Properties

Label 3750.2.a.u.1.6
Level $3750$
Weight $2$
Character 3750.1
Self dual yes
Analytic conductor $29.944$
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] = [3750,2,Mod(1,3750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3750.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3750.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: \(29.9439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.71684000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 18x^{6} + 10x^{5} + 101x^{4} + 40x^{3} - 132x^{2} - 96x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 150)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.74919\) of defining polynomial
Character \(\chi\) \(=\) 3750.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +0.533559 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +0.533559 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.43425 q^{11} -1.00000 q^{12} -6.57392 q^{13} -0.533559 q^{14} +1.00000 q^{16} +0.958413 q^{17} -1.00000 q^{18} +0.212889 q^{19} -0.533559 q^{21} +1.43425 q^{22} -3.76401 q^{23} +1.00000 q^{24} +6.57392 q^{26} -1.00000 q^{27} +0.533559 q^{28} +6.19448 q^{29} -2.33773 q^{31} -1.00000 q^{32} +1.43425 q^{33} -0.958413 q^{34} +1.00000 q^{36} -4.06291 q^{37} -0.212889 q^{38} +6.57392 q^{39} -7.94156 q^{41} +0.533559 q^{42} +11.3607 q^{43} -1.43425 q^{44} +3.76401 q^{46} -10.1489 q^{47} -1.00000 q^{48} -6.71531 q^{49} -0.958413 q^{51} -6.57392 q^{52} +3.23354 q^{53} +1.00000 q^{54} -0.533559 q^{56} -0.212889 q^{57} -6.19448 q^{58} +7.52455 q^{59} +12.5721 q^{61} +2.33773 q^{62} +0.533559 q^{63} +1.00000 q^{64} -1.43425 q^{66} +6.91285 q^{67} +0.958413 q^{68} +3.76401 q^{69} +10.1247 q^{71} -1.00000 q^{72} +13.9771 q^{73} +4.06291 q^{74} +0.212889 q^{76} -0.765259 q^{77} -6.57392 q^{78} +15.5279 q^{79} +1.00000 q^{81} +7.94156 q^{82} -16.3308 q^{83} -0.533559 q^{84} -11.3607 q^{86} -6.19448 q^{87} +1.43425 q^{88} +5.62220 q^{89} -3.50758 q^{91} -3.76401 q^{92} +2.33773 q^{93} +10.1489 q^{94} +1.00000 q^{96} +5.56558 q^{97} +6.71531 q^{98} -1.43425 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{6} - 4 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{6} - 4 q^{7} - 8 q^{8} + 8 q^{9} + 6 q^{11} - 8 q^{12} - 2 q^{13} + 4 q^{14} + 8 q^{16} - 14 q^{17} - 8 q^{18} + 10 q^{19} + 4 q^{21} - 6 q^{22} - 12 q^{23} + 8 q^{24} + 2 q^{26} - 8 q^{27} - 4 q^{28} + 10 q^{29} + 16 q^{31} - 8 q^{32} - 6 q^{33} + 14 q^{34} + 8 q^{36} + 6 q^{37} - 10 q^{38} + 2 q^{39} + 6 q^{41} - 4 q^{42} - 2 q^{43} + 6 q^{44} + 12 q^{46} - 14 q^{47} - 8 q^{48} + 26 q^{49} + 14 q^{51} - 2 q^{52} - 12 q^{53} + 8 q^{54} + 4 q^{56} - 10 q^{57} - 10 q^{58} + 16 q^{61} - 16 q^{62} - 4 q^{63} + 8 q^{64} + 6 q^{66} + 6 q^{67} - 14 q^{68} + 12 q^{69} + 6 q^{71} - 8 q^{72} + 8 q^{73} - 6 q^{74} + 10 q^{76} - 8 q^{77} - 2 q^{78} + 10 q^{79} + 8 q^{81} - 6 q^{82} - 22 q^{83} + 4 q^{84} + 2 q^{86} - 10 q^{87} - 6 q^{88} + 20 q^{89} + 6 q^{91} - 12 q^{92} - 16 q^{93} + 14 q^{94} + 8 q^{96} + 16 q^{97} - 26 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0.533559 0.201666 0.100833 0.994903i \(-0.467849\pi\)
0.100833 + 0.994903i \(0.467849\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.43425 −0.432444 −0.216222 0.976344i \(-0.569373\pi\)
−0.216222 + 0.976344i \(0.569373\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.57392 −1.82328 −0.911639 0.410991i \(-0.865183\pi\)
−0.911639 + 0.410991i \(0.865183\pi\)
\(14\) −0.533559 −0.142600
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.958413 0.232449 0.116225 0.993223i \(-0.462921\pi\)
0.116225 + 0.993223i \(0.462921\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.212889 0.0488401 0.0244201 0.999702i \(-0.492226\pi\)
0.0244201 + 0.999702i \(0.492226\pi\)
\(20\) 0 0
\(21\) −0.533559 −0.116432
\(22\) 1.43425 0.305784
\(23\) −3.76401 −0.784850 −0.392425 0.919784i \(-0.628364\pi\)
−0.392425 + 0.919784i \(0.628364\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.57392 1.28925
\(27\) −1.00000 −0.192450
\(28\) 0.533559 0.100833
\(29\) 6.19448 1.15029 0.575143 0.818053i \(-0.304946\pi\)
0.575143 + 0.818053i \(0.304946\pi\)
\(30\) 0 0
\(31\) −2.33773 −0.419869 −0.209934 0.977716i \(-0.567325\pi\)
−0.209934 + 0.977716i \(0.567325\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.43425 0.249671
\(34\) −0.958413 −0.164367
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.06291 −0.667938 −0.333969 0.942584i \(-0.608388\pi\)
−0.333969 + 0.942584i \(0.608388\pi\)
\(38\) −0.212889 −0.0345352
\(39\) 6.57392 1.05267
\(40\) 0 0
\(41\) −7.94156 −1.24026 −0.620132 0.784497i \(-0.712921\pi\)
−0.620132 + 0.784497i \(0.712921\pi\)
\(42\) 0.533559 0.0823300
\(43\) 11.3607 1.73250 0.866248 0.499614i \(-0.166525\pi\)
0.866248 + 0.499614i \(0.166525\pi\)
\(44\) −1.43425 −0.216222
\(45\) 0 0
\(46\) 3.76401 0.554973
\(47\) −10.1489 −1.48037 −0.740186 0.672402i \(-0.765263\pi\)
−0.740186 + 0.672402i \(0.765263\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.71531 −0.959331
\(50\) 0 0
\(51\) −0.958413 −0.134205
\(52\) −6.57392 −0.911639
\(53\) 3.23354 0.444162 0.222081 0.975028i \(-0.428715\pi\)
0.222081 + 0.975028i \(0.428715\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −0.533559 −0.0712998
\(57\) −0.212889 −0.0281978
\(58\) −6.19448 −0.813375
\(59\) 7.52455 0.979613 0.489806 0.871831i \(-0.337067\pi\)
0.489806 + 0.871831i \(0.337067\pi\)
\(60\) 0 0
\(61\) 12.5721 1.60969 0.804844 0.593486i \(-0.202249\pi\)
0.804844 + 0.593486i \(0.202249\pi\)
\(62\) 2.33773 0.296892
\(63\) 0.533559 0.0672221
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.43425 −0.176544
\(67\) 6.91285 0.844539 0.422269 0.906470i \(-0.361234\pi\)
0.422269 + 0.906470i \(0.361234\pi\)
\(68\) 0.958413 0.116225
\(69\) 3.76401 0.453134
\(70\) 0 0
\(71\) 10.1247 1.20158 0.600789 0.799408i \(-0.294853\pi\)
0.600789 + 0.799408i \(0.294853\pi\)
\(72\) −1.00000 −0.117851
\(73\) 13.9771 1.63589 0.817947 0.575294i \(-0.195112\pi\)
0.817947 + 0.575294i \(0.195112\pi\)
\(74\) 4.06291 0.472304
\(75\) 0 0
\(76\) 0.212889 0.0244201
\(77\) −0.765259 −0.0872093
\(78\) −6.57392 −0.744350
\(79\) 15.5279 1.74703 0.873515 0.486798i \(-0.161835\pi\)
0.873515 + 0.486798i \(0.161835\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.94156 0.876999
\(83\) −16.3308 −1.79254 −0.896270 0.443508i \(-0.853734\pi\)
−0.896270 + 0.443508i \(0.853734\pi\)
\(84\) −0.533559 −0.0582161
\(85\) 0 0
\(86\) −11.3607 −1.22506
\(87\) −6.19448 −0.664118
\(88\) 1.43425 0.152892
\(89\) 5.62220 0.595951 0.297976 0.954573i \(-0.403689\pi\)
0.297976 + 0.954573i \(0.403689\pi\)
\(90\) 0 0
\(91\) −3.50758 −0.367694
\(92\) −3.76401 −0.392425
\(93\) 2.33773 0.242411
\(94\) 10.1489 1.04678
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 5.56558 0.565099 0.282549 0.959253i \(-0.408820\pi\)
0.282549 + 0.959253i \(0.408820\pi\)
\(98\) 6.71531 0.678349
\(99\) −1.43425 −0.144148
\(100\) 0 0
\(101\) −9.42708 −0.938029 −0.469015 0.883190i \(-0.655391\pi\)
−0.469015 + 0.883190i \(0.655391\pi\)
\(102\) 0.958413 0.0948971
\(103\) −7.99871 −0.788137 −0.394068 0.919081i \(-0.628933\pi\)
−0.394068 + 0.919081i \(0.628933\pi\)
\(104\) 6.57392 0.644626
\(105\) 0 0
\(106\) −3.23354 −0.314070
\(107\) −18.9260 −1.82964 −0.914822 0.403857i \(-0.867669\pi\)
−0.914822 + 0.403857i \(0.867669\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.65524 −0.254326 −0.127163 0.991882i \(-0.540587\pi\)
−0.127163 + 0.991882i \(0.540587\pi\)
\(110\) 0 0
\(111\) 4.06291 0.385634
\(112\) 0.533559 0.0504166
\(113\) −3.06283 −0.288127 −0.144063 0.989568i \(-0.546017\pi\)
−0.144063 + 0.989568i \(0.546017\pi\)
\(114\) 0.212889 0.0199389
\(115\) 0 0
\(116\) 6.19448 0.575143
\(117\) −6.57392 −0.607759
\(118\) −7.52455 −0.692691
\(119\) 0.511370 0.0468772
\(120\) 0 0
\(121\) −8.94292 −0.812993
\(122\) −12.5721 −1.13822
\(123\) 7.94156 0.716067
\(124\) −2.33773 −0.209934
\(125\) 0 0
\(126\) −0.533559 −0.0475332
\(127\) 6.48257 0.575235 0.287618 0.957745i \(-0.407137\pi\)
0.287618 + 0.957745i \(0.407137\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −11.3607 −1.00026
\(130\) 0 0
\(131\) −9.45794 −0.826344 −0.413172 0.910653i \(-0.635579\pi\)
−0.413172 + 0.910653i \(0.635579\pi\)
\(132\) 1.43425 0.124836
\(133\) 0.113589 0.00984941
\(134\) −6.91285 −0.597179
\(135\) 0 0
\(136\) −0.958413 −0.0821833
\(137\) 12.3361 1.05394 0.526971 0.849883i \(-0.323328\pi\)
0.526971 + 0.849883i \(0.323328\pi\)
\(138\) −3.76401 −0.320414
\(139\) 4.53918 0.385008 0.192504 0.981296i \(-0.438339\pi\)
0.192504 + 0.981296i \(0.438339\pi\)
\(140\) 0 0
\(141\) 10.1489 0.854693
\(142\) −10.1247 −0.849643
\(143\) 9.42867 0.788465
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −13.9771 −1.15675
\(147\) 6.71531 0.553870
\(148\) −4.06291 −0.333969
\(149\) 11.0750 0.907303 0.453651 0.891179i \(-0.350121\pi\)
0.453651 + 0.891179i \(0.350121\pi\)
\(150\) 0 0
\(151\) −1.63387 −0.132962 −0.0664812 0.997788i \(-0.521177\pi\)
−0.0664812 + 0.997788i \(0.521177\pi\)
\(152\) −0.212889 −0.0172676
\(153\) 0.958413 0.0774831
\(154\) 0.765259 0.0616663
\(155\) 0 0
\(156\) 6.57392 0.526335
\(157\) −6.64544 −0.530364 −0.265182 0.964198i \(-0.585432\pi\)
−0.265182 + 0.964198i \(0.585432\pi\)
\(158\) −15.5279 −1.23534
\(159\) −3.23354 −0.256437
\(160\) 0 0
\(161\) −2.00832 −0.158278
\(162\) −1.00000 −0.0785674
\(163\) 10.1134 0.792142 0.396071 0.918220i \(-0.370373\pi\)
0.396071 + 0.918220i \(0.370373\pi\)
\(164\) −7.94156 −0.620132
\(165\) 0 0
\(166\) 16.3308 1.26752
\(167\) −12.7885 −0.989600 −0.494800 0.869007i \(-0.664759\pi\)
−0.494800 + 0.869007i \(0.664759\pi\)
\(168\) 0.533559 0.0411650
\(169\) 30.2165 2.32434
\(170\) 0 0
\(171\) 0.212889 0.0162800
\(172\) 11.3607 0.866248
\(173\) 13.6575 1.03836 0.519182 0.854664i \(-0.326237\pi\)
0.519182 + 0.854664i \(0.326237\pi\)
\(174\) 6.19448 0.469602
\(175\) 0 0
\(176\) −1.43425 −0.108111
\(177\) −7.52455 −0.565580
\(178\) −5.62220 −0.421401
\(179\) 2.99142 0.223589 0.111795 0.993731i \(-0.464340\pi\)
0.111795 + 0.993731i \(0.464340\pi\)
\(180\) 0 0
\(181\) 7.62285 0.566602 0.283301 0.959031i \(-0.408570\pi\)
0.283301 + 0.959031i \(0.408570\pi\)
\(182\) 3.50758 0.259999
\(183\) −12.5721 −0.929354
\(184\) 3.76401 0.277487
\(185\) 0 0
\(186\) −2.33773 −0.171411
\(187\) −1.37461 −0.100521
\(188\) −10.1489 −0.740186
\(189\) −0.533559 −0.0388107
\(190\) 0 0
\(191\) −8.40045 −0.607835 −0.303917 0.952698i \(-0.598295\pi\)
−0.303917 + 0.952698i \(0.598295\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.5266 0.757723 0.378861 0.925453i \(-0.376316\pi\)
0.378861 + 0.925453i \(0.376316\pi\)
\(194\) −5.56558 −0.399585
\(195\) 0 0
\(196\) −6.71531 −0.479665
\(197\) 10.2080 0.727293 0.363647 0.931537i \(-0.381532\pi\)
0.363647 + 0.931537i \(0.381532\pi\)
\(198\) 1.43425 0.101928
\(199\) 3.84318 0.272436 0.136218 0.990679i \(-0.456505\pi\)
0.136218 + 0.990679i \(0.456505\pi\)
\(200\) 0 0
\(201\) −6.91285 −0.487595
\(202\) 9.42708 0.663287
\(203\) 3.30512 0.231974
\(204\) −0.958413 −0.0671023
\(205\) 0 0
\(206\) 7.99871 0.557297
\(207\) −3.76401 −0.261617
\(208\) −6.57392 −0.455820
\(209\) −0.305337 −0.0211206
\(210\) 0 0
\(211\) 5.24920 0.361370 0.180685 0.983541i \(-0.442169\pi\)
0.180685 + 0.983541i \(0.442169\pi\)
\(212\) 3.23354 0.222081
\(213\) −10.1247 −0.693731
\(214\) 18.9260 1.29375
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −1.24732 −0.0846734
\(218\) 2.65524 0.179836
\(219\) −13.9771 −0.944484
\(220\) 0 0
\(221\) −6.30054 −0.423820
\(222\) −4.06291 −0.272685
\(223\) 19.2782 1.29097 0.645483 0.763774i \(-0.276656\pi\)
0.645483 + 0.763774i \(0.276656\pi\)
\(224\) −0.533559 −0.0356499
\(225\) 0 0
\(226\) 3.06283 0.203736
\(227\) −8.43957 −0.560154 −0.280077 0.959978i \(-0.590360\pi\)
−0.280077 + 0.959978i \(0.590360\pi\)
\(228\) −0.212889 −0.0140989
\(229\) 5.95568 0.393563 0.196781 0.980447i \(-0.436951\pi\)
0.196781 + 0.980447i \(0.436951\pi\)
\(230\) 0 0
\(231\) 0.765259 0.0503503
\(232\) −6.19448 −0.406688
\(233\) 2.09293 0.137112 0.0685561 0.997647i \(-0.478161\pi\)
0.0685561 + 0.997647i \(0.478161\pi\)
\(234\) 6.57392 0.429751
\(235\) 0 0
\(236\) 7.52455 0.489806
\(237\) −15.5279 −1.00865
\(238\) −0.511370 −0.0331472
\(239\) 9.89840 0.640274 0.320137 0.947371i \(-0.396271\pi\)
0.320137 + 0.947371i \(0.396271\pi\)
\(240\) 0 0
\(241\) −21.4567 −1.38215 −0.691074 0.722784i \(-0.742862\pi\)
−0.691074 + 0.722784i \(0.742862\pi\)
\(242\) 8.94292 0.574873
\(243\) −1.00000 −0.0641500
\(244\) 12.5721 0.804844
\(245\) 0 0
\(246\) −7.94156 −0.506336
\(247\) −1.39952 −0.0890491
\(248\) 2.33773 0.148446
\(249\) 16.3308 1.03492
\(250\) 0 0
\(251\) −4.10753 −0.259265 −0.129632 0.991562i \(-0.541380\pi\)
−0.129632 + 0.991562i \(0.541380\pi\)
\(252\) 0.533559 0.0336111
\(253\) 5.39854 0.339404
\(254\) −6.48257 −0.406753
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 30.7748 1.91968 0.959839 0.280552i \(-0.0905176\pi\)
0.959839 + 0.280552i \(0.0905176\pi\)
\(258\) 11.3607 0.707289
\(259\) −2.16780 −0.134701
\(260\) 0 0
\(261\) 6.19448 0.383429
\(262\) 9.45794 0.584313
\(263\) 28.3729 1.74955 0.874773 0.484533i \(-0.161011\pi\)
0.874773 + 0.484533i \(0.161011\pi\)
\(264\) −1.43425 −0.0882722
\(265\) 0 0
\(266\) −0.113589 −0.00696458
\(267\) −5.62220 −0.344073
\(268\) 6.91285 0.422269
\(269\) 10.6668 0.650364 0.325182 0.945651i \(-0.394574\pi\)
0.325182 + 0.945651i \(0.394574\pi\)
\(270\) 0 0
\(271\) 22.8439 1.38767 0.693835 0.720134i \(-0.255920\pi\)
0.693835 + 0.720134i \(0.255920\pi\)
\(272\) 0.958413 0.0581123
\(273\) 3.50758 0.212288
\(274\) −12.3361 −0.745250
\(275\) 0 0
\(276\) 3.76401 0.226567
\(277\) 31.4592 1.89020 0.945099 0.326784i \(-0.105965\pi\)
0.945099 + 0.326784i \(0.105965\pi\)
\(278\) −4.53918 −0.272242
\(279\) −2.33773 −0.139956
\(280\) 0 0
\(281\) −10.7785 −0.642992 −0.321496 0.946911i \(-0.604186\pi\)
−0.321496 + 0.946911i \(0.604186\pi\)
\(282\) −10.1489 −0.604359
\(283\) 4.16428 0.247541 0.123770 0.992311i \(-0.460501\pi\)
0.123770 + 0.992311i \(0.460501\pi\)
\(284\) 10.1247 0.600789
\(285\) 0 0
\(286\) −9.42867 −0.557529
\(287\) −4.23729 −0.250120
\(288\) −1.00000 −0.0589256
\(289\) −16.0814 −0.945967
\(290\) 0 0
\(291\) −5.56558 −0.326260
\(292\) 13.9771 0.817947
\(293\) 17.9603 1.04925 0.524626 0.851333i \(-0.324205\pi\)
0.524626 + 0.851333i \(0.324205\pi\)
\(294\) −6.71531 −0.391645
\(295\) 0 0
\(296\) 4.06291 0.236152
\(297\) 1.43425 0.0832238
\(298\) −11.0750 −0.641560
\(299\) 24.7443 1.43100
\(300\) 0 0
\(301\) 6.06163 0.349386
\(302\) 1.63387 0.0940187
\(303\) 9.42708 0.541571
\(304\) 0.212889 0.0122100
\(305\) 0 0
\(306\) −0.958413 −0.0547888
\(307\) 20.8174 1.18811 0.594056 0.804423i \(-0.297526\pi\)
0.594056 + 0.804423i \(0.297526\pi\)
\(308\) −0.765259 −0.0436047
\(309\) 7.99871 0.455031
\(310\) 0 0
\(311\) −19.4526 −1.10305 −0.551527 0.834157i \(-0.685955\pi\)
−0.551527 + 0.834157i \(0.685955\pi\)
\(312\) −6.57392 −0.372175
\(313\) −5.88310 −0.332532 −0.166266 0.986081i \(-0.553171\pi\)
−0.166266 + 0.986081i \(0.553171\pi\)
\(314\) 6.64544 0.375024
\(315\) 0 0
\(316\) 15.5279 0.873515
\(317\) 10.7610 0.604400 0.302200 0.953245i \(-0.402279\pi\)
0.302200 + 0.953245i \(0.402279\pi\)
\(318\) 3.23354 0.181328
\(319\) −8.88445 −0.497434
\(320\) 0 0
\(321\) 18.9260 1.05635
\(322\) 2.00832 0.111919
\(323\) 0.204036 0.0113528
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −10.1134 −0.560129
\(327\) 2.65524 0.146835
\(328\) 7.94156 0.438500
\(329\) −5.41505 −0.298541
\(330\) 0 0
\(331\) −0.617092 −0.0339184 −0.0169592 0.999856i \(-0.505399\pi\)
−0.0169592 + 0.999856i \(0.505399\pi\)
\(332\) −16.3308 −0.896270
\(333\) −4.06291 −0.222646
\(334\) 12.7885 0.699753
\(335\) 0 0
\(336\) −0.533559 −0.0291080
\(337\) 15.4106 0.839471 0.419735 0.907647i \(-0.362123\pi\)
0.419735 + 0.907647i \(0.362123\pi\)
\(338\) −30.2165 −1.64356
\(339\) 3.06283 0.166350
\(340\) 0 0
\(341\) 3.35289 0.181569
\(342\) −0.212889 −0.0115117
\(343\) −7.31793 −0.395131
\(344\) −11.3607 −0.612530
\(345\) 0 0
\(346\) −13.6575 −0.734234
\(347\) −22.6565 −1.21627 −0.608133 0.793835i \(-0.708081\pi\)
−0.608133 + 0.793835i \(0.708081\pi\)
\(348\) −6.19448 −0.332059
\(349\) −16.9543 −0.907544 −0.453772 0.891118i \(-0.649922\pi\)
−0.453772 + 0.891118i \(0.649922\pi\)
\(350\) 0 0
\(351\) 6.57392 0.350890
\(352\) 1.43425 0.0764459
\(353\) −6.06255 −0.322677 −0.161339 0.986899i \(-0.551581\pi\)
−0.161339 + 0.986899i \(0.551581\pi\)
\(354\) 7.52455 0.399925
\(355\) 0 0
\(356\) 5.62220 0.297976
\(357\) −0.511370 −0.0270646
\(358\) −2.99142 −0.158101
\(359\) 18.8987 0.997437 0.498718 0.866764i \(-0.333804\pi\)
0.498718 + 0.866764i \(0.333804\pi\)
\(360\) 0 0
\(361\) −18.9547 −0.997615
\(362\) −7.62285 −0.400648
\(363\) 8.94292 0.469381
\(364\) −3.50758 −0.183847
\(365\) 0 0
\(366\) 12.5721 0.657152
\(367\) 25.0483 1.30751 0.653756 0.756705i \(-0.273192\pi\)
0.653756 + 0.756705i \(0.273192\pi\)
\(368\) −3.76401 −0.196213
\(369\) −7.94156 −0.413421
\(370\) 0 0
\(371\) 1.72529 0.0895725
\(372\) 2.33773 0.121206
\(373\) 11.8954 0.615922 0.307961 0.951399i \(-0.400353\pi\)
0.307961 + 0.951399i \(0.400353\pi\)
\(374\) 1.37461 0.0710792
\(375\) 0 0
\(376\) 10.1489 0.523390
\(377\) −40.7220 −2.09729
\(378\) 0.533559 0.0274433
\(379\) 27.5184 1.41353 0.706763 0.707450i \(-0.250155\pi\)
0.706763 + 0.707450i \(0.250155\pi\)
\(380\) 0 0
\(381\) −6.48257 −0.332112
\(382\) 8.40045 0.429804
\(383\) 22.0314 1.12575 0.562877 0.826541i \(-0.309695\pi\)
0.562877 + 0.826541i \(0.309695\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.5266 −0.535791
\(387\) 11.3607 0.577499
\(388\) 5.56558 0.282549
\(389\) 17.4892 0.886738 0.443369 0.896339i \(-0.353783\pi\)
0.443369 + 0.896339i \(0.353783\pi\)
\(390\) 0 0
\(391\) −3.60748 −0.182438
\(392\) 6.71531 0.339175
\(393\) 9.45794 0.477090
\(394\) −10.2080 −0.514274
\(395\) 0 0
\(396\) −1.43425 −0.0720739
\(397\) 24.4990 1.22957 0.614786 0.788694i \(-0.289242\pi\)
0.614786 + 0.788694i \(0.289242\pi\)
\(398\) −3.84318 −0.192641
\(399\) −0.113589 −0.00568656
\(400\) 0 0
\(401\) 1.04105 0.0519875 0.0259937 0.999662i \(-0.491725\pi\)
0.0259937 + 0.999662i \(0.491725\pi\)
\(402\) 6.91285 0.344782
\(403\) 15.3681 0.765537
\(404\) −9.42708 −0.469015
\(405\) 0 0
\(406\) −3.30512 −0.164030
\(407\) 5.82724 0.288845
\(408\) 0.958413 0.0474485
\(409\) 22.2568 1.10053 0.550263 0.834991i \(-0.314527\pi\)
0.550263 + 0.834991i \(0.314527\pi\)
\(410\) 0 0
\(411\) −12.3361 −0.608494
\(412\) −7.99871 −0.394068
\(413\) 4.01479 0.197555
\(414\) 3.76401 0.184991
\(415\) 0 0
\(416\) 6.57392 0.322313
\(417\) −4.53918 −0.222284
\(418\) 0.305337 0.0149345
\(419\) −23.1294 −1.12994 −0.564972 0.825110i \(-0.691113\pi\)
−0.564972 + 0.825110i \(0.691113\pi\)
\(420\) 0 0
\(421\) 9.76139 0.475741 0.237870 0.971297i \(-0.423551\pi\)
0.237870 + 0.971297i \(0.423551\pi\)
\(422\) −5.24920 −0.255527
\(423\) −10.1489 −0.493457
\(424\) −3.23354 −0.157035
\(425\) 0 0
\(426\) 10.1247 0.490542
\(427\) 6.70794 0.324620
\(428\) −18.9260 −0.914822
\(429\) −9.42867 −0.455220
\(430\) 0 0
\(431\) 8.49298 0.409093 0.204546 0.978857i \(-0.434428\pi\)
0.204546 + 0.978857i \(0.434428\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −22.0042 −1.05745 −0.528727 0.848792i \(-0.677330\pi\)
−0.528727 + 0.848792i \(0.677330\pi\)
\(434\) 1.24732 0.0598731
\(435\) 0 0
\(436\) −2.65524 −0.127163
\(437\) −0.801317 −0.0383322
\(438\) 13.9771 0.667851
\(439\) −38.3582 −1.83074 −0.915369 0.402615i \(-0.868101\pi\)
−0.915369 + 0.402615i \(0.868101\pi\)
\(440\) 0 0
\(441\) −6.71531 −0.319777
\(442\) 6.30054 0.299686
\(443\) −12.7478 −0.605666 −0.302833 0.953044i \(-0.597932\pi\)
−0.302833 + 0.953044i \(0.597932\pi\)
\(444\) 4.06291 0.192817
\(445\) 0 0
\(446\) −19.2782 −0.912851
\(447\) −11.0750 −0.523831
\(448\) 0.533559 0.0252083
\(449\) 18.0358 0.851161 0.425580 0.904921i \(-0.360070\pi\)
0.425580 + 0.904921i \(0.360070\pi\)
\(450\) 0 0
\(451\) 11.3902 0.536344
\(452\) −3.06283 −0.144063
\(453\) 1.63387 0.0767659
\(454\) 8.43957 0.396089
\(455\) 0 0
\(456\) 0.212889 0.00996944
\(457\) 21.1495 0.989334 0.494667 0.869083i \(-0.335290\pi\)
0.494667 + 0.869083i \(0.335290\pi\)
\(458\) −5.95568 −0.278291
\(459\) −0.958413 −0.0447349
\(460\) 0 0
\(461\) 32.0085 1.49078 0.745391 0.666627i \(-0.232263\pi\)
0.745391 + 0.666627i \(0.232263\pi\)
\(462\) −0.765259 −0.0356031
\(463\) −28.7776 −1.33741 −0.668703 0.743529i \(-0.733150\pi\)
−0.668703 + 0.743529i \(0.733150\pi\)
\(464\) 6.19448 0.287572
\(465\) 0 0
\(466\) −2.09293 −0.0969529
\(467\) −5.20326 −0.240778 −0.120389 0.992727i \(-0.538414\pi\)
−0.120389 + 0.992727i \(0.538414\pi\)
\(468\) −6.57392 −0.303880
\(469\) 3.68841 0.170315
\(470\) 0 0
\(471\) 6.64544 0.306206
\(472\) −7.52455 −0.346345
\(473\) −16.2942 −0.749207
\(474\) 15.5279 0.713222
\(475\) 0 0
\(476\) 0.511370 0.0234386
\(477\) 3.23354 0.148054
\(478\) −9.89840 −0.452742
\(479\) −39.6385 −1.81113 −0.905565 0.424206i \(-0.860553\pi\)
−0.905565 + 0.424206i \(0.860553\pi\)
\(480\) 0 0
\(481\) 26.7093 1.21784
\(482\) 21.4567 0.977326
\(483\) 2.00832 0.0913818
\(484\) −8.94292 −0.406496
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 5.44308 0.246649 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(488\) −12.5721 −0.569111
\(489\) −10.1134 −0.457344
\(490\) 0 0
\(491\) 0.873965 0.0394415 0.0197207 0.999806i \(-0.493722\pi\)
0.0197207 + 0.999806i \(0.493722\pi\)
\(492\) 7.94156 0.358033
\(493\) 5.93687 0.267383
\(494\) 1.39952 0.0629672
\(495\) 0 0
\(496\) −2.33773 −0.104967
\(497\) 5.40211 0.242318
\(498\) −16.3308 −0.731802
\(499\) −34.9604 −1.56504 −0.782522 0.622623i \(-0.786067\pi\)
−0.782522 + 0.622623i \(0.786067\pi\)
\(500\) 0 0
\(501\) 12.7885 0.571346
\(502\) 4.10753 0.183328
\(503\) 0.319724 0.0142558 0.00712790 0.999975i \(-0.497731\pi\)
0.00712790 + 0.999975i \(0.497731\pi\)
\(504\) −0.533559 −0.0237666
\(505\) 0 0
\(506\) −5.39854 −0.239995
\(507\) −30.2165 −1.34196
\(508\) 6.48257 0.287618
\(509\) 20.6259 0.914225 0.457113 0.889409i \(-0.348884\pi\)
0.457113 + 0.889409i \(0.348884\pi\)
\(510\) 0 0
\(511\) 7.45760 0.329905
\(512\) −1.00000 −0.0441942
\(513\) −0.212889 −0.00939928
\(514\) −30.7748 −1.35742
\(515\) 0 0
\(516\) −11.3607 −0.500129
\(517\) 14.5561 0.640177
\(518\) 2.16780 0.0952477
\(519\) −13.6575 −0.599500
\(520\) 0 0
\(521\) −4.22039 −0.184899 −0.0924493 0.995717i \(-0.529470\pi\)
−0.0924493 + 0.995717i \(0.529470\pi\)
\(522\) −6.19448 −0.271125
\(523\) −21.5579 −0.942659 −0.471330 0.881957i \(-0.656226\pi\)
−0.471330 + 0.881957i \(0.656226\pi\)
\(524\) −9.45794 −0.413172
\(525\) 0 0
\(526\) −28.3729 −1.23712
\(527\) −2.24051 −0.0975982
\(528\) 1.43425 0.0624178
\(529\) −8.83222 −0.384010
\(530\) 0 0
\(531\) 7.52455 0.326538
\(532\) 0.113589 0.00492470
\(533\) 52.2072 2.26135
\(534\) 5.62220 0.243296
\(535\) 0 0
\(536\) −6.91285 −0.298590
\(537\) −2.99142 −0.129089
\(538\) −10.6668 −0.459877
\(539\) 9.63146 0.414856
\(540\) 0 0
\(541\) −8.93662 −0.384215 −0.192108 0.981374i \(-0.561532\pi\)
−0.192108 + 0.981374i \(0.561532\pi\)
\(542\) −22.8439 −0.981231
\(543\) −7.62285 −0.327128
\(544\) −0.958413 −0.0410916
\(545\) 0 0
\(546\) −3.50758 −0.150110
\(547\) 12.0786 0.516443 0.258222 0.966086i \(-0.416864\pi\)
0.258222 + 0.966086i \(0.416864\pi\)
\(548\) 12.3361 0.526971
\(549\) 12.5721 0.536563
\(550\) 0 0
\(551\) 1.31874 0.0561801
\(552\) −3.76401 −0.160207
\(553\) 8.28507 0.352317
\(554\) −31.4592 −1.33657
\(555\) 0 0
\(556\) 4.53918 0.192504
\(557\) −8.23596 −0.348969 −0.174484 0.984660i \(-0.555826\pi\)
−0.174484 + 0.984660i \(0.555826\pi\)
\(558\) 2.33773 0.0989640
\(559\) −74.6846 −3.15882
\(560\) 0 0
\(561\) 1.37461 0.0580360
\(562\) 10.7785 0.454664
\(563\) 14.6358 0.616825 0.308413 0.951253i \(-0.400202\pi\)
0.308413 + 0.951253i \(0.400202\pi\)
\(564\) 10.1489 0.427347
\(565\) 0 0
\(566\) −4.16428 −0.175038
\(567\) 0.533559 0.0224074
\(568\) −10.1247 −0.424822
\(569\) 31.2471 1.30995 0.654974 0.755651i \(-0.272679\pi\)
0.654974 + 0.755651i \(0.272679\pi\)
\(570\) 0 0
\(571\) 26.2261 1.09753 0.548763 0.835978i \(-0.315099\pi\)
0.548763 + 0.835978i \(0.315099\pi\)
\(572\) 9.42867 0.394233
\(573\) 8.40045 0.350934
\(574\) 4.23729 0.176861
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −8.93340 −0.371902 −0.185951 0.982559i \(-0.559537\pi\)
−0.185951 + 0.982559i \(0.559537\pi\)
\(578\) 16.0814 0.668900
\(579\) −10.5266 −0.437471
\(580\) 0 0
\(581\) −8.71346 −0.361495
\(582\) 5.56558 0.230701
\(583\) −4.63772 −0.192075
\(584\) −13.9771 −0.578376
\(585\) 0 0
\(586\) −17.9603 −0.741933
\(587\) −3.99360 −0.164833 −0.0824167 0.996598i \(-0.526264\pi\)
−0.0824167 + 0.996598i \(0.526264\pi\)
\(588\) 6.71531 0.276935
\(589\) −0.497677 −0.0205064
\(590\) 0 0
\(591\) −10.2080 −0.419903
\(592\) −4.06291 −0.166985
\(593\) −11.2114 −0.460396 −0.230198 0.973144i \(-0.573937\pi\)
−0.230198 + 0.973144i \(0.573937\pi\)
\(594\) −1.43425 −0.0588481
\(595\) 0 0
\(596\) 11.0750 0.453651
\(597\) −3.84318 −0.157291
\(598\) −24.7443 −1.01187
\(599\) −6.64762 −0.271614 −0.135807 0.990735i \(-0.543363\pi\)
−0.135807 + 0.990735i \(0.543363\pi\)
\(600\) 0 0
\(601\) −10.7465 −0.438359 −0.219179 0.975685i \(-0.570338\pi\)
−0.219179 + 0.975685i \(0.570338\pi\)
\(602\) −6.06163 −0.247053
\(603\) 6.91285 0.281513
\(604\) −1.63387 −0.0664812
\(605\) 0 0
\(606\) −9.42708 −0.382949
\(607\) −15.4591 −0.627466 −0.313733 0.949511i \(-0.601580\pi\)
−0.313733 + 0.949511i \(0.601580\pi\)
\(608\) −0.212889 −0.00863379
\(609\) −3.30512 −0.133930
\(610\) 0 0
\(611\) 66.7182 2.69913
\(612\) 0.958413 0.0387416
\(613\) 24.4867 0.989007 0.494503 0.869176i \(-0.335350\pi\)
0.494503 + 0.869176i \(0.335350\pi\)
\(614\) −20.8174 −0.840122
\(615\) 0 0
\(616\) 0.765259 0.0308332
\(617\) 18.9157 0.761515 0.380758 0.924675i \(-0.375663\pi\)
0.380758 + 0.924675i \(0.375663\pi\)
\(618\) −7.99871 −0.321755
\(619\) 2.55005 0.102495 0.0512476 0.998686i \(-0.483680\pi\)
0.0512476 + 0.998686i \(0.483680\pi\)
\(620\) 0 0
\(621\) 3.76401 0.151045
\(622\) 19.4526 0.779978
\(623\) 2.99977 0.120183
\(624\) 6.57392 0.263168
\(625\) 0 0
\(626\) 5.88310 0.235136
\(627\) 0.305337 0.0121940
\(628\) −6.64544 −0.265182
\(629\) −3.89395 −0.155262
\(630\) 0 0
\(631\) −36.4530 −1.45117 −0.725585 0.688133i \(-0.758431\pi\)
−0.725585 + 0.688133i \(0.758431\pi\)
\(632\) −15.5279 −0.617668
\(633\) −5.24920 −0.208637
\(634\) −10.7610 −0.427375
\(635\) 0 0
\(636\) −3.23354 −0.128218
\(637\) 44.1460 1.74913
\(638\) 8.88445 0.351739
\(639\) 10.1247 0.400526
\(640\) 0 0
\(641\) −20.3669 −0.804444 −0.402222 0.915542i \(-0.631762\pi\)
−0.402222 + 0.915542i \(0.631762\pi\)
\(642\) −18.9260 −0.746949
\(643\) 11.4218 0.450433 0.225217 0.974309i \(-0.427691\pi\)
0.225217 + 0.974309i \(0.427691\pi\)
\(644\) −2.00832 −0.0791390
\(645\) 0 0
\(646\) −0.204036 −0.00802768
\(647\) −5.89217 −0.231645 −0.115823 0.993270i \(-0.536950\pi\)
−0.115823 + 0.993270i \(0.536950\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −10.7921 −0.423627
\(650\) 0 0
\(651\) 1.24732 0.0488862
\(652\) 10.1134 0.396071
\(653\) −17.1063 −0.669422 −0.334711 0.942321i \(-0.608639\pi\)
−0.334711 + 0.942321i \(0.608639\pi\)
\(654\) −2.65524 −0.103828
\(655\) 0 0
\(656\) −7.94156 −0.310066
\(657\) 13.9771 0.545298
\(658\) 5.41505 0.211101
\(659\) −29.7191 −1.15769 −0.578846 0.815437i \(-0.696497\pi\)
−0.578846 + 0.815437i \(0.696497\pi\)
\(660\) 0 0
\(661\) 46.3036 1.80100 0.900501 0.434854i \(-0.143200\pi\)
0.900501 + 0.434854i \(0.143200\pi\)
\(662\) 0.617092 0.0239840
\(663\) 6.30054 0.244693
\(664\) 16.3308 0.633759
\(665\) 0 0
\(666\) 4.06291 0.157435
\(667\) −23.3161 −0.902803
\(668\) −12.7885 −0.494800
\(669\) −19.2782 −0.745340
\(670\) 0 0
\(671\) −18.0315 −0.696099
\(672\) 0.533559 0.0205825
\(673\) −19.7080 −0.759689 −0.379845 0.925050i \(-0.624023\pi\)
−0.379845 + 0.925050i \(0.624023\pi\)
\(674\) −15.4106 −0.593595
\(675\) 0 0
\(676\) 30.2165 1.16217
\(677\) 24.7774 0.952273 0.476137 0.879371i \(-0.342037\pi\)
0.476137 + 0.879371i \(0.342037\pi\)
\(678\) −3.06283 −0.117627
\(679\) 2.96957 0.113961
\(680\) 0 0
\(681\) 8.43957 0.323405
\(682\) −3.35289 −0.128389
\(683\) 16.5874 0.634699 0.317349 0.948309i \(-0.397207\pi\)
0.317349 + 0.948309i \(0.397207\pi\)
\(684\) 0.212889 0.00814002
\(685\) 0 0
\(686\) 7.31793 0.279400
\(687\) −5.95568 −0.227224
\(688\) 11.3607 0.433124
\(689\) −21.2571 −0.809830
\(690\) 0 0
\(691\) 42.2893 1.60876 0.804381 0.594114i \(-0.202497\pi\)
0.804381 + 0.594114i \(0.202497\pi\)
\(692\) 13.6575 0.519182
\(693\) −0.765259 −0.0290698
\(694\) 22.6565 0.860031
\(695\) 0 0
\(696\) 6.19448 0.234801
\(697\) −7.61130 −0.288299
\(698\) 16.9543 0.641731
\(699\) −2.09293 −0.0791617
\(700\) 0 0
\(701\) −22.7240 −0.858273 −0.429137 0.903240i \(-0.641182\pi\)
−0.429137 + 0.903240i \(0.641182\pi\)
\(702\) −6.57392 −0.248117
\(703\) −0.864949 −0.0326222
\(704\) −1.43425 −0.0540554
\(705\) 0 0
\(706\) 6.06255 0.228167
\(707\) −5.02990 −0.189169
\(708\) −7.52455 −0.282790
\(709\) 46.4122 1.74305 0.871524 0.490353i \(-0.163132\pi\)
0.871524 + 0.490353i \(0.163132\pi\)
\(710\) 0 0
\(711\) 15.5279 0.582343
\(712\) −5.62220 −0.210701
\(713\) 8.79924 0.329534
\(714\) 0.511370 0.0191375
\(715\) 0 0
\(716\) 2.99142 0.111795
\(717\) −9.89840 −0.369662
\(718\) −18.8987 −0.705294
\(719\) −28.2066 −1.05193 −0.525964 0.850507i \(-0.676295\pi\)
−0.525964 + 0.850507i \(0.676295\pi\)
\(720\) 0 0
\(721\) −4.26779 −0.158941
\(722\) 18.9547 0.705420
\(723\) 21.4567 0.797983
\(724\) 7.62285 0.283301
\(725\) 0 0
\(726\) −8.94292 −0.331903
\(727\) 0.851592 0.0315838 0.0157919 0.999875i \(-0.494973\pi\)
0.0157919 + 0.999875i \(0.494973\pi\)
\(728\) 3.50758 0.129999
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.8883 0.402718
\(732\) −12.5721 −0.464677
\(733\) 30.7299 1.13504 0.567518 0.823361i \(-0.307904\pi\)
0.567518 + 0.823361i \(0.307904\pi\)
\(734\) −25.0483 −0.924551
\(735\) 0 0
\(736\) 3.76401 0.138743
\(737\) −9.91477 −0.365215
\(738\) 7.94156 0.292333
\(739\) 2.06485 0.0759566 0.0379783 0.999279i \(-0.487908\pi\)
0.0379783 + 0.999279i \(0.487908\pi\)
\(740\) 0 0
\(741\) 1.39952 0.0514125
\(742\) −1.72529 −0.0633373
\(743\) −37.8972 −1.39031 −0.695157 0.718858i \(-0.744665\pi\)
−0.695157 + 0.718858i \(0.744665\pi\)
\(744\) −2.33773 −0.0857053
\(745\) 0 0
\(746\) −11.8954 −0.435522
\(747\) −16.3308 −0.597514
\(748\) −1.37461 −0.0502606
\(749\) −10.0981 −0.368978
\(750\) 0 0
\(751\) −40.6331 −1.48272 −0.741361 0.671106i \(-0.765819\pi\)
−0.741361 + 0.671106i \(0.765819\pi\)
\(752\) −10.1489 −0.370093
\(753\) 4.10753 0.149687
\(754\) 40.7220 1.48301
\(755\) 0 0
\(756\) −0.533559 −0.0194054
\(757\) 10.0032 0.363572 0.181786 0.983338i \(-0.441812\pi\)
0.181786 + 0.983338i \(0.441812\pi\)
\(758\) −27.5184 −0.999514
\(759\) −5.39854 −0.195955
\(760\) 0 0
\(761\) −30.1813 −1.09407 −0.547035 0.837110i \(-0.684244\pi\)
−0.547035 + 0.837110i \(0.684244\pi\)
\(762\) 6.48257 0.234839
\(763\) −1.41673 −0.0512890
\(764\) −8.40045 −0.303917
\(765\) 0 0
\(766\) −22.0314 −0.796028
\(767\) −49.4658 −1.78611
\(768\) −1.00000 −0.0360844
\(769\) 21.1468 0.762571 0.381286 0.924457i \(-0.375481\pi\)
0.381286 + 0.924457i \(0.375481\pi\)
\(770\) 0 0
\(771\) −30.7748 −1.10833
\(772\) 10.5266 0.378861
\(773\) −29.4470 −1.05913 −0.529567 0.848268i \(-0.677646\pi\)
−0.529567 + 0.848268i \(0.677646\pi\)
\(774\) −11.3607 −0.408353
\(775\) 0 0
\(776\) −5.56558 −0.199793
\(777\) 2.16780 0.0777695
\(778\) −17.4892 −0.627019
\(779\) −1.69067 −0.0605746
\(780\) 0 0
\(781\) −14.5213 −0.519614
\(782\) 3.60748 0.129003
\(783\) −6.19448 −0.221373
\(784\) −6.71531 −0.239833
\(785\) 0 0
\(786\) −9.45794 −0.337353
\(787\) −5.16875 −0.184246 −0.0921230 0.995748i \(-0.529365\pi\)
−0.0921230 + 0.995748i \(0.529365\pi\)
\(788\) 10.2080 0.363647
\(789\) −28.3729 −1.01010
\(790\) 0 0
\(791\) −1.63420 −0.0581055
\(792\) 1.43425 0.0509640
\(793\) −82.6478 −2.93491
\(794\) −24.4990 −0.869439
\(795\) 0 0
\(796\) 3.84318 0.136218
\(797\) −3.75807 −0.133118 −0.0665588 0.997783i \(-0.521202\pi\)
−0.0665588 + 0.997783i \(0.521202\pi\)
\(798\) 0.113589 0.00402100
\(799\) −9.72686 −0.344111
\(800\) 0 0
\(801\) 5.62220 0.198650
\(802\) −1.04105 −0.0367607
\(803\) −20.0467 −0.707432
\(804\) −6.91285 −0.243797
\(805\) 0 0
\(806\) −15.3681 −0.541317
\(807\) −10.6668 −0.375488
\(808\) 9.42708 0.331643
\(809\) −20.8847 −0.734266 −0.367133 0.930168i \(-0.619661\pi\)
−0.367133 + 0.930168i \(0.619661\pi\)
\(810\) 0 0
\(811\) −36.5032 −1.28180 −0.640900 0.767625i \(-0.721439\pi\)
−0.640900 + 0.767625i \(0.721439\pi\)
\(812\) 3.30512 0.115987
\(813\) −22.8439 −0.801172
\(814\) −5.82724 −0.204245
\(815\) 0 0
\(816\) −0.958413 −0.0335512
\(817\) 2.41858 0.0846153
\(818\) −22.2568 −0.778190
\(819\) −3.50758 −0.122565
\(820\) 0 0
\(821\) 33.2221 1.15946 0.579730 0.814809i \(-0.303158\pi\)
0.579730 + 0.814809i \(0.303158\pi\)
\(822\) 12.3361 0.430270
\(823\) −32.7898 −1.14298 −0.571490 0.820609i \(-0.693634\pi\)
−0.571490 + 0.820609i \(0.693634\pi\)
\(824\) 7.99871 0.278648
\(825\) 0 0
\(826\) −4.01479 −0.139692
\(827\) 5.18915 0.180444 0.0902222 0.995922i \(-0.471242\pi\)
0.0902222 + 0.995922i \(0.471242\pi\)
\(828\) −3.76401 −0.130808
\(829\) −5.55820 −0.193044 −0.0965221 0.995331i \(-0.530772\pi\)
−0.0965221 + 0.995331i \(0.530772\pi\)
\(830\) 0 0
\(831\) −31.4592 −1.09131
\(832\) −6.57392 −0.227910
\(833\) −6.43605 −0.222996
\(834\) 4.53918 0.157179
\(835\) 0 0
\(836\) −0.305337 −0.0105603
\(837\) 2.33773 0.0808037
\(838\) 23.1294 0.798991
\(839\) 43.4383 1.49966 0.749828 0.661633i \(-0.230136\pi\)
0.749828 + 0.661633i \(0.230136\pi\)
\(840\) 0 0
\(841\) 9.37160 0.323159
\(842\) −9.76139 −0.336400
\(843\) 10.7785 0.371232
\(844\) 5.24920 0.180685
\(845\) 0 0
\(846\) 10.1489 0.348927
\(847\) −4.77158 −0.163953
\(848\) 3.23354 0.111040
\(849\) −4.16428 −0.142918
\(850\) 0 0
\(851\) 15.2928 0.524231
\(852\) −10.1247 −0.346865
\(853\) 3.34342 0.114477 0.0572383 0.998361i \(-0.481771\pi\)
0.0572383 + 0.998361i \(0.481771\pi\)
\(854\) −6.70794 −0.229541
\(855\) 0 0
\(856\) 18.9260 0.646877
\(857\) −34.5415 −1.17991 −0.589957 0.807434i \(-0.700855\pi\)
−0.589957 + 0.807434i \(0.700855\pi\)
\(858\) 9.42867 0.321889
\(859\) −23.8513 −0.813796 −0.406898 0.913474i \(-0.633390\pi\)
−0.406898 + 0.913474i \(0.633390\pi\)
\(860\) 0 0
\(861\) 4.23729 0.144407
\(862\) −8.49298 −0.289272
\(863\) 5.73061 0.195072 0.0975361 0.995232i \(-0.468904\pi\)
0.0975361 + 0.995232i \(0.468904\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 22.0042 0.747733
\(867\) 16.0814 0.546154
\(868\) −1.24732 −0.0423367
\(869\) −22.2710 −0.755492
\(870\) 0 0
\(871\) −45.4445 −1.53983
\(872\) 2.65524 0.0899178
\(873\) 5.56558 0.188366
\(874\) 0.801317 0.0271049
\(875\) 0 0
\(876\) −13.9771 −0.472242
\(877\) 49.8482 1.68325 0.841627 0.540059i \(-0.181598\pi\)
0.841627 + 0.540059i \(0.181598\pi\)
\(878\) 38.3582 1.29453
\(879\) −17.9603 −0.605786
\(880\) 0 0
\(881\) −7.65609 −0.257940 −0.128970 0.991648i \(-0.541167\pi\)
−0.128970 + 0.991648i \(0.541167\pi\)
\(882\) 6.71531 0.226116
\(883\) 38.9959 1.31232 0.656158 0.754623i \(-0.272180\pi\)
0.656158 + 0.754623i \(0.272180\pi\)
\(884\) −6.30054 −0.211910
\(885\) 0 0
\(886\) 12.7478 0.428270
\(887\) −32.5338 −1.09238 −0.546189 0.837662i \(-0.683922\pi\)
−0.546189 + 0.837662i \(0.683922\pi\)
\(888\) −4.06291 −0.136342
\(889\) 3.45884 0.116006
\(890\) 0 0
\(891\) −1.43425 −0.0480493
\(892\) 19.2782 0.645483
\(893\) −2.16059 −0.0723015
\(894\) 11.0750 0.370405
\(895\) 0 0
\(896\) −0.533559 −0.0178250
\(897\) −24.7443 −0.826189
\(898\) −18.0358 −0.601862
\(899\) −14.4810 −0.482969
\(900\) 0 0
\(901\) 3.09907 0.103245
\(902\) −11.3902 −0.379253
\(903\) −6.06163 −0.201718
\(904\) 3.06283 0.101868
\(905\) 0 0
\(906\) −1.63387 −0.0542817
\(907\) 16.5820 0.550595 0.275297 0.961359i \(-0.411224\pi\)
0.275297 + 0.961359i \(0.411224\pi\)
\(908\) −8.43957 −0.280077
\(909\) −9.42708 −0.312676
\(910\) 0 0
\(911\) −28.5720 −0.946633 −0.473316 0.880893i \(-0.656943\pi\)
−0.473316 + 0.880893i \(0.656943\pi\)
\(912\) −0.212889 −0.00704946
\(913\) 23.4225 0.775173
\(914\) −21.1495 −0.699565
\(915\) 0 0
\(916\) 5.95568 0.196781
\(917\) −5.04637 −0.166646
\(918\) 0.958413 0.0316324
\(919\) −0.382861 −0.0126294 −0.00631471 0.999980i \(-0.502010\pi\)
−0.00631471 + 0.999980i \(0.502010\pi\)
\(920\) 0 0
\(921\) −20.8174 −0.685957
\(922\) −32.0085 −1.05414
\(923\) −66.5588 −2.19081
\(924\) 0.765259 0.0251752
\(925\) 0 0
\(926\) 28.7776 0.945689
\(927\) −7.99871 −0.262712
\(928\) −6.19448 −0.203344
\(929\) −6.25711 −0.205289 −0.102645 0.994718i \(-0.532730\pi\)
−0.102645 + 0.994718i \(0.532730\pi\)
\(930\) 0 0
\(931\) −1.42962 −0.0468538
\(932\) 2.09293 0.0685561
\(933\) 19.4526 0.636849
\(934\) 5.20326 0.170256
\(935\) 0 0
\(936\) 6.57392 0.214875
\(937\) −4.93214 −0.161126 −0.0805630 0.996750i \(-0.525672\pi\)
−0.0805630 + 0.996750i \(0.525672\pi\)
\(938\) −3.68841 −0.120431
\(939\) 5.88310 0.191988
\(940\) 0 0
\(941\) 29.8107 0.971801 0.485901 0.874014i \(-0.338492\pi\)
0.485901 + 0.874014i \(0.338492\pi\)
\(942\) −6.64544 −0.216520
\(943\) 29.8921 0.973422
\(944\) 7.52455 0.244903
\(945\) 0 0
\(946\) 16.2942 0.529769
\(947\) −26.9125 −0.874539 −0.437270 0.899330i \(-0.644054\pi\)
−0.437270 + 0.899330i \(0.644054\pi\)
\(948\) −15.5279 −0.504324
\(949\) −91.8843 −2.98269
\(950\) 0 0
\(951\) −10.7610 −0.348950
\(952\) −0.511370 −0.0165736
\(953\) 37.2337 1.20612 0.603059 0.797697i \(-0.293949\pi\)
0.603059 + 0.797697i \(0.293949\pi\)
\(954\) −3.23354 −0.104690
\(955\) 0 0
\(956\) 9.89840 0.320137
\(957\) 8.88445 0.287194
\(958\) 39.6385 1.28066
\(959\) 6.58203 0.212545
\(960\) 0 0
\(961\) −25.5350 −0.823710
\(962\) −26.7093 −0.861141
\(963\) −18.9260 −0.609882
\(964\) −21.4567 −0.691074
\(965\) 0 0
\(966\) −2.00832 −0.0646167
\(967\) 16.0845 0.517244 0.258622 0.965979i \(-0.416732\pi\)
0.258622 + 0.965979i \(0.416732\pi\)
\(968\) 8.94292 0.287436
\(969\) −0.204036 −0.00655457
\(970\) 0 0
\(971\) 12.2569 0.393342 0.196671 0.980470i \(-0.436987\pi\)
0.196671 + 0.980470i \(0.436987\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 2.42192 0.0776431
\(974\) −5.44308 −0.174407
\(975\) 0 0
\(976\) 12.5721 0.402422
\(977\) 18.0218 0.576569 0.288285 0.957545i \(-0.406915\pi\)
0.288285 + 0.957545i \(0.406915\pi\)
\(978\) 10.1134 0.323391
\(979\) −8.06365 −0.257715
\(980\) 0 0
\(981\) −2.65524 −0.0847753
\(982\) −0.873965 −0.0278893
\(983\) 49.4029 1.57571 0.787854 0.615862i \(-0.211192\pi\)
0.787854 + 0.615862i \(0.211192\pi\)
\(984\) −7.94156 −0.253168
\(985\) 0 0
\(986\) −5.93687 −0.189069
\(987\) 5.41505 0.172363
\(988\) −1.39952 −0.0445246
\(989\) −42.7619 −1.35975
\(990\) 0 0
\(991\) −30.5664 −0.970972 −0.485486 0.874244i \(-0.661357\pi\)
−0.485486 + 0.874244i \(0.661357\pi\)
\(992\) 2.33773 0.0742230
\(993\) 0.617092 0.0195828
\(994\) −5.40211 −0.171345
\(995\) 0 0
\(996\) 16.3308 0.517462
\(997\) 6.81050 0.215691 0.107845 0.994168i \(-0.465605\pi\)
0.107845 + 0.994168i \(0.465605\pi\)
\(998\) 34.9604 1.10665
\(999\) 4.06291 0.128545
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 3750.2.a.u.1.6 8
5.2 odd 4 3750.2.c.k.1249.6 16
5.3 odd 4 3750.2.c.k.1249.11 16
5.4 even 2 3750.2.a.v.1.3 8
25.3 odd 20 150.2.h.b.109.3 16
25.4 even 10 750.2.g.f.451.2 16
25.6 even 5 750.2.g.g.301.3 16
25.8 odd 20 750.2.h.d.199.1 16
25.17 odd 20 150.2.h.b.139.3 yes 16
25.19 even 10 750.2.g.f.301.2 16
25.21 even 5 750.2.g.g.451.3 16
25.22 odd 20 750.2.h.d.49.2 16
75.17 even 20 450.2.l.c.289.2 16
75.53 even 20 450.2.l.c.109.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.2.h.b.109.3 16 25.3 odd 20
150.2.h.b.139.3 yes 16 25.17 odd 20
450.2.l.c.109.2 16 75.53 even 20
450.2.l.c.289.2 16 75.17 even 20
750.2.g.f.301.2 16 25.19 even 10
750.2.g.f.451.2 16 25.4 even 10
750.2.g.g.301.3 16 25.6 even 5
750.2.g.g.451.3 16 25.21 even 5
750.2.h.d.49.2 16 25.22 odd 20
750.2.h.d.199.1 16 25.8 odd 20
3750.2.a.u.1.6 8 1.1 even 1 trivial
3750.2.a.v.1.3 8 5.4 even 2
3750.2.c.k.1249.6 16 5.2 odd 4
3750.2.c.k.1249.11 16 5.3 odd 4