Properties

Label 6825.2.a.bg.1.4
Level $6825$
Weight $2$
Character 6825.1
Self dual yes
Analytic conductor $54.498$
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] = [6825,2,Mod(1,6825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6825.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: \(54.4978993795\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.787711\) of defining polynomial
Character \(\chi\) \(=\) 6825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.37951 q^{2} -1.00000 q^{3} +3.66208 q^{4} -2.37951 q^{6} -1.00000 q^{7} +3.95493 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.37951 q^{2} -1.00000 q^{3} +3.66208 q^{4} -2.37951 q^{6} -1.00000 q^{7} +3.95493 q^{8} +1.00000 q^{9} +1.57542 q^{11} -3.66208 q^{12} -1.00000 q^{13} -2.37951 q^{14} +2.08666 q^{16} -4.75902 q^{17} +2.37951 q^{18} -2.23750 q^{19} +1.00000 q^{21} +3.74873 q^{22} -5.84568 q^{23} -3.95493 q^{24} -2.37951 q^{26} -1.00000 q^{27} -3.66208 q^{28} +4.23750 q^{29} +7.28055 q^{31} -2.94464 q^{32} -1.57542 q^{33} -11.3242 q^{34} +3.66208 q^{36} -10.4750 q^{37} -5.32415 q^{38} +1.00000 q^{39} -2.25127 q^{41} +2.37951 q^{42} -0.913344 q^{43} +5.76931 q^{44} -13.9099 q^{46} +2.09695 q^{47} -2.08666 q^{48} +1.00000 q^{49} +4.75902 q^{51} -3.66208 q^{52} -1.08666 q^{53} -2.37951 q^{54} -3.95493 q^{56} +2.23750 q^{57} +10.0832 q^{58} -12.3344 q^{59} -7.51805 q^{61} +17.3242 q^{62} -1.00000 q^{63} -11.1801 q^{64} -3.74873 q^{66} +15.6914 q^{67} -17.4279 q^{68} +5.84568 q^{69} +10.0504 q^{71} +3.95493 q^{72} -15.0797 q^{73} -24.9254 q^{74} -8.19389 q^{76} -1.57542 q^{77} +2.37951 q^{78} -11.7555 q^{79} +1.00000 q^{81} -5.35692 q^{82} -7.42110 q^{83} +3.66208 q^{84} -2.17331 q^{86} -4.23750 q^{87} +6.23069 q^{88} -11.1371 q^{89} +1.00000 q^{91} -21.4073 q^{92} -7.28055 q^{93} +4.98971 q^{94} +2.94464 q^{96} +14.6047 q^{97} +2.37951 q^{98} +1.57542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24} + q^{26} - 4 q^{27} - 7 q^{28} + q^{29} + 3 q^{31} - 7 q^{32} + 2 q^{33} - 30 q^{34} + 7 q^{36} - 10 q^{37} - 6 q^{38} + 4 q^{39} - 16 q^{41} - q^{42} - 3 q^{43} - 12 q^{44} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 4 q^{49} - 2 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 3 q^{56} - 7 q^{57} + 4 q^{58} - 20 q^{59} + 12 q^{61} + 54 q^{62} - 4 q^{63} + 5 q^{64} - 8 q^{66} + 22 q^{67} + 10 q^{68} + 3 q^{69} - 3 q^{72} + 13 q^{73} - 6 q^{74} - 6 q^{76} + 2 q^{77} - q^{78} + 11 q^{79} + 4 q^{81} - 10 q^{82} - q^{83} + 7 q^{84} - 10 q^{86} - q^{87} + 60 q^{88} - 5 q^{89} + 4 q^{91} - 34 q^{92} - 3 q^{93} + 34 q^{94} + 7 q^{96} + 17 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37951 1.68257 0.841285 0.540593i \(-0.181800\pi\)
0.841285 + 0.540593i \(0.181800\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.66208 1.83104
\(5\) 0 0
\(6\) −2.37951 −0.971432
\(7\) −1.00000 −0.377964
\(8\) 3.95493 1.39828
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.57542 0.475007 0.237504 0.971387i \(-0.423671\pi\)
0.237504 + 0.971387i \(0.423671\pi\)
\(12\) −3.66208 −1.05715
\(13\) −1.00000 −0.277350
\(14\) −2.37951 −0.635951
\(15\) 0 0
\(16\) 2.08666 0.521664
\(17\) −4.75902 −1.15423 −0.577116 0.816662i \(-0.695822\pi\)
−0.577116 + 0.816662i \(0.695822\pi\)
\(18\) 2.37951 0.560856
\(19\) −2.23750 −0.513317 −0.256659 0.966502i \(-0.582622\pi\)
−0.256659 + 0.966502i \(0.582622\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 3.74873 0.799233
\(23\) −5.84568 −1.21891 −0.609454 0.792821i \(-0.708611\pi\)
−0.609454 + 0.792821i \(0.708611\pi\)
\(24\) −3.95493 −0.807297
\(25\) 0 0
\(26\) −2.37951 −0.466661
\(27\) −1.00000 −0.192450
\(28\) −3.66208 −0.692068
\(29\) 4.23750 0.786884 0.393442 0.919349i \(-0.371284\pi\)
0.393442 + 0.919349i \(0.371284\pi\)
\(30\) 0 0
\(31\) 7.28055 1.30763 0.653813 0.756656i \(-0.273168\pi\)
0.653813 + 0.756656i \(0.273168\pi\)
\(32\) −2.94464 −0.520544
\(33\) −1.57542 −0.274246
\(34\) −11.3242 −1.94208
\(35\) 0 0
\(36\) 3.66208 0.610346
\(37\) −10.4750 −1.72208 −0.861039 0.508538i \(-0.830186\pi\)
−0.861039 + 0.508538i \(0.830186\pi\)
\(38\) −5.32415 −0.863692
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −2.25127 −0.351589 −0.175794 0.984427i \(-0.556249\pi\)
−0.175794 + 0.984427i \(0.556249\pi\)
\(42\) 2.37951 0.367167
\(43\) −0.913344 −0.139284 −0.0696418 0.997572i \(-0.522186\pi\)
−0.0696418 + 0.997572i \(0.522186\pi\)
\(44\) 5.76931 0.869757
\(45\) 0 0
\(46\) −13.9099 −2.05090
\(47\) 2.09695 0.305871 0.152936 0.988236i \(-0.451127\pi\)
0.152936 + 0.988236i \(0.451127\pi\)
\(48\) −2.08666 −0.301183
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.75902 0.666397
\(52\) −3.66208 −0.507839
\(53\) −1.08666 −0.149264 −0.0746319 0.997211i \(-0.523778\pi\)
−0.0746319 + 0.997211i \(0.523778\pi\)
\(54\) −2.37951 −0.323811
\(55\) 0 0
\(56\) −3.95493 −0.528500
\(57\) 2.23750 0.296364
\(58\) 10.0832 1.32399
\(59\) −12.3344 −1.60581 −0.802904 0.596108i \(-0.796713\pi\)
−0.802904 + 0.596108i \(0.796713\pi\)
\(60\) 0 0
\(61\) −7.51805 −0.962587 −0.481294 0.876559i \(-0.659833\pi\)
−0.481294 + 0.876559i \(0.659833\pi\)
\(62\) 17.3242 2.20017
\(63\) −1.00000 −0.125988
\(64\) −11.1801 −1.39752
\(65\) 0 0
\(66\) −3.74873 −0.461437
\(67\) 15.6914 1.91700 0.958502 0.285084i \(-0.0920217\pi\)
0.958502 + 0.285084i \(0.0920217\pi\)
\(68\) −17.4279 −2.11345
\(69\) 5.84568 0.703737
\(70\) 0 0
\(71\) 10.0504 1.19277 0.596383 0.802700i \(-0.296604\pi\)
0.596383 + 0.802700i \(0.296604\pi\)
\(72\) 3.95493 0.466093
\(73\) −15.0797 −1.76495 −0.882473 0.470363i \(-0.844123\pi\)
−0.882473 + 0.470363i \(0.844123\pi\)
\(74\) −24.9254 −2.89752
\(75\) 0 0
\(76\) −8.19389 −0.939904
\(77\) −1.57542 −0.179536
\(78\) 2.37951 0.269427
\(79\) −11.7555 −1.32260 −0.661301 0.750121i \(-0.729995\pi\)
−0.661301 + 0.750121i \(0.729995\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −5.35692 −0.591572
\(83\) −7.42110 −0.814572 −0.407286 0.913301i \(-0.633525\pi\)
−0.407286 + 0.913301i \(0.633525\pi\)
\(84\) 3.66208 0.399565
\(85\) 0 0
\(86\) −2.17331 −0.234354
\(87\) −4.23750 −0.454308
\(88\) 6.23069 0.664193
\(89\) −11.1371 −1.18053 −0.590264 0.807210i \(-0.700976\pi\)
−0.590264 + 0.807210i \(0.700976\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −21.4073 −2.23187
\(93\) −7.28055 −0.754958
\(94\) 4.98971 0.514649
\(95\) 0 0
\(96\) 2.94464 0.300536
\(97\) 14.6047 1.48288 0.741442 0.671017i \(-0.234142\pi\)
0.741442 + 0.671017i \(0.234142\pi\)
\(98\) 2.37951 0.240367
\(99\) 1.57542 0.158336
\(100\) 0 0
\(101\) −11.4349 −1.13781 −0.568906 0.822403i \(-0.692633\pi\)
−0.568906 + 0.822403i \(0.692633\pi\)
\(102\) 11.3242 1.12126
\(103\) −11.1508 −1.09873 −0.549363 0.835584i \(-0.685129\pi\)
−0.549363 + 0.835584i \(0.685129\pi\)
\(104\) −3.95493 −0.387813
\(105\) 0 0
\(106\) −2.58571 −0.251147
\(107\) −2.28403 −0.220805 −0.110403 0.993887i \(-0.535214\pi\)
−0.110403 + 0.993887i \(0.535214\pi\)
\(108\) −3.66208 −0.352384
\(109\) 14.6689 1.40502 0.702512 0.711671i \(-0.252062\pi\)
0.702512 + 0.711671i \(0.252062\pi\)
\(110\) 0 0
\(111\) 10.4750 0.994243
\(112\) −2.08666 −0.197170
\(113\) 4.23750 0.398630 0.199315 0.979935i \(-0.436128\pi\)
0.199315 + 0.979935i \(0.436128\pi\)
\(114\) 5.32415 0.498653
\(115\) 0 0
\(116\) 15.5180 1.44081
\(117\) −1.00000 −0.0924500
\(118\) −29.3500 −2.70188
\(119\) 4.75902 0.436259
\(120\) 0 0
\(121\) −8.51805 −0.774368
\(122\) −17.8893 −1.61962
\(123\) 2.25127 0.202990
\(124\) 26.6619 2.39431
\(125\) 0 0
\(126\) −2.37951 −0.211984
\(127\) −4.84916 −0.430293 −0.215147 0.976582i \(-0.569023\pi\)
−0.215147 + 0.976582i \(0.569023\pi\)
\(128\) −20.7140 −1.83087
\(129\) 0.913344 0.0804154
\(130\) 0 0
\(131\) −6.36721 −0.556305 −0.278153 0.960537i \(-0.589722\pi\)
−0.278153 + 0.960537i \(0.589722\pi\)
\(132\) −5.76931 −0.502154
\(133\) 2.23750 0.194016
\(134\) 37.3378 3.22549
\(135\) 0 0
\(136\) −18.8216 −1.61394
\(137\) 18.0258 1.54005 0.770024 0.638015i \(-0.220244\pi\)
0.770024 + 0.638015i \(0.220244\pi\)
\(138\) 13.9099 1.18409
\(139\) 2.67585 0.226962 0.113481 0.993540i \(-0.463800\pi\)
0.113481 + 0.993540i \(0.463800\pi\)
\(140\) 0 0
\(141\) −2.09695 −0.176595
\(142\) 23.9151 2.00691
\(143\) −1.57542 −0.131743
\(144\) 2.08666 0.173888
\(145\) 0 0
\(146\) −35.8823 −2.96964
\(147\) −1.00000 −0.0824786
\(148\) −38.3603 −3.15319
\(149\) −7.46471 −0.611533 −0.305766 0.952107i \(-0.598913\pi\)
−0.305766 + 0.952107i \(0.598913\pi\)
\(150\) 0 0
\(151\) 13.0155 1.05919 0.529594 0.848251i \(-0.322344\pi\)
0.529594 + 0.848251i \(0.322344\pi\)
\(152\) −8.84916 −0.717761
\(153\) −4.75902 −0.384744
\(154\) −3.74873 −0.302082
\(155\) 0 0
\(156\) 3.66208 0.293201
\(157\) 17.1233 1.36659 0.683294 0.730143i \(-0.260547\pi\)
0.683294 + 0.730143i \(0.260547\pi\)
\(158\) −27.9725 −2.22537
\(159\) 1.08666 0.0861774
\(160\) 0 0
\(161\) 5.84568 0.460704
\(162\) 2.37951 0.186952
\(163\) 0.849158 0.0665112 0.0332556 0.999447i \(-0.489412\pi\)
0.0332556 + 0.999447i \(0.489412\pi\)
\(164\) −8.24431 −0.643773
\(165\) 0 0
\(166\) −17.6586 −1.37057
\(167\) −18.3781 −1.42214 −0.711068 0.703123i \(-0.751788\pi\)
−0.711068 + 0.703123i \(0.751788\pi\)
\(168\) 3.95493 0.305130
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.23750 −0.171106
\(172\) −3.34474 −0.255034
\(173\) 12.2565 0.931844 0.465922 0.884826i \(-0.345723\pi\)
0.465922 + 0.884826i \(0.345723\pi\)
\(174\) −10.0832 −0.764404
\(175\) 0 0
\(176\) 3.28736 0.247794
\(177\) 12.3344 0.927114
\(178\) −26.5008 −1.98632
\(179\) 18.5146 1.38384 0.691922 0.721972i \(-0.256764\pi\)
0.691922 + 0.721972i \(0.256764\pi\)
\(180\) 0 0
\(181\) −18.1664 −1.35029 −0.675147 0.737683i \(-0.735920\pi\)
−0.675147 + 0.737683i \(0.735920\pi\)
\(182\) 2.37951 0.176381
\(183\) 7.51805 0.555750
\(184\) −23.1193 −1.70438
\(185\) 0 0
\(186\) −17.3242 −1.27027
\(187\) −7.49747 −0.548269
\(188\) 7.67918 0.560062
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 23.5151 1.70149 0.850747 0.525575i \(-0.176150\pi\)
0.850747 + 0.525575i \(0.176150\pi\)
\(192\) 11.1801 0.806856
\(193\) −14.7344 −1.06061 −0.530303 0.847808i \(-0.677922\pi\)
−0.530303 + 0.847808i \(0.677922\pi\)
\(194\) 34.7521 2.49505
\(195\) 0 0
\(196\) 3.66208 0.261577
\(197\) −14.5008 −1.03314 −0.516570 0.856245i \(-0.672791\pi\)
−0.516570 + 0.856245i \(0.672791\pi\)
\(198\) 3.74873 0.266411
\(199\) −13.5180 −0.958269 −0.479135 0.877741i \(-0.659049\pi\)
−0.479135 + 0.877741i \(0.659049\pi\)
\(200\) 0 0
\(201\) −15.6914 −1.10678
\(202\) −27.2094 −1.91445
\(203\) −4.23750 −0.297414
\(204\) 17.4279 1.22020
\(205\) 0 0
\(206\) −26.5336 −1.84868
\(207\) −5.84568 −0.406303
\(208\) −2.08666 −0.144684
\(209\) −3.52500 −0.243830
\(210\) 0 0
\(211\) 4.06419 0.279790 0.139895 0.990166i \(-0.455323\pi\)
0.139895 + 0.990166i \(0.455323\pi\)
\(212\) −3.97942 −0.273308
\(213\) −10.0504 −0.688643
\(214\) −5.43487 −0.371520
\(215\) 0 0
\(216\) −3.95493 −0.269099
\(217\) −7.28055 −0.494236
\(218\) 34.9048 2.36405
\(219\) 15.0797 1.01899
\(220\) 0 0
\(221\) 4.75902 0.320127
\(222\) 24.9254 1.67288
\(223\) 0.0641862 0.00429822 0.00214911 0.999998i \(-0.499316\pi\)
0.00214911 + 0.999998i \(0.499316\pi\)
\(224\) 2.94464 0.196747
\(225\) 0 0
\(226\) 10.0832 0.670723
\(227\) 1.68614 0.111913 0.0559564 0.998433i \(-0.482179\pi\)
0.0559564 + 0.998433i \(0.482179\pi\)
\(228\) 8.19389 0.542654
\(229\) 0.648310 0.0428415 0.0214208 0.999771i \(-0.493181\pi\)
0.0214208 + 0.999771i \(0.493181\pi\)
\(230\) 0 0
\(231\) 1.57542 0.103655
\(232\) 16.7590 1.10028
\(233\) −3.10724 −0.203562 −0.101781 0.994807i \(-0.532454\pi\)
−0.101781 + 0.994807i \(0.532454\pi\)
\(234\) −2.37951 −0.155554
\(235\) 0 0
\(236\) −45.1697 −2.94030
\(237\) 11.7555 0.763605
\(238\) 11.3242 0.734036
\(239\) −15.0659 −0.974534 −0.487267 0.873253i \(-0.662006\pi\)
−0.487267 + 0.873253i \(0.662006\pi\)
\(240\) 0 0
\(241\) 25.7280 1.65729 0.828643 0.559777i \(-0.189113\pi\)
0.828643 + 0.559777i \(0.189113\pi\)
\(242\) −20.2688 −1.30293
\(243\) −1.00000 −0.0641500
\(244\) −27.5317 −1.76253
\(245\) 0 0
\(246\) 5.35692 0.341544
\(247\) 2.23750 0.142369
\(248\) 28.7941 1.82843
\(249\) 7.42110 0.470293
\(250\) 0 0
\(251\) −11.6258 −0.733816 −0.366908 0.930257i \(-0.619584\pi\)
−0.366908 + 0.930257i \(0.619584\pi\)
\(252\) −3.66208 −0.230689
\(253\) −9.20941 −0.578991
\(254\) −11.5386 −0.723998
\(255\) 0 0
\(256\) −26.9289 −1.68305
\(257\) 12.5651 0.783791 0.391896 0.920010i \(-0.371819\pi\)
0.391896 + 0.920010i \(0.371819\pi\)
\(258\) 2.17331 0.135305
\(259\) 10.4750 0.650885
\(260\) 0 0
\(261\) 4.23750 0.262295
\(262\) −15.1508 −0.936022
\(263\) −9.37068 −0.577821 −0.288911 0.957356i \(-0.593293\pi\)
−0.288911 + 0.957356i \(0.593293\pi\)
\(264\) −6.23069 −0.383472
\(265\) 0 0
\(266\) 5.32415 0.326445
\(267\) 11.1371 0.681578
\(268\) 57.4630 3.51011
\(269\) −10.3918 −0.633600 −0.316800 0.948492i \(-0.602608\pi\)
−0.316800 + 0.948492i \(0.602608\pi\)
\(270\) 0 0
\(271\) −5.13026 −0.311641 −0.155821 0.987785i \(-0.549802\pi\)
−0.155821 + 0.987785i \(0.549802\pi\)
\(272\) −9.93045 −0.602122
\(273\) −1.00000 −0.0605228
\(274\) 42.8926 2.59124
\(275\) 0 0
\(276\) 21.4073 1.28857
\(277\) −8.04361 −0.483293 −0.241647 0.970364i \(-0.577688\pi\)
−0.241647 + 0.970364i \(0.577688\pi\)
\(278\) 6.36721 0.381880
\(279\) 7.28055 0.435875
\(280\) 0 0
\(281\) −27.8525 −1.66154 −0.830770 0.556616i \(-0.812100\pi\)
−0.830770 + 0.556616i \(0.812100\pi\)
\(282\) −4.98971 −0.297133
\(283\) −23.8197 −1.41594 −0.707968 0.706244i \(-0.750388\pi\)
−0.707968 + 0.706244i \(0.750388\pi\)
\(284\) 36.8054 2.18400
\(285\) 0 0
\(286\) −3.74873 −0.221667
\(287\) 2.25127 0.132888
\(288\) −2.94464 −0.173515
\(289\) 5.64831 0.332254
\(290\) 0 0
\(291\) −14.6047 −0.856143
\(292\) −55.2230 −3.23168
\(293\) 16.4612 0.961675 0.480838 0.876810i \(-0.340333\pi\)
0.480838 + 0.876810i \(0.340333\pi\)
\(294\) −2.37951 −0.138776
\(295\) 0 0
\(296\) −41.4279 −2.40795
\(297\) −1.57542 −0.0914152
\(298\) −17.7624 −1.02895
\(299\) 5.84568 0.338064
\(300\) 0 0
\(301\) 0.913344 0.0526443
\(302\) 30.9706 1.78216
\(303\) 11.4349 0.656916
\(304\) −4.66889 −0.267779
\(305\) 0 0
\(306\) −11.3242 −0.647359
\(307\) −1.95639 −0.111657 −0.0558287 0.998440i \(-0.517780\pi\)
−0.0558287 + 0.998440i \(0.517780\pi\)
\(308\) −5.76931 −0.328737
\(309\) 11.1508 0.634349
\(310\) 0 0
\(311\) −6.67585 −0.378552 −0.189276 0.981924i \(-0.560614\pi\)
−0.189276 + 0.981924i \(0.560614\pi\)
\(312\) 3.95493 0.223904
\(313\) 6.78364 0.383434 0.191717 0.981450i \(-0.438594\pi\)
0.191717 + 0.981450i \(0.438594\pi\)
\(314\) 40.7451 2.29938
\(315\) 0 0
\(316\) −43.0497 −2.42174
\(317\) 30.6947 1.72399 0.861993 0.506920i \(-0.169216\pi\)
0.861993 + 0.506920i \(0.169216\pi\)
\(318\) 2.58571 0.145000
\(319\) 6.67585 0.373776
\(320\) 0 0
\(321\) 2.28403 0.127482
\(322\) 13.9099 0.775167
\(323\) 10.6483 0.592488
\(324\) 3.66208 0.203449
\(325\) 0 0
\(326\) 2.02058 0.111910
\(327\) −14.6689 −0.811191
\(328\) −8.90361 −0.491620
\(329\) −2.09695 −0.115608
\(330\) 0 0
\(331\) −13.8267 −0.759983 −0.379992 0.924990i \(-0.624073\pi\)
−0.379992 + 0.924990i \(0.624073\pi\)
\(332\) −27.1766 −1.49151
\(333\) −10.4750 −0.574026
\(334\) −43.7308 −2.39284
\(335\) 0 0
\(336\) 2.08666 0.113836
\(337\) −23.8633 −1.29992 −0.649959 0.759969i \(-0.725214\pi\)
−0.649959 + 0.759969i \(0.725214\pi\)
\(338\) 2.37951 0.129428
\(339\) −4.23750 −0.230149
\(340\) 0 0
\(341\) 11.4699 0.621132
\(342\) −5.32415 −0.287897
\(343\) −1.00000 −0.0539949
\(344\) −3.61221 −0.194758
\(345\) 0 0
\(346\) 29.1645 1.56789
\(347\) 13.6012 0.730152 0.365076 0.930978i \(-0.381043\pi\)
0.365076 + 0.930978i \(0.381043\pi\)
\(348\) −15.5180 −0.831855
\(349\) −16.4039 −0.878078 −0.439039 0.898468i \(-0.644681\pi\)
−0.439039 + 0.898468i \(0.644681\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −4.63905 −0.247262
\(353\) 4.73322 0.251924 0.125962 0.992035i \(-0.459798\pi\)
0.125962 + 0.992035i \(0.459798\pi\)
\(354\) 29.3500 1.55993
\(355\) 0 0
\(356\) −40.7848 −2.16159
\(357\) −4.75902 −0.251874
\(358\) 44.0556 2.32841
\(359\) −17.5479 −0.926142 −0.463071 0.886321i \(-0.653253\pi\)
−0.463071 + 0.886321i \(0.653253\pi\)
\(360\) 0 0
\(361\) −13.9936 −0.736505
\(362\) −43.2271 −2.27196
\(363\) 8.51805 0.447082
\(364\) 3.66208 0.191945
\(365\) 0 0
\(366\) 17.8893 0.935088
\(367\) 11.1888 0.584052 0.292026 0.956410i \(-0.405671\pi\)
0.292026 + 0.956410i \(0.405671\pi\)
\(368\) −12.1979 −0.635861
\(369\) −2.25127 −0.117196
\(370\) 0 0
\(371\) 1.08666 0.0564164
\(372\) −26.6619 −1.38236
\(373\) 5.43195 0.281256 0.140628 0.990063i \(-0.455088\pi\)
0.140628 + 0.990063i \(0.455088\pi\)
\(374\) −17.8403 −0.922501
\(375\) 0 0
\(376\) 8.29328 0.427693
\(377\) −4.23750 −0.218242
\(378\) 2.37951 0.122389
\(379\) 10.8422 0.556927 0.278463 0.960447i \(-0.410175\pi\)
0.278463 + 0.960447i \(0.410175\pi\)
\(380\) 0 0
\(381\) 4.84916 0.248430
\(382\) 55.9545 2.86288
\(383\) −10.5353 −0.538328 −0.269164 0.963094i \(-0.586747\pi\)
−0.269164 + 0.963094i \(0.586747\pi\)
\(384\) 20.7140 1.05705
\(385\) 0 0
\(386\) −35.0607 −1.78454
\(387\) −0.913344 −0.0464279
\(388\) 53.4836 2.71522
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 27.8197 1.40690
\(392\) 3.95493 0.199754
\(393\) 6.36721 0.321183
\(394\) −34.5048 −1.73833
\(395\) 0 0
\(396\) 5.76931 0.289919
\(397\) 35.4400 1.77868 0.889340 0.457246i \(-0.151164\pi\)
0.889340 + 0.457246i \(0.151164\pi\)
\(398\) −32.1664 −1.61235
\(399\) −2.23750 −0.112015
\(400\) 0 0
\(401\) 23.7241 1.18473 0.592363 0.805671i \(-0.298195\pi\)
0.592363 + 0.805671i \(0.298195\pi\)
\(402\) −37.3378 −1.86224
\(403\) −7.28055 −0.362670
\(404\) −41.8754 −2.08338
\(405\) 0 0
\(406\) −10.0832 −0.500420
\(407\) −16.5025 −0.818000
\(408\) 18.8216 0.931809
\(409\) −3.95639 −0.195631 −0.0978156 0.995205i \(-0.531186\pi\)
−0.0978156 + 0.995205i \(0.531186\pi\)
\(410\) 0 0
\(411\) −18.0258 −0.889147
\(412\) −40.8352 −2.01181
\(413\) 12.3344 0.606938
\(414\) −13.9099 −0.683633
\(415\) 0 0
\(416\) 2.94464 0.144373
\(417\) −2.67585 −0.131037
\(418\) −8.38779 −0.410260
\(419\) −18.0586 −0.882219 −0.441109 0.897453i \(-0.645415\pi\)
−0.441109 + 0.897453i \(0.645415\pi\)
\(420\) 0 0
\(421\) 24.4956 1.19384 0.596921 0.802300i \(-0.296391\pi\)
0.596921 + 0.802300i \(0.296391\pi\)
\(422\) 9.67078 0.470766
\(423\) 2.09695 0.101957
\(424\) −4.29765 −0.208712
\(425\) 0 0
\(426\) −23.9151 −1.15869
\(427\) 7.51805 0.363824
\(428\) −8.36428 −0.404303
\(429\) 1.57542 0.0760621
\(430\) 0 0
\(431\) −40.7779 −1.96420 −0.982101 0.188357i \(-0.939684\pi\)
−0.982101 + 0.188357i \(0.939684\pi\)
\(432\) −2.08666 −0.100394
\(433\) 18.0791 0.868828 0.434414 0.900713i \(-0.356955\pi\)
0.434414 + 0.900713i \(0.356955\pi\)
\(434\) −17.3242 −0.831586
\(435\) 0 0
\(436\) 53.7186 2.57265
\(437\) 13.0797 0.625687
\(438\) 35.8823 1.71452
\(439\) 20.7561 0.990635 0.495317 0.868712i \(-0.335052\pi\)
0.495317 + 0.868712i \(0.335052\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 11.3242 0.538635
\(443\) 8.24042 0.391514 0.195757 0.980652i \(-0.437284\pi\)
0.195757 + 0.980652i \(0.437284\pi\)
\(444\) 38.3603 1.82050
\(445\) 0 0
\(446\) 0.152732 0.00723206
\(447\) 7.46471 0.353069
\(448\) 11.1801 0.528211
\(449\) 36.1061 1.70395 0.851975 0.523582i \(-0.175405\pi\)
0.851975 + 0.523582i \(0.175405\pi\)
\(450\) 0 0
\(451\) −3.54669 −0.167007
\(452\) 15.5180 0.729908
\(453\) −13.0155 −0.611522
\(454\) 4.01218 0.188301
\(455\) 0 0
\(456\) 8.84916 0.414400
\(457\) 6.54052 0.305953 0.152976 0.988230i \(-0.451114\pi\)
0.152976 + 0.988230i \(0.451114\pi\)
\(458\) 1.54266 0.0720838
\(459\) 4.75902 0.222132
\(460\) 0 0
\(461\) −14.6987 −0.684588 −0.342294 0.939593i \(-0.611204\pi\)
−0.342294 + 0.939593i \(0.611204\pi\)
\(462\) 3.74873 0.174407
\(463\) −21.7775 −1.01208 −0.506042 0.862509i \(-0.668892\pi\)
−0.506042 + 0.862509i \(0.668892\pi\)
\(464\) 8.84220 0.410489
\(465\) 0 0
\(466\) −7.39371 −0.342507
\(467\) −35.0086 −1.62000 −0.810001 0.586428i \(-0.800534\pi\)
−0.810001 + 0.586428i \(0.800534\pi\)
\(468\) −3.66208 −0.169280
\(469\) −15.6914 −0.724560
\(470\) 0 0
\(471\) −17.1233 −0.789000
\(472\) −48.7819 −2.24537
\(473\) −1.43890 −0.0661607
\(474\) 27.9725 1.28482
\(475\) 0 0
\(476\) 17.4279 0.798807
\(477\) −1.08666 −0.0497546
\(478\) −35.8496 −1.63972
\(479\) 31.7814 1.45213 0.726064 0.687628i \(-0.241348\pi\)
0.726064 + 0.687628i \(0.241348\pi\)
\(480\) 0 0
\(481\) 10.4750 0.477619
\(482\) 61.2201 2.78850
\(483\) −5.84568 −0.265988
\(484\) −31.1938 −1.41790
\(485\) 0 0
\(486\) −2.37951 −0.107937
\(487\) 28.5817 1.29516 0.647580 0.761998i \(-0.275781\pi\)
0.647580 + 0.761998i \(0.275781\pi\)
\(488\) −29.7334 −1.34597
\(489\) −0.849158 −0.0384002
\(490\) 0 0
\(491\) 21.7160 0.980028 0.490014 0.871715i \(-0.336992\pi\)
0.490014 + 0.871715i \(0.336992\pi\)
\(492\) 8.24431 0.371682
\(493\) −20.1664 −0.908247
\(494\) 5.32415 0.239545
\(495\) 0 0
\(496\) 15.1920 0.682141
\(497\) −10.0504 −0.450823
\(498\) 17.6586 0.791301
\(499\) −42.6895 −1.91104 −0.955522 0.294921i \(-0.904707\pi\)
−0.955522 + 0.294921i \(0.904707\pi\)
\(500\) 0 0
\(501\) 18.3781 0.821071
\(502\) −27.6638 −1.23470
\(503\) −21.8922 −0.976125 −0.488063 0.872809i \(-0.662296\pi\)
−0.488063 + 0.872809i \(0.662296\pi\)
\(504\) −3.95493 −0.176167
\(505\) 0 0
\(506\) −21.9139 −0.974192
\(507\) −1.00000 −0.0444116
\(508\) −17.7580 −0.787883
\(509\) −5.36546 −0.237820 −0.118910 0.992905i \(-0.537940\pi\)
−0.118910 + 0.992905i \(0.537940\pi\)
\(510\) 0 0
\(511\) 15.0797 0.667087
\(512\) −22.6496 −1.00098
\(513\) 2.23750 0.0987880
\(514\) 29.8989 1.31878
\(515\) 0 0
\(516\) 3.34474 0.147244
\(517\) 3.30357 0.145291
\(518\) 24.9254 1.09516
\(519\) −12.2565 −0.538000
\(520\) 0 0
\(521\) −11.5221 −0.504791 −0.252396 0.967624i \(-0.581218\pi\)
−0.252396 + 0.967624i \(0.581218\pi\)
\(522\) 10.0832 0.441329
\(523\) 3.62584 0.158547 0.0792734 0.996853i \(-0.474740\pi\)
0.0792734 + 0.996853i \(0.474740\pi\)
\(524\) −23.3172 −1.01862
\(525\) 0 0
\(526\) −22.2977 −0.972224
\(527\) −34.6483 −1.50930
\(528\) −3.28736 −0.143064
\(529\) 11.1720 0.485738
\(530\) 0 0
\(531\) −12.3344 −0.535269
\(532\) 8.19389 0.355250
\(533\) 2.25127 0.0975132
\(534\) 26.5008 1.14680
\(535\) 0 0
\(536\) 62.0583 2.68051
\(537\) −18.5146 −0.798963
\(538\) −24.7275 −1.06608
\(539\) 1.57542 0.0678582
\(540\) 0 0
\(541\) 37.2300 1.60064 0.800321 0.599572i \(-0.204662\pi\)
0.800321 + 0.599572i \(0.204662\pi\)
\(542\) −12.2075 −0.524358
\(543\) 18.1664 0.779593
\(544\) 14.0136 0.600829
\(545\) 0 0
\(546\) −2.37951 −0.101834
\(547\) 4.34529 0.185791 0.0928956 0.995676i \(-0.470388\pi\)
0.0928956 + 0.995676i \(0.470388\pi\)
\(548\) 66.0119 2.81989
\(549\) −7.51805 −0.320862
\(550\) 0 0
\(551\) −9.48140 −0.403921
\(552\) 23.1193 0.984022
\(553\) 11.7555 0.499897
\(554\) −19.1399 −0.813175
\(555\) 0 0
\(556\) 9.79915 0.415577
\(557\) −28.0052 −1.18662 −0.593310 0.804974i \(-0.702179\pi\)
−0.593310 + 0.804974i \(0.702179\pi\)
\(558\) 17.3242 0.733390
\(559\) 0.913344 0.0386303
\(560\) 0 0
\(561\) 7.49747 0.316543
\(562\) −66.2753 −2.79566
\(563\) 22.2525 0.937829 0.468915 0.883243i \(-0.344645\pi\)
0.468915 + 0.883243i \(0.344645\pi\)
\(564\) −7.67918 −0.323352
\(565\) 0 0
\(566\) −56.6793 −2.38241
\(567\) −1.00000 −0.0419961
\(568\) 39.7487 1.66782
\(569\) 3.10724 0.130262 0.0651311 0.997877i \(-0.479253\pi\)
0.0651311 + 0.997877i \(0.479253\pi\)
\(570\) 0 0
\(571\) 16.5772 0.693733 0.346866 0.937915i \(-0.387246\pi\)
0.346866 + 0.937915i \(0.387246\pi\)
\(572\) −5.76931 −0.241227
\(573\) −23.5151 −0.982358
\(574\) 5.35692 0.223593
\(575\) 0 0
\(576\) −11.1801 −0.465839
\(577\) 7.45253 0.310253 0.155126 0.987895i \(-0.450422\pi\)
0.155126 + 0.987895i \(0.450422\pi\)
\(578\) 13.4402 0.559039
\(579\) 14.7344 0.612341
\(580\) 0 0
\(581\) 7.42110 0.307879
\(582\) −34.7521 −1.44052
\(583\) −1.71194 −0.0709014
\(584\) −59.6392 −2.46789
\(585\) 0 0
\(586\) 39.1697 1.61809
\(587\) −11.9892 −0.494845 −0.247423 0.968908i \(-0.579584\pi\)
−0.247423 + 0.968908i \(0.579584\pi\)
\(588\) −3.66208 −0.151022
\(589\) −16.2902 −0.671227
\(590\) 0 0
\(591\) 14.5008 0.596483
\(592\) −21.8577 −0.898347
\(593\) 33.4767 1.37473 0.687363 0.726314i \(-0.258768\pi\)
0.687363 + 0.726314i \(0.258768\pi\)
\(594\) −3.74873 −0.153812
\(595\) 0 0
\(596\) −27.3363 −1.11974
\(597\) 13.5180 0.553257
\(598\) 13.9099 0.568817
\(599\) −13.5863 −0.555120 −0.277560 0.960708i \(-0.589526\pi\)
−0.277560 + 0.960708i \(0.589526\pi\)
\(600\) 0 0
\(601\) −6.78364 −0.276710 −0.138355 0.990383i \(-0.544182\pi\)
−0.138355 + 0.990383i \(0.544182\pi\)
\(602\) 2.17331 0.0885776
\(603\) 15.6914 0.639002
\(604\) 47.6638 1.93941
\(605\) 0 0
\(606\) 27.2094 1.10531
\(607\) 17.4905 0.709918 0.354959 0.934882i \(-0.384495\pi\)
0.354959 + 0.934882i \(0.384495\pi\)
\(608\) 6.58863 0.267204
\(609\) 4.23750 0.171712
\(610\) 0 0
\(611\) −2.09695 −0.0848334
\(612\) −17.4279 −0.704482
\(613\) 38.2455 1.54472 0.772361 0.635184i \(-0.219076\pi\)
0.772361 + 0.635184i \(0.219076\pi\)
\(614\) −4.65526 −0.187871
\(615\) 0 0
\(616\) −6.23069 −0.251041
\(617\) 34.1542 1.37500 0.687498 0.726187i \(-0.258709\pi\)
0.687498 + 0.726187i \(0.258709\pi\)
\(618\) 26.5336 1.06734
\(619\) −8.56805 −0.344379 −0.172190 0.985064i \(-0.555084\pi\)
−0.172190 + 0.985064i \(0.555084\pi\)
\(620\) 0 0
\(621\) 5.84568 0.234579
\(622\) −15.8853 −0.636941
\(623\) 11.1371 0.446197
\(624\) 2.08666 0.0835331
\(625\) 0 0
\(626\) 16.1417 0.645154
\(627\) 3.52500 0.140775
\(628\) 62.7069 2.50228
\(629\) 49.8508 1.98768
\(630\) 0 0
\(631\) −17.7119 −0.705101 −0.352551 0.935793i \(-0.614686\pi\)
−0.352551 + 0.935793i \(0.614686\pi\)
\(632\) −46.4924 −1.84937
\(633\) −4.06419 −0.161537
\(634\) 73.0384 2.90073
\(635\) 0 0
\(636\) 3.97942 0.157794
\(637\) −1.00000 −0.0396214
\(638\) 15.8853 0.628903
\(639\) 10.0504 0.397588
\(640\) 0 0
\(641\) 16.6117 0.656121 0.328061 0.944657i \(-0.393605\pi\)
0.328061 + 0.944657i \(0.393605\pi\)
\(642\) 5.43487 0.214497
\(643\) −12.9500 −0.510698 −0.255349 0.966849i \(-0.582190\pi\)
−0.255349 + 0.966849i \(0.582190\pi\)
\(644\) 21.4073 0.843567
\(645\) 0 0
\(646\) 25.3378 0.996902
\(647\) 23.6775 0.930857 0.465428 0.885086i \(-0.345900\pi\)
0.465428 + 0.885086i \(0.345900\pi\)
\(648\) 3.95493 0.155364
\(649\) −19.4319 −0.762771
\(650\) 0 0
\(651\) 7.28055 0.285347
\(652\) 3.10968 0.121785
\(653\) −22.2811 −0.871927 −0.435963 0.899964i \(-0.643592\pi\)
−0.435963 + 0.899964i \(0.643592\pi\)
\(654\) −34.9048 −1.36489
\(655\) 0 0
\(656\) −4.69762 −0.183411
\(657\) −15.0797 −0.588315
\(658\) −4.98971 −0.194519
\(659\) 15.5165 0.604435 0.302218 0.953239i \(-0.402273\pi\)
0.302218 + 0.953239i \(0.402273\pi\)
\(660\) 0 0
\(661\) 28.1444 1.09469 0.547346 0.836906i \(-0.315638\pi\)
0.547346 + 0.836906i \(0.315638\pi\)
\(662\) −32.9008 −1.27872
\(663\) −4.75902 −0.184825
\(664\) −29.3500 −1.13900
\(665\) 0 0
\(666\) −24.9254 −0.965839
\(667\) −24.7711 −0.959139
\(668\) −67.3018 −2.60399
\(669\) −0.0641862 −0.00248158
\(670\) 0 0
\(671\) −11.8441 −0.457236
\(672\) −2.94464 −0.113592
\(673\) −7.13477 −0.275025 −0.137513 0.990500i \(-0.543911\pi\)
−0.137513 + 0.990500i \(0.543911\pi\)
\(674\) −56.7831 −2.18720
\(675\) 0 0
\(676\) 3.66208 0.140849
\(677\) 10.7660 0.413770 0.206885 0.978365i \(-0.433667\pi\)
0.206885 + 0.978365i \(0.433667\pi\)
\(678\) −10.0832 −0.387242
\(679\) −14.6047 −0.560477
\(680\) 0 0
\(681\) −1.68614 −0.0646129
\(682\) 27.2928 1.04510
\(683\) 12.2020 0.466898 0.233449 0.972369i \(-0.424999\pi\)
0.233449 + 0.972369i \(0.424999\pi\)
\(684\) −8.19389 −0.313301
\(685\) 0 0
\(686\) −2.37951 −0.0908502
\(687\) −0.648310 −0.0247346
\(688\) −1.90583 −0.0726593
\(689\) 1.08666 0.0413983
\(690\) 0 0
\(691\) 32.0367 1.21873 0.609366 0.792889i \(-0.291424\pi\)
0.609366 + 0.792889i \(0.291424\pi\)
\(692\) 44.8842 1.70624
\(693\) −1.57542 −0.0598453
\(694\) 32.3643 1.22853
\(695\) 0 0
\(696\) −16.7590 −0.635249
\(697\) 10.7138 0.405815
\(698\) −39.0332 −1.47743
\(699\) 3.10724 0.117526
\(700\) 0 0
\(701\) −14.9513 −0.564704 −0.282352 0.959311i \(-0.591115\pi\)
−0.282352 + 0.959311i \(0.591115\pi\)
\(702\) 2.37951 0.0898089
\(703\) 23.4378 0.883973
\(704\) −17.6134 −0.663830
\(705\) 0 0
\(706\) 11.2627 0.423879
\(707\) 11.4349 0.430053
\(708\) 45.1697 1.69758
\(709\) −8.58279 −0.322333 −0.161167 0.986927i \(-0.551526\pi\)
−0.161167 + 0.986927i \(0.551526\pi\)
\(710\) 0 0
\(711\) −11.7555 −0.440867
\(712\) −44.0464 −1.65071
\(713\) −42.5598 −1.59388
\(714\) −11.3242 −0.423796
\(715\) 0 0
\(716\) 67.8018 2.53387
\(717\) 15.0659 0.562648
\(718\) −41.7554 −1.55830
\(719\) 35.9644 1.34125 0.670623 0.741798i \(-0.266027\pi\)
0.670623 + 0.741798i \(0.266027\pi\)
\(720\) 0 0
\(721\) 11.1508 0.415279
\(722\) −33.2979 −1.23922
\(723\) −25.7280 −0.956835
\(724\) −66.5266 −2.47244
\(725\) 0 0
\(726\) 20.2688 0.752246
\(727\) 31.5111 1.16868 0.584341 0.811508i \(-0.301353\pi\)
0.584341 + 0.811508i \(0.301353\pi\)
\(728\) 3.95493 0.146580
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.34662 0.160766
\(732\) 27.5317 1.01760
\(733\) 26.6047 0.982667 0.491334 0.870971i \(-0.336510\pi\)
0.491334 + 0.870971i \(0.336510\pi\)
\(734\) 26.6240 0.982708
\(735\) 0 0
\(736\) 17.2134 0.634496
\(737\) 24.7205 0.910591
\(738\) −5.35692 −0.197191
\(739\) −50.3397 −1.85177 −0.925887 0.377800i \(-0.876681\pi\)
−0.925887 + 0.377800i \(0.876681\pi\)
\(740\) 0 0
\(741\) −2.23750 −0.0821966
\(742\) 2.58571 0.0949245
\(743\) −23.4021 −0.858540 −0.429270 0.903176i \(-0.641229\pi\)
−0.429270 + 0.903176i \(0.641229\pi\)
\(744\) −28.7941 −1.05564
\(745\) 0 0
\(746\) 12.9254 0.473232
\(747\) −7.42110 −0.271524
\(748\) −27.4563 −1.00390
\(749\) 2.28403 0.0834565
\(750\) 0 0
\(751\) 49.0305 1.78915 0.894574 0.446920i \(-0.147479\pi\)
0.894574 + 0.446920i \(0.147479\pi\)
\(752\) 4.37561 0.159562
\(753\) 11.6258 0.423669
\(754\) −10.0832 −0.367208
\(755\) 0 0
\(756\) 3.66208 0.133188
\(757\) 35.3883 1.28621 0.643106 0.765778i \(-0.277646\pi\)
0.643106 + 0.765778i \(0.277646\pi\)
\(758\) 25.7992 0.937067
\(759\) 9.20941 0.334280
\(760\) 0 0
\(761\) 27.7051 1.00431 0.502155 0.864778i \(-0.332541\pi\)
0.502155 + 0.864778i \(0.332541\pi\)
\(762\) 11.5386 0.418000
\(763\) −14.6689 −0.531049
\(764\) 86.1142 3.11550
\(765\) 0 0
\(766\) −25.0689 −0.905775
\(767\) 12.3344 0.445371
\(768\) 26.9289 0.971712
\(769\) 21.1383 0.762265 0.381133 0.924520i \(-0.375534\pi\)
0.381133 + 0.924520i \(0.375534\pi\)
\(770\) 0 0
\(771\) −12.5651 −0.452522
\(772\) −53.9586 −1.94201
\(773\) 41.4537 1.49099 0.745493 0.666513i \(-0.232214\pi\)
0.745493 + 0.666513i \(0.232214\pi\)
\(774\) −2.17331 −0.0781181
\(775\) 0 0
\(776\) 57.7606 2.07349
\(777\) −10.4750 −0.375788
\(778\) 14.2771 0.511858
\(779\) 5.03721 0.180477
\(780\) 0 0
\(781\) 15.8336 0.566572
\(782\) 66.1974 2.36721
\(783\) −4.23750 −0.151436
\(784\) 2.08666 0.0745234
\(785\) 0 0
\(786\) 15.1508 0.540413
\(787\) 28.7711 1.02558 0.512789 0.858515i \(-0.328612\pi\)
0.512789 + 0.858515i \(0.328612\pi\)
\(788\) −53.1031 −1.89172
\(789\) 9.37068 0.333605
\(790\) 0 0
\(791\) −4.23750 −0.150668
\(792\) 6.23069 0.221398
\(793\) 7.51805 0.266974
\(794\) 84.3298 2.99275
\(795\) 0 0
\(796\) −49.5041 −1.75463
\(797\) −19.5357 −0.691990 −0.345995 0.938236i \(-0.612459\pi\)
−0.345995 + 0.938236i \(0.612459\pi\)
\(798\) −5.32415 −0.188473
\(799\) −9.97942 −0.353046
\(800\) 0 0
\(801\) −11.1371 −0.393509
\(802\) 56.4518 1.99338
\(803\) −23.7569 −0.838362
\(804\) −57.4630 −2.02656
\(805\) 0 0
\(806\) −17.3242 −0.610217
\(807\) 10.3918 0.365809
\(808\) −45.2241 −1.59098
\(809\) −36.1824 −1.27211 −0.636053 0.771645i \(-0.719434\pi\)
−0.636053 + 0.771645i \(0.719434\pi\)
\(810\) 0 0
\(811\) 0.794087 0.0278842 0.0139421 0.999903i \(-0.495562\pi\)
0.0139421 + 0.999903i \(0.495562\pi\)
\(812\) −15.5180 −0.544577
\(813\) 5.13026 0.179926
\(814\) −39.2680 −1.37634
\(815\) 0 0
\(816\) 9.93045 0.347635
\(817\) 2.04361 0.0714967
\(818\) −9.41429 −0.329163
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −15.8525 −0.553256 −0.276628 0.960977i \(-0.589217\pi\)
−0.276628 + 0.960977i \(0.589217\pi\)
\(822\) −42.8926 −1.49605
\(823\) 20.0412 0.698591 0.349295 0.937013i \(-0.386421\pi\)
0.349295 + 0.937013i \(0.386421\pi\)
\(824\) −44.1008 −1.53633
\(825\) 0 0
\(826\) 29.3500 1.02122
\(827\) −34.7848 −1.20959 −0.604794 0.796382i \(-0.706744\pi\)
−0.604794 + 0.796382i \(0.706744\pi\)
\(828\) −21.4073 −0.743956
\(829\) −30.6793 −1.06554 −0.532769 0.846261i \(-0.678848\pi\)
−0.532769 + 0.846261i \(0.678848\pi\)
\(830\) 0 0
\(831\) 8.04361 0.279030
\(832\) 11.1801 0.387601
\(833\) −4.75902 −0.164890
\(834\) −6.36721 −0.220478
\(835\) 0 0
\(836\) −12.9088 −0.446461
\(837\) −7.28055 −0.251653
\(838\) −42.9706 −1.48439
\(839\) 39.8238 1.37487 0.687436 0.726245i \(-0.258736\pi\)
0.687436 + 0.726245i \(0.258736\pi\)
\(840\) 0 0
\(841\) −11.0436 −0.380814
\(842\) 58.2875 2.00872
\(843\) 27.8525 0.959291
\(844\) 14.8834 0.512307
\(845\) 0 0
\(846\) 4.98971 0.171550
\(847\) 8.51805 0.292684
\(848\) −2.26748 −0.0778655
\(849\) 23.8197 0.817491
\(850\) 0 0
\(851\) 61.2335 2.09906
\(852\) −36.8054 −1.26093
\(853\) −24.7505 −0.847440 −0.423720 0.905793i \(-0.639276\pi\)
−0.423720 + 0.905793i \(0.639276\pi\)
\(854\) 17.8893 0.612159
\(855\) 0 0
\(856\) −9.03317 −0.308748
\(857\) −6.47097 −0.221044 −0.110522 0.993874i \(-0.535252\pi\)
−0.110522 + 0.993874i \(0.535252\pi\)
\(858\) 3.74873 0.127980
\(859\) 31.0741 1.06023 0.530117 0.847925i \(-0.322148\pi\)
0.530117 + 0.847925i \(0.322148\pi\)
\(860\) 0 0
\(861\) −2.25127 −0.0767230
\(862\) −97.0314 −3.30490
\(863\) −28.6391 −0.974885 −0.487442 0.873155i \(-0.662070\pi\)
−0.487442 + 0.873155i \(0.662070\pi\)
\(864\) 2.94464 0.100179
\(865\) 0 0
\(866\) 43.0195 1.46186
\(867\) −5.64831 −0.191827
\(868\) −26.6619 −0.904965
\(869\) −18.5199 −0.628246
\(870\) 0 0
\(871\) −15.6914 −0.531681
\(872\) 58.0145 1.96462
\(873\) 14.6047 0.494294
\(874\) 31.1233 1.05276
\(875\) 0 0
\(876\) 55.2230 1.86581
\(877\) −0.00110921 −3.74555e−5 0 −1.87278e−5 1.00000i \(-0.500006\pi\)
−1.87278e−5 1.00000i \(0.500006\pi\)
\(878\) 49.3894 1.66681
\(879\) −16.4612 −0.555223
\(880\) 0 0
\(881\) 23.9479 0.806824 0.403412 0.915019i \(-0.367824\pi\)
0.403412 + 0.915019i \(0.367824\pi\)
\(882\) 2.37951 0.0801223
\(883\) −51.9965 −1.74982 −0.874911 0.484283i \(-0.839081\pi\)
−0.874911 + 0.484283i \(0.839081\pi\)
\(884\) 17.4279 0.586164
\(885\) 0 0
\(886\) 19.6082 0.658750
\(887\) −10.8697 −0.364970 −0.182485 0.983209i \(-0.558414\pi\)
−0.182485 + 0.983209i \(0.558414\pi\)
\(888\) 41.4279 1.39023
\(889\) 4.84916 0.162636
\(890\) 0 0
\(891\) 1.57542 0.0527786
\(892\) 0.235055 0.00787021
\(893\) −4.69192 −0.157009
\(894\) 17.7624 0.594062
\(895\) 0 0
\(896\) 20.7140 0.692005
\(897\) −5.84568 −0.195182
\(898\) 85.9148 2.86701
\(899\) 30.8513 1.02895
\(900\) 0 0
\(901\) 5.17142 0.172285
\(902\) −8.43940 −0.281001
\(903\) −0.913344 −0.0303942
\(904\) 16.7590 0.557397
\(905\) 0 0
\(906\) −30.9706 −1.02893
\(907\) 25.4892 0.846354 0.423177 0.906047i \(-0.360915\pi\)
0.423177 + 0.906047i \(0.360915\pi\)
\(908\) 6.17476 0.204917
\(909\) −11.4349 −0.379271
\(910\) 0 0
\(911\) 42.2059 1.39834 0.699172 0.714953i \(-0.253552\pi\)
0.699172 + 0.714953i \(0.253552\pi\)
\(912\) 4.66889 0.154602
\(913\) −11.6914 −0.386928
\(914\) 15.5632 0.514786
\(915\) 0 0
\(916\) 2.37416 0.0784445
\(917\) 6.36721 0.210264
\(918\) 11.3242 0.373753
\(919\) −29.4033 −0.969925 −0.484963 0.874535i \(-0.661167\pi\)
−0.484963 + 0.874535i \(0.661167\pi\)
\(920\) 0 0
\(921\) 1.95639 0.0644654
\(922\) −34.9758 −1.15187
\(923\) −10.0504 −0.330814
\(924\) 5.76931 0.189797
\(925\) 0 0
\(926\) −51.8197 −1.70290
\(927\) −11.1508 −0.366242
\(928\) −12.4779 −0.409608
\(929\) −13.3759 −0.438849 −0.219425 0.975629i \(-0.570418\pi\)
−0.219425 + 0.975629i \(0.570418\pi\)
\(930\) 0 0
\(931\) −2.23750 −0.0733311
\(932\) −11.3789 −0.372730
\(933\) 6.67585 0.218557
\(934\) −83.3033 −2.72577
\(935\) 0 0
\(936\) −3.95493 −0.129271
\(937\) 24.9981 0.816653 0.408326 0.912836i \(-0.366113\pi\)
0.408326 + 0.912836i \(0.366113\pi\)
\(938\) −37.3378 −1.21912
\(939\) −6.78364 −0.221376
\(940\) 0 0
\(941\) 10.5369 0.343493 0.171746 0.985141i \(-0.445059\pi\)
0.171746 + 0.985141i \(0.445059\pi\)
\(942\) −40.7451 −1.32755
\(943\) 13.1602 0.428555
\(944\) −25.7377 −0.837692
\(945\) 0 0
\(946\) −3.42388 −0.111320
\(947\) 52.7779 1.71505 0.857525 0.514442i \(-0.172001\pi\)
0.857525 + 0.514442i \(0.172001\pi\)
\(948\) 43.0497 1.39819
\(949\) 15.0797 0.489508
\(950\) 0 0
\(951\) −30.6947 −0.995344
\(952\) 18.8216 0.610012
\(953\) −21.8486 −0.707746 −0.353873 0.935294i \(-0.615135\pi\)
−0.353873 + 0.935294i \(0.615135\pi\)
\(954\) −2.58571 −0.0837155
\(955\) 0 0
\(956\) −55.1726 −1.78441
\(957\) −6.67585 −0.215799
\(958\) 75.6241 2.44330
\(959\) −18.0258 −0.582084
\(960\) 0 0
\(961\) 22.0064 0.709884
\(962\) 24.9254 0.803627
\(963\) −2.28403 −0.0736017
\(964\) 94.2180 3.03456
\(965\) 0 0
\(966\) −13.9099 −0.447543
\(967\) −1.94143 −0.0624323 −0.0312162 0.999513i \(-0.509938\pi\)
−0.0312162 + 0.999513i \(0.509938\pi\)
\(968\) −33.6883 −1.08278
\(969\) −10.6483 −0.342073
\(970\) 0 0
\(971\) −14.9077 −0.478412 −0.239206 0.970969i \(-0.576887\pi\)
−0.239206 + 0.970969i \(0.576887\pi\)
\(972\) −3.66208 −0.117461
\(973\) −2.67585 −0.0857837
\(974\) 68.0105 2.17920
\(975\) 0 0
\(976\) −15.6876 −0.502147
\(977\) 16.5353 0.529011 0.264505 0.964384i \(-0.414791\pi\)
0.264505 + 0.964384i \(0.414791\pi\)
\(978\) −2.02058 −0.0646110
\(979\) −17.5456 −0.560759
\(980\) 0 0
\(981\) 14.6689 0.468342
\(982\) 51.6734 1.64897
\(983\) 30.0694 0.959065 0.479533 0.877524i \(-0.340806\pi\)
0.479533 + 0.877524i \(0.340806\pi\)
\(984\) 8.90361 0.283837
\(985\) 0 0
\(986\) −47.9861 −1.52819
\(987\) 2.09695 0.0667465
\(988\) 8.19389 0.260682
\(989\) 5.33912 0.169774
\(990\) 0 0
\(991\) 43.5902 1.38469 0.692345 0.721567i \(-0.256578\pi\)
0.692345 + 0.721567i \(0.256578\pi\)
\(992\) −21.4386 −0.680677
\(993\) 13.8267 0.438777
\(994\) −23.9151 −0.758541
\(995\) 0 0
\(996\) 27.1766 0.861125
\(997\) −57.8577 −1.83237 −0.916186 0.400753i \(-0.868749\pi\)
−0.916186 + 0.400753i \(0.868749\pi\)
\(998\) −101.580 −3.21546
\(999\) 10.4750 0.331414
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 6825.2.a.bg.1.4 4
5.4 even 2 273.2.a.e.1.1 4
15.14 odd 2 819.2.a.k.1.4 4
20.19 odd 2 4368.2.a.br.1.2 4
35.34 odd 2 1911.2.a.s.1.1 4
65.64 even 2 3549.2.a.w.1.4 4
105.104 even 2 5733.2.a.bf.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.1 4 5.4 even 2
819.2.a.k.1.4 4 15.14 odd 2
1911.2.a.s.1.1 4 35.34 odd 2
3549.2.a.w.1.4 4 65.64 even 2
4368.2.a.br.1.2 4 20.19 odd 2
5733.2.a.bf.1.4 4 105.104 even 2
6825.2.a.bg.1.4 4 1.1 even 1 trivial