Properties

Label 3750.2.a.v.1.3
Level $3750$
Weight $2$
Character 3750.1
Self dual yes
Analytic conductor $29.944$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.3
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 \(-0.532151\pi\)
−0.100833 + 0.994903i \(0.532151\pi\)
\(8\) 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 \(-0.134817\pi\)
0.911639 + 0.410991i \(0.134817\pi\)
\(14\) −0.533559 −0.142600
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.958413 −0.232449 −0.116225 0.993223i \(-0.537079\pi\)
−0.116225 + 0.993223i \(0.537079\pi\)
\(18\) 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 \(-0.371636\pi\)
0.392425 + 0.919784i \(0.371636\pi\)
\(24\) 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 \(-0.391612\pi\)
0.333969 + 0.942584i \(0.391612\pi\)
\(38\) 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 \(-0.833475\pi\)
−0.866248 + 0.499614i \(0.833475\pi\)
\(44\) −1.43425 −0.216222
\(45\) 0 0
\(46\) 3.76401 0.554973
\(47\) 10.1489 1.48037 0.740186 0.672402i \(-0.234737\pi\)
0.740186 + 0.672402i \(0.234737\pi\)
\(48\) 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 \(-0.571285\pi\)
−0.222081 + 0.975028i \(0.571285\pi\)
\(54\) 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 \(-0.638766\pi\)
−0.422269 + 0.906470i \(0.638766\pi\)
\(68\) −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 \(-0.804888\pi\)
−0.817947 + 0.575294i \(0.804888\pi\)
\(74\) 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 \(-0.146266\pi\)
0.896270 + 0.443508i \(0.146266\pi\)
\(84\) −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 \(-0.591180\pi\)
−0.282549 + 0.959253i \(0.591180\pi\)
\(98\) −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 \(-0.371067\pi\)
0.394068 + 0.919081i \(0.371067\pi\)
\(104\) 6.57392 0.644626
\(105\) 0 0
\(106\) −3.23354 −0.314070
\(107\) 18.9260 1.82964 0.914822 0.403857i \(-0.132331\pi\)
0.914822 + 0.403857i \(0.132331\pi\)
\(108\) 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 \(-0.453983\pi\)
0.144063 + 0.989568i \(0.453983\pi\)
\(114\) 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 \(-0.592863\pi\)
−0.287618 + 0.957745i \(0.592863\pi\)
\(128\) 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 \(-0.676672\pi\)
−0.526971 + 0.849883i \(0.676672\pi\)
\(138\) 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 \(-0.414568\pi\)
0.265182 + 0.964198i \(0.414568\pi\)
\(158\) 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 \(-0.629627\pi\)
−0.396071 + 0.918220i \(0.629627\pi\)
\(164\) −7.94156 −0.620132
\(165\) 0 0
\(166\) 16.3308 1.26752
\(167\) 12.7885 0.989600 0.494800 0.869007i \(-0.335241\pi\)
0.494800 + 0.869007i \(0.335241\pi\)
\(168\) −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 \(-0.673763\pi\)
−0.519182 + 0.854664i \(0.673763\pi\)
\(174\) 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 \(-0.623684\pi\)
−0.378861 + 0.925453i \(0.623684\pi\)
\(194\) −5.56558 −0.399585
\(195\) 0 0
\(196\) −6.71531 −0.479665
\(197\) −10.2080 −0.727293 −0.363647 0.931537i \(-0.618468\pi\)
−0.363647 + 0.931537i \(0.618468\pi\)
\(198\) −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 \(-0.723344\pi\)
−0.645483 + 0.763774i \(0.723344\pi\)
\(224\) −0.533559 −0.0356499
\(225\) 0 0
\(226\) 3.06283 0.203736
\(227\) 8.43957 0.560154 0.280077 0.959978i \(-0.409640\pi\)
0.280077 + 0.959978i \(0.409640\pi\)
\(228\) 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 \(-0.521839\pi\)
−0.0685561 + 0.997647i \(0.521839\pi\)
\(234\) 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 \(-0.909482\pi\)
−0.959839 + 0.280552i \(0.909482\pi\)
\(258\) −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 \(-0.838989\pi\)
−0.874773 + 0.484533i \(0.838989\pi\)
\(264\) −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 \(-0.894035\pi\)
−0.945099 + 0.326784i \(0.894035\pi\)
\(278\) 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 \(-0.539499\pi\)
−0.123770 + 0.992311i \(0.539499\pi\)
\(284\) 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 \(-0.675795\pi\)
−0.524626 + 0.851333i \(0.675795\pi\)
\(294\) −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 \(-0.702474\pi\)
−0.594056 + 0.804423i \(0.702474\pi\)
\(308\) 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 \(-0.446829\pi\)
0.166266 + 0.986081i \(0.446829\pi\)
\(314\) 6.64544 0.375024
\(315\) 0 0
\(316\) 15.5279 0.873515
\(317\) −10.7610 −0.604400 −0.302200 0.953245i \(-0.597721\pi\)
−0.302200 + 0.953245i \(0.597721\pi\)
\(318\) −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 \(-0.637877\pi\)
−0.419735 + 0.907647i \(0.637877\pi\)
\(338\) 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 \(-0.291919\pi\)
0.608133 + 0.793835i \(0.291919\pi\)
\(348\) 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 \(-0.448419\pi\)
0.161339 + 0.986899i \(0.448419\pi\)
\(354\) 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 \(-0.726808\pi\)
−0.653756 + 0.756705i \(0.726808\pi\)
\(368\) 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 \(-0.599647\pi\)
−0.307961 + 0.951399i \(0.599647\pi\)
\(374\) 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 \(-0.690305\pi\)
−0.562877 + 0.826541i \(0.690305\pi\)
\(384\) 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 \(-0.710758\pi\)
−0.614786 + 0.788694i \(0.710758\pi\)
\(398\) 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 \(-0.322670\pi\)
0.528727 + 0.848792i \(0.322670\pi\)
\(434\) 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 \(-0.402068\pi\)
0.302833 + 0.953044i \(0.402068\pi\)
\(444\) 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 \(-0.664710\pi\)
−0.494667 + 0.869083i \(0.664710\pi\)
\(458\) 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 \(-0.266850\pi\)
0.668703 + 0.743529i \(0.266850\pi\)
\(464\) 6.19448 0.287572
\(465\) 0 0
\(466\) −2.09293 −0.0969529
\(467\) 5.20326 0.240778 0.120389 0.992727i \(-0.461586\pi\)
0.120389 + 0.992727i \(0.461586\pi\)
\(468\) 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 \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(488\) 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 \(-0.502269\pi\)
−0.00712790 + 0.999975i \(0.502269\pi\)
\(504\) −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 \(-0.343774\pi\)
0.471330 + 0.881957i \(0.343774\pi\)
\(524\) −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 \(-0.583136\pi\)
−0.258222 + 0.966086i \(0.583136\pi\)
\(548\) −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 \(-0.444174\pi\)
0.174484 + 0.984660i \(0.444174\pi\)
\(558\) −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 \(-0.599798\pi\)
−0.308413 + 0.951253i \(0.599798\pi\)
\(564\) 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 \(-0.440463\pi\)
0.185951 + 0.982559i \(0.440463\pi\)
\(578\) −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 \(-0.473736\pi\)
0.0824167 + 0.996598i \(0.473736\pi\)
\(588\) −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 \(-0.426063\pi\)
0.230198 + 0.973144i \(0.426063\pi\)
\(594\) −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 \(-0.398420\pi\)
0.313733 + 0.949511i \(0.398420\pi\)
\(608\) 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 \(-0.664650\pi\)
−0.494503 + 0.869176i \(0.664650\pi\)
\(614\) −20.8174 −0.840122
\(615\) 0 0
\(616\) 0.765259 0.0308332
\(617\) −18.9157 −0.761515 −0.380758 0.924675i \(-0.624337\pi\)
−0.380758 + 0.924675i \(0.624337\pi\)
\(618\) 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 \(-0.572309\pi\)
−0.225217 + 0.974309i \(0.572309\pi\)
\(644\) −2.00832 −0.0791390
\(645\) 0 0
\(646\) −0.204036 −0.00802768
\(647\) 5.89217 0.231645 0.115823 0.993270i \(-0.463050\pi\)
0.115823 + 0.993270i \(0.463050\pi\)
\(648\) 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 \(-0.391361\pi\)
0.334711 + 0.942321i \(0.391361\pi\)
\(654\) −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 \(-0.375977\pi\)
0.379845 + 0.925050i \(0.375977\pi\)
\(674\) −15.4106 −0.593595
\(675\) 0 0
\(676\) 30.2165 1.16217
\(677\) −24.7774 −0.952273 −0.476137 0.879371i \(-0.657963\pi\)
−0.476137 + 0.879371i \(0.657963\pi\)
\(678\) 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 \(-0.602793\pi\)
−0.317349 + 0.948309i \(0.602793\pi\)
\(684\) 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 \(-0.505027\pi\)
−0.0157919 + 0.999875i \(0.505027\pi\)
\(728\) −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 \(-0.692096\pi\)
−0.567518 + 0.823361i \(0.692096\pi\)
\(734\) −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 \(-0.255335\pi\)
0.695157 + 0.718858i \(0.255335\pi\)
\(744\) −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 \(-0.558188\pi\)
−0.181786 + 0.983338i \(0.558188\pi\)
\(758\) 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 \(-0.322354\pi\)
0.529567 + 0.848268i \(0.322354\pi\)
\(774\) −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 \(-0.470635\pi\)
0.0921230 + 0.995748i \(0.470635\pi\)
\(788\) −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 \(-0.478798\pi\)
0.0665588 + 0.997783i \(0.478798\pi\)
\(798\) −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 \(-0.306366\pi\)
0.571490 + 0.820609i \(0.306366\pi\)
\(824\) 7.99871 0.278648
\(825\) 0 0
\(826\) −4.01479 −0.139692
\(827\) −5.18915 −0.180444 −0.0902222 0.995922i \(-0.528758\pi\)
−0.0902222 + 0.995922i \(0.528758\pi\)
\(828\) 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 \(-0.518229\pi\)
−0.0572383 + 0.998361i \(0.518229\pi\)
\(854\) −6.70794 −0.229541
\(855\) 0 0
\(856\) 18.9260 0.646877
\(857\) 34.5415 1.17991 0.589957 0.807434i \(-0.299145\pi\)
0.589957 + 0.807434i \(0.299145\pi\)
\(858\) −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 \(-0.531096\pi\)
−0.0975361 + 0.995232i \(0.531096\pi\)
\(864\) 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 \(-0.818402\pi\)
−0.841627 + 0.540059i \(0.818402\pi\)
\(878\) −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 \(-0.727820\pi\)
−0.656158 + 0.754623i \(0.727820\pi\)
\(884\) −6.30054 −0.211910
\(885\) 0 0
\(886\) 12.7478 0.428270
\(887\) 32.5338 1.09238 0.546189 0.837662i \(-0.316078\pi\)
0.546189 + 0.837662i \(0.316078\pi\)
\(888\) 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 \(-0.588776\pi\)
−0.275297 + 0.961359i \(0.588776\pi\)
\(908\) 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 \(-0.474328\pi\)
0.0805630 + 0.996750i \(0.474328\pi\)
\(938\) 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 \(-0.355946\pi\)
0.437270 + 0.899330i \(0.355946\pi\)
\(948\) 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 \(-0.706051\pi\)
−0.603059 + 0.797697i \(0.706051\pi\)
\(954\) −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 \(-0.583268\pi\)
−0.258622 + 0.965979i \(0.583268\pi\)
\(968\) −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 \(-0.593085\pi\)
−0.288285 + 0.957545i \(0.593085\pi\)
\(978\) −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 \(-0.788808\pi\)
−0.787854 + 0.615862i \(0.788808\pi\)
\(984\) −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 \(-0.534395\pi\)
−0.107845 + 0.994168i \(0.534395\pi\)
\(998\) −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.v.1.3 8
5.2 odd 4 3750.2.c.k.1249.11 16
5.3 odd 4 3750.2.c.k.1249.6 16
5.4 even 2 3750.2.a.u.1.6 8
25.3 odd 20 750.2.h.d.49.2 16
25.4 even 10 750.2.g.g.451.3 16
25.6 even 5 750.2.g.f.301.2 16
25.8 odd 20 150.2.h.b.139.3 yes 16
25.17 odd 20 750.2.h.d.199.1 16
25.19 even 10 750.2.g.g.301.3 16
25.21 even 5 750.2.g.f.451.2 16
25.22 odd 20 150.2.h.b.109.3 16
75.8 even 20 450.2.l.c.289.2 16
75.47 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 25.22 odd 20
150.2.h.b.139.3 yes 16 25.8 odd 20
450.2.l.c.109.2 16 75.47 even 20
450.2.l.c.289.2 16 75.8 even 20
750.2.g.f.301.2 16 25.6 even 5
750.2.g.f.451.2 16 25.21 even 5
750.2.g.g.301.3 16 25.19 even 10
750.2.g.g.451.3 16 25.4 even 10
750.2.h.d.49.2 16 25.3 odd 20
750.2.h.d.199.1 16 25.17 odd 20
3750.2.a.u.1.6 8 5.4 even 2
3750.2.a.v.1.3 8 1.1 even 1 trivial
3750.2.c.k.1249.6 16 5.3 odd 4
3750.2.c.k.1249.11 16 5.2 odd 4