Properties

Label 2475.4.a.v.1.1
Level $2475$
Weight $4$
Character 2475.1
Self dual yes
Analytic conductor $146.030$
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] = [2475,4,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(146.029727264\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.03932\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.03932 q^{2} +8.31608 q^{4} -6.49923 q^{7} -1.27677 q^{8} +O(q^{10})\) \(q-4.03932 q^{2} +8.31608 q^{4} -6.49923 q^{7} -1.27677 q^{8} -11.0000 q^{11} +30.5385 q^{13} +26.2524 q^{14} -61.3714 q^{16} +43.1314 q^{17} -6.46044 q^{19} +44.4325 q^{22} -108.104 q^{23} -123.355 q^{26} -54.0481 q^{28} +274.857 q^{29} -68.5519 q^{31} +258.113 q^{32} -174.221 q^{34} -402.200 q^{37} +26.0958 q^{38} +268.450 q^{41} -30.5334 q^{43} -91.4769 q^{44} +436.666 q^{46} -31.5850 q^{47} -300.760 q^{49} +253.961 q^{52} -252.497 q^{53} +8.29800 q^{56} -1110.24 q^{58} +558.331 q^{59} +335.270 q^{61} +276.903 q^{62} -551.628 q^{64} +28.7521 q^{67} +358.684 q^{68} -89.0081 q^{71} -717.622 q^{73} +1624.61 q^{74} -53.7256 q^{76} +71.4915 q^{77} +200.702 q^{79} -1084.36 q^{82} -243.681 q^{83} +123.334 q^{86} +14.0444 q^{88} +312.832 q^{89} -198.477 q^{91} -899.002 q^{92} +127.582 q^{94} +27.9996 q^{97} +1214.87 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8} - 33 q^{11} + 45 q^{13} + 116 q^{14} - 85 q^{16} + 58 q^{17} - 169 q^{19} + 11 q^{22} - 155 q^{23} - 167 q^{26} + 100 q^{28} + 277 q^{29} - 173 q^{31} - 97 q^{32} - 146 q^{34} + 60 q^{37} - 169 q^{38} - 44 q^{41} - 109 q^{43} - 77 q^{44} + 425 q^{46} - 270 q^{47} - 427 q^{49} + 45 q^{52} + 148 q^{53} - 168 q^{56} - 783 q^{58} + 684 q^{59} - 1038 q^{61} - 953 q^{62} - 1129 q^{64} + 314 q^{67} + 366 q^{68} + 1459 q^{71} - 1170 q^{73} + 1764 q^{74} + 211 q^{76} - 176 q^{77} - 506 q^{79} - 1040 q^{82} + 347 q^{83} + 527 q^{86} - 33 q^{88} + 607 q^{89} - 398 q^{91} - 687 q^{92} - 1822 q^{94} + 1263 q^{97} + 2273 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.03932 −1.42811 −0.714057 0.700087i \(-0.753144\pi\)
−0.714057 + 0.700087i \(0.753144\pi\)
\(3\) 0 0
\(4\) 8.31608 1.03951
\(5\) 0 0
\(6\) 0 0
\(7\) −6.49923 −0.350925 −0.175463 0.984486i \(-0.556142\pi\)
−0.175463 + 0.984486i \(0.556142\pi\)
\(8\) −1.27677 −0.0564257
\(9\) 0 0
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 30.5385 0.651528 0.325764 0.945451i \(-0.394378\pi\)
0.325764 + 0.945451i \(0.394378\pi\)
\(14\) 26.2524 0.501161
\(15\) 0 0
\(16\) −61.3714 −0.958928
\(17\) 43.1314 0.615347 0.307674 0.951492i \(-0.400450\pi\)
0.307674 + 0.951492i \(0.400450\pi\)
\(18\) 0 0
\(19\) −6.46044 −0.0780066 −0.0390033 0.999239i \(-0.512418\pi\)
−0.0390033 + 0.999239i \(0.512418\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 44.4325 0.430593
\(23\) −108.104 −0.980054 −0.490027 0.871707i \(-0.663013\pi\)
−0.490027 + 0.871707i \(0.663013\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −123.355 −0.930457
\(27\) 0 0
\(28\) −54.0481 −0.364791
\(29\) 274.857 1.75999 0.879995 0.474984i \(-0.157546\pi\)
0.879995 + 0.474984i \(0.157546\pi\)
\(30\) 0 0
\(31\) −68.5519 −0.397170 −0.198585 0.980084i \(-0.563635\pi\)
−0.198585 + 0.980084i \(0.563635\pi\)
\(32\) 258.113 1.42588
\(33\) 0 0
\(34\) −174.221 −0.878786
\(35\) 0 0
\(36\) 0 0
\(37\) −402.200 −1.78706 −0.893530 0.449004i \(-0.851779\pi\)
−0.893530 + 0.449004i \(0.851779\pi\)
\(38\) 26.0958 0.111402
\(39\) 0 0
\(40\) 0 0
\(41\) 268.450 1.02256 0.511280 0.859414i \(-0.329172\pi\)
0.511280 + 0.859414i \(0.329172\pi\)
\(42\) 0 0
\(43\) −30.5334 −0.108286 −0.0541431 0.998533i \(-0.517243\pi\)
−0.0541431 + 0.998533i \(0.517243\pi\)
\(44\) −91.4769 −0.313424
\(45\) 0 0
\(46\) 436.666 1.39963
\(47\) −31.5850 −0.0980243 −0.0490121 0.998798i \(-0.515607\pi\)
−0.0490121 + 0.998798i \(0.515607\pi\)
\(48\) 0 0
\(49\) −300.760 −0.876851
\(50\) 0 0
\(51\) 0 0
\(52\) 253.961 0.677271
\(53\) −252.497 −0.654398 −0.327199 0.944956i \(-0.606105\pi\)
−0.327199 + 0.944956i \(0.606105\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.29800 0.0198012
\(57\) 0 0
\(58\) −1110.24 −2.51347
\(59\) 558.331 1.23201 0.616004 0.787743i \(-0.288750\pi\)
0.616004 + 0.787743i \(0.288750\pi\)
\(60\) 0 0
\(61\) 335.270 0.703719 0.351860 0.936053i \(-0.385549\pi\)
0.351860 + 0.936053i \(0.385549\pi\)
\(62\) 276.903 0.567205
\(63\) 0 0
\(64\) −551.628 −1.07740
\(65\) 0 0
\(66\) 0 0
\(67\) 28.7521 0.0524273 0.0262136 0.999656i \(-0.491655\pi\)
0.0262136 + 0.999656i \(0.491655\pi\)
\(68\) 358.684 0.639660
\(69\) 0 0
\(70\) 0 0
\(71\) −89.0081 −0.148779 −0.0743895 0.997229i \(-0.523701\pi\)
−0.0743895 + 0.997229i \(0.523701\pi\)
\(72\) 0 0
\(73\) −717.622 −1.15057 −0.575283 0.817954i \(-0.695108\pi\)
−0.575283 + 0.817954i \(0.695108\pi\)
\(74\) 1624.61 2.55213
\(75\) 0 0
\(76\) −53.7256 −0.0810887
\(77\) 71.4915 0.105808
\(78\) 0 0
\(79\) 200.702 0.285832 0.142916 0.989735i \(-0.454352\pi\)
0.142916 + 0.989735i \(0.454352\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1084.36 −1.46033
\(83\) −243.681 −0.322258 −0.161129 0.986933i \(-0.551514\pi\)
−0.161129 + 0.986933i \(0.551514\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 123.334 0.154645
\(87\) 0 0
\(88\) 14.0444 0.0170130
\(89\) 312.832 0.372585 0.186293 0.982494i \(-0.440353\pi\)
0.186293 + 0.982494i \(0.440353\pi\)
\(90\) 0 0
\(91\) −198.477 −0.228638
\(92\) −899.002 −1.01878
\(93\) 0 0
\(94\) 127.582 0.139990
\(95\) 0 0
\(96\) 0 0
\(97\) 27.9996 0.0293085 0.0146543 0.999893i \(-0.495335\pi\)
0.0146543 + 0.999893i \(0.495335\pi\)
\(98\) 1214.87 1.25224
\(99\) 0 0
\(100\) 0 0
\(101\) 228.278 0.224896 0.112448 0.993658i \(-0.464131\pi\)
0.112448 + 0.993658i \(0.464131\pi\)
\(102\) 0 0
\(103\) −800.328 −0.765618 −0.382809 0.923827i \(-0.625043\pi\)
−0.382809 + 0.923827i \(0.625043\pi\)
\(104\) −38.9906 −0.0367629
\(105\) 0 0
\(106\) 1019.91 0.934555
\(107\) 1046.89 0.945857 0.472929 0.881101i \(-0.343197\pi\)
0.472929 + 0.881101i \(0.343197\pi\)
\(108\) 0 0
\(109\) 317.023 0.278581 0.139291 0.990252i \(-0.455518\pi\)
0.139291 + 0.990252i \(0.455518\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 398.867 0.336512
\(113\) −1333.82 −1.11040 −0.555198 0.831718i \(-0.687358\pi\)
−0.555198 + 0.831718i \(0.687358\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2285.74 1.82953
\(117\) 0 0
\(118\) −2255.28 −1.75945
\(119\) −280.321 −0.215941
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −1354.26 −1.00499
\(123\) 0 0
\(124\) −570.083 −0.412863
\(125\) 0 0
\(126\) 0 0
\(127\) 564.888 0.394691 0.197345 0.980334i \(-0.436768\pi\)
0.197345 + 0.980334i \(0.436768\pi\)
\(128\) 163.299 0.112763
\(129\) 0 0
\(130\) 0 0
\(131\) 214.082 0.142782 0.0713908 0.997448i \(-0.477256\pi\)
0.0713908 + 0.997448i \(0.477256\pi\)
\(132\) 0 0
\(133\) 41.9879 0.0273745
\(134\) −116.139 −0.0748721
\(135\) 0 0
\(136\) −55.0688 −0.0347214
\(137\) 65.2288 0.0406779 0.0203389 0.999793i \(-0.493525\pi\)
0.0203389 + 0.999793i \(0.493525\pi\)
\(138\) 0 0
\(139\) −407.084 −0.248406 −0.124203 0.992257i \(-0.539637\pi\)
−0.124203 + 0.992257i \(0.539637\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 359.532 0.212474
\(143\) −335.924 −0.196443
\(144\) 0 0
\(145\) 0 0
\(146\) 2898.70 1.64314
\(147\) 0 0
\(148\) −3344.73 −1.85767
\(149\) −792.092 −0.435508 −0.217754 0.976004i \(-0.569873\pi\)
−0.217754 + 0.976004i \(0.569873\pi\)
\(150\) 0 0
\(151\) −2861.80 −1.54232 −0.771159 0.636642i \(-0.780323\pi\)
−0.771159 + 0.636642i \(0.780323\pi\)
\(152\) 8.24848 0.00440158
\(153\) 0 0
\(154\) −288.777 −0.151106
\(155\) 0 0
\(156\) 0 0
\(157\) 2991.55 1.52071 0.760355 0.649507i \(-0.225025\pi\)
0.760355 + 0.649507i \(0.225025\pi\)
\(158\) −810.698 −0.408201
\(159\) 0 0
\(160\) 0 0
\(161\) 702.592 0.343926
\(162\) 0 0
\(163\) 2907.60 1.39718 0.698592 0.715520i \(-0.253810\pi\)
0.698592 + 0.715520i \(0.253810\pi\)
\(164\) 2232.46 1.06296
\(165\) 0 0
\(166\) 984.304 0.460222
\(167\) −1100.39 −0.509885 −0.254942 0.966956i \(-0.582056\pi\)
−0.254942 + 0.966956i \(0.582056\pi\)
\(168\) 0 0
\(169\) −1264.40 −0.575511
\(170\) 0 0
\(171\) 0 0
\(172\) −253.919 −0.112565
\(173\) −3754.75 −1.65011 −0.825054 0.565054i \(-0.808855\pi\)
−0.825054 + 0.565054i \(0.808855\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 675.086 0.289128
\(177\) 0 0
\(178\) −1263.63 −0.532094
\(179\) 3218.79 1.34404 0.672021 0.740532i \(-0.265426\pi\)
0.672021 + 0.740532i \(0.265426\pi\)
\(180\) 0 0
\(181\) −456.885 −0.187624 −0.0938121 0.995590i \(-0.529905\pi\)
−0.0938121 + 0.995590i \(0.529905\pi\)
\(182\) 801.711 0.326521
\(183\) 0 0
\(184\) 138.024 0.0553002
\(185\) 0 0
\(186\) 0 0
\(187\) −474.445 −0.185534
\(188\) −262.663 −0.101897
\(189\) 0 0
\(190\) 0 0
\(191\) 4907.11 1.85899 0.929493 0.368841i \(-0.120245\pi\)
0.929493 + 0.368841i \(0.120245\pi\)
\(192\) 0 0
\(193\) 1462.17 0.545333 0.272667 0.962109i \(-0.412094\pi\)
0.272667 + 0.962109i \(0.412094\pi\)
\(194\) −113.099 −0.0418559
\(195\) 0 0
\(196\) −2501.15 −0.911496
\(197\) −2210.10 −0.799307 −0.399653 0.916666i \(-0.630869\pi\)
−0.399653 + 0.916666i \(0.630869\pi\)
\(198\) 0 0
\(199\) −395.272 −0.140804 −0.0704022 0.997519i \(-0.522428\pi\)
−0.0704022 + 0.997519i \(0.522428\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −922.086 −0.321177
\(203\) −1786.36 −0.617625
\(204\) 0 0
\(205\) 0 0
\(206\) 3232.78 1.09339
\(207\) 0 0
\(208\) −1874.19 −0.624769
\(209\) 71.0648 0.0235199
\(210\) 0 0
\(211\) 1559.98 0.508974 0.254487 0.967076i \(-0.418093\pi\)
0.254487 + 0.967076i \(0.418093\pi\)
\(212\) −2099.78 −0.680253
\(213\) 0 0
\(214\) −4228.72 −1.35079
\(215\) 0 0
\(216\) 0 0
\(217\) 445.534 0.139377
\(218\) −1280.56 −0.397846
\(219\) 0 0
\(220\) 0 0
\(221\) 1317.17 0.400916
\(222\) 0 0
\(223\) 756.215 0.227085 0.113542 0.993533i \(-0.463780\pi\)
0.113542 + 0.993533i \(0.463780\pi\)
\(224\) −1677.53 −0.500379
\(225\) 0 0
\(226\) 5387.70 1.58577
\(227\) 5758.69 1.68378 0.841890 0.539649i \(-0.181443\pi\)
0.841890 + 0.539649i \(0.181443\pi\)
\(228\) 0 0
\(229\) −1912.63 −0.551921 −0.275961 0.961169i \(-0.588996\pi\)
−0.275961 + 0.961169i \(0.588996\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −350.929 −0.0993086
\(233\) −1034.86 −0.290970 −0.145485 0.989360i \(-0.546474\pi\)
−0.145485 + 0.989360i \(0.546474\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4643.13 1.28069
\(237\) 0 0
\(238\) 1132.30 0.308388
\(239\) −2647.23 −0.716465 −0.358232 0.933632i \(-0.616620\pi\)
−0.358232 + 0.933632i \(0.616620\pi\)
\(240\) 0 0
\(241\) 6734.00 1.79990 0.899948 0.435996i \(-0.143604\pi\)
0.899948 + 0.435996i \(0.143604\pi\)
\(242\) −488.757 −0.129829
\(243\) 0 0
\(244\) 2788.13 0.731523
\(245\) 0 0
\(246\) 0 0
\(247\) −197.292 −0.0508235
\(248\) 87.5248 0.0224106
\(249\) 0 0
\(250\) 0 0
\(251\) 5377.57 1.35231 0.676154 0.736761i \(-0.263646\pi\)
0.676154 + 0.736761i \(0.263646\pi\)
\(252\) 0 0
\(253\) 1189.14 0.295497
\(254\) −2281.76 −0.563663
\(255\) 0 0
\(256\) 3753.41 0.916360
\(257\) −5882.33 −1.42774 −0.713871 0.700278i \(-0.753060\pi\)
−0.713871 + 0.700278i \(0.753060\pi\)
\(258\) 0 0
\(259\) 2613.99 0.627124
\(260\) 0 0
\(261\) 0 0
\(262\) −864.744 −0.203909
\(263\) 4001.67 0.938227 0.469113 0.883138i \(-0.344574\pi\)
0.469113 + 0.883138i \(0.344574\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −169.602 −0.0390939
\(267\) 0 0
\(268\) 239.105 0.0544987
\(269\) 1758.29 0.398530 0.199265 0.979946i \(-0.436144\pi\)
0.199265 + 0.979946i \(0.436144\pi\)
\(270\) 0 0
\(271\) 6018.44 1.34906 0.674528 0.738249i \(-0.264347\pi\)
0.674528 + 0.738249i \(0.264347\pi\)
\(272\) −2647.03 −0.590074
\(273\) 0 0
\(274\) −263.480 −0.0580927
\(275\) 0 0
\(276\) 0 0
\(277\) 2910.30 0.631275 0.315637 0.948880i \(-0.397782\pi\)
0.315637 + 0.948880i \(0.397782\pi\)
\(278\) 1644.34 0.354752
\(279\) 0 0
\(280\) 0 0
\(281\) −6893.19 −1.46339 −0.731697 0.681631i \(-0.761271\pi\)
−0.731697 + 0.681631i \(0.761271\pi\)
\(282\) 0 0
\(283\) 3120.16 0.655385 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(284\) −740.199 −0.154657
\(285\) 0 0
\(286\) 1356.90 0.280543
\(287\) −1744.72 −0.358842
\(288\) 0 0
\(289\) −3052.68 −0.621348
\(290\) 0 0
\(291\) 0 0
\(292\) −5967.81 −1.19603
\(293\) −8869.62 −1.76849 −0.884247 0.467020i \(-0.845328\pi\)
−0.884247 + 0.467020i \(0.845328\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 513.515 0.100836
\(297\) 0 0
\(298\) 3199.51 0.621955
\(299\) −3301.34 −0.638533
\(300\) 0 0
\(301\) 198.444 0.0380003
\(302\) 11559.7 2.20261
\(303\) 0 0
\(304\) 396.486 0.0748028
\(305\) 0 0
\(306\) 0 0
\(307\) −5713.42 −1.06216 −0.531078 0.847323i \(-0.678213\pi\)
−0.531078 + 0.847323i \(0.678213\pi\)
\(308\) 594.529 0.109988
\(309\) 0 0
\(310\) 0 0
\(311\) −1984.01 −0.361746 −0.180873 0.983506i \(-0.557892\pi\)
−0.180873 + 0.983506i \(0.557892\pi\)
\(312\) 0 0
\(313\) −2675.50 −0.483156 −0.241578 0.970381i \(-0.577665\pi\)
−0.241578 + 0.970381i \(0.577665\pi\)
\(314\) −12083.8 −2.17175
\(315\) 0 0
\(316\) 1669.05 0.297125
\(317\) 7083.76 1.25509 0.627545 0.778580i \(-0.284060\pi\)
0.627545 + 0.778580i \(0.284060\pi\)
\(318\) 0 0
\(319\) −3023.43 −0.530657
\(320\) 0 0
\(321\) 0 0
\(322\) −2837.99 −0.491165
\(323\) −278.648 −0.0480012
\(324\) 0 0
\(325\) 0 0
\(326\) −11744.7 −1.99534
\(327\) 0 0
\(328\) −342.749 −0.0576986
\(329\) 205.278 0.0343992
\(330\) 0 0
\(331\) 7812.39 1.29730 0.648652 0.761085i \(-0.275333\pi\)
0.648652 + 0.761085i \(0.275333\pi\)
\(332\) −2026.47 −0.334991
\(333\) 0 0
\(334\) 4444.82 0.728173
\(335\) 0 0
\(336\) 0 0
\(337\) −11866.4 −1.91811 −0.959056 0.283218i \(-0.908598\pi\)
−0.959056 + 0.283218i \(0.908598\pi\)
\(338\) 5107.30 0.821895
\(339\) 0 0
\(340\) 0 0
\(341\) 754.071 0.119751
\(342\) 0 0
\(343\) 4183.94 0.658635
\(344\) 38.9841 0.00611012
\(345\) 0 0
\(346\) 15166.6 2.35654
\(347\) −3485.50 −0.539225 −0.269613 0.962969i \(-0.586896\pi\)
−0.269613 + 0.962969i \(0.586896\pi\)
\(348\) 0 0
\(349\) −6366.76 −0.976518 −0.488259 0.872699i \(-0.662368\pi\)
−0.488259 + 0.872699i \(0.662368\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2839.24 −0.429920
\(353\) −8223.15 −1.23987 −0.619935 0.784653i \(-0.712841\pi\)
−0.619935 + 0.784653i \(0.712841\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2601.54 0.387306
\(357\) 0 0
\(358\) −13001.7 −1.91945
\(359\) 9377.54 1.37863 0.689314 0.724462i \(-0.257912\pi\)
0.689314 + 0.724462i \(0.257912\pi\)
\(360\) 0 0
\(361\) −6817.26 −0.993915
\(362\) 1845.50 0.267949
\(363\) 0 0
\(364\) −1650.55 −0.237671
\(365\) 0 0
\(366\) 0 0
\(367\) −310.895 −0.0442196 −0.0221098 0.999756i \(-0.507038\pi\)
−0.0221098 + 0.999756i \(0.507038\pi\)
\(368\) 6634.49 0.939801
\(369\) 0 0
\(370\) 0 0
\(371\) 1641.03 0.229645
\(372\) 0 0
\(373\) −4738.51 −0.657777 −0.328889 0.944369i \(-0.606674\pi\)
−0.328889 + 0.944369i \(0.606674\pi\)
\(374\) 1916.44 0.264964
\(375\) 0 0
\(376\) 40.3267 0.00553109
\(377\) 8393.74 1.14668
\(378\) 0 0
\(379\) −4735.47 −0.641806 −0.320903 0.947112i \(-0.603986\pi\)
−0.320903 + 0.947112i \(0.603986\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −19821.4 −2.65484
\(383\) −3632.14 −0.484579 −0.242289 0.970204i \(-0.577898\pi\)
−0.242289 + 0.970204i \(0.577898\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5906.17 −0.778798
\(387\) 0 0
\(388\) 232.847 0.0304665
\(389\) −6557.50 −0.854701 −0.427350 0.904086i \(-0.640553\pi\)
−0.427350 + 0.904086i \(0.640553\pi\)
\(390\) 0 0
\(391\) −4662.68 −0.603073
\(392\) 384.001 0.0494769
\(393\) 0 0
\(394\) 8927.31 1.14150
\(395\) 0 0
\(396\) 0 0
\(397\) 4084.97 0.516420 0.258210 0.966089i \(-0.416867\pi\)
0.258210 + 0.966089i \(0.416867\pi\)
\(398\) 1596.63 0.201085
\(399\) 0 0
\(400\) 0 0
\(401\) 6676.04 0.831386 0.415693 0.909505i \(-0.363539\pi\)
0.415693 + 0.909505i \(0.363539\pi\)
\(402\) 0 0
\(403\) −2093.47 −0.258768
\(404\) 1898.38 0.233782
\(405\) 0 0
\(406\) 7215.67 0.882039
\(407\) 4424.20 0.538819
\(408\) 0 0
\(409\) 1718.56 0.207769 0.103884 0.994589i \(-0.466873\pi\)
0.103884 + 0.994589i \(0.466873\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6655.60 −0.795868
\(413\) −3628.72 −0.432343
\(414\) 0 0
\(415\) 0 0
\(416\) 7882.39 0.929004
\(417\) 0 0
\(418\) −287.053 −0.0335891
\(419\) −5732.23 −0.668348 −0.334174 0.942511i \(-0.608457\pi\)
−0.334174 + 0.942511i \(0.608457\pi\)
\(420\) 0 0
\(421\) −9647.38 −1.11683 −0.558414 0.829563i \(-0.688590\pi\)
−0.558414 + 0.829563i \(0.688590\pi\)
\(422\) −6301.26 −0.726873
\(423\) 0 0
\(424\) 322.379 0.0369248
\(425\) 0 0
\(426\) 0 0
\(427\) −2178.99 −0.246953
\(428\) 8706.03 0.983229
\(429\) 0 0
\(430\) 0 0
\(431\) 3054.57 0.341376 0.170688 0.985325i \(-0.445401\pi\)
0.170688 + 0.985325i \(0.445401\pi\)
\(432\) 0 0
\(433\) 11022.8 1.22337 0.611686 0.791100i \(-0.290491\pi\)
0.611686 + 0.791100i \(0.290491\pi\)
\(434\) −1799.65 −0.199046
\(435\) 0 0
\(436\) 2636.39 0.289588
\(437\) 698.399 0.0764507
\(438\) 0 0
\(439\) −7589.30 −0.825097 −0.412548 0.910936i \(-0.635361\pi\)
−0.412548 + 0.910936i \(0.635361\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5320.47 −0.572554
\(443\) −4043.30 −0.433641 −0.216821 0.976211i \(-0.569569\pi\)
−0.216821 + 0.976211i \(0.569569\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3054.59 −0.324303
\(447\) 0 0
\(448\) 3585.16 0.378086
\(449\) −14619.3 −1.53658 −0.768291 0.640101i \(-0.778893\pi\)
−0.768291 + 0.640101i \(0.778893\pi\)
\(450\) 0 0
\(451\) −2952.96 −0.308313
\(452\) −11092.1 −1.15427
\(453\) 0 0
\(454\) −23261.2 −2.40463
\(455\) 0 0
\(456\) 0 0
\(457\) −10910.8 −1.11682 −0.558411 0.829565i \(-0.688589\pi\)
−0.558411 + 0.829565i \(0.688589\pi\)
\(458\) 7725.71 0.788207
\(459\) 0 0
\(460\) 0 0
\(461\) −16371.0 −1.65395 −0.826977 0.562236i \(-0.809941\pi\)
−0.826977 + 0.562236i \(0.809941\pi\)
\(462\) 0 0
\(463\) −16023.5 −1.60837 −0.804186 0.594377i \(-0.797399\pi\)
−0.804186 + 0.594377i \(0.797399\pi\)
\(464\) −16868.4 −1.68770
\(465\) 0 0
\(466\) 4180.13 0.415538
\(467\) 15097.3 1.49597 0.747985 0.663716i \(-0.231022\pi\)
0.747985 + 0.663716i \(0.231022\pi\)
\(468\) 0 0
\(469\) −186.866 −0.0183981
\(470\) 0 0
\(471\) 0 0
\(472\) −712.859 −0.0695170
\(473\) 335.868 0.0326495
\(474\) 0 0
\(475\) 0 0
\(476\) −2331.17 −0.224473
\(477\) 0 0
\(478\) 10693.0 1.02319
\(479\) 10459.0 0.997667 0.498834 0.866698i \(-0.333762\pi\)
0.498834 + 0.866698i \(0.333762\pi\)
\(480\) 0 0
\(481\) −12282.6 −1.16432
\(482\) −27200.8 −2.57046
\(483\) 0 0
\(484\) 1006.25 0.0945010
\(485\) 0 0
\(486\) 0 0
\(487\) −7939.86 −0.738788 −0.369394 0.929273i \(-0.620435\pi\)
−0.369394 + 0.929273i \(0.620435\pi\)
\(488\) −428.061 −0.0397078
\(489\) 0 0
\(490\) 0 0
\(491\) −11492.7 −1.05633 −0.528167 0.849141i \(-0.677120\pi\)
−0.528167 + 0.849141i \(0.677120\pi\)
\(492\) 0 0
\(493\) 11855.0 1.08300
\(494\) 796.927 0.0725818
\(495\) 0 0
\(496\) 4207.13 0.380858
\(497\) 578.484 0.0522103
\(498\) 0 0
\(499\) 4813.54 0.431831 0.215915 0.976412i \(-0.430726\pi\)
0.215915 + 0.976412i \(0.430726\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −21721.7 −1.93125
\(503\) −17756.3 −1.57399 −0.786993 0.616962i \(-0.788363\pi\)
−0.786993 + 0.616962i \(0.788363\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −4803.33 −0.422004
\(507\) 0 0
\(508\) 4697.66 0.410285
\(509\) −3183.05 −0.277183 −0.138592 0.990350i \(-0.544258\pi\)
−0.138592 + 0.990350i \(0.544258\pi\)
\(510\) 0 0
\(511\) 4663.99 0.403763
\(512\) −16467.6 −1.42143
\(513\) 0 0
\(514\) 23760.6 2.03898
\(515\) 0 0
\(516\) 0 0
\(517\) 347.435 0.0295554
\(518\) −10558.7 −0.895605
\(519\) 0 0
\(520\) 0 0
\(521\) −7246.60 −0.609365 −0.304683 0.952454i \(-0.598550\pi\)
−0.304683 + 0.952454i \(0.598550\pi\)
\(522\) 0 0
\(523\) −17985.0 −1.50369 −0.751845 0.659340i \(-0.770836\pi\)
−0.751845 + 0.659340i \(0.770836\pi\)
\(524\) 1780.32 0.148423
\(525\) 0 0
\(526\) −16164.0 −1.33990
\(527\) −2956.74 −0.244398
\(528\) 0 0
\(529\) −480.530 −0.0394945
\(530\) 0 0
\(531\) 0 0
\(532\) 349.175 0.0284561
\(533\) 8198.09 0.666226
\(534\) 0 0
\(535\) 0 0
\(536\) −36.7097 −0.00295825
\(537\) 0 0
\(538\) −7102.28 −0.569147
\(539\) 3308.36 0.264381
\(540\) 0 0
\(541\) −14549.3 −1.15624 −0.578119 0.815952i \(-0.696213\pi\)
−0.578119 + 0.815952i \(0.696213\pi\)
\(542\) −24310.4 −1.92661
\(543\) 0 0
\(544\) 11132.8 0.877414
\(545\) 0 0
\(546\) 0 0
\(547\) 5487.29 0.428920 0.214460 0.976733i \(-0.431201\pi\)
0.214460 + 0.976733i \(0.431201\pi\)
\(548\) 542.448 0.0422851
\(549\) 0 0
\(550\) 0 0
\(551\) −1775.70 −0.137291
\(552\) 0 0
\(553\) −1304.41 −0.100306
\(554\) −11755.6 −0.901532
\(555\) 0 0
\(556\) −3385.35 −0.258221
\(557\) 7366.60 0.560382 0.280191 0.959944i \(-0.409602\pi\)
0.280191 + 0.959944i \(0.409602\pi\)
\(558\) 0 0
\(559\) −932.446 −0.0705515
\(560\) 0 0
\(561\) 0 0
\(562\) 27843.8 2.08989
\(563\) 4516.14 0.338068 0.169034 0.985610i \(-0.445935\pi\)
0.169034 + 0.985610i \(0.445935\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −12603.3 −0.935965
\(567\) 0 0
\(568\) 113.643 0.00839496
\(569\) −2174.35 −0.160200 −0.0800999 0.996787i \(-0.525524\pi\)
−0.0800999 + 0.996787i \(0.525524\pi\)
\(570\) 0 0
\(571\) −25339.7 −1.85715 −0.928575 0.371145i \(-0.878965\pi\)
−0.928575 + 0.371145i \(0.878965\pi\)
\(572\) −2793.57 −0.204205
\(573\) 0 0
\(574\) 7047.48 0.512467
\(575\) 0 0
\(576\) 0 0
\(577\) −21909.1 −1.58074 −0.790370 0.612630i \(-0.790112\pi\)
−0.790370 + 0.612630i \(0.790112\pi\)
\(578\) 12330.8 0.887356
\(579\) 0 0
\(580\) 0 0
\(581\) 1583.74 0.113089
\(582\) 0 0
\(583\) 2777.46 0.197308
\(584\) 916.237 0.0649215
\(585\) 0 0
\(586\) 35827.2 2.52561
\(587\) 922.235 0.0648462 0.0324231 0.999474i \(-0.489678\pi\)
0.0324231 + 0.999474i \(0.489678\pi\)
\(588\) 0 0
\(589\) 442.875 0.0309819
\(590\) 0 0
\(591\) 0 0
\(592\) 24683.6 1.71366
\(593\) 9075.86 0.628501 0.314250 0.949340i \(-0.398247\pi\)
0.314250 + 0.949340i \(0.398247\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6587.10 −0.452715
\(597\) 0 0
\(598\) 13335.2 0.911898
\(599\) −19987.0 −1.36335 −0.681675 0.731655i \(-0.738748\pi\)
−0.681675 + 0.731655i \(0.738748\pi\)
\(600\) 0 0
\(601\) 11116.7 0.754509 0.377255 0.926110i \(-0.376868\pi\)
0.377255 + 0.926110i \(0.376868\pi\)
\(602\) −801.577 −0.0542688
\(603\) 0 0
\(604\) −23799.0 −1.60326
\(605\) 0 0
\(606\) 0 0
\(607\) 19355.0 1.29423 0.647114 0.762393i \(-0.275976\pi\)
0.647114 + 0.762393i \(0.275976\pi\)
\(608\) −1667.52 −0.111228
\(609\) 0 0
\(610\) 0 0
\(611\) −964.559 −0.0638656
\(612\) 0 0
\(613\) −12325.5 −0.812111 −0.406056 0.913848i \(-0.633096\pi\)
−0.406056 + 0.913848i \(0.633096\pi\)
\(614\) 23078.3 1.51688
\(615\) 0 0
\(616\) −91.2780 −0.00597029
\(617\) −28207.0 −1.84047 −0.920236 0.391363i \(-0.872004\pi\)
−0.920236 + 0.391363i \(0.872004\pi\)
\(618\) 0 0
\(619\) 17236.9 1.11924 0.559621 0.828748i \(-0.310947\pi\)
0.559621 + 0.828748i \(0.310947\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8014.06 0.516615
\(623\) −2033.16 −0.130750
\(624\) 0 0
\(625\) 0 0
\(626\) 10807.2 0.690003
\(627\) 0 0
\(628\) 24878.0 1.58080
\(629\) −17347.4 −1.09966
\(630\) 0 0
\(631\) −23582.7 −1.48782 −0.743908 0.668282i \(-0.767030\pi\)
−0.743908 + 0.668282i \(0.767030\pi\)
\(632\) −256.250 −0.0161283
\(633\) 0 0
\(634\) −28613.6 −1.79241
\(635\) 0 0
\(636\) 0 0
\(637\) −9184.77 −0.571294
\(638\) 12212.6 0.757839
\(639\) 0 0
\(640\) 0 0
\(641\) 13003.1 0.801233 0.400617 0.916246i \(-0.368796\pi\)
0.400617 + 0.916246i \(0.368796\pi\)
\(642\) 0 0
\(643\) −21269.1 −1.30446 −0.652232 0.758019i \(-0.726167\pi\)
−0.652232 + 0.758019i \(0.726167\pi\)
\(644\) 5842.82 0.357514
\(645\) 0 0
\(646\) 1125.55 0.0685511
\(647\) −29247.2 −1.77716 −0.888582 0.458717i \(-0.848309\pi\)
−0.888582 + 0.458717i \(0.848309\pi\)
\(648\) 0 0
\(649\) −6141.64 −0.371465
\(650\) 0 0
\(651\) 0 0
\(652\) 24179.9 1.45239
\(653\) 1678.30 0.100577 0.0502885 0.998735i \(-0.483986\pi\)
0.0502885 + 0.998735i \(0.483986\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −16475.2 −0.980561
\(657\) 0 0
\(658\) −829.182 −0.0491260
\(659\) 7620.41 0.450454 0.225227 0.974306i \(-0.427688\pi\)
0.225227 + 0.974306i \(0.427688\pi\)
\(660\) 0 0
\(661\) −22762.2 −1.33941 −0.669703 0.742629i \(-0.733579\pi\)
−0.669703 + 0.742629i \(0.733579\pi\)
\(662\) −31556.7 −1.85270
\(663\) 0 0
\(664\) 311.124 0.0181837
\(665\) 0 0
\(666\) 0 0
\(667\) −29713.2 −1.72488
\(668\) −9150.93 −0.530030
\(669\) 0 0
\(670\) 0 0
\(671\) −3687.96 −0.212179
\(672\) 0 0
\(673\) 698.528 0.0400093 0.0200047 0.999800i \(-0.493632\pi\)
0.0200047 + 0.999800i \(0.493632\pi\)
\(674\) 47932.1 2.73928
\(675\) 0 0
\(676\) −10514.8 −0.598250
\(677\) 10860.1 0.616524 0.308262 0.951302i \(-0.400253\pi\)
0.308262 + 0.951302i \(0.400253\pi\)
\(678\) 0 0
\(679\) −181.976 −0.0102851
\(680\) 0 0
\(681\) 0 0
\(682\) −3045.93 −0.171019
\(683\) 6534.95 0.366110 0.183055 0.983103i \(-0.441401\pi\)
0.183055 + 0.983103i \(0.441401\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −16900.3 −0.940606
\(687\) 0 0
\(688\) 1873.88 0.103839
\(689\) −7710.88 −0.426359
\(690\) 0 0
\(691\) −10915.3 −0.600924 −0.300462 0.953794i \(-0.597141\pi\)
−0.300462 + 0.953794i \(0.597141\pi\)
\(692\) −31224.9 −1.71530
\(693\) 0 0
\(694\) 14079.0 0.770076
\(695\) 0 0
\(696\) 0 0
\(697\) 11578.6 0.629229
\(698\) 25717.3 1.39458
\(699\) 0 0
\(700\) 0 0
\(701\) −18405.4 −0.991674 −0.495837 0.868416i \(-0.665139\pi\)
−0.495837 + 0.868416i \(0.665139\pi\)
\(702\) 0 0
\(703\) 2598.39 0.139403
\(704\) 6067.91 0.324848
\(705\) 0 0
\(706\) 33215.9 1.77068
\(707\) −1483.63 −0.0789217
\(708\) 0 0
\(709\) −21436.2 −1.13547 −0.567737 0.823210i \(-0.692181\pi\)
−0.567737 + 0.823210i \(0.692181\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −399.413 −0.0210234
\(713\) 7410.73 0.389248
\(714\) 0 0
\(715\) 0 0
\(716\) 26767.7 1.39715
\(717\) 0 0
\(718\) −37878.9 −1.96884
\(719\) −28641.7 −1.48561 −0.742807 0.669506i \(-0.766506\pi\)
−0.742807 + 0.669506i \(0.766506\pi\)
\(720\) 0 0
\(721\) 5201.51 0.268675
\(722\) 27537.1 1.41942
\(723\) 0 0
\(724\) −3799.49 −0.195037
\(725\) 0 0
\(726\) 0 0
\(727\) −1723.13 −0.0879055 −0.0439527 0.999034i \(-0.513995\pi\)
−0.0439527 + 0.999034i \(0.513995\pi\)
\(728\) 253.409 0.0129010
\(729\) 0 0
\(730\) 0 0
\(731\) −1316.95 −0.0666335
\(732\) 0 0
\(733\) −18813.4 −0.948005 −0.474003 0.880523i \(-0.657191\pi\)
−0.474003 + 0.880523i \(0.657191\pi\)
\(734\) 1255.80 0.0631506
\(735\) 0 0
\(736\) −27903.0 −1.39744
\(737\) −316.273 −0.0158074
\(738\) 0 0
\(739\) −10893.3 −0.542242 −0.271121 0.962545i \(-0.587394\pi\)
−0.271121 + 0.962545i \(0.587394\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6628.65 −0.327959
\(743\) −11888.6 −0.587013 −0.293507 0.955957i \(-0.594822\pi\)
−0.293507 + 0.955957i \(0.594822\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 19140.4 0.939381
\(747\) 0 0
\(748\) −3945.53 −0.192865
\(749\) −6803.98 −0.331925
\(750\) 0 0
\(751\) −32134.8 −1.56140 −0.780702 0.624903i \(-0.785138\pi\)
−0.780702 + 0.624903i \(0.785138\pi\)
\(752\) 1938.41 0.0939982
\(753\) 0 0
\(754\) −33905.0 −1.63759
\(755\) 0 0
\(756\) 0 0
\(757\) 17366.1 0.833794 0.416897 0.908954i \(-0.363118\pi\)
0.416897 + 0.908954i \(0.363118\pi\)
\(758\) 19128.1 0.916573
\(759\) 0 0
\(760\) 0 0
\(761\) 27393.9 1.30490 0.652450 0.757832i \(-0.273741\pi\)
0.652450 + 0.757832i \(0.273741\pi\)
\(762\) 0 0
\(763\) −2060.41 −0.0977611
\(764\) 40808.0 1.93243
\(765\) 0 0
\(766\) 14671.4 0.692034
\(767\) 17050.6 0.802689
\(768\) 0 0
\(769\) −12091.3 −0.567000 −0.283500 0.958972i \(-0.591496\pi\)
−0.283500 + 0.958972i \(0.591496\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 12159.5 0.566879
\(773\) −18895.8 −0.879216 −0.439608 0.898190i \(-0.644883\pi\)
−0.439608 + 0.898190i \(0.644883\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −35.7490 −0.00165375
\(777\) 0 0
\(778\) 26487.8 1.22061
\(779\) −1734.31 −0.0797664
\(780\) 0 0
\(781\) 979.089 0.0448586
\(782\) 18834.0 0.861258
\(783\) 0 0
\(784\) 18458.1 0.840838
\(785\) 0 0
\(786\) 0 0
\(787\) 23705.9 1.07373 0.536863 0.843669i \(-0.319609\pi\)
0.536863 + 0.843669i \(0.319609\pi\)
\(788\) −18379.4 −0.830888
\(789\) 0 0
\(790\) 0 0
\(791\) 8668.77 0.389666
\(792\) 0 0
\(793\) 10238.6 0.458493
\(794\) −16500.5 −0.737507
\(795\) 0 0
\(796\) −3287.11 −0.146368
\(797\) 37248.0 1.65545 0.827725 0.561135i \(-0.189635\pi\)
0.827725 + 0.561135i \(0.189635\pi\)
\(798\) 0 0
\(799\) −1362.30 −0.0603189
\(800\) 0 0
\(801\) 0 0
\(802\) −26966.7 −1.18731
\(803\) 7893.84 0.346909
\(804\) 0 0
\(805\) 0 0
\(806\) 8456.21 0.369550
\(807\) 0 0
\(808\) −291.458 −0.0126899
\(809\) 24861.0 1.08043 0.540214 0.841528i \(-0.318343\pi\)
0.540214 + 0.841528i \(0.318343\pi\)
\(810\) 0 0
\(811\) 4131.24 0.178875 0.0894375 0.995992i \(-0.471493\pi\)
0.0894375 + 0.995992i \(0.471493\pi\)
\(812\) −14855.5 −0.642027
\(813\) 0 0
\(814\) −17870.7 −0.769495
\(815\) 0 0
\(816\) 0 0
\(817\) 197.259 0.00844704
\(818\) −6941.81 −0.296717
\(819\) 0 0
\(820\) 0 0
\(821\) −25297.3 −1.07537 −0.537687 0.843145i \(-0.680702\pi\)
−0.537687 + 0.843145i \(0.680702\pi\)
\(822\) 0 0
\(823\) −18155.5 −0.768969 −0.384484 0.923131i \(-0.625621\pi\)
−0.384484 + 0.923131i \(0.625621\pi\)
\(824\) 1021.83 0.0432005
\(825\) 0 0
\(826\) 14657.6 0.617435
\(827\) 29566.8 1.24321 0.621607 0.783329i \(-0.286480\pi\)
0.621607 + 0.783329i \(0.286480\pi\)
\(828\) 0 0
\(829\) 27249.4 1.14163 0.570814 0.821079i \(-0.306628\pi\)
0.570814 + 0.821079i \(0.306628\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −16845.9 −0.701956
\(833\) −12972.2 −0.539568
\(834\) 0 0
\(835\) 0 0
\(836\) 590.981 0.0244492
\(837\) 0 0
\(838\) 23154.3 0.954478
\(839\) 15506.1 0.638058 0.319029 0.947745i \(-0.396643\pi\)
0.319029 + 0.947745i \(0.396643\pi\)
\(840\) 0 0
\(841\) 51157.5 2.09756
\(842\) 38968.8 1.59496
\(843\) 0 0
\(844\) 12972.9 0.529084
\(845\) 0 0
\(846\) 0 0
\(847\) −786.406 −0.0319023
\(848\) 15496.1 0.627520
\(849\) 0 0
\(850\) 0 0
\(851\) 43479.4 1.75141
\(852\) 0 0
\(853\) 8698.16 0.349143 0.174572 0.984644i \(-0.444146\pi\)
0.174572 + 0.984644i \(0.444146\pi\)
\(854\) 8801.64 0.352677
\(855\) 0 0
\(856\) −1336.64 −0.0533707
\(857\) 28468.8 1.13474 0.567372 0.823462i \(-0.307960\pi\)
0.567372 + 0.823462i \(0.307960\pi\)
\(858\) 0 0
\(859\) −21154.3 −0.840251 −0.420125 0.907466i \(-0.638014\pi\)
−0.420125 + 0.907466i \(0.638014\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −12338.4 −0.487525
\(863\) 45959.5 1.81284 0.906420 0.422378i \(-0.138805\pi\)
0.906420 + 0.422378i \(0.138805\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −44524.5 −1.74712
\(867\) 0 0
\(868\) 3705.10 0.144884
\(869\) −2207.72 −0.0861815
\(870\) 0 0
\(871\) 878.047 0.0341579
\(872\) −404.765 −0.0157191
\(873\) 0 0
\(874\) −2821.06 −0.109180
\(875\) 0 0
\(876\) 0 0
\(877\) −7315.10 −0.281657 −0.140829 0.990034i \(-0.544977\pi\)
−0.140829 + 0.990034i \(0.544977\pi\)
\(878\) 30655.6 1.17833
\(879\) 0 0
\(880\) 0 0
\(881\) −4652.19 −0.177907 −0.0889536 0.996036i \(-0.528352\pi\)
−0.0889536 + 0.996036i \(0.528352\pi\)
\(882\) 0 0
\(883\) −38954.3 −1.48462 −0.742309 0.670058i \(-0.766269\pi\)
−0.742309 + 0.670058i \(0.766269\pi\)
\(884\) 10953.7 0.416756
\(885\) 0 0
\(886\) 16332.2 0.619289
\(887\) 16917.5 0.640398 0.320199 0.947350i \(-0.396250\pi\)
0.320199 + 0.947350i \(0.396250\pi\)
\(888\) 0 0
\(889\) −3671.33 −0.138507
\(890\) 0 0
\(891\) 0 0
\(892\) 6288.75 0.236057
\(893\) 204.053 0.00764654
\(894\) 0 0
\(895\) 0 0
\(896\) −1061.32 −0.0395715
\(897\) 0 0
\(898\) 59051.8 2.19441
\(899\) −18842.0 −0.699016
\(900\) 0 0
\(901\) −10890.5 −0.402682
\(902\) 11927.9 0.440306
\(903\) 0 0
\(904\) 1702.97 0.0626549
\(905\) 0 0
\(906\) 0 0
\(907\) 38985.4 1.42722 0.713610 0.700543i \(-0.247059\pi\)
0.713610 + 0.700543i \(0.247059\pi\)
\(908\) 47889.8 1.75031
\(909\) 0 0
\(910\) 0 0
\(911\) −27759.8 −1.00957 −0.504787 0.863244i \(-0.668429\pi\)
−0.504787 + 0.863244i \(0.668429\pi\)
\(912\) 0 0
\(913\) 2680.49 0.0971646
\(914\) 44072.3 1.59495
\(915\) 0 0
\(916\) −15905.6 −0.573728
\(917\) −1391.37 −0.0501057
\(918\) 0 0
\(919\) 29705.5 1.06626 0.533130 0.846033i \(-0.321016\pi\)
0.533130 + 0.846033i \(0.321016\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 66127.6 2.36204
\(923\) −2718.18 −0.0969338
\(924\) 0 0
\(925\) 0 0
\(926\) 64724.1 2.29694
\(927\) 0 0
\(928\) 70944.1 2.50954
\(929\) −2940.01 −0.103831 −0.0519153 0.998651i \(-0.516533\pi\)
−0.0519153 + 0.998651i \(0.516533\pi\)
\(930\) 0 0
\(931\) 1943.04 0.0684002
\(932\) −8605.99 −0.302466
\(933\) 0 0
\(934\) −60982.6 −2.13642
\(935\) 0 0
\(936\) 0 0
\(937\) −20041.9 −0.698762 −0.349381 0.936981i \(-0.613608\pi\)
−0.349381 + 0.936981i \(0.613608\pi\)
\(938\) 754.813 0.0262745
\(939\) 0 0
\(940\) 0 0
\(941\) 14446.1 0.500457 0.250229 0.968187i \(-0.419494\pi\)
0.250229 + 0.968187i \(0.419494\pi\)
\(942\) 0 0
\(943\) −29020.6 −1.00216
\(944\) −34265.6 −1.18141
\(945\) 0 0
\(946\) −1356.68 −0.0466272
\(947\) −3901.33 −0.133871 −0.0669357 0.997757i \(-0.521322\pi\)
−0.0669357 + 0.997757i \(0.521322\pi\)
\(948\) 0 0
\(949\) −21915.1 −0.749626
\(950\) 0 0
\(951\) 0 0
\(952\) 357.904 0.0121846
\(953\) −613.963 −0.0208691 −0.0104345 0.999946i \(-0.503321\pi\)
−0.0104345 + 0.999946i \(0.503321\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −22014.6 −0.744773
\(957\) 0 0
\(958\) −42247.1 −1.42478
\(959\) −423.937 −0.0142749
\(960\) 0 0
\(961\) −25091.6 −0.842256
\(962\) 49613.3 1.66278
\(963\) 0 0
\(964\) 56000.5 1.87101
\(965\) 0 0
\(966\) 0 0
\(967\) −20617.4 −0.685637 −0.342819 0.939402i \(-0.611382\pi\)
−0.342819 + 0.939402i \(0.611382\pi\)
\(968\) −154.489 −0.00512961
\(969\) 0 0
\(970\) 0 0
\(971\) −22816.0 −0.754069 −0.377035 0.926199i \(-0.623056\pi\)
−0.377035 + 0.926199i \(0.623056\pi\)
\(972\) 0 0
\(973\) 2645.73 0.0871719
\(974\) 32071.6 1.05507
\(975\) 0 0
\(976\) −20576.0 −0.674816
\(977\) −22650.2 −0.741702 −0.370851 0.928692i \(-0.620934\pi\)
−0.370851 + 0.928692i \(0.620934\pi\)
\(978\) 0 0
\(979\) −3441.15 −0.112339
\(980\) 0 0
\(981\) 0 0
\(982\) 46422.8 1.50857
\(983\) 8963.01 0.290820 0.145410 0.989372i \(-0.453550\pi\)
0.145410 + 0.989372i \(0.453550\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −47886.0 −1.54665
\(987\) 0 0
\(988\) −1640.70 −0.0528316
\(989\) 3300.78 0.106126
\(990\) 0 0
\(991\) 422.204 0.0135336 0.00676678 0.999977i \(-0.497846\pi\)
0.00676678 + 0.999977i \(0.497846\pi\)
\(992\) −17694.1 −0.566319
\(993\) 0 0
\(994\) −2336.68 −0.0745623
\(995\) 0 0
\(996\) 0 0
\(997\) −12577.3 −0.399524 −0.199762 0.979844i \(-0.564017\pi\)
−0.199762 + 0.979844i \(0.564017\pi\)
\(998\) −19443.4 −0.616704
\(999\) 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 2475.4.a.v.1.1 3
3.2 odd 2 825.4.a.q.1.3 yes 3
5.4 even 2 2475.4.a.y.1.3 3
15.2 even 4 825.4.c.n.199.6 6
15.8 even 4 825.4.c.n.199.1 6
15.14 odd 2 825.4.a.o.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.o.1.1 3 15.14 odd 2
825.4.a.q.1.3 yes 3 3.2 odd 2
825.4.c.n.199.1 6 15.8 even 4
825.4.c.n.199.6 6 15.2 even 4
2475.4.a.v.1.1 3 1.1 even 1 trivial
2475.4.a.y.1.3 3 5.4 even 2