Properties

Label 9405.2.a.w.1.4
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
Dimension $6$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.577704\) of defining polynomial
Character \(\chi\) \(=\) 9405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.577704 q^{2} -1.66626 q^{4} -1.00000 q^{5} -4.19297 q^{7} -2.11801 q^{8} +O(q^{10})\) \(q+0.577704 q^{2} -1.66626 q^{4} -1.00000 q^{5} -4.19297 q^{7} -2.11801 q^{8} -0.577704 q^{10} +1.00000 q^{11} +0.324413 q^{13} -2.42230 q^{14} +2.10893 q^{16} +5.00810 q^{17} -1.00000 q^{19} +1.66626 q^{20} +0.577704 q^{22} -8.11252 q^{23} +1.00000 q^{25} +0.187415 q^{26} +6.98657 q^{28} +3.53579 q^{29} -3.45986 q^{31} +5.45436 q^{32} +2.89320 q^{34} +4.19297 q^{35} +5.77902 q^{37} -0.577704 q^{38} +2.11801 q^{40} -0.484585 q^{41} +5.86472 q^{43} -1.66626 q^{44} -4.68664 q^{46} -6.22580 q^{47} +10.5810 q^{49} +0.577704 q^{50} -0.540556 q^{52} +10.2794 q^{53} -1.00000 q^{55} +8.88076 q^{56} +2.04264 q^{58} -7.16900 q^{59} +5.33749 q^{61} -1.99877 q^{62} -1.06685 q^{64} -0.324413 q^{65} +8.20951 q^{67} -8.34479 q^{68} +2.42230 q^{70} +10.0184 q^{71} +2.88049 q^{73} +3.33857 q^{74} +1.66626 q^{76} -4.19297 q^{77} +3.95809 q^{79} -2.10893 q^{80} -0.279947 q^{82} -10.2837 q^{83} -5.00810 q^{85} +3.38807 q^{86} -2.11801 q^{88} -7.47277 q^{89} -1.36025 q^{91} +13.5175 q^{92} -3.59667 q^{94} +1.00000 q^{95} -13.9617 q^{97} +6.11268 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34} - 5 q^{35} + 9 q^{37} - 25 q^{41} + 15 q^{43} + 8 q^{44} - 16 q^{46} - 24 q^{47} + 13 q^{49} - 27 q^{52} - 5 q^{53} - 6 q^{55} + 12 q^{56} + 13 q^{58} - 39 q^{59} - 11 q^{61} + 42 q^{62} - 14 q^{64} + 9 q^{65} + 24 q^{67} - 45 q^{68} + 18 q^{70} + 24 q^{71} - 26 q^{73} - q^{74} - 8 q^{76} + 5 q^{77} + 11 q^{79} - 4 q^{80} + 8 q^{82} - 39 q^{83} - 5 q^{85} - 18 q^{86} - 22 q^{89} - 26 q^{91} + 11 q^{92} - 30 q^{94} + 6 q^{95} + 22 q^{97} - 33 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.577704 0.408499 0.204249 0.978919i \(-0.434525\pi\)
0.204249 + 0.978919i \(0.434525\pi\)
\(3\) 0 0
\(4\) −1.66626 −0.833129
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.19297 −1.58479 −0.792397 0.610006i \(-0.791167\pi\)
−0.792397 + 0.610006i \(0.791167\pi\)
\(8\) −2.11801 −0.748830
\(9\) 0 0
\(10\) −0.577704 −0.182686
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0.324413 0.0899760 0.0449880 0.998988i \(-0.485675\pi\)
0.0449880 + 0.998988i \(0.485675\pi\)
\(14\) −2.42230 −0.647386
\(15\) 0 0
\(16\) 2.10893 0.527233
\(17\) 5.00810 1.21464 0.607322 0.794456i \(-0.292244\pi\)
0.607322 + 0.794456i \(0.292244\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 1.66626 0.372587
\(21\) 0 0
\(22\) 0.577704 0.123167
\(23\) −8.11252 −1.69158 −0.845789 0.533518i \(-0.820870\pi\)
−0.845789 + 0.533518i \(0.820870\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.187415 0.0367551
\(27\) 0 0
\(28\) 6.98657 1.32034
\(29\) 3.53579 0.656580 0.328290 0.944577i \(-0.393528\pi\)
0.328290 + 0.944577i \(0.393528\pi\)
\(30\) 0 0
\(31\) −3.45986 −0.621409 −0.310704 0.950507i \(-0.600565\pi\)
−0.310704 + 0.950507i \(0.600565\pi\)
\(32\) 5.45436 0.964204
\(33\) 0 0
\(34\) 2.89320 0.496180
\(35\) 4.19297 0.708741
\(36\) 0 0
\(37\) 5.77902 0.950066 0.475033 0.879968i \(-0.342436\pi\)
0.475033 + 0.879968i \(0.342436\pi\)
\(38\) −0.577704 −0.0937160
\(39\) 0 0
\(40\) 2.11801 0.334887
\(41\) −0.484585 −0.0756794 −0.0378397 0.999284i \(-0.512048\pi\)
−0.0378397 + 0.999284i \(0.512048\pi\)
\(42\) 0 0
\(43\) 5.86472 0.894362 0.447181 0.894444i \(-0.352428\pi\)
0.447181 + 0.894444i \(0.352428\pi\)
\(44\) −1.66626 −0.251198
\(45\) 0 0
\(46\) −4.68664 −0.691007
\(47\) −6.22580 −0.908126 −0.454063 0.890970i \(-0.650026\pi\)
−0.454063 + 0.890970i \(0.650026\pi\)
\(48\) 0 0
\(49\) 10.5810 1.51157
\(50\) 0.577704 0.0816997
\(51\) 0 0
\(52\) −0.540556 −0.0749616
\(53\) 10.2794 1.41198 0.705991 0.708221i \(-0.250502\pi\)
0.705991 + 0.708221i \(0.250502\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 8.88076 1.18674
\(57\) 0 0
\(58\) 2.04264 0.268212
\(59\) −7.16900 −0.933325 −0.466662 0.884435i \(-0.654544\pi\)
−0.466662 + 0.884435i \(0.654544\pi\)
\(60\) 0 0
\(61\) 5.33749 0.683396 0.341698 0.939810i \(-0.388998\pi\)
0.341698 + 0.939810i \(0.388998\pi\)
\(62\) −1.99877 −0.253845
\(63\) 0 0
\(64\) −1.06685 −0.133357
\(65\) −0.324413 −0.0402385
\(66\) 0 0
\(67\) 8.20951 1.00295 0.501476 0.865172i \(-0.332791\pi\)
0.501476 + 0.865172i \(0.332791\pi\)
\(68\) −8.34479 −1.01195
\(69\) 0 0
\(70\) 2.42230 0.289520
\(71\) 10.0184 1.18897 0.594484 0.804108i \(-0.297356\pi\)
0.594484 + 0.804108i \(0.297356\pi\)
\(72\) 0 0
\(73\) 2.88049 0.337136 0.168568 0.985690i \(-0.446086\pi\)
0.168568 + 0.985690i \(0.446086\pi\)
\(74\) 3.33857 0.388100
\(75\) 0 0
\(76\) 1.66626 0.191133
\(77\) −4.19297 −0.477833
\(78\) 0 0
\(79\) 3.95809 0.445320 0.222660 0.974896i \(-0.428526\pi\)
0.222660 + 0.974896i \(0.428526\pi\)
\(80\) −2.10893 −0.235786
\(81\) 0 0
\(82\) −0.279947 −0.0309149
\(83\) −10.2837 −1.12878 −0.564389 0.825509i \(-0.690888\pi\)
−0.564389 + 0.825509i \(0.690888\pi\)
\(84\) 0 0
\(85\) −5.00810 −0.543205
\(86\) 3.38807 0.365345
\(87\) 0 0
\(88\) −2.11801 −0.225781
\(89\) −7.47277 −0.792112 −0.396056 0.918226i \(-0.629621\pi\)
−0.396056 + 0.918226i \(0.629621\pi\)
\(90\) 0 0
\(91\) −1.36025 −0.142593
\(92\) 13.5175 1.40930
\(93\) 0 0
\(94\) −3.59667 −0.370968
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −13.9617 −1.41759 −0.708797 0.705412i \(-0.750762\pi\)
−0.708797 + 0.705412i \(0.750762\pi\)
\(98\) 6.11268 0.617474
\(99\) 0 0
\(100\) −1.66626 −0.166626
\(101\) −0.481481 −0.0479091 −0.0239546 0.999713i \(-0.507626\pi\)
−0.0239546 + 0.999713i \(0.507626\pi\)
\(102\) 0 0
\(103\) 14.0759 1.38694 0.693468 0.720488i \(-0.256082\pi\)
0.693468 + 0.720488i \(0.256082\pi\)
\(104\) −0.687111 −0.0673768
\(105\) 0 0
\(106\) 5.93845 0.576793
\(107\) 0.995650 0.0962531 0.0481265 0.998841i \(-0.484675\pi\)
0.0481265 + 0.998841i \(0.484675\pi\)
\(108\) 0 0
\(109\) 5.36197 0.513584 0.256792 0.966467i \(-0.417334\pi\)
0.256792 + 0.966467i \(0.417334\pi\)
\(110\) −0.577704 −0.0550819
\(111\) 0 0
\(112\) −8.84268 −0.835555
\(113\) −2.57028 −0.241792 −0.120896 0.992665i \(-0.538577\pi\)
−0.120896 + 0.992665i \(0.538577\pi\)
\(114\) 0 0
\(115\) 8.11252 0.756496
\(116\) −5.89154 −0.547016
\(117\) 0 0
\(118\) −4.14156 −0.381262
\(119\) −20.9988 −1.92496
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.08349 0.279166
\(123\) 0 0
\(124\) 5.76501 0.517714
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 19.9231 1.76789 0.883947 0.467588i \(-0.154877\pi\)
0.883947 + 0.467588i \(0.154877\pi\)
\(128\) −11.5251 −1.01868
\(129\) 0 0
\(130\) −0.187415 −0.0164374
\(131\) 1.71963 0.150245 0.0751224 0.997174i \(-0.476065\pi\)
0.0751224 + 0.997174i \(0.476065\pi\)
\(132\) 0 0
\(133\) 4.19297 0.363577
\(134\) 4.74267 0.409704
\(135\) 0 0
\(136\) −10.6072 −0.909562
\(137\) −12.6481 −1.08060 −0.540298 0.841473i \(-0.681689\pi\)
−0.540298 + 0.841473i \(0.681689\pi\)
\(138\) 0 0
\(139\) −17.9111 −1.51920 −0.759601 0.650390i \(-0.774606\pi\)
−0.759601 + 0.650390i \(0.774606\pi\)
\(140\) −6.98657 −0.590473
\(141\) 0 0
\(142\) 5.78768 0.485692
\(143\) 0.324413 0.0271288
\(144\) 0 0
\(145\) −3.53579 −0.293632
\(146\) 1.66407 0.137720
\(147\) 0 0
\(148\) −9.62934 −0.791527
\(149\) 17.1692 1.40656 0.703279 0.710914i \(-0.251718\pi\)
0.703279 + 0.710914i \(0.251718\pi\)
\(150\) 0 0
\(151\) −15.6004 −1.26954 −0.634772 0.772699i \(-0.718906\pi\)
−0.634772 + 0.772699i \(0.718906\pi\)
\(152\) 2.11801 0.171793
\(153\) 0 0
\(154\) −2.42230 −0.195194
\(155\) 3.45986 0.277902
\(156\) 0 0
\(157\) 5.17981 0.413394 0.206697 0.978405i \(-0.433729\pi\)
0.206697 + 0.978405i \(0.433729\pi\)
\(158\) 2.28660 0.181912
\(159\) 0 0
\(160\) −5.45436 −0.431205
\(161\) 34.0155 2.68080
\(162\) 0 0
\(163\) −2.66829 −0.208997 −0.104498 0.994525i \(-0.533324\pi\)
−0.104498 + 0.994525i \(0.533324\pi\)
\(164\) 0.807443 0.0630507
\(165\) 0 0
\(166\) −5.94091 −0.461104
\(167\) −12.7357 −0.985516 −0.492758 0.870166i \(-0.664011\pi\)
−0.492758 + 0.870166i \(0.664011\pi\)
\(168\) 0 0
\(169\) −12.8948 −0.991904
\(170\) −2.89320 −0.221898
\(171\) 0 0
\(172\) −9.77214 −0.745119
\(173\) 3.51189 0.267004 0.133502 0.991049i \(-0.457378\pi\)
0.133502 + 0.991049i \(0.457378\pi\)
\(174\) 0 0
\(175\) −4.19297 −0.316959
\(176\) 2.10893 0.158967
\(177\) 0 0
\(178\) −4.31705 −0.323577
\(179\) 0.129487 0.00967830 0.00483915 0.999988i \(-0.498460\pi\)
0.00483915 + 0.999988i \(0.498460\pi\)
\(180\) 0 0
\(181\) −1.43867 −0.106936 −0.0534679 0.998570i \(-0.517027\pi\)
−0.0534679 + 0.998570i \(0.517027\pi\)
\(182\) −0.785825 −0.0582492
\(183\) 0 0
\(184\) 17.1824 1.26670
\(185\) −5.77902 −0.424882
\(186\) 0 0
\(187\) 5.00810 0.366229
\(188\) 10.3738 0.756586
\(189\) 0 0
\(190\) 0.577704 0.0419111
\(191\) −9.83771 −0.711832 −0.355916 0.934518i \(-0.615831\pi\)
−0.355916 + 0.934518i \(0.615831\pi\)
\(192\) 0 0
\(193\) −21.0622 −1.51609 −0.758046 0.652201i \(-0.773846\pi\)
−0.758046 + 0.652201i \(0.773846\pi\)
\(194\) −8.06572 −0.579085
\(195\) 0 0
\(196\) −17.6307 −1.25933
\(197\) 15.7435 1.12168 0.560840 0.827925i \(-0.310478\pi\)
0.560840 + 0.827925i \(0.310478\pi\)
\(198\) 0 0
\(199\) −10.0628 −0.713330 −0.356665 0.934232i \(-0.616086\pi\)
−0.356665 + 0.934232i \(0.616086\pi\)
\(200\) −2.11801 −0.149766
\(201\) 0 0
\(202\) −0.278153 −0.0195708
\(203\) −14.8255 −1.04054
\(204\) 0 0
\(205\) 0.484585 0.0338449
\(206\) 8.13168 0.566561
\(207\) 0 0
\(208\) 0.684165 0.0474383
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −11.0673 −0.761906 −0.380953 0.924594i \(-0.624404\pi\)
−0.380953 + 0.924594i \(0.624404\pi\)
\(212\) −17.1281 −1.17636
\(213\) 0 0
\(214\) 0.575191 0.0393192
\(215\) −5.86472 −0.399971
\(216\) 0 0
\(217\) 14.5071 0.984804
\(218\) 3.09763 0.209798
\(219\) 0 0
\(220\) 1.66626 0.112339
\(221\) 1.62469 0.109289
\(222\) 0 0
\(223\) 23.5560 1.57743 0.788713 0.614762i \(-0.210748\pi\)
0.788713 + 0.614762i \(0.210748\pi\)
\(224\) −22.8700 −1.52806
\(225\) 0 0
\(226\) −1.48486 −0.0987716
\(227\) −9.43858 −0.626461 −0.313230 0.949677i \(-0.601411\pi\)
−0.313230 + 0.949677i \(0.601411\pi\)
\(228\) 0 0
\(229\) −17.0790 −1.12861 −0.564305 0.825566i \(-0.690856\pi\)
−0.564305 + 0.825566i \(0.690856\pi\)
\(230\) 4.68664 0.309028
\(231\) 0 0
\(232\) −7.48885 −0.491667
\(233\) 2.52469 0.165398 0.0826991 0.996575i \(-0.473646\pi\)
0.0826991 + 0.996575i \(0.473646\pi\)
\(234\) 0 0
\(235\) 6.22580 0.406126
\(236\) 11.9454 0.777580
\(237\) 0 0
\(238\) −12.1311 −0.786343
\(239\) −16.4351 −1.06310 −0.531550 0.847027i \(-0.678390\pi\)
−0.531550 + 0.847027i \(0.678390\pi\)
\(240\) 0 0
\(241\) 22.8885 1.47438 0.737189 0.675687i \(-0.236153\pi\)
0.737189 + 0.675687i \(0.236153\pi\)
\(242\) 0.577704 0.0371362
\(243\) 0 0
\(244\) −8.89364 −0.569357
\(245\) −10.5810 −0.675995
\(246\) 0 0
\(247\) −0.324413 −0.0206419
\(248\) 7.32802 0.465330
\(249\) 0 0
\(250\) −0.577704 −0.0365372
\(251\) −13.7873 −0.870246 −0.435123 0.900371i \(-0.643295\pi\)
−0.435123 + 0.900371i \(0.643295\pi\)
\(252\) 0 0
\(253\) −8.11252 −0.510030
\(254\) 11.5097 0.722182
\(255\) 0 0
\(256\) −4.52436 −0.282773
\(257\) 22.2128 1.38560 0.692800 0.721130i \(-0.256377\pi\)
0.692800 + 0.721130i \(0.256377\pi\)
\(258\) 0 0
\(259\) −24.2313 −1.50566
\(260\) 0.540556 0.0335239
\(261\) 0 0
\(262\) 0.993438 0.0613748
\(263\) −23.6220 −1.45660 −0.728298 0.685260i \(-0.759688\pi\)
−0.728298 + 0.685260i \(0.759688\pi\)
\(264\) 0 0
\(265\) −10.2794 −0.631458
\(266\) 2.42230 0.148520
\(267\) 0 0
\(268\) −13.6792 −0.835588
\(269\) −10.3081 −0.628497 −0.314249 0.949341i \(-0.601753\pi\)
−0.314249 + 0.949341i \(0.601753\pi\)
\(270\) 0 0
\(271\) −11.0923 −0.673808 −0.336904 0.941539i \(-0.609380\pi\)
−0.336904 + 0.941539i \(0.609380\pi\)
\(272\) 10.5617 0.640400
\(273\) 0 0
\(274\) −7.30684 −0.441422
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 24.0828 1.44700 0.723498 0.690326i \(-0.242533\pi\)
0.723498 + 0.690326i \(0.242533\pi\)
\(278\) −10.3473 −0.620591
\(279\) 0 0
\(280\) −8.88076 −0.530727
\(281\) 22.2918 1.32982 0.664908 0.746926i \(-0.268471\pi\)
0.664908 + 0.746926i \(0.268471\pi\)
\(282\) 0 0
\(283\) −16.0712 −0.955333 −0.477666 0.878541i \(-0.658517\pi\)
−0.477666 + 0.878541i \(0.658517\pi\)
\(284\) −16.6933 −0.990564
\(285\) 0 0
\(286\) 0.187415 0.0110821
\(287\) 2.03185 0.119936
\(288\) 0 0
\(289\) 8.08109 0.475358
\(290\) −2.04264 −0.119948
\(291\) 0 0
\(292\) −4.79965 −0.280878
\(293\) −22.1899 −1.29635 −0.648175 0.761491i \(-0.724468\pi\)
−0.648175 + 0.761491i \(0.724468\pi\)
\(294\) 0 0
\(295\) 7.16900 0.417396
\(296\) −12.2400 −0.711438
\(297\) 0 0
\(298\) 9.91874 0.574577
\(299\) −2.63181 −0.152201
\(300\) 0 0
\(301\) −24.5906 −1.41738
\(302\) −9.01243 −0.518607
\(303\) 0 0
\(304\) −2.10893 −0.120956
\(305\) −5.33749 −0.305624
\(306\) 0 0
\(307\) −13.5034 −0.770680 −0.385340 0.922775i \(-0.625916\pi\)
−0.385340 + 0.922775i \(0.625916\pi\)
\(308\) 6.98657 0.398097
\(309\) 0 0
\(310\) 1.99877 0.113523
\(311\) −26.8854 −1.52453 −0.762265 0.647265i \(-0.775913\pi\)
−0.762265 + 0.647265i \(0.775913\pi\)
\(312\) 0 0
\(313\) −24.0161 −1.35747 −0.678735 0.734383i \(-0.737471\pi\)
−0.678735 + 0.734383i \(0.737471\pi\)
\(314\) 2.99240 0.168871
\(315\) 0 0
\(316\) −6.59520 −0.371009
\(317\) 3.97085 0.223025 0.111513 0.993763i \(-0.464430\pi\)
0.111513 + 0.993763i \(0.464430\pi\)
\(318\) 0 0
\(319\) 3.53579 0.197966
\(320\) 1.06685 0.0596390
\(321\) 0 0
\(322\) 19.6509 1.09510
\(323\) −5.00810 −0.278658
\(324\) 0 0
\(325\) 0.324413 0.0179952
\(326\) −1.54148 −0.0853748
\(327\) 0 0
\(328\) 1.02636 0.0566711
\(329\) 26.1046 1.43919
\(330\) 0 0
\(331\) 10.2915 0.565672 0.282836 0.959168i \(-0.408725\pi\)
0.282836 + 0.959168i \(0.408725\pi\)
\(332\) 17.1352 0.940418
\(333\) 0 0
\(334\) −7.35745 −0.402582
\(335\) −8.20951 −0.448534
\(336\) 0 0
\(337\) 14.8946 0.811358 0.405679 0.914016i \(-0.367035\pi\)
0.405679 + 0.914016i \(0.367035\pi\)
\(338\) −7.44935 −0.405191
\(339\) 0 0
\(340\) 8.34479 0.452560
\(341\) −3.45986 −0.187362
\(342\) 0 0
\(343\) −15.0150 −0.810734
\(344\) −12.4216 −0.669725
\(345\) 0 0
\(346\) 2.02883 0.109071
\(347\) 30.9460 1.66127 0.830635 0.556817i \(-0.187978\pi\)
0.830635 + 0.556817i \(0.187978\pi\)
\(348\) 0 0
\(349\) −4.85236 −0.259741 −0.129870 0.991531i \(-0.541456\pi\)
−0.129870 + 0.991531i \(0.541456\pi\)
\(350\) −2.42230 −0.129477
\(351\) 0 0
\(352\) 5.45436 0.290719
\(353\) −35.5761 −1.89353 −0.946764 0.321930i \(-0.895669\pi\)
−0.946764 + 0.321930i \(0.895669\pi\)
\(354\) 0 0
\(355\) −10.0184 −0.531723
\(356\) 12.4516 0.659932
\(357\) 0 0
\(358\) 0.0748051 0.00395357
\(359\) −18.1615 −0.958525 −0.479263 0.877672i \(-0.659096\pi\)
−0.479263 + 0.877672i \(0.659096\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −0.831127 −0.0436831
\(363\) 0 0
\(364\) 2.26654 0.118799
\(365\) −2.88049 −0.150772
\(366\) 0 0
\(367\) −12.6666 −0.661189 −0.330595 0.943773i \(-0.607249\pi\)
−0.330595 + 0.943773i \(0.607249\pi\)
\(368\) −17.1087 −0.891855
\(369\) 0 0
\(370\) −3.33857 −0.173564
\(371\) −43.1012 −2.23770
\(372\) 0 0
\(373\) 27.6947 1.43398 0.716988 0.697086i \(-0.245520\pi\)
0.716988 + 0.697086i \(0.245520\pi\)
\(374\) 2.89320 0.149604
\(375\) 0 0
\(376\) 13.1863 0.680033
\(377\) 1.14706 0.0590765
\(378\) 0 0
\(379\) −5.84097 −0.300031 −0.150015 0.988684i \(-0.547932\pi\)
−0.150015 + 0.988684i \(0.547932\pi\)
\(380\) −1.66626 −0.0854772
\(381\) 0 0
\(382\) −5.68329 −0.290782
\(383\) −23.4576 −1.19863 −0.599313 0.800515i \(-0.704559\pi\)
−0.599313 + 0.800515i \(0.704559\pi\)
\(384\) 0 0
\(385\) 4.19297 0.213694
\(386\) −12.1677 −0.619321
\(387\) 0 0
\(388\) 23.2638 1.18104
\(389\) 5.61130 0.284504 0.142252 0.989830i \(-0.454566\pi\)
0.142252 + 0.989830i \(0.454566\pi\)
\(390\) 0 0
\(391\) −40.6283 −2.05466
\(392\) −22.4107 −1.13191
\(393\) 0 0
\(394\) 9.09509 0.458204
\(395\) −3.95809 −0.199153
\(396\) 0 0
\(397\) 2.81880 0.141472 0.0707358 0.997495i \(-0.477465\pi\)
0.0707358 + 0.997495i \(0.477465\pi\)
\(398\) −5.81330 −0.291394
\(399\) 0 0
\(400\) 2.10893 0.105447
\(401\) −32.8017 −1.63804 −0.819019 0.573767i \(-0.805482\pi\)
−0.819019 + 0.573767i \(0.805482\pi\)
\(402\) 0 0
\(403\) −1.12242 −0.0559119
\(404\) 0.802271 0.0399145
\(405\) 0 0
\(406\) −8.56474 −0.425061
\(407\) 5.77902 0.286456
\(408\) 0 0
\(409\) −20.0663 −0.992214 −0.496107 0.868262i \(-0.665238\pi\)
−0.496107 + 0.868262i \(0.665238\pi\)
\(410\) 0.279947 0.0138256
\(411\) 0 0
\(412\) −23.4540 −1.15550
\(413\) 30.0594 1.47913
\(414\) 0 0
\(415\) 10.2837 0.504805
\(416\) 1.76947 0.0867553
\(417\) 0 0
\(418\) −0.577704 −0.0282564
\(419\) 7.51827 0.367292 0.183646 0.982992i \(-0.441210\pi\)
0.183646 + 0.982992i \(0.441210\pi\)
\(420\) 0 0
\(421\) 6.89149 0.335870 0.167935 0.985798i \(-0.446290\pi\)
0.167935 + 0.985798i \(0.446290\pi\)
\(422\) −6.39364 −0.311238
\(423\) 0 0
\(424\) −21.7719 −1.05734
\(425\) 5.00810 0.242929
\(426\) 0 0
\(427\) −22.3800 −1.08304
\(428\) −1.65901 −0.0801912
\(429\) 0 0
\(430\) −3.38807 −0.163387
\(431\) −13.0438 −0.628297 −0.314149 0.949374i \(-0.601719\pi\)
−0.314149 + 0.949374i \(0.601719\pi\)
\(432\) 0 0
\(433\) 9.38251 0.450895 0.225447 0.974255i \(-0.427616\pi\)
0.225447 + 0.974255i \(0.427616\pi\)
\(434\) 8.38080 0.402291
\(435\) 0 0
\(436\) −8.93443 −0.427882
\(437\) 8.11252 0.388074
\(438\) 0 0
\(439\) −34.4559 −1.64449 −0.822244 0.569135i \(-0.807278\pi\)
−0.822244 + 0.569135i \(0.807278\pi\)
\(440\) 2.11801 0.100972
\(441\) 0 0
\(442\) 0.938593 0.0446443
\(443\) −15.6033 −0.741336 −0.370668 0.928765i \(-0.620871\pi\)
−0.370668 + 0.928765i \(0.620871\pi\)
\(444\) 0 0
\(445\) 7.47277 0.354243
\(446\) 13.6084 0.644376
\(447\) 0 0
\(448\) 4.47329 0.211343
\(449\) −40.4269 −1.90786 −0.953932 0.300022i \(-0.903006\pi\)
−0.953932 + 0.300022i \(0.903006\pi\)
\(450\) 0 0
\(451\) −0.484585 −0.0228182
\(452\) 4.28275 0.201444
\(453\) 0 0
\(454\) −5.45271 −0.255908
\(455\) 1.36025 0.0637697
\(456\) 0 0
\(457\) 13.7522 0.643301 0.321650 0.946858i \(-0.395762\pi\)
0.321650 + 0.946858i \(0.395762\pi\)
\(458\) −9.86660 −0.461036
\(459\) 0 0
\(460\) −13.5175 −0.630259
\(461\) −26.0534 −1.21343 −0.606715 0.794919i \(-0.707513\pi\)
−0.606715 + 0.794919i \(0.707513\pi\)
\(462\) 0 0
\(463\) −5.34837 −0.248560 −0.124280 0.992247i \(-0.539662\pi\)
−0.124280 + 0.992247i \(0.539662\pi\)
\(464\) 7.45674 0.346171
\(465\) 0 0
\(466\) 1.45853 0.0675649
\(467\) −8.54901 −0.395601 −0.197800 0.980242i \(-0.563380\pi\)
−0.197800 + 0.980242i \(0.563380\pi\)
\(468\) 0 0
\(469\) −34.4222 −1.58947
\(470\) 3.59667 0.165902
\(471\) 0 0
\(472\) 15.1840 0.698902
\(473\) 5.86472 0.269660
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 34.9895 1.60374
\(477\) 0 0
\(478\) −9.49464 −0.434275
\(479\) 7.72446 0.352939 0.176470 0.984306i \(-0.443532\pi\)
0.176470 + 0.984306i \(0.443532\pi\)
\(480\) 0 0
\(481\) 1.87479 0.0854831
\(482\) 13.2228 0.602281
\(483\) 0 0
\(484\) −1.66626 −0.0757390
\(485\) 13.9617 0.633968
\(486\) 0 0
\(487\) 31.5919 1.43157 0.715784 0.698322i \(-0.246070\pi\)
0.715784 + 0.698322i \(0.246070\pi\)
\(488\) −11.3049 −0.511748
\(489\) 0 0
\(490\) −6.11268 −0.276143
\(491\) 22.8113 1.02946 0.514729 0.857353i \(-0.327892\pi\)
0.514729 + 0.857353i \(0.327892\pi\)
\(492\) 0 0
\(493\) 17.7076 0.797511
\(494\) −0.187415 −0.00843219
\(495\) 0 0
\(496\) −7.29660 −0.327627
\(497\) −42.0069 −1.88427
\(498\) 0 0
\(499\) 41.7098 1.86719 0.933593 0.358336i \(-0.116656\pi\)
0.933593 + 0.358336i \(0.116656\pi\)
\(500\) 1.66626 0.0745173
\(501\) 0 0
\(502\) −7.96498 −0.355494
\(503\) 5.43339 0.242263 0.121131 0.992636i \(-0.461348\pi\)
0.121131 + 0.992636i \(0.461348\pi\)
\(504\) 0 0
\(505\) 0.481481 0.0214256
\(506\) −4.68664 −0.208346
\(507\) 0 0
\(508\) −33.1971 −1.47288
\(509\) 20.8742 0.925233 0.462616 0.886559i \(-0.346911\pi\)
0.462616 + 0.886559i \(0.346911\pi\)
\(510\) 0 0
\(511\) −12.0778 −0.534291
\(512\) 20.4364 0.903168
\(513\) 0 0
\(514\) 12.8325 0.566015
\(515\) −14.0759 −0.620256
\(516\) 0 0
\(517\) −6.22580 −0.273810
\(518\) −13.9985 −0.615059
\(519\) 0 0
\(520\) 0.687111 0.0301318
\(521\) −40.4540 −1.77232 −0.886161 0.463377i \(-0.846638\pi\)
−0.886161 + 0.463377i \(0.846638\pi\)
\(522\) 0 0
\(523\) 13.3864 0.585345 0.292672 0.956213i \(-0.405455\pi\)
0.292672 + 0.956213i \(0.405455\pi\)
\(524\) −2.86535 −0.125173
\(525\) 0 0
\(526\) −13.6465 −0.595017
\(527\) −17.3273 −0.754790
\(528\) 0 0
\(529\) 42.8130 1.86143
\(530\) −5.93845 −0.257950
\(531\) 0 0
\(532\) −6.98657 −0.302906
\(533\) −0.157206 −0.00680934
\(534\) 0 0
\(535\) −0.995650 −0.0430457
\(536\) −17.3878 −0.751041
\(537\) 0 0
\(538\) −5.95505 −0.256740
\(539\) 10.5810 0.455756
\(540\) 0 0
\(541\) 33.5953 1.44437 0.722186 0.691699i \(-0.243137\pi\)
0.722186 + 0.691699i \(0.243137\pi\)
\(542\) −6.40805 −0.275249
\(543\) 0 0
\(544\) 27.3160 1.17116
\(545\) −5.36197 −0.229682
\(546\) 0 0
\(547\) 16.5276 0.706671 0.353335 0.935497i \(-0.385047\pi\)
0.353335 + 0.935497i \(0.385047\pi\)
\(548\) 21.0749 0.900277
\(549\) 0 0
\(550\) 0.577704 0.0246334
\(551\) −3.53579 −0.150630
\(552\) 0 0
\(553\) −16.5961 −0.705740
\(554\) 13.9127 0.591096
\(555\) 0 0
\(556\) 29.8445 1.26569
\(557\) 39.5660 1.67647 0.838234 0.545311i \(-0.183589\pi\)
0.838234 + 0.545311i \(0.183589\pi\)
\(558\) 0 0
\(559\) 1.90259 0.0804711
\(560\) 8.84268 0.373672
\(561\) 0 0
\(562\) 12.8780 0.543228
\(563\) −27.8205 −1.17249 −0.586246 0.810133i \(-0.699395\pi\)
−0.586246 + 0.810133i \(0.699395\pi\)
\(564\) 0 0
\(565\) 2.57028 0.108133
\(566\) −9.28439 −0.390252
\(567\) 0 0
\(568\) −21.2191 −0.890335
\(569\) −25.3277 −1.06179 −0.530897 0.847436i \(-0.678145\pi\)
−0.530897 + 0.847436i \(0.678145\pi\)
\(570\) 0 0
\(571\) 27.5290 1.15205 0.576027 0.817431i \(-0.304602\pi\)
0.576027 + 0.817431i \(0.304602\pi\)
\(572\) −0.540556 −0.0226018
\(573\) 0 0
\(574\) 1.17381 0.0489938
\(575\) −8.11252 −0.338315
\(576\) 0 0
\(577\) 28.5982 1.19056 0.595280 0.803519i \(-0.297041\pi\)
0.595280 + 0.803519i \(0.297041\pi\)
\(578\) 4.66848 0.194183
\(579\) 0 0
\(580\) 5.89154 0.244633
\(581\) 43.1191 1.78888
\(582\) 0 0
\(583\) 10.2794 0.425729
\(584\) −6.10092 −0.252458
\(585\) 0 0
\(586\) −12.8192 −0.529557
\(587\) −27.7094 −1.14369 −0.571844 0.820362i \(-0.693772\pi\)
−0.571844 + 0.820362i \(0.693772\pi\)
\(588\) 0 0
\(589\) 3.45986 0.142561
\(590\) 4.14156 0.170505
\(591\) 0 0
\(592\) 12.1876 0.500906
\(593\) 15.3303 0.629539 0.314769 0.949168i \(-0.398073\pi\)
0.314769 + 0.949168i \(0.398073\pi\)
\(594\) 0 0
\(595\) 20.9988 0.860868
\(596\) −28.6084 −1.17184
\(597\) 0 0
\(598\) −1.52041 −0.0621740
\(599\) 6.66497 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(600\) 0 0
\(601\) 2.25703 0.0920661 0.0460331 0.998940i \(-0.485342\pi\)
0.0460331 + 0.998940i \(0.485342\pi\)
\(602\) −14.2061 −0.578997
\(603\) 0 0
\(604\) 25.9943 1.05769
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 1.76278 0.0715491 0.0357745 0.999360i \(-0.488610\pi\)
0.0357745 + 0.999360i \(0.488610\pi\)
\(608\) −5.45436 −0.221204
\(609\) 0 0
\(610\) −3.08349 −0.124847
\(611\) −2.01973 −0.0817096
\(612\) 0 0
\(613\) −35.6831 −1.44123 −0.720614 0.693337i \(-0.756140\pi\)
−0.720614 + 0.693337i \(0.756140\pi\)
\(614\) −7.80097 −0.314822
\(615\) 0 0
\(616\) 8.88076 0.357816
\(617\) −3.86204 −0.155480 −0.0777400 0.996974i \(-0.524770\pi\)
−0.0777400 + 0.996974i \(0.524770\pi\)
\(618\) 0 0
\(619\) 40.3816 1.62308 0.811538 0.584300i \(-0.198631\pi\)
0.811538 + 0.584300i \(0.198631\pi\)
\(620\) −5.76501 −0.231529
\(621\) 0 0
\(622\) −15.5318 −0.622768
\(623\) 31.3331 1.25533
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −13.8742 −0.554524
\(627\) 0 0
\(628\) −8.63089 −0.344410
\(629\) 28.9419 1.15399
\(630\) 0 0
\(631\) −31.9100 −1.27032 −0.635159 0.772382i \(-0.719065\pi\)
−0.635159 + 0.772382i \(0.719065\pi\)
\(632\) −8.38328 −0.333469
\(633\) 0 0
\(634\) 2.29398 0.0911054
\(635\) −19.9231 −0.790626
\(636\) 0 0
\(637\) 3.43261 0.136005
\(638\) 2.04264 0.0808690
\(639\) 0 0
\(640\) 11.5251 0.455568
\(641\) 42.9789 1.69757 0.848783 0.528741i \(-0.177336\pi\)
0.848783 + 0.528741i \(0.177336\pi\)
\(642\) 0 0
\(643\) 5.15134 0.203149 0.101574 0.994828i \(-0.467612\pi\)
0.101574 + 0.994828i \(0.467612\pi\)
\(644\) −56.6787 −2.23345
\(645\) 0 0
\(646\) −2.89320 −0.113831
\(647\) 11.7329 0.461268 0.230634 0.973041i \(-0.425920\pi\)
0.230634 + 0.973041i \(0.425920\pi\)
\(648\) 0 0
\(649\) −7.16900 −0.281408
\(650\) 0.187415 0.00735102
\(651\) 0 0
\(652\) 4.44606 0.174121
\(653\) −24.3265 −0.951971 −0.475986 0.879453i \(-0.657909\pi\)
−0.475986 + 0.879453i \(0.657909\pi\)
\(654\) 0 0
\(655\) −1.71963 −0.0671916
\(656\) −1.02196 −0.0399007
\(657\) 0 0
\(658\) 15.0807 0.587908
\(659\) −24.4562 −0.952680 −0.476340 0.879261i \(-0.658037\pi\)
−0.476340 + 0.879261i \(0.658037\pi\)
\(660\) 0 0
\(661\) 6.67388 0.259584 0.129792 0.991541i \(-0.458569\pi\)
0.129792 + 0.991541i \(0.458569\pi\)
\(662\) 5.94544 0.231076
\(663\) 0 0
\(664\) 21.7809 0.845263
\(665\) −4.19297 −0.162596
\(666\) 0 0
\(667\) −28.6842 −1.11066
\(668\) 21.2209 0.821062
\(669\) 0 0
\(670\) −4.74267 −0.183225
\(671\) 5.33749 0.206052
\(672\) 0 0
\(673\) 2.22527 0.0857778 0.0428889 0.999080i \(-0.486344\pi\)
0.0428889 + 0.999080i \(0.486344\pi\)
\(674\) 8.60465 0.331439
\(675\) 0 0
\(676\) 21.4860 0.826384
\(677\) −22.6964 −0.872292 −0.436146 0.899876i \(-0.643657\pi\)
−0.436146 + 0.899876i \(0.643657\pi\)
\(678\) 0 0
\(679\) 58.5409 2.24659
\(680\) 10.6072 0.406768
\(681\) 0 0
\(682\) −1.99877 −0.0765370
\(683\) 39.2539 1.50201 0.751005 0.660296i \(-0.229569\pi\)
0.751005 + 0.660296i \(0.229569\pi\)
\(684\) 0 0
\(685\) 12.6481 0.483258
\(686\) −8.67423 −0.331183
\(687\) 0 0
\(688\) 12.3683 0.471537
\(689\) 3.33477 0.127045
\(690\) 0 0
\(691\) 28.4393 1.08188 0.540941 0.841060i \(-0.318068\pi\)
0.540941 + 0.841060i \(0.318068\pi\)
\(692\) −5.85171 −0.222449
\(693\) 0 0
\(694\) 17.8777 0.678627
\(695\) 17.9111 0.679407
\(696\) 0 0
\(697\) −2.42685 −0.0919235
\(698\) −2.80323 −0.106104
\(699\) 0 0
\(700\) 6.98657 0.264067
\(701\) −13.1038 −0.494925 −0.247463 0.968897i \(-0.579597\pi\)
−0.247463 + 0.968897i \(0.579597\pi\)
\(702\) 0 0
\(703\) −5.77902 −0.217960
\(704\) −1.06685 −0.0402086
\(705\) 0 0
\(706\) −20.5525 −0.773503
\(707\) 2.01883 0.0759260
\(708\) 0 0
\(709\) 18.3440 0.688924 0.344462 0.938800i \(-0.388061\pi\)
0.344462 + 0.938800i \(0.388061\pi\)
\(710\) −5.78768 −0.217208
\(711\) 0 0
\(712\) 15.8274 0.593158
\(713\) 28.0682 1.05116
\(714\) 0 0
\(715\) −0.324413 −0.0121324
\(716\) −0.215759 −0.00806327
\(717\) 0 0
\(718\) −10.4920 −0.391556
\(719\) 1.88586 0.0703309 0.0351654 0.999382i \(-0.488804\pi\)
0.0351654 + 0.999382i \(0.488804\pi\)
\(720\) 0 0
\(721\) −59.0196 −2.19801
\(722\) 0.577704 0.0214999
\(723\) 0 0
\(724\) 2.39720 0.0890913
\(725\) 3.53579 0.131316
\(726\) 0 0
\(727\) −16.9917 −0.630185 −0.315093 0.949061i \(-0.602036\pi\)
−0.315093 + 0.949061i \(0.602036\pi\)
\(728\) 2.88104 0.106778
\(729\) 0 0
\(730\) −1.66407 −0.0615901
\(731\) 29.3711 1.08633
\(732\) 0 0
\(733\) −17.8984 −0.661091 −0.330545 0.943790i \(-0.607233\pi\)
−0.330545 + 0.943790i \(0.607233\pi\)
\(734\) −7.31753 −0.270095
\(735\) 0 0
\(736\) −44.2486 −1.63103
\(737\) 8.20951 0.302401
\(738\) 0 0
\(739\) −51.7970 −1.90538 −0.952691 0.303940i \(-0.901698\pi\)
−0.952691 + 0.303940i \(0.901698\pi\)
\(740\) 9.62934 0.353982
\(741\) 0 0
\(742\) −24.8997 −0.914097
\(743\) 38.8105 1.42382 0.711909 0.702272i \(-0.247831\pi\)
0.711909 + 0.702272i \(0.247831\pi\)
\(744\) 0 0
\(745\) −17.1692 −0.629032
\(746\) 15.9993 0.585777
\(747\) 0 0
\(748\) −8.34479 −0.305116
\(749\) −4.17473 −0.152541
\(750\) 0 0
\(751\) 16.2035 0.591276 0.295638 0.955300i \(-0.404468\pi\)
0.295638 + 0.955300i \(0.404468\pi\)
\(752\) −13.1298 −0.478794
\(753\) 0 0
\(754\) 0.662660 0.0241327
\(755\) 15.6004 0.567758
\(756\) 0 0
\(757\) 2.83100 0.102894 0.0514471 0.998676i \(-0.483617\pi\)
0.0514471 + 0.998676i \(0.483617\pi\)
\(758\) −3.37435 −0.122562
\(759\) 0 0
\(760\) −2.11801 −0.0768284
\(761\) 17.0317 0.617399 0.308700 0.951160i \(-0.400106\pi\)
0.308700 + 0.951160i \(0.400106\pi\)
\(762\) 0 0
\(763\) −22.4826 −0.813925
\(764\) 16.3922 0.593048
\(765\) 0 0
\(766\) −13.5515 −0.489637
\(767\) −2.32572 −0.0839769
\(768\) 0 0
\(769\) −24.5256 −0.884415 −0.442208 0.896913i \(-0.645805\pi\)
−0.442208 + 0.896913i \(0.645805\pi\)
\(770\) 2.42230 0.0872935
\(771\) 0 0
\(772\) 35.0951 1.26310
\(773\) −2.29735 −0.0826299 −0.0413150 0.999146i \(-0.513155\pi\)
−0.0413150 + 0.999146i \(0.513155\pi\)
\(774\) 0 0
\(775\) −3.45986 −0.124282
\(776\) 29.5710 1.06154
\(777\) 0 0
\(778\) 3.24167 0.116220
\(779\) 0.484585 0.0173621
\(780\) 0 0
\(781\) 10.0184 0.358487
\(782\) −23.4712 −0.839327
\(783\) 0 0
\(784\) 22.3146 0.796950
\(785\) −5.17981 −0.184875
\(786\) 0 0
\(787\) −5.33806 −0.190281 −0.0951406 0.995464i \(-0.530330\pi\)
−0.0951406 + 0.995464i \(0.530330\pi\)
\(788\) −26.2328 −0.934503
\(789\) 0 0
\(790\) −2.28660 −0.0813537
\(791\) 10.7771 0.383190
\(792\) 0 0
\(793\) 1.73155 0.0614893
\(794\) 1.62843 0.0577909
\(795\) 0 0
\(796\) 16.7672 0.594296
\(797\) 51.5514 1.82604 0.913022 0.407911i \(-0.133743\pi\)
0.913022 + 0.407911i \(0.133743\pi\)
\(798\) 0 0
\(799\) −31.1794 −1.10305
\(800\) 5.45436 0.192841
\(801\) 0 0
\(802\) −18.9497 −0.669136
\(803\) 2.88049 0.101650
\(804\) 0 0
\(805\) −34.0155 −1.19889
\(806\) −0.648429 −0.0228399
\(807\) 0 0
\(808\) 1.01978 0.0358758
\(809\) −20.4992 −0.720712 −0.360356 0.932815i \(-0.617345\pi\)
−0.360356 + 0.932815i \(0.617345\pi\)
\(810\) 0 0
\(811\) 39.0473 1.37114 0.685568 0.728009i \(-0.259554\pi\)
0.685568 + 0.728009i \(0.259554\pi\)
\(812\) 24.7031 0.866907
\(813\) 0 0
\(814\) 3.33857 0.117017
\(815\) 2.66829 0.0934661
\(816\) 0 0
\(817\) −5.86472 −0.205181
\(818\) −11.5924 −0.405318
\(819\) 0 0
\(820\) −0.807443 −0.0281971
\(821\) −6.52431 −0.227700 −0.113850 0.993498i \(-0.536318\pi\)
−0.113850 + 0.993498i \(0.536318\pi\)
\(822\) 0 0
\(823\) −13.0628 −0.455339 −0.227670 0.973738i \(-0.573111\pi\)
−0.227670 + 0.973738i \(0.573111\pi\)
\(824\) −29.8128 −1.03858
\(825\) 0 0
\(826\) 17.3655 0.604221
\(827\) −22.8198 −0.793522 −0.396761 0.917922i \(-0.629866\pi\)
−0.396761 + 0.917922i \(0.629866\pi\)
\(828\) 0 0
\(829\) −43.7012 −1.51780 −0.758902 0.651205i \(-0.774264\pi\)
−0.758902 + 0.651205i \(0.774264\pi\)
\(830\) 5.94091 0.206212
\(831\) 0 0
\(832\) −0.346102 −0.0119989
\(833\) 52.9907 1.83602
\(834\) 0 0
\(835\) 12.7357 0.440736
\(836\) 1.66626 0.0576287
\(837\) 0 0
\(838\) 4.34334 0.150038
\(839\) 12.2493 0.422892 0.211446 0.977390i \(-0.432183\pi\)
0.211446 + 0.977390i \(0.432183\pi\)
\(840\) 0 0
\(841\) −16.4982 −0.568902
\(842\) 3.98124 0.137203
\(843\) 0 0
\(844\) 18.4410 0.634766
\(845\) 12.8948 0.443593
\(846\) 0 0
\(847\) −4.19297 −0.144072
\(848\) 21.6785 0.744444
\(849\) 0 0
\(850\) 2.89320 0.0992360
\(851\) −46.8824 −1.60711
\(852\) 0 0
\(853\) 2.70591 0.0926487 0.0463243 0.998926i \(-0.485249\pi\)
0.0463243 + 0.998926i \(0.485249\pi\)
\(854\) −12.9290 −0.442421
\(855\) 0 0
\(856\) −2.10880 −0.0720773
\(857\) −6.16547 −0.210608 −0.105304 0.994440i \(-0.533582\pi\)
−0.105304 + 0.994440i \(0.533582\pi\)
\(858\) 0 0
\(859\) 48.5324 1.65590 0.827952 0.560800i \(-0.189506\pi\)
0.827952 + 0.560800i \(0.189506\pi\)
\(860\) 9.77214 0.333227
\(861\) 0 0
\(862\) −7.53545 −0.256659
\(863\) 41.0881 1.39866 0.699328 0.714801i \(-0.253483\pi\)
0.699328 + 0.714801i \(0.253483\pi\)
\(864\) 0 0
\(865\) −3.51189 −0.119408
\(866\) 5.42032 0.184190
\(867\) 0 0
\(868\) −24.1725 −0.820469
\(869\) 3.95809 0.134269
\(870\) 0 0
\(871\) 2.66327 0.0902416
\(872\) −11.3567 −0.384587
\(873\) 0 0
\(874\) 4.68664 0.158528
\(875\) 4.19297 0.141748
\(876\) 0 0
\(877\) −6.98501 −0.235867 −0.117934 0.993021i \(-0.537627\pi\)
−0.117934 + 0.993021i \(0.537627\pi\)
\(878\) −19.9053 −0.671771
\(879\) 0 0
\(880\) −2.10893 −0.0710921
\(881\) −34.5166 −1.16289 −0.581447 0.813584i \(-0.697513\pi\)
−0.581447 + 0.813584i \(0.697513\pi\)
\(882\) 0 0
\(883\) −42.9328 −1.44480 −0.722401 0.691474i \(-0.756962\pi\)
−0.722401 + 0.691474i \(0.756962\pi\)
\(884\) −2.70716 −0.0910517
\(885\) 0 0
\(886\) −9.01410 −0.302835
\(887\) 31.2234 1.04838 0.524189 0.851602i \(-0.324369\pi\)
0.524189 + 0.851602i \(0.324369\pi\)
\(888\) 0 0
\(889\) −83.5371 −2.80175
\(890\) 4.31705 0.144708
\(891\) 0 0
\(892\) −39.2503 −1.31420
\(893\) 6.22580 0.208338
\(894\) 0 0
\(895\) −0.129487 −0.00432827
\(896\) 48.3242 1.61440
\(897\) 0 0
\(898\) −23.3548 −0.779360
\(899\) −12.2333 −0.408005
\(900\) 0 0
\(901\) 51.4802 1.71506
\(902\) −0.279947 −0.00932121
\(903\) 0 0
\(904\) 5.44389 0.181061
\(905\) 1.43867 0.0478231
\(906\) 0 0
\(907\) 31.2576 1.03789 0.518946 0.854807i \(-0.326324\pi\)
0.518946 + 0.854807i \(0.326324\pi\)
\(908\) 15.7271 0.521923
\(909\) 0 0
\(910\) 0.785825 0.0260498
\(911\) −48.9833 −1.62289 −0.811445 0.584429i \(-0.801319\pi\)
−0.811445 + 0.584429i \(0.801319\pi\)
\(912\) 0 0
\(913\) −10.2837 −0.340339
\(914\) 7.94470 0.262787
\(915\) 0 0
\(916\) 28.4580 0.940278
\(917\) −7.21036 −0.238107
\(918\) 0 0
\(919\) −20.9524 −0.691157 −0.345578 0.938390i \(-0.612317\pi\)
−0.345578 + 0.938390i \(0.612317\pi\)
\(920\) −17.1824 −0.566487
\(921\) 0 0
\(922\) −15.0512 −0.495684
\(923\) 3.25011 0.106979
\(924\) 0 0
\(925\) 5.77902 0.190013
\(926\) −3.08977 −0.101536
\(927\) 0 0
\(928\) 19.2855 0.633077
\(929\) −56.0533 −1.83905 −0.919525 0.393031i \(-0.871426\pi\)
−0.919525 + 0.393031i \(0.871426\pi\)
\(930\) 0 0
\(931\) −10.5810 −0.346778
\(932\) −4.20679 −0.137798
\(933\) 0 0
\(934\) −4.93880 −0.161602
\(935\) −5.00810 −0.163782
\(936\) 0 0
\(937\) 7.53790 0.246253 0.123126 0.992391i \(-0.460708\pi\)
0.123126 + 0.992391i \(0.460708\pi\)
\(938\) −19.8859 −0.649297
\(939\) 0 0
\(940\) −10.3738 −0.338356
\(941\) −38.3962 −1.25168 −0.625840 0.779951i \(-0.715244\pi\)
−0.625840 + 0.779951i \(0.715244\pi\)
\(942\) 0 0
\(943\) 3.93120 0.128018
\(944\) −15.1189 −0.492080
\(945\) 0 0
\(946\) 3.38807 0.110156
\(947\) 10.9350 0.355340 0.177670 0.984090i \(-0.443144\pi\)
0.177670 + 0.984090i \(0.443144\pi\)
\(948\) 0 0
\(949\) 0.934470 0.0303342
\(950\) −0.577704 −0.0187432
\(951\) 0 0
\(952\) 44.4758 1.44147
\(953\) −45.9324 −1.48790 −0.743948 0.668238i \(-0.767049\pi\)
−0.743948 + 0.668238i \(0.767049\pi\)
\(954\) 0 0
\(955\) 9.83771 0.318341
\(956\) 27.3852 0.885700
\(957\) 0 0
\(958\) 4.46245 0.144175
\(959\) 53.0329 1.71252
\(960\) 0 0
\(961\) −19.0294 −0.613851
\(962\) 1.08307 0.0349197
\(963\) 0 0
\(964\) −38.1381 −1.22835
\(965\) 21.0622 0.678017
\(966\) 0 0
\(967\) −36.8982 −1.18657 −0.593283 0.804994i \(-0.702168\pi\)
−0.593283 + 0.804994i \(0.702168\pi\)
\(968\) −2.11801 −0.0680755
\(969\) 0 0
\(970\) 8.06572 0.258975
\(971\) 49.1483 1.57724 0.788622 0.614878i \(-0.210795\pi\)
0.788622 + 0.614878i \(0.210795\pi\)
\(972\) 0 0
\(973\) 75.1008 2.40762
\(974\) 18.2508 0.584793
\(975\) 0 0
\(976\) 11.2564 0.360309
\(977\) −27.8361 −0.890557 −0.445279 0.895392i \(-0.646895\pi\)
−0.445279 + 0.895392i \(0.646895\pi\)
\(978\) 0 0
\(979\) −7.47277 −0.238831
\(980\) 17.6307 0.563191
\(981\) 0 0
\(982\) 13.1782 0.420532
\(983\) 12.7794 0.407598 0.203799 0.979013i \(-0.434671\pi\)
0.203799 + 0.979013i \(0.434671\pi\)
\(984\) 0 0
\(985\) −15.7435 −0.501630
\(986\) 10.2298 0.325782
\(987\) 0 0
\(988\) 0.540556 0.0171974
\(989\) −47.5777 −1.51288
\(990\) 0 0
\(991\) 62.7704 1.99397 0.996984 0.0776029i \(-0.0247266\pi\)
0.996984 + 0.0776029i \(0.0247266\pi\)
\(992\) −18.8713 −0.599165
\(993\) 0 0
\(994\) −24.2676 −0.769721
\(995\) 10.0628 0.319011
\(996\) 0 0
\(997\) 13.0365 0.412871 0.206435 0.978460i \(-0.433814\pi\)
0.206435 + 0.978460i \(0.433814\pi\)
\(998\) 24.0959 0.762743
\(999\) 0 0
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 9405.2.a.w.1.4 6
3.2 odd 2 1045.2.a.g.1.3 6
15.14 odd 2 5225.2.a.k.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.3 6 3.2 odd 2
5225.2.a.k.1.4 6 15.14 odd 2
9405.2.a.w.1.4 6 1.1 even 1 trivial