Properties

Label 2541.2.a.bn.1.4
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $1$
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] = [2541,2,Mod(1,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2541.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.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: \(20.2899871536\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.477260\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.77222 q^{2} -1.00000 q^{3} +1.14077 q^{4} +0.589926 q^{5} -1.77222 q^{6} -1.00000 q^{7} -1.52274 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.77222 q^{2} -1.00000 q^{3} +1.14077 q^{4} +0.589926 q^{5} -1.77222 q^{6} -1.00000 q^{7} -1.52274 q^{8} +1.00000 q^{9} +1.04548 q^{10} -1.14077 q^{12} -1.54445 q^{13} -1.77222 q^{14} -0.589926 q^{15} -4.98018 q^{16} +7.14511 q^{17} +1.77222 q^{18} -6.19059 q^{19} +0.672972 q^{20} +1.00000 q^{21} +3.76314 q^{23} +1.52274 q^{24} -4.65199 q^{25} -2.73710 q^{26} -1.00000 q^{27} -1.14077 q^{28} +0.607298 q^{29} -1.04548 q^{30} -6.87147 q^{31} -5.78051 q^{32} +12.6627 q^{34} -0.589926 q^{35} +1.14077 q^{36} -7.70693 q^{37} -10.9711 q^{38} +1.54445 q^{39} -0.898304 q^{40} +7.48287 q^{41} +1.77222 q^{42} -10.9537 q^{43} +0.589926 q^{45} +6.66913 q^{46} -10.2072 q^{47} +4.98018 q^{48} +1.00000 q^{49} -8.24436 q^{50} -7.14511 q^{51} -1.76186 q^{52} +7.04469 q^{53} -1.77222 q^{54} +1.52274 q^{56} +6.19059 q^{57} +1.07627 q^{58} -3.35386 q^{59} -0.672972 q^{60} -3.37044 q^{61} -12.1778 q^{62} -1.00000 q^{63} -0.283993 q^{64} -0.911108 q^{65} -2.19138 q^{67} +8.15095 q^{68} -3.76314 q^{69} -1.04548 q^{70} -13.9430 q^{71} -1.52274 q^{72} -2.58993 q^{73} -13.6584 q^{74} +4.65199 q^{75} -7.06206 q^{76} +2.73710 q^{78} -4.66351 q^{79} -2.93794 q^{80} +1.00000 q^{81} +13.2613 q^{82} -3.50970 q^{83} +1.14077 q^{84} +4.21508 q^{85} -19.4125 q^{86} -0.607298 q^{87} +8.15095 q^{89} +1.04548 q^{90} +1.54445 q^{91} +4.29289 q^{92} +6.87147 q^{93} -18.0894 q^{94} -3.65199 q^{95} +5.78051 q^{96} -4.79710 q^{97} +1.77222 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9} + 10 q^{10} - 3 q^{12} + 6 q^{13} - q^{14} + 4 q^{15} - 3 q^{16} + 8 q^{17} + q^{18} - 10 q^{19} + 4 q^{21} - 10 q^{23} + 9 q^{24} + 12 q^{25} - 20 q^{26} - 4 q^{27} - 3 q^{28} - 10 q^{30} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} + 3 q^{36} - 2 q^{37} - 8 q^{38} - 6 q^{39} + 6 q^{40} + 10 q^{41} + q^{42} - 4 q^{43} - 4 q^{45} + 11 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{49} - 9 q^{50} - 8 q^{51} + 20 q^{52} - q^{54} + 9 q^{56} + 10 q^{57} + 14 q^{58} - 16 q^{59} + 14 q^{61} - 4 q^{63} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} + 10 q^{69} - 10 q^{70} - 18 q^{71} - 9 q^{72} - 4 q^{73} - 41 q^{74} - 12 q^{75} - 4 q^{76} + 20 q^{78} - 20 q^{79} - 36 q^{80} + 4 q^{81} - 24 q^{82} + 6 q^{83} + 3 q^{84} - 20 q^{85} - 20 q^{86} - 16 q^{89} + 10 q^{90} - 6 q^{91} - 22 q^{92} + 18 q^{93} - 16 q^{94} + 16 q^{95} + 2 q^{96} + 32 q^{97} + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.77222 1.25315 0.626575 0.779361i \(-0.284456\pi\)
0.626575 + 0.779361i \(0.284456\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.14077 0.570387
\(5\) 0.589926 0.263823 0.131911 0.991262i \(-0.457889\pi\)
0.131911 + 0.991262i \(0.457889\pi\)
\(6\) −1.77222 −0.723507
\(7\) −1.00000 −0.377964
\(8\) −1.52274 −0.538370
\(9\) 1.00000 0.333333
\(10\) 1.04548 0.330610
\(11\) 0 0
\(12\) −1.14077 −0.329313
\(13\) −1.54445 −0.428352 −0.214176 0.976795i \(-0.568707\pi\)
−0.214176 + 0.976795i \(0.568707\pi\)
\(14\) −1.77222 −0.473646
\(15\) −0.589926 −0.152318
\(16\) −4.98018 −1.24505
\(17\) 7.14511 1.73294 0.866472 0.499226i \(-0.166382\pi\)
0.866472 + 0.499226i \(0.166382\pi\)
\(18\) 1.77222 0.417717
\(19\) −6.19059 −1.42022 −0.710109 0.704092i \(-0.751354\pi\)
−0.710109 + 0.704092i \(0.751354\pi\)
\(20\) 0.672972 0.150481
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 3.76314 0.784669 0.392335 0.919823i \(-0.371668\pi\)
0.392335 + 0.919823i \(0.371668\pi\)
\(24\) 1.52274 0.310828
\(25\) −4.65199 −0.930398
\(26\) −2.73710 −0.536790
\(27\) −1.00000 −0.192450
\(28\) −1.14077 −0.215586
\(29\) 0.607298 0.112772 0.0563862 0.998409i \(-0.482042\pi\)
0.0563862 + 0.998409i \(0.482042\pi\)
\(30\) −1.04548 −0.190878
\(31\) −6.87147 −1.23415 −0.617077 0.786903i \(-0.711683\pi\)
−0.617077 + 0.786903i \(0.711683\pi\)
\(32\) −5.78051 −1.02186
\(33\) 0 0
\(34\) 12.6627 2.17164
\(35\) −0.589926 −0.0997157
\(36\) 1.14077 0.190129
\(37\) −7.70693 −1.26701 −0.633505 0.773738i \(-0.718385\pi\)
−0.633505 + 0.773738i \(0.718385\pi\)
\(38\) −10.9711 −1.77975
\(39\) 1.54445 0.247309
\(40\) −0.898304 −0.142034
\(41\) 7.48287 1.16863 0.584314 0.811528i \(-0.301364\pi\)
0.584314 + 0.811528i \(0.301364\pi\)
\(42\) 1.77222 0.273460
\(43\) −10.9537 −1.67043 −0.835214 0.549925i \(-0.814656\pi\)
−0.835214 + 0.549925i \(0.814656\pi\)
\(44\) 0 0
\(45\) 0.589926 0.0879409
\(46\) 6.66913 0.983309
\(47\) −10.2072 −1.48887 −0.744434 0.667696i \(-0.767281\pi\)
−0.744434 + 0.667696i \(0.767281\pi\)
\(48\) 4.98018 0.718827
\(49\) 1.00000 0.142857
\(50\) −8.24436 −1.16593
\(51\) −7.14511 −1.00052
\(52\) −1.76186 −0.244327
\(53\) 7.04469 0.967663 0.483831 0.875161i \(-0.339245\pi\)
0.483831 + 0.875161i \(0.339245\pi\)
\(54\) −1.77222 −0.241169
\(55\) 0 0
\(56\) 1.52274 0.203485
\(57\) 6.19059 0.819963
\(58\) 1.07627 0.141321
\(59\) −3.35386 −0.436635 −0.218318 0.975878i \(-0.570057\pi\)
−0.218318 + 0.975878i \(0.570057\pi\)
\(60\) −0.672972 −0.0868803
\(61\) −3.37044 −0.431541 −0.215770 0.976444i \(-0.569226\pi\)
−0.215770 + 0.976444i \(0.569226\pi\)
\(62\) −12.1778 −1.54658
\(63\) −1.00000 −0.125988
\(64\) −0.283993 −0.0354992
\(65\) −0.911108 −0.113009
\(66\) 0 0
\(67\) −2.19138 −0.267719 −0.133860 0.991000i \(-0.542737\pi\)
−0.133860 + 0.991000i \(0.542737\pi\)
\(68\) 8.15095 0.988448
\(69\) −3.76314 −0.453029
\(70\) −1.04548 −0.124959
\(71\) −13.9430 −1.65473 −0.827364 0.561665i \(-0.810161\pi\)
−0.827364 + 0.561665i \(0.810161\pi\)
\(72\) −1.52274 −0.179457
\(73\) −2.58993 −0.303128 −0.151564 0.988447i \(-0.548431\pi\)
−0.151564 + 0.988447i \(0.548431\pi\)
\(74\) −13.6584 −1.58776
\(75\) 4.65199 0.537165
\(76\) −7.06206 −0.810074
\(77\) 0 0
\(78\) 2.73710 0.309916
\(79\) −4.66351 −0.524686 −0.262343 0.964975i \(-0.584495\pi\)
−0.262343 + 0.964975i \(0.584495\pi\)
\(80\) −2.93794 −0.328471
\(81\) 1.00000 0.111111
\(82\) 13.2613 1.46447
\(83\) −3.50970 −0.385240 −0.192620 0.981273i \(-0.561698\pi\)
−0.192620 + 0.981273i \(0.561698\pi\)
\(84\) 1.14077 0.124469
\(85\) 4.21508 0.457190
\(86\) −19.4125 −2.09330
\(87\) −0.607298 −0.0651091
\(88\) 0 0
\(89\) 8.15095 0.863999 0.432000 0.901874i \(-0.357808\pi\)
0.432000 + 0.901874i \(0.357808\pi\)
\(90\) 1.04548 0.110203
\(91\) 1.54445 0.161902
\(92\) 4.29289 0.447565
\(93\) 6.87147 0.712539
\(94\) −18.0894 −1.86578
\(95\) −3.65199 −0.374686
\(96\) 5.78051 0.589971
\(97\) −4.79710 −0.487071 −0.243536 0.969892i \(-0.578307\pi\)
−0.243536 + 0.969892i \(0.578307\pi\)
\(98\) 1.77222 0.179022
\(99\) 0 0
\(100\) −5.30687 −0.530687
\(101\) −1.42666 −0.141958 −0.0709788 0.997478i \(-0.522612\pi\)
−0.0709788 + 0.997478i \(0.522612\pi\)
\(102\) −12.6627 −1.25380
\(103\) 17.6172 1.73588 0.867939 0.496670i \(-0.165444\pi\)
0.867939 + 0.496670i \(0.165444\pi\)
\(104\) 2.35179 0.230612
\(105\) 0.589926 0.0575709
\(106\) 12.4848 1.21263
\(107\) −7.20005 −0.696055 −0.348027 0.937484i \(-0.613148\pi\)
−0.348027 + 0.937484i \(0.613148\pi\)
\(108\) −1.14077 −0.109771
\(109\) −8.88678 −0.851199 −0.425599 0.904912i \(-0.639937\pi\)
−0.425599 + 0.904912i \(0.639937\pi\)
\(110\) 0 0
\(111\) 7.70693 0.731509
\(112\) 4.98018 0.470583
\(113\) −7.80862 −0.734573 −0.367287 0.930108i \(-0.619713\pi\)
−0.367287 + 0.930108i \(0.619713\pi\)
\(114\) 10.9711 1.02754
\(115\) 2.21997 0.207014
\(116\) 0.692789 0.0643239
\(117\) −1.54445 −0.142784
\(118\) −5.94378 −0.547170
\(119\) −7.14511 −0.654991
\(120\) 0.898304 0.0820035
\(121\) 0 0
\(122\) −5.97317 −0.540785
\(123\) −7.48287 −0.674708
\(124\) −7.83880 −0.703945
\(125\) −5.69396 −0.509283
\(126\) −1.77222 −0.157882
\(127\) 8.25186 0.732234 0.366117 0.930569i \(-0.380687\pi\)
0.366117 + 0.930569i \(0.380687\pi\)
\(128\) 11.0577 0.977374
\(129\) 10.9537 0.964422
\(130\) −1.61469 −0.141617
\(131\) 12.1162 1.05860 0.529299 0.848435i \(-0.322455\pi\)
0.529299 + 0.848435i \(0.322455\pi\)
\(132\) 0 0
\(133\) 6.19059 0.536792
\(134\) −3.88361 −0.335493
\(135\) −0.589926 −0.0507727
\(136\) −10.8801 −0.932964
\(137\) −17.4548 −1.49126 −0.745631 0.666360i \(-0.767852\pi\)
−0.745631 + 0.666360i \(0.767852\pi\)
\(138\) −6.66913 −0.567714
\(139\) 14.7624 1.25213 0.626063 0.779772i \(-0.284665\pi\)
0.626063 + 0.779772i \(0.284665\pi\)
\(140\) −0.672972 −0.0568765
\(141\) 10.2072 0.859599
\(142\) −24.7101 −2.07362
\(143\) 0 0
\(144\) −4.98018 −0.415015
\(145\) 0.358261 0.0297519
\(146\) −4.58993 −0.379865
\(147\) −1.00000 −0.0824786
\(148\) −8.79186 −0.722686
\(149\) 12.5258 1.02615 0.513077 0.858343i \(-0.328506\pi\)
0.513077 + 0.858343i \(0.328506\pi\)
\(150\) 8.24436 0.673149
\(151\) 0.328572 0.0267388 0.0133694 0.999911i \(-0.495744\pi\)
0.0133694 + 0.999911i \(0.495744\pi\)
\(152\) 9.42666 0.764603
\(153\) 7.14511 0.577648
\(154\) 0 0
\(155\) −4.05366 −0.325598
\(156\) 1.76186 0.141062
\(157\) −2.68089 −0.213958 −0.106979 0.994261i \(-0.534118\pi\)
−0.106979 + 0.994261i \(0.534118\pi\)
\(158\) −8.26479 −0.657511
\(159\) −7.04469 −0.558680
\(160\) −3.41007 −0.269590
\(161\) −3.76314 −0.296577
\(162\) 1.77222 0.139239
\(163\) −0.891668 −0.0698408 −0.0349204 0.999390i \(-0.511118\pi\)
−0.0349204 + 0.999390i \(0.511118\pi\)
\(164\) 8.53627 0.666570
\(165\) 0 0
\(166\) −6.21997 −0.482764
\(167\) 11.5610 0.894619 0.447310 0.894379i \(-0.352382\pi\)
0.447310 + 0.894379i \(0.352382\pi\)
\(168\) −1.52274 −0.117482
\(169\) −10.6147 −0.816514
\(170\) 7.47007 0.572928
\(171\) −6.19059 −0.473406
\(172\) −12.4957 −0.952790
\(173\) 8.98926 0.683441 0.341721 0.939802i \(-0.388990\pi\)
0.341721 + 0.939802i \(0.388990\pi\)
\(174\) −1.07627 −0.0815916
\(175\) 4.65199 0.351657
\(176\) 0 0
\(177\) 3.35386 0.252091
\(178\) 14.4453 1.08272
\(179\) −6.14590 −0.459366 −0.229683 0.973265i \(-0.573769\pi\)
−0.229683 + 0.973265i \(0.573769\pi\)
\(180\) 0.672972 0.0501604
\(181\) 14.9662 1.11243 0.556215 0.831039i \(-0.312253\pi\)
0.556215 + 0.831039i \(0.312253\pi\)
\(182\) 2.73710 0.202888
\(183\) 3.37044 0.249150
\(184\) −5.73029 −0.422442
\(185\) −4.54651 −0.334266
\(186\) 12.1778 0.892918
\(187\) 0 0
\(188\) −11.6441 −0.849231
\(189\) 1.00000 0.0727393
\(190\) −6.47214 −0.469538
\(191\) 9.87351 0.714422 0.357211 0.934024i \(-0.383728\pi\)
0.357211 + 0.934024i \(0.383728\pi\)
\(192\) 0.283993 0.0204955
\(193\) −6.18113 −0.444927 −0.222464 0.974941i \(-0.571410\pi\)
−0.222464 + 0.974941i \(0.571410\pi\)
\(194\) −8.50152 −0.610374
\(195\) 0.911108 0.0652458
\(196\) 1.14077 0.0814839
\(197\) −19.0720 −1.35882 −0.679412 0.733757i \(-0.737765\pi\)
−0.679412 + 0.733757i \(0.737765\pi\)
\(198\) 0 0
\(199\) −11.8479 −0.839878 −0.419939 0.907552i \(-0.637948\pi\)
−0.419939 + 0.907552i \(0.637948\pi\)
\(200\) 7.08377 0.500898
\(201\) 2.19138 0.154568
\(202\) −2.52835 −0.177894
\(203\) −0.607298 −0.0426239
\(204\) −8.15095 −0.570681
\(205\) 4.41434 0.308311
\(206\) 31.2217 2.17532
\(207\) 3.76314 0.261556
\(208\) 7.69162 0.533318
\(209\) 0 0
\(210\) 1.04548 0.0721450
\(211\) −20.9810 −1.44439 −0.722197 0.691687i \(-0.756868\pi\)
−0.722197 + 0.691687i \(0.756868\pi\)
\(212\) 8.03640 0.551942
\(213\) 13.9430 0.955358
\(214\) −12.7601 −0.872262
\(215\) −6.46189 −0.440697
\(216\) 1.52274 0.103609
\(217\) 6.87147 0.466466
\(218\) −15.7493 −1.06668
\(219\) 2.58993 0.175011
\(220\) 0 0
\(221\) −11.0352 −0.742310
\(222\) 13.6584 0.916691
\(223\) 3.38860 0.226918 0.113459 0.993543i \(-0.463807\pi\)
0.113459 + 0.993543i \(0.463807\pi\)
\(224\) 5.78051 0.386227
\(225\) −4.65199 −0.310133
\(226\) −13.8386 −0.920531
\(227\) 19.3973 1.28744 0.643721 0.765260i \(-0.277390\pi\)
0.643721 + 0.765260i \(0.277390\pi\)
\(228\) 7.06206 0.467696
\(229\) −9.86405 −0.651835 −0.325917 0.945398i \(-0.605673\pi\)
−0.325917 + 0.945398i \(0.605673\pi\)
\(230\) 3.93429 0.259419
\(231\) 0 0
\(232\) −0.924756 −0.0607132
\(233\) 7.12619 0.466852 0.233426 0.972375i \(-0.425006\pi\)
0.233426 + 0.972375i \(0.425006\pi\)
\(234\) −2.73710 −0.178930
\(235\) −6.02147 −0.392798
\(236\) −3.82599 −0.249051
\(237\) 4.66351 0.302928
\(238\) −12.6627 −0.820802
\(239\) 22.2894 1.44178 0.720892 0.693048i \(-0.243733\pi\)
0.720892 + 0.693048i \(0.243733\pi\)
\(240\) 2.93794 0.189643
\(241\) 14.8753 0.958199 0.479100 0.877761i \(-0.340963\pi\)
0.479100 + 0.877761i \(0.340963\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −3.84491 −0.246145
\(245\) 0.589926 0.0376890
\(246\) −13.2613 −0.845511
\(247\) 9.56103 0.608354
\(248\) 10.4635 0.664431
\(249\) 3.50970 0.222418
\(250\) −10.0910 −0.638208
\(251\) −25.9233 −1.63626 −0.818132 0.575031i \(-0.804990\pi\)
−0.818132 + 0.575031i \(0.804990\pi\)
\(252\) −1.14077 −0.0718620
\(253\) 0 0
\(254\) 14.6241 0.917600
\(255\) −4.21508 −0.263959
\(256\) 20.1647 1.26030
\(257\) 0.871474 0.0543610 0.0271805 0.999631i \(-0.491347\pi\)
0.0271805 + 0.999631i \(0.491347\pi\)
\(258\) 19.4125 1.20857
\(259\) 7.70693 0.478885
\(260\) −1.03937 −0.0644589
\(261\) 0.607298 0.0375908
\(262\) 21.4726 1.32658
\(263\) 26.8873 1.65794 0.828970 0.559293i \(-0.188927\pi\)
0.828970 + 0.559293i \(0.188927\pi\)
\(264\) 0 0
\(265\) 4.15584 0.255291
\(266\) 10.9711 0.672681
\(267\) −8.15095 −0.498830
\(268\) −2.49987 −0.152704
\(269\) −0.992574 −0.0605183 −0.0302592 0.999542i \(-0.509633\pi\)
−0.0302592 + 0.999542i \(0.509633\pi\)
\(270\) −1.04548 −0.0636259
\(271\) 26.8650 1.63193 0.815966 0.578099i \(-0.196205\pi\)
0.815966 + 0.578099i \(0.196205\pi\)
\(272\) −35.5839 −2.15759
\(273\) −1.54445 −0.0934741
\(274\) −30.9337 −1.86878
\(275\) 0 0
\(276\) −4.29289 −0.258402
\(277\) −27.6302 −1.66014 −0.830069 0.557660i \(-0.811699\pi\)
−0.830069 + 0.557660i \(0.811699\pi\)
\(278\) 26.1622 1.56910
\(279\) −6.87147 −0.411384
\(280\) 0.898304 0.0536839
\(281\) 23.8418 1.42228 0.711141 0.703050i \(-0.248179\pi\)
0.711141 + 0.703050i \(0.248179\pi\)
\(282\) 18.0894 1.07721
\(283\) 8.74735 0.519976 0.259988 0.965612i \(-0.416281\pi\)
0.259988 + 0.965612i \(0.416281\pi\)
\(284\) −15.9058 −0.943836
\(285\) 3.65199 0.216325
\(286\) 0 0
\(287\) −7.48287 −0.441700
\(288\) −5.78051 −0.340620
\(289\) 34.0526 2.00309
\(290\) 0.634918 0.0372836
\(291\) 4.79710 0.281211
\(292\) −2.95452 −0.172900
\(293\) −19.6252 −1.14651 −0.573257 0.819376i \(-0.694320\pi\)
−0.573257 + 0.819376i \(0.694320\pi\)
\(294\) −1.77222 −0.103358
\(295\) −1.97853 −0.115194
\(296\) 11.7356 0.682120
\(297\) 0 0
\(298\) 22.1985 1.28592
\(299\) −5.81197 −0.336115
\(300\) 5.30687 0.306392
\(301\) 10.9537 0.631362
\(302\) 0.582303 0.0335078
\(303\) 1.42666 0.0819592
\(304\) 30.8303 1.76824
\(305\) −1.98831 −0.113850
\(306\) 12.6627 0.723880
\(307\) −17.0835 −0.975009 −0.487504 0.873121i \(-0.662093\pi\)
−0.487504 + 0.873121i \(0.662093\pi\)
\(308\) 0 0
\(309\) −17.6172 −1.00221
\(310\) −7.18399 −0.408023
\(311\) −7.23351 −0.410175 −0.205087 0.978744i \(-0.565748\pi\)
−0.205087 + 0.978744i \(0.565748\pi\)
\(312\) −2.35179 −0.133144
\(313\) 13.7784 0.778804 0.389402 0.921068i \(-0.372682\pi\)
0.389402 + 0.921068i \(0.372682\pi\)
\(314\) −4.75113 −0.268122
\(315\) −0.589926 −0.0332386
\(316\) −5.32002 −0.299274
\(317\) −6.95629 −0.390704 −0.195352 0.980733i \(-0.562585\pi\)
−0.195352 + 0.980733i \(0.562585\pi\)
\(318\) −12.4848 −0.700111
\(319\) 0 0
\(320\) −0.167535 −0.00936550
\(321\) 7.20005 0.401867
\(322\) −6.66913 −0.371656
\(323\) −44.2324 −2.46116
\(324\) 1.14077 0.0633763
\(325\) 7.18474 0.398538
\(326\) −1.58023 −0.0875211
\(327\) 8.88678 0.491440
\(328\) −11.3945 −0.629154
\(329\) 10.2072 0.562739
\(330\) 0 0
\(331\) −29.6698 −1.63080 −0.815401 0.578896i \(-0.803484\pi\)
−0.815401 + 0.578896i \(0.803484\pi\)
\(332\) −4.00378 −0.219736
\(333\) −7.70693 −0.422337
\(334\) 20.4887 1.12109
\(335\) −1.29275 −0.0706305
\(336\) −4.98018 −0.271691
\(337\) 19.5714 1.06612 0.533062 0.846076i \(-0.321041\pi\)
0.533062 + 0.846076i \(0.321041\pi\)
\(338\) −18.8116 −1.02322
\(339\) 7.80862 0.424106
\(340\) 4.80846 0.260775
\(341\) 0 0
\(342\) −10.9711 −0.593249
\(343\) −1.00000 −0.0539949
\(344\) 16.6797 0.899308
\(345\) −2.21997 −0.119519
\(346\) 15.9310 0.856455
\(347\) −6.49184 −0.348500 −0.174250 0.984701i \(-0.555750\pi\)
−0.174250 + 0.984701i \(0.555750\pi\)
\(348\) −0.692789 −0.0371374
\(349\) 7.87965 0.421788 0.210894 0.977509i \(-0.432362\pi\)
0.210894 + 0.977509i \(0.432362\pi\)
\(350\) 8.24436 0.440680
\(351\) 1.54445 0.0824364
\(352\) 0 0
\(353\) −5.34594 −0.284536 −0.142268 0.989828i \(-0.545439\pi\)
−0.142268 + 0.989828i \(0.545439\pi\)
\(354\) 5.94378 0.315909
\(355\) −8.22533 −0.436555
\(356\) 9.29840 0.492814
\(357\) 7.14511 0.378159
\(358\) −10.8919 −0.575655
\(359\) −17.8515 −0.942169 −0.471084 0.882088i \(-0.656137\pi\)
−0.471084 + 0.882088i \(0.656137\pi\)
\(360\) −0.898304 −0.0473448
\(361\) 19.3234 1.01702
\(362\) 26.5235 1.39404
\(363\) 0 0
\(364\) 1.76186 0.0923467
\(365\) −1.52786 −0.0799721
\(366\) 5.97317 0.312223
\(367\) −29.9357 −1.56263 −0.781316 0.624135i \(-0.785451\pi\)
−0.781316 + 0.624135i \(0.785451\pi\)
\(368\) −18.7411 −0.976949
\(369\) 7.48287 0.389543
\(370\) −8.05744 −0.418886
\(371\) −7.04469 −0.365742
\(372\) 7.83880 0.406423
\(373\) 23.4915 1.21635 0.608173 0.793805i \(-0.291903\pi\)
0.608173 + 0.793805i \(0.291903\pi\)
\(374\) 0 0
\(375\) 5.69396 0.294035
\(376\) 15.5429 0.801562
\(377\) −0.937938 −0.0483063
\(378\) 1.77222 0.0911533
\(379\) −31.4532 −1.61564 −0.807821 0.589428i \(-0.799353\pi\)
−0.807821 + 0.589428i \(0.799353\pi\)
\(380\) −4.16609 −0.213716
\(381\) −8.25186 −0.422756
\(382\) 17.4981 0.895278
\(383\) −5.62749 −0.287551 −0.143776 0.989610i \(-0.545924\pi\)
−0.143776 + 0.989610i \(0.545924\pi\)
\(384\) −11.0577 −0.564287
\(385\) 0 0
\(386\) −10.9543 −0.557561
\(387\) −10.9537 −0.556809
\(388\) −5.47240 −0.277819
\(389\) 26.1072 1.32369 0.661845 0.749641i \(-0.269774\pi\)
0.661845 + 0.749641i \(0.269774\pi\)
\(390\) 1.61469 0.0817629
\(391\) 26.8881 1.35979
\(392\) −1.52274 −0.0769100
\(393\) −12.1162 −0.611182
\(394\) −33.7998 −1.70281
\(395\) −2.75113 −0.138424
\(396\) 0 0
\(397\) 10.3828 0.521096 0.260548 0.965461i \(-0.416097\pi\)
0.260548 + 0.965461i \(0.416097\pi\)
\(398\) −20.9972 −1.05249
\(399\) −6.19059 −0.309917
\(400\) 23.1677 1.15839
\(401\) 3.68876 0.184208 0.0921040 0.995749i \(-0.470641\pi\)
0.0921040 + 0.995749i \(0.470641\pi\)
\(402\) 3.88361 0.193697
\(403\) 10.6126 0.528652
\(404\) −1.62749 −0.0809708
\(405\) 0.589926 0.0293136
\(406\) −1.07627 −0.0534142
\(407\) 0 0
\(408\) 10.8801 0.538647
\(409\) 1.08022 0.0534136 0.0267068 0.999643i \(-0.491498\pi\)
0.0267068 + 0.999643i \(0.491498\pi\)
\(410\) 7.82319 0.386360
\(411\) 17.4548 0.860980
\(412\) 20.0973 0.990123
\(413\) 3.35386 0.165033
\(414\) 6.66913 0.327770
\(415\) −2.07046 −0.101635
\(416\) 8.92769 0.437716
\(417\) −14.7624 −0.722915
\(418\) 0 0
\(419\) 10.0766 0.492273 0.246136 0.969235i \(-0.420839\pi\)
0.246136 + 0.969235i \(0.420839\pi\)
\(420\) 0.672972 0.0328377
\(421\) 11.9757 0.583658 0.291829 0.956470i \(-0.405736\pi\)
0.291829 + 0.956470i \(0.405736\pi\)
\(422\) −37.1831 −1.81004
\(423\) −10.2072 −0.496290
\(424\) −10.7272 −0.520960
\(425\) −33.2390 −1.61233
\(426\) 24.7101 1.19721
\(427\) 3.37044 0.163107
\(428\) −8.21363 −0.397021
\(429\) 0 0
\(430\) −11.4519 −0.552260
\(431\) −0.548547 −0.0264226 −0.0132113 0.999913i \(-0.504205\pi\)
−0.0132113 + 0.999913i \(0.504205\pi\)
\(432\) 4.98018 0.239609
\(433\) 19.5879 0.941332 0.470666 0.882311i \(-0.344014\pi\)
0.470666 + 0.882311i \(0.344014\pi\)
\(434\) 12.1778 0.584552
\(435\) −0.358261 −0.0171773
\(436\) −10.1378 −0.485513
\(437\) −23.2961 −1.11440
\(438\) 4.58993 0.219315
\(439\) 37.4069 1.78533 0.892667 0.450716i \(-0.148831\pi\)
0.892667 + 0.450716i \(0.148831\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −19.5569 −0.930226
\(443\) 21.4478 1.01902 0.509508 0.860466i \(-0.329827\pi\)
0.509508 + 0.860466i \(0.329827\pi\)
\(444\) 8.79186 0.417243
\(445\) 4.80846 0.227943
\(446\) 6.00536 0.284362
\(447\) −12.5258 −0.592450
\(448\) 0.283993 0.0134174
\(449\) 6.39904 0.301989 0.150995 0.988535i \(-0.451752\pi\)
0.150995 + 0.988535i \(0.451752\pi\)
\(450\) −8.24436 −0.388643
\(451\) 0 0
\(452\) −8.90787 −0.418991
\(453\) −0.328572 −0.0154377
\(454\) 34.3763 1.61336
\(455\) 0.911108 0.0427134
\(456\) −9.42666 −0.441444
\(457\) −36.9430 −1.72812 −0.864060 0.503389i \(-0.832086\pi\)
−0.864060 + 0.503389i \(0.832086\pi\)
\(458\) −17.4813 −0.816847
\(459\) −7.14511 −0.333505
\(460\) 2.53249 0.118078
\(461\) 7.39825 0.344571 0.172285 0.985047i \(-0.444885\pi\)
0.172285 + 0.985047i \(0.444885\pi\)
\(462\) 0 0
\(463\) −14.3717 −0.667910 −0.333955 0.942589i \(-0.608383\pi\)
−0.333955 + 0.942589i \(0.608383\pi\)
\(464\) −3.02445 −0.140407
\(465\) 4.05366 0.187984
\(466\) 12.6292 0.585036
\(467\) 36.1626 1.67341 0.836704 0.547656i \(-0.184480\pi\)
0.836704 + 0.547656i \(0.184480\pi\)
\(468\) −1.76186 −0.0814422
\(469\) 2.19138 0.101188
\(470\) −10.6714 −0.492235
\(471\) 2.68089 0.123529
\(472\) 5.10705 0.235071
\(473\) 0 0
\(474\) 8.26479 0.379614
\(475\) 28.7985 1.32137
\(476\) −8.15095 −0.373598
\(477\) 7.04469 0.322554
\(478\) 39.5018 1.80677
\(479\) 16.0252 0.732212 0.366106 0.930573i \(-0.380691\pi\)
0.366106 + 0.930573i \(0.380691\pi\)
\(480\) 3.41007 0.155648
\(481\) 11.9029 0.542727
\(482\) 26.3623 1.20077
\(483\) 3.76314 0.171229
\(484\) 0 0
\(485\) −2.82993 −0.128501
\(486\) −1.77222 −0.0803897
\(487\) 1.55059 0.0702641 0.0351320 0.999383i \(-0.488815\pi\)
0.0351320 + 0.999383i \(0.488815\pi\)
\(488\) 5.13230 0.232328
\(489\) 0.891668 0.0403226
\(490\) 1.04548 0.0472300
\(491\) −0.0331635 −0.00149665 −0.000748324 1.00000i \(-0.500238\pi\)
−0.000748324 1.00000i \(0.500238\pi\)
\(492\) −8.53627 −0.384845
\(493\) 4.33921 0.195428
\(494\) 16.9443 0.762359
\(495\) 0 0
\(496\) 34.2212 1.53658
\(497\) 13.9430 0.625429
\(498\) 6.21997 0.278724
\(499\) −27.0314 −1.21009 −0.605046 0.796190i \(-0.706845\pi\)
−0.605046 + 0.796190i \(0.706845\pi\)
\(500\) −6.49552 −0.290488
\(501\) −11.5610 −0.516509
\(502\) −45.9418 −2.05048
\(503\) −29.3341 −1.30794 −0.653972 0.756519i \(-0.726899\pi\)
−0.653972 + 0.756519i \(0.726899\pi\)
\(504\) 1.52274 0.0678282
\(505\) −0.841621 −0.0374516
\(506\) 0 0
\(507\) 10.6147 0.471415
\(508\) 9.41351 0.417657
\(509\) −16.6329 −0.737239 −0.368620 0.929580i \(-0.620169\pi\)
−0.368620 + 0.929580i \(0.620169\pi\)
\(510\) −7.47007 −0.330780
\(511\) 2.58993 0.114572
\(512\) 13.6210 0.601967
\(513\) 6.19059 0.273321
\(514\) 1.54445 0.0681226
\(515\) 10.3929 0.457964
\(516\) 12.4957 0.550094
\(517\) 0 0
\(518\) 13.6584 0.600115
\(519\) −8.98926 −0.394585
\(520\) 1.38738 0.0608407
\(521\) 35.8923 1.57247 0.786236 0.617927i \(-0.212027\pi\)
0.786236 + 0.617927i \(0.212027\pi\)
\(522\) 1.07627 0.0471069
\(523\) 7.27873 0.318276 0.159138 0.987256i \(-0.449128\pi\)
0.159138 + 0.987256i \(0.449128\pi\)
\(524\) 13.8219 0.603811
\(525\) −4.65199 −0.203029
\(526\) 47.6502 2.07765
\(527\) −49.0974 −2.13872
\(528\) 0 0
\(529\) −8.83876 −0.384294
\(530\) 7.36508 0.319919
\(531\) −3.35386 −0.145545
\(532\) 7.06206 0.306179
\(533\) −11.5569 −0.500585
\(534\) −14.4453 −0.625110
\(535\) −4.24749 −0.183635
\(536\) 3.33690 0.144132
\(537\) 6.14590 0.265215
\(538\) −1.75906 −0.0758386
\(539\) 0 0
\(540\) −0.672972 −0.0289601
\(541\) −10.4709 −0.450177 −0.225089 0.974338i \(-0.572267\pi\)
−0.225089 + 0.974338i \(0.572267\pi\)
\(542\) 47.6108 2.04506
\(543\) −14.9662 −0.642262
\(544\) −41.3024 −1.77083
\(545\) −5.24254 −0.224566
\(546\) −2.73710 −0.117137
\(547\) 8.16609 0.349157 0.174578 0.984643i \(-0.444144\pi\)
0.174578 + 0.984643i \(0.444144\pi\)
\(548\) −19.9119 −0.850596
\(549\) −3.37044 −0.143847
\(550\) 0 0
\(551\) −3.75953 −0.160161
\(552\) 5.73029 0.243897
\(553\) 4.66351 0.198313
\(554\) −48.9669 −2.08040
\(555\) 4.54651 0.192989
\(556\) 16.8405 0.714197
\(557\) 11.8765 0.503225 0.251612 0.967828i \(-0.419039\pi\)
0.251612 + 0.967828i \(0.419039\pi\)
\(558\) −12.1778 −0.515527
\(559\) 16.9174 0.715532
\(560\) 2.93794 0.124151
\(561\) 0 0
\(562\) 42.2530 1.78233
\(563\) 4.83929 0.203952 0.101976 0.994787i \(-0.467484\pi\)
0.101976 + 0.994787i \(0.467484\pi\)
\(564\) 11.6441 0.490304
\(565\) −4.60651 −0.193797
\(566\) 15.5023 0.651608
\(567\) −1.00000 −0.0419961
\(568\) 21.2316 0.890856
\(569\) 6.52993 0.273749 0.136875 0.990588i \(-0.456294\pi\)
0.136875 + 0.990588i \(0.456294\pi\)
\(570\) 6.47214 0.271088
\(571\) −33.8691 −1.41738 −0.708689 0.705521i \(-0.750713\pi\)
−0.708689 + 0.705521i \(0.750713\pi\)
\(572\) 0 0
\(573\) −9.87351 −0.412472
\(574\) −13.2613 −0.553517
\(575\) −17.5061 −0.730054
\(576\) −0.283993 −0.0118331
\(577\) −37.3609 −1.55536 −0.777678 0.628662i \(-0.783603\pi\)
−0.777678 + 0.628662i \(0.783603\pi\)
\(578\) 60.3487 2.51018
\(579\) 6.18113 0.256879
\(580\) 0.408694 0.0169701
\(581\) 3.50970 0.145607
\(582\) 8.50152 0.352399
\(583\) 0 0
\(584\) 3.94378 0.163195
\(585\) −0.911108 −0.0376697
\(586\) −34.7802 −1.43675
\(587\) −2.07974 −0.0858399 −0.0429199 0.999079i \(-0.513666\pi\)
−0.0429199 + 0.999079i \(0.513666\pi\)
\(588\) −1.14077 −0.0470447
\(589\) 42.5385 1.75277
\(590\) −3.50639 −0.144356
\(591\) 19.0720 0.784518
\(592\) 38.3819 1.57749
\(593\) 6.54983 0.268969 0.134485 0.990916i \(-0.457062\pi\)
0.134485 + 0.990916i \(0.457062\pi\)
\(594\) 0 0
\(595\) −4.21508 −0.172802
\(596\) 14.2891 0.585304
\(597\) 11.8479 0.484904
\(598\) −10.3001 −0.421203
\(599\) −4.98187 −0.203554 −0.101777 0.994807i \(-0.532453\pi\)
−0.101777 + 0.994807i \(0.532453\pi\)
\(600\) −7.08377 −0.289194
\(601\) −32.4713 −1.32453 −0.662266 0.749269i \(-0.730405\pi\)
−0.662266 + 0.749269i \(0.730405\pi\)
\(602\) 19.4125 0.791192
\(603\) −2.19138 −0.0892398
\(604\) 0.374827 0.0152515
\(605\) 0 0
\(606\) 2.52835 0.102707
\(607\) 3.13388 0.127200 0.0636002 0.997975i \(-0.479742\pi\)
0.0636002 + 0.997975i \(0.479742\pi\)
\(608\) 35.7848 1.45126
\(609\) 0.607298 0.0246089
\(610\) −3.52373 −0.142672
\(611\) 15.7644 0.637760
\(612\) 8.15095 0.329483
\(613\) 46.8484 1.89219 0.946094 0.323892i \(-0.104991\pi\)
0.946094 + 0.323892i \(0.104991\pi\)
\(614\) −30.2758 −1.22183
\(615\) −4.41434 −0.178003
\(616\) 0 0
\(617\) 26.5924 1.07057 0.535286 0.844671i \(-0.320204\pi\)
0.535286 + 0.844671i \(0.320204\pi\)
\(618\) −31.2217 −1.25592
\(619\) −32.9765 −1.32544 −0.662718 0.748869i \(-0.730597\pi\)
−0.662718 + 0.748869i \(0.730597\pi\)
\(620\) −4.62431 −0.185717
\(621\) −3.76314 −0.151010
\(622\) −12.8194 −0.514011
\(623\) −8.15095 −0.326561
\(624\) −7.69162 −0.307911
\(625\) 19.9009 0.796037
\(626\) 24.4185 0.975959
\(627\) 0 0
\(628\) −3.05828 −0.122039
\(629\) −55.0668 −2.19566
\(630\) −1.04548 −0.0416529
\(631\) 11.1721 0.444754 0.222377 0.974961i \(-0.428618\pi\)
0.222377 + 0.974961i \(0.428618\pi\)
\(632\) 7.10132 0.282475
\(633\) 20.9810 0.833922
\(634\) −12.3281 −0.489611
\(635\) 4.86798 0.193180
\(636\) −8.03640 −0.318664
\(637\) −1.54445 −0.0611932
\(638\) 0 0
\(639\) −13.9430 −0.551576
\(640\) 6.52324 0.257854
\(641\) −41.9153 −1.65556 −0.827778 0.561056i \(-0.810395\pi\)
−0.827778 + 0.561056i \(0.810395\pi\)
\(642\) 12.7601 0.503600
\(643\) 29.0873 1.14709 0.573546 0.819173i \(-0.305567\pi\)
0.573546 + 0.819173i \(0.305567\pi\)
\(644\) −4.29289 −0.169164
\(645\) 6.46189 0.254437
\(646\) −78.3897 −3.08420
\(647\) 1.89804 0.0746196 0.0373098 0.999304i \(-0.488121\pi\)
0.0373098 + 0.999304i \(0.488121\pi\)
\(648\) −1.52274 −0.0598189
\(649\) 0 0
\(650\) 12.7330 0.499428
\(651\) −6.87147 −0.269314
\(652\) −1.01719 −0.0398363
\(653\) 20.2519 0.792517 0.396258 0.918139i \(-0.370308\pi\)
0.396258 + 0.918139i \(0.370308\pi\)
\(654\) 15.7493 0.615848
\(655\) 7.14766 0.279282
\(656\) −37.2661 −1.45500
\(657\) −2.58993 −0.101043
\(658\) 18.0894 0.705197
\(659\) −11.5620 −0.450392 −0.225196 0.974314i \(-0.572302\pi\)
−0.225196 + 0.974314i \(0.572302\pi\)
\(660\) 0 0
\(661\) −32.6894 −1.27147 −0.635735 0.771907i \(-0.719303\pi\)
−0.635735 + 0.771907i \(0.719303\pi\)
\(662\) −52.5816 −2.04364
\(663\) 11.0352 0.428573
\(664\) 5.34436 0.207402
\(665\) 3.65199 0.141618
\(666\) −13.6584 −0.529252
\(667\) 2.28535 0.0884890
\(668\) 13.1885 0.510279
\(669\) −3.38860 −0.131011
\(670\) −2.29104 −0.0885107
\(671\) 0 0
\(672\) −5.78051 −0.222988
\(673\) −20.1249 −0.775758 −0.387879 0.921710i \(-0.626792\pi\)
−0.387879 + 0.921710i \(0.626792\pi\)
\(674\) 34.6850 1.33601
\(675\) 4.65199 0.179055
\(676\) −12.1090 −0.465729
\(677\) 23.5746 0.906045 0.453022 0.891499i \(-0.350346\pi\)
0.453022 + 0.891499i \(0.350346\pi\)
\(678\) 13.8386 0.531469
\(679\) 4.79710 0.184096
\(680\) −6.41848 −0.246137
\(681\) −19.3973 −0.743305
\(682\) 0 0
\(683\) 11.5812 0.443143 0.221572 0.975144i \(-0.428881\pi\)
0.221572 + 0.975144i \(0.428881\pi\)
\(684\) −7.06206 −0.270025
\(685\) −10.2970 −0.393429
\(686\) −1.77222 −0.0676638
\(687\) 9.86405 0.376337
\(688\) 54.5516 2.07976
\(689\) −10.8801 −0.414500
\(690\) −3.93429 −0.149776
\(691\) 43.8164 1.66685 0.833426 0.552630i \(-0.186376\pi\)
0.833426 + 0.552630i \(0.186376\pi\)
\(692\) 10.2547 0.389826
\(693\) 0 0
\(694\) −11.5050 −0.436723
\(695\) 8.70869 0.330340
\(696\) 0.924756 0.0350528
\(697\) 53.4659 2.02517
\(698\) 13.9645 0.528564
\(699\) −7.12619 −0.269537
\(700\) 5.30687 0.200581
\(701\) −22.8165 −0.861769 −0.430884 0.902407i \(-0.641798\pi\)
−0.430884 + 0.902407i \(0.641798\pi\)
\(702\) 2.73710 0.103305
\(703\) 47.7104 1.79943
\(704\) 0 0
\(705\) 6.02147 0.226782
\(706\) −9.47420 −0.356566
\(707\) 1.42666 0.0536549
\(708\) 3.82599 0.143790
\(709\) 39.9746 1.50128 0.750638 0.660713i \(-0.229746\pi\)
0.750638 + 0.660713i \(0.229746\pi\)
\(710\) −14.5771 −0.547070
\(711\) −4.66351 −0.174895
\(712\) −12.4118 −0.465151
\(713\) −25.8583 −0.968402
\(714\) 12.6627 0.473891
\(715\) 0 0
\(716\) −7.01108 −0.262016
\(717\) −22.2894 −0.832414
\(718\) −31.6369 −1.18068
\(719\) −26.2200 −0.977840 −0.488920 0.872329i \(-0.662609\pi\)
−0.488920 + 0.872329i \(0.662609\pi\)
\(720\) −2.93794 −0.109490
\(721\) −17.6172 −0.656100
\(722\) 34.2453 1.27448
\(723\) −14.8753 −0.553217
\(724\) 17.0731 0.634515
\(725\) −2.82514 −0.104923
\(726\) 0 0
\(727\) −22.2366 −0.824708 −0.412354 0.911024i \(-0.635293\pi\)
−0.412354 + 0.911024i \(0.635293\pi\)
\(728\) −2.35179 −0.0871631
\(729\) 1.00000 0.0370370
\(730\) −2.70772 −0.100217
\(731\) −78.2656 −2.89476
\(732\) 3.84491 0.142112
\(733\) −21.1284 −0.780395 −0.390197 0.920731i \(-0.627593\pi\)
−0.390197 + 0.920731i \(0.627593\pi\)
\(734\) −53.0528 −1.95821
\(735\) −0.589926 −0.0217597
\(736\) −21.7529 −0.801822
\(737\) 0 0
\(738\) 13.2613 0.488156
\(739\) 37.5036 1.37959 0.689795 0.724004i \(-0.257701\pi\)
0.689795 + 0.724004i \(0.257701\pi\)
\(740\) −5.18655 −0.190661
\(741\) −9.56103 −0.351233
\(742\) −12.4848 −0.458330
\(743\) −36.1464 −1.32608 −0.663041 0.748583i \(-0.730735\pi\)
−0.663041 + 0.748583i \(0.730735\pi\)
\(744\) −10.4635 −0.383609
\(745\) 7.38929 0.270723
\(746\) 41.6322 1.52426
\(747\) −3.50970 −0.128413
\(748\) 0 0
\(749\) 7.20005 0.263084
\(750\) 10.0910 0.368470
\(751\) −43.5533 −1.58928 −0.794641 0.607080i \(-0.792341\pi\)
−0.794641 + 0.607080i \(0.792341\pi\)
\(752\) 50.8336 1.85371
\(753\) 25.9233 0.944697
\(754\) −1.66224 −0.0605351
\(755\) 0.193833 0.00705431
\(756\) 1.14077 0.0414895
\(757\) −44.4442 −1.61535 −0.807675 0.589628i \(-0.799274\pi\)
−0.807675 + 0.589628i \(0.799274\pi\)
\(758\) −55.7421 −2.02464
\(759\) 0 0
\(760\) 5.56103 0.201720
\(761\) −15.9613 −0.578598 −0.289299 0.957239i \(-0.593422\pi\)
−0.289299 + 0.957239i \(0.593422\pi\)
\(762\) −14.6241 −0.529777
\(763\) 8.88678 0.321723
\(764\) 11.2634 0.407497
\(765\) 4.21508 0.152397
\(766\) −9.97317 −0.360345
\(767\) 5.17985 0.187034
\(768\) −20.1647 −0.727633
\(769\) 5.19899 0.187480 0.0937402 0.995597i \(-0.470118\pi\)
0.0937402 + 0.995597i \(0.470118\pi\)
\(770\) 0 0
\(771\) −0.871474 −0.0313854
\(772\) −7.05127 −0.253781
\(773\) −12.1614 −0.437416 −0.218708 0.975790i \(-0.570184\pi\)
−0.218708 + 0.975790i \(0.570184\pi\)
\(774\) −19.4125 −0.697766
\(775\) 31.9660 1.14825
\(776\) 7.30473 0.262224
\(777\) −7.70693 −0.276484
\(778\) 46.2678 1.65878
\(779\) −46.3234 −1.65971
\(780\) 1.03937 0.0372154
\(781\) 0 0
\(782\) 47.6516 1.70402
\(783\) −0.607298 −0.0217030
\(784\) −4.98018 −0.177864
\(785\) −1.58152 −0.0564470
\(786\) −21.4726 −0.765903
\(787\) −39.6430 −1.41312 −0.706559 0.707654i \(-0.749754\pi\)
−0.706559 + 0.707654i \(0.749754\pi\)
\(788\) −21.7569 −0.775056
\(789\) −26.8873 −0.957212
\(790\) −4.87561 −0.173466
\(791\) 7.80862 0.277643
\(792\) 0 0
\(793\) 5.20546 0.184851
\(794\) 18.4006 0.653011
\(795\) −4.15584 −0.147393
\(796\) −13.5158 −0.479055
\(797\) −35.1118 −1.24372 −0.621862 0.783127i \(-0.713624\pi\)
−0.621862 + 0.783127i \(0.713624\pi\)
\(798\) −10.9711 −0.388373
\(799\) −72.9313 −2.58012
\(800\) 26.8909 0.950736
\(801\) 8.15095 0.288000
\(802\) 6.53731 0.230840
\(803\) 0 0
\(804\) 2.49987 0.0881635
\(805\) −2.21997 −0.0782438
\(806\) 18.8079 0.662481
\(807\) 0.992574 0.0349403
\(808\) 2.17243 0.0764257
\(809\) −41.7873 −1.46916 −0.734581 0.678521i \(-0.762621\pi\)
−0.734581 + 0.678521i \(0.762621\pi\)
\(810\) 1.04548 0.0367344
\(811\) −9.89879 −0.347594 −0.173797 0.984782i \(-0.555604\pi\)
−0.173797 + 0.984782i \(0.555604\pi\)
\(812\) −0.692789 −0.0243121
\(813\) −26.8650 −0.942197
\(814\) 0 0
\(815\) −0.526018 −0.0184256
\(816\) 35.5839 1.24569
\(817\) 67.8100 2.37237
\(818\) 1.91440 0.0669353
\(819\) 1.54445 0.0539673
\(820\) 5.03576 0.175856
\(821\) 16.1936 0.565161 0.282580 0.959244i \(-0.408810\pi\)
0.282580 + 0.959244i \(0.408810\pi\)
\(822\) 30.9337 1.07894
\(823\) −3.86642 −0.134775 −0.0673874 0.997727i \(-0.521466\pi\)
−0.0673874 + 0.997727i \(0.521466\pi\)
\(824\) −26.8265 −0.934545
\(825\) 0 0
\(826\) 5.94378 0.206811
\(827\) −46.8179 −1.62802 −0.814009 0.580852i \(-0.802720\pi\)
−0.814009 + 0.580852i \(0.802720\pi\)
\(828\) 4.29289 0.149188
\(829\) −41.5599 −1.44344 −0.721718 0.692187i \(-0.756647\pi\)
−0.721718 + 0.692187i \(0.756647\pi\)
\(830\) −3.66932 −0.127364
\(831\) 27.6302 0.958481
\(832\) 0.438613 0.0152062
\(833\) 7.14511 0.247563
\(834\) −26.1622 −0.905922
\(835\) 6.82015 0.236021
\(836\) 0 0
\(837\) 6.87147 0.237513
\(838\) 17.8579 0.616892
\(839\) 5.14119 0.177494 0.0887469 0.996054i \(-0.471714\pi\)
0.0887469 + 0.996054i \(0.471714\pi\)
\(840\) −0.898304 −0.0309944
\(841\) −28.6312 −0.987282
\(842\) 21.2236 0.731412
\(843\) −23.8418 −0.821155
\(844\) −23.9346 −0.823864
\(845\) −6.26188 −0.215415
\(846\) −18.0894 −0.621926
\(847\) 0 0
\(848\) −35.0838 −1.20478
\(849\) −8.74735 −0.300208
\(850\) −58.9068 −2.02049
\(851\) −29.0023 −0.994184
\(852\) 15.9058 0.544924
\(853\) 6.69649 0.229283 0.114642 0.993407i \(-0.463428\pi\)
0.114642 + 0.993407i \(0.463428\pi\)
\(854\) 5.97317 0.204398
\(855\) −3.65199 −0.124895
\(856\) 10.9638 0.374735
\(857\) −36.1064 −1.23337 −0.616686 0.787209i \(-0.711525\pi\)
−0.616686 + 0.787209i \(0.711525\pi\)
\(858\) 0 0
\(859\) −37.1034 −1.26595 −0.632976 0.774171i \(-0.718167\pi\)
−0.632976 + 0.774171i \(0.718167\pi\)
\(860\) −7.37155 −0.251368
\(861\) 7.48287 0.255016
\(862\) −0.972147 −0.0331115
\(863\) 38.3724 1.30621 0.653105 0.757267i \(-0.273466\pi\)
0.653105 + 0.757267i \(0.273466\pi\)
\(864\) 5.78051 0.196657
\(865\) 5.30300 0.180307
\(866\) 34.7140 1.17963
\(867\) −34.0526 −1.15649
\(868\) 7.83880 0.266066
\(869\) 0 0
\(870\) −0.634918 −0.0215257
\(871\) 3.38446 0.114678
\(872\) 13.5323 0.458260
\(873\) −4.79710 −0.162357
\(874\) −41.2858 −1.39651
\(875\) 5.69396 0.192491
\(876\) 2.95452 0.0998240
\(877\) −0.811156 −0.0273908 −0.0136954 0.999906i \(-0.504360\pi\)
−0.0136954 + 0.999906i \(0.504360\pi\)
\(878\) 66.2934 2.23729
\(879\) 19.6252 0.661940
\(880\) 0 0
\(881\) −32.5076 −1.09521 −0.547605 0.836737i \(-0.684460\pi\)
−0.547605 + 0.836737i \(0.684460\pi\)
\(882\) 1.77222 0.0596738
\(883\) 48.0995 1.61868 0.809338 0.587343i \(-0.199826\pi\)
0.809338 + 0.587343i \(0.199826\pi\)
\(884\) −12.5887 −0.423404
\(885\) 1.97853 0.0665075
\(886\) 38.0103 1.27698
\(887\) 11.3847 0.382260 0.191130 0.981565i \(-0.438785\pi\)
0.191130 + 0.981565i \(0.438785\pi\)
\(888\) −11.7356 −0.393822
\(889\) −8.25186 −0.276758
\(890\) 8.52166 0.285647
\(891\) 0 0
\(892\) 3.86563 0.129431
\(893\) 63.1884 2.11452
\(894\) −22.1985 −0.742429
\(895\) −3.62562 −0.121191
\(896\) −11.0577 −0.369413
\(897\) 5.81197 0.194056
\(898\) 11.3405 0.378438
\(899\) −4.17303 −0.139178
\(900\) −5.30687 −0.176896
\(901\) 50.3351 1.67690
\(902\) 0 0
\(903\) −10.9537 −0.364517
\(904\) 11.8905 0.395472
\(905\) 8.82895 0.293484
\(906\) −0.582303 −0.0193457
\(907\) 13.4259 0.445799 0.222899 0.974841i \(-0.428448\pi\)
0.222899 + 0.974841i \(0.428448\pi\)
\(908\) 22.1279 0.734340
\(909\) −1.42666 −0.0473192
\(910\) 1.61469 0.0535264
\(911\) 3.39303 0.112416 0.0562080 0.998419i \(-0.482099\pi\)
0.0562080 + 0.998419i \(0.482099\pi\)
\(912\) −30.8303 −1.02089
\(913\) 0 0
\(914\) −65.4712 −2.16560
\(915\) 1.98831 0.0657315
\(916\) −11.2526 −0.371798
\(917\) −12.1162 −0.400112
\(918\) −12.6627 −0.417932
\(919\) −44.5668 −1.47012 −0.735062 0.678000i \(-0.762847\pi\)
−0.735062 + 0.678000i \(0.762847\pi\)
\(920\) −3.38044 −0.111450
\(921\) 17.0835 0.562922
\(922\) 13.1113 0.431799
\(923\) 21.5342 0.708807
\(924\) 0 0
\(925\) 35.8525 1.17882
\(926\) −25.4699 −0.836992
\(927\) 17.6172 0.578626
\(928\) −3.51049 −0.115238
\(929\) −7.15205 −0.234651 −0.117325 0.993094i \(-0.537432\pi\)
−0.117325 + 0.993094i \(0.537432\pi\)
\(930\) 7.18399 0.235572
\(931\) −6.19059 −0.202888
\(932\) 8.12937 0.266287
\(933\) 7.23351 0.236815
\(934\) 64.0883 2.09703
\(935\) 0 0
\(936\) 2.35179 0.0768706
\(937\) −1.30811 −0.0427342 −0.0213671 0.999772i \(-0.506802\pi\)
−0.0213671 + 0.999772i \(0.506802\pi\)
\(938\) 3.88361 0.126804
\(939\) −13.7784 −0.449643
\(940\) −6.86914 −0.224047
\(941\) −22.1225 −0.721174 −0.360587 0.932726i \(-0.617424\pi\)
−0.360587 + 0.932726i \(0.617424\pi\)
\(942\) 4.75113 0.154800
\(943\) 28.1591 0.916987
\(944\) 16.7028 0.543631
\(945\) 0.589926 0.0191903
\(946\) 0 0
\(947\) −17.6842 −0.574658 −0.287329 0.957832i \(-0.592767\pi\)
−0.287329 + 0.957832i \(0.592767\pi\)
\(948\) 5.32002 0.172786
\(949\) 4.00000 0.129845
\(950\) 51.0374 1.65587
\(951\) 6.95629 0.225573
\(952\) 10.8801 0.352627
\(953\) −23.7741 −0.770118 −0.385059 0.922892i \(-0.625819\pi\)
−0.385059 + 0.922892i \(0.625819\pi\)
\(954\) 12.4848 0.404209
\(955\) 5.82464 0.188481
\(956\) 25.4272 0.822374
\(957\) 0 0
\(958\) 28.4003 0.917573
\(959\) 17.4548 0.563644
\(960\) 0.167535 0.00540717
\(961\) 16.2172 0.523134
\(962\) 21.0946 0.680119
\(963\) −7.20005 −0.232018
\(964\) 16.9693 0.546544
\(965\) −3.64641 −0.117382
\(966\) 6.66913 0.214576
\(967\) 21.1829 0.681195 0.340597 0.940209i \(-0.389371\pi\)
0.340597 + 0.940209i \(0.389371\pi\)
\(968\) 0 0
\(969\) 44.2324 1.42095
\(970\) −5.01527 −0.161031
\(971\) −47.4360 −1.52229 −0.761147 0.648579i \(-0.775363\pi\)
−0.761147 + 0.648579i \(0.775363\pi\)
\(972\) −1.14077 −0.0365903
\(973\) −14.7624 −0.473259
\(974\) 2.74800 0.0880515
\(975\) −7.18474 −0.230096
\(976\) 16.7854 0.537288
\(977\) −13.1860 −0.421858 −0.210929 0.977501i \(-0.567649\pi\)
−0.210929 + 0.977501i \(0.567649\pi\)
\(978\) 1.58023 0.0505303
\(979\) 0 0
\(980\) 0.672972 0.0214973
\(981\) −8.88678 −0.283733
\(982\) −0.0587731 −0.00187553
\(983\) −4.07740 −0.130049 −0.0650244 0.997884i \(-0.520713\pi\)
−0.0650244 + 0.997884i \(0.520713\pi\)
\(984\) 11.3945 0.363242
\(985\) −11.2511 −0.358489
\(986\) 7.69004 0.244901
\(987\) −10.2072 −0.324898
\(988\) 10.9070 0.346997
\(989\) −41.2204 −1.31073
\(990\) 0 0
\(991\) −5.49154 −0.174445 −0.0872223 0.996189i \(-0.527799\pi\)
−0.0872223 + 0.996189i \(0.527799\pi\)
\(992\) 39.7206 1.26113
\(993\) 29.6698 0.941544
\(994\) 24.7101 0.783757
\(995\) −6.98940 −0.221579
\(996\) 4.00378 0.126865
\(997\) 22.9725 0.727548 0.363774 0.931487i \(-0.381488\pi\)
0.363774 + 0.931487i \(0.381488\pi\)
\(998\) −47.9057 −1.51643
\(999\) 7.70693 0.243836
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 2541.2.a.bn.1.4 4
3.2 odd 2 7623.2.a.ci.1.1 4
11.5 even 5 231.2.j.f.190.1 yes 8
11.9 even 5 231.2.j.f.169.1 8
11.10 odd 2 2541.2.a.bm.1.1 4
33.5 odd 10 693.2.m.f.190.2 8
33.20 odd 10 693.2.m.f.631.2 8
33.32 even 2 7623.2.a.cl.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.f.169.1 8 11.9 even 5
231.2.j.f.190.1 yes 8 11.5 even 5
693.2.m.f.190.2 8 33.5 odd 10
693.2.m.f.631.2 8 33.20 odd 10
2541.2.a.bm.1.1 4 11.10 odd 2
2541.2.a.bn.1.4 4 1.1 even 1 trivial
7623.2.a.ci.1.1 4 3.2 odd 2
7623.2.a.cl.1.4 4 33.32 even 2