Properties

Label 1521.4.a.s.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $2$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.74166\) 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)
 
\(f(q)\) \(=\) \(q-2.74166 q^{2} -0.483315 q^{4} +19.4833 q^{5} -7.48331 q^{7} +23.2583 q^{8} +O(q^{10})\) \(q-2.74166 q^{2} -0.483315 q^{4} +19.4833 q^{5} -7.48331 q^{7} +23.2583 q^{8} -53.4166 q^{10} +22.8999 q^{11} +20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} -16.5167 q^{19} -9.41657 q^{20} -62.7836 q^{22} +175.600 q^{23} +254.600 q^{25} +3.61680 q^{28} -291.800 q^{29} -117.283 q^{31} -21.8418 q^{32} +183.783 q^{34} -145.800 q^{35} +154.766 q^{37} +45.2831 q^{38} +453.150 q^{40} -251.716 q^{41} -502.566 q^{43} -11.0679 q^{44} -481.434 q^{46} -281.733 q^{47} -287.000 q^{49} -698.025 q^{50} -366.999 q^{53} +446.166 q^{55} -174.049 q^{56} +800.015 q^{58} -79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} +539.082 q^{64} -400.082 q^{67} +32.3982 q^{68} +399.733 q^{70} +528.299 q^{71} +734.366 q^{73} -424.316 q^{74} +7.98276 q^{76} -171.367 q^{77} +113.266 q^{79} -1167.05 q^{80} +690.118 q^{82} -933.466 q^{83} -1306.03 q^{85} +1377.86 q^{86} +532.613 q^{88} +1190.91 q^{89} -84.8699 q^{92} +772.415 q^{94} -321.800 q^{95} -557.165 q^{97} +786.856 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 14 q^{4} + 24 q^{5} + 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 14 q^{4} + 24 q^{5} + 54 q^{8} - 32 q^{10} - 44 q^{11} + 56 q^{14} - 30 q^{16} - 164 q^{17} - 48 q^{19} + 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 112 q^{28} - 404 q^{29} - 40 q^{31} - 126 q^{32} - 276 q^{34} - 112 q^{35} + 100 q^{37} - 104 q^{38} + 592 q^{40} + 200 q^{41} - 616 q^{43} - 980 q^{44} - 1352 q^{46} - 324 q^{47} - 574 q^{49} - 1194 q^{50} + 164 q^{53} + 144 q^{55} + 56 q^{56} + 268 q^{58} + 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 472 q^{67} - 1372 q^{68} + 560 q^{70} + 428 q^{71} + 900 q^{73} - 684 q^{74} - 448 q^{76} - 672 q^{77} - 432 q^{79} - 1032 q^{80} + 2832 q^{82} - 1388 q^{83} - 1744 q^{85} + 840 q^{86} - 1524 q^{88} + 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} + 532 q^{97} - 574 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74166 −0.969322 −0.484661 0.874702i \(-0.661057\pi\)
−0.484661 + 0.874702i \(0.661057\pi\)
\(3\) 0 0
\(4\) −0.483315 −0.0604143
\(5\) 19.4833 1.74264 0.871320 0.490715i \(-0.163264\pi\)
0.871320 + 0.490715i \(0.163264\pi\)
\(6\) 0 0
\(7\) −7.48331 −0.404061 −0.202031 0.979379i \(-0.564754\pi\)
−0.202031 + 0.979379i \(0.564754\pi\)
\(8\) 23.2583 1.02788
\(9\) 0 0
\(10\) −53.4166 −1.68918
\(11\) 22.8999 0.627689 0.313844 0.949474i \(-0.398383\pi\)
0.313844 + 0.949474i \(0.398383\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 20.5167 0.391665
\(15\) 0 0
\(16\) −59.8999 −0.935936
\(17\) −67.0334 −0.956352 −0.478176 0.878264i \(-0.658702\pi\)
−0.478176 + 0.878264i \(0.658702\pi\)
\(18\) 0 0
\(19\) −16.5167 −0.199431 −0.0997155 0.995016i \(-0.531793\pi\)
−0.0997155 + 0.995016i \(0.531793\pi\)
\(20\) −9.41657 −0.105280
\(21\) 0 0
\(22\) −62.7836 −0.608433
\(23\) 175.600 1.59196 0.795979 0.605324i \(-0.206956\pi\)
0.795979 + 0.605324i \(0.206956\pi\)
\(24\) 0 0
\(25\) 254.600 2.03680
\(26\) 0 0
\(27\) 0 0
\(28\) 3.61680 0.0244111
\(29\) −291.800 −1.86848 −0.934239 0.356648i \(-0.883920\pi\)
−0.934239 + 0.356648i \(0.883920\pi\)
\(30\) 0 0
\(31\) −117.283 −0.679505 −0.339753 0.940515i \(-0.610343\pi\)
−0.339753 + 0.940515i \(0.610343\pi\)
\(32\) −21.8418 −0.120660
\(33\) 0 0
\(34\) 183.783 0.927013
\(35\) −145.800 −0.704133
\(36\) 0 0
\(37\) 154.766 0.687661 0.343830 0.939032i \(-0.388276\pi\)
0.343830 + 0.939032i \(0.388276\pi\)
\(38\) 45.2831 0.193313
\(39\) 0 0
\(40\) 453.150 1.79123
\(41\) −251.716 −0.958815 −0.479407 0.877592i \(-0.659148\pi\)
−0.479407 + 0.877592i \(0.659148\pi\)
\(42\) 0 0
\(43\) −502.566 −1.78234 −0.891170 0.453669i \(-0.850115\pi\)
−0.891170 + 0.453669i \(0.850115\pi\)
\(44\) −11.0679 −0.0379214
\(45\) 0 0
\(46\) −481.434 −1.54312
\(47\) −281.733 −0.874361 −0.437181 0.899374i \(-0.644023\pi\)
−0.437181 + 0.899374i \(0.644023\pi\)
\(48\) 0 0
\(49\) −287.000 −0.836735
\(50\) −698.025 −1.97431
\(51\) 0 0
\(52\) 0 0
\(53\) −366.999 −0.951154 −0.475577 0.879674i \(-0.657761\pi\)
−0.475577 + 0.879674i \(0.657761\pi\)
\(54\) 0 0
\(55\) 446.166 1.09384
\(56\) −174.049 −0.415328
\(57\) 0 0
\(58\) 800.015 1.81116
\(59\) −79.6663 −0.175791 −0.0878955 0.996130i \(-0.528014\pi\)
−0.0878955 + 0.996130i \(0.528014\pi\)
\(60\) 0 0
\(61\) −194.865 −0.409016 −0.204508 0.978865i \(-0.565559\pi\)
−0.204508 + 0.978865i \(0.565559\pi\)
\(62\) 321.550 0.658660
\(63\) 0 0
\(64\) 539.082 1.05289
\(65\) 0 0
\(66\) 0 0
\(67\) −400.082 −0.729519 −0.364759 0.931102i \(-0.618849\pi\)
−0.364759 + 0.931102i \(0.618849\pi\)
\(68\) 32.3982 0.0577774
\(69\) 0 0
\(70\) 399.733 0.682532
\(71\) 528.299 0.883065 0.441532 0.897245i \(-0.354435\pi\)
0.441532 + 0.897245i \(0.354435\pi\)
\(72\) 0 0
\(73\) 734.366 1.17741 0.588706 0.808347i \(-0.299638\pi\)
0.588706 + 0.808347i \(0.299638\pi\)
\(74\) −424.316 −0.666565
\(75\) 0 0
\(76\) 7.98276 0.0120485
\(77\) −171.367 −0.253625
\(78\) 0 0
\(79\) 113.266 0.161309 0.0806545 0.996742i \(-0.474299\pi\)
0.0806545 + 0.996742i \(0.474299\pi\)
\(80\) −1167.05 −1.63100
\(81\) 0 0
\(82\) 690.118 0.929400
\(83\) −933.466 −1.23447 −0.617236 0.786778i \(-0.711748\pi\)
−0.617236 + 0.786778i \(0.711748\pi\)
\(84\) 0 0
\(85\) −1306.03 −1.66658
\(86\) 1377.86 1.72766
\(87\) 0 0
\(88\) 532.613 0.645191
\(89\) 1190.91 1.41839 0.709195 0.705012i \(-0.249059\pi\)
0.709195 + 0.705012i \(0.249059\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −84.8699 −0.0961771
\(93\) 0 0
\(94\) 772.415 0.847538
\(95\) −321.800 −0.347536
\(96\) 0 0
\(97\) −557.165 −0.583211 −0.291606 0.956539i \(-0.594189\pi\)
−0.291606 + 0.956539i \(0.594189\pi\)
\(98\) 786.856 0.811066
\(99\) 0 0
\(100\) −123.052 −0.123052
\(101\) 286.766 0.282518 0.141259 0.989973i \(-0.454885\pi\)
0.141259 + 0.989973i \(0.454885\pi\)
\(102\) 0 0
\(103\) −1911.36 −1.82847 −0.914234 0.405187i \(-0.867206\pi\)
−0.914234 + 0.405187i \(0.867206\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1006.19 0.921975
\(107\) −834.334 −0.753814 −0.376907 0.926251i \(-0.623012\pi\)
−0.376907 + 0.926251i \(0.623012\pi\)
\(108\) 0 0
\(109\) 1077.66 0.946986 0.473493 0.880798i \(-0.342993\pi\)
0.473493 + 0.880798i \(0.342993\pi\)
\(110\) −1223.23 −1.06028
\(111\) 0 0
\(112\) 448.250 0.378175
\(113\) 166.065 0.138248 0.0691241 0.997608i \(-0.477980\pi\)
0.0691241 + 0.997608i \(0.477980\pi\)
\(114\) 0 0
\(115\) 3421.26 2.77421
\(116\) 141.031 0.112883
\(117\) 0 0
\(118\) 218.418 0.170398
\(119\) 501.632 0.386424
\(120\) 0 0
\(121\) −806.595 −0.606007
\(122\) 534.254 0.396468
\(123\) 0 0
\(124\) 56.6847 0.0410519
\(125\) 2525.03 1.80676
\(126\) 0 0
\(127\) 1296.16 0.905637 0.452819 0.891603i \(-0.350419\pi\)
0.452819 + 0.891603i \(0.350419\pi\)
\(128\) −1303.24 −0.899934
\(129\) 0 0
\(130\) 0 0
\(131\) 197.201 0.131523 0.0657617 0.997835i \(-0.479052\pi\)
0.0657617 + 0.997835i \(0.479052\pi\)
\(132\) 0 0
\(133\) 123.600 0.0805823
\(134\) 1096.89 0.707139
\(135\) 0 0
\(136\) −1559.09 −0.983018
\(137\) −546.915 −0.341066 −0.170533 0.985352i \(-0.554549\pi\)
−0.170533 + 0.985352i \(0.554549\pi\)
\(138\) 0 0
\(139\) 609.666 0.372023 0.186012 0.982548i \(-0.440444\pi\)
0.186012 + 0.982548i \(0.440444\pi\)
\(140\) 70.4672 0.0425397
\(141\) 0 0
\(142\) −1448.42 −0.855974
\(143\) 0 0
\(144\) 0 0
\(145\) −5685.23 −3.25609
\(146\) −2013.38 −1.14129
\(147\) 0 0
\(148\) −74.8009 −0.0415446
\(149\) −2165.08 −1.19040 −0.595202 0.803576i \(-0.702928\pi\)
−0.595202 + 0.803576i \(0.702928\pi\)
\(150\) 0 0
\(151\) 846.549 0.456233 0.228116 0.973634i \(-0.426743\pi\)
0.228116 + 0.973634i \(0.426743\pi\)
\(152\) −384.151 −0.204992
\(153\) 0 0
\(154\) 469.830 0.245844
\(155\) −2285.06 −1.18413
\(156\) 0 0
\(157\) 1653.60 0.840581 0.420291 0.907390i \(-0.361928\pi\)
0.420291 + 0.907390i \(0.361928\pi\)
\(158\) −310.536 −0.156360
\(159\) 0 0
\(160\) −425.550 −0.210267
\(161\) −1314.07 −0.643248
\(162\) 0 0
\(163\) 2866.51 1.37744 0.688720 0.725027i \(-0.258173\pi\)
0.688720 + 0.725027i \(0.258173\pi\)
\(164\) 121.658 0.0579262
\(165\) 0 0
\(166\) 2559.24 1.19660
\(167\) 729.066 0.337825 0.168913 0.985631i \(-0.445974\pi\)
0.168913 + 0.985631i \(0.445974\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 3580.69 1.61545
\(171\) 0 0
\(172\) 242.898 0.107679
\(173\) 3834.83 1.68530 0.842650 0.538462i \(-0.180995\pi\)
0.842650 + 0.538462i \(0.180995\pi\)
\(174\) 0 0
\(175\) −1905.25 −0.822990
\(176\) −1371.70 −0.587476
\(177\) 0 0
\(178\) −3265.08 −1.37488
\(179\) 283.862 0.118530 0.0592649 0.998242i \(-0.481124\pi\)
0.0592649 + 0.998242i \(0.481124\pi\)
\(180\) 0 0
\(181\) 2363.60 0.970634 0.485317 0.874338i \(-0.338704\pi\)
0.485317 + 0.874338i \(0.338704\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4084.15 1.63635
\(185\) 3015.36 1.19835
\(186\) 0 0
\(187\) −1535.06 −0.600291
\(188\) 136.166 0.0528240
\(189\) 0 0
\(190\) 882.265 0.336875
\(191\) −2514.26 −0.952491 −0.476246 0.879312i \(-0.658003\pi\)
−0.476246 + 0.879312i \(0.658003\pi\)
\(192\) 0 0
\(193\) −2420.73 −0.902839 −0.451420 0.892312i \(-0.649082\pi\)
−0.451420 + 0.892312i \(0.649082\pi\)
\(194\) 1527.55 0.565320
\(195\) 0 0
\(196\) 138.711 0.0505508
\(197\) −4633.65 −1.67581 −0.837903 0.545819i \(-0.816219\pi\)
−0.837903 + 0.545819i \(0.816219\pi\)
\(198\) 0 0
\(199\) 3054.17 1.08796 0.543980 0.839098i \(-0.316917\pi\)
0.543980 + 0.839098i \(0.316917\pi\)
\(200\) 5921.56 2.09359
\(201\) 0 0
\(202\) −786.215 −0.273851
\(203\) 2183.63 0.754979
\(204\) 0 0
\(205\) −4904.26 −1.67087
\(206\) 5240.30 1.77237
\(207\) 0 0
\(208\) 0 0
\(209\) −378.230 −0.125181
\(210\) 0 0
\(211\) −4031.60 −1.31539 −0.657694 0.753285i \(-0.728468\pi\)
−0.657694 + 0.753285i \(0.728468\pi\)
\(212\) 177.376 0.0574634
\(213\) 0 0
\(214\) 2287.46 0.730689
\(215\) −9791.66 −3.10598
\(216\) 0 0
\(217\) 877.666 0.274562
\(218\) −2954.59 −0.917935
\(219\) 0 0
\(220\) −215.638 −0.0660834
\(221\) 0 0
\(222\) 0 0
\(223\) −3784.95 −1.13659 −0.568294 0.822826i \(-0.692396\pi\)
−0.568294 + 0.822826i \(0.692396\pi\)
\(224\) 163.449 0.0487539
\(225\) 0 0
\(226\) −455.292 −0.134007
\(227\) 2013.83 0.588821 0.294411 0.955679i \(-0.404877\pi\)
0.294411 + 0.955679i \(0.404877\pi\)
\(228\) 0 0
\(229\) 3050.73 0.880340 0.440170 0.897915i \(-0.354918\pi\)
0.440170 + 0.897915i \(0.354918\pi\)
\(230\) −9379.93 −2.68910
\(231\) 0 0
\(232\) −6786.78 −1.92058
\(233\) −5587.49 −1.57103 −0.785513 0.618846i \(-0.787601\pi\)
−0.785513 + 0.618846i \(0.787601\pi\)
\(234\) 0 0
\(235\) −5489.09 −1.52370
\(236\) 38.5039 0.0106203
\(237\) 0 0
\(238\) −1375.30 −0.374570
\(239\) −1335.69 −0.361501 −0.180750 0.983529i \(-0.557853\pi\)
−0.180750 + 0.983529i \(0.557853\pi\)
\(240\) 0 0
\(241\) 571.558 0.152769 0.0763845 0.997078i \(-0.475662\pi\)
0.0763845 + 0.997078i \(0.475662\pi\)
\(242\) 2211.41 0.587416
\(243\) 0 0
\(244\) 94.1813 0.0247104
\(245\) −5591.71 −1.45813
\(246\) 0 0
\(247\) 0 0
\(248\) −2727.81 −0.698452
\(249\) 0 0
\(250\) −6922.76 −1.75134
\(251\) −4088.60 −1.02817 −0.514084 0.857740i \(-0.671868\pi\)
−0.514084 + 0.857740i \(0.671868\pi\)
\(252\) 0 0
\(253\) 4021.21 0.999254
\(254\) −3553.64 −0.877854
\(255\) 0 0
\(256\) −739.607 −0.180568
\(257\) −3050.23 −0.740342 −0.370171 0.928964i \(-0.620701\pi\)
−0.370171 + 0.928964i \(0.620701\pi\)
\(258\) 0 0
\(259\) −1158.17 −0.277857
\(260\) 0 0
\(261\) 0 0
\(262\) −540.659 −0.127489
\(263\) −5770.99 −1.35306 −0.676530 0.736415i \(-0.736517\pi\)
−0.676530 + 0.736415i \(0.736517\pi\)
\(264\) 0 0
\(265\) −7150.35 −1.65752
\(266\) −338.868 −0.0781102
\(267\) 0 0
\(268\) 193.365 0.0440734
\(269\) 2079.40 0.471314 0.235657 0.971836i \(-0.424276\pi\)
0.235657 + 0.971836i \(0.424276\pi\)
\(270\) 0 0
\(271\) −6012.00 −1.34761 −0.673807 0.738908i \(-0.735342\pi\)
−0.673807 + 0.738908i \(0.735342\pi\)
\(272\) 4015.29 0.895084
\(273\) 0 0
\(274\) 1499.45 0.330603
\(275\) 5830.30 1.27847
\(276\) 0 0
\(277\) −735.201 −0.159473 −0.0797364 0.996816i \(-0.525408\pi\)
−0.0797364 + 0.996816i \(0.525408\pi\)
\(278\) −1671.50 −0.360610
\(279\) 0 0
\(280\) −3391.06 −0.723767
\(281\) −1902.92 −0.403981 −0.201990 0.979387i \(-0.564741\pi\)
−0.201990 + 0.979387i \(0.564741\pi\)
\(282\) 0 0
\(283\) 2125.71 0.446502 0.223251 0.974761i \(-0.428333\pi\)
0.223251 + 0.974761i \(0.428333\pi\)
\(284\) −255.335 −0.0533498
\(285\) 0 0
\(286\) 0 0
\(287\) 1883.67 0.387420
\(288\) 0 0
\(289\) −419.527 −0.0853913
\(290\) 15586.9 3.15620
\(291\) 0 0
\(292\) −354.930 −0.0711325
\(293\) −1641.03 −0.327200 −0.163600 0.986527i \(-0.552311\pi\)
−0.163600 + 0.986527i \(0.552311\pi\)
\(294\) 0 0
\(295\) −1552.16 −0.306341
\(296\) 3599.61 0.706835
\(297\) 0 0
\(298\) 5935.91 1.15389
\(299\) 0 0
\(300\) 0 0
\(301\) 3760.86 0.720174
\(302\) −2320.95 −0.442237
\(303\) 0 0
\(304\) 989.348 0.186655
\(305\) −3796.62 −0.712767
\(306\) 0 0
\(307\) 3373.27 0.627111 0.313555 0.949570i \(-0.398480\pi\)
0.313555 + 0.949570i \(0.398480\pi\)
\(308\) 82.8242 0.0153226
\(309\) 0 0
\(310\) 6264.86 1.14781
\(311\) 868.525 0.158359 0.0791793 0.996860i \(-0.474770\pi\)
0.0791793 + 0.996860i \(0.474770\pi\)
\(312\) 0 0
\(313\) −4343.19 −0.784319 −0.392159 0.919897i \(-0.628272\pi\)
−0.392159 + 0.919897i \(0.628272\pi\)
\(314\) −4533.59 −0.814794
\(315\) 0 0
\(316\) −54.7431 −0.00974537
\(317\) −3277.65 −0.580730 −0.290365 0.956916i \(-0.593777\pi\)
−0.290365 + 0.956916i \(0.593777\pi\)
\(318\) 0 0
\(319\) −6682.18 −1.17282
\(320\) 10503.1 1.83482
\(321\) 0 0
\(322\) 3602.72 0.623515
\(323\) 1107.17 0.190726
\(324\) 0 0
\(325\) 0 0
\(326\) −7859.00 −1.33518
\(327\) 0 0
\(328\) −5854.49 −0.985550
\(329\) 2108.30 0.353295
\(330\) 0 0
\(331\) −5589.62 −0.928197 −0.464099 0.885784i \(-0.653622\pi\)
−0.464099 + 0.885784i \(0.653622\pi\)
\(332\) 451.158 0.0745798
\(333\) 0 0
\(334\) −1998.85 −0.327461
\(335\) −7794.92 −1.27129
\(336\) 0 0
\(337\) 901.544 0.145728 0.0728638 0.997342i \(-0.476786\pi\)
0.0728638 + 0.997342i \(0.476786\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 631.225 0.100685
\(341\) −2685.77 −0.426518
\(342\) 0 0
\(343\) 4714.49 0.742153
\(344\) −11688.9 −1.83204
\(345\) 0 0
\(346\) −10513.8 −1.63360
\(347\) 812.318 0.125670 0.0628350 0.998024i \(-0.479986\pi\)
0.0628350 + 0.998024i \(0.479986\pi\)
\(348\) 0 0
\(349\) −4437.96 −0.680683 −0.340342 0.940302i \(-0.610543\pi\)
−0.340342 + 0.940302i \(0.610543\pi\)
\(350\) 5223.54 0.797743
\(351\) 0 0
\(352\) −500.174 −0.0757368
\(353\) 7115.35 1.07284 0.536419 0.843952i \(-0.319777\pi\)
0.536419 + 0.843952i \(0.319777\pi\)
\(354\) 0 0
\(355\) 10293.0 1.53886
\(356\) −575.587 −0.0856911
\(357\) 0 0
\(358\) −778.253 −0.114894
\(359\) 4693.98 0.690081 0.345040 0.938588i \(-0.387865\pi\)
0.345040 + 0.938588i \(0.387865\pi\)
\(360\) 0 0
\(361\) −6586.20 −0.960227
\(362\) −6480.17 −0.940857
\(363\) 0 0
\(364\) 0 0
\(365\) 14307.9 2.05181
\(366\) 0 0
\(367\) 9243.98 1.31480 0.657400 0.753542i \(-0.271656\pi\)
0.657400 + 0.753542i \(0.271656\pi\)
\(368\) −10518.4 −1.48997
\(369\) 0 0
\(370\) −8267.09 −1.16158
\(371\) 2746.37 0.384324
\(372\) 0 0
\(373\) −4311.99 −0.598569 −0.299285 0.954164i \(-0.596748\pi\)
−0.299285 + 0.954164i \(0.596748\pi\)
\(374\) 4208.60 0.581876
\(375\) 0 0
\(376\) −6552.64 −0.898741
\(377\) 0 0
\(378\) 0 0
\(379\) 2382.73 0.322936 0.161468 0.986878i \(-0.448377\pi\)
0.161468 + 0.986878i \(0.448377\pi\)
\(380\) 155.531 0.0209962
\(381\) 0 0
\(382\) 6893.25 0.923271
\(383\) 4845.81 0.646499 0.323250 0.946314i \(-0.395225\pi\)
0.323250 + 0.946314i \(0.395225\pi\)
\(384\) 0 0
\(385\) −3338.80 −0.441976
\(386\) 6636.81 0.875142
\(387\) 0 0
\(388\) 269.286 0.0352343
\(389\) −9561.50 −1.24624 −0.623120 0.782127i \(-0.714135\pi\)
−0.623120 + 0.782127i \(0.714135\pi\)
\(390\) 0 0
\(391\) −11771.0 −1.52247
\(392\) −6675.14 −0.860066
\(393\) 0 0
\(394\) 12703.9 1.62440
\(395\) 2206.79 0.281103
\(396\) 0 0
\(397\) 7440.11 0.940575 0.470287 0.882513i \(-0.344150\pi\)
0.470287 + 0.882513i \(0.344150\pi\)
\(398\) −8373.48 −1.05458
\(399\) 0 0
\(400\) −15250.5 −1.90631
\(401\) −8687.80 −1.08192 −0.540958 0.841050i \(-0.681938\pi\)
−0.540958 + 0.841050i \(0.681938\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −138.598 −0.0170681
\(405\) 0 0
\(406\) −5986.76 −0.731818
\(407\) 3544.13 0.431637
\(408\) 0 0
\(409\) −2556.10 −0.309024 −0.154512 0.987991i \(-0.549381\pi\)
−0.154512 + 0.987991i \(0.549381\pi\)
\(410\) 13445.8 1.61961
\(411\) 0 0
\(412\) 923.790 0.110466
\(413\) 596.168 0.0710303
\(414\) 0 0
\(415\) −18187.0 −2.15124
\(416\) 0 0
\(417\) 0 0
\(418\) 1036.98 0.121340
\(419\) 3347.46 0.390296 0.195148 0.980774i \(-0.437481\pi\)
0.195148 + 0.980774i \(0.437481\pi\)
\(420\) 0 0
\(421\) 1854.48 0.214684 0.107342 0.994222i \(-0.465766\pi\)
0.107342 + 0.994222i \(0.465766\pi\)
\(422\) 11053.3 1.27503
\(423\) 0 0
\(424\) −8535.79 −0.977675
\(425\) −17066.7 −1.94789
\(426\) 0 0
\(427\) 1458.24 0.165267
\(428\) 403.246 0.0455412
\(429\) 0 0
\(430\) 26845.4 3.01069
\(431\) −14043.1 −1.56945 −0.784725 0.619844i \(-0.787196\pi\)
−0.784725 + 0.619844i \(0.787196\pi\)
\(432\) 0 0
\(433\) 3086.47 0.342555 0.171278 0.985223i \(-0.445210\pi\)
0.171278 + 0.985223i \(0.445210\pi\)
\(434\) −2406.26 −0.266139
\(435\) 0 0
\(436\) −520.851 −0.0572116
\(437\) −2900.32 −0.317486
\(438\) 0 0
\(439\) 2837.68 0.308508 0.154254 0.988031i \(-0.450703\pi\)
0.154254 + 0.988031i \(0.450703\pi\)
\(440\) 10377.1 1.12434
\(441\) 0 0
\(442\) 0 0
\(443\) −18309.4 −1.96367 −0.981834 0.189744i \(-0.939234\pi\)
−0.981834 + 0.189744i \(0.939234\pi\)
\(444\) 0 0
\(445\) 23203.0 2.47174
\(446\) 10377.0 1.10172
\(447\) 0 0
\(448\) −4034.12 −0.425433
\(449\) 13861.2 1.45690 0.728451 0.685098i \(-0.240241\pi\)
0.728451 + 0.685098i \(0.240241\pi\)
\(450\) 0 0
\(451\) −5764.26 −0.601837
\(452\) −80.2614 −0.00835217
\(453\) 0 0
\(454\) −5521.23 −0.570758
\(455\) 0 0
\(456\) 0 0
\(457\) 8990.36 0.920243 0.460122 0.887856i \(-0.347806\pi\)
0.460122 + 0.887856i \(0.347806\pi\)
\(458\) −8364.05 −0.853333
\(459\) 0 0
\(460\) −1653.55 −0.167602
\(461\) −3406.90 −0.344198 −0.172099 0.985080i \(-0.555055\pi\)
−0.172099 + 0.985080i \(0.555055\pi\)
\(462\) 0 0
\(463\) 7498.45 0.752662 0.376331 0.926485i \(-0.377186\pi\)
0.376331 + 0.926485i \(0.377186\pi\)
\(464\) 17478.8 1.74878
\(465\) 0 0
\(466\) 15319.0 1.52283
\(467\) −7711.38 −0.764112 −0.382056 0.924139i \(-0.624784\pi\)
−0.382056 + 0.924139i \(0.624784\pi\)
\(468\) 0 0
\(469\) 2993.94 0.294770
\(470\) 15049.2 1.47695
\(471\) 0 0
\(472\) −1852.91 −0.180693
\(473\) −11508.7 −1.11875
\(474\) 0 0
\(475\) −4205.14 −0.406200
\(476\) −242.446 −0.0233456
\(477\) 0 0
\(478\) 3662.01 0.350411
\(479\) −9439.82 −0.900451 −0.450226 0.892915i \(-0.648656\pi\)
−0.450226 + 0.892915i \(0.648656\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1567.02 −0.148082
\(483\) 0 0
\(484\) 389.839 0.0366115
\(485\) −10855.4 −1.01633
\(486\) 0 0
\(487\) 6156.20 0.572821 0.286411 0.958107i \(-0.407538\pi\)
0.286411 + 0.958107i \(0.407538\pi\)
\(488\) −4532.25 −0.420420
\(489\) 0 0
\(490\) 15330.6 1.41340
\(491\) −3842.74 −0.353198 −0.176599 0.984283i \(-0.556510\pi\)
−0.176599 + 0.984283i \(0.556510\pi\)
\(492\) 0 0
\(493\) 19560.3 1.78692
\(494\) 0 0
\(495\) 0 0
\(496\) 7025.24 0.635973
\(497\) −3953.43 −0.356812
\(498\) 0 0
\(499\) 12842.4 1.15211 0.576056 0.817410i \(-0.304591\pi\)
0.576056 + 0.817410i \(0.304591\pi\)
\(500\) −1220.38 −0.109154
\(501\) 0 0
\(502\) 11209.5 0.996627
\(503\) −8580.11 −0.760573 −0.380287 0.924869i \(-0.624175\pi\)
−0.380287 + 0.924869i \(0.624175\pi\)
\(504\) 0 0
\(505\) 5587.16 0.492327
\(506\) −11024.8 −0.968599
\(507\) 0 0
\(508\) −626.455 −0.0547135
\(509\) −43.5957 −0.00379635 −0.00189818 0.999998i \(-0.500604\pi\)
−0.00189818 + 0.999998i \(0.500604\pi\)
\(510\) 0 0
\(511\) −5495.49 −0.475746
\(512\) 12453.7 1.07496
\(513\) 0 0
\(514\) 8362.68 0.717630
\(515\) −37239.7 −3.18636
\(516\) 0 0
\(517\) −6451.66 −0.548827
\(518\) 3175.29 0.269333
\(519\) 0 0
\(520\) 0 0
\(521\) −11368.1 −0.955939 −0.477969 0.878377i \(-0.658627\pi\)
−0.477969 + 0.878377i \(0.658627\pi\)
\(522\) 0 0
\(523\) −5229.53 −0.437230 −0.218615 0.975811i \(-0.570154\pi\)
−0.218615 + 0.975811i \(0.570154\pi\)
\(524\) −95.3103 −0.00794590
\(525\) 0 0
\(526\) 15822.1 1.31155
\(527\) 7861.88 0.649846
\(528\) 0 0
\(529\) 18668.2 1.53433
\(530\) 19603.8 1.60667
\(531\) 0 0
\(532\) −59.7375 −0.00486832
\(533\) 0 0
\(534\) 0 0
\(535\) −16255.6 −1.31363
\(536\) −9305.24 −0.749860
\(537\) 0 0
\(538\) −5701.01 −0.456855
\(539\) −6572.27 −0.525209
\(540\) 0 0
\(541\) 6567.99 0.521959 0.260980 0.965344i \(-0.415954\pi\)
0.260980 + 0.965344i \(0.415954\pi\)
\(542\) 16482.9 1.30627
\(543\) 0 0
\(544\) 1464.13 0.115393
\(545\) 20996.5 1.65026
\(546\) 0 0
\(547\) −13675.7 −1.06897 −0.534487 0.845177i \(-0.679495\pi\)
−0.534487 + 0.845177i \(0.679495\pi\)
\(548\) 264.332 0.0206053
\(549\) 0 0
\(550\) −15984.7 −1.23925
\(551\) 4819.57 0.372632
\(552\) 0 0
\(553\) −847.604 −0.0651786
\(554\) 2015.67 0.154581
\(555\) 0 0
\(556\) −294.661 −0.0224755
\(557\) 4527.96 0.344445 0.172222 0.985058i \(-0.444905\pi\)
0.172222 + 0.985058i \(0.444905\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 8733.39 0.659023
\(561\) 0 0
\(562\) 5217.15 0.391588
\(563\) −18441.8 −1.38051 −0.690256 0.723566i \(-0.742502\pi\)
−0.690256 + 0.723566i \(0.742502\pi\)
\(564\) 0 0
\(565\) 3235.49 0.240917
\(566\) −5827.96 −0.432804
\(567\) 0 0
\(568\) 12287.4 0.907687
\(569\) 13553.5 0.998578 0.499289 0.866436i \(-0.333595\pi\)
0.499289 + 0.866436i \(0.333595\pi\)
\(570\) 0 0
\(571\) 14815.5 1.08583 0.542915 0.839788i \(-0.317321\pi\)
0.542915 + 0.839788i \(0.317321\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −5164.37 −0.375534
\(575\) 44707.6 3.24249
\(576\) 0 0
\(577\) −21596.2 −1.55816 −0.779081 0.626923i \(-0.784314\pi\)
−0.779081 + 0.626923i \(0.784314\pi\)
\(578\) 1150.20 0.0827716
\(579\) 0 0
\(580\) 2747.75 0.196714
\(581\) 6985.42 0.498802
\(582\) 0 0
\(583\) −8404.23 −0.597029
\(584\) 17080.1 1.21024
\(585\) 0 0
\(586\) 4499.13 0.317163
\(587\) −918.801 −0.0646047 −0.0323024 0.999478i \(-0.510284\pi\)
−0.0323024 + 0.999478i \(0.510284\pi\)
\(588\) 0 0
\(589\) 1937.13 0.135514
\(590\) 4255.50 0.296943
\(591\) 0 0
\(592\) −9270.49 −0.643606
\(593\) 19816.0 1.37226 0.686128 0.727481i \(-0.259309\pi\)
0.686128 + 0.727481i \(0.259309\pi\)
\(594\) 0 0
\(595\) 9773.45 0.673399
\(596\) 1046.42 0.0719175
\(597\) 0 0
\(598\) 0 0
\(599\) 5141.86 0.350736 0.175368 0.984503i \(-0.443889\pi\)
0.175368 + 0.984503i \(0.443889\pi\)
\(600\) 0 0
\(601\) 12380.9 0.840312 0.420156 0.907452i \(-0.361975\pi\)
0.420156 + 0.907452i \(0.361975\pi\)
\(602\) −10311.0 −0.698081
\(603\) 0 0
\(604\) −409.150 −0.0275630
\(605\) −15715.1 −1.05605
\(606\) 0 0
\(607\) −23717.0 −1.58590 −0.792951 0.609286i \(-0.791456\pi\)
−0.792951 + 0.609286i \(0.791456\pi\)
\(608\) 360.754 0.0240633
\(609\) 0 0
\(610\) 10409.0 0.690901
\(611\) 0 0
\(612\) 0 0
\(613\) 26157.1 1.72345 0.861726 0.507373i \(-0.169383\pi\)
0.861726 + 0.507373i \(0.169383\pi\)
\(614\) −9248.36 −0.607872
\(615\) 0 0
\(616\) −3985.71 −0.260696
\(617\) 23613.9 1.54077 0.770387 0.637576i \(-0.220063\pi\)
0.770387 + 0.637576i \(0.220063\pi\)
\(618\) 0 0
\(619\) −23345.4 −1.51588 −0.757940 0.652324i \(-0.773794\pi\)
−0.757940 + 0.652324i \(0.773794\pi\)
\(620\) 1104.40 0.0715387
\(621\) 0 0
\(622\) −2381.20 −0.153501
\(623\) −8911.99 −0.573116
\(624\) 0 0
\(625\) 17371.0 1.11174
\(626\) 11907.5 0.760258
\(627\) 0 0
\(628\) −799.207 −0.0507832
\(629\) −10374.5 −0.657645
\(630\) 0 0
\(631\) −15245.7 −0.961841 −0.480921 0.876764i \(-0.659698\pi\)
−0.480921 + 0.876764i \(0.659698\pi\)
\(632\) 2634.38 0.165807
\(633\) 0 0
\(634\) 8986.20 0.562914
\(635\) 25253.6 1.57820
\(636\) 0 0
\(637\) 0 0
\(638\) 18320.3 1.13684
\(639\) 0 0
\(640\) −25391.5 −1.56826
\(641\) −10192.7 −0.628063 −0.314032 0.949413i \(-0.601680\pi\)
−0.314032 + 0.949413i \(0.601680\pi\)
\(642\) 0 0
\(643\) 5506.31 0.337710 0.168855 0.985641i \(-0.445993\pi\)
0.168855 + 0.985641i \(0.445993\pi\)
\(644\) 635.108 0.0388614
\(645\) 0 0
\(646\) −3035.48 −0.184875
\(647\) 13297.5 0.808005 0.404003 0.914758i \(-0.367619\pi\)
0.404003 + 0.914758i \(0.367619\pi\)
\(648\) 0 0
\(649\) −1824.35 −0.110342
\(650\) 0 0
\(651\) 0 0
\(652\) −1385.43 −0.0832171
\(653\) 12440.2 0.745519 0.372760 0.927928i \(-0.378412\pi\)
0.372760 + 0.927928i \(0.378412\pi\)
\(654\) 0 0
\(655\) 3842.14 0.229198
\(656\) 15077.7 0.897389
\(657\) 0 0
\(658\) −5780.23 −0.342457
\(659\) 9562.87 0.565276 0.282638 0.959227i \(-0.408791\pi\)
0.282638 + 0.959227i \(0.408791\pi\)
\(660\) 0 0
\(661\) −2409.69 −0.141795 −0.0708973 0.997484i \(-0.522586\pi\)
−0.0708973 + 0.997484i \(0.522586\pi\)
\(662\) 15324.8 0.899722
\(663\) 0 0
\(664\) −21710.9 −1.26889
\(665\) 2408.13 0.140426
\(666\) 0 0
\(667\) −51239.9 −2.97454
\(668\) −352.368 −0.0204095
\(669\) 0 0
\(670\) 21371.0 1.23229
\(671\) −4462.40 −0.256735
\(672\) 0 0
\(673\) 7929.02 0.454147 0.227074 0.973878i \(-0.427084\pi\)
0.227074 + 0.973878i \(0.427084\pi\)
\(674\) −2471.72 −0.141257
\(675\) 0 0
\(676\) 0 0
\(677\) 2628.26 0.149206 0.0746030 0.997213i \(-0.476231\pi\)
0.0746030 + 0.997213i \(0.476231\pi\)
\(678\) 0 0
\(679\) 4169.44 0.235653
\(680\) −30376.1 −1.71305
\(681\) 0 0
\(682\) 7363.46 0.413433
\(683\) 10021.5 0.561437 0.280719 0.959790i \(-0.409427\pi\)
0.280719 + 0.959790i \(0.409427\pi\)
\(684\) 0 0
\(685\) −10655.7 −0.594356
\(686\) −12925.5 −0.719385
\(687\) 0 0
\(688\) 30103.7 1.66816
\(689\) 0 0
\(690\) 0 0
\(691\) −23987.2 −1.32057 −0.660286 0.751014i \(-0.729565\pi\)
−0.660286 + 0.751014i \(0.729565\pi\)
\(692\) −1853.43 −0.101816
\(693\) 0 0
\(694\) −2227.10 −0.121815
\(695\) 11878.3 0.648303
\(696\) 0 0
\(697\) 16873.4 0.916964
\(698\) 12167.4 0.659801
\(699\) 0 0
\(700\) 920.835 0.0497204
\(701\) 3763.71 0.202787 0.101393 0.994846i \(-0.467670\pi\)
0.101393 + 0.994846i \(0.467670\pi\)
\(702\) 0 0
\(703\) −2556.23 −0.137141
\(704\) 12344.9 0.660890
\(705\) 0 0
\(706\) −19507.8 −1.03993
\(707\) −2145.96 −0.114155
\(708\) 0 0
\(709\) 36047.8 1.90946 0.954728 0.297479i \(-0.0961458\pi\)
0.954728 + 0.297479i \(0.0961458\pi\)
\(710\) −28219.9 −1.49166
\(711\) 0 0
\(712\) 27698.7 1.45794
\(713\) −20594.9 −1.08174
\(714\) 0 0
\(715\) 0 0
\(716\) −137.195 −0.00716090
\(717\) 0 0
\(718\) −12869.3 −0.668910
\(719\) 3944.18 0.204580 0.102290 0.994755i \(-0.467383\pi\)
0.102290 + 0.994755i \(0.467383\pi\)
\(720\) 0 0
\(721\) 14303.3 0.738812
\(722\) 18057.1 0.930770
\(723\) 0 0
\(724\) −1142.36 −0.0586402
\(725\) −74292.1 −3.80571
\(726\) 0 0
\(727\) −20447.8 −1.04315 −0.521573 0.853206i \(-0.674655\pi\)
−0.521573 + 0.853206i \(0.674655\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −39227.3 −1.98886
\(731\) 33688.7 1.70454
\(732\) 0 0
\(733\) 13536.2 0.682089 0.341045 0.940047i \(-0.389219\pi\)
0.341045 + 0.940047i \(0.389219\pi\)
\(734\) −25343.8 −1.27447
\(735\) 0 0
\(736\) −3835.40 −0.192085
\(737\) −9161.83 −0.457911
\(738\) 0 0
\(739\) −15839.1 −0.788433 −0.394217 0.919018i \(-0.628984\pi\)
−0.394217 + 0.919018i \(0.628984\pi\)
\(740\) −1457.37 −0.0723972
\(741\) 0 0
\(742\) −7529.60 −0.372534
\(743\) −1664.92 −0.0822075 −0.0411037 0.999155i \(-0.513087\pi\)
−0.0411037 + 0.999155i \(0.513087\pi\)
\(744\) 0 0
\(745\) −42182.9 −2.07445
\(746\) 11822.0 0.580206
\(747\) 0 0
\(748\) 741.916 0.0362662
\(749\) 6243.58 0.304587
\(750\) 0 0
\(751\) 22399.1 1.08835 0.544177 0.838970i \(-0.316842\pi\)
0.544177 + 0.838970i \(0.316842\pi\)
\(752\) 16875.8 0.818346
\(753\) 0 0
\(754\) 0 0
\(755\) 16493.6 0.795050
\(756\) 0 0
\(757\) 23798.9 1.14265 0.571326 0.820723i \(-0.306429\pi\)
0.571326 + 0.820723i \(0.306429\pi\)
\(758\) −6532.64 −0.313029
\(759\) 0 0
\(760\) −7484.53 −0.357227
\(761\) 13693.5 0.652285 0.326142 0.945321i \(-0.394251\pi\)
0.326142 + 0.945321i \(0.394251\pi\)
\(762\) 0 0
\(763\) −8064.50 −0.382640
\(764\) 1215.18 0.0575441
\(765\) 0 0
\(766\) −13285.5 −0.626666
\(767\) 0 0
\(768\) 0 0
\(769\) −16299.9 −0.764358 −0.382179 0.924088i \(-0.624826\pi\)
−0.382179 + 0.924088i \(0.624826\pi\)
\(770\) 9153.84 0.428418
\(771\) 0 0
\(772\) 1169.97 0.0545444
\(773\) 33532.2 1.56024 0.780122 0.625628i \(-0.215157\pi\)
0.780122 + 0.625628i \(0.215157\pi\)
\(774\) 0 0
\(775\) −29860.2 −1.38401
\(776\) −12958.7 −0.599473
\(777\) 0 0
\(778\) 26214.3 1.20801
\(779\) 4157.51 0.191217
\(780\) 0 0
\(781\) 12098.0 0.554290
\(782\) 32272.1 1.47577
\(783\) 0 0
\(784\) 17191.3 0.783130
\(785\) 32217.5 1.46483
\(786\) 0 0
\(787\) −16163.3 −0.732097 −0.366049 0.930596i \(-0.619290\pi\)
−0.366049 + 0.930596i \(0.619290\pi\)
\(788\) 2239.51 0.101243
\(789\) 0 0
\(790\) −6050.27 −0.272480
\(791\) −1242.71 −0.0558607
\(792\) 0 0
\(793\) 0 0
\(794\) −20398.2 −0.911720
\(795\) 0 0
\(796\) −1476.12 −0.0657284
\(797\) 39636.4 1.76160 0.880798 0.473492i \(-0.157007\pi\)
0.880798 + 0.473492i \(0.157007\pi\)
\(798\) 0 0
\(799\) 18885.5 0.836197
\(800\) −5560.90 −0.245760
\(801\) 0 0
\(802\) 23819.0 1.04872
\(803\) 16816.9 0.739048
\(804\) 0 0
\(805\) −25602.4 −1.12095
\(806\) 0 0
\(807\) 0 0
\(808\) 6669.71 0.290396
\(809\) 23811.2 1.03481 0.517403 0.855742i \(-0.326899\pi\)
0.517403 + 0.855742i \(0.326899\pi\)
\(810\) 0 0
\(811\) −27218.6 −1.17851 −0.589256 0.807946i \(-0.700579\pi\)
−0.589256 + 0.807946i \(0.700579\pi\)
\(812\) −1055.38 −0.0456116
\(813\) 0 0
\(814\) −9716.80 −0.418395
\(815\) 55849.2 2.40038
\(816\) 0 0
\(817\) 8300.73 0.355454
\(818\) 7007.94 0.299544
\(819\) 0 0
\(820\) 2370.30 0.100944
\(821\) −43094.8 −1.83193 −0.915967 0.401253i \(-0.868575\pi\)
−0.915967 + 0.401253i \(0.868575\pi\)
\(822\) 0 0
\(823\) 26541.1 1.12414 0.562068 0.827091i \(-0.310006\pi\)
0.562068 + 0.827091i \(0.310006\pi\)
\(824\) −44455.1 −1.87945
\(825\) 0 0
\(826\) −1634.49 −0.0688512
\(827\) −44898.7 −1.88788 −0.943942 0.330112i \(-0.892913\pi\)
−0.943942 + 0.330112i \(0.892913\pi\)
\(828\) 0 0
\(829\) −7137.48 −0.299029 −0.149514 0.988760i \(-0.547771\pi\)
−0.149514 + 0.988760i \(0.547771\pi\)
\(830\) 49862.6 2.08525
\(831\) 0 0
\(832\) 0 0
\(833\) 19238.6 0.800213
\(834\) 0 0
\(835\) 14204.6 0.588708
\(836\) 182.804 0.00756270
\(837\) 0 0
\(838\) −9177.59 −0.378323
\(839\) −4387.17 −0.180527 −0.0902634 0.995918i \(-0.528771\pi\)
−0.0902634 + 0.995918i \(0.528771\pi\)
\(840\) 0 0
\(841\) 60758.1 2.49121
\(842\) −5084.36 −0.208098
\(843\) 0 0
\(844\) 1948.53 0.0794683
\(845\) 0 0
\(846\) 0 0
\(847\) 6036.01 0.244864
\(848\) 21983.2 0.890219
\(849\) 0 0
\(850\) 46791.0 1.88814
\(851\) 27176.9 1.09473
\(852\) 0 0
\(853\) 9328.85 0.374459 0.187230 0.982316i \(-0.440049\pi\)
0.187230 + 0.982316i \(0.440049\pi\)
\(854\) −3997.99 −0.160197
\(855\) 0 0
\(856\) −19405.2 −0.774833
\(857\) 5010.39 0.199710 0.0998552 0.995002i \(-0.468162\pi\)
0.0998552 + 0.995002i \(0.468162\pi\)
\(858\) 0 0
\(859\) 30233.4 1.20088 0.600438 0.799672i \(-0.294993\pi\)
0.600438 + 0.799672i \(0.294993\pi\)
\(860\) 4732.45 0.187646
\(861\) 0 0
\(862\) 38501.4 1.52130
\(863\) 4334.93 0.170988 0.0854940 0.996339i \(-0.472753\pi\)
0.0854940 + 0.996339i \(0.472753\pi\)
\(864\) 0 0
\(865\) 74715.2 2.93687
\(866\) −8462.05 −0.332047
\(867\) 0 0
\(868\) −424.189 −0.0165875
\(869\) 2593.78 0.101252
\(870\) 0 0
\(871\) 0 0
\(872\) 25064.7 0.973391
\(873\) 0 0
\(874\) 7951.69 0.307746
\(875\) −18895.6 −0.730043
\(876\) 0 0
\(877\) −34683.3 −1.33543 −0.667716 0.744416i \(-0.732728\pi\)
−0.667716 + 0.744416i \(0.732728\pi\)
\(878\) −7779.95 −0.299044
\(879\) 0 0
\(880\) −26725.3 −1.02376
\(881\) 18269.2 0.698642 0.349321 0.937003i \(-0.386412\pi\)
0.349321 + 0.937003i \(0.386412\pi\)
\(882\) 0 0
\(883\) −14592.0 −0.556128 −0.278064 0.960563i \(-0.589693\pi\)
−0.278064 + 0.960563i \(0.589693\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 50198.0 1.90343
\(887\) −30459.3 −1.15301 −0.576507 0.817092i \(-0.695585\pi\)
−0.576507 + 0.817092i \(0.695585\pi\)
\(888\) 0 0
\(889\) −9699.60 −0.365933
\(890\) −63614.6 −2.39592
\(891\) 0 0
\(892\) 1829.32 0.0686662
\(893\) 4653.30 0.174375
\(894\) 0 0
\(895\) 5530.57 0.206555
\(896\) 9752.58 0.363628
\(897\) 0 0
\(898\) −38002.6 −1.41221
\(899\) 34223.2 1.26964
\(900\) 0 0
\(901\) 24601.2 0.909638
\(902\) 15803.6 0.583374
\(903\) 0 0
\(904\) 3862.39 0.142103
\(905\) 46050.7 1.69147
\(906\) 0 0
\(907\) −9364.89 −0.342840 −0.171420 0.985198i \(-0.554836\pi\)
−0.171420 + 0.985198i \(0.554836\pi\)
\(908\) −973.313 −0.0355733
\(909\) 0 0
\(910\) 0 0
\(911\) −32479.8 −1.18123 −0.590616 0.806952i \(-0.701115\pi\)
−0.590616 + 0.806952i \(0.701115\pi\)
\(912\) 0 0
\(913\) −21376.3 −0.774864
\(914\) −24648.5 −0.892012
\(915\) 0 0
\(916\) −1474.46 −0.0531851
\(917\) −1475.72 −0.0531435
\(918\) 0 0
\(919\) 295.958 0.0106232 0.00531161 0.999986i \(-0.498309\pi\)
0.00531161 + 0.999986i \(0.498309\pi\)
\(920\) 79572.9 2.85157
\(921\) 0 0
\(922\) 9340.56 0.333639
\(923\) 0 0
\(924\) 0 0
\(925\) 39403.5 1.40062
\(926\) −20558.2 −0.729572
\(927\) 0 0
\(928\) 6373.42 0.225450
\(929\) −5620.38 −0.198492 −0.0992458 0.995063i \(-0.531643\pi\)
−0.0992458 + 0.995063i \(0.531643\pi\)
\(930\) 0 0
\(931\) 4740.29 0.166871
\(932\) 2700.52 0.0949125
\(933\) 0 0
\(934\) 21142.0 0.740670
\(935\) −29908.0 −1.04609
\(936\) 0 0
\(937\) −32583.1 −1.13601 −0.568006 0.823024i \(-0.692285\pi\)
−0.568006 + 0.823024i \(0.692285\pi\)
\(938\) −8208.35 −0.285727
\(939\) 0 0
\(940\) 2652.96 0.0920532
\(941\) 8812.99 0.305308 0.152654 0.988280i \(-0.451218\pi\)
0.152654 + 0.988280i \(0.451218\pi\)
\(942\) 0 0
\(943\) −44201.2 −1.52639
\(944\) 4772.00 0.164529
\(945\) 0 0
\(946\) 31552.9 1.08443
\(947\) 13426.8 0.460732 0.230366 0.973104i \(-0.426008\pi\)
0.230366 + 0.973104i \(0.426008\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 11529.1 0.393739
\(951\) 0 0
\(952\) 11667.1 0.397199
\(953\) 13394.6 0.455293 0.227647 0.973744i \(-0.426897\pi\)
0.227647 + 0.973744i \(0.426897\pi\)
\(954\) 0 0
\(955\) −48986.2 −1.65985
\(956\) 645.560 0.0218398
\(957\) 0 0
\(958\) 25880.7 0.872828
\(959\) 4092.74 0.137812
\(960\) 0 0
\(961\) −16035.7 −0.538273
\(962\) 0 0
\(963\) 0 0
\(964\) −276.243 −0.00922944
\(965\) −47163.8 −1.57332
\(966\) 0 0
\(967\) −45590.8 −1.51613 −0.758066 0.652178i \(-0.773856\pi\)
−0.758066 + 0.652178i \(0.773856\pi\)
\(968\) −18760.1 −0.622904
\(969\) 0 0
\(970\) 29761.8 0.985149
\(971\) −264.763 −0.00875041 −0.00437521 0.999990i \(-0.501393\pi\)
−0.00437521 + 0.999990i \(0.501393\pi\)
\(972\) 0 0
\(973\) −4562.32 −0.150320
\(974\) −16878.2 −0.555248
\(975\) 0 0
\(976\) 11672.4 0.382812
\(977\) 610.521 0.0199921 0.00999606 0.999950i \(-0.496818\pi\)
0.00999606 + 0.999950i \(0.496818\pi\)
\(978\) 0 0
\(979\) 27271.8 0.890308
\(980\) 2702.56 0.0880918
\(981\) 0 0
\(982\) 10535.5 0.342363
\(983\) −57829.7 −1.87638 −0.938190 0.346121i \(-0.887499\pi\)
−0.938190 + 0.346121i \(0.887499\pi\)
\(984\) 0 0
\(985\) −90278.8 −2.92033
\(986\) −53627.7 −1.73210
\(987\) 0 0
\(988\) 0 0
\(989\) −88250.4 −2.83741
\(990\) 0 0
\(991\) −56780.7 −1.82008 −0.910039 0.414522i \(-0.863949\pi\)
−0.910039 + 0.414522i \(0.863949\pi\)
\(992\) 2561.67 0.0819890
\(993\) 0 0
\(994\) 10838.9 0.345866
\(995\) 59505.3 1.89592
\(996\) 0 0
\(997\) 18616.6 0.591369 0.295684 0.955286i \(-0.404452\pi\)
0.295684 + 0.955286i \(0.404452\pi\)
\(998\) −35209.4 −1.11677
\(999\) 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.s.1.1 2
3.2 odd 2 507.4.a.f.1.2 2
13.12 even 2 117.4.a.c.1.2 2
39.5 even 4 507.4.b.f.337.2 4
39.8 even 4 507.4.b.f.337.3 4
39.38 odd 2 39.4.a.b.1.1 2
52.51 odd 2 1872.4.a.t.1.1 2
156.155 even 2 624.4.a.r.1.2 2
195.194 odd 2 975.4.a.j.1.2 2
273.272 even 2 1911.4.a.h.1.1 2
312.77 odd 2 2496.4.a.bc.1.1 2
312.155 even 2 2496.4.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 2 39.38 odd 2
117.4.a.c.1.2 2 13.12 even 2
507.4.a.f.1.2 2 3.2 odd 2
507.4.b.f.337.2 4 39.5 even 4
507.4.b.f.337.3 4 39.8 even 4
624.4.a.r.1.2 2 156.155 even 2
975.4.a.j.1.2 2 195.194 odd 2
1521.4.a.s.1.1 2 1.1 even 1 trivial
1872.4.a.t.1.1 2 52.51 odd 2
1911.4.a.h.1.1 2 273.272 even 2
2496.4.a.s.1.1 2 312.155 even 2
2496.4.a.bc.1.1 2 312.77 odd 2