Properties

Label 3381.2.a.be.1.4
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $8$
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] = [3381,2,Mod(1,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, 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("3381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.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: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 15x^{6} + 11x^{5} + 75x^{4} - 35x^{3} - 141x^{2} + 37x + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.818544\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.818544 q^{2} -1.00000 q^{3} -1.32999 q^{4} -3.72041 q^{5} +0.818544 q^{6} +2.72574 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.818544 q^{2} -1.00000 q^{3} -1.32999 q^{4} -3.72041 q^{5} +0.818544 q^{6} +2.72574 q^{8} +1.00000 q^{9} +3.04532 q^{10} +3.70004 q^{11} +1.32999 q^{12} +6.90735 q^{13} +3.72041 q^{15} +0.428834 q^{16} +3.21772 q^{17} -0.818544 q^{18} -6.07700 q^{19} +4.94809 q^{20} -3.02865 q^{22} -1.00000 q^{23} -2.72574 q^{24} +8.84143 q^{25} -5.65397 q^{26} -1.00000 q^{27} -3.33137 q^{29} -3.04532 q^{30} -0.638468 q^{31} -5.80250 q^{32} -3.70004 q^{33} -2.63384 q^{34} -1.32999 q^{36} +7.78032 q^{37} +4.97429 q^{38} -6.90735 q^{39} -10.1409 q^{40} -10.4188 q^{41} +4.34017 q^{43} -4.92100 q^{44} -3.72041 q^{45} +0.818544 q^{46} +6.02770 q^{47} -0.428834 q^{48} -7.23710 q^{50} -3.21772 q^{51} -9.18668 q^{52} +7.27207 q^{53} +0.818544 q^{54} -13.7657 q^{55} +6.07700 q^{57} +2.72687 q^{58} +5.29661 q^{59} -4.94809 q^{60} -8.11466 q^{61} +0.522615 q^{62} +3.89193 q^{64} -25.6981 q^{65} +3.02865 q^{66} -4.01771 q^{67} -4.27952 q^{68} +1.00000 q^{69} -2.92567 q^{71} +2.72574 q^{72} -9.14983 q^{73} -6.36854 q^{74} -8.84143 q^{75} +8.08232 q^{76} +5.65397 q^{78} +8.17142 q^{79} -1.59544 q^{80} +1.00000 q^{81} +8.52824 q^{82} -6.01247 q^{83} -11.9712 q^{85} -3.55262 q^{86} +3.33137 q^{87} +10.0853 q^{88} -14.4303 q^{89} +3.04532 q^{90} +1.32999 q^{92} +0.638468 q^{93} -4.93394 q^{94} +22.6089 q^{95} +5.80250 q^{96} +14.8716 q^{97} +3.70004 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 8 q^{3} + 15 q^{4} - 5 q^{5} - q^{6} + 9 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 8 q^{3} + 15 q^{4} - 5 q^{5} - q^{6} + 9 q^{8} + 8 q^{9} + 3 q^{10} + 10 q^{11} - 15 q^{12} + 6 q^{13} + 5 q^{15} + 13 q^{16} - 21 q^{17} + q^{18} - 5 q^{19} + q^{20} + 18 q^{22} - 8 q^{23} - 9 q^{24} + 27 q^{25} - 3 q^{26} - 8 q^{27} + 2 q^{29} - 3 q^{30} + 13 q^{31} + 29 q^{32} - 10 q^{33} + 19 q^{34} + 15 q^{36} + 13 q^{37} + 6 q^{38} - 6 q^{39} - 7 q^{40} - 16 q^{41} + 15 q^{43} + 24 q^{44} - 5 q^{45} - q^{46} + q^{47} - 13 q^{48} + 16 q^{50} + 21 q^{51} + 19 q^{52} + 3 q^{53} - q^{54} + 10 q^{55} + 5 q^{57} - 40 q^{58} - 26 q^{59} - q^{60} + 14 q^{61} + 14 q^{62} + 49 q^{64} - 3 q^{65} - 18 q^{66} + 38 q^{67} - 43 q^{68} + 8 q^{69} + 9 q^{71} + 9 q^{72} + 6 q^{73} + 32 q^{74} - 27 q^{75} + 14 q^{76} + 3 q^{78} + 23 q^{79} + 17 q^{80} + 8 q^{81} - 20 q^{82} - 30 q^{83} - 37 q^{85} - 28 q^{86} - 2 q^{87} + 86 q^{88} - 12 q^{89} + 3 q^{90} - 15 q^{92} - 13 q^{93} + 45 q^{94} + 16 q^{95} - 29 q^{96} - 14 q^{97} + 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.818544 −0.578798 −0.289399 0.957209i \(-0.593455\pi\)
−0.289399 + 0.957209i \(0.593455\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.32999 −0.664993
\(5\) −3.72041 −1.66382 −0.831908 0.554913i \(-0.812751\pi\)
−0.831908 + 0.554913i \(0.812751\pi\)
\(6\) 0.818544 0.334169
\(7\) 0 0
\(8\) 2.72574 0.963695
\(9\) 1.00000 0.333333
\(10\) 3.04532 0.963014
\(11\) 3.70004 1.11560 0.557802 0.829974i \(-0.311645\pi\)
0.557802 + 0.829974i \(0.311645\pi\)
\(12\) 1.32999 0.383934
\(13\) 6.90735 1.91575 0.957877 0.287179i \(-0.0927174\pi\)
0.957877 + 0.287179i \(0.0927174\pi\)
\(14\) 0 0
\(15\) 3.72041 0.960605
\(16\) 0.428834 0.107208
\(17\) 3.21772 0.780411 0.390206 0.920728i \(-0.372404\pi\)
0.390206 + 0.920728i \(0.372404\pi\)
\(18\) −0.818544 −0.192933
\(19\) −6.07700 −1.39416 −0.697080 0.716994i \(-0.745518\pi\)
−0.697080 + 0.716994i \(0.745518\pi\)
\(20\) 4.94809 1.10643
\(21\) 0 0
\(22\) −3.02865 −0.645710
\(23\) −1.00000 −0.208514
\(24\) −2.72574 −0.556389
\(25\) 8.84143 1.76829
\(26\) −5.65397 −1.10883
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.33137 −0.618619 −0.309310 0.950961i \(-0.600098\pi\)
−0.309310 + 0.950961i \(0.600098\pi\)
\(30\) −3.04532 −0.555996
\(31\) −0.638468 −0.114672 −0.0573362 0.998355i \(-0.518261\pi\)
−0.0573362 + 0.998355i \(0.518261\pi\)
\(32\) −5.80250 −1.02575
\(33\) −3.70004 −0.644094
\(34\) −2.63384 −0.451700
\(35\) 0 0
\(36\) −1.32999 −0.221664
\(37\) 7.78032 1.27908 0.639539 0.768759i \(-0.279125\pi\)
0.639539 + 0.768759i \(0.279125\pi\)
\(38\) 4.97429 0.806937
\(39\) −6.90735 −1.10606
\(40\) −10.1409 −1.60341
\(41\) −10.4188 −1.62714 −0.813571 0.581466i \(-0.802479\pi\)
−0.813571 + 0.581466i \(0.802479\pi\)
\(42\) 0 0
\(43\) 4.34017 0.661870 0.330935 0.943654i \(-0.392636\pi\)
0.330935 + 0.943654i \(0.392636\pi\)
\(44\) −4.92100 −0.741869
\(45\) −3.72041 −0.554606
\(46\) 0.818544 0.120688
\(47\) 6.02770 0.879230 0.439615 0.898186i \(-0.355115\pi\)
0.439615 + 0.898186i \(0.355115\pi\)
\(48\) −0.428834 −0.0618968
\(49\) 0 0
\(50\) −7.23710 −1.02348
\(51\) −3.21772 −0.450571
\(52\) −9.18668 −1.27396
\(53\) 7.27207 0.998896 0.499448 0.866344i \(-0.333536\pi\)
0.499448 + 0.866344i \(0.333536\pi\)
\(54\) 0.818544 0.111390
\(55\) −13.7657 −1.85616
\(56\) 0 0
\(57\) 6.07700 0.804918
\(58\) 2.72687 0.358056
\(59\) 5.29661 0.689560 0.344780 0.938683i \(-0.387953\pi\)
0.344780 + 0.938683i \(0.387953\pi\)
\(60\) −4.94809 −0.638795
\(61\) −8.11466 −1.03898 −0.519488 0.854478i \(-0.673877\pi\)
−0.519488 + 0.854478i \(0.673877\pi\)
\(62\) 0.522615 0.0663721
\(63\) 0 0
\(64\) 3.89193 0.486492
\(65\) −25.6981 −3.18746
\(66\) 3.02865 0.372801
\(67\) −4.01771 −0.490841 −0.245420 0.969417i \(-0.578926\pi\)
−0.245420 + 0.969417i \(0.578926\pi\)
\(68\) −4.27952 −0.518968
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −2.92567 −0.347213 −0.173606 0.984815i \(-0.555542\pi\)
−0.173606 + 0.984815i \(0.555542\pi\)
\(72\) 2.72574 0.321232
\(73\) −9.14983 −1.07091 −0.535453 0.844565i \(-0.679859\pi\)
−0.535453 + 0.844565i \(0.679859\pi\)
\(74\) −6.36854 −0.740327
\(75\) −8.84143 −1.02092
\(76\) 8.08232 0.927106
\(77\) 0 0
\(78\) 5.65397 0.640186
\(79\) 8.17142 0.919357 0.459678 0.888085i \(-0.347965\pi\)
0.459678 + 0.888085i \(0.347965\pi\)
\(80\) −1.59544 −0.178375
\(81\) 1.00000 0.111111
\(82\) 8.52824 0.941786
\(83\) −6.01247 −0.659954 −0.329977 0.943989i \(-0.607041\pi\)
−0.329977 + 0.943989i \(0.607041\pi\)
\(84\) 0 0
\(85\) −11.9712 −1.29846
\(86\) −3.55262 −0.383089
\(87\) 3.33137 0.357160
\(88\) 10.0853 1.07510
\(89\) −14.4303 −1.52961 −0.764805 0.644262i \(-0.777165\pi\)
−0.764805 + 0.644262i \(0.777165\pi\)
\(90\) 3.04532 0.321005
\(91\) 0 0
\(92\) 1.32999 0.138661
\(93\) 0.638468 0.0662061
\(94\) −4.93394 −0.508897
\(95\) 22.6089 2.31963
\(96\) 5.80250 0.592215
\(97\) 14.8716 1.50998 0.754991 0.655735i \(-0.227641\pi\)
0.754991 + 0.655735i \(0.227641\pi\)
\(98\) 0 0
\(99\) 3.70004 0.371868
\(100\) −11.7590 −1.17590
\(101\) −4.20008 −0.417924 −0.208962 0.977924i \(-0.567008\pi\)
−0.208962 + 0.977924i \(0.567008\pi\)
\(102\) 2.63384 0.260789
\(103\) 17.4541 1.71980 0.859900 0.510462i \(-0.170526\pi\)
0.859900 + 0.510462i \(0.170526\pi\)
\(104\) 18.8276 1.84620
\(105\) 0 0
\(106\) −5.95251 −0.578159
\(107\) −0.974217 −0.0941811 −0.0470905 0.998891i \(-0.514995\pi\)
−0.0470905 + 0.998891i \(0.514995\pi\)
\(108\) 1.32999 0.127978
\(109\) 0.186541 0.0178674 0.00893371 0.999960i \(-0.497156\pi\)
0.00893371 + 0.999960i \(0.497156\pi\)
\(110\) 11.2678 1.07434
\(111\) −7.78032 −0.738476
\(112\) 0 0
\(113\) 2.70857 0.254801 0.127400 0.991851i \(-0.459337\pi\)
0.127400 + 0.991851i \(0.459337\pi\)
\(114\) −4.97429 −0.465885
\(115\) 3.72041 0.346930
\(116\) 4.43067 0.411377
\(117\) 6.90735 0.638585
\(118\) −4.33551 −0.399116
\(119\) 0 0
\(120\) 10.1409 0.925730
\(121\) 2.69030 0.244573
\(122\) 6.64220 0.601357
\(123\) 10.4188 0.939431
\(124\) 0.849154 0.0762563
\(125\) −14.2917 −1.27829
\(126\) 0 0
\(127\) 6.67778 0.592557 0.296278 0.955102i \(-0.404254\pi\)
0.296278 + 0.955102i \(0.404254\pi\)
\(128\) 8.41928 0.744166
\(129\) −4.34017 −0.382131
\(130\) 21.0351 1.84490
\(131\) −7.70604 −0.673280 −0.336640 0.941633i \(-0.609291\pi\)
−0.336640 + 0.941633i \(0.609291\pi\)
\(132\) 4.92100 0.428318
\(133\) 0 0
\(134\) 3.28867 0.284098
\(135\) 3.72041 0.320202
\(136\) 8.77066 0.752078
\(137\) −20.3951 −1.74247 −0.871234 0.490867i \(-0.836680\pi\)
−0.871234 + 0.490867i \(0.836680\pi\)
\(138\) −0.818544 −0.0696791
\(139\) 12.1820 1.03326 0.516632 0.856208i \(-0.327186\pi\)
0.516632 + 0.856208i \(0.327186\pi\)
\(140\) 0 0
\(141\) −6.02770 −0.507624
\(142\) 2.39479 0.200966
\(143\) 25.5575 2.13722
\(144\) 0.428834 0.0357361
\(145\) 12.3940 1.02927
\(146\) 7.48953 0.619838
\(147\) 0 0
\(148\) −10.3477 −0.850577
\(149\) −1.00394 −0.0822459 −0.0411229 0.999154i \(-0.513094\pi\)
−0.0411229 + 0.999154i \(0.513094\pi\)
\(150\) 7.23710 0.590907
\(151\) −2.71063 −0.220588 −0.110294 0.993899i \(-0.535179\pi\)
−0.110294 + 0.993899i \(0.535179\pi\)
\(152\) −16.5643 −1.34354
\(153\) 3.21772 0.260137
\(154\) 0 0
\(155\) 2.37536 0.190794
\(156\) 9.18668 0.735523
\(157\) −3.02533 −0.241448 −0.120724 0.992686i \(-0.538522\pi\)
−0.120724 + 0.992686i \(0.538522\pi\)
\(158\) −6.68867 −0.532122
\(159\) −7.27207 −0.576713
\(160\) 21.5877 1.70665
\(161\) 0 0
\(162\) −0.818544 −0.0643109
\(163\) 7.68199 0.601700 0.300850 0.953671i \(-0.402730\pi\)
0.300850 + 0.953671i \(0.402730\pi\)
\(164\) 13.8568 1.08204
\(165\) 13.7657 1.07166
\(166\) 4.92147 0.381980
\(167\) −4.58319 −0.354658 −0.177329 0.984152i \(-0.556746\pi\)
−0.177329 + 0.984152i \(0.556746\pi\)
\(168\) 0 0
\(169\) 34.7115 2.67011
\(170\) 9.79897 0.751547
\(171\) −6.07700 −0.464720
\(172\) −5.77237 −0.440139
\(173\) 20.0190 1.52201 0.761006 0.648744i \(-0.224706\pi\)
0.761006 + 0.648744i \(0.224706\pi\)
\(174\) −2.72687 −0.206723
\(175\) 0 0
\(176\) 1.58670 0.119602
\(177\) −5.29661 −0.398118
\(178\) 11.8118 0.885336
\(179\) −4.71460 −0.352385 −0.176193 0.984356i \(-0.556378\pi\)
−0.176193 + 0.984356i \(0.556378\pi\)
\(180\) 4.94809 0.368809
\(181\) 1.17108 0.0870460 0.0435230 0.999052i \(-0.486142\pi\)
0.0435230 + 0.999052i \(0.486142\pi\)
\(182\) 0 0
\(183\) 8.11466 0.599853
\(184\) −2.72574 −0.200944
\(185\) −28.9460 −2.12815
\(186\) −0.522615 −0.0383200
\(187\) 11.9057 0.870630
\(188\) −8.01676 −0.584682
\(189\) 0 0
\(190\) −18.5064 −1.34259
\(191\) 24.1866 1.75008 0.875039 0.484052i \(-0.160836\pi\)
0.875039 + 0.484052i \(0.160836\pi\)
\(192\) −3.89193 −0.280876
\(193\) 12.3253 0.887194 0.443597 0.896226i \(-0.353702\pi\)
0.443597 + 0.896226i \(0.353702\pi\)
\(194\) −12.1731 −0.873975
\(195\) 25.6981 1.84028
\(196\) 0 0
\(197\) −2.76210 −0.196792 −0.0983958 0.995147i \(-0.531371\pi\)
−0.0983958 + 0.995147i \(0.531371\pi\)
\(198\) −3.02865 −0.215237
\(199\) −16.5812 −1.17541 −0.587704 0.809076i \(-0.699968\pi\)
−0.587704 + 0.809076i \(0.699968\pi\)
\(200\) 24.0994 1.70409
\(201\) 4.01771 0.283387
\(202\) 3.43795 0.241893
\(203\) 0 0
\(204\) 4.27952 0.299626
\(205\) 38.7621 2.70727
\(206\) −14.2869 −0.995417
\(207\) −1.00000 −0.0695048
\(208\) 2.96210 0.205385
\(209\) −22.4851 −1.55533
\(210\) 0 0
\(211\) 2.75812 0.189877 0.0949386 0.995483i \(-0.469735\pi\)
0.0949386 + 0.995483i \(0.469735\pi\)
\(212\) −9.67175 −0.664259
\(213\) 2.92567 0.200463
\(214\) 0.797439 0.0545118
\(215\) −16.1472 −1.10123
\(216\) −2.72574 −0.185463
\(217\) 0 0
\(218\) −0.152692 −0.0103416
\(219\) 9.14983 0.618288
\(220\) 18.3081 1.23433
\(221\) 22.2259 1.49508
\(222\) 6.36854 0.427428
\(223\) 25.8192 1.72898 0.864490 0.502650i \(-0.167642\pi\)
0.864490 + 0.502650i \(0.167642\pi\)
\(224\) 0 0
\(225\) 8.84143 0.589429
\(226\) −2.21708 −0.147478
\(227\) −8.37096 −0.555600 −0.277800 0.960639i \(-0.589605\pi\)
−0.277800 + 0.960639i \(0.589605\pi\)
\(228\) −8.08232 −0.535265
\(229\) −19.5187 −1.28983 −0.644915 0.764254i \(-0.723107\pi\)
−0.644915 + 0.764254i \(0.723107\pi\)
\(230\) −3.04532 −0.200802
\(231\) 0 0
\(232\) −9.08044 −0.596160
\(233\) 6.25868 0.410020 0.205010 0.978760i \(-0.434277\pi\)
0.205010 + 0.978760i \(0.434277\pi\)
\(234\) −5.65397 −0.369612
\(235\) −22.4255 −1.46288
\(236\) −7.04442 −0.458553
\(237\) −8.17142 −0.530791
\(238\) 0 0
\(239\) −4.53876 −0.293588 −0.146794 0.989167i \(-0.546895\pi\)
−0.146794 + 0.989167i \(0.546895\pi\)
\(240\) 1.59544 0.102985
\(241\) 19.6771 1.26751 0.633756 0.773533i \(-0.281512\pi\)
0.633756 + 0.773533i \(0.281512\pi\)
\(242\) −2.20213 −0.141558
\(243\) −1.00000 −0.0641500
\(244\) 10.7924 0.690911
\(245\) 0 0
\(246\) −8.52824 −0.543741
\(247\) −41.9760 −2.67087
\(248\) −1.74030 −0.110509
\(249\) 6.01247 0.381025
\(250\) 11.6984 0.739869
\(251\) 10.3015 0.650226 0.325113 0.945675i \(-0.394598\pi\)
0.325113 + 0.945675i \(0.394598\pi\)
\(252\) 0 0
\(253\) −3.70004 −0.232620
\(254\) −5.46606 −0.342971
\(255\) 11.9712 0.749667
\(256\) −14.6754 −0.917214
\(257\) −15.3148 −0.955312 −0.477656 0.878547i \(-0.658513\pi\)
−0.477656 + 0.878547i \(0.658513\pi\)
\(258\) 3.55262 0.221177
\(259\) 0 0
\(260\) 34.1782 2.11964
\(261\) −3.33137 −0.206206
\(262\) 6.30773 0.389693
\(263\) −5.31749 −0.327891 −0.163945 0.986469i \(-0.552422\pi\)
−0.163945 + 0.986469i \(0.552422\pi\)
\(264\) −10.0853 −0.620710
\(265\) −27.0551 −1.66198
\(266\) 0 0
\(267\) 14.4303 0.883121
\(268\) 5.34349 0.326406
\(269\) 1.76465 0.107592 0.0537962 0.998552i \(-0.482868\pi\)
0.0537962 + 0.998552i \(0.482868\pi\)
\(270\) −3.04532 −0.185332
\(271\) −16.6166 −1.00938 −0.504692 0.863300i \(-0.668394\pi\)
−0.504692 + 0.863300i \(0.668394\pi\)
\(272\) 1.37987 0.0836666
\(273\) 0 0
\(274\) 16.6943 1.00854
\(275\) 32.7136 1.97271
\(276\) −1.32999 −0.0800557
\(277\) 1.95295 0.117342 0.0586708 0.998277i \(-0.481314\pi\)
0.0586708 + 0.998277i \(0.481314\pi\)
\(278\) −9.97150 −0.598051
\(279\) −0.638468 −0.0382241
\(280\) 0 0
\(281\) 13.1183 0.782575 0.391287 0.920268i \(-0.372030\pi\)
0.391287 + 0.920268i \(0.372030\pi\)
\(282\) 4.93394 0.293812
\(283\) −0.979872 −0.0582473 −0.0291237 0.999576i \(-0.509272\pi\)
−0.0291237 + 0.999576i \(0.509272\pi\)
\(284\) 3.89110 0.230894
\(285\) −22.6089 −1.33924
\(286\) −20.9199 −1.23702
\(287\) 0 0
\(288\) −5.80250 −0.341916
\(289\) −6.64630 −0.390959
\(290\) −10.1451 −0.595739
\(291\) −14.8716 −0.871789
\(292\) 12.1691 0.712145
\(293\) 22.4505 1.31157 0.655786 0.754947i \(-0.272337\pi\)
0.655786 + 0.754947i \(0.272337\pi\)
\(294\) 0 0
\(295\) −19.7056 −1.14730
\(296\) 21.2071 1.23264
\(297\) −3.70004 −0.214698
\(298\) 0.821768 0.0476038
\(299\) −6.90735 −0.399462
\(300\) 11.7590 0.678905
\(301\) 0 0
\(302\) 2.21877 0.127676
\(303\) 4.20008 0.241288
\(304\) −2.60602 −0.149466
\(305\) 30.1898 1.72866
\(306\) −2.63384 −0.150567
\(307\) −29.2019 −1.66664 −0.833322 0.552789i \(-0.813564\pi\)
−0.833322 + 0.552789i \(0.813564\pi\)
\(308\) 0 0
\(309\) −17.4541 −0.992927
\(310\) −1.94434 −0.110431
\(311\) −1.79193 −0.101611 −0.0508055 0.998709i \(-0.516179\pi\)
−0.0508055 + 0.998709i \(0.516179\pi\)
\(312\) −18.8276 −1.06591
\(313\) 8.83165 0.499194 0.249597 0.968350i \(-0.419702\pi\)
0.249597 + 0.968350i \(0.419702\pi\)
\(314\) 2.47637 0.139750
\(315\) 0 0
\(316\) −10.8679 −0.611366
\(317\) −21.5495 −1.21034 −0.605170 0.796096i \(-0.706895\pi\)
−0.605170 + 0.796096i \(0.706895\pi\)
\(318\) 5.95251 0.333800
\(319\) −12.3262 −0.690134
\(320\) −14.4796 −0.809433
\(321\) 0.974217 0.0543755
\(322\) 0 0
\(323\) −19.5541 −1.08802
\(324\) −1.32999 −0.0738881
\(325\) 61.0708 3.38760
\(326\) −6.28805 −0.348263
\(327\) −0.186541 −0.0103158
\(328\) −28.3989 −1.56807
\(329\) 0 0
\(330\) −11.2678 −0.620272
\(331\) 23.1086 1.27016 0.635082 0.772444i \(-0.280966\pi\)
0.635082 + 0.772444i \(0.280966\pi\)
\(332\) 7.99650 0.438865
\(333\) 7.78032 0.426359
\(334\) 3.75154 0.205275
\(335\) 14.9475 0.816669
\(336\) 0 0
\(337\) 4.83315 0.263279 0.131639 0.991298i \(-0.457976\pi\)
0.131639 + 0.991298i \(0.457976\pi\)
\(338\) −28.4129 −1.54546
\(339\) −2.70857 −0.147109
\(340\) 15.9215 0.863467
\(341\) −2.36236 −0.127929
\(342\) 4.97429 0.268979
\(343\) 0 0
\(344\) 11.8302 0.637841
\(345\) −3.72041 −0.200300
\(346\) −16.3864 −0.880938
\(347\) 2.31584 0.124321 0.0621605 0.998066i \(-0.480201\pi\)
0.0621605 + 0.998066i \(0.480201\pi\)
\(348\) −4.43067 −0.237509
\(349\) −2.05541 −0.110024 −0.0550119 0.998486i \(-0.517520\pi\)
−0.0550119 + 0.998486i \(0.517520\pi\)
\(350\) 0 0
\(351\) −6.90735 −0.368687
\(352\) −21.4695 −1.14433
\(353\) −18.5639 −0.988058 −0.494029 0.869445i \(-0.664476\pi\)
−0.494029 + 0.869445i \(0.664476\pi\)
\(354\) 4.33551 0.230430
\(355\) 10.8847 0.577698
\(356\) 19.1921 1.01718
\(357\) 0 0
\(358\) 3.85910 0.203960
\(359\) −7.27408 −0.383911 −0.191956 0.981404i \(-0.561483\pi\)
−0.191956 + 0.981404i \(0.561483\pi\)
\(360\) −10.1409 −0.534470
\(361\) 17.9299 0.943681
\(362\) −0.958584 −0.0503821
\(363\) −2.69030 −0.141204
\(364\) 0 0
\(365\) 34.0411 1.78179
\(366\) −6.64220 −0.347194
\(367\) 20.2397 1.05651 0.528253 0.849087i \(-0.322847\pi\)
0.528253 + 0.849087i \(0.322847\pi\)
\(368\) −0.428834 −0.0223545
\(369\) −10.4188 −0.542381
\(370\) 23.6935 1.23177
\(371\) 0 0
\(372\) −0.849154 −0.0440266
\(373\) 13.3237 0.689876 0.344938 0.938626i \(-0.387900\pi\)
0.344938 + 0.938626i \(0.387900\pi\)
\(374\) −9.74533 −0.503919
\(375\) 14.2917 0.738019
\(376\) 16.4299 0.847310
\(377\) −23.0109 −1.18512
\(378\) 0 0
\(379\) 9.15684 0.470355 0.235178 0.971952i \(-0.424433\pi\)
0.235178 + 0.971952i \(0.424433\pi\)
\(380\) −30.0695 −1.54253
\(381\) −6.67778 −0.342113
\(382\) −19.7978 −1.01294
\(383\) −8.98554 −0.459140 −0.229570 0.973292i \(-0.573732\pi\)
−0.229570 + 0.973292i \(0.573732\pi\)
\(384\) −8.41928 −0.429645
\(385\) 0 0
\(386\) −10.0888 −0.513506
\(387\) 4.34017 0.220623
\(388\) −19.7790 −1.00413
\(389\) −16.3724 −0.830115 −0.415057 0.909795i \(-0.636238\pi\)
−0.415057 + 0.909795i \(0.636238\pi\)
\(390\) −21.0351 −1.06515
\(391\) −3.21772 −0.162727
\(392\) 0 0
\(393\) 7.70604 0.388718
\(394\) 2.26090 0.113903
\(395\) −30.4010 −1.52964
\(396\) −4.92100 −0.247290
\(397\) −9.68491 −0.486072 −0.243036 0.970017i \(-0.578143\pi\)
−0.243036 + 0.970017i \(0.578143\pi\)
\(398\) 13.5724 0.680323
\(399\) 0 0
\(400\) 3.79150 0.189575
\(401\) 11.4662 0.572593 0.286297 0.958141i \(-0.407576\pi\)
0.286297 + 0.958141i \(0.407576\pi\)
\(402\) −3.28867 −0.164024
\(403\) −4.41012 −0.219684
\(404\) 5.58605 0.277916
\(405\) −3.72041 −0.184869
\(406\) 0 0
\(407\) 28.7875 1.42694
\(408\) −8.77066 −0.434212
\(409\) 38.6532 1.91128 0.955639 0.294541i \(-0.0951666\pi\)
0.955639 + 0.294541i \(0.0951666\pi\)
\(410\) −31.7285 −1.56696
\(411\) 20.3951 1.00601
\(412\) −23.2137 −1.14365
\(413\) 0 0
\(414\) 0.818544 0.0402292
\(415\) 22.3688 1.09804
\(416\) −40.0799 −1.96508
\(417\) −12.1820 −0.596555
\(418\) 18.4051 0.900222
\(419\) 6.70860 0.327736 0.163868 0.986482i \(-0.447603\pi\)
0.163868 + 0.986482i \(0.447603\pi\)
\(420\) 0 0
\(421\) −34.0515 −1.65957 −0.829783 0.558086i \(-0.811536\pi\)
−0.829783 + 0.558086i \(0.811536\pi\)
\(422\) −2.25765 −0.109900
\(423\) 6.02770 0.293077
\(424\) 19.8218 0.962630
\(425\) 28.4492 1.37999
\(426\) −2.39479 −0.116028
\(427\) 0 0
\(428\) 1.29569 0.0626298
\(429\) −25.5575 −1.23393
\(430\) 13.2172 0.637390
\(431\) 7.46897 0.359768 0.179884 0.983688i \(-0.442428\pi\)
0.179884 + 0.983688i \(0.442428\pi\)
\(432\) −0.428834 −0.0206323
\(433\) 25.8698 1.24322 0.621611 0.783326i \(-0.286478\pi\)
0.621611 + 0.783326i \(0.286478\pi\)
\(434\) 0 0
\(435\) −12.3940 −0.594249
\(436\) −0.248097 −0.0118817
\(437\) 6.07700 0.290702
\(438\) −7.48953 −0.357864
\(439\) 15.9723 0.762318 0.381159 0.924509i \(-0.375525\pi\)
0.381159 + 0.924509i \(0.375525\pi\)
\(440\) −37.5216 −1.78877
\(441\) 0 0
\(442\) −18.1929 −0.865347
\(443\) 30.6283 1.45520 0.727598 0.686004i \(-0.240637\pi\)
0.727598 + 0.686004i \(0.240637\pi\)
\(444\) 10.3477 0.491081
\(445\) 53.6867 2.54499
\(446\) −21.1341 −1.00073
\(447\) 1.00394 0.0474847
\(448\) 0 0
\(449\) −7.12584 −0.336289 −0.168144 0.985762i \(-0.553778\pi\)
−0.168144 + 0.985762i \(0.553778\pi\)
\(450\) −7.23710 −0.341160
\(451\) −38.5500 −1.81525
\(452\) −3.60236 −0.169441
\(453\) 2.71063 0.127357
\(454\) 6.85200 0.321580
\(455\) 0 0
\(456\) 16.5643 0.775695
\(457\) 20.6851 0.967609 0.483805 0.875176i \(-0.339254\pi\)
0.483805 + 0.875176i \(0.339254\pi\)
\(458\) 15.9769 0.746551
\(459\) −3.21772 −0.150190
\(460\) −4.94809 −0.230706
\(461\) −12.1611 −0.566401 −0.283200 0.959061i \(-0.591396\pi\)
−0.283200 + 0.959061i \(0.591396\pi\)
\(462\) 0 0
\(463\) −1.96022 −0.0910992 −0.0455496 0.998962i \(-0.514504\pi\)
−0.0455496 + 0.998962i \(0.514504\pi\)
\(464\) −1.42860 −0.0663212
\(465\) −2.37536 −0.110155
\(466\) −5.12301 −0.237319
\(467\) 39.3922 1.82285 0.911427 0.411463i \(-0.134982\pi\)
0.911427 + 0.411463i \(0.134982\pi\)
\(468\) −9.18668 −0.424654
\(469\) 0 0
\(470\) 18.3563 0.846711
\(471\) 3.02533 0.139400
\(472\) 14.4372 0.664525
\(473\) 16.0588 0.738385
\(474\) 6.68867 0.307221
\(475\) −53.7294 −2.46527
\(476\) 0 0
\(477\) 7.27207 0.332965
\(478\) 3.71517 0.169928
\(479\) 2.82732 0.129183 0.0645917 0.997912i \(-0.479425\pi\)
0.0645917 + 0.997912i \(0.479425\pi\)
\(480\) −21.5877 −0.985337
\(481\) 53.7414 2.45040
\(482\) −16.1066 −0.733634
\(483\) 0 0
\(484\) −3.57806 −0.162639
\(485\) −55.3284 −2.51233
\(486\) 0.818544 0.0371299
\(487\) −24.7848 −1.12311 −0.561554 0.827440i \(-0.689796\pi\)
−0.561554 + 0.827440i \(0.689796\pi\)
\(488\) −22.1184 −1.00125
\(489\) −7.68199 −0.347392
\(490\) 0 0
\(491\) 13.1356 0.592802 0.296401 0.955064i \(-0.404214\pi\)
0.296401 + 0.955064i \(0.404214\pi\)
\(492\) −13.8568 −0.624715
\(493\) −10.7194 −0.482777
\(494\) 34.3592 1.54589
\(495\) −13.7657 −0.618720
\(496\) −0.273797 −0.0122938
\(497\) 0 0
\(498\) −4.92147 −0.220536
\(499\) 16.4810 0.737790 0.368895 0.929471i \(-0.379736\pi\)
0.368895 + 0.929471i \(0.379736\pi\)
\(500\) 19.0077 0.850051
\(501\) 4.58319 0.204762
\(502\) −8.43224 −0.376349
\(503\) −17.3522 −0.773696 −0.386848 0.922144i \(-0.626436\pi\)
−0.386848 + 0.922144i \(0.626436\pi\)
\(504\) 0 0
\(505\) 15.6260 0.695348
\(506\) 3.02865 0.134640
\(507\) −34.7115 −1.54159
\(508\) −8.88135 −0.394046
\(509\) 34.8099 1.54292 0.771461 0.636277i \(-0.219527\pi\)
0.771461 + 0.636277i \(0.219527\pi\)
\(510\) −9.79897 −0.433906
\(511\) 0 0
\(512\) −4.82609 −0.213285
\(513\) 6.07700 0.268306
\(514\) 12.5358 0.552933
\(515\) −64.9362 −2.86143
\(516\) 5.77237 0.254114
\(517\) 22.3027 0.980873
\(518\) 0 0
\(519\) −20.0190 −0.878735
\(520\) −70.0465 −3.07174
\(521\) −16.9148 −0.741052 −0.370526 0.928822i \(-0.620823\pi\)
−0.370526 + 0.928822i \(0.620823\pi\)
\(522\) 2.72687 0.119352
\(523\) 29.0471 1.27014 0.635071 0.772454i \(-0.280971\pi\)
0.635071 + 0.772454i \(0.280971\pi\)
\(524\) 10.2489 0.447726
\(525\) 0 0
\(526\) 4.35260 0.189783
\(527\) −2.05441 −0.0894915
\(528\) −1.58670 −0.0690524
\(529\) 1.00000 0.0434783
\(530\) 22.1458 0.961950
\(531\) 5.29661 0.229853
\(532\) 0 0
\(533\) −71.9662 −3.11720
\(534\) −11.8118 −0.511149
\(535\) 3.62448 0.156700
\(536\) −10.9512 −0.473021
\(537\) 4.71460 0.203450
\(538\) −1.44444 −0.0622743
\(539\) 0 0
\(540\) −4.94809 −0.212932
\(541\) −3.79565 −0.163188 −0.0815938 0.996666i \(-0.526001\pi\)
−0.0815938 + 0.996666i \(0.526001\pi\)
\(542\) 13.6014 0.584229
\(543\) −1.17108 −0.0502560
\(544\) −18.6708 −0.800504
\(545\) −0.694010 −0.0297281
\(546\) 0 0
\(547\) 3.31002 0.141526 0.0707630 0.997493i \(-0.477457\pi\)
0.0707630 + 0.997493i \(0.477457\pi\)
\(548\) 27.1252 1.15873
\(549\) −8.11466 −0.346325
\(550\) −26.7776 −1.14180
\(551\) 20.2447 0.862454
\(552\) 2.72574 0.116015
\(553\) 0 0
\(554\) −1.59858 −0.0679171
\(555\) 28.9460 1.22869
\(556\) −16.2019 −0.687113
\(557\) −7.75267 −0.328491 −0.164246 0.986419i \(-0.552519\pi\)
−0.164246 + 0.986419i \(0.552519\pi\)
\(558\) 0.522615 0.0221240
\(559\) 29.9791 1.26798
\(560\) 0 0
\(561\) −11.9057 −0.502658
\(562\) −10.7379 −0.452953
\(563\) 16.2180 0.683508 0.341754 0.939789i \(-0.388979\pi\)
0.341754 + 0.939789i \(0.388979\pi\)
\(564\) 8.01676 0.337566
\(565\) −10.0770 −0.423942
\(566\) 0.802068 0.0337134
\(567\) 0 0
\(568\) −7.97461 −0.334607
\(569\) −24.0539 −1.00839 −0.504196 0.863589i \(-0.668211\pi\)
−0.504196 + 0.863589i \(0.668211\pi\)
\(570\) 18.5064 0.775147
\(571\) 43.9333 1.83855 0.919276 0.393613i \(-0.128775\pi\)
0.919276 + 0.393613i \(0.128775\pi\)
\(572\) −33.9911 −1.42124
\(573\) −24.1866 −1.01041
\(574\) 0 0
\(575\) −8.84143 −0.368713
\(576\) 3.89193 0.162164
\(577\) 18.0483 0.751359 0.375679 0.926750i \(-0.377409\pi\)
0.375679 + 0.926750i \(0.377409\pi\)
\(578\) 5.44029 0.226286
\(579\) −12.3253 −0.512222
\(580\) −16.4839 −0.684456
\(581\) 0 0
\(582\) 12.1731 0.504590
\(583\) 26.9070 1.11437
\(584\) −24.9400 −1.03203
\(585\) −25.6981 −1.06249
\(586\) −18.3767 −0.759136
\(587\) 0.826642 0.0341192 0.0170596 0.999854i \(-0.494569\pi\)
0.0170596 + 0.999854i \(0.494569\pi\)
\(588\) 0 0
\(589\) 3.87997 0.159871
\(590\) 16.1299 0.664056
\(591\) 2.76210 0.113618
\(592\) 3.33646 0.137128
\(593\) 0.685463 0.0281486 0.0140743 0.999901i \(-0.495520\pi\)
0.0140743 + 0.999901i \(0.495520\pi\)
\(594\) 3.02865 0.124267
\(595\) 0 0
\(596\) 1.33522 0.0546929
\(597\) 16.5812 0.678622
\(598\) 5.65397 0.231208
\(599\) −27.3692 −1.11827 −0.559137 0.829075i \(-0.688867\pi\)
−0.559137 + 0.829075i \(0.688867\pi\)
\(600\) −24.0994 −0.983855
\(601\) 10.3969 0.424096 0.212048 0.977259i \(-0.431987\pi\)
0.212048 + 0.977259i \(0.431987\pi\)
\(602\) 0 0
\(603\) −4.01771 −0.163614
\(604\) 3.60510 0.146689
\(605\) −10.0090 −0.406925
\(606\) −3.43795 −0.139657
\(607\) −15.8582 −0.643663 −0.321832 0.946797i \(-0.604299\pi\)
−0.321832 + 0.946797i \(0.604299\pi\)
\(608\) 35.2618 1.43005
\(609\) 0 0
\(610\) −24.7117 −1.00055
\(611\) 41.6354 1.68439
\(612\) −4.27952 −0.172989
\(613\) 8.49341 0.343046 0.171523 0.985180i \(-0.445131\pi\)
0.171523 + 0.985180i \(0.445131\pi\)
\(614\) 23.9031 0.964650
\(615\) −38.7621 −1.56304
\(616\) 0 0
\(617\) −20.0404 −0.806797 −0.403399 0.915024i \(-0.632171\pi\)
−0.403399 + 0.915024i \(0.632171\pi\)
\(618\) 14.2869 0.574704
\(619\) 27.3390 1.09885 0.549424 0.835544i \(-0.314847\pi\)
0.549424 + 0.835544i \(0.314847\pi\)
\(620\) −3.15920 −0.126876
\(621\) 1.00000 0.0401286
\(622\) 1.46677 0.0588123
\(623\) 0 0
\(624\) −2.96210 −0.118579
\(625\) 8.96370 0.358548
\(626\) −7.22909 −0.288933
\(627\) 22.4851 0.897970
\(628\) 4.02365 0.160561
\(629\) 25.0349 0.998206
\(630\) 0 0
\(631\) −12.7286 −0.506717 −0.253358 0.967373i \(-0.581535\pi\)
−0.253358 + 0.967373i \(0.581535\pi\)
\(632\) 22.2732 0.885979
\(633\) −2.75812 −0.109626
\(634\) 17.6392 0.700543
\(635\) −24.8441 −0.985906
\(636\) 9.67175 0.383510
\(637\) 0 0
\(638\) 10.0895 0.399448
\(639\) −2.92567 −0.115738
\(640\) −31.3231 −1.23816
\(641\) 14.0195 0.553738 0.276869 0.960908i \(-0.410703\pi\)
0.276869 + 0.960908i \(0.410703\pi\)
\(642\) −0.797439 −0.0314724
\(643\) −10.2819 −0.405477 −0.202738 0.979233i \(-0.564984\pi\)
−0.202738 + 0.979233i \(0.564984\pi\)
\(644\) 0 0
\(645\) 16.1472 0.635796
\(646\) 16.0059 0.629742
\(647\) −35.5195 −1.39641 −0.698207 0.715896i \(-0.746018\pi\)
−0.698207 + 0.715896i \(0.746018\pi\)
\(648\) 2.72574 0.107077
\(649\) 19.5977 0.769276
\(650\) −49.9892 −1.96074
\(651\) 0 0
\(652\) −10.2169 −0.400126
\(653\) 27.4136 1.07278 0.536389 0.843971i \(-0.319788\pi\)
0.536389 + 0.843971i \(0.319788\pi\)
\(654\) 0.152692 0.00597074
\(655\) 28.6696 1.12021
\(656\) −4.46793 −0.174443
\(657\) −9.14983 −0.356969
\(658\) 0 0
\(659\) 32.7484 1.27570 0.637848 0.770162i \(-0.279825\pi\)
0.637848 + 0.770162i \(0.279825\pi\)
\(660\) −18.3081 −0.712643
\(661\) 38.3045 1.48987 0.744936 0.667136i \(-0.232480\pi\)
0.744936 + 0.667136i \(0.232480\pi\)
\(662\) −18.9154 −0.735169
\(663\) −22.2259 −0.863182
\(664\) −16.3884 −0.635994
\(665\) 0 0
\(666\) −6.36854 −0.246776
\(667\) 3.33137 0.128991
\(668\) 6.09557 0.235845
\(669\) −25.8192 −0.998227
\(670\) −12.2352 −0.472686
\(671\) −30.0246 −1.15909
\(672\) 0 0
\(673\) −0.487689 −0.0187990 −0.00939951 0.999956i \(-0.502992\pi\)
−0.00939951 + 0.999956i \(0.502992\pi\)
\(674\) −3.95615 −0.152385
\(675\) −8.84143 −0.340307
\(676\) −46.1658 −1.77561
\(677\) 47.3426 1.81953 0.909763 0.415129i \(-0.136264\pi\)
0.909763 + 0.415129i \(0.136264\pi\)
\(678\) 2.21708 0.0851466
\(679\) 0 0
\(680\) −32.6304 −1.25132
\(681\) 8.37096 0.320776
\(682\) 1.93370 0.0740450
\(683\) 38.7091 1.48116 0.740582 0.671966i \(-0.234550\pi\)
0.740582 + 0.671966i \(0.234550\pi\)
\(684\) 8.08232 0.309035
\(685\) 75.8780 2.89915
\(686\) 0 0
\(687\) 19.5187 0.744684
\(688\) 1.86121 0.0709581
\(689\) 50.2307 1.91364
\(690\) 3.04532 0.115933
\(691\) 21.9568 0.835277 0.417639 0.908613i \(-0.362858\pi\)
0.417639 + 0.908613i \(0.362858\pi\)
\(692\) −26.6249 −1.01213
\(693\) 0 0
\(694\) −1.89562 −0.0719567
\(695\) −45.3220 −1.71916
\(696\) 9.08044 0.344193
\(697\) −33.5247 −1.26984
\(698\) 1.68245 0.0636815
\(699\) −6.25868 −0.236725
\(700\) 0 0
\(701\) 7.52091 0.284061 0.142031 0.989862i \(-0.454637\pi\)
0.142031 + 0.989862i \(0.454637\pi\)
\(702\) 5.65397 0.213395
\(703\) −47.2810 −1.78324
\(704\) 14.4003 0.542732
\(705\) 22.4255 0.844593
\(706\) 15.1954 0.571886
\(707\) 0 0
\(708\) 7.04442 0.264745
\(709\) 0.693914 0.0260605 0.0130302 0.999915i \(-0.495852\pi\)
0.0130302 + 0.999915i \(0.495852\pi\)
\(710\) −8.90958 −0.334371
\(711\) 8.17142 0.306452
\(712\) −39.3333 −1.47408
\(713\) 0.638468 0.0239108
\(714\) 0 0
\(715\) −95.0842 −3.55595
\(716\) 6.27034 0.234334
\(717\) 4.53876 0.169503
\(718\) 5.95415 0.222207
\(719\) 9.37630 0.349677 0.174838 0.984597i \(-0.444060\pi\)
0.174838 + 0.984597i \(0.444060\pi\)
\(720\) −1.59544 −0.0594584
\(721\) 0 0
\(722\) −14.6764 −0.546200
\(723\) −19.6771 −0.731799
\(724\) −1.55753 −0.0578850
\(725\) −29.4540 −1.09390
\(726\) 2.20213 0.0817288
\(727\) 18.9568 0.703069 0.351535 0.936175i \(-0.385660\pi\)
0.351535 + 0.936175i \(0.385660\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −27.8641 −1.03130
\(731\) 13.9654 0.516531
\(732\) −10.7924 −0.398898
\(733\) 6.39615 0.236247 0.118124 0.992999i \(-0.462312\pi\)
0.118124 + 0.992999i \(0.462312\pi\)
\(734\) −16.5671 −0.611504
\(735\) 0 0
\(736\) 5.80250 0.213883
\(737\) −14.8657 −0.547584
\(738\) 8.52824 0.313929
\(739\) 29.8857 1.09936 0.549682 0.835374i \(-0.314749\pi\)
0.549682 + 0.835374i \(0.314749\pi\)
\(740\) 38.4977 1.41520
\(741\) 41.9760 1.54203
\(742\) 0 0
\(743\) 16.7564 0.614731 0.307366 0.951591i \(-0.400553\pi\)
0.307366 + 0.951591i \(0.400553\pi\)
\(744\) 1.74030 0.0638025
\(745\) 3.73506 0.136842
\(746\) −10.9060 −0.399299
\(747\) −6.01247 −0.219985
\(748\) −15.8344 −0.578963
\(749\) 0 0
\(750\) −11.6984 −0.427164
\(751\) −19.1378 −0.698348 −0.349174 0.937058i \(-0.613538\pi\)
−0.349174 + 0.937058i \(0.613538\pi\)
\(752\) 2.58488 0.0942609
\(753\) −10.3015 −0.375408
\(754\) 18.8354 0.685946
\(755\) 10.0846 0.367018
\(756\) 0 0
\(757\) −43.3744 −1.57647 −0.788234 0.615376i \(-0.789004\pi\)
−0.788234 + 0.615376i \(0.789004\pi\)
\(758\) −7.49527 −0.272241
\(759\) 3.70004 0.134303
\(760\) 61.6260 2.23541
\(761\) −6.16432 −0.223456 −0.111728 0.993739i \(-0.535639\pi\)
−0.111728 + 0.993739i \(0.535639\pi\)
\(762\) 5.46606 0.198014
\(763\) 0 0
\(764\) −32.1678 −1.16379
\(765\) −11.9712 −0.432820
\(766\) 7.35506 0.265749
\(767\) 36.5855 1.32103
\(768\) 14.6754 0.529553
\(769\) −2.61523 −0.0943076 −0.0471538 0.998888i \(-0.515015\pi\)
−0.0471538 + 0.998888i \(0.515015\pi\)
\(770\) 0 0
\(771\) 15.3148 0.551550
\(772\) −16.3925 −0.589977
\(773\) 52.4555 1.88669 0.943346 0.331810i \(-0.107659\pi\)
0.943346 + 0.331810i \(0.107659\pi\)
\(774\) −3.55262 −0.127696
\(775\) −5.64497 −0.202773
\(776\) 40.5361 1.45516
\(777\) 0 0
\(778\) 13.4015 0.480469
\(779\) 63.3150 2.26850
\(780\) −34.1782 −1.22377
\(781\) −10.8251 −0.387352
\(782\) 2.63384 0.0941860
\(783\) 3.33137 0.119053
\(784\) 0 0
\(785\) 11.2555 0.401725
\(786\) −6.30773 −0.224989
\(787\) 33.2853 1.18649 0.593246 0.805021i \(-0.297846\pi\)
0.593246 + 0.805021i \(0.297846\pi\)
\(788\) 3.67356 0.130865
\(789\) 5.31749 0.189308
\(790\) 24.8846 0.885353
\(791\) 0 0
\(792\) 10.0853 0.358367
\(793\) −56.0508 −1.99042
\(794\) 7.92752 0.281337
\(795\) 27.0551 0.959544
\(796\) 22.0527 0.781638
\(797\) 26.0207 0.921702 0.460851 0.887477i \(-0.347544\pi\)
0.460851 + 0.887477i \(0.347544\pi\)
\(798\) 0 0
\(799\) 19.3954 0.686161
\(800\) −51.3024 −1.81381
\(801\) −14.4303 −0.509870
\(802\) −9.38557 −0.331416
\(803\) −33.8547 −1.19471
\(804\) −5.34349 −0.188450
\(805\) 0 0
\(806\) 3.60988 0.127153
\(807\) −1.76465 −0.0621185
\(808\) −11.4483 −0.402751
\(809\) −41.8504 −1.47138 −0.735691 0.677317i \(-0.763142\pi\)
−0.735691 + 0.677317i \(0.763142\pi\)
\(810\) 3.04532 0.107002
\(811\) 17.1370 0.601760 0.300880 0.953662i \(-0.402720\pi\)
0.300880 + 0.953662i \(0.402720\pi\)
\(812\) 0 0
\(813\) 16.6166 0.582768
\(814\) −23.5638 −0.825912
\(815\) −28.5801 −1.00112
\(816\) −1.37987 −0.0483050
\(817\) −26.3752 −0.922752
\(818\) −31.6393 −1.10624
\(819\) 0 0
\(820\) −51.5531 −1.80031
\(821\) −4.76075 −0.166151 −0.0830757 0.996543i \(-0.526474\pi\)
−0.0830757 + 0.996543i \(0.526474\pi\)
\(822\) −16.6943 −0.582279
\(823\) −46.9815 −1.63767 −0.818835 0.574029i \(-0.805380\pi\)
−0.818835 + 0.574029i \(0.805380\pi\)
\(824\) 47.5752 1.65736
\(825\) −32.7136 −1.13894
\(826\) 0 0
\(827\) −13.3820 −0.465336 −0.232668 0.972556i \(-0.574746\pi\)
−0.232668 + 0.972556i \(0.574746\pi\)
\(828\) 1.32999 0.0462202
\(829\) 0.371697 0.0129096 0.00645478 0.999979i \(-0.497945\pi\)
0.00645478 + 0.999979i \(0.497945\pi\)
\(830\) −18.3099 −0.635545
\(831\) −1.95295 −0.0677472
\(832\) 26.8829 0.931998
\(833\) 0 0
\(834\) 9.97150 0.345285
\(835\) 17.0513 0.590085
\(836\) 29.9049 1.03428
\(837\) 0.638468 0.0220687
\(838\) −5.49128 −0.189693
\(839\) −45.6631 −1.57647 −0.788233 0.615377i \(-0.789004\pi\)
−0.788233 + 0.615377i \(0.789004\pi\)
\(840\) 0 0
\(841\) −17.9020 −0.617310
\(842\) 27.8726 0.960554
\(843\) −13.1183 −0.451820
\(844\) −3.66827 −0.126267
\(845\) −129.141 −4.44258
\(846\) −4.93394 −0.169632
\(847\) 0 0
\(848\) 3.11851 0.107090
\(849\) 0.979872 0.0336291
\(850\) −23.2869 −0.798735
\(851\) −7.78032 −0.266706
\(852\) −3.89110 −0.133307
\(853\) −42.5768 −1.45780 −0.728901 0.684619i \(-0.759969\pi\)
−0.728901 + 0.684619i \(0.759969\pi\)
\(854\) 0 0
\(855\) 22.6089 0.773209
\(856\) −2.65546 −0.0907618
\(857\) −41.3922 −1.41393 −0.706965 0.707248i \(-0.749936\pi\)
−0.706965 + 0.707248i \(0.749936\pi\)
\(858\) 20.9199 0.714194
\(859\) 52.7547 1.79997 0.899984 0.435923i \(-0.143578\pi\)
0.899984 + 0.435923i \(0.143578\pi\)
\(860\) 21.4756 0.732310
\(861\) 0 0
\(862\) −6.11368 −0.208233
\(863\) 22.7441 0.774217 0.387109 0.922034i \(-0.373474\pi\)
0.387109 + 0.922034i \(0.373474\pi\)
\(864\) 5.80250 0.197405
\(865\) −74.4787 −2.53235
\(866\) −21.1756 −0.719575
\(867\) 6.64630 0.225720
\(868\) 0 0
\(869\) 30.2346 1.02564
\(870\) 10.1451 0.343950
\(871\) −27.7517 −0.940330
\(872\) 0.508463 0.0172187
\(873\) 14.8716 0.503328
\(874\) −4.97429 −0.168258
\(875\) 0 0
\(876\) −12.1691 −0.411157
\(877\) 5.28592 0.178493 0.0892464 0.996010i \(-0.471554\pi\)
0.0892464 + 0.996010i \(0.471554\pi\)
\(878\) −13.0741 −0.441228
\(879\) −22.4505 −0.757237
\(880\) −5.90318 −0.198996
\(881\) 51.9769 1.75115 0.875573 0.483086i \(-0.160484\pi\)
0.875573 + 0.483086i \(0.160484\pi\)
\(882\) 0 0
\(883\) 13.0607 0.439526 0.219763 0.975553i \(-0.429472\pi\)
0.219763 + 0.975553i \(0.429472\pi\)
\(884\) −29.5601 −0.994215
\(885\) 19.7056 0.662395
\(886\) −25.0706 −0.842264
\(887\) 49.1688 1.65093 0.825464 0.564455i \(-0.190913\pi\)
0.825464 + 0.564455i \(0.190913\pi\)
\(888\) −21.2071 −0.711665
\(889\) 0 0
\(890\) −43.9449 −1.47304
\(891\) 3.70004 0.123956
\(892\) −34.3391 −1.14976
\(893\) −36.6303 −1.22579
\(894\) −0.821768 −0.0274840
\(895\) 17.5402 0.586305
\(896\) 0 0
\(897\) 6.90735 0.230630
\(898\) 5.83281 0.194643
\(899\) 2.12697 0.0709385
\(900\) −11.7590 −0.391966
\(901\) 23.3995 0.779549
\(902\) 31.5548 1.05066
\(903\) 0 0
\(904\) 7.38286 0.245550
\(905\) −4.35691 −0.144829
\(906\) −2.21877 −0.0737137
\(907\) −51.1121 −1.69715 −0.848575 0.529074i \(-0.822539\pi\)
−0.848575 + 0.529074i \(0.822539\pi\)
\(908\) 11.1333 0.369470
\(909\) −4.20008 −0.139308
\(910\) 0 0
\(911\) 10.9073 0.361376 0.180688 0.983540i \(-0.442168\pi\)
0.180688 + 0.983540i \(0.442168\pi\)
\(912\) 2.60602 0.0862940
\(913\) −22.2464 −0.736248
\(914\) −16.9317 −0.560050
\(915\) −30.1898 −0.998045
\(916\) 25.9596 0.857728
\(917\) 0 0
\(918\) 2.63384 0.0869298
\(919\) −34.8033 −1.14805 −0.574027 0.818836i \(-0.694619\pi\)
−0.574027 + 0.818836i \(0.694619\pi\)
\(920\) 10.1409 0.334334
\(921\) 29.2019 0.962237
\(922\) 9.95443 0.327832
\(923\) −20.2086 −0.665174
\(924\) 0 0
\(925\) 68.7892 2.26177
\(926\) 1.60453 0.0527280
\(927\) 17.4541 0.573267
\(928\) 19.3302 0.634547
\(929\) 7.12498 0.233763 0.116882 0.993146i \(-0.462710\pi\)
0.116882 + 0.993146i \(0.462710\pi\)
\(930\) 1.94434 0.0637574
\(931\) 0 0
\(932\) −8.32396 −0.272660
\(933\) 1.79193 0.0586652
\(934\) −32.2442 −1.05506
\(935\) −44.2940 −1.44857
\(936\) 18.8276 0.615401
\(937\) 2.12786 0.0695142 0.0347571 0.999396i \(-0.488934\pi\)
0.0347571 + 0.999396i \(0.488934\pi\)
\(938\) 0 0
\(939\) −8.83165 −0.288210
\(940\) 29.8256 0.972804
\(941\) 46.7565 1.52422 0.762109 0.647449i \(-0.224164\pi\)
0.762109 + 0.647449i \(0.224164\pi\)
\(942\) −2.47637 −0.0806844
\(943\) 10.4188 0.339283
\(944\) 2.27137 0.0739267
\(945\) 0 0
\(946\) −13.1448 −0.427376
\(947\) 29.9575 0.973487 0.486744 0.873545i \(-0.338185\pi\)
0.486744 + 0.873545i \(0.338185\pi\)
\(948\) 10.8679 0.352972
\(949\) −63.2010 −2.05159
\(950\) 43.9798 1.42689
\(951\) 21.5495 0.698791
\(952\) 0 0
\(953\) 60.6214 1.96372 0.981860 0.189605i \(-0.0607207\pi\)
0.981860 + 0.189605i \(0.0607207\pi\)
\(954\) −5.95251 −0.192720
\(955\) −89.9838 −2.91181
\(956\) 6.03648 0.195234
\(957\) 12.3262 0.398449
\(958\) −2.31428 −0.0747711
\(959\) 0 0
\(960\) 14.4796 0.467326
\(961\) −30.5924 −0.986850
\(962\) −43.9897 −1.41828
\(963\) −0.974217 −0.0313937
\(964\) −26.1703 −0.842887
\(965\) −45.8551 −1.47613
\(966\) 0 0
\(967\) 6.34634 0.204084 0.102042 0.994780i \(-0.467462\pi\)
0.102042 + 0.994780i \(0.467462\pi\)
\(968\) 7.33307 0.235694
\(969\) 19.5541 0.628167
\(970\) 45.2888 1.45413
\(971\) 31.3344 1.00557 0.502784 0.864412i \(-0.332309\pi\)
0.502784 + 0.864412i \(0.332309\pi\)
\(972\) 1.32999 0.0426593
\(973\) 0 0
\(974\) 20.2875 0.650052
\(975\) −61.0708 −1.95583
\(976\) −3.47984 −0.111387
\(977\) −17.6550 −0.564833 −0.282417 0.959292i \(-0.591136\pi\)
−0.282417 + 0.959292i \(0.591136\pi\)
\(978\) 6.28805 0.201070
\(979\) −53.3928 −1.70644
\(980\) 0 0
\(981\) 0.186541 0.00595581
\(982\) −10.7521 −0.343113
\(983\) −3.40755 −0.108684 −0.0543420 0.998522i \(-0.517306\pi\)
−0.0543420 + 0.998522i \(0.517306\pi\)
\(984\) 28.3989 0.905324
\(985\) 10.2761 0.327425
\(986\) 8.77430 0.279431
\(987\) 0 0
\(988\) 55.8274 1.77611
\(989\) −4.34017 −0.138009
\(990\) 11.2678 0.358114
\(991\) 1.05296 0.0334485 0.0167242 0.999860i \(-0.494676\pi\)
0.0167242 + 0.999860i \(0.494676\pi\)
\(992\) 3.70471 0.117625
\(993\) −23.1086 −0.733330
\(994\) 0 0
\(995\) 61.6887 1.95566
\(996\) −7.99650 −0.253379
\(997\) 54.9471 1.74019 0.870095 0.492883i \(-0.164057\pi\)
0.870095 + 0.492883i \(0.164057\pi\)
\(998\) −13.4904 −0.427032
\(999\) −7.78032 −0.246159
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 3381.2.a.be.1.4 8
7.3 odd 6 483.2.i.g.415.5 yes 16
7.5 odd 6 483.2.i.g.277.5 16
7.6 odd 2 3381.2.a.bf.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.g.277.5 16 7.5 odd 6
483.2.i.g.415.5 yes 16 7.3 odd 6
3381.2.a.be.1.4 8 1.1 even 1 trivial
3381.2.a.bf.1.4 8 7.6 odd 2