Properties

Label 1521.4.a.bg.1.8
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, 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("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(4.83438\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.83438 q^{2} +6.70249 q^{4} -11.3710 q^{5} +31.0623 q^{7} -4.97517 q^{8} +O(q^{10})\) \(q+3.83438 q^{2} +6.70249 q^{4} -11.3710 q^{5} +31.0623 q^{7} -4.97517 q^{8} -43.6008 q^{10} +20.9478 q^{11} +119.105 q^{14} -72.6966 q^{16} -114.389 q^{17} -45.1768 q^{19} -76.2140 q^{20} +80.3219 q^{22} -73.9590 q^{23} +4.29980 q^{25} +208.194 q^{28} +27.2113 q^{29} +179.587 q^{31} -238.945 q^{32} -438.610 q^{34} -353.209 q^{35} +354.736 q^{37} -173.225 q^{38} +56.5727 q^{40} -81.4187 q^{41} -256.018 q^{43} +140.402 q^{44} -283.587 q^{46} -463.501 q^{47} +621.865 q^{49} +16.4871 q^{50} -76.6055 q^{53} -238.198 q^{55} -154.540 q^{56} +104.338 q^{58} -54.4676 q^{59} -494.496 q^{61} +688.606 q^{62} -334.634 q^{64} -611.991 q^{67} -766.688 q^{68} -1354.34 q^{70} -16.3056 q^{71} +321.825 q^{73} +1360.19 q^{74} -302.797 q^{76} +650.686 q^{77} +385.324 q^{79} +826.633 q^{80} -312.190 q^{82} -663.760 q^{83} +1300.71 q^{85} -981.671 q^{86} -104.219 q^{88} -545.723 q^{89} -495.709 q^{92} -1777.24 q^{94} +513.706 q^{95} -689.189 q^{97} +2384.47 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8} - 147 q^{10} - 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} + 161 q^{19} - 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} + 344 q^{28} - 280 q^{29} + 706 q^{31} - 680 q^{32} + 216 q^{34} - 20 q^{35} + 298 q^{37} + 739 q^{38} + 13 q^{40} - 1201 q^{41} - 533 q^{43} - 355 q^{44} - 840 q^{46} - 956 q^{47} + 403 q^{49} + 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} - 2877 q^{58} - 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} - 931 q^{67} + 1536 q^{68} - 4854 q^{70} - 2046 q^{71} - 45 q^{73} + 1990 q^{74} - 3608 q^{76} + 718 q^{77} + 412 q^{79} + 787 q^{80} + 2757 q^{82} - 3709 q^{83} - 2106 q^{85} - 125 q^{86} - 636 q^{88} - 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} - 1087 q^{97} + 282 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.83438 1.35566 0.677829 0.735219i \(-0.262921\pi\)
0.677829 + 0.735219i \(0.262921\pi\)
\(3\) 0 0
\(4\) 6.70249 0.837811
\(5\) −11.3710 −1.01705 −0.508527 0.861046i \(-0.669810\pi\)
−0.508527 + 0.861046i \(0.669810\pi\)
\(6\) 0 0
\(7\) 31.0623 1.67721 0.838603 0.544744i \(-0.183373\pi\)
0.838603 + 0.544744i \(0.183373\pi\)
\(8\) −4.97517 −0.219873
\(9\) 0 0
\(10\) −43.6008 −1.37878
\(11\) 20.9478 0.574182 0.287091 0.957903i \(-0.407312\pi\)
0.287091 + 0.957903i \(0.407312\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 119.105 2.27372
\(15\) 0 0
\(16\) −72.6966 −1.13588
\(17\) −114.389 −1.63196 −0.815980 0.578080i \(-0.803802\pi\)
−0.815980 + 0.578080i \(0.803802\pi\)
\(18\) 0 0
\(19\) −45.1768 −0.545488 −0.272744 0.962087i \(-0.587931\pi\)
−0.272744 + 0.962087i \(0.587931\pi\)
\(20\) −76.2140 −0.852098
\(21\) 0 0
\(22\) 80.3219 0.778395
\(23\) −73.9590 −0.670501 −0.335250 0.942129i \(-0.608821\pi\)
−0.335250 + 0.942129i \(0.608821\pi\)
\(24\) 0 0
\(25\) 4.29980 0.0343984
\(26\) 0 0
\(27\) 0 0
\(28\) 208.194 1.40518
\(29\) 27.2113 0.174242 0.0871208 0.996198i \(-0.472233\pi\)
0.0871208 + 0.996198i \(0.472233\pi\)
\(30\) 0 0
\(31\) 179.587 1.04048 0.520239 0.854021i \(-0.325843\pi\)
0.520239 + 0.854021i \(0.325843\pi\)
\(32\) −238.945 −1.32000
\(33\) 0 0
\(34\) −438.610 −2.21238
\(35\) −353.209 −1.70581
\(36\) 0 0
\(37\) 354.736 1.57617 0.788085 0.615567i \(-0.211073\pi\)
0.788085 + 0.615567i \(0.211073\pi\)
\(38\) −173.225 −0.739496
\(39\) 0 0
\(40\) 56.5727 0.223623
\(41\) −81.4187 −0.310133 −0.155067 0.987904i \(-0.549559\pi\)
−0.155067 + 0.987904i \(0.549559\pi\)
\(42\) 0 0
\(43\) −256.018 −0.907962 −0.453981 0.891011i \(-0.649997\pi\)
−0.453981 + 0.891011i \(0.649997\pi\)
\(44\) 140.402 0.481056
\(45\) 0 0
\(46\) −283.587 −0.908970
\(47\) −463.501 −1.43848 −0.719240 0.694762i \(-0.755510\pi\)
−0.719240 + 0.694762i \(0.755510\pi\)
\(48\) 0 0
\(49\) 621.865 1.81302
\(50\) 16.4871 0.0466325
\(51\) 0 0
\(52\) 0 0
\(53\) −76.6055 −0.198539 −0.0992695 0.995061i \(-0.531651\pi\)
−0.0992695 + 0.995061i \(0.531651\pi\)
\(54\) 0 0
\(55\) −238.198 −0.583974
\(56\) −154.540 −0.368773
\(57\) 0 0
\(58\) 104.338 0.236212
\(59\) −54.4676 −0.120188 −0.0600939 0.998193i \(-0.519140\pi\)
−0.0600939 + 0.998193i \(0.519140\pi\)
\(60\) 0 0
\(61\) −494.496 −1.03793 −0.518965 0.854795i \(-0.673683\pi\)
−0.518965 + 0.854795i \(0.673683\pi\)
\(62\) 688.606 1.41053
\(63\) 0 0
\(64\) −334.634 −0.653582
\(65\) 0 0
\(66\) 0 0
\(67\) −611.991 −1.11592 −0.557960 0.829868i \(-0.688416\pi\)
−0.557960 + 0.829868i \(0.688416\pi\)
\(68\) −766.688 −1.36727
\(69\) 0 0
\(70\) −1354.34 −2.31249
\(71\) −16.3056 −0.0272551 −0.0136276 0.999907i \(-0.504338\pi\)
−0.0136276 + 0.999907i \(0.504338\pi\)
\(72\) 0 0
\(73\) 321.825 0.515983 0.257992 0.966147i \(-0.416939\pi\)
0.257992 + 0.966147i \(0.416939\pi\)
\(74\) 1360.19 2.13675
\(75\) 0 0
\(76\) −302.797 −0.457016
\(77\) 650.686 0.963021
\(78\) 0 0
\(79\) 385.324 0.548764 0.274382 0.961621i \(-0.411527\pi\)
0.274382 + 0.961621i \(0.411527\pi\)
\(80\) 826.633 1.15526
\(81\) 0 0
\(82\) −312.190 −0.420435
\(83\) −663.760 −0.877797 −0.438899 0.898537i \(-0.644631\pi\)
−0.438899 + 0.898537i \(0.644631\pi\)
\(84\) 0 0
\(85\) 1300.71 1.65979
\(86\) −981.671 −1.23089
\(87\) 0 0
\(88\) −104.219 −0.126247
\(89\) −545.723 −0.649961 −0.324980 0.945721i \(-0.605358\pi\)
−0.324980 + 0.945721i \(0.605358\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −495.709 −0.561753
\(93\) 0 0
\(94\) −1777.24 −1.95009
\(95\) 513.706 0.554791
\(96\) 0 0
\(97\) −689.189 −0.721408 −0.360704 0.932680i \(-0.617464\pi\)
−0.360704 + 0.932680i \(0.617464\pi\)
\(98\) 2384.47 2.45783
\(99\) 0 0
\(100\) 28.8194 0.0288194
\(101\) −471.398 −0.464414 −0.232207 0.972666i \(-0.574595\pi\)
−0.232207 + 0.972666i \(0.574595\pi\)
\(102\) 0 0
\(103\) −335.521 −0.320970 −0.160485 0.987038i \(-0.551306\pi\)
−0.160485 + 0.987038i \(0.551306\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −293.735 −0.269151
\(107\) −1629.26 −1.47203 −0.736014 0.676967i \(-0.763294\pi\)
−0.736014 + 0.676967i \(0.763294\pi\)
\(108\) 0 0
\(109\) −1518.95 −1.33476 −0.667381 0.744717i \(-0.732585\pi\)
−0.667381 + 0.744717i \(0.732585\pi\)
\(110\) −913.341 −0.791669
\(111\) 0 0
\(112\) −2258.12 −1.90511
\(113\) −1195.41 −0.995173 −0.497586 0.867414i \(-0.665780\pi\)
−0.497586 + 0.867414i \(0.665780\pi\)
\(114\) 0 0
\(115\) 840.988 0.681935
\(116\) 182.383 0.145982
\(117\) 0 0
\(118\) −208.850 −0.162934
\(119\) −3553.17 −2.73713
\(120\) 0 0
\(121\) −892.190 −0.670315
\(122\) −1896.09 −1.40708
\(123\) 0 0
\(124\) 1203.68 0.871723
\(125\) 1372.48 0.982069
\(126\) 0 0
\(127\) 526.425 0.367816 0.183908 0.982943i \(-0.441125\pi\)
0.183908 + 0.982943i \(0.441125\pi\)
\(128\) 628.446 0.433963
\(129\) 0 0
\(130\) 0 0
\(131\) 834.024 0.556252 0.278126 0.960545i \(-0.410287\pi\)
0.278126 + 0.960545i \(0.410287\pi\)
\(132\) 0 0
\(133\) −1403.30 −0.914896
\(134\) −2346.61 −1.51281
\(135\) 0 0
\(136\) 569.103 0.358825
\(137\) 466.762 0.291082 0.145541 0.989352i \(-0.453508\pi\)
0.145541 + 0.989352i \(0.453508\pi\)
\(138\) 0 0
\(139\) −713.080 −0.435127 −0.217563 0.976046i \(-0.569811\pi\)
−0.217563 + 0.976046i \(0.569811\pi\)
\(140\) −2367.38 −1.42914
\(141\) 0 0
\(142\) −62.5218 −0.0369487
\(143\) 0 0
\(144\) 0 0
\(145\) −309.420 −0.177213
\(146\) 1234.00 0.699497
\(147\) 0 0
\(148\) 2377.61 1.32053
\(149\) −662.814 −0.364428 −0.182214 0.983259i \(-0.558326\pi\)
−0.182214 + 0.983259i \(0.558326\pi\)
\(150\) 0 0
\(151\) 190.862 0.102862 0.0514310 0.998677i \(-0.483622\pi\)
0.0514310 + 0.998677i \(0.483622\pi\)
\(152\) 224.762 0.119938
\(153\) 0 0
\(154\) 2494.98 1.30553
\(155\) −2042.09 −1.05822
\(156\) 0 0
\(157\) −2833.96 −1.44060 −0.720302 0.693661i \(-0.755997\pi\)
−0.720302 + 0.693661i \(0.755997\pi\)
\(158\) 1477.48 0.743937
\(159\) 0 0
\(160\) 2717.05 1.34251
\(161\) −2297.34 −1.12457
\(162\) 0 0
\(163\) 3565.55 1.71334 0.856672 0.515861i \(-0.172528\pi\)
0.856672 + 0.515861i \(0.172528\pi\)
\(164\) −545.708 −0.259833
\(165\) 0 0
\(166\) −2545.11 −1.18999
\(167\) 120.630 0.0558959 0.0279480 0.999609i \(-0.491103\pi\)
0.0279480 + 0.999609i \(0.491103\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 4987.44 2.25011
\(171\) 0 0
\(172\) −1715.96 −0.760700
\(173\) 2080.28 0.914223 0.457112 0.889409i \(-0.348884\pi\)
0.457112 + 0.889409i \(0.348884\pi\)
\(174\) 0 0
\(175\) 133.562 0.0576932
\(176\) −1522.83 −0.652204
\(177\) 0 0
\(178\) −2092.51 −0.881125
\(179\) 553.140 0.230970 0.115485 0.993309i \(-0.463158\pi\)
0.115485 + 0.993309i \(0.463158\pi\)
\(180\) 0 0
\(181\) −3305.19 −1.35731 −0.678655 0.734457i \(-0.737437\pi\)
−0.678655 + 0.734457i \(0.737437\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 367.958 0.147425
\(185\) −4033.71 −1.60305
\(186\) 0 0
\(187\) −2396.19 −0.937042
\(188\) −3106.61 −1.20517
\(189\) 0 0
\(190\) 1969.75 0.752107
\(191\) 3318.14 1.25703 0.628514 0.777798i \(-0.283664\pi\)
0.628514 + 0.777798i \(0.283664\pi\)
\(192\) 0 0
\(193\) −3902.18 −1.45536 −0.727681 0.685916i \(-0.759402\pi\)
−0.727681 + 0.685916i \(0.759402\pi\)
\(194\) −2642.61 −0.977983
\(195\) 0 0
\(196\) 4168.04 1.51897
\(197\) 3632.99 1.31391 0.656953 0.753932i \(-0.271845\pi\)
0.656953 + 0.753932i \(0.271845\pi\)
\(198\) 0 0
\(199\) 2957.23 1.05343 0.526715 0.850042i \(-0.323424\pi\)
0.526715 + 0.850042i \(0.323424\pi\)
\(200\) −21.3922 −0.00756330
\(201\) 0 0
\(202\) −1807.52 −0.629587
\(203\) 845.245 0.292239
\(204\) 0 0
\(205\) 925.813 0.315422
\(206\) −1286.52 −0.435125
\(207\) 0 0
\(208\) 0 0
\(209\) −946.355 −0.313209
\(210\) 0 0
\(211\) −3194.05 −1.04212 −0.521061 0.853520i \(-0.674464\pi\)
−0.521061 + 0.853520i \(0.674464\pi\)
\(212\) −513.447 −0.166338
\(213\) 0 0
\(214\) −6247.22 −1.99557
\(215\) 2911.18 0.923446
\(216\) 0 0
\(217\) 5578.39 1.74509
\(218\) −5824.23 −1.80948
\(219\) 0 0
\(220\) −1596.52 −0.489259
\(221\) 0 0
\(222\) 0 0
\(223\) 75.1092 0.0225546 0.0112773 0.999936i \(-0.496410\pi\)
0.0112773 + 0.999936i \(0.496410\pi\)
\(224\) −7422.18 −2.21391
\(225\) 0 0
\(226\) −4583.65 −1.34911
\(227\) −4322.79 −1.26394 −0.631968 0.774995i \(-0.717752\pi\)
−0.631968 + 0.774995i \(0.717752\pi\)
\(228\) 0 0
\(229\) −677.923 −0.195626 −0.0978132 0.995205i \(-0.531185\pi\)
−0.0978132 + 0.995205i \(0.531185\pi\)
\(230\) 3224.67 0.924472
\(231\) 0 0
\(232\) −135.381 −0.0383111
\(233\) −92.9061 −0.0261222 −0.0130611 0.999915i \(-0.504158\pi\)
−0.0130611 + 0.999915i \(0.504158\pi\)
\(234\) 0 0
\(235\) 5270.47 1.46301
\(236\) −365.068 −0.100695
\(237\) 0 0
\(238\) −13624.2 −3.71062
\(239\) 2082.56 0.563638 0.281819 0.959468i \(-0.409062\pi\)
0.281819 + 0.959468i \(0.409062\pi\)
\(240\) 0 0
\(241\) −6541.25 −1.74838 −0.874189 0.485586i \(-0.838606\pi\)
−0.874189 + 0.485586i \(0.838606\pi\)
\(242\) −3421.00 −0.908719
\(243\) 0 0
\(244\) −3314.35 −0.869589
\(245\) −7071.23 −1.84394
\(246\) 0 0
\(247\) 0 0
\(248\) −893.476 −0.228773
\(249\) 0 0
\(250\) 5262.62 1.33135
\(251\) 2409.29 0.605868 0.302934 0.953012i \(-0.402034\pi\)
0.302934 + 0.953012i \(0.402034\pi\)
\(252\) 0 0
\(253\) −1549.28 −0.384989
\(254\) 2018.51 0.498633
\(255\) 0 0
\(256\) 5086.77 1.24189
\(257\) 263.587 0.0639770 0.0319885 0.999488i \(-0.489816\pi\)
0.0319885 + 0.999488i \(0.489816\pi\)
\(258\) 0 0
\(259\) 11018.9 2.64356
\(260\) 0 0
\(261\) 0 0
\(262\) 3197.97 0.754088
\(263\) 7324.62 1.71732 0.858660 0.512545i \(-0.171297\pi\)
0.858660 + 0.512545i \(0.171297\pi\)
\(264\) 0 0
\(265\) 871.081 0.201925
\(266\) −5380.77 −1.24029
\(267\) 0 0
\(268\) −4101.86 −0.934930
\(269\) 4683.37 1.06152 0.530762 0.847521i \(-0.321906\pi\)
0.530762 + 0.847521i \(0.321906\pi\)
\(270\) 0 0
\(271\) −1629.37 −0.365231 −0.182615 0.983184i \(-0.558456\pi\)
−0.182615 + 0.983184i \(0.558456\pi\)
\(272\) 8315.66 1.85372
\(273\) 0 0
\(274\) 1789.75 0.394608
\(275\) 90.0714 0.0197510
\(276\) 0 0
\(277\) 6200.66 1.34499 0.672494 0.740103i \(-0.265223\pi\)
0.672494 + 0.740103i \(0.265223\pi\)
\(278\) −2734.22 −0.589883
\(279\) 0 0
\(280\) 1757.28 0.375062
\(281\) −2951.26 −0.626538 −0.313269 0.949664i \(-0.601424\pi\)
−0.313269 + 0.949664i \(0.601424\pi\)
\(282\) 0 0
\(283\) 5249.17 1.10258 0.551291 0.834313i \(-0.314135\pi\)
0.551291 + 0.834313i \(0.314135\pi\)
\(284\) −109.288 −0.0228346
\(285\) 0 0
\(286\) 0 0
\(287\) −2529.05 −0.520157
\(288\) 0 0
\(289\) 8171.77 1.66330
\(290\) −1186.43 −0.240241
\(291\) 0 0
\(292\) 2157.03 0.432296
\(293\) 2912.33 0.580684 0.290342 0.956923i \(-0.406231\pi\)
0.290342 + 0.956923i \(0.406231\pi\)
\(294\) 0 0
\(295\) 619.352 0.122237
\(296\) −1764.87 −0.346558
\(297\) 0 0
\(298\) −2541.48 −0.494040
\(299\) 0 0
\(300\) 0 0
\(301\) −7952.50 −1.52284
\(302\) 731.840 0.139446
\(303\) 0 0
\(304\) 3284.20 0.619611
\(305\) 5622.92 1.05563
\(306\) 0 0
\(307\) 5000.10 0.929546 0.464773 0.885430i \(-0.346136\pi\)
0.464773 + 0.885430i \(0.346136\pi\)
\(308\) 4361.22 0.806829
\(309\) 0 0
\(310\) −7830.14 −1.43459
\(311\) 7840.94 1.42964 0.714822 0.699307i \(-0.246508\pi\)
0.714822 + 0.699307i \(0.246508\pi\)
\(312\) 0 0
\(313\) −7518.43 −1.35772 −0.678860 0.734267i \(-0.737526\pi\)
−0.678860 + 0.734267i \(0.737526\pi\)
\(314\) −10866.5 −1.95297
\(315\) 0 0
\(316\) 2582.63 0.459761
\(317\) −485.238 −0.0859738 −0.0429869 0.999076i \(-0.513687\pi\)
−0.0429869 + 0.999076i \(0.513687\pi\)
\(318\) 0 0
\(319\) 570.017 0.100046
\(320\) 3805.13 0.664728
\(321\) 0 0
\(322\) −8808.86 −1.52453
\(323\) 5167.72 0.890215
\(324\) 0 0
\(325\) 0 0
\(326\) 13671.7 2.32271
\(327\) 0 0
\(328\) 405.072 0.0681901
\(329\) −14397.4 −2.41263
\(330\) 0 0
\(331\) 8222.88 1.36547 0.682735 0.730666i \(-0.260790\pi\)
0.682735 + 0.730666i \(0.260790\pi\)
\(332\) −4448.84 −0.735428
\(333\) 0 0
\(334\) 462.541 0.0757758
\(335\) 6958.96 1.13495
\(336\) 0 0
\(337\) 7744.78 1.25188 0.625942 0.779870i \(-0.284715\pi\)
0.625942 + 0.779870i \(0.284715\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 8718.02 1.39059
\(341\) 3761.96 0.597423
\(342\) 0 0
\(343\) 8662.19 1.36360
\(344\) 1273.73 0.199637
\(345\) 0 0
\(346\) 7976.58 1.23937
\(347\) 3532.61 0.546514 0.273257 0.961941i \(-0.411899\pi\)
0.273257 + 0.961941i \(0.411899\pi\)
\(348\) 0 0
\(349\) 3366.19 0.516298 0.258149 0.966105i \(-0.416888\pi\)
0.258149 + 0.966105i \(0.416888\pi\)
\(350\) 512.127 0.0782123
\(351\) 0 0
\(352\) −5005.37 −0.757919
\(353\) 9967.98 1.50295 0.751476 0.659761i \(-0.229342\pi\)
0.751476 + 0.659761i \(0.229342\pi\)
\(354\) 0 0
\(355\) 185.411 0.0277199
\(356\) −3657.70 −0.544544
\(357\) 0 0
\(358\) 2120.95 0.313117
\(359\) −2742.72 −0.403219 −0.201609 0.979466i \(-0.564617\pi\)
−0.201609 + 0.979466i \(0.564617\pi\)
\(360\) 0 0
\(361\) −4818.05 −0.702443
\(362\) −12673.4 −1.84005
\(363\) 0 0
\(364\) 0 0
\(365\) −3659.47 −0.524783
\(366\) 0 0
\(367\) −1641.86 −0.233527 −0.116763 0.993160i \(-0.537252\pi\)
−0.116763 + 0.993160i \(0.537252\pi\)
\(368\) 5376.57 0.761611
\(369\) 0 0
\(370\) −15466.8 −2.17319
\(371\) −2379.54 −0.332991
\(372\) 0 0
\(373\) 11207.5 1.55577 0.777887 0.628404i \(-0.216291\pi\)
0.777887 + 0.628404i \(0.216291\pi\)
\(374\) −9187.91 −1.27031
\(375\) 0 0
\(376\) 2306.00 0.316284
\(377\) 0 0
\(378\) 0 0
\(379\) 3844.47 0.521048 0.260524 0.965467i \(-0.416105\pi\)
0.260524 + 0.965467i \(0.416105\pi\)
\(380\) 3443.11 0.464810
\(381\) 0 0
\(382\) 12723.0 1.70410
\(383\) −12154.3 −1.62156 −0.810778 0.585354i \(-0.800955\pi\)
−0.810778 + 0.585354i \(0.800955\pi\)
\(384\) 0 0
\(385\) −7398.96 −0.979444
\(386\) −14962.4 −1.97297
\(387\) 0 0
\(388\) −4619.28 −0.604403
\(389\) 5269.41 0.686812 0.343406 0.939187i \(-0.388419\pi\)
0.343406 + 0.939187i \(0.388419\pi\)
\(390\) 0 0
\(391\) 8460.07 1.09423
\(392\) −3093.88 −0.398634
\(393\) 0 0
\(394\) 13930.3 1.78121
\(395\) −4381.53 −0.558123
\(396\) 0 0
\(397\) 655.483 0.0828659 0.0414329 0.999141i \(-0.486808\pi\)
0.0414329 + 0.999141i \(0.486808\pi\)
\(398\) 11339.2 1.42809
\(399\) 0 0
\(400\) −312.581 −0.0390726
\(401\) −7699.97 −0.958898 −0.479449 0.877570i \(-0.659163\pi\)
−0.479449 + 0.877570i \(0.659163\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −3159.53 −0.389091
\(405\) 0 0
\(406\) 3240.99 0.396177
\(407\) 7430.94 0.905008
\(408\) 0 0
\(409\) −555.201 −0.0671220 −0.0335610 0.999437i \(-0.510685\pi\)
−0.0335610 + 0.999437i \(0.510685\pi\)
\(410\) 3549.92 0.427605
\(411\) 0 0
\(412\) −2248.83 −0.268912
\(413\) −1691.89 −0.201580
\(414\) 0 0
\(415\) 7547.62 0.892767
\(416\) 0 0
\(417\) 0 0
\(418\) −3628.69 −0.424605
\(419\) −3316.99 −0.386744 −0.193372 0.981126i \(-0.561942\pi\)
−0.193372 + 0.981126i \(0.561942\pi\)
\(420\) 0 0
\(421\) −13270.3 −1.53623 −0.768117 0.640310i \(-0.778806\pi\)
−0.768117 + 0.640310i \(0.778806\pi\)
\(422\) −12247.2 −1.41276
\(423\) 0 0
\(424\) 381.125 0.0436535
\(425\) −491.849 −0.0561369
\(426\) 0 0
\(427\) −15360.2 −1.74082
\(428\) −10920.1 −1.23328
\(429\) 0 0
\(430\) 11162.6 1.25188
\(431\) −13692.2 −1.53023 −0.765117 0.643892i \(-0.777319\pi\)
−0.765117 + 0.643892i \(0.777319\pi\)
\(432\) 0 0
\(433\) −15189.1 −1.68578 −0.842888 0.538089i \(-0.819146\pi\)
−0.842888 + 0.538089i \(0.819146\pi\)
\(434\) 21389.7 2.36575
\(435\) 0 0
\(436\) −10180.7 −1.11828
\(437\) 3341.23 0.365750
\(438\) 0 0
\(439\) −2802.04 −0.304634 −0.152317 0.988332i \(-0.548673\pi\)
−0.152317 + 0.988332i \(0.548673\pi\)
\(440\) 1185.07 0.128400
\(441\) 0 0
\(442\) 0 0
\(443\) 7539.78 0.808636 0.404318 0.914618i \(-0.367509\pi\)
0.404318 + 0.914618i \(0.367509\pi\)
\(444\) 0 0
\(445\) 6205.42 0.661045
\(446\) 287.997 0.0305764
\(447\) 0 0
\(448\) −10394.5 −1.09619
\(449\) −5558.82 −0.584269 −0.292135 0.956377i \(-0.594366\pi\)
−0.292135 + 0.956377i \(0.594366\pi\)
\(450\) 0 0
\(451\) −1705.54 −0.178073
\(452\) −8012.20 −0.833766
\(453\) 0 0
\(454\) −16575.2 −1.71347
\(455\) 0 0
\(456\) 0 0
\(457\) −4327.71 −0.442980 −0.221490 0.975163i \(-0.571092\pi\)
−0.221490 + 0.975163i \(0.571092\pi\)
\(458\) −2599.42 −0.265203
\(459\) 0 0
\(460\) 5636.71 0.571333
\(461\) 11292.7 1.14090 0.570450 0.821333i \(-0.306769\pi\)
0.570450 + 0.821333i \(0.306769\pi\)
\(462\) 0 0
\(463\) 8582.54 0.861478 0.430739 0.902477i \(-0.358253\pi\)
0.430739 + 0.902477i \(0.358253\pi\)
\(464\) −1978.17 −0.197918
\(465\) 0 0
\(466\) −356.238 −0.0354128
\(467\) −17543.0 −1.73831 −0.869157 0.494536i \(-0.835338\pi\)
−0.869157 + 0.494536i \(0.835338\pi\)
\(468\) 0 0
\(469\) −19009.8 −1.87163
\(470\) 20209.0 1.98334
\(471\) 0 0
\(472\) 270.986 0.0264261
\(473\) −5363.01 −0.521335
\(474\) 0 0
\(475\) −194.252 −0.0187639
\(476\) −23815.1 −2.29320
\(477\) 0 0
\(478\) 7985.32 0.764101
\(479\) −6299.15 −0.600867 −0.300434 0.953803i \(-0.597131\pi\)
−0.300434 + 0.953803i \(0.597131\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −25081.7 −2.37020
\(483\) 0 0
\(484\) −5979.89 −0.561597
\(485\) 7836.78 0.733711
\(486\) 0 0
\(487\) 3571.73 0.332342 0.166171 0.986097i \(-0.446860\pi\)
0.166171 + 0.986097i \(0.446860\pi\)
\(488\) 2460.20 0.228213
\(489\) 0 0
\(490\) −27113.8 −2.49975
\(491\) 13385.2 1.23028 0.615139 0.788419i \(-0.289100\pi\)
0.615139 + 0.788419i \(0.289100\pi\)
\(492\) 0 0
\(493\) −3112.66 −0.284356
\(494\) 0 0
\(495\) 0 0
\(496\) −13055.4 −1.18186
\(497\) −506.488 −0.0457125
\(498\) 0 0
\(499\) −19227.7 −1.72495 −0.862476 0.506098i \(-0.831087\pi\)
−0.862476 + 0.506098i \(0.831087\pi\)
\(500\) 9199.05 0.822788
\(501\) 0 0
\(502\) 9238.13 0.821350
\(503\) 18651.2 1.65332 0.826658 0.562705i \(-0.190239\pi\)
0.826658 + 0.562705i \(0.190239\pi\)
\(504\) 0 0
\(505\) 5360.26 0.472334
\(506\) −5940.53 −0.521914
\(507\) 0 0
\(508\) 3528.35 0.308160
\(509\) 15886.1 1.38338 0.691688 0.722196i \(-0.256867\pi\)
0.691688 + 0.722196i \(0.256867\pi\)
\(510\) 0 0
\(511\) 9996.62 0.865410
\(512\) 14477.1 1.24961
\(513\) 0 0
\(514\) 1010.69 0.0867310
\(515\) 3815.21 0.326443
\(516\) 0 0
\(517\) −9709.33 −0.825949
\(518\) 42250.7 3.58377
\(519\) 0 0
\(520\) 0 0
\(521\) −1824.67 −0.153436 −0.0767179 0.997053i \(-0.524444\pi\)
−0.0767179 + 0.997053i \(0.524444\pi\)
\(522\) 0 0
\(523\) 16206.3 1.35498 0.677488 0.735534i \(-0.263069\pi\)
0.677488 + 0.735534i \(0.263069\pi\)
\(524\) 5590.04 0.466034
\(525\) 0 0
\(526\) 28085.4 2.32810
\(527\) −20542.7 −1.69802
\(528\) 0 0
\(529\) −6697.07 −0.550429
\(530\) 3340.06 0.273741
\(531\) 0 0
\(532\) −9405.57 −0.766510
\(533\) 0 0
\(534\) 0 0
\(535\) 18526.4 1.49713
\(536\) 3044.76 0.245361
\(537\) 0 0
\(538\) 17957.8 1.43906
\(539\) 13026.7 1.04100
\(540\) 0 0
\(541\) 2429.83 0.193099 0.0965496 0.995328i \(-0.469219\pi\)
0.0965496 + 0.995328i \(0.469219\pi\)
\(542\) −6247.65 −0.495128
\(543\) 0 0
\(544\) 27332.6 2.15418
\(545\) 17272.0 1.35752
\(546\) 0 0
\(547\) −2409.16 −0.188315 −0.0941573 0.995557i \(-0.530016\pi\)
−0.0941573 + 0.995557i \(0.530016\pi\)
\(548\) 3128.47 0.243871
\(549\) 0 0
\(550\) 345.368 0.0267756
\(551\) −1229.32 −0.0950468
\(552\) 0 0
\(553\) 11969.1 0.920390
\(554\) 23775.7 1.82334
\(555\) 0 0
\(556\) −4779.41 −0.364554
\(557\) 8668.70 0.659434 0.329717 0.944080i \(-0.393047\pi\)
0.329717 + 0.944080i \(0.393047\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 25677.1 1.93760
\(561\) 0 0
\(562\) −11316.2 −0.849371
\(563\) 7818.35 0.585265 0.292632 0.956225i \(-0.405469\pi\)
0.292632 + 0.956225i \(0.405469\pi\)
\(564\) 0 0
\(565\) 13593.0 1.01214
\(566\) 20127.3 1.49473
\(567\) 0 0
\(568\) 81.1229 0.00599268
\(569\) −9117.24 −0.671730 −0.335865 0.941910i \(-0.609029\pi\)
−0.335865 + 0.941910i \(0.609029\pi\)
\(570\) 0 0
\(571\) −11842.4 −0.867932 −0.433966 0.900929i \(-0.642886\pi\)
−0.433966 + 0.900929i \(0.642886\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −9697.35 −0.705156
\(575\) −318.009 −0.0230642
\(576\) 0 0
\(577\) −10958.1 −0.790623 −0.395312 0.918547i \(-0.629363\pi\)
−0.395312 + 0.918547i \(0.629363\pi\)
\(578\) 31333.7 2.25486
\(579\) 0 0
\(580\) −2073.88 −0.148471
\(581\) −20617.9 −1.47225
\(582\) 0 0
\(583\) −1604.72 −0.113997
\(584\) −1601.13 −0.113451
\(585\) 0 0
\(586\) 11167.0 0.787209
\(587\) −23057.8 −1.62129 −0.810644 0.585539i \(-0.800883\pi\)
−0.810644 + 0.585539i \(0.800883\pi\)
\(588\) 0 0
\(589\) −8113.18 −0.567568
\(590\) 2374.83 0.165712
\(591\) 0 0
\(592\) −25788.1 −1.79035
\(593\) 9904.56 0.685888 0.342944 0.939356i \(-0.388576\pi\)
0.342944 + 0.939356i \(0.388576\pi\)
\(594\) 0 0
\(595\) 40403.2 2.78381
\(596\) −4442.50 −0.305322
\(597\) 0 0
\(598\) 0 0
\(599\) −21334.5 −1.45527 −0.727633 0.685967i \(-0.759380\pi\)
−0.727633 + 0.685967i \(0.759380\pi\)
\(600\) 0 0
\(601\) −19484.4 −1.32244 −0.661219 0.750193i \(-0.729960\pi\)
−0.661219 + 0.750193i \(0.729960\pi\)
\(602\) −30492.9 −2.06445
\(603\) 0 0
\(604\) 1279.25 0.0861789
\(605\) 10145.1 0.681747
\(606\) 0 0
\(607\) 19855.7 1.32770 0.663852 0.747864i \(-0.268921\pi\)
0.663852 + 0.747864i \(0.268921\pi\)
\(608\) 10794.8 0.720043
\(609\) 0 0
\(610\) 21560.4 1.43108
\(611\) 0 0
\(612\) 0 0
\(613\) 5153.57 0.339561 0.169781 0.985482i \(-0.445694\pi\)
0.169781 + 0.985482i \(0.445694\pi\)
\(614\) 19172.3 1.26015
\(615\) 0 0
\(616\) −3237.27 −0.211743
\(617\) 924.218 0.0603040 0.0301520 0.999545i \(-0.490401\pi\)
0.0301520 + 0.999545i \(0.490401\pi\)
\(618\) 0 0
\(619\) −15690.6 −1.01883 −0.509417 0.860520i \(-0.670139\pi\)
−0.509417 + 0.860520i \(0.670139\pi\)
\(620\) −13687.1 −0.886589
\(621\) 0 0
\(622\) 30065.2 1.93811
\(623\) −16951.4 −1.09012
\(624\) 0 0
\(625\) −16144.0 −1.03322
\(626\) −28828.5 −1.84061
\(627\) 0 0
\(628\) −18994.6 −1.20695
\(629\) −40577.8 −2.57225
\(630\) 0 0
\(631\) 21103.6 1.33141 0.665705 0.746215i \(-0.268131\pi\)
0.665705 + 0.746215i \(0.268131\pi\)
\(632\) −1917.05 −0.120659
\(633\) 0 0
\(634\) −1860.59 −0.116551
\(635\) −5985.98 −0.374089
\(636\) 0 0
\(637\) 0 0
\(638\) 2185.66 0.135629
\(639\) 0 0
\(640\) −7146.06 −0.441364
\(641\) 6695.38 0.412561 0.206280 0.978493i \(-0.433864\pi\)
0.206280 + 0.978493i \(0.433864\pi\)
\(642\) 0 0
\(643\) −9647.69 −0.591707 −0.295854 0.955233i \(-0.595604\pi\)
−0.295854 + 0.955233i \(0.595604\pi\)
\(644\) −15397.9 −0.942175
\(645\) 0 0
\(646\) 19815.0 1.20683
\(647\) 23747.5 1.44299 0.721493 0.692422i \(-0.243456\pi\)
0.721493 + 0.692422i \(0.243456\pi\)
\(648\) 0 0
\(649\) −1140.98 −0.0690096
\(650\) 0 0
\(651\) 0 0
\(652\) 23898.0 1.43546
\(653\) −19833.5 −1.18858 −0.594292 0.804249i \(-0.702568\pi\)
−0.594292 + 0.804249i \(0.702568\pi\)
\(654\) 0 0
\(655\) −9483.70 −0.565738
\(656\) 5918.86 0.352275
\(657\) 0 0
\(658\) −55205.1 −3.27070
\(659\) −10807.4 −0.638841 −0.319421 0.947613i \(-0.603488\pi\)
−0.319421 + 0.947613i \(0.603488\pi\)
\(660\) 0 0
\(661\) 13431.5 0.790356 0.395178 0.918605i \(-0.370683\pi\)
0.395178 + 0.918605i \(0.370683\pi\)
\(662\) 31529.7 1.85111
\(663\) 0 0
\(664\) 3302.32 0.193004
\(665\) 15956.9 0.930498
\(666\) 0 0
\(667\) −2012.52 −0.116829
\(668\) 808.520 0.0468302
\(669\) 0 0
\(670\) 26683.3 1.53861
\(671\) −10358.6 −0.595961
\(672\) 0 0
\(673\) 21581.4 1.23611 0.618054 0.786136i \(-0.287921\pi\)
0.618054 + 0.786136i \(0.287921\pi\)
\(674\) 29696.4 1.69713
\(675\) 0 0
\(676\) 0 0
\(677\) −17482.4 −0.992474 −0.496237 0.868187i \(-0.665285\pi\)
−0.496237 + 0.868187i \(0.665285\pi\)
\(678\) 0 0
\(679\) −21407.8 −1.20995
\(680\) −6471.27 −0.364944
\(681\) 0 0
\(682\) 14424.8 0.809902
\(683\) 2842.23 0.159231 0.0796155 0.996826i \(-0.474631\pi\)
0.0796155 + 0.996826i \(0.474631\pi\)
\(684\) 0 0
\(685\) −5307.56 −0.296046
\(686\) 33214.1 1.84857
\(687\) 0 0
\(688\) 18611.6 1.03134
\(689\) 0 0
\(690\) 0 0
\(691\) −22875.5 −1.25937 −0.629686 0.776850i \(-0.716816\pi\)
−0.629686 + 0.776850i \(0.716816\pi\)
\(692\) 13943.0 0.765946
\(693\) 0 0
\(694\) 13545.4 0.740886
\(695\) 8108.43 0.442547
\(696\) 0 0
\(697\) 9313.38 0.506125
\(698\) 12907.2 0.699923
\(699\) 0 0
\(700\) 895.195 0.0483360
\(701\) −11980.3 −0.645492 −0.322746 0.946486i \(-0.604606\pi\)
−0.322746 + 0.946486i \(0.604606\pi\)
\(702\) 0 0
\(703\) −16025.9 −0.859782
\(704\) −7009.85 −0.375275
\(705\) 0 0
\(706\) 38221.0 2.03749
\(707\) −14642.7 −0.778918
\(708\) 0 0
\(709\) −4204.77 −0.222727 −0.111363 0.993780i \(-0.535522\pi\)
−0.111363 + 0.993780i \(0.535522\pi\)
\(710\) 710.936 0.0375788
\(711\) 0 0
\(712\) 2715.06 0.142909
\(713\) −13282.1 −0.697641
\(714\) 0 0
\(715\) 0 0
\(716\) 3707.41 0.193509
\(717\) 0 0
\(718\) −10516.7 −0.546627
\(719\) 11537.0 0.598411 0.299205 0.954189i \(-0.403278\pi\)
0.299205 + 0.954189i \(0.403278\pi\)
\(720\) 0 0
\(721\) −10422.0 −0.538332
\(722\) −18474.3 −0.952272
\(723\) 0 0
\(724\) −22153.0 −1.13717
\(725\) 117.003 0.00599364
\(726\) 0 0
\(727\) −33899.7 −1.72940 −0.864698 0.502293i \(-0.832490\pi\)
−0.864698 + 0.502293i \(0.832490\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −14031.8 −0.711426
\(731\) 29285.6 1.48176
\(732\) 0 0
\(733\) 378.221 0.0190585 0.00952927 0.999955i \(-0.496967\pi\)
0.00952927 + 0.999955i \(0.496967\pi\)
\(734\) −6295.51 −0.316582
\(735\) 0 0
\(736\) 17672.1 0.885059
\(737\) −12819.9 −0.640741
\(738\) 0 0
\(739\) −22663.6 −1.12814 −0.564069 0.825728i \(-0.690765\pi\)
−0.564069 + 0.825728i \(0.690765\pi\)
\(740\) −27035.9 −1.34305
\(741\) 0 0
\(742\) −9124.06 −0.451422
\(743\) 16407.3 0.810130 0.405065 0.914288i \(-0.367249\pi\)
0.405065 + 0.914288i \(0.367249\pi\)
\(744\) 0 0
\(745\) 7536.86 0.370643
\(746\) 42974.0 2.10910
\(747\) 0 0
\(748\) −16060.4 −0.785064
\(749\) −50608.7 −2.46889
\(750\) 0 0
\(751\) 21957.9 1.06692 0.533458 0.845827i \(-0.320892\pi\)
0.533458 + 0.845827i \(0.320892\pi\)
\(752\) 33694.9 1.63395
\(753\) 0 0
\(754\) 0 0
\(755\) −2170.30 −0.104616
\(756\) 0 0
\(757\) 20823.5 0.999792 0.499896 0.866085i \(-0.333371\pi\)
0.499896 + 0.866085i \(0.333371\pi\)
\(758\) 14741.2 0.706364
\(759\) 0 0
\(760\) −2555.77 −0.121984
\(761\) −28456.1 −1.35550 −0.677749 0.735294i \(-0.737044\pi\)
−0.677749 + 0.735294i \(0.737044\pi\)
\(762\) 0 0
\(763\) −47182.0 −2.23867
\(764\) 22239.8 1.05315
\(765\) 0 0
\(766\) −46604.2 −2.19828
\(767\) 0 0
\(768\) 0 0
\(769\) 13450.0 0.630713 0.315356 0.948973i \(-0.397876\pi\)
0.315356 + 0.948973i \(0.397876\pi\)
\(770\) −28370.4 −1.32779
\(771\) 0 0
\(772\) −26154.3 −1.21932
\(773\) 26577.8 1.23666 0.618329 0.785919i \(-0.287810\pi\)
0.618329 + 0.785919i \(0.287810\pi\)
\(774\) 0 0
\(775\) 772.190 0.0357908
\(776\) 3428.83 0.158618
\(777\) 0 0
\(778\) 20204.9 0.931083
\(779\) 3678.24 0.169174
\(780\) 0 0
\(781\) −341.566 −0.0156494
\(782\) 32439.1 1.48340
\(783\) 0 0
\(784\) −45207.5 −2.05938
\(785\) 32225.0 1.46517
\(786\) 0 0
\(787\) −8281.40 −0.375095 −0.187548 0.982256i \(-0.560054\pi\)
−0.187548 + 0.982256i \(0.560054\pi\)
\(788\) 24350.0 1.10080
\(789\) 0 0
\(790\) −16800.4 −0.756624
\(791\) −37132.1 −1.66911
\(792\) 0 0
\(793\) 0 0
\(794\) 2513.37 0.112338
\(795\) 0 0
\(796\) 19820.8 0.882575
\(797\) −20668.8 −0.918605 −0.459302 0.888280i \(-0.651901\pi\)
−0.459302 + 0.888280i \(0.651901\pi\)
\(798\) 0 0
\(799\) 53019.3 2.34754
\(800\) −1027.42 −0.0454059
\(801\) 0 0
\(802\) −29524.6 −1.29994
\(803\) 6741.52 0.296268
\(804\) 0 0
\(805\) 26123.0 1.14375
\(806\) 0 0
\(807\) 0 0
\(808\) 2345.28 0.102112
\(809\) −7238.85 −0.314591 −0.157296 0.987552i \(-0.550278\pi\)
−0.157296 + 0.987552i \(0.550278\pi\)
\(810\) 0 0
\(811\) 13101.3 0.567260 0.283630 0.958934i \(-0.408461\pi\)
0.283630 + 0.958934i \(0.408461\pi\)
\(812\) 5665.24 0.244841
\(813\) 0 0
\(814\) 28493.1 1.22688
\(815\) −40543.9 −1.74256
\(816\) 0 0
\(817\) 11566.1 0.495283
\(818\) −2128.85 −0.0909946
\(819\) 0 0
\(820\) 6205.25 0.264264
\(821\) −12092.9 −0.514062 −0.257031 0.966403i \(-0.582744\pi\)
−0.257031 + 0.966403i \(0.582744\pi\)
\(822\) 0 0
\(823\) 27041.2 1.14532 0.572659 0.819794i \(-0.305912\pi\)
0.572659 + 0.819794i \(0.305912\pi\)
\(824\) 1669.27 0.0705727
\(825\) 0 0
\(826\) −6487.35 −0.273273
\(827\) 25572.3 1.07526 0.537628 0.843182i \(-0.319320\pi\)
0.537628 + 0.843182i \(0.319320\pi\)
\(828\) 0 0
\(829\) −17505.8 −0.733415 −0.366707 0.930336i \(-0.619515\pi\)
−0.366707 + 0.930336i \(0.619515\pi\)
\(830\) 28940.5 1.21029
\(831\) 0 0
\(832\) 0 0
\(833\) −71134.3 −2.95877
\(834\) 0 0
\(835\) −1371.68 −0.0568491
\(836\) −6342.93 −0.262410
\(837\) 0 0
\(838\) −12718.6 −0.524293
\(839\) 28293.4 1.16424 0.582119 0.813103i \(-0.302224\pi\)
0.582119 + 0.813103i \(0.302224\pi\)
\(840\) 0 0
\(841\) −23648.5 −0.969640
\(842\) −50883.3 −2.08261
\(843\) 0 0
\(844\) −21408.1 −0.873100
\(845\) 0 0
\(846\) 0 0
\(847\) −27713.4 −1.12426
\(848\) 5568.95 0.225517
\(849\) 0 0
\(850\) −1885.94 −0.0761025
\(851\) −26235.9 −1.05682
\(852\) 0 0
\(853\) −15866.0 −0.636861 −0.318430 0.947946i \(-0.603156\pi\)
−0.318430 + 0.947946i \(0.603156\pi\)
\(854\) −58896.8 −2.35996
\(855\) 0 0
\(856\) 8105.86 0.323660
\(857\) −23623.3 −0.941607 −0.470804 0.882238i \(-0.656036\pi\)
−0.470804 + 0.882238i \(0.656036\pi\)
\(858\) 0 0
\(859\) 33403.1 1.32677 0.663387 0.748277i \(-0.269118\pi\)
0.663387 + 0.748277i \(0.269118\pi\)
\(860\) 19512.2 0.773673
\(861\) 0 0
\(862\) −52501.2 −2.07447
\(863\) 45490.5 1.79434 0.897169 0.441687i \(-0.145620\pi\)
0.897169 + 0.441687i \(0.145620\pi\)
\(864\) 0 0
\(865\) −23654.9 −0.929814
\(866\) −58240.8 −2.28534
\(867\) 0 0
\(868\) 37389.0 1.46206
\(869\) 8071.70 0.315090
\(870\) 0 0
\(871\) 0 0
\(872\) 7557.03 0.293479
\(873\) 0 0
\(874\) 12811.6 0.495833
\(875\) 42632.4 1.64713
\(876\) 0 0
\(877\) −19570.1 −0.753518 −0.376759 0.926311i \(-0.622961\pi\)
−0.376759 + 0.926311i \(0.622961\pi\)
\(878\) −10744.1 −0.412979
\(879\) 0 0
\(880\) 17316.1 0.663326
\(881\) 3016.95 0.115373 0.0576864 0.998335i \(-0.481628\pi\)
0.0576864 + 0.998335i \(0.481628\pi\)
\(882\) 0 0
\(883\) 17163.8 0.654141 0.327071 0.945000i \(-0.393939\pi\)
0.327071 + 0.945000i \(0.393939\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 28910.4 1.09623
\(887\) −28464.7 −1.07751 −0.538754 0.842463i \(-0.681105\pi\)
−0.538754 + 0.842463i \(0.681105\pi\)
\(888\) 0 0
\(889\) 16352.0 0.616903
\(890\) 23793.9 0.896151
\(891\) 0 0
\(892\) 503.418 0.0188965
\(893\) 20939.5 0.784674
\(894\) 0 0
\(895\) −6289.76 −0.234909
\(896\) 19520.9 0.727845
\(897\) 0 0
\(898\) −21314.6 −0.792070
\(899\) 4886.80 0.181295
\(900\) 0 0
\(901\) 8762.80 0.324008
\(902\) −6539.70 −0.241406
\(903\) 0 0
\(904\) 5947.36 0.218812
\(905\) 37583.4 1.38046
\(906\) 0 0
\(907\) −26610.1 −0.974171 −0.487085 0.873354i \(-0.661940\pi\)
−0.487085 + 0.873354i \(0.661940\pi\)
\(908\) −28973.4 −1.05894
\(909\) 0 0
\(910\) 0 0
\(911\) 52648.2 1.91472 0.957362 0.288890i \(-0.0932863\pi\)
0.957362 + 0.288890i \(0.0932863\pi\)
\(912\) 0 0
\(913\) −13904.3 −0.504015
\(914\) −16594.1 −0.600529
\(915\) 0 0
\(916\) −4543.77 −0.163898
\(917\) 25906.7 0.932949
\(918\) 0 0
\(919\) 46923.5 1.68429 0.842146 0.539250i \(-0.181292\pi\)
0.842146 + 0.539250i \(0.181292\pi\)
\(920\) −4184.06 −0.149939
\(921\) 0 0
\(922\) 43300.6 1.54667
\(923\) 0 0
\(924\) 0 0
\(925\) 1525.30 0.0542178
\(926\) 32908.7 1.16787
\(927\) 0 0
\(928\) −6502.00 −0.229999
\(929\) −41163.7 −1.45376 −0.726878 0.686767i \(-0.759029\pi\)
−0.726878 + 0.686767i \(0.759029\pi\)
\(930\) 0 0
\(931\) −28093.9 −0.988980
\(932\) −622.702 −0.0218855
\(933\) 0 0
\(934\) −67266.5 −2.35656
\(935\) 27247.1 0.953022
\(936\) 0 0
\(937\) −23380.3 −0.815156 −0.407578 0.913170i \(-0.633627\pi\)
−0.407578 + 0.913170i \(0.633627\pi\)
\(938\) −72891.0 −2.53729
\(939\) 0 0
\(940\) 35325.3 1.22573
\(941\) 34506.2 1.19540 0.597700 0.801720i \(-0.296082\pi\)
0.597700 + 0.801720i \(0.296082\pi\)
\(942\) 0 0
\(943\) 6021.65 0.207945
\(944\) 3959.61 0.136519
\(945\) 0 0
\(946\) −20563.8 −0.706753
\(947\) 483.439 0.0165889 0.00829443 0.999966i \(-0.497360\pi\)
0.00829443 + 0.999966i \(0.497360\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −744.835 −0.0254375
\(951\) 0 0
\(952\) 17677.6 0.601823
\(953\) 11962.7 0.406619 0.203310 0.979114i \(-0.434830\pi\)
0.203310 + 0.979114i \(0.434830\pi\)
\(954\) 0 0
\(955\) −37730.6 −1.27846
\(956\) 13958.3 0.472222
\(957\) 0 0
\(958\) −24153.3 −0.814571
\(959\) 14498.7 0.488204
\(960\) 0 0
\(961\) 2460.54 0.0825934
\(962\) 0 0
\(963\) 0 0
\(964\) −43842.6 −1.46481
\(965\) 44371.7 1.48018
\(966\) 0 0
\(967\) 7695.43 0.255913 0.127957 0.991780i \(-0.459158\pi\)
0.127957 + 0.991780i \(0.459158\pi\)
\(968\) 4438.79 0.147385
\(969\) 0 0
\(970\) 30049.2 0.994661
\(971\) 44177.9 1.46008 0.730039 0.683406i \(-0.239502\pi\)
0.730039 + 0.683406i \(0.239502\pi\)
\(972\) 0 0
\(973\) −22149.9 −0.729797
\(974\) 13695.4 0.450542
\(975\) 0 0
\(976\) 35948.2 1.17897
\(977\) −3412.61 −0.111749 −0.0558746 0.998438i \(-0.517795\pi\)
−0.0558746 + 0.998438i \(0.517795\pi\)
\(978\) 0 0
\(979\) −11431.7 −0.373196
\(980\) −47394.8 −1.54487
\(981\) 0 0
\(982\) 51324.0 1.66784
\(983\) −6313.78 −0.204861 −0.102431 0.994740i \(-0.532662\pi\)
−0.102431 + 0.994740i \(0.532662\pi\)
\(984\) 0 0
\(985\) −41310.7 −1.33631
\(986\) −11935.1 −0.385489
\(987\) 0 0
\(988\) 0 0
\(989\) 18934.8 0.608789
\(990\) 0 0
\(991\) 12177.5 0.390344 0.195172 0.980769i \(-0.437474\pi\)
0.195172 + 0.980769i \(0.437474\pi\)
\(992\) −42911.5 −1.37343
\(993\) 0 0
\(994\) −1942.07 −0.0619705
\(995\) −33626.7 −1.07139
\(996\) 0 0
\(997\) 17506.2 0.556096 0.278048 0.960567i \(-0.410313\pi\)
0.278048 + 0.960567i \(0.410313\pi\)
\(998\) −73726.4 −2.33845
\(999\) 0 0
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 1521.4.a.bg.1.8 9
3.2 odd 2 169.4.a.l.1.2 yes 9
13.12 even 2 1521.4.a.bh.1.2 9
39.2 even 12 169.4.e.h.147.4 36
39.5 even 4 169.4.b.g.168.15 18
39.8 even 4 169.4.b.g.168.4 18
39.11 even 12 169.4.e.h.147.15 36
39.17 odd 6 169.4.c.l.146.2 18
39.20 even 12 169.4.e.h.23.4 36
39.23 odd 6 169.4.c.l.22.2 18
39.29 odd 6 169.4.c.k.22.8 18
39.32 even 12 169.4.e.h.23.15 36
39.35 odd 6 169.4.c.k.146.8 18
39.38 odd 2 169.4.a.k.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.4.a.k.1.8 9 39.38 odd 2
169.4.a.l.1.2 yes 9 3.2 odd 2
169.4.b.g.168.4 18 39.8 even 4
169.4.b.g.168.15 18 39.5 even 4
169.4.c.k.22.8 18 39.29 odd 6
169.4.c.k.146.8 18 39.35 odd 6
169.4.c.l.22.2 18 39.23 odd 6
169.4.c.l.146.2 18 39.17 odd 6
169.4.e.h.23.4 36 39.20 even 12
169.4.e.h.23.15 36 39.32 even 12
169.4.e.h.147.4 36 39.2 even 12
169.4.e.h.147.15 36 39.11 even 12
1521.4.a.bg.1.8 9 1.1 even 1 trivial
1521.4.a.bh.1.2 9 13.12 even 2