) −3.33673 −0.0127100 −0.00635500 0.999980i \(-0.502023\pi\)
−0.00635500 + 0.999980i \(0.502023\pi\)
\(42\) −33.5996 −0.123441
\(43\) −122.798 −0.435502 −0.217751 0.976004i \(-0.569872\pi\)
−0.217751 + 0.976004i \(0.569872\pi\)
\(44\) −59.8407 −0.205030
\(45\) −155.486 −0.515077
\(46\) −261.125 −0.836975
\(47\) 390.972 1.21338 0.606692 0.794937i \(-0.292496\pi\)
0.606692 + 0.794937i \(0.292496\pi\)
\(48\) −27.3445 −0.0822258
\(49\) 49.0000 0.142857
\(50\) 277.545 0.785016
\(51\) −59.6255 −0.163711
\(52\) −250.821 −0.668896
\(53\) −410.957 −1.06508 −0.532541 0.846404i \(-0.678763\pi\)
−0.532541 + 0.846404i \(0.678763\pi\)
\(54\) −43.1995 −0.108865
\(55\) −190.038 −0.465905
\(56\) −150.527 −0.359196
\(57\) −229.712 −0.533791
\(58\) 254.169 0.575414
\(59\) 408.774 0.901998 0.450999 0.892524i \(-0.351068\pi\)
0.450999 + 0.892524i \(0.351068\pi\)
\(60\) −281.951 −0.606663
\(61\) −21.9747 −0.0461241 −0.0230621 0.999734i \(-0.507342\pi\)
−0.0230621 + 0.999734i \(0.507342\pi\)
\(62\) 273.225 0.559671
\(63\) 63.0000 0.125988
\(64\) 225.660 0.440743
\(65\) −796.541 −1.51998
\(66\) −52.7993 −0.0984719
\(67\) 618.424 1.12765 0.563824 0.825895i \(-0.309329\pi\)
0.563824 + 0.825895i \(0.309329\pi\)
\(68\) −108.122 −0.192820
\(69\) 489.616 0.854245
\(70\) −193.491 −0.330380
\(71\) 929.041 1.55291 0.776457 0.630171i \(-0.217015\pi\)
0.776457 + 0.630171i \(0.217015\pi\)
\(72\) −193.534 −0.316781
\(73\) 868.090 1.39181 0.695906 0.718133i \(-0.255003\pi\)
0.695906 + 0.718133i \(0.255003\pi\)
\(74\) −393.549 −0.618232
\(75\) −520.404 −0.801214
\(76\) −416.549 −0.628704
\(77\) 77.0000 0.113961
\(78\) −221.307 −0.321257
\(79\) 152.702 0.217472 0.108736 0.994071i \(-0.465320\pi\)
0.108736 + 0.994071i \(0.465320\pi\)
\(80\) −157.470 −0.220071
\(81\) 81.0000 0.111111
\(82\) −5.33871 −0.00718977
\(83\) −100.924 −0.133469 −0.0667343 0.997771i \(-0.521258\pi\)
−0.0667343 + 0.997771i \(0.521258\pi\)
\(84\) 114.241 0.148390
\(85\) −343.368 −0.438159
\(86\) −196.475 −0.246354
\(87\) −476.573 −0.587287
\(88\) −236.542 −0.286540
\(89\) −1063.23 −1.26631 −0.633156 0.774024i \(-0.718241\pi\)
−0.633156 + 0.774024i \(0.718241\pi\)
\(90\) −248.774 −0.291368
\(91\) 322.743 0.371788
\(92\) 887.848 1.00614
\(93\) −512.303 −0.571219
\(94\) 625.547 0.686385
\(95\) −1322.85 −1.42865
\(96\) −559.843 −0.595195
\(97\) 1415.50 1.48167 0.740836 0.671686i \(-0.234430\pi\)
0.740836 + 0.671686i \(0.234430\pi\)
\(98\) 78.3990 0.0808112
\(99\) 99.0000 0.100504
\(100\) −943.677 −0.943677
\(101\) −774.140 −0.762671 −0.381336 0.924437i \(-0.624536\pi\)
−0.381336 + 0.924437i \(0.624536\pi\)
\(102\) −95.3997 −0.0926076
\(103\) 759.159 0.726235 0.363118 0.931743i \(-0.381712\pi\)
0.363118 + 0.931743i \(0.381712\pi\)
\(104\) −991.460 −0.934814
\(105\) 362.801 0.337197
\(106\) −657.523 −0.602493
\(107\) −1181.22 −1.06722 −0.533612 0.845729i \(-0.679166\pi\)
−0.533612 + 0.845729i \(0.679166\pi\)
\(108\) 146.882 0.130868
\(109\) 738.091 0.648590 0.324295 0.945956i \(-0.394873\pi\)
0.324295 + 0.945956i \(0.394873\pi\)
\(110\) −304.058 −0.263552
\(111\) 737.914 0.630989
\(112\) 63.8038 0.0538294
\(113\) 620.181 0.516298 0.258149 0.966105i \(-0.416887\pi\)
0.258149 + 0.966105i \(0.416887\pi\)
\(114\) −367.534 −0.301954
\(115\) 2819.57 2.28632
\(116\) −864.195 −0.691711
\(117\) 414.956 0.327886
\(118\) 654.031 0.510241
\(119\) 139.126 0.107174
\(120\) −1114.51 −0.847840
\(121\) 121.000 0.0909091
\(122\) −35.1591 −0.0260914
\(123\) 10.0102 0.00733813
\(124\) −928.988 −0.672787
\(125\) −837.342 −0.599153
\(126\) 100.799 0.0712688
\(127\) −2291.79 −1.60129 −0.800645 0.599139i \(-0.795510\pi\)
−0.800645 + 0.599139i \(0.795510\pi\)
\(128\) −1131.86 −0.781589
\(129\) 368.395 0.251437
\(130\) −1274.45 −0.859820
\(131\) −1491.77 −0.994938 −0.497469 0.867482i \(-0.665737\pi\)
−0.497469 + 0.867482i \(0.665737\pi\)
\(132\) 179.522 0.118374
\(133\) 535.995 0.349448
\(134\) 989.465 0.637886
\(135\) 466.458 0.297380
\(136\) −427.393 −0.269475
\(137\) 2590.79 1.61566 0.807831 0.589414i \(-0.200641\pi\)
0.807831 + 0.589414i \(0.200641\pi\)
\(138\) 783.376 0.483228
\(139\) 2674.14 1.63178 0.815890 0.578207i \(-0.196248\pi\)
0.815890 + 0.578207i \(0.196248\pi\)
\(140\) 657.886 0.397154
\(141\) −1172.92 −0.700548
\(142\) 1486.45 0.878449
\(143\) 507.168 0.296584
\(144\) 82.0335 0.0474731
\(145\) −2744.46 −1.57183
\(146\) 1388.93 0.787318
\(147\) −147.000 −0.0824786
\(148\) 1338.10 0.743184
\(149\) 822.355 0.452147 0.226074 0.974110i \(-0.427411\pi\)
0.226074 + 0.974110i \(0.427411\pi\)
\(150\) −832.635 −0.453229
\(151\) −138.803 −0.0748053 −0.0374027 0.999300i \(-0.511908\pi\)
−0.0374027 + 0.999300i \(0.511908\pi\)
\(152\) −1646.56 −0.878644
\(153\) 178.877 0.0945184
\(154\) 123.198 0.0644650
\(155\) −2950.22 −1.52882
\(156\) 752.462 0.386187
\(157\) −634.746 −0.322664 −0.161332 0.986900i \(-0.551579\pi\)
−0.161332 + 0.986900i \(0.551579\pi\)
\(158\) 244.320 0.123019
\(159\) 1232.87 0.614925
\(160\) −3223.99 −1.59299
\(161\) −1142.44 −0.559235
\(162\) 129.598 0.0628531
\(163\) −608.367 −0.292337 −0.146169 0.989260i \(-0.546694\pi\)
−0.146169 + 0.989260i \(0.546694\pi\)
\(164\) 18.1520 0.00864291
\(165\) 570.115 0.268990
\(166\) −161.477 −0.0755003
\(167\) −1545.50 −0.716135 −0.358067 0.933696i \(-0.616564\pi\)
−0.358067 + 0.933696i \(0.616564\pi\)
\(168\) 451.580 0.207382
\(169\) −71.2185 −0.0324163
\(170\) −549.382 −0.247857
\(171\) 689.136 0.308184
\(172\) 668.031 0.296145
\(173\) 126.598 0.0556362 0.0278181 0.999613i \(-0.491144\pi\)
0.0278181 + 0.999613i \(0.491144\pi\)
\(174\) −762.506 −0.332215
\(175\) 1214.28 0.524518
\(176\) 100.263 0.0429410
\(177\) −1226.32 −0.520769
\(178\) −1701.14 −0.716325
\(179\) −144.263 −0.0602387 −0.0301194 0.999546i \(-0.509589\pi\)
−0.0301194 + 0.999546i \(0.509589\pi\)
\(180\) 845.854 0.350257
\(181\) 2925.01 1.20118 0.600591 0.799556i \(-0.294932\pi\)
0.600591 + 0.799556i \(0.294932\pi\)
\(182\) 516.383 0.210312
\(183\) 65.9241 0.0266298
\(184\) 3509.54 1.40612
\(185\) 4249.46 1.68879
\(186\) −819.675 −0.323126
\(187\) 218.627 0.0854951
\(188\) −2126.91 −0.825111
\(189\) −189.000 −0.0727393
\(190\) −2116.54 −0.808156
\(191\) −3771.70 −1.42885 −0.714426 0.699711i \(-0.753312\pi\)
−0.714426 + 0.699711i \(0.753312\pi\)
\(192\) −676.981 −0.254463
\(193\) −1282.92 −0.478480 −0.239240 0.970960i \(-0.576898\pi\)
−0.239240 + 0.970960i \(0.576898\pi\)
\(194\) 2264.77 0.838149
\(195\) 2389.62 0.877561
\(196\) −266.563 −0.0971440
\(197\) 3052.31 1.10390 0.551949 0.833878i \(-0.313884\pi\)
0.551949 + 0.833878i \(0.313884\pi\)
\(198\) 158.398 0.0568528
\(199\) 5040.28 1.79546 0.897729 0.440547i \(-0.145216\pi\)
0.897729 + 0.440547i \(0.145216\pi\)
\(200\) −3730.22 −1.31883
\(201\) −1855.27 −0.651048
\(202\) −1238.61 −0.431426
\(203\) 1112.00 0.384469
\(204\) 324.367 0.111325
\(205\) 57.6462 0.0196399
\(206\) 1214.64 0.410815
\(207\) −1468.85 −0.493198
\(208\) 420.250 0.140092
\(209\) 842.277 0.278763
\(210\) 580.474 0.190745
\(211\) 3556.28 1.16031 0.580153 0.814508i \(-0.302993\pi\)
0.580153 + 0.814508i \(0.302993\pi\)
\(212\) 2235.63 0.724264
\(213\) −2787.12 −0.896575
\(214\) −1889.93 −0.603706
\(215\) 2121.49 0.672952
\(216\) 580.603 0.182894
\(217\) 1195.37 0.373951
\(218\) 1180.93 0.366893
\(219\) −2604.27 −0.803563
\(220\) 1033.82 0.316819
\(221\) 916.369 0.278922
\(222\) 1180.65 0.356937
\(223\) 1767.85 0.530870 0.265435 0.964129i \(-0.414484\pi\)
0.265435 + 0.964129i \(0.414484\pi\)
\(224\) 1306.30 0.389646
\(225\) 1561.21 0.462581
\(226\) 992.277 0.292059
\(227\) 4836.74 1.41421 0.707105 0.707109i \(-0.250001\pi\)
0.707105 + 0.707109i \(0.250001\pi\)
\(228\) 1249.65 0.362982
\(229\) 4941.30 1.42590 0.712949 0.701216i \(-0.247359\pi\)
0.712949 + 0.701216i \(0.247359\pi\)
\(230\) 4511.26 1.29332
\(231\) −231.000 −0.0657952
\(232\) −3416.04 −0.966700
\(233\) 4822.14 1.35583 0.677916 0.735140i \(-0.262883\pi\)
0.677916 + 0.735140i \(0.262883\pi\)
\(234\) 663.921 0.185478
\(235\) −6754.51 −1.87496
\(236\) −2223.76 −0.613366
\(237\) −458.106 −0.125558
\(238\) 222.599 0.0606259
\(239\) 4821.80 1.30501 0.652503 0.757786i \(-0.273719\pi\)
0.652503 + 0.757786i \(0.273719\pi\)
\(240\) 472.409 0.127058
\(241\) 1544.41 0.412798 0.206399 0.978468i \(-0.433825\pi\)
0.206399 + 0.978468i \(0.433825\pi\)
\(242\) 193.598 0.0514253
\(243\) −243.000 −0.0641500
\(244\) 119.544 0.0313648
\(245\) −846.535 −0.220747
\(246\) 16.0161 0.00415102
\(247\) 3530.38 0.909445
\(248\) −3672.16 −0.940252
\(249\) 302.773 0.0770581
\(250\) −1339.73 −0.338928
\(251\) −7568.20 −1.90319 −0.951595 0.307354i \(-0.900556\pi\)
−0.951595 + 0.307354i \(0.900556\pi\)
\(252\) −342.724 −0.0856730
\(253\) −1795.26 −0.446115
\(254\) −3666.82 −0.905815
\(255\) 1030.10 0.252971
\(256\) −3616.24 −0.882871
\(257\) 1415.19 0.343490 0.171745 0.985141i \(-0.445059\pi\)
0.171745 + 0.985141i \(0.445059\pi\)
\(258\) 589.425 0.142233
\(259\) −1721.80 −0.413079
\(260\) 4333.23 1.03360
\(261\) 1429.72 0.339070
\(262\) −2386.81 −0.562815
\(263\) −7875.57 −1.84650 −0.923248 0.384204i \(-0.874476\pi\)
−0.923248 + 0.384204i \(0.874476\pi\)
\(264\) 709.626 0.165434
\(265\) 7099.79 1.64580
\(266\) 857.580 0.197675
\(267\) 3189.68 0.731105
\(268\) −3364.27 −0.766810
\(269\) −2062.01 −0.467371 −0.233685 0.972312i \(-0.575079\pi\)
−0.233685 + 0.972312i \(0.575079\pi\)
\(270\) 746.323 0.168221
\(271\) −8247.07 −1.84861 −0.924306 0.381651i \(-0.875355\pi\)
−0.924306 + 0.381651i \(0.875355\pi\)
\(272\) 181.159 0.0403837
\(273\) −968.230 −0.214652
\(274\) 4145.21 0.913945
\(275\) 1908.15 0.418420
\(276\) −2663.55 −0.580893
\(277\) −6637.33 −1.43971 −0.719853 0.694126i \(-0.755791\pi\)
−0.719853 + 0.694126i \(0.755791\pi\)
\(278\) 4278.57 0.923062
\(279\) 1536.91 0.329793
\(280\) 2600.53 0.555042
\(281\) 5600.82 1.18903 0.594515 0.804085i \(-0.297344\pi\)
0.594515 + 0.804085i \(0.297344\pi\)
\(282\) −1876.64 −0.396285
\(283\) −934.745 −0.196342 −0.0981711 0.995170i \(-0.531299\pi\)
−0.0981711 + 0.995170i \(0.531299\pi\)
\(284\) −5054.04 −1.05599
\(285\) 3968.56 0.824831
\(286\) 811.459 0.167771
\(287\) −23.3571 −0.00480393
\(288\) 1679.53 0.343636
\(289\) −4517.98 −0.919596
\(290\) −4391.08 −0.889148
\(291\) −4246.50 −0.855444
\(292\) −4722.47 −0.946444
\(293\) 9169.86 1.82836 0.914179 0.405311i \(-0.132837\pi\)
0.914179 + 0.405311i \(0.132837\pi\)
\(294\) −235.197 −0.0466564
\(295\) −7062.08 −1.39380
\(296\) 5289.33 1.03864
\(297\) −297.000 −0.0580259
\(298\) 1315.75 0.255770
\(299\) −7524.78 −1.45542
\(300\) 2831.03 0.544832
\(301\) −859.589 −0.164604
\(302\) −222.081 −0.0423157
\(303\) 2322.42 0.440328
\(304\) 697.929 0.131674
\(305\) 379.640 0.0712725
\(306\) 286.199 0.0534670
\(307\) −6193.71 −1.15144 −0.575722 0.817645i \(-0.695279\pi\)
−0.575722 + 0.817645i \(0.695279\pi\)
\(308\) −418.885 −0.0774941
\(309\) −2277.48 −0.419292
\(310\) −4720.29 −0.864822
\(311\) −3373.58 −0.615106 −0.307553 0.951531i \(-0.599510\pi\)
−0.307553 + 0.951531i \(0.599510\pi\)
\(312\) 2974.38 0.539715
\(313\) 7175.09 1.29572 0.647860 0.761760i \(-0.275664\pi\)
0.647860 + 0.761760i \(0.275664\pi\)
\(314\) −1015.58 −0.182524
\(315\) −1088.40 −0.194681
\(316\) −830.709 −0.147883
\(317\) −6302.06 −1.11659 −0.558295 0.829643i \(-0.688544\pi\)
−0.558295 + 0.829643i \(0.688544\pi\)
\(318\) 1972.57 0.347850
\(319\) 1747.43 0.306700
\(320\) −3898.56 −0.681050
\(321\) 3543.66 0.616162
\(322\) −1827.88 −0.316347
\(323\) 1521.86 0.262162
\(324\) −440.645 −0.0755565
\(325\) 7997.94 1.36506
\(326\) −973.375 −0.165369
\(327\) −2214.27 −0.374463
\(328\) 71.7526 0.0120789
\(329\) 2736.80 0.458616
\(330\) 912.173 0.152162
\(331\) −7404.03 −1.22949 −0.614747 0.788725i \(-0.710742\pi\)
−0.614747 + 0.788725i \(0.710742\pi\)
\(332\) 549.035 0.0907597
\(333\) −2213.74 −0.364301
\(334\) −2472.77 −0.405102
\(335\) −10684.0 −1.74248
\(336\) −191.411 −0.0310784
\(337\) 7008.68 1.13290 0.566450 0.824096i \(-0.308316\pi\)
0.566450 + 0.824096i \(0.308316\pi\)
\(338\) −113.948 −0.0183372
\(339\) −1860.54 −0.298085
\(340\) 1867.94 0.297952
\(341\) 1878.45 0.298309
\(342\) 1102.60 0.174333
\(343\) 343.000 0.0539949
\(344\) 2640.64 0.413877
\(345\) −8458.72 −1.32001
\(346\) 202.554 0.0314722
\(347\) −10025.8 −1.55104 −0.775521 0.631322i \(-0.782513\pi\)
−0.775521 + 0.631322i \(0.782513\pi\)
\(348\) 2592.59 0.399360
\(349\) −10746.5 −1.64827 −0.824136 0.566391i \(-0.808339\pi\)
−0.824136 + 0.566391i \(0.808339\pi\)
\(350\) 1942.82 0.296708
\(351\) −1244.87 −0.189305
\(352\) 2052.76 0.310830
\(353\) −606.136 −0.0913919 −0.0456960 0.998955i \(-0.514551\pi\)
−0.0456960 + 0.998955i \(0.514551\pi\)
\(354\) −1962.09 −0.294588
\(355\) −16050.3 −2.39961
\(356\) 5784.02 0.861102
\(357\) −417.379 −0.0618768
\(358\) −230.818 −0.0340757
\(359\) 8527.32 1.25363 0.626817 0.779166i \(-0.284357\pi\)
0.626817 + 0.779166i \(0.284357\pi\)
\(360\) 3343.54 0.489501
\(361\) −995.934 −0.145201
\(362\) 4679.95 0.679483
\(363\) −363.000 −0.0524864
\(364\) −1755.75 −0.252819
\(365\) −14997.3 −2.15067
\(366\) 105.477 0.0150639
\(367\) 5504.16 0.782874 0.391437 0.920205i \(-0.371978\pi\)
0.391437 + 0.920205i \(0.371978\pi\)
\(368\) −1487.59 −0.210723
\(369\) −30.0306 −0.00423667
\(370\) 6799.05 0.955312
\(371\) −2876.70 −0.402563
\(372\) 2786.96 0.388434
\(373\) −7091.36 −0.984387 −0.492194 0.870486i \(-0.663805\pi\)
−0.492194 + 0.870486i \(0.663805\pi\)
\(374\) 349.799 0.0483627
\(375\) 2512.03 0.345921
\(376\) −8407.39 −1.15313
\(377\) 7324.32 1.00059
\(378\) −302.396 −0.0411470
\(379\) −77.3370 −0.0104816 −0.00524081 0.999986i \(-0.501668\pi\)
−0.00524081 + 0.999986i \(0.501668\pi\)
\(380\) 7196.40 0.971493
\(381\) 6875.38 0.924505
\(382\) −6034.65 −0.808271
\(383\) 4843.75 0.646224 0.323112 0.946361i \(-0.395271\pi\)
0.323112 + 0.946361i \(0.395271\pi\)
\(384\) 3395.59 0.451251
\(385\) −1330.27 −0.176096
\(386\) −2052.65 −0.270666
\(387\) −1105.19 −0.145167
\(388\) −7700.41 −1.00755
\(389\) 9882.00 1.28801 0.644007 0.765020i \(-0.277271\pi\)
0.644007 + 0.765020i \(0.277271\pi\)
\(390\) 3823.35 0.496417
\(391\) −3243.74 −0.419547
\(392\) −1053.69 −0.135763
\(393\) 4475.32 0.574428
\(394\) 4883.63 0.624451
\(395\) −2638.11 −0.336045
\(396\) −538.566 −0.0683434
\(397\) −10135.5 −1.28133 −0.640663 0.767823i \(-0.721340\pi\)
−0.640663 + 0.767823i \(0.721340\pi\)
\(398\) 8064.35 1.01565
\(399\) −1607.98 −0.201754
\(400\) 1581.13 0.197641
\(401\) 9220.96 1.14831 0.574155 0.818746i \(-0.305331\pi\)
0.574155 + 0.818746i \(0.305331\pi\)
\(402\) −2968.40 −0.368284
\(403\) 7873.45 0.973212
\(404\) 4211.37 0.518623
\(405\) −1399.37 −0.171692
\(406\) 1779.18 0.217486
\(407\) −2705.69 −0.329523
\(408\) 1282.18 0.155582
\(409\) −5710.89 −0.690429 −0.345214 0.938524i \(-0.612194\pi\)
−0.345214 + 0.938524i \(0.612194\pi\)
\(410\) 92.2327 0.0111099
\(411\) −7772.36 −0.932803
\(412\) −4129.88 −0.493846
\(413\) 2861.42 0.340923
\(414\) −2350.13 −0.278992
\(415\) 1743.59 0.206240
\(416\) 8604.07 1.01406
\(417\) −8022.41 −0.942108
\(418\) 1347.63 0.157690
\(419\) 4520.73 0.527093 0.263547 0.964647i \(-0.415108\pi\)
0.263547 + 0.964647i \(0.415108\pi\)
\(420\) −1973.66 −0.229297
\(421\) 3798.26 0.439705 0.219853 0.975533i \(-0.429442\pi\)
0.219853 + 0.975533i \(0.429442\pi\)
\(422\) 5689.98 0.656359
\(423\) 3518.75 0.404462
\(424\) 8837.15 1.01219
\(425\) 3447.71 0.393502
\(426\) −4459.34 −0.507173
\(427\) −153.823 −0.0174333
\(428\) 6425.92 0.725721
\(429\) −1521.50 −0.171233
\(430\) 3394.35 0.380674
\(431\) −3468.51 −0.387638 −0.193819 0.981037i \(-0.562088\pi\)
−0.193819 + 0.981037i \(0.562088\pi\)
\(432\) −246.100 −0.0274086
\(433\) −9859.98 −1.09432 −0.547160 0.837028i \(-0.684291\pi\)
−0.547160 + 0.837028i \(0.684291\pi\)
\(434\) 1912.57 0.211536
\(435\) 8233.37 0.907494
\(436\) −4015.26 −0.441046
\(437\) −12496.8 −1.36796
\(438\) −4166.78 −0.454558
\(439\) 8464.33 0.920228 0.460114 0.887860i \(-0.347808\pi\)
0.460114 + 0.887860i \(0.347808\pi\)
\(440\) 4086.55 0.442770
\(441\) 441.000 0.0476190
\(442\) 1466.17 0.157780
\(443\) −7825.41 −0.839270 −0.419635 0.907693i \(-0.637842\pi\)
−0.419635 + 0.907693i \(0.637842\pi\)
\(444\) −4014.30 −0.429077
\(445\) 18368.5 1.95675
\(446\) 2828.52 0.300302
\(447\) −2467.06 −0.261047
\(448\) 1579.62 0.166585
\(449\) 1050.94 0.110461 0.0552303 0.998474i \(-0.482411\pi\)
0.0552303 + 0.998474i \(0.482411\pi\)
\(450\) 2497.91 0.261672
\(451\) −36.7041 −0.00383221
\(452\) −3373.83 −0.351087
\(453\) 416.408 0.0431889
\(454\) 7738.68 0.799988
\(455\) −5575.79 −0.574499
\(456\) 4939.69 0.507285
\(457\) 14397.2 1.47369 0.736843 0.676064i \(-0.236316\pi\)
0.736843 + 0.676064i \(0.236316\pi\)
\(458\) 7905.98 0.806599
\(459\) −536.630 −0.0545702
\(460\) −15338.7 −1.55472
\(461\) −5755.83 −0.581509 −0.290755 0.956798i \(-0.593906\pi\)
−0.290755 + 0.956798i \(0.593906\pi\)
\(462\) −369.595 −0.0372189
\(463\) −10056.0 −1.00938 −0.504689 0.863302i \(-0.668393\pi\)
−0.504689 + 0.863302i \(0.668393\pi\)
\(464\) 1447.96 0.144870
\(465\) 8850.67 0.882666
\(466\) 7715.32 0.766964
\(467\) 2607.32 0.258356 0.129178 0.991621i \(-0.458766\pi\)
0.129178 + 0.991621i \(0.458766\pi\)
\(468\) −2257.39 −0.222965
\(469\) 4328.97 0.426211
\(470\) −10807.1 −1.06062
\(471\) 1904.24 0.186290
\(472\) −8790.21 −0.857208
\(473\) −1350.78 −0.131309
\(474\) −732.960 −0.0710253
\(475\) 13282.5 1.28304
\(476\) −756.856 −0.0728791
\(477\) −3698.61 −0.355027
\(478\) 7714.79 0.738214
\(479\) −5944.97 −0.567083 −0.283541 0.958960i \(-0.591509\pi\)
−0.283541 + 0.958960i \(0.591509\pi\)
\(480\) 9671.96 0.919714
\(481\) −11340.8 −1.07504
\(482\) 2471.03 0.233511
\(483\) 3427.32 0.322874
\(484\) −658.248 −0.0618189
\(485\) −24454.5 −2.28953
\(486\) −388.795 −0.0362883
\(487\) 15895.1 1.47901 0.739504 0.673152i \(-0.235060\pi\)
0.739504 + 0.673152i \(0.235060\pi\)
\(488\) 472.540 0.0438338
\(489\) 1825.10 0.168781
\(490\) −1354.44 −0.124872
\(491\) 3655.59 0.335997 0.167999 0.985787i \(-0.446270\pi\)
0.167999 + 0.985787i \(0.446270\pi\)
\(492\) −54.4561 −0.00498998
\(493\) 3157.32 0.288435
\(494\) 5648.54 0.514453
\(495\) −1710.35 −0.155302
\(496\) 1556.52 0.140907
\(497\) 6503.28 0.586946
\(498\) 484.431 0.0435901
\(499\) 21171.9 1.89937 0.949684 0.313208i \(-0.101404\pi\)
0.949684 + 0.313208i \(0.101404\pi\)
\(500\) 4555.19 0.407429
\(501\) 4636.50 0.413460
\(502\) −12109.0 −1.07659
\(503\) −12270.3 −1.08768 −0.543842 0.839188i \(-0.683031\pi\)
−0.543842 + 0.839188i \(0.683031\pi\)
\(504\) −1354.74 −0.119732
\(505\) 13374.2 1.17850
\(506\) −2872.38 −0.252357
\(507\) 213.656 0.0187155
\(508\) 12467.5 1.08889
\(509\) −3168.41 −0.275908 −0.137954 0.990439i \(-0.544053\pi\)
−0.137954 + 0.990439i \(0.544053\pi\)
\(510\) 1648.15 0.143100
\(511\) 6076.63 0.526056
\(512\) 3268.99 0.282168
\(513\) −2067.41 −0.177930
\(514\) 2264.27 0.194305
\(515\) −13115.4 −1.12220
\(516\) −2004.09 −0.170979
\(517\) 4300.69 0.365849
\(518\) −2754.85 −0.233670
\(519\) −379.794 −0.0321216
\(520\) 17128.7 1.44450
\(521\) −1305.16 −0.109751 −0.0548755 0.998493i \(-0.517476\pi\)
−0.0548755 + 0.998493i \(0.517476\pi\)
\(522\) 2287.52 0.191805
\(523\) 707.998 0.0591943 0.0295971 0.999562i \(-0.490578\pi\)
0.0295971 + 0.999562i \(0.490578\pi\)
\(524\) 8115.35 0.676566
\(525\) −3642.83 −0.302830
\(526\) −12600.8 −1.04452
\(527\) 3394.04 0.280544
\(528\) −300.789 −0.0247920
\(529\) 14469.0 1.18920
\(530\) 11359.5 0.930992
\(531\) 3678.97 0.300666
\(532\) −2915.85 −0.237628
\(533\) −153.844 −0.0125023
\(534\) 5103.42 0.413570
\(535\) 20407.0 1.64911
\(536\) −13298.5 −1.07165
\(537\) 432.789 0.0347788
\(538\) −3299.17 −0.264382
\(539\) 539.000 0.0430730
\(540\) −2537.56 −0.202221
\(541\) −5424.05 −0.431050 −0.215525 0.976498i \(-0.569146\pi\)
−0.215525 + 0.976498i \(0.569146\pi\)
\(542\) −13195.1 −1.04572
\(543\) −8775.02 −0.693503
\(544\) 3708.99 0.292319
\(545\) −12751.4 −1.00222
\(546\) −1549.15 −0.121424
\(547\) −15353.7 −1.20014 −0.600071 0.799947i \(-0.704861\pi\)
−0.600071 + 0.799947i \(0.704861\pi\)
\(548\) −14094.0 −1.09866
\(549\) −197.772 −0.0153747
\(550\) 3053.00 0.236691
\(551\) 12163.8 0.940465
\(552\) −10528.6 −0.811826
\(553\) 1068.91 0.0821968
\(554\) −10619.6 −0.814410
\(555\) −12748.4 −0.975024
\(556\) −14547.5 −1.10962
\(557\) 13685.2 1.04104 0.520519 0.853850i \(-0.325738\pi\)
0.520519 + 0.853850i \(0.325738\pi\)
\(558\) 2459.02 0.186557
\(559\) −5661.77 −0.428385
\(560\) −1102.29 −0.0831789
\(561\) −655.881 −0.0493606
\(562\) 8961.21 0.672608
\(563\) −1565.67 −0.117203 −0.0586015 0.998281i \(-0.518664\pi\)
−0.0586015 + 0.998281i \(0.518664\pi\)
\(564\) 6380.73 0.476378
\(565\) −10714.4 −0.797801
\(566\) −1495.57 −0.111066
\(567\) 567.000 0.0419961
\(568\) −19977.9 −1.47580
\(569\) 20508.1 1.51097 0.755487 0.655164i \(-0.227400\pi\)
0.755487 + 0.655164i \(0.227400\pi\)
\(570\) 6349.61 0.466589
\(571\) 971.535 0.0712040 0.0356020 0.999366i \(-0.488665\pi\)
0.0356020 + 0.999366i \(0.488665\pi\)
\(572\) −2759.03 −0.201680
\(573\) 11315.1 0.824949
\(574\) −37.3709 −0.00271748
\(575\) −28310.9 −2.05330
\(576\) 2030.94 0.146914
\(577\) 13961.1 1.00729 0.503647 0.863910i \(-0.331991\pi\)
0.503647 + 0.863910i \(0.331991\pi\)
\(578\) −7228.67 −0.520196
\(579\) 3848.77 0.276251
\(580\) 14930.0 1.06885
\(581\) −706.471 −0.0504464
\(582\) −6794.31 −0.483906
\(583\) −4520.53 −0.321134
\(584\) −18667.3 −1.32270
\(585\) −7168.87 −0.506660
\(586\) 14671.6 1.03426
\(587\) −23834.3 −1.67589 −0.837946 0.545753i \(-0.816244\pi\)
−0.837946 + 0.545753i \(0.816244\pi\)
\(588\) 799.690 0.0560861
\(589\) 13075.8 0.914735
\(590\) −11299.2 −0.788440
\(591\) −9156.93 −0.637336
\(592\) −2241.99 −0.155651
\(593\) 25102.9 1.73837 0.869184 0.494489i \(-0.164645\pi\)
0.869184 + 0.494489i \(0.164645\pi\)
\(594\) −475.194 −0.0328240
\(595\) −2403.58 −0.165608
\(596\) −4473.66 −0.307464
\(597\) −15120.9 −1.03661
\(598\) −12039.5 −0.823297
\(599\) −12122.2 −0.826875 −0.413437 0.910533i \(-0.635672\pi\)
−0.413437 + 0.910533i \(0.635672\pi\)
\(600\) 11190.7 0.761429
\(601\) 20672.9 1.40310 0.701552 0.712619i \(-0.252491\pi\)
0.701552 + 0.712619i \(0.252491\pi\)
\(602\) −1375.32 −0.0931130
\(603\) 5565.81 0.375883
\(604\) 755.096 0.0508682
\(605\) −2090.42 −0.140476
\(606\) 3715.82 0.249084
\(607\) −3571.58 −0.238824 −0.119412 0.992845i \(-0.538101\pi\)
−0.119412 + 0.992845i \(0.538101\pi\)
\(608\) 14289.2 0.953129
\(609\) −3336.01 −0.221973
\(610\) 607.416 0.0403173
\(611\) 18026.2 1.19356
\(612\) −973.101 −0.0642733
\(613\) −25996.1 −1.71284 −0.856420 0.516280i \(-0.827317\pi\)
−0.856420 + 0.516280i \(0.827317\pi\)
\(614\) −9909.81 −0.651347
\(615\) −172.938 −0.0113391
\(616\) −1655.79 −0.108302
\(617\) 7266.40 0.474123 0.237062 0.971495i \(-0.423816\pi\)
0.237062 + 0.971495i \(0.423816\pi\)
\(618\) −3643.92 −0.237184
\(619\) 16227.6 1.05371 0.526853 0.849957i \(-0.323372\pi\)
0.526853 + 0.849957i \(0.323372\pi\)
\(620\) 16049.4 1.03961
\(621\) 4406.55 0.284748
\(622\) −5397.65 −0.347952
\(623\) −7442.58 −0.478621
\(624\) −1260.75 −0.0808821
\(625\) −7217.38 −0.461912
\(626\) 11480.0 0.732960
\(627\) −2526.83 −0.160944
\(628\) 3453.06 0.219414
\(629\) −4888.73 −0.309899
\(630\) −1741.42 −0.110127
\(631\) −3162.02 −0.199490 −0.0997450 0.995013i \(-0.531803\pi\)
−0.0997450 + 0.995013i \(0.531803\pi\)
\(632\) −3283.68 −0.206674
\(633\) −10668.8 −0.669903
\(634\) −10083.2 −0.631630
\(635\) 39593.5 2.47436
\(636\) −6706.90 −0.418154
\(637\) 2259.20 0.140523
\(638\) 2795.86 0.173494
\(639\) 8361.36 0.517638
\(640\) 19554.3 1.20774
\(641\) −8679.48 −0.534819 −0.267409 0.963583i \(-0.586168\pi\)
−0.267409 + 0.963583i \(0.586168\pi\)
\(642\) 5669.79 0.348550
\(643\) −19226.2 −1.17917 −0.589586 0.807705i \(-0.700709\pi\)
−0.589586 + 0.807705i \(0.700709\pi\)
\(644\) 6214.94 0.380284
\(645\) −6364.48 −0.388529
\(646\) 2434.94 0.148299
\(647\) 11211.6 0.681258 0.340629 0.940198i \(-0.389360\pi\)
0.340629 + 0.940198i \(0.389360\pi\)
\(648\) −1741.81 −0.105594
\(649\) 4496.52 0.271963
\(650\) 12796.5 0.772188
\(651\) −3586.12 −0.215901
\(652\) 3309.56 0.198792
\(653\) −12404.5 −0.743378 −0.371689 0.928357i \(-0.621221\pi\)
−0.371689 + 0.928357i \(0.621221\pi\)
\(654\) −3542.79 −0.211826
\(655\) 25772.2 1.53741
\(656\) −30.4138 −0.00181015
\(657\) 7812.81 0.463937
\(658\) 4378.83 0.259429
\(659\) 20954.9 1.23868 0.619339 0.785124i \(-0.287401\pi\)
0.619339 + 0.785124i \(0.287401\pi\)
\(660\) −3101.46 −0.182916
\(661\) −20349.0 −1.19741 −0.598703 0.800971i \(-0.704317\pi\)
−0.598703 + 0.800971i \(0.704317\pi\)
\(662\) −11846.3 −0.695498
\(663\) −2749.11 −0.161035
\(664\) 2170.26 0.126841
\(665\) −9259.96 −0.539979
\(666\) −3541.94 −0.206077
\(667\) −25926.4 −1.50506
\(668\) 8407.63 0.486977
\(669\) −5303.55 −0.306498
\(670\) −17094.2 −0.985683
\(671\) −241.722 −0.0139070
\(672\) −3918.90 −0.224962
\(673\) 31712.5 1.81639 0.908193 0.418553i \(-0.137462\pi\)
0.908193 + 0.418553i \(0.137462\pi\)
\(674\) 11213.7 0.640857
\(675\) −4683.63 −0.267071
\(676\) 387.433 0.0220433
\(677\) 1195.61 0.0678744 0.0339372 0.999424i \(-0.489195\pi\)
0.0339372 + 0.999424i \(0.489195\pi\)
\(678\) −2976.83 −0.168620
\(679\) 9908.49 0.560019
\(680\) 7383.73 0.416402
\(681\) −14510.2 −0.816494
\(682\) 3005.47 0.168747
\(683\) 16709.8 0.936140 0.468070 0.883691i \(-0.344949\pi\)
0.468070 + 0.883691i \(0.344949\pi\)
\(684\) −3748.94 −0.209568
\(685\) −44759.0 −2.49657
\(686\) 548.793 0.0305438
\(687\) −14823.9 −0.823242
\(688\) −1119.29 −0.0620238
\(689\) −18947.7 −1.04768
\(690\) −13533.8 −0.746699
\(691\) −11154.4 −0.614088 −0.307044 0.951695i \(-0.599340\pi\)
−0.307044 + 0.951695i \(0.599340\pi\)
\(692\) −688.701 −0.0378331
\(693\) 693.000 0.0379869
\(694\) −16041.0 −0.877391
\(695\) −46199.0 −2.52148
\(696\) 10248.1 0.558124
\(697\) −66.3182 −0.00360399
\(698\) −17194.2 −0.932392
\(699\) −14466.4 −0.782790
\(700\) −6605.74 −0.356676
\(701\) 10388.9 0.559749 0.279874 0.960037i \(-0.409707\pi\)
0.279874 + 0.960037i \(0.409707\pi\)
\(702\) −1991.76 −0.107086
\(703\) −18834.2 −1.01045
\(704\) 2482.26 0.132889
\(705\) 20263.5 1.08251
\(706\) −969.805 −0.0516984
\(707\) −5418.98 −0.288263
\(708\) 6671.28 0.354127
\(709\) −4288.90 −0.227183 −0.113592 0.993528i \(-0.536236\pi\)
−0.113592 + 0.993528i \(0.536236\pi\)
\(710\) −25680.2 −1.35741
\(711\) 1374.32 0.0724908
\(712\) 22863.4 1.20343
\(713\) −27870.2 −1.46388
\(714\) −667.798 −0.0350024
\(715\) −8761.95 −0.458291
\(716\) 784.800 0.0409628
\(717\) −14465.4 −0.753446
\(718\) 13643.5 0.709153
\(719\) 31876.4 1.65339 0.826697 0.562647i \(-0.190217\pi\)
0.826697 + 0.562647i \(0.190217\pi\)
\(720\) −1417.23 −0.0733569
\(721\) 5314.12 0.274491
\(722\) −1593.47 −0.0821371
\(723\) −4633.24 −0.238329
\(724\) −15912.2 −0.816814
\(725\) 27556.7 1.41163
\(726\) −580.793 −0.0296904
\(727\) 14030.2 0.715754 0.357877 0.933769i \(-0.383501\pi\)
0.357877 + 0.933769i \(0.383501\pi\)
\(728\) −6940.22 −0.353326
\(729\) 729.000 0.0370370
\(730\) −23995.4 −1.21659
\(731\) −2440.64 −0.123489
\(732\) −358.631 −0.0181085
\(733\) −5466.09 −0.275436 −0.137718 0.990471i \(-0.543977\pi\)
−0.137718 + 0.990471i \(0.543977\pi\)
\(734\) 8806.54 0.442855
\(735\) 2539.60 0.127449
\(736\) −30456.5 −1.52533
\(737\) 6802.66 0.339999
\(738\) −48.0484 −0.00239659
\(739\) 28957.0 1.44141 0.720704 0.693243i \(-0.243819\pi\)
0.720704 + 0.693243i \(0.243819\pi\)
\(740\) −23117.3 −1.14839
\(741\) −10591.1 −0.525068
\(742\) −4602.66 −0.227721
\(743\) −17742.1 −0.876037 −0.438019 0.898966i \(-0.644320\pi\)
−0.438019 + 0.898966i \(0.644320\pi\)
\(744\) 11016.5 0.542855
\(745\) −14207.2 −0.698672
\(746\) −11346.0 −0.556846
\(747\) −908.320 −0.0444895
\(748\) −1189.35 −0.0581374
\(749\) −8268.55 −0.403373
\(750\) 4019.19 0.195680
\(751\) 30053.7 1.46029 0.730143 0.683294i \(-0.239453\pi\)
0.730143 + 0.683294i \(0.239453\pi\)
\(752\) 3563.64 0.172809
\(753\) 22704.6 1.09881
\(754\) 11718.8 0.566011
\(755\) 2397.99 0.115592
\(756\) 1028.17 0.0494633
\(757\) −8581.32 −0.412012 −0.206006 0.978551i \(-0.566047\pi\)
−0.206006 + 0.978551i \(0.566047\pi\)
\(758\) −123.738 −0.00592923
\(759\) 5385.78 0.257565
\(760\) 28446.4 1.35771
\(761\) 30413.3 1.44873 0.724363 0.689419i \(-0.242134\pi\)
0.724363 + 0.689419i \(0.242134\pi\)
\(762\) 11000.5 0.522972
\(763\) 5166.63 0.245144
\(764\) 20518.3 0.971632
\(765\) −3090.31 −0.146053
\(766\) 7749.90 0.365555
\(767\) 18847.0 0.887258
\(768\) 10848.7 0.509726
\(769\) −24735.2 −1.15992 −0.579958 0.814646i \(-0.696931\pi\)
−0.579958 + 0.814646i \(0.696931\pi\)
\(770\) −2128.40 −0.0996134
\(771\) −4245.56 −0.198314
\(772\) 6979.18 0.325371
\(773\) 25108.1 1.16827 0.584137 0.811655i \(-0.301433\pi\)
0.584137 + 0.811655i \(0.301433\pi\)
\(774\) −1768.27 −0.0821180
\(775\) 29622.7 1.37301
\(776\) −30438.7 −1.40810
\(777\) 5165.40 0.238491
\(778\) 15811.0 0.728601
\(779\) −255.496 −0.0117511
\(780\) −12999.7 −0.596749
\(781\) 10219.4 0.468221
\(782\) −5189.92 −0.237329
\(783\) −4289.15 −0.195762
\(784\) 446.627 0.0203456
\(785\) 10966.0 0.498591
\(786\) 7160.42 0.324941
\(787\) −36364.0 −1.64706 −0.823531 0.567271i \(-0.807999\pi\)
−0.823531 + 0.567271i \(0.807999\pi\)
\(788\) −16604.8 −0.750660
\(789\) 23626.7 1.06608
\(790\) −4220.93 −0.190094
\(791\) 4341.27 0.195142
\(792\) −2128.88 −0.0955132
\(793\) −1013.17 −0.0453704
\(794\) −16216.6 −0.724818
\(795\) −21299.4 −0.950202
\(796\) −27419.5 −1.22093
\(797\) −18605.4 −0.826897 −0.413448 0.910528i \(-0.635676\pi\)
−0.413448 + 0.910528i \(0.635676\pi\)
\(798\) −2572.74 −0.114128
\(799\) 7770.63 0.344062
\(800\) 32371.6 1.43063
\(801\) −9569.04 −0.422104
\(802\) 14753.3 0.649574
\(803\) 9548.99 0.419647
\(804\) 10092.8 0.442718
\(805\) 19737.0 0.864147
\(806\) 12597.4 0.550525
\(807\) 6186.02 0.269837
\(808\) 16647.0 0.724800
\(809\) 26846.8 1.16673 0.583364 0.812211i \(-0.301736\pi\)
0.583364 + 0.812211i \(0.301736\pi\)
\(810\) −2238.97 −0.0971227
\(811\) −32344.3 −1.40045 −0.700224 0.713924i \(-0.746916\pi\)
−0.700224 + 0.713924i \(0.746916\pi\)
\(812\) −6049.37 −0.261442
\(813\) 24741.2 1.06730
\(814\) −4329.04 −0.186404
\(815\) 10510.3 0.451729
\(816\) −543.477 −0.0233156
\(817\) −9402.76 −0.402645
\(818\) −9137.31 −0.390561
\(819\) 2904.69 0.123929
\(820\) −313.599 −0.0133553
\(821\) 16953.3 0.720677 0.360338 0.932822i \(-0.382661\pi\)
0.360338 + 0.932822i \(0.382661\pi\)
\(822\) −12435.6 −0.527667
\(823\) 37814.0 1.60159 0.800797 0.598936i \(-0.204410\pi\)
0.800797 + 0.598936i \(0.204410\pi\)
\(824\) −16324.8 −0.690173
\(825\) −5724.44 −0.241575
\(826\) 4578.21 0.192853
\(827\) −11894.6 −0.500141 −0.250071 0.968228i \(-0.580454\pi\)
−0.250071 + 0.968228i \(0.580454\pi\)
\(828\) 7990.64 0.335379
\(829\) 30744.7 1.28807 0.644033 0.764998i \(-0.277260\pi\)
0.644033 + 0.764998i \(0.277260\pi\)
\(830\) 2789.71 0.116665
\(831\) 19912.0 0.831215
\(832\) 10404.3 0.433540
\(833\) 973.884 0.0405079
\(834\) −12835.7 −0.532930
\(835\) 26700.4 1.10659
\(836\) −4582.04 −0.189561
\(837\) −4610.73 −0.190406
\(838\) 7233.08 0.298165
\(839\) −24677.8 −1.01546 −0.507731 0.861516i \(-0.669516\pi\)
−0.507731 + 0.861516i \(0.669516\pi\)
\(840\) −7801.60 −0.320453
\(841\) 846.708 0.0347168
\(842\) 6077.14 0.248732
\(843\) −16802.5 −0.686486
\(844\) −19346.4 −0.789017
\(845\) 1230.39 0.0500907
\(846\) 5629.92 0.228795
\(847\) 847.000 0.0343604
\(848\) −3745.80 −0.151688
\(849\) 2804.24 0.113358
\(850\) 5516.26 0.222595
\(851\) 40143.9 1.61706
\(852\) 15162.1 0.609678
\(853\) −1855.69 −0.0744873 −0.0372437 0.999306i \(-0.511858\pi\)
−0.0372437 + 0.999306i \(0.511858\pi\)
\(854\) −246.114 −0.00986163
\(855\) −11905.7 −0.476217
\(856\) 25400.8 1.01423
\(857\) 3984.84 0.158832 0.0794162 0.996842i \(-0.474694\pi\)
0.0794162 + 0.996842i \(0.474694\pi\)
\(858\) −2434.38 −0.0968627
\(859\) −12526.1 −0.497538 −0.248769 0.968563i \(-0.580026\pi\)
−0.248769 + 0.968563i \(0.580026\pi\)
\(860\) −11541.1 −0.457613
\(861\) 70.0714 0.00277355
\(862\) −5549.54 −0.219279
\(863\) −22040.9 −0.869387 −0.434693 0.900579i \(-0.643143\pi\)
−0.434693 + 0.900579i \(0.643143\pi\)
\(864\) −5038.58 −0.198398
\(865\) −2187.14 −0.0859709
\(866\) −15775.8 −0.619033
\(867\) 13553.9 0.530929
\(868\) −6502.91 −0.254290
\(869\) 1679.72 0.0655704
\(870\) 13173.2 0.513350
\(871\) 28513.2 1.10922
\(872\) −15871.8 −0.616383
\(873\) 12739.5 0.493891
\(874\) −19994.5 −0.773828
\(875\) −5861.39 −0.226459
\(876\) 14167.4 0.546430
\(877\) 21039.5 0.810096 0.405048 0.914295i \(-0.367255\pi\)
0.405048 + 0.914295i \(0.367255\pi\)
\(878\) 13542.7 0.520553
\(879\) −27509.6 −1.05560
\(880\) −1732.17 −0.0663538
\(881\) 33610.9 1.28533 0.642667 0.766146i \(-0.277828\pi\)
0.642667 + 0.766146i \(0.277828\pi\)
\(882\) 705.591 0.0269371
\(883\) −4618.52 −0.176020 −0.0880099 0.996120i \(-0.528051\pi\)
−0.0880099 + 0.996120i \(0.528051\pi\)
\(884\) −4985.11 −0.189669
\(885\) 21186.2 0.804709
\(886\) −12520.5 −0.474757
\(887\) −40095.6 −1.51779 −0.758895 0.651213i \(-0.774260\pi\)
−0.758895 + 0.651213i \(0.774260\pi\)
\(888\) −15868.0 −0.599656
\(889\) −16042.6 −0.605231
\(890\) 29389.3 1.10689
\(891\) 891.000 0.0335013
\(892\) −9617.22 −0.360996
\(893\) 29937.0 1.12184
\(894\) −3947.25 −0.147669
\(895\) 2492.32 0.0930828
\(896\) −7923.03 −0.295413
\(897\) 22574.4 0.840285
\(898\) 1681.48 0.0624852
\(899\) 27127.7 1.00641
\(900\) −8493.09 −0.314559
\(901\) −8167.85 −0.302009
\(902\) −58.7258 −0.00216780
\(903\) 2578.77 0.0950343
\(904\) −13336.3 −0.490661
\(905\) −50533.1 −1.85611
\(906\) 666.244 0.0244310
\(907\) −52281.2 −1.91397 −0.956984 0.290141i \(-0.906298\pi\)
−0.956984 + 0.290141i \(0.906298\pi\)
\(908\) −26312.2 −0.961674
\(909\) −6967.26 −0.254224
\(910\) −8921.15 −0.324981
\(911\) 30635.6 1.11416 0.557081 0.830458i \(-0.311921\pi\)
0.557081 + 0.830458i \(0.311921\pi\)
\(912\) −2093.79 −0.0760221
\(913\) −1110.17 −0.0402423
\(914\) 23035.3 0.833632
\(915\) −1138.92 −0.0411492
\(916\) −26881.0 −0.969622
\(917\) −10442.4 −0.376051
\(918\) −858.597 −0.0308692
\(919\) −14574.9 −0.523159 −0.261579 0.965182i \(-0.584243\pi\)
−0.261579 + 0.965182i \(0.584243\pi\)
\(920\) −60631.6 −2.17279
\(921\) 18581.1 0.664787
\(922\) −9209.21 −0.328947
\(923\) 42834.5 1.52754
\(924\) 1256.66 0.0447413
\(925\) −42668.1 −1.51667
\(926\) −16089.4 −0.570983
\(927\) 6832.44 0.242078
\(928\) 29645.1 1.04865
\(929\) −18955.8 −0.669451 −0.334726 0.942316i \(-0.608644\pi\)
−0.334726 + 0.942316i \(0.608644\pi\)
\(930\) 14160.9 0.499305
\(931\) 3751.96 0.132079
\(932\) −26232.7 −0.921976
\(933\) 10120.7 0.355132
\(934\) 4171.66 0.146147
\(935\) −3777.05 −0.132110
\(936\) −8923.14 −0.311605
\(937\) 6520.68 0.227344 0.113672 0.993518i \(-0.463739\pi\)
0.113672 + 0.993518i \(0.463739\pi\)
\(938\) 6926.26 0.241098
\(939\) −21525.3 −0.748084
\(940\) 36745.0 1.27499
\(941\) 12631.4 0.437589 0.218795 0.975771i \(-0.429788\pi\)
0.218795 + 0.975771i \(0.429788\pi\)
\(942\) 3046.74 0.105380
\(943\) 544.573 0.0188057
\(944\) 3725.91 0.128462
\(945\) 3265.21 0.112399
\(946\) −2161.22 −0.0742785
\(947\) −35170.9 −1.20686 −0.603432 0.797415i \(-0.706200\pi\)
−0.603432 + 0.797415i \(0.706200\pi\)
\(948\) 2492.13 0.0853803
\(949\) 40024.3 1.36907
\(950\) 21251.8 0.725789
\(951\) 18906.2 0.644663
\(952\) −2991.75 −0.101852
\(953\) 39539.0 1.34396 0.671980 0.740570i \(-0.265444\pi\)
0.671980 + 0.740570i \(0.265444\pi\)
\(954\) −5917.71 −0.200831
\(955\) 65160.8 2.20791
\(956\) −26230.9 −0.887415
\(957\) −5242.30 −0.177074
\(958\) −9511.83 −0.320786
\(959\) 18135.5 0.610663
\(960\) 11695.7 0.393204
\(961\) −629.373 −0.0211263
\(962\) −18145.1 −0.608129
\(963\) −10631.0 −0.355741
\(964\) −8401.70 −0.280706
\(965\) 22164.0 0.739363
\(966\) 5483.63 0.182643
\(967\) −29986.0 −0.997194 −0.498597 0.866834i \(-0.666151\pi\)
−0.498597 + 0.866834i \(0.666151\pi\)
\(968\) −2601.96 −0.0863949
\(969\) −4565.57 −0.151359
\(970\) −39126.7 −1.29514
\(971\) −14879.8 −0.491777 −0.245889 0.969298i \(-0.579080\pi\)
−0.245889 + 0.969298i \(0.579080\pi\)
\(972\) 1321.94 0.0436225
\(973\) 18719.0 0.616755
\(974\) 25431.9 0.836643
\(975\) −23993.8 −0.788121
\(976\) −200.296 −0.00656896
\(977\) 40864.1 1.33813 0.669067 0.743202i \(-0.266694\pi\)
0.669067 + 0.743202i \(0.266694\pi\)
\(978\) 2920.13 0.0954758
\(979\) −11695.5 −0.381807
\(980\) 4605.21 0.150110
\(981\) 6642.82 0.216197
\(982\) 5848.88 0.190066
\(983\) 48180.6 1.56330 0.781650 0.623718i \(-0.214379\pi\)
0.781650 + 0.623718i \(0.214379\pi\)
\(984\) −215.258 −0.00697374
\(985\) −52732.4 −1.70578
\(986\) 5051.65 0.163162
\(987\) −8210.41 −0.264782
\(988\) −19205.5 −0.618430
\(989\) 20041.4 0.644367
\(990\) −2736.52 −0.0878508
\(991\) −22380.4 −0.717392 −0.358696 0.933454i \(-0.616779\pi\)
−0.358696 + 0.933454i \(0.616779\pi\)
\(992\) 31867.7 1.01996
\(993\) 22212.1 0.709848
\(994\) 10405.1 0.332023
\(995\) −87077.1 −2.77440
\(996\) −1647.11 −0.0524002
\(997\) −12425.1 −0.394691 −0.197346 0.980334i \(-0.563232\pi\)
−0.197346 + 0.980334i \(0.563232\pi\)
\(998\) 33874.6 1.07443
\(999\) 6641.23 0.210330
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 231.4.a.k.1.4 5
3.2 odd 2 693.4.a.p.1.2 5
7.6 odd 2 1617.4.a.n.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.4.a.k.1.4 5 1.1 even 1 trivial
693.4.a.p.1.2 5 3.2 odd 2
1617.4.a.n.1.4 5 7.6 odd 2