Properties

Label 825.4.a.t.1.3
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, 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("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1540841.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 27x^{2} - 18x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.60719\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.607192 q^{2} -3.00000 q^{3} -7.63132 q^{4} -1.82158 q^{6} -8.95080 q^{7} -9.49121 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+0.607192 q^{2} -3.00000 q^{3} -7.63132 q^{4} -1.82158 q^{6} -8.95080 q^{7} -9.49121 q^{8} +9.00000 q^{9} -11.0000 q^{11} +22.8940 q^{12} +0.460387 q^{13} -5.43485 q^{14} +55.2876 q^{16} -128.395 q^{17} +5.46473 q^{18} -0.0245858 q^{19} +26.8524 q^{21} -6.67911 q^{22} -171.528 q^{23} +28.4736 q^{24} +0.279543 q^{26} -27.0000 q^{27} +68.3064 q^{28} -226.938 q^{29} +195.637 q^{31} +109.500 q^{32} +33.0000 q^{33} -77.9604 q^{34} -68.6819 q^{36} -338.584 q^{37} -0.0149283 q^{38} -1.38116 q^{39} +136.972 q^{41} +16.3046 q^{42} -336.083 q^{43} +83.9445 q^{44} -104.151 q^{46} +540.292 q^{47} -165.863 q^{48} -262.883 q^{49} +385.185 q^{51} -3.51336 q^{52} +622.387 q^{53} -16.3942 q^{54} +84.9539 q^{56} +0.0737574 q^{57} -137.795 q^{58} -9.86955 q^{59} +902.712 q^{61} +118.789 q^{62} -80.5572 q^{63} -375.813 q^{64} +20.0373 q^{66} -146.979 q^{67} +979.823 q^{68} +514.585 q^{69} -893.798 q^{71} -85.4209 q^{72} +1149.71 q^{73} -205.585 q^{74} +0.187622 q^{76} +98.4588 q^{77} -0.838629 q^{78} -459.528 q^{79} +81.0000 q^{81} +83.1686 q^{82} +125.876 q^{83} -204.919 q^{84} -204.067 q^{86} +680.813 q^{87} +104.403 q^{88} +150.461 q^{89} -4.12083 q^{91} +1308.99 q^{92} -586.912 q^{93} +328.061 q^{94} -328.499 q^{96} +1264.58 q^{97} -159.621 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 12 q^{3} + 26 q^{4} + 12 q^{6} - 34 q^{7} - 48 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 12 q^{3} + 26 q^{4} + 12 q^{6} - 34 q^{7} - 48 q^{8} + 36 q^{9} - 44 q^{11} - 78 q^{12} - 2 q^{13} - 52 q^{14} + 66 q^{16} - 74 q^{17} - 36 q^{18} + 136 q^{19} + 102 q^{21} + 44 q^{22} + 64 q^{23} + 144 q^{24} - 320 q^{26} - 108 q^{27} + 20 q^{28} + 52 q^{29} + 492 q^{31} - 208 q^{32} + 132 q^{33} + 244 q^{34} + 234 q^{36} + 4 q^{37} + 404 q^{38} + 6 q^{39} + 268 q^{41} + 156 q^{42} - 546 q^{43} - 286 q^{44} + 368 q^{46} + 276 q^{47} - 198 q^{48} - 496 q^{49} + 222 q^{51} + 1084 q^{52} + 184 q^{53} + 108 q^{54} - 852 q^{56} - 408 q^{57} + 444 q^{58} - 1032 q^{59} + 116 q^{61} + 1240 q^{62} - 306 q^{63} - 918 q^{64} - 132 q^{66} + 552 q^{67} + 720 q^{68} - 192 q^{69} - 920 q^{71} - 432 q^{72} - 926 q^{73} - 2856 q^{74} + 1572 q^{76} + 374 q^{77} + 960 q^{78} + 1152 q^{79} + 324 q^{81} + 1924 q^{82} + 134 q^{83} - 60 q^{84} + 236 q^{86} - 156 q^{87} + 528 q^{88} - 1064 q^{89} + 2780 q^{91} + 4896 q^{92} - 1476 q^{93} - 1432 q^{94} + 624 q^{96} + 1648 q^{97} + 188 q^{98} - 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.607192 0.214675 0.107337 0.994223i \(-0.465768\pi\)
0.107337 + 0.994223i \(0.465768\pi\)
\(3\) −3.00000 −0.577350
\(4\) −7.63132 −0.953915
\(5\) 0 0
\(6\) −1.82158 −0.123943
\(7\) −8.95080 −0.483298 −0.241649 0.970364i \(-0.577688\pi\)
−0.241649 + 0.970364i \(0.577688\pi\)
\(8\) −9.49121 −0.419456
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 22.8940 0.550743
\(13\) 0.460387 0.00982218 0.00491109 0.999988i \(-0.498437\pi\)
0.00491109 + 0.999988i \(0.498437\pi\)
\(14\) −5.43485 −0.103752
\(15\) 0 0
\(16\) 55.2876 0.863868
\(17\) −128.395 −1.83179 −0.915893 0.401423i \(-0.868516\pi\)
−0.915893 + 0.401423i \(0.868516\pi\)
\(18\) 5.46473 0.0715582
\(19\) −0.0245858 −0.000296862 0 −0.000148431 1.00000i \(-0.500047\pi\)
−0.000148431 1.00000i \(0.500047\pi\)
\(20\) 0 0
\(21\) 26.8524 0.279032
\(22\) −6.67911 −0.0647269
\(23\) −171.528 −1.55505 −0.777525 0.628852i \(-0.783525\pi\)
−0.777525 + 0.628852i \(0.783525\pi\)
\(24\) 28.4736 0.242173
\(25\) 0 0
\(26\) 0.279543 0.00210857
\(27\) −27.0000 −0.192450
\(28\) 68.3064 0.461025
\(29\) −226.938 −1.45315 −0.726574 0.687088i \(-0.758889\pi\)
−0.726574 + 0.687088i \(0.758889\pi\)
\(30\) 0 0
\(31\) 195.637 1.13347 0.566734 0.823901i \(-0.308207\pi\)
0.566734 + 0.823901i \(0.308207\pi\)
\(32\) 109.500 0.604907
\(33\) 33.0000 0.174078
\(34\) −77.9604 −0.393238
\(35\) 0 0
\(36\) −68.6819 −0.317972
\(37\) −338.584 −1.50440 −0.752200 0.658935i \(-0.771007\pi\)
−0.752200 + 0.658935i \(0.771007\pi\)
\(38\) −0.0149283 −6.37287e−5 0
\(39\) −1.38116 −0.00567084
\(40\) 0 0
\(41\) 136.972 0.521744 0.260872 0.965373i \(-0.415990\pi\)
0.260872 + 0.965373i \(0.415990\pi\)
\(42\) 16.3046 0.0599011
\(43\) −336.083 −1.19191 −0.595956 0.803017i \(-0.703227\pi\)
−0.595956 + 0.803017i \(0.703227\pi\)
\(44\) 83.9445 0.287616
\(45\) 0 0
\(46\) −104.151 −0.333830
\(47\) 540.292 1.67680 0.838400 0.545055i \(-0.183491\pi\)
0.838400 + 0.545055i \(0.183491\pi\)
\(48\) −165.863 −0.498754
\(49\) −262.883 −0.766423
\(50\) 0 0
\(51\) 385.185 1.05758
\(52\) −3.51336 −0.00936952
\(53\) 622.387 1.61305 0.806523 0.591203i \(-0.201347\pi\)
0.806523 + 0.591203i \(0.201347\pi\)
\(54\) −16.3942 −0.0413142
\(55\) 0 0
\(56\) 84.9539 0.202722
\(57\) 0.0737574 0.000171393 0
\(58\) −137.795 −0.311954
\(59\) −9.86955 −0.0217781 −0.0108890 0.999941i \(-0.503466\pi\)
−0.0108890 + 0.999941i \(0.503466\pi\)
\(60\) 0 0
\(61\) 902.712 1.89476 0.947380 0.320110i \(-0.103720\pi\)
0.947380 + 0.320110i \(0.103720\pi\)
\(62\) 118.789 0.243327
\(63\) −80.5572 −0.161099
\(64\) −375.813 −0.734010
\(65\) 0 0
\(66\) 20.0373 0.0373701
\(67\) −146.979 −0.268005 −0.134003 0.990981i \(-0.542783\pi\)
−0.134003 + 0.990981i \(0.542783\pi\)
\(68\) 979.823 1.74737
\(69\) 514.585 0.897809
\(70\) 0 0
\(71\) −893.798 −1.49400 −0.747002 0.664821i \(-0.768508\pi\)
−0.747002 + 0.664821i \(0.768508\pi\)
\(72\) −85.4209 −0.139819
\(73\) 1149.71 1.84333 0.921664 0.387988i \(-0.126830\pi\)
0.921664 + 0.387988i \(0.126830\pi\)
\(74\) −205.585 −0.322957
\(75\) 0 0
\(76\) 0.187622 0.000283181 0
\(77\) 98.4588 0.145720
\(78\) −0.838629 −0.00121739
\(79\) −459.528 −0.654443 −0.327221 0.944948i \(-0.606112\pi\)
−0.327221 + 0.944948i \(0.606112\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 83.1686 0.112005
\(83\) 125.876 0.166466 0.0832331 0.996530i \(-0.473475\pi\)
0.0832331 + 0.996530i \(0.473475\pi\)
\(84\) −204.919 −0.266173
\(85\) 0 0
\(86\) −204.067 −0.255873
\(87\) 680.813 0.838975
\(88\) 104.403 0.126471
\(89\) 150.461 0.179201 0.0896003 0.995978i \(-0.471441\pi\)
0.0896003 + 0.995978i \(0.471441\pi\)
\(90\) 0 0
\(91\) −4.12083 −0.00474704
\(92\) 1308.99 1.48339
\(93\) −586.912 −0.654408
\(94\) 328.061 0.359967
\(95\) 0 0
\(96\) −328.499 −0.349243
\(97\) 1264.58 1.32370 0.661851 0.749635i \(-0.269771\pi\)
0.661851 + 0.749635i \(0.269771\pi\)
\(98\) −159.621 −0.164532
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 690.792 0.680558 0.340279 0.940324i \(-0.389478\pi\)
0.340279 + 0.940324i \(0.389478\pi\)
\(102\) 233.881 0.227036
\(103\) 11.7823 0.0112713 0.00563563 0.999984i \(-0.498206\pi\)
0.00563563 + 0.999984i \(0.498206\pi\)
\(104\) −4.36963 −0.00411997
\(105\) 0 0
\(106\) 377.908 0.346280
\(107\) 462.884 0.418212 0.209106 0.977893i \(-0.432945\pi\)
0.209106 + 0.977893i \(0.432945\pi\)
\(108\) 206.046 0.183581
\(109\) 330.525 0.290446 0.145223 0.989399i \(-0.453610\pi\)
0.145223 + 0.989399i \(0.453610\pi\)
\(110\) 0 0
\(111\) 1015.75 0.868566
\(112\) −494.868 −0.417506
\(113\) −1142.31 −0.950965 −0.475483 0.879725i \(-0.657727\pi\)
−0.475483 + 0.879725i \(0.657727\pi\)
\(114\) 0.0447849 3.67938e−5 0
\(115\) 0 0
\(116\) 1731.83 1.38618
\(117\) 4.14348 0.00327406
\(118\) −5.99271 −0.00467520
\(119\) 1149.24 0.885298
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 548.119 0.406757
\(123\) −410.917 −0.301229
\(124\) −1492.97 −1.08123
\(125\) 0 0
\(126\) −48.9137 −0.0345839
\(127\) −629.589 −0.439898 −0.219949 0.975511i \(-0.570589\pi\)
−0.219949 + 0.975511i \(0.570589\pi\)
\(128\) −1104.19 −0.762480
\(129\) 1008.25 0.688151
\(130\) 0 0
\(131\) 572.566 0.381873 0.190937 0.981602i \(-0.438848\pi\)
0.190937 + 0.981602i \(0.438848\pi\)
\(132\) −251.833 −0.166055
\(133\) 0.220063 0.000143473 0
\(134\) −89.2446 −0.0575340
\(135\) 0 0
\(136\) 1218.62 0.768354
\(137\) 948.680 0.591615 0.295807 0.955248i \(-0.404411\pi\)
0.295807 + 0.955248i \(0.404411\pi\)
\(138\) 312.452 0.192737
\(139\) 2488.86 1.51872 0.759362 0.650668i \(-0.225511\pi\)
0.759362 + 0.650668i \(0.225511\pi\)
\(140\) 0 0
\(141\) −1620.87 −0.968101
\(142\) −542.707 −0.320725
\(143\) −5.06425 −0.00296150
\(144\) 497.588 0.287956
\(145\) 0 0
\(146\) 698.093 0.395716
\(147\) 788.650 0.442495
\(148\) 2583.84 1.43507
\(149\) −2186.43 −1.20214 −0.601072 0.799195i \(-0.705259\pi\)
−0.601072 + 0.799195i \(0.705259\pi\)
\(150\) 0 0
\(151\) 3668.21 1.97692 0.988459 0.151488i \(-0.0484064\pi\)
0.988459 + 0.151488i \(0.0484064\pi\)
\(152\) 0.233349 0.000124520 0
\(153\) −1155.55 −0.610595
\(154\) 59.7834 0.0312824
\(155\) 0 0
\(156\) 10.5401 0.00540950
\(157\) −1418.28 −0.720960 −0.360480 0.932767i \(-0.617387\pi\)
−0.360480 + 0.932767i \(0.617387\pi\)
\(158\) −279.022 −0.140492
\(159\) −1867.16 −0.931293
\(160\) 0 0
\(161\) 1535.32 0.751552
\(162\) 49.1825 0.0238527
\(163\) 601.009 0.288802 0.144401 0.989519i \(-0.453875\pi\)
0.144401 + 0.989519i \(0.453875\pi\)
\(164\) −1045.28 −0.497699
\(165\) 0 0
\(166\) 76.4309 0.0357361
\(167\) 2221.19 1.02923 0.514614 0.857422i \(-0.327935\pi\)
0.514614 + 0.857422i \(0.327935\pi\)
\(168\) −254.862 −0.117042
\(169\) −2196.79 −0.999904
\(170\) 0 0
\(171\) −0.221272 −9.89539e−5 0
\(172\) 2564.76 1.13698
\(173\) 237.280 0.104278 0.0521390 0.998640i \(-0.483396\pi\)
0.0521390 + 0.998640i \(0.483396\pi\)
\(174\) 413.384 0.180107
\(175\) 0 0
\(176\) −608.163 −0.260466
\(177\) 29.6086 0.0125736
\(178\) 91.3588 0.0384698
\(179\) 1065.77 0.445023 0.222512 0.974930i \(-0.428574\pi\)
0.222512 + 0.974930i \(0.428574\pi\)
\(180\) 0 0
\(181\) −1241.00 −0.509627 −0.254813 0.966990i \(-0.582014\pi\)
−0.254813 + 0.966990i \(0.582014\pi\)
\(182\) −2.50213 −0.00101907
\(183\) −2708.14 −1.09394
\(184\) 1628.01 0.652275
\(185\) 0 0
\(186\) −356.368 −0.140485
\(187\) 1412.34 0.552304
\(188\) −4123.14 −1.59952
\(189\) 241.672 0.0930107
\(190\) 0 0
\(191\) −1956.80 −0.741305 −0.370653 0.928772i \(-0.620866\pi\)
−0.370653 + 0.928772i \(0.620866\pi\)
\(192\) 1127.44 0.423781
\(193\) −2778.02 −1.03610 −0.518048 0.855352i \(-0.673341\pi\)
−0.518048 + 0.855352i \(0.673341\pi\)
\(194\) 767.846 0.284165
\(195\) 0 0
\(196\) 2006.15 0.731102
\(197\) −800.242 −0.289416 −0.144708 0.989474i \(-0.546224\pi\)
−0.144708 + 0.989474i \(0.546224\pi\)
\(198\) −60.1120 −0.0215756
\(199\) −2536.78 −0.903657 −0.451829 0.892105i \(-0.649228\pi\)
−0.451829 + 0.892105i \(0.649228\pi\)
\(200\) 0 0
\(201\) 440.938 0.154733
\(202\) 419.443 0.146099
\(203\) 2031.27 0.702303
\(204\) −2939.47 −1.00884
\(205\) 0 0
\(206\) 7.15409 0.00241966
\(207\) −1543.76 −0.518350
\(208\) 25.4537 0.00848507
\(209\) 0.270444 8.95071e−5 0
\(210\) 0 0
\(211\) 1598.95 0.521688 0.260844 0.965381i \(-0.415999\pi\)
0.260844 + 0.965381i \(0.415999\pi\)
\(212\) −4749.63 −1.53871
\(213\) 2681.40 0.862564
\(214\) 281.060 0.0897797
\(215\) 0 0
\(216\) 256.263 0.0807244
\(217\) −1751.11 −0.547802
\(218\) 200.692 0.0623513
\(219\) −3449.12 −1.06425
\(220\) 0 0
\(221\) −59.1113 −0.0179921
\(222\) 616.756 0.186459
\(223\) 5074.55 1.52384 0.761922 0.647669i \(-0.224256\pi\)
0.761922 + 0.647669i \(0.224256\pi\)
\(224\) −980.111 −0.292350
\(225\) 0 0
\(226\) −693.599 −0.204148
\(227\) −1048.19 −0.306480 −0.153240 0.988189i \(-0.548971\pi\)
−0.153240 + 0.988189i \(0.548971\pi\)
\(228\) −0.562866 −0.000163494 0
\(229\) −734.662 −0.211999 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(230\) 0 0
\(231\) −295.376 −0.0841313
\(232\) 2153.91 0.609532
\(233\) 2012.78 0.565929 0.282965 0.959130i \(-0.408682\pi\)
0.282965 + 0.959130i \(0.408682\pi\)
\(234\) 2.51589 0.000702858 0
\(235\) 0 0
\(236\) 75.3176 0.0207744
\(237\) 1378.59 0.377843
\(238\) 697.808 0.190051
\(239\) 5566.41 1.50653 0.753267 0.657715i \(-0.228477\pi\)
0.753267 + 0.657715i \(0.228477\pi\)
\(240\) 0 0
\(241\) −3361.25 −0.898412 −0.449206 0.893428i \(-0.648293\pi\)
−0.449206 + 0.893428i \(0.648293\pi\)
\(242\) 73.4702 0.0195159
\(243\) −243.000 −0.0641500
\(244\) −6888.88 −1.80744
\(245\) 0 0
\(246\) −249.506 −0.0646663
\(247\) −0.0113190 −2.91583e−6 0
\(248\) −1856.83 −0.475440
\(249\) −377.628 −0.0961093
\(250\) 0 0
\(251\) −2930.46 −0.736929 −0.368464 0.929642i \(-0.620116\pi\)
−0.368464 + 0.929642i \(0.620116\pi\)
\(252\) 614.758 0.153675
\(253\) 1886.81 0.468865
\(254\) −382.281 −0.0944349
\(255\) 0 0
\(256\) 2336.05 0.570325
\(257\) 3412.89 0.828367 0.414184 0.910193i \(-0.364067\pi\)
0.414184 + 0.910193i \(0.364067\pi\)
\(258\) 612.201 0.147729
\(259\) 3030.59 0.727073
\(260\) 0 0
\(261\) −2042.44 −0.484383
\(262\) 347.658 0.0819785
\(263\) −3709.06 −0.869622 −0.434811 0.900522i \(-0.643185\pi\)
−0.434811 + 0.900522i \(0.643185\pi\)
\(264\) −313.210 −0.0730179
\(265\) 0 0
\(266\) 0.133620 3.07999e−5 0
\(267\) −451.384 −0.103462
\(268\) 1121.64 0.255654
\(269\) −5764.39 −1.30655 −0.653273 0.757122i \(-0.726605\pi\)
−0.653273 + 0.757122i \(0.726605\pi\)
\(270\) 0 0
\(271\) −1886.33 −0.422827 −0.211414 0.977397i \(-0.567807\pi\)
−0.211414 + 0.977397i \(0.567807\pi\)
\(272\) −7098.64 −1.58242
\(273\) 12.3625 0.00274070
\(274\) 576.031 0.127005
\(275\) 0 0
\(276\) −3926.97 −0.856433
\(277\) 4095.99 0.888462 0.444231 0.895912i \(-0.353477\pi\)
0.444231 + 0.895912i \(0.353477\pi\)
\(278\) 1511.22 0.326032
\(279\) 1760.73 0.377822
\(280\) 0 0
\(281\) −8788.69 −1.86580 −0.932899 0.360138i \(-0.882729\pi\)
−0.932899 + 0.360138i \(0.882729\pi\)
\(282\) −984.182 −0.207827
\(283\) 1767.75 0.371314 0.185657 0.982615i \(-0.440559\pi\)
0.185657 + 0.982615i \(0.440559\pi\)
\(284\) 6820.86 1.42515
\(285\) 0 0
\(286\) −3.07497 −0.000635759 0
\(287\) −1226.01 −0.252158
\(288\) 985.498 0.201636
\(289\) 11572.3 2.35544
\(290\) 0 0
\(291\) −3793.75 −0.764240
\(292\) −8773.78 −1.75838
\(293\) −3079.85 −0.614085 −0.307043 0.951696i \(-0.599339\pi\)
−0.307043 + 0.951696i \(0.599339\pi\)
\(294\) 478.862 0.0949924
\(295\) 0 0
\(296\) 3213.57 0.631030
\(297\) 297.000 0.0580259
\(298\) −1327.58 −0.258070
\(299\) −78.9695 −0.0152740
\(300\) 0 0
\(301\) 3008.21 0.576048
\(302\) 2227.31 0.424394
\(303\) −2072.38 −0.392921
\(304\) −1.35929 −0.000256449 0
\(305\) 0 0
\(306\) −701.643 −0.131079
\(307\) −2872.16 −0.533950 −0.266975 0.963703i \(-0.586024\pi\)
−0.266975 + 0.963703i \(0.586024\pi\)
\(308\) −751.370 −0.139004
\(309\) −35.3468 −0.00650747
\(310\) 0 0
\(311\) −317.978 −0.0579771 −0.0289885 0.999580i \(-0.509229\pi\)
−0.0289885 + 0.999580i \(0.509229\pi\)
\(312\) 13.1089 0.00237867
\(313\) 8274.81 1.49431 0.747156 0.664648i \(-0.231419\pi\)
0.747156 + 0.664648i \(0.231419\pi\)
\(314\) −861.166 −0.154772
\(315\) 0 0
\(316\) 3506.81 0.624283
\(317\) −1024.34 −0.181492 −0.0907459 0.995874i \(-0.528925\pi\)
−0.0907459 + 0.995874i \(0.528925\pi\)
\(318\) −1133.73 −0.199925
\(319\) 2496.32 0.438141
\(320\) 0 0
\(321\) −1388.65 −0.241455
\(322\) 932.232 0.161339
\(323\) 3.15669 0.000543787 0
\(324\) −618.137 −0.105991
\(325\) 0 0
\(326\) 364.928 0.0619984
\(327\) −991.575 −0.167689
\(328\) −1300.03 −0.218849
\(329\) −4836.04 −0.810394
\(330\) 0 0
\(331\) 4575.83 0.759850 0.379925 0.925017i \(-0.375950\pi\)
0.379925 + 0.925017i \(0.375950\pi\)
\(332\) −960.600 −0.158795
\(333\) −3047.25 −0.501467
\(334\) 1348.69 0.220949
\(335\) 0 0
\(336\) 1484.60 0.241047
\(337\) 917.888 0.148370 0.0741848 0.997245i \(-0.476365\pi\)
0.0741848 + 0.997245i \(0.476365\pi\)
\(338\) −1333.87 −0.214654
\(339\) 3426.92 0.549040
\(340\) 0 0
\(341\) −2152.01 −0.341753
\(342\) −0.134355 −2.12429e−5 0
\(343\) 5423.14 0.853708
\(344\) 3189.84 0.499955
\(345\) 0 0
\(346\) 144.075 0.0223859
\(347\) −5402.26 −0.835760 −0.417880 0.908502i \(-0.637227\pi\)
−0.417880 + 0.908502i \(0.637227\pi\)
\(348\) −5195.50 −0.800311
\(349\) 5115.85 0.784656 0.392328 0.919825i \(-0.371670\pi\)
0.392328 + 0.919825i \(0.371670\pi\)
\(350\) 0 0
\(351\) −12.4304 −0.00189028
\(352\) −1204.50 −0.182386
\(353\) 3102.03 0.467718 0.233859 0.972270i \(-0.424865\pi\)
0.233859 + 0.972270i \(0.424865\pi\)
\(354\) 17.9781 0.00269923
\(355\) 0 0
\(356\) −1148.22 −0.170942
\(357\) −3447.71 −0.511127
\(358\) 647.125 0.0955352
\(359\) 3496.32 0.514008 0.257004 0.966410i \(-0.417265\pi\)
0.257004 + 0.966410i \(0.417265\pi\)
\(360\) 0 0
\(361\) −6859.00 −1.00000
\(362\) −753.522 −0.109404
\(363\) −363.000 −0.0524864
\(364\) 31.4474 0.00452827
\(365\) 0 0
\(366\) −1644.36 −0.234841
\(367\) −7644.54 −1.08731 −0.543654 0.839309i \(-0.682960\pi\)
−0.543654 + 0.839309i \(0.682960\pi\)
\(368\) −9483.39 −1.34336
\(369\) 1232.75 0.173915
\(370\) 0 0
\(371\) −5570.86 −0.779582
\(372\) 4478.91 0.624249
\(373\) 5189.90 0.720437 0.360218 0.932868i \(-0.382702\pi\)
0.360218 + 0.932868i \(0.382702\pi\)
\(374\) 857.564 0.118566
\(375\) 0 0
\(376\) −5128.02 −0.703344
\(377\) −104.479 −0.0142731
\(378\) 146.741 0.0199670
\(379\) 8573.91 1.16204 0.581019 0.813890i \(-0.302654\pi\)
0.581019 + 0.813890i \(0.302654\pi\)
\(380\) 0 0
\(381\) 1888.77 0.253975
\(382\) −1188.15 −0.159140
\(383\) 11740.2 1.56631 0.783157 0.621824i \(-0.213608\pi\)
0.783157 + 0.621824i \(0.213608\pi\)
\(384\) 3312.57 0.440218
\(385\) 0 0
\(386\) −1686.79 −0.222424
\(387\) −3024.75 −0.397304
\(388\) −9650.45 −1.26270
\(389\) −3763.68 −0.490556 −0.245278 0.969453i \(-0.578879\pi\)
−0.245278 + 0.969453i \(0.578879\pi\)
\(390\) 0 0
\(391\) 22023.4 2.84852
\(392\) 2495.08 0.321481
\(393\) −1717.70 −0.220474
\(394\) −485.900 −0.0621302
\(395\) 0 0
\(396\) 755.500 0.0958720
\(397\) −1151.35 −0.145554 −0.0727768 0.997348i \(-0.523186\pi\)
−0.0727768 + 0.997348i \(0.523186\pi\)
\(398\) −1540.31 −0.193992
\(399\) −0.660188 −8.28339e−5 0
\(400\) 0 0
\(401\) −10051.0 −1.25168 −0.625842 0.779950i \(-0.715245\pi\)
−0.625842 + 0.779950i \(0.715245\pi\)
\(402\) 267.734 0.0332173
\(403\) 90.0688 0.0111331
\(404\) −5271.66 −0.649195
\(405\) 0 0
\(406\) 1233.37 0.150767
\(407\) 3724.42 0.453594
\(408\) −3655.87 −0.443609
\(409\) −6263.39 −0.757225 −0.378612 0.925555i \(-0.623599\pi\)
−0.378612 + 0.925555i \(0.623599\pi\)
\(410\) 0 0
\(411\) −2846.04 −0.341569
\(412\) −89.9142 −0.0107518
\(413\) 88.3403 0.0105253
\(414\) −937.356 −0.111277
\(415\) 0 0
\(416\) 50.4123 0.00594150
\(417\) −7466.59 −0.876836
\(418\) 0.164211 1.92149e−5 0
\(419\) 8625.67 1.00571 0.502854 0.864372i \(-0.332283\pi\)
0.502854 + 0.864372i \(0.332283\pi\)
\(420\) 0 0
\(421\) −9095.24 −1.05291 −0.526455 0.850203i \(-0.676479\pi\)
−0.526455 + 0.850203i \(0.676479\pi\)
\(422\) 970.868 0.111993
\(423\) 4862.62 0.558934
\(424\) −5907.21 −0.676602
\(425\) 0 0
\(426\) 1628.12 0.185171
\(427\) −8079.99 −0.915733
\(428\) −3532.42 −0.398939
\(429\) 15.1928 0.00170982
\(430\) 0 0
\(431\) −4008.72 −0.448013 −0.224006 0.974588i \(-0.571914\pi\)
−0.224006 + 0.974588i \(0.571914\pi\)
\(432\) −1492.76 −0.166251
\(433\) −2715.38 −0.301369 −0.150684 0.988582i \(-0.548148\pi\)
−0.150684 + 0.988582i \(0.548148\pi\)
\(434\) −1063.26 −0.117599
\(435\) 0 0
\(436\) −2522.34 −0.277060
\(437\) 4.21717 0.000461635 0
\(438\) −2094.28 −0.228467
\(439\) −11087.5 −1.20541 −0.602707 0.797962i \(-0.705911\pi\)
−0.602707 + 0.797962i \(0.705911\pi\)
\(440\) 0 0
\(441\) −2365.95 −0.255474
\(442\) −35.8919 −0.00386245
\(443\) −7132.98 −0.765007 −0.382503 0.923954i \(-0.624938\pi\)
−0.382503 + 0.923954i \(0.624938\pi\)
\(444\) −7751.52 −0.828538
\(445\) 0 0
\(446\) 3081.23 0.327131
\(447\) 6559.29 0.694058
\(448\) 3363.83 0.354745
\(449\) 6291.28 0.661255 0.330628 0.943761i \(-0.392740\pi\)
0.330628 + 0.943761i \(0.392740\pi\)
\(450\) 0 0
\(451\) −1506.70 −0.157312
\(452\) 8717.30 0.907140
\(453\) −11004.6 −1.14137
\(454\) −636.455 −0.0657936
\(455\) 0 0
\(456\) −0.700047 −7.18919e−5 0
\(457\) −7291.98 −0.746400 −0.373200 0.927751i \(-0.621739\pi\)
−0.373200 + 0.927751i \(0.621739\pi\)
\(458\) −446.081 −0.0455109
\(459\) 3466.66 0.352527
\(460\) 0 0
\(461\) −16805.4 −1.69784 −0.848922 0.528519i \(-0.822748\pi\)
−0.848922 + 0.528519i \(0.822748\pi\)
\(462\) −179.350 −0.0180609
\(463\) −7478.48 −0.750658 −0.375329 0.926892i \(-0.622470\pi\)
−0.375329 + 0.926892i \(0.622470\pi\)
\(464\) −12546.8 −1.25533
\(465\) 0 0
\(466\) 1222.14 0.121491
\(467\) 7526.83 0.745824 0.372912 0.927867i \(-0.378359\pi\)
0.372912 + 0.927867i \(0.378359\pi\)
\(468\) −31.6202 −0.00312317
\(469\) 1315.58 0.129526
\(470\) 0 0
\(471\) 4254.83 0.416247
\(472\) 93.6739 0.00913494
\(473\) 3696.92 0.359375
\(474\) 837.066 0.0811133
\(475\) 0 0
\(476\) −8770.20 −0.844499
\(477\) 5601.48 0.537682
\(478\) 3379.88 0.323415
\(479\) 1258.80 0.120075 0.0600376 0.998196i \(-0.480878\pi\)
0.0600376 + 0.998196i \(0.480878\pi\)
\(480\) 0 0
\(481\) −155.879 −0.0147765
\(482\) −2040.93 −0.192866
\(483\) −4605.95 −0.433909
\(484\) −923.389 −0.0867195
\(485\) 0 0
\(486\) −147.548 −0.0137714
\(487\) −5127.97 −0.477147 −0.238574 0.971124i \(-0.576680\pi\)
−0.238574 + 0.971124i \(0.576680\pi\)
\(488\) −8567.83 −0.794769
\(489\) −1803.03 −0.166740
\(490\) 0 0
\(491\) 11932.3 1.09673 0.548366 0.836238i \(-0.315250\pi\)
0.548366 + 0.836238i \(0.315250\pi\)
\(492\) 3135.84 0.287347
\(493\) 29137.7 2.66185
\(494\) −0.00687279 −6.25955e−7 0
\(495\) 0 0
\(496\) 10816.3 0.979166
\(497\) 8000.21 0.722049
\(498\) −229.293 −0.0206322
\(499\) 14716.3 1.32022 0.660111 0.751168i \(-0.270509\pi\)
0.660111 + 0.751168i \(0.270509\pi\)
\(500\) 0 0
\(501\) −6663.57 −0.594225
\(502\) −1779.35 −0.158200
\(503\) 5598.95 0.496312 0.248156 0.968720i \(-0.420175\pi\)
0.248156 + 0.968720i \(0.420175\pi\)
\(504\) 764.585 0.0675741
\(505\) 0 0
\(506\) 1145.66 0.100654
\(507\) 6590.36 0.577295
\(508\) 4804.60 0.419625
\(509\) −16720.0 −1.45600 −0.727998 0.685579i \(-0.759549\pi\)
−0.727998 + 0.685579i \(0.759549\pi\)
\(510\) 0 0
\(511\) −10290.8 −0.890877
\(512\) 10251.9 0.884915
\(513\) 0.663817 5.71310e−5 0
\(514\) 2072.28 0.177830
\(515\) 0 0
\(516\) −7694.27 −0.656437
\(517\) −5943.21 −0.505574
\(518\) 1840.15 0.156084
\(519\) −711.841 −0.0602050
\(520\) 0 0
\(521\) 3498.81 0.294214 0.147107 0.989121i \(-0.453004\pi\)
0.147107 + 0.989121i \(0.453004\pi\)
\(522\) −1240.15 −0.103985
\(523\) 5681.71 0.475036 0.237518 0.971383i \(-0.423666\pi\)
0.237518 + 0.971383i \(0.423666\pi\)
\(524\) −4369.44 −0.364274
\(525\) 0 0
\(526\) −2252.11 −0.186686
\(527\) −25118.8 −2.07627
\(528\) 1824.49 0.150380
\(529\) 17255.0 1.41818
\(530\) 0 0
\(531\) −88.8259 −0.00725935
\(532\) −1.67937 −0.000136861 0
\(533\) 63.0603 0.00512466
\(534\) −274.076 −0.0222106
\(535\) 0 0
\(536\) 1395.01 0.112417
\(537\) −3197.30 −0.256934
\(538\) −3500.09 −0.280482
\(539\) 2891.72 0.231085
\(540\) 0 0
\(541\) 11641.7 0.925169 0.462585 0.886575i \(-0.346922\pi\)
0.462585 + 0.886575i \(0.346922\pi\)
\(542\) −1145.36 −0.0907704
\(543\) 3722.99 0.294233
\(544\) −14059.2 −1.10806
\(545\) 0 0
\(546\) 7.50640 0.000588360 0
\(547\) 16460.2 1.28663 0.643315 0.765602i \(-0.277559\pi\)
0.643315 + 0.765602i \(0.277559\pi\)
\(548\) −7239.68 −0.564350
\(549\) 8124.41 0.631587
\(550\) 0 0
\(551\) 5.57945 0.000431384 0
\(552\) −4884.04 −0.376591
\(553\) 4113.15 0.316291
\(554\) 2487.05 0.190730
\(555\) 0 0
\(556\) −18993.3 −1.44873
\(557\) 11769.7 0.895331 0.447666 0.894201i \(-0.352255\pi\)
0.447666 + 0.894201i \(0.352255\pi\)
\(558\) 1069.10 0.0811089
\(559\) −154.728 −0.0117072
\(560\) 0 0
\(561\) −4237.03 −0.318873
\(562\) −5336.42 −0.400540
\(563\) −21123.9 −1.58129 −0.790644 0.612276i \(-0.790254\pi\)
−0.790644 + 0.612276i \(0.790254\pi\)
\(564\) 12369.4 0.923486
\(565\) 0 0
\(566\) 1073.36 0.0797118
\(567\) −725.015 −0.0536997
\(568\) 8483.23 0.626670
\(569\) −21890.4 −1.61282 −0.806410 0.591357i \(-0.798592\pi\)
−0.806410 + 0.591357i \(0.798592\pi\)
\(570\) 0 0
\(571\) 6630.99 0.485986 0.242993 0.970028i \(-0.421871\pi\)
0.242993 + 0.970028i \(0.421871\pi\)
\(572\) 38.6469 0.00282502
\(573\) 5870.41 0.427993
\(574\) −744.425 −0.0541319
\(575\) 0 0
\(576\) −3382.32 −0.244670
\(577\) −13361.3 −0.964015 −0.482007 0.876167i \(-0.660092\pi\)
−0.482007 + 0.876167i \(0.660092\pi\)
\(578\) 7026.58 0.505653
\(579\) 8334.07 0.598190
\(580\) 0 0
\(581\) −1126.69 −0.0804527
\(582\) −2303.54 −0.164063
\(583\) −6846.26 −0.486352
\(584\) −10912.1 −0.773196
\(585\) 0 0
\(586\) −1870.06 −0.131829
\(587\) 9451.34 0.664563 0.332282 0.943180i \(-0.392182\pi\)
0.332282 + 0.943180i \(0.392182\pi\)
\(588\) −6018.44 −0.422102
\(589\) −4.80990 −0.000336483 0
\(590\) 0 0
\(591\) 2400.73 0.167094
\(592\) −18719.5 −1.29960
\(593\) −712.344 −0.0493296 −0.0246648 0.999696i \(-0.507852\pi\)
−0.0246648 + 0.999696i \(0.507852\pi\)
\(594\) 180.336 0.0124567
\(595\) 0 0
\(596\) 16685.3 1.14674
\(597\) 7610.35 0.521727
\(598\) −47.9496 −0.00327894
\(599\) 23951.9 1.63380 0.816901 0.576777i \(-0.195690\pi\)
0.816901 + 0.576777i \(0.195690\pi\)
\(600\) 0 0
\(601\) 18577.3 1.26087 0.630434 0.776243i \(-0.282877\pi\)
0.630434 + 0.776243i \(0.282877\pi\)
\(602\) 1826.56 0.123663
\(603\) −1322.81 −0.0893351
\(604\) −27993.3 −1.88581
\(605\) 0 0
\(606\) −1258.33 −0.0843501
\(607\) 6230.72 0.416634 0.208317 0.978061i \(-0.433201\pi\)
0.208317 + 0.978061i \(0.433201\pi\)
\(608\) −2.69214 −0.000179574 0
\(609\) −6093.82 −0.405475
\(610\) 0 0
\(611\) 248.743 0.0164698
\(612\) 8818.40 0.582456
\(613\) 17744.8 1.16918 0.584588 0.811330i \(-0.301256\pi\)
0.584588 + 0.811330i \(0.301256\pi\)
\(614\) −1743.95 −0.114626
\(615\) 0 0
\(616\) −934.493 −0.0611230
\(617\) −15305.4 −0.998656 −0.499328 0.866413i \(-0.666420\pi\)
−0.499328 + 0.866413i \(0.666420\pi\)
\(618\) −21.4623 −0.00139699
\(619\) 17372.3 1.12803 0.564015 0.825764i \(-0.309256\pi\)
0.564015 + 0.825764i \(0.309256\pi\)
\(620\) 0 0
\(621\) 4631.27 0.299270
\(622\) −193.074 −0.0124462
\(623\) −1346.75 −0.0866073
\(624\) −76.3610 −0.00489886
\(625\) 0 0
\(626\) 5024.40 0.320791
\(627\) −0.811331 −5.16770e−5 0
\(628\) 10823.3 0.687735
\(629\) 43472.4 2.75574
\(630\) 0 0
\(631\) 23699.3 1.49517 0.747586 0.664165i \(-0.231213\pi\)
0.747586 + 0.664165i \(0.231213\pi\)
\(632\) 4361.48 0.274510
\(633\) −4796.85 −0.301197
\(634\) −621.973 −0.0389617
\(635\) 0 0
\(636\) 14248.9 0.888374
\(637\) −121.028 −0.00752795
\(638\) 1515.74 0.0940577
\(639\) −8044.19 −0.498002
\(640\) 0 0
\(641\) −15423.9 −0.950400 −0.475200 0.879878i \(-0.657624\pi\)
−0.475200 + 0.879878i \(0.657624\pi\)
\(642\) −843.179 −0.0518343
\(643\) −13152.3 −0.806652 −0.403326 0.915056i \(-0.632146\pi\)
−0.403326 + 0.915056i \(0.632146\pi\)
\(644\) −11716.5 −0.716917
\(645\) 0 0
\(646\) 1.91672 0.000116737 0
\(647\) 5601.87 0.340390 0.170195 0.985410i \(-0.445560\pi\)
0.170195 + 0.985410i \(0.445560\pi\)
\(648\) −768.788 −0.0466062
\(649\) 108.565 0.00656633
\(650\) 0 0
\(651\) 5253.33 0.316274
\(652\) −4586.49 −0.275492
\(653\) 27505.9 1.64837 0.824186 0.566319i \(-0.191633\pi\)
0.824186 + 0.566319i \(0.191633\pi\)
\(654\) −602.076 −0.0359986
\(655\) 0 0
\(656\) 7572.87 0.450718
\(657\) 10347.4 0.614443
\(658\) −2936.41 −0.173971
\(659\) 16490.6 0.974785 0.487392 0.873183i \(-0.337948\pi\)
0.487392 + 0.873183i \(0.337948\pi\)
\(660\) 0 0
\(661\) 12184.4 0.716973 0.358487 0.933535i \(-0.383293\pi\)
0.358487 + 0.933535i \(0.383293\pi\)
\(662\) 2778.41 0.163121
\(663\) 177.334 0.0103878
\(664\) −1194.72 −0.0698253
\(665\) 0 0
\(666\) −1850.27 −0.107652
\(667\) 38926.3 2.25972
\(668\) −16950.6 −0.981795
\(669\) −15223.7 −0.879791
\(670\) 0 0
\(671\) −9929.83 −0.571292
\(672\) 2940.33 0.168788
\(673\) 13598.5 0.778876 0.389438 0.921053i \(-0.372669\pi\)
0.389438 + 0.921053i \(0.372669\pi\)
\(674\) 557.334 0.0318512
\(675\) 0 0
\(676\) 16764.4 0.953823
\(677\) −23318.1 −1.32377 −0.661883 0.749607i \(-0.730242\pi\)
−0.661883 + 0.749607i \(0.730242\pi\)
\(678\) 2080.80 0.117865
\(679\) −11319.0 −0.639742
\(680\) 0 0
\(681\) 3144.58 0.176947
\(682\) −1306.68 −0.0733658
\(683\) −11493.5 −0.643906 −0.321953 0.946756i \(-0.604339\pi\)
−0.321953 + 0.946756i \(0.604339\pi\)
\(684\) 1.68860 9.43935e−5 0
\(685\) 0 0
\(686\) 3292.89 0.183270
\(687\) 2203.99 0.122398
\(688\) −18581.2 −1.02965
\(689\) 286.539 0.0158436
\(690\) 0 0
\(691\) −20253.1 −1.11500 −0.557501 0.830176i \(-0.688240\pi\)
−0.557501 + 0.830176i \(0.688240\pi\)
\(692\) −1810.76 −0.0994724
\(693\) 886.129 0.0485733
\(694\) −3280.21 −0.179416
\(695\) 0 0
\(696\) −6461.74 −0.351913
\(697\) −17586.6 −0.955723
\(698\) 3106.30 0.168446
\(699\) −6038.34 −0.326739
\(700\) 0 0
\(701\) 26895.3 1.44910 0.724551 0.689221i \(-0.242047\pi\)
0.724551 + 0.689221i \(0.242047\pi\)
\(702\) −7.54766 −0.000405795 0
\(703\) 8.32435 0.000446599 0
\(704\) 4133.94 0.221312
\(705\) 0 0
\(706\) 1883.53 0.100407
\(707\) −6183.14 −0.328912
\(708\) −225.953 −0.0119941
\(709\) −18567.3 −0.983512 −0.491756 0.870733i \(-0.663645\pi\)
−0.491756 + 0.870733i \(0.663645\pi\)
\(710\) 0 0
\(711\) −4135.76 −0.218148
\(712\) −1428.06 −0.0751668
\(713\) −33557.4 −1.76260
\(714\) −2093.42 −0.109726
\(715\) 0 0
\(716\) −8133.21 −0.424514
\(717\) −16699.2 −0.869797
\(718\) 2122.94 0.110344
\(719\) −19223.5 −0.997104 −0.498552 0.866860i \(-0.666135\pi\)
−0.498552 + 0.866860i \(0.666135\pi\)
\(720\) 0 0
\(721\) −105.461 −0.00544738
\(722\) −4164.73 −0.214675
\(723\) 10083.8 0.518698
\(724\) 9470.43 0.486140
\(725\) 0 0
\(726\) −220.411 −0.0112675
\(727\) 33855.9 1.72716 0.863581 0.504210i \(-0.168216\pi\)
0.863581 + 0.504210i \(0.168216\pi\)
\(728\) 39.1117 0.00199117
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 43151.4 2.18333
\(732\) 20666.6 1.04353
\(733\) −26532.1 −1.33695 −0.668477 0.743733i \(-0.733053\pi\)
−0.668477 + 0.743733i \(0.733053\pi\)
\(734\) −4641.71 −0.233418
\(735\) 0 0
\(736\) −18782.3 −0.940661
\(737\) 1616.77 0.0808067
\(738\) 748.517 0.0373351
\(739\) −15358.9 −0.764528 −0.382264 0.924053i \(-0.624856\pi\)
−0.382264 + 0.924053i \(0.624856\pi\)
\(740\) 0 0
\(741\) 0.0339569 1.68345e−6 0
\(742\) −3382.58 −0.167356
\(743\) 30995.4 1.53043 0.765216 0.643774i \(-0.222632\pi\)
0.765216 + 0.643774i \(0.222632\pi\)
\(744\) 5570.50 0.274495
\(745\) 0 0
\(746\) 3151.27 0.154660
\(747\) 1132.88 0.0554887
\(748\) −10778.0 −0.526851
\(749\) −4143.19 −0.202121
\(750\) 0 0
\(751\) −32222.4 −1.56566 −0.782831 0.622234i \(-0.786225\pi\)
−0.782831 + 0.622234i \(0.786225\pi\)
\(752\) 29871.4 1.44853
\(753\) 8791.38 0.425466
\(754\) −63.4389 −0.00306407
\(755\) 0 0
\(756\) −1844.27 −0.0887243
\(757\) 36596.5 1.75710 0.878549 0.477651i \(-0.158512\pi\)
0.878549 + 0.477651i \(0.158512\pi\)
\(758\) 5206.01 0.249460
\(759\) −5660.44 −0.270700
\(760\) 0 0
\(761\) 24718.0 1.17743 0.588716 0.808340i \(-0.299634\pi\)
0.588716 + 0.808340i \(0.299634\pi\)
\(762\) 1146.84 0.0545220
\(763\) −2958.46 −0.140372
\(764\) 14933.0 0.707142
\(765\) 0 0
\(766\) 7128.58 0.336248
\(767\) −4.54381 −0.000213908 0
\(768\) −7008.15 −0.329277
\(769\) 36376.7 1.70582 0.852910 0.522058i \(-0.174836\pi\)
0.852910 + 0.522058i \(0.174836\pi\)
\(770\) 0 0
\(771\) −10238.7 −0.478258
\(772\) 21200.0 0.988347
\(773\) −7525.40 −0.350155 −0.175078 0.984555i \(-0.556018\pi\)
−0.175078 + 0.984555i \(0.556018\pi\)
\(774\) −1836.60 −0.0852911
\(775\) 0 0
\(776\) −12002.4 −0.555235
\(777\) −9091.78 −0.419776
\(778\) −2285.28 −0.105310
\(779\) −3.36758 −0.000154886 0
\(780\) 0 0
\(781\) 9831.78 0.450459
\(782\) 13372.4 0.611505
\(783\) 6127.32 0.279658
\(784\) −14534.2 −0.662089
\(785\) 0 0
\(786\) −1042.97 −0.0473303
\(787\) 41044.5 1.85906 0.929530 0.368747i \(-0.120213\pi\)
0.929530 + 0.368747i \(0.120213\pi\)
\(788\) 6106.90 0.276078
\(789\) 11127.2 0.502077
\(790\) 0 0
\(791\) 10224.5 0.459599
\(792\) 939.630 0.0421569
\(793\) 415.597 0.0186107
\(794\) −699.093 −0.0312467
\(795\) 0 0
\(796\) 19359.0 0.862012
\(797\) 10836.9 0.481635 0.240817 0.970570i \(-0.422584\pi\)
0.240817 + 0.970570i \(0.422584\pi\)
\(798\) −0.400861 −1.77823e−5 0
\(799\) −69370.7 −3.07154
\(800\) 0 0
\(801\) 1354.15 0.0597335
\(802\) −6102.91 −0.268705
\(803\) −12646.8 −0.555785
\(804\) −3364.93 −0.147602
\(805\) 0 0
\(806\) 54.6890 0.00239000
\(807\) 17293.2 0.754335
\(808\) −6556.45 −0.285464
\(809\) −16093.5 −0.699402 −0.349701 0.936861i \(-0.613717\pi\)
−0.349701 + 0.936861i \(0.613717\pi\)
\(810\) 0 0
\(811\) 7289.26 0.315611 0.157805 0.987470i \(-0.449558\pi\)
0.157805 + 0.987470i \(0.449558\pi\)
\(812\) −15501.3 −0.669937
\(813\) 5658.98 0.244120
\(814\) 2261.44 0.0973751
\(815\) 0 0
\(816\) 21295.9 0.913611
\(817\) 8.26288 0.000353833 0
\(818\) −3803.08 −0.162557
\(819\) −37.0875 −0.00158235
\(820\) 0 0
\(821\) −33153.4 −1.40933 −0.704666 0.709539i \(-0.748903\pi\)
−0.704666 + 0.709539i \(0.748903\pi\)
\(822\) −1728.09 −0.0733262
\(823\) 2781.57 0.117812 0.0589060 0.998264i \(-0.481239\pi\)
0.0589060 + 0.998264i \(0.481239\pi\)
\(824\) −111.828 −0.00472780
\(825\) 0 0
\(826\) 53.6395 0.00225951
\(827\) 4498.61 0.189156 0.0945780 0.995517i \(-0.469850\pi\)
0.0945780 + 0.995517i \(0.469850\pi\)
\(828\) 11780.9 0.494462
\(829\) 15630.3 0.654843 0.327421 0.944878i \(-0.393820\pi\)
0.327421 + 0.944878i \(0.393820\pi\)
\(830\) 0 0
\(831\) −12288.0 −0.512954
\(832\) −173.019 −0.00720958
\(833\) 33752.9 1.40392
\(834\) −4533.66 −0.188235
\(835\) 0 0
\(836\) −2.06384 −8.53822e−5 0
\(837\) −5282.20 −0.218136
\(838\) 5237.43 0.215900
\(839\) −6182.34 −0.254396 −0.127198 0.991877i \(-0.540598\pi\)
−0.127198 + 0.991877i \(0.540598\pi\)
\(840\) 0 0
\(841\) 27111.8 1.11164
\(842\) −5522.56 −0.226033
\(843\) 26366.1 1.07722
\(844\) −12202.1 −0.497646
\(845\) 0 0
\(846\) 2952.55 0.119989
\(847\) −1083.05 −0.0439362
\(848\) 34410.3 1.39346
\(849\) −5303.26 −0.214378
\(850\) 0 0
\(851\) 58076.7 2.33942
\(852\) −20462.6 −0.822813
\(853\) −31979.3 −1.28365 −0.641824 0.766852i \(-0.721822\pi\)
−0.641824 + 0.766852i \(0.721822\pi\)
\(854\) −4906.11 −0.196585
\(855\) 0 0
\(856\) −4393.33 −0.175422
\(857\) 8439.63 0.336397 0.168199 0.985753i \(-0.446205\pi\)
0.168199 + 0.985753i \(0.446205\pi\)
\(858\) 9.22492 0.000367056 0
\(859\) 47640.1 1.89227 0.946134 0.323774i \(-0.104952\pi\)
0.946134 + 0.323774i \(0.104952\pi\)
\(860\) 0 0
\(861\) 3678.04 0.145583
\(862\) −2434.06 −0.0961770
\(863\) −27963.3 −1.10299 −0.551496 0.834177i \(-0.685943\pi\)
−0.551496 + 0.834177i \(0.685943\pi\)
\(864\) −2956.50 −0.116414
\(865\) 0 0
\(866\) −1648.75 −0.0646962
\(867\) −34716.8 −1.35991
\(868\) 13363.3 0.522557
\(869\) 5054.81 0.197322
\(870\) 0 0
\(871\) −67.6673 −0.00263240
\(872\) −3137.08 −0.121829
\(873\) 11381.3 0.441234
\(874\) 2.56063 9.91013e−5 0
\(875\) 0 0
\(876\) 26321.3 1.01520
\(877\) 30584.0 1.17759 0.588796 0.808282i \(-0.299602\pi\)
0.588796 + 0.808282i \(0.299602\pi\)
\(878\) −6732.23 −0.258772
\(879\) 9239.56 0.354542
\(880\) 0 0
\(881\) 14444.3 0.552372 0.276186 0.961104i \(-0.410929\pi\)
0.276186 + 0.961104i \(0.410929\pi\)
\(882\) −1436.58 −0.0548439
\(883\) 19921.4 0.759241 0.379620 0.925142i \(-0.376055\pi\)
0.379620 + 0.925142i \(0.376055\pi\)
\(884\) 451.097 0.0171630
\(885\) 0 0
\(886\) −4331.09 −0.164228
\(887\) 13172.5 0.498635 0.249317 0.968422i \(-0.419794\pi\)
0.249317 + 0.968422i \(0.419794\pi\)
\(888\) −9640.71 −0.364325
\(889\) 5635.33 0.212602
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) −38725.5 −1.45362
\(893\) −13.2835 −0.000497778 0
\(894\) 3982.75 0.148997
\(895\) 0 0
\(896\) 9883.38 0.368505
\(897\) 236.908 0.00881844
\(898\) 3820.01 0.141955
\(899\) −44397.5 −1.64710
\(900\) 0 0
\(901\) −79911.4 −2.95475
\(902\) −914.854 −0.0337709
\(903\) −9024.64 −0.332582
\(904\) 10841.9 0.398888
\(905\) 0 0
\(906\) −6681.92 −0.245024
\(907\) −16309.7 −0.597084 −0.298542 0.954396i \(-0.596500\pi\)
−0.298542 + 0.954396i \(0.596500\pi\)
\(908\) 7999.10 0.292356
\(909\) 6217.13 0.226853
\(910\) 0 0
\(911\) −14994.4 −0.545320 −0.272660 0.962110i \(-0.587903\pi\)
−0.272660 + 0.962110i \(0.587903\pi\)
\(912\) 4.07787 0.000148061 0
\(913\) −1384.64 −0.0501914
\(914\) −4427.63 −0.160233
\(915\) 0 0
\(916\) 5606.44 0.202229
\(917\) −5124.93 −0.184558
\(918\) 2104.93 0.0756787
\(919\) 40263.5 1.44524 0.722618 0.691248i \(-0.242939\pi\)
0.722618 + 0.691248i \(0.242939\pi\)
\(920\) 0 0
\(921\) 8616.48 0.308276
\(922\) −10204.1 −0.364484
\(923\) −411.493 −0.0146744
\(924\) 2254.11 0.0802541
\(925\) 0 0
\(926\) −4540.87 −0.161147
\(927\) 106.040 0.00375709
\(928\) −24849.6 −0.879019
\(929\) −9469.52 −0.334429 −0.167215 0.985921i \(-0.553477\pi\)
−0.167215 + 0.985921i \(0.553477\pi\)
\(930\) 0 0
\(931\) 6.46319 0.000227522 0
\(932\) −15360.2 −0.539848
\(933\) 953.933 0.0334731
\(934\) 4570.23 0.160110
\(935\) 0 0
\(936\) −39.3266 −0.00137332
\(937\) −6356.51 −0.221620 −0.110810 0.993842i \(-0.535345\pi\)
−0.110810 + 0.993842i \(0.535345\pi\)
\(938\) 798.810 0.0278061
\(939\) −24824.4 −0.862742
\(940\) 0 0
\(941\) −5764.45 −0.199698 −0.0998489 0.995003i \(-0.531836\pi\)
−0.0998489 + 0.995003i \(0.531836\pi\)
\(942\) 2583.50 0.0893577
\(943\) −23494.7 −0.811338
\(944\) −545.663 −0.0188134
\(945\) 0 0
\(946\) 2244.74 0.0771487
\(947\) 2284.74 0.0783993 0.0391997 0.999231i \(-0.487519\pi\)
0.0391997 + 0.999231i \(0.487519\pi\)
\(948\) −10520.4 −0.360430
\(949\) 529.310 0.0181055
\(950\) 0 0
\(951\) 3073.03 0.104784
\(952\) −10907.7 −0.371344
\(953\) 17209.0 0.584947 0.292474 0.956274i \(-0.405522\pi\)
0.292474 + 0.956274i \(0.405522\pi\)
\(954\) 3401.18 0.115427
\(955\) 0 0
\(956\) −42479.1 −1.43710
\(957\) −7488.95 −0.252961
\(958\) 764.333 0.0257771
\(959\) −8491.45 −0.285926
\(960\) 0 0
\(961\) 8482.92 0.284748
\(962\) −94.6487 −0.00317214
\(963\) 4165.96 0.139404
\(964\) 25650.8 0.857008
\(965\) 0 0
\(966\) −2796.70 −0.0931493
\(967\) 7966.25 0.264920 0.132460 0.991188i \(-0.457712\pi\)
0.132460 + 0.991188i \(0.457712\pi\)
\(968\) −1148.44 −0.0381324
\(969\) −9.47008 −0.000313955 0
\(970\) 0 0
\(971\) −25547.5 −0.844343 −0.422172 0.906516i \(-0.638732\pi\)
−0.422172 + 0.906516i \(0.638732\pi\)
\(972\) 1854.41 0.0611937
\(973\) −22277.3 −0.733996
\(974\) −3113.66 −0.102431
\(975\) 0 0
\(976\) 49908.7 1.63682
\(977\) −12899.8 −0.422417 −0.211209 0.977441i \(-0.567740\pi\)
−0.211209 + 0.977441i \(0.567740\pi\)
\(978\) −1094.78 −0.0357948
\(979\) −1655.07 −0.0540310
\(980\) 0 0
\(981\) 2974.73 0.0968152
\(982\) 7245.18 0.235441
\(983\) 7399.28 0.240082 0.120041 0.992769i \(-0.461697\pi\)
0.120041 + 0.992769i \(0.461697\pi\)
\(984\) 3900.10 0.126352
\(985\) 0 0
\(986\) 17692.2 0.571433
\(987\) 14508.1 0.467881
\(988\) 0.0863787 2.78145e−6 0
\(989\) 57647.8 1.85348
\(990\) 0 0
\(991\) −38466.4 −1.23302 −0.616511 0.787346i \(-0.711454\pi\)
−0.616511 + 0.787346i \(0.711454\pi\)
\(992\) 21422.2 0.685642
\(993\) −13727.5 −0.438700
\(994\) 4857.66 0.155006
\(995\) 0 0
\(996\) 2881.80 0.0916801
\(997\) 62092.5 1.97241 0.986203 0.165538i \(-0.0529361\pi\)
0.986203 + 0.165538i \(0.0529361\pi\)
\(998\) 8935.60 0.283418
\(999\) 9141.76 0.289522
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 825.4.a.t.1.3 4
3.2 odd 2 2475.4.a.be.1.2 4
5.2 odd 4 825.4.c.p.199.5 8
5.3 odd 4 825.4.c.p.199.4 8
5.4 even 2 165.4.a.h.1.2 4
15.14 odd 2 495.4.a.m.1.3 4
55.54 odd 2 1815.4.a.t.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.h.1.2 4 5.4 even 2
495.4.a.m.1.3 4 15.14 odd 2
825.4.a.t.1.3 4 1.1 even 1 trivial
825.4.c.p.199.4 8 5.3 odd 4
825.4.c.p.199.5 8 5.2 odd 4
1815.4.a.t.1.3 4 55.54 odd 2
2475.4.a.be.1.2 4 3.2 odd 2