Properties

Label 1336.2.a.c.1.4
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $1$
Dimension $9$
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] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.460631\) of defining polynomial
Character \(\chi\) \(=\) 1336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.460631 q^{3} +0.528623 q^{5} -1.88578 q^{7} -2.78782 q^{9} +O(q^{10})\) \(q-0.460631 q^{3} +0.528623 q^{5} -1.88578 q^{7} -2.78782 q^{9} +4.52080 q^{11} -4.67304 q^{13} -0.243500 q^{15} +7.22703 q^{17} -2.05086 q^{19} +0.868646 q^{21} -0.329593 q^{23} -4.72056 q^{25} +2.66605 q^{27} -3.05453 q^{29} -7.53069 q^{31} -2.08242 q^{33} -0.996865 q^{35} -8.39234 q^{37} +2.15254 q^{39} -5.74458 q^{41} +3.97486 q^{43} -1.47371 q^{45} -3.13084 q^{47} -3.44385 q^{49} -3.32899 q^{51} -4.78055 q^{53} +2.38980 q^{55} +0.944691 q^{57} +8.57453 q^{59} -4.95580 q^{61} +5.25720 q^{63} -2.47028 q^{65} -4.60533 q^{67} +0.151821 q^{69} +11.2444 q^{71} -8.43916 q^{73} +2.17443 q^{75} -8.52522 q^{77} -10.3162 q^{79} +7.13539 q^{81} +4.86634 q^{83} +3.82038 q^{85} +1.40701 q^{87} -11.6367 q^{89} +8.81229 q^{91} +3.46887 q^{93} -1.08413 q^{95} +1.79262 q^{97} -12.6032 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} - 8 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{3} - 8 q^{5} + 2 q^{7} - 10 q^{11} - 13 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 10 q^{27} - 25 q^{29} - q^{31} - 12 q^{33} - 17 q^{35} - 35 q^{37} - 4 q^{39} - 16 q^{41} + 9 q^{43} - 24 q^{45} - q^{47} - q^{49} - 10 q^{51} - 29 q^{53} + 9 q^{55} - 17 q^{57} - 14 q^{59} - 28 q^{61} + 4 q^{63} - 31 q^{65} + 19 q^{67} - 19 q^{69} - 9 q^{71} - 7 q^{75} - 33 q^{77} - 18 q^{79} - 27 q^{81} - 13 q^{83} - 36 q^{85} + 18 q^{87} - 21 q^{89} + 20 q^{91} - 35 q^{93} - 12 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.460631 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(4\) 0 0
\(5\) 0.528623 0.236408 0.118204 0.992989i \(-0.462286\pi\)
0.118204 + 0.992989i \(0.462286\pi\)
\(6\) 0 0
\(7\) −1.88578 −0.712756 −0.356378 0.934342i \(-0.615988\pi\)
−0.356378 + 0.934342i \(0.615988\pi\)
\(8\) 0 0
\(9\) −2.78782 −0.929273
\(10\) 0 0
\(11\) 4.52080 1.36307 0.681537 0.731784i \(-0.261312\pi\)
0.681537 + 0.731784i \(0.261312\pi\)
\(12\) 0 0
\(13\) −4.67304 −1.29607 −0.648033 0.761612i \(-0.724408\pi\)
−0.648033 + 0.761612i \(0.724408\pi\)
\(14\) 0 0
\(15\) −0.243500 −0.0628715
\(16\) 0 0
\(17\) 7.22703 1.75281 0.876406 0.481573i \(-0.159935\pi\)
0.876406 + 0.481573i \(0.159935\pi\)
\(18\) 0 0
\(19\) −2.05086 −0.470500 −0.235250 0.971935i \(-0.575591\pi\)
−0.235250 + 0.971935i \(0.575591\pi\)
\(20\) 0 0
\(21\) 0.868646 0.189554
\(22\) 0 0
\(23\) −0.329593 −0.0687248 −0.0343624 0.999409i \(-0.510940\pi\)
−0.0343624 + 0.999409i \(0.510940\pi\)
\(24\) 0 0
\(25\) −4.72056 −0.944111
\(26\) 0 0
\(27\) 2.66605 0.513081
\(28\) 0 0
\(29\) −3.05453 −0.567211 −0.283606 0.958941i \(-0.591531\pi\)
−0.283606 + 0.958941i \(0.591531\pi\)
\(30\) 0 0
\(31\) −7.53069 −1.35255 −0.676276 0.736649i \(-0.736407\pi\)
−0.676276 + 0.736649i \(0.736407\pi\)
\(32\) 0 0
\(33\) −2.08242 −0.362503
\(34\) 0 0
\(35\) −0.996865 −0.168501
\(36\) 0 0
\(37\) −8.39234 −1.37969 −0.689846 0.723956i \(-0.742322\pi\)
−0.689846 + 0.723956i \(0.742322\pi\)
\(38\) 0 0
\(39\) 2.15254 0.344683
\(40\) 0 0
\(41\) −5.74458 −0.897153 −0.448576 0.893744i \(-0.648069\pi\)
−0.448576 + 0.893744i \(0.648069\pi\)
\(42\) 0 0
\(43\) 3.97486 0.606160 0.303080 0.952965i \(-0.401985\pi\)
0.303080 + 0.952965i \(0.401985\pi\)
\(44\) 0 0
\(45\) −1.47371 −0.219687
\(46\) 0 0
\(47\) −3.13084 −0.456680 −0.228340 0.973581i \(-0.573330\pi\)
−0.228340 + 0.973581i \(0.573330\pi\)
\(48\) 0 0
\(49\) −3.44385 −0.491979
\(50\) 0 0
\(51\) −3.32899 −0.466152
\(52\) 0 0
\(53\) −4.78055 −0.656659 −0.328330 0.944563i \(-0.606486\pi\)
−0.328330 + 0.944563i \(0.606486\pi\)
\(54\) 0 0
\(55\) 2.38980 0.322241
\(56\) 0 0
\(57\) 0.944691 0.125127
\(58\) 0 0
\(59\) 8.57453 1.11631 0.558154 0.829737i \(-0.311510\pi\)
0.558154 + 0.829737i \(0.311510\pi\)
\(60\) 0 0
\(61\) −4.95580 −0.634526 −0.317263 0.948338i \(-0.602764\pi\)
−0.317263 + 0.948338i \(0.602764\pi\)
\(62\) 0 0
\(63\) 5.25720 0.662345
\(64\) 0 0
\(65\) −2.47028 −0.306400
\(66\) 0 0
\(67\) −4.60533 −0.562630 −0.281315 0.959615i \(-0.590771\pi\)
−0.281315 + 0.959615i \(0.590771\pi\)
\(68\) 0 0
\(69\) 0.151821 0.0182771
\(70\) 0 0
\(71\) 11.2444 1.33446 0.667231 0.744851i \(-0.267479\pi\)
0.667231 + 0.744851i \(0.267479\pi\)
\(72\) 0 0
\(73\) −8.43916 −0.987729 −0.493864 0.869539i \(-0.664416\pi\)
−0.493864 + 0.869539i \(0.664416\pi\)
\(74\) 0 0
\(75\) 2.17443 0.251082
\(76\) 0 0
\(77\) −8.52522 −0.971539
\(78\) 0 0
\(79\) −10.3162 −1.16066 −0.580332 0.814380i \(-0.697077\pi\)
−0.580332 + 0.814380i \(0.697077\pi\)
\(80\) 0 0
\(81\) 7.13539 0.792821
\(82\) 0 0
\(83\) 4.86634 0.534150 0.267075 0.963676i \(-0.413943\pi\)
0.267075 + 0.963676i \(0.413943\pi\)
\(84\) 0 0
\(85\) 3.82038 0.414378
\(86\) 0 0
\(87\) 1.40701 0.150847
\(88\) 0 0
\(89\) −11.6367 −1.23348 −0.616742 0.787165i \(-0.711548\pi\)
−0.616742 + 0.787165i \(0.711548\pi\)
\(90\) 0 0
\(91\) 8.81229 0.923780
\(92\) 0 0
\(93\) 3.46887 0.359705
\(94\) 0 0
\(95\) −1.08413 −0.111230
\(96\) 0 0
\(97\) 1.79262 0.182013 0.0910067 0.995850i \(-0.470992\pi\)
0.0910067 + 0.995850i \(0.470992\pi\)
\(98\) 0 0
\(99\) −12.6032 −1.26667
\(100\) 0 0
\(101\) 2.88795 0.287362 0.143681 0.989624i \(-0.454106\pi\)
0.143681 + 0.989624i \(0.454106\pi\)
\(102\) 0 0
\(103\) −2.19062 −0.215848 −0.107924 0.994159i \(-0.534420\pi\)
−0.107924 + 0.994159i \(0.534420\pi\)
\(104\) 0 0
\(105\) 0.459187 0.0448120
\(106\) 0 0
\(107\) −12.2271 −1.18204 −0.591020 0.806657i \(-0.701274\pi\)
−0.591020 + 0.806657i \(0.701274\pi\)
\(108\) 0 0
\(109\) −11.4503 −1.09674 −0.548372 0.836234i \(-0.684752\pi\)
−0.548372 + 0.836234i \(0.684752\pi\)
\(110\) 0 0
\(111\) 3.86577 0.366923
\(112\) 0 0
\(113\) −2.56004 −0.240829 −0.120414 0.992724i \(-0.538422\pi\)
−0.120414 + 0.992724i \(0.538422\pi\)
\(114\) 0 0
\(115\) −0.174230 −0.0162471
\(116\) 0 0
\(117\) 13.0276 1.20440
\(118\) 0 0
\(119\) −13.6285 −1.24933
\(120\) 0 0
\(121\) 9.43766 0.857969
\(122\) 0 0
\(123\) 2.64613 0.238594
\(124\) 0 0
\(125\) −5.13851 −0.459603
\(126\) 0 0
\(127\) 9.00412 0.798987 0.399493 0.916736i \(-0.369186\pi\)
0.399493 + 0.916736i \(0.369186\pi\)
\(128\) 0 0
\(129\) −1.83094 −0.161205
\(130\) 0 0
\(131\) −5.08184 −0.444003 −0.222001 0.975046i \(-0.571259\pi\)
−0.222001 + 0.975046i \(0.571259\pi\)
\(132\) 0 0
\(133\) 3.86747 0.335352
\(134\) 0 0
\(135\) 1.40934 0.121296
\(136\) 0 0
\(137\) 17.0117 1.45340 0.726702 0.686953i \(-0.241052\pi\)
0.726702 + 0.686953i \(0.241052\pi\)
\(138\) 0 0
\(139\) 4.58826 0.389171 0.194586 0.980886i \(-0.437664\pi\)
0.194586 + 0.980886i \(0.437664\pi\)
\(140\) 0 0
\(141\) 1.44216 0.121452
\(142\) 0 0
\(143\) −21.1259 −1.76663
\(144\) 0 0
\(145\) −1.61469 −0.134093
\(146\) 0 0
\(147\) 1.58634 0.130840
\(148\) 0 0
\(149\) −8.45447 −0.692617 −0.346309 0.938121i \(-0.612565\pi\)
−0.346309 + 0.938121i \(0.612565\pi\)
\(150\) 0 0
\(151\) 12.0737 0.982543 0.491272 0.871006i \(-0.336532\pi\)
0.491272 + 0.871006i \(0.336532\pi\)
\(152\) 0 0
\(153\) −20.1476 −1.62884
\(154\) 0 0
\(155\) −3.98090 −0.319753
\(156\) 0 0
\(157\) 8.87818 0.708556 0.354278 0.935140i \(-0.384727\pi\)
0.354278 + 0.935140i \(0.384727\pi\)
\(158\) 0 0
\(159\) 2.20207 0.174635
\(160\) 0 0
\(161\) 0.621538 0.0489840
\(162\) 0 0
\(163\) −4.29588 −0.336480 −0.168240 0.985746i \(-0.553808\pi\)
−0.168240 + 0.985746i \(0.553808\pi\)
\(164\) 0 0
\(165\) −1.10082 −0.0856985
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 8.83726 0.679790
\(170\) 0 0
\(171\) 5.71744 0.437223
\(172\) 0 0
\(173\) 0.589144 0.0447918 0.0223959 0.999749i \(-0.492871\pi\)
0.0223959 + 0.999749i \(0.492871\pi\)
\(174\) 0 0
\(175\) 8.90191 0.672921
\(176\) 0 0
\(177\) −3.94969 −0.296877
\(178\) 0 0
\(179\) −18.6810 −1.39629 −0.698143 0.715958i \(-0.745990\pi\)
−0.698143 + 0.715958i \(0.745990\pi\)
\(180\) 0 0
\(181\) 1.55333 0.115458 0.0577291 0.998332i \(-0.481614\pi\)
0.0577291 + 0.998332i \(0.481614\pi\)
\(182\) 0 0
\(183\) 2.28280 0.168749
\(184\) 0 0
\(185\) −4.43639 −0.326170
\(186\) 0 0
\(187\) 32.6720 2.38921
\(188\) 0 0
\(189\) −5.02757 −0.365702
\(190\) 0 0
\(191\) −0.264735 −0.0191556 −0.00957778 0.999954i \(-0.503049\pi\)
−0.00957778 + 0.999954i \(0.503049\pi\)
\(192\) 0 0
\(193\) −12.6341 −0.909426 −0.454713 0.890638i \(-0.650258\pi\)
−0.454713 + 0.890638i \(0.650258\pi\)
\(194\) 0 0
\(195\) 1.13789 0.0814857
\(196\) 0 0
\(197\) 18.3493 1.30733 0.653666 0.756783i \(-0.273230\pi\)
0.653666 + 0.756783i \(0.273230\pi\)
\(198\) 0 0
\(199\) 7.54692 0.534987 0.267494 0.963560i \(-0.413805\pi\)
0.267494 + 0.963560i \(0.413805\pi\)
\(200\) 0 0
\(201\) 2.12136 0.149629
\(202\) 0 0
\(203\) 5.76015 0.404283
\(204\) 0 0
\(205\) −3.03672 −0.212094
\(206\) 0 0
\(207\) 0.918845 0.0638641
\(208\) 0 0
\(209\) −9.27155 −0.641327
\(210\) 0 0
\(211\) −7.43774 −0.512035 −0.256018 0.966672i \(-0.582410\pi\)
−0.256018 + 0.966672i \(0.582410\pi\)
\(212\) 0 0
\(213\) −5.17951 −0.354894
\(214\) 0 0
\(215\) 2.10120 0.143301
\(216\) 0 0
\(217\) 14.2012 0.964039
\(218\) 0 0
\(219\) 3.88734 0.262682
\(220\) 0 0
\(221\) −33.7722 −2.27176
\(222\) 0 0
\(223\) 26.1035 1.74802 0.874009 0.485910i \(-0.161512\pi\)
0.874009 + 0.485910i \(0.161512\pi\)
\(224\) 0 0
\(225\) 13.1601 0.877337
\(226\) 0 0
\(227\) 7.56577 0.502158 0.251079 0.967967i \(-0.419215\pi\)
0.251079 + 0.967967i \(0.419215\pi\)
\(228\) 0 0
\(229\) −23.3347 −1.54200 −0.770999 0.636837i \(-0.780243\pi\)
−0.770999 + 0.636837i \(0.780243\pi\)
\(230\) 0 0
\(231\) 3.92698 0.258376
\(232\) 0 0
\(233\) 5.01646 0.328639 0.164320 0.986407i \(-0.447457\pi\)
0.164320 + 0.986407i \(0.447457\pi\)
\(234\) 0 0
\(235\) −1.65503 −0.107963
\(236\) 0 0
\(237\) 4.75196 0.308673
\(238\) 0 0
\(239\) 16.9238 1.09471 0.547355 0.836900i \(-0.315635\pi\)
0.547355 + 0.836900i \(0.315635\pi\)
\(240\) 0 0
\(241\) 0.454933 0.0293048 0.0146524 0.999893i \(-0.495336\pi\)
0.0146524 + 0.999893i \(0.495336\pi\)
\(242\) 0 0
\(243\) −11.2849 −0.723928
\(244\) 0 0
\(245\) −1.82050 −0.116308
\(246\) 0 0
\(247\) 9.58376 0.609800
\(248\) 0 0
\(249\) −2.24159 −0.142055
\(250\) 0 0
\(251\) 16.3483 1.03190 0.515948 0.856620i \(-0.327440\pi\)
0.515948 + 0.856620i \(0.327440\pi\)
\(252\) 0 0
\(253\) −1.49002 −0.0936770
\(254\) 0 0
\(255\) −1.75978 −0.110202
\(256\) 0 0
\(257\) 10.5775 0.659807 0.329903 0.944015i \(-0.392984\pi\)
0.329903 + 0.944015i \(0.392984\pi\)
\(258\) 0 0
\(259\) 15.8261 0.983384
\(260\) 0 0
\(261\) 8.51547 0.527094
\(262\) 0 0
\(263\) 13.0632 0.805512 0.402756 0.915307i \(-0.368052\pi\)
0.402756 + 0.915307i \(0.368052\pi\)
\(264\) 0 0
\(265\) −2.52711 −0.155239
\(266\) 0 0
\(267\) 5.36021 0.328039
\(268\) 0 0
\(269\) −10.0094 −0.610282 −0.305141 0.952307i \(-0.598704\pi\)
−0.305141 + 0.952307i \(0.598704\pi\)
\(270\) 0 0
\(271\) 9.49026 0.576492 0.288246 0.957556i \(-0.406928\pi\)
0.288246 + 0.957556i \(0.406928\pi\)
\(272\) 0 0
\(273\) −4.05922 −0.245675
\(274\) 0 0
\(275\) −21.3407 −1.28689
\(276\) 0 0
\(277\) 1.81055 0.108785 0.0543927 0.998520i \(-0.482678\pi\)
0.0543927 + 0.998520i \(0.482678\pi\)
\(278\) 0 0
\(279\) 20.9942 1.25689
\(280\) 0 0
\(281\) 24.7931 1.47903 0.739515 0.673140i \(-0.235055\pi\)
0.739515 + 0.673140i \(0.235055\pi\)
\(282\) 0 0
\(283\) 9.67137 0.574903 0.287452 0.957795i \(-0.407192\pi\)
0.287452 + 0.957795i \(0.407192\pi\)
\(284\) 0 0
\(285\) 0.499386 0.0295811
\(286\) 0 0
\(287\) 10.8330 0.639451
\(288\) 0 0
\(289\) 35.2299 2.07235
\(290\) 0 0
\(291\) −0.825738 −0.0484056
\(292\) 0 0
\(293\) 20.1006 1.17429 0.587144 0.809482i \(-0.300252\pi\)
0.587144 + 0.809482i \(0.300252\pi\)
\(294\) 0 0
\(295\) 4.53270 0.263904
\(296\) 0 0
\(297\) 12.0527 0.699367
\(298\) 0 0
\(299\) 1.54020 0.0890720
\(300\) 0 0
\(301\) −7.49568 −0.432044
\(302\) 0 0
\(303\) −1.33028 −0.0764225
\(304\) 0 0
\(305\) −2.61975 −0.150007
\(306\) 0 0
\(307\) −19.2967 −1.10132 −0.550660 0.834730i \(-0.685624\pi\)
−0.550660 + 0.834730i \(0.685624\pi\)
\(308\) 0 0
\(309\) 1.00907 0.0574039
\(310\) 0 0
\(311\) −23.9474 −1.35793 −0.678966 0.734169i \(-0.737572\pi\)
−0.678966 + 0.734169i \(0.737572\pi\)
\(312\) 0 0
\(313\) 23.7438 1.34208 0.671041 0.741420i \(-0.265848\pi\)
0.671041 + 0.741420i \(0.265848\pi\)
\(314\) 0 0
\(315\) 2.77908 0.156583
\(316\) 0 0
\(317\) 0.0507140 0.00284838 0.00142419 0.999999i \(-0.499547\pi\)
0.00142419 + 0.999999i \(0.499547\pi\)
\(318\) 0 0
\(319\) −13.8089 −0.773151
\(320\) 0 0
\(321\) 5.63219 0.314358
\(322\) 0 0
\(323\) −14.8216 −0.824698
\(324\) 0 0
\(325\) 22.0593 1.22363
\(326\) 0 0
\(327\) 5.27438 0.291674
\(328\) 0 0
\(329\) 5.90406 0.325501
\(330\) 0 0
\(331\) −32.8008 −1.80289 −0.901447 0.432889i \(-0.857494\pi\)
−0.901447 + 0.432889i \(0.857494\pi\)
\(332\) 0 0
\(333\) 23.3963 1.28211
\(334\) 0 0
\(335\) −2.43448 −0.133010
\(336\) 0 0
\(337\) 23.5480 1.28274 0.641370 0.767231i \(-0.278366\pi\)
0.641370 + 0.767231i \(0.278366\pi\)
\(338\) 0 0
\(339\) 1.17923 0.0640472
\(340\) 0 0
\(341\) −34.0448 −1.84363
\(342\) 0 0
\(343\) 19.6948 1.06342
\(344\) 0 0
\(345\) 0.0802559 0.00432083
\(346\) 0 0
\(347\) −14.0160 −0.752419 −0.376210 0.926535i \(-0.622773\pi\)
−0.376210 + 0.926535i \(0.622773\pi\)
\(348\) 0 0
\(349\) −30.1235 −1.61247 −0.806236 0.591594i \(-0.798499\pi\)
−0.806236 + 0.591594i \(0.798499\pi\)
\(350\) 0 0
\(351\) −12.4585 −0.664988
\(352\) 0 0
\(353\) 17.9936 0.957701 0.478850 0.877897i \(-0.341054\pi\)
0.478850 + 0.877897i \(0.341054\pi\)
\(354\) 0 0
\(355\) 5.94404 0.315477
\(356\) 0 0
\(357\) 6.27773 0.332253
\(358\) 0 0
\(359\) −21.7908 −1.15007 −0.575036 0.818128i \(-0.695012\pi\)
−0.575036 + 0.818128i \(0.695012\pi\)
\(360\) 0 0
\(361\) −14.7940 −0.778629
\(362\) 0 0
\(363\) −4.34728 −0.228173
\(364\) 0 0
\(365\) −4.46114 −0.233507
\(366\) 0 0
\(367\) −6.33533 −0.330702 −0.165351 0.986235i \(-0.552876\pi\)
−0.165351 + 0.986235i \(0.552876\pi\)
\(368\) 0 0
\(369\) 16.0149 0.833700
\(370\) 0 0
\(371\) 9.01504 0.468038
\(372\) 0 0
\(373\) −5.67690 −0.293939 −0.146969 0.989141i \(-0.546952\pi\)
−0.146969 + 0.989141i \(0.546952\pi\)
\(374\) 0 0
\(375\) 2.36696 0.122229
\(376\) 0 0
\(377\) 14.2739 0.735144
\(378\) 0 0
\(379\) 0.295498 0.0151787 0.00758935 0.999971i \(-0.497584\pi\)
0.00758935 + 0.999971i \(0.497584\pi\)
\(380\) 0 0
\(381\) −4.14758 −0.212487
\(382\) 0 0
\(383\) −30.9211 −1.58000 −0.789998 0.613109i \(-0.789919\pi\)
−0.789998 + 0.613109i \(0.789919\pi\)
\(384\) 0 0
\(385\) −4.50663 −0.229679
\(386\) 0 0
\(387\) −11.0812 −0.563288
\(388\) 0 0
\(389\) −23.4419 −1.18855 −0.594275 0.804262i \(-0.702561\pi\)
−0.594275 + 0.804262i \(0.702561\pi\)
\(390\) 0 0
\(391\) −2.38198 −0.120462
\(392\) 0 0
\(393\) 2.34085 0.118080
\(394\) 0 0
\(395\) −5.45338 −0.274390
\(396\) 0 0
\(397\) −10.3509 −0.519498 −0.259749 0.965676i \(-0.583640\pi\)
−0.259749 + 0.965676i \(0.583640\pi\)
\(398\) 0 0
\(399\) −1.78148 −0.0891853
\(400\) 0 0
\(401\) 13.1228 0.655322 0.327661 0.944795i \(-0.393740\pi\)
0.327661 + 0.944795i \(0.393740\pi\)
\(402\) 0 0
\(403\) 35.1912 1.75300
\(404\) 0 0
\(405\) 3.77194 0.187429
\(406\) 0 0
\(407\) −37.9401 −1.88062
\(408\) 0 0
\(409\) 14.7190 0.727806 0.363903 0.931437i \(-0.381444\pi\)
0.363903 + 0.931437i \(0.381444\pi\)
\(410\) 0 0
\(411\) −7.83610 −0.386526
\(412\) 0 0
\(413\) −16.1696 −0.795656
\(414\) 0 0
\(415\) 2.57246 0.126277
\(416\) 0 0
\(417\) −2.11350 −0.103498
\(418\) 0 0
\(419\) 9.53165 0.465651 0.232826 0.972518i \(-0.425203\pi\)
0.232826 + 0.972518i \(0.425203\pi\)
\(420\) 0 0
\(421\) −19.3126 −0.941240 −0.470620 0.882336i \(-0.655970\pi\)
−0.470620 + 0.882336i \(0.655970\pi\)
\(422\) 0 0
\(423\) 8.72821 0.424380
\(424\) 0 0
\(425\) −34.1156 −1.65485
\(426\) 0 0
\(427\) 9.34553 0.452262
\(428\) 0 0
\(429\) 9.73123 0.469828
\(430\) 0 0
\(431\) 23.2355 1.11921 0.559606 0.828758i \(-0.310952\pi\)
0.559606 + 0.828758i \(0.310952\pi\)
\(432\) 0 0
\(433\) −31.6576 −1.52137 −0.760684 0.649123i \(-0.775136\pi\)
−0.760684 + 0.649123i \(0.775136\pi\)
\(434\) 0 0
\(435\) 0.743778 0.0356614
\(436\) 0 0
\(437\) 0.675950 0.0323351
\(438\) 0 0
\(439\) 23.9084 1.14109 0.570543 0.821268i \(-0.306733\pi\)
0.570543 + 0.821268i \(0.306733\pi\)
\(440\) 0 0
\(441\) 9.60084 0.457183
\(442\) 0 0
\(443\) −21.6729 −1.02971 −0.514855 0.857277i \(-0.672154\pi\)
−0.514855 + 0.857277i \(0.672154\pi\)
\(444\) 0 0
\(445\) −6.15141 −0.291605
\(446\) 0 0
\(447\) 3.89439 0.184198
\(448\) 0 0
\(449\) 35.1288 1.65783 0.828915 0.559374i \(-0.188959\pi\)
0.828915 + 0.559374i \(0.188959\pi\)
\(450\) 0 0
\(451\) −25.9701 −1.22289
\(452\) 0 0
\(453\) −5.56152 −0.261303
\(454\) 0 0
\(455\) 4.65839 0.218388
\(456\) 0 0
\(457\) 22.6679 1.06036 0.530181 0.847885i \(-0.322124\pi\)
0.530181 + 0.847885i \(0.322124\pi\)
\(458\) 0 0
\(459\) 19.2676 0.899335
\(460\) 0 0
\(461\) −0.831182 −0.0387120 −0.0193560 0.999813i \(-0.506162\pi\)
−0.0193560 + 0.999813i \(0.506162\pi\)
\(462\) 0 0
\(463\) −15.0478 −0.699331 −0.349665 0.936875i \(-0.613705\pi\)
−0.349665 + 0.936875i \(0.613705\pi\)
\(464\) 0 0
\(465\) 1.83372 0.0850369
\(466\) 0 0
\(467\) 36.9603 1.71032 0.855160 0.518364i \(-0.173459\pi\)
0.855160 + 0.518364i \(0.173459\pi\)
\(468\) 0 0
\(469\) 8.68461 0.401018
\(470\) 0 0
\(471\) −4.08956 −0.188437
\(472\) 0 0
\(473\) 17.9695 0.826240
\(474\) 0 0
\(475\) 9.68122 0.444205
\(476\) 0 0
\(477\) 13.3273 0.610216
\(478\) 0 0
\(479\) 21.0720 0.962802 0.481401 0.876500i \(-0.340128\pi\)
0.481401 + 0.876500i \(0.340128\pi\)
\(480\) 0 0
\(481\) 39.2177 1.78817
\(482\) 0 0
\(483\) −0.286300 −0.0130271
\(484\) 0 0
\(485\) 0.947623 0.0430293
\(486\) 0 0
\(487\) −13.9444 −0.631880 −0.315940 0.948779i \(-0.602320\pi\)
−0.315940 + 0.948779i \(0.602320\pi\)
\(488\) 0 0
\(489\) 1.97882 0.0894852
\(490\) 0 0
\(491\) −19.9232 −0.899124 −0.449562 0.893249i \(-0.648420\pi\)
−0.449562 + 0.893249i \(0.648420\pi\)
\(492\) 0 0
\(493\) −22.0751 −0.994214
\(494\) 0 0
\(495\) −6.66234 −0.299450
\(496\) 0 0
\(497\) −21.2044 −0.951146
\(498\) 0 0
\(499\) 21.6053 0.967186 0.483593 0.875293i \(-0.339332\pi\)
0.483593 + 0.875293i \(0.339332\pi\)
\(500\) 0 0
\(501\) −0.460631 −0.0205795
\(502\) 0 0
\(503\) 5.69154 0.253773 0.126887 0.991917i \(-0.459502\pi\)
0.126887 + 0.991917i \(0.459502\pi\)
\(504\) 0 0
\(505\) 1.52664 0.0679344
\(506\) 0 0
\(507\) −4.07072 −0.180787
\(508\) 0 0
\(509\) 15.3992 0.682559 0.341279 0.939962i \(-0.389140\pi\)
0.341279 + 0.939962i \(0.389140\pi\)
\(510\) 0 0
\(511\) 15.9144 0.704010
\(512\) 0 0
\(513\) −5.46770 −0.241405
\(514\) 0 0
\(515\) −1.15801 −0.0510282
\(516\) 0 0
\(517\) −14.1539 −0.622488
\(518\) 0 0
\(519\) −0.271378 −0.0119122
\(520\) 0 0
\(521\) 33.4568 1.46577 0.732885 0.680353i \(-0.238174\pi\)
0.732885 + 0.680353i \(0.238174\pi\)
\(522\) 0 0
\(523\) −25.0006 −1.09320 −0.546600 0.837394i \(-0.684078\pi\)
−0.546600 + 0.837394i \(0.684078\pi\)
\(524\) 0 0
\(525\) −4.10049 −0.178960
\(526\) 0 0
\(527\) −54.4245 −2.37077
\(528\) 0 0
\(529\) −22.8914 −0.995277
\(530\) 0 0
\(531\) −23.9042 −1.03736
\(532\) 0 0
\(533\) 26.8446 1.16277
\(534\) 0 0
\(535\) −6.46354 −0.279443
\(536\) 0 0
\(537\) 8.60506 0.371336
\(538\) 0 0
\(539\) −15.5690 −0.670603
\(540\) 0 0
\(541\) −36.2501 −1.55851 −0.779257 0.626704i \(-0.784403\pi\)
−0.779257 + 0.626704i \(0.784403\pi\)
\(542\) 0 0
\(543\) −0.715513 −0.0307056
\(544\) 0 0
\(545\) −6.05292 −0.259279
\(546\) 0 0
\(547\) 45.5824 1.94896 0.974480 0.224473i \(-0.0720661\pi\)
0.974480 + 0.224473i \(0.0720661\pi\)
\(548\) 0 0
\(549\) 13.8159 0.589648
\(550\) 0 0
\(551\) 6.26442 0.266873
\(552\) 0 0
\(553\) 19.4540 0.827270
\(554\) 0 0
\(555\) 2.04354 0.0867433
\(556\) 0 0
\(557\) 36.6207 1.55167 0.775835 0.630936i \(-0.217329\pi\)
0.775835 + 0.630936i \(0.217329\pi\)
\(558\) 0 0
\(559\) −18.5746 −0.785624
\(560\) 0 0
\(561\) −15.0497 −0.635400
\(562\) 0 0
\(563\) −3.00687 −0.126725 −0.0633623 0.997991i \(-0.520182\pi\)
−0.0633623 + 0.997991i \(0.520182\pi\)
\(564\) 0 0
\(565\) −1.35330 −0.0569337
\(566\) 0 0
\(567\) −13.4557 −0.565088
\(568\) 0 0
\(569\) −42.3071 −1.77361 −0.886804 0.462146i \(-0.847080\pi\)
−0.886804 + 0.462146i \(0.847080\pi\)
\(570\) 0 0
\(571\) −7.85312 −0.328643 −0.164321 0.986407i \(-0.552543\pi\)
−0.164321 + 0.986407i \(0.552543\pi\)
\(572\) 0 0
\(573\) 0.121945 0.00509433
\(574\) 0 0
\(575\) 1.55586 0.0648839
\(576\) 0 0
\(577\) −7.80955 −0.325116 −0.162558 0.986699i \(-0.551974\pi\)
−0.162558 + 0.986699i \(0.551974\pi\)
\(578\) 0 0
\(579\) 5.81968 0.241858
\(580\) 0 0
\(581\) −9.17683 −0.380719
\(582\) 0 0
\(583\) −21.6119 −0.895075
\(584\) 0 0
\(585\) 6.88668 0.284729
\(586\) 0 0
\(587\) 30.9133 1.27593 0.637964 0.770066i \(-0.279777\pi\)
0.637964 + 0.770066i \(0.279777\pi\)
\(588\) 0 0
\(589\) 15.4444 0.636376
\(590\) 0 0
\(591\) −8.45225 −0.347679
\(592\) 0 0
\(593\) −35.1986 −1.44543 −0.722716 0.691145i \(-0.757106\pi\)
−0.722716 + 0.691145i \(0.757106\pi\)
\(594\) 0 0
\(595\) −7.20437 −0.295350
\(596\) 0 0
\(597\) −3.47635 −0.142277
\(598\) 0 0
\(599\) −44.4328 −1.81548 −0.907738 0.419538i \(-0.862192\pi\)
−0.907738 + 0.419538i \(0.862192\pi\)
\(600\) 0 0
\(601\) −30.0351 −1.22516 −0.612578 0.790410i \(-0.709867\pi\)
−0.612578 + 0.790410i \(0.709867\pi\)
\(602\) 0 0
\(603\) 12.8388 0.522837
\(604\) 0 0
\(605\) 4.98897 0.202830
\(606\) 0 0
\(607\) −27.5148 −1.11679 −0.558395 0.829575i \(-0.688583\pi\)
−0.558395 + 0.829575i \(0.688583\pi\)
\(608\) 0 0
\(609\) −2.65330 −0.107517
\(610\) 0 0
\(611\) 14.6305 0.591888
\(612\) 0 0
\(613\) −8.13449 −0.328549 −0.164275 0.986415i \(-0.552528\pi\)
−0.164275 + 0.986415i \(0.552528\pi\)
\(614\) 0 0
\(615\) 1.39881 0.0564053
\(616\) 0 0
\(617\) −14.8846 −0.599232 −0.299616 0.954060i \(-0.596858\pi\)
−0.299616 + 0.954060i \(0.596858\pi\)
\(618\) 0 0
\(619\) 8.59265 0.345368 0.172684 0.984977i \(-0.444756\pi\)
0.172684 + 0.984977i \(0.444756\pi\)
\(620\) 0 0
\(621\) −0.878710 −0.0352614
\(622\) 0 0
\(623\) 21.9441 0.879173
\(624\) 0 0
\(625\) 20.8864 0.835458
\(626\) 0 0
\(627\) 4.27076 0.170558
\(628\) 0 0
\(629\) −60.6516 −2.41834
\(630\) 0 0
\(631\) −31.2593 −1.24441 −0.622207 0.782853i \(-0.713764\pi\)
−0.622207 + 0.782853i \(0.713764\pi\)
\(632\) 0 0
\(633\) 3.42605 0.136173
\(634\) 0 0
\(635\) 4.75979 0.188886
\(636\) 0 0
\(637\) 16.0932 0.637638
\(638\) 0 0
\(639\) −31.3473 −1.24008
\(640\) 0 0
\(641\) −31.2323 −1.23360 −0.616800 0.787120i \(-0.711571\pi\)
−0.616800 + 0.787120i \(0.711571\pi\)
\(642\) 0 0
\(643\) −44.1159 −1.73976 −0.869880 0.493263i \(-0.835804\pi\)
−0.869880 + 0.493263i \(0.835804\pi\)
\(644\) 0 0
\(645\) −0.967879 −0.0381102
\(646\) 0 0
\(647\) 8.15533 0.320619 0.160310 0.987067i \(-0.448751\pi\)
0.160310 + 0.987067i \(0.448751\pi\)
\(648\) 0 0
\(649\) 38.7638 1.52161
\(650\) 0 0
\(651\) −6.54151 −0.256382
\(652\) 0 0
\(653\) 2.81732 0.110250 0.0551252 0.998479i \(-0.482444\pi\)
0.0551252 + 0.998479i \(0.482444\pi\)
\(654\) 0 0
\(655\) −2.68638 −0.104966
\(656\) 0 0
\(657\) 23.5269 0.917870
\(658\) 0 0
\(659\) −31.2713 −1.21816 −0.609079 0.793110i \(-0.708461\pi\)
−0.609079 + 0.793110i \(0.708461\pi\)
\(660\) 0 0
\(661\) 43.2560 1.68246 0.841231 0.540675i \(-0.181831\pi\)
0.841231 + 0.540675i \(0.181831\pi\)
\(662\) 0 0
\(663\) 15.5565 0.604164
\(664\) 0 0
\(665\) 2.04443 0.0792797
\(666\) 0 0
\(667\) 1.00675 0.0389815
\(668\) 0 0
\(669\) −12.0241 −0.464877
\(670\) 0 0
\(671\) −22.4042 −0.864905
\(672\) 0 0
\(673\) −2.67949 −0.103287 −0.0516433 0.998666i \(-0.516446\pi\)
−0.0516433 + 0.998666i \(0.516446\pi\)
\(674\) 0 0
\(675\) −12.5852 −0.484406
\(676\) 0 0
\(677\) 3.57007 0.137209 0.0686045 0.997644i \(-0.478145\pi\)
0.0686045 + 0.997644i \(0.478145\pi\)
\(678\) 0 0
\(679\) −3.38049 −0.129731
\(680\) 0 0
\(681\) −3.48503 −0.133547
\(682\) 0 0
\(683\) −40.7566 −1.55951 −0.779755 0.626085i \(-0.784656\pi\)
−0.779755 + 0.626085i \(0.784656\pi\)
\(684\) 0 0
\(685\) 8.99276 0.343596
\(686\) 0 0
\(687\) 10.7487 0.410087
\(688\) 0 0
\(689\) 22.3397 0.851074
\(690\) 0 0
\(691\) 44.3926 1.68878 0.844388 0.535733i \(-0.179965\pi\)
0.844388 + 0.535733i \(0.179965\pi\)
\(692\) 0 0
\(693\) 23.7668 0.902825
\(694\) 0 0
\(695\) 2.42546 0.0920031
\(696\) 0 0
\(697\) −41.5162 −1.57254
\(698\) 0 0
\(699\) −2.31074 −0.0874001
\(700\) 0 0
\(701\) −8.39607 −0.317115 −0.158558 0.987350i \(-0.550684\pi\)
−0.158558 + 0.987350i \(0.550684\pi\)
\(702\) 0 0
\(703\) 17.2115 0.649145
\(704\) 0 0
\(705\) 0.762360 0.0287121
\(706\) 0 0
\(707\) −5.44602 −0.204819
\(708\) 0 0
\(709\) 24.0828 0.904448 0.452224 0.891904i \(-0.350631\pi\)
0.452224 + 0.891904i \(0.350631\pi\)
\(710\) 0 0
\(711\) 28.7597 1.07857
\(712\) 0 0
\(713\) 2.48206 0.0929539
\(714\) 0 0
\(715\) −11.1676 −0.417646
\(716\) 0 0
\(717\) −7.79563 −0.291133
\(718\) 0 0
\(719\) −34.6056 −1.29057 −0.645285 0.763942i \(-0.723261\pi\)
−0.645285 + 0.763942i \(0.723261\pi\)
\(720\) 0 0
\(721\) 4.13102 0.153847
\(722\) 0 0
\(723\) −0.209556 −0.00779348
\(724\) 0 0
\(725\) 14.4191 0.535511
\(726\) 0 0
\(727\) 1.40361 0.0520569 0.0260284 0.999661i \(-0.491714\pi\)
0.0260284 + 0.999661i \(0.491714\pi\)
\(728\) 0 0
\(729\) −16.2080 −0.600296
\(730\) 0 0
\(731\) 28.7264 1.06248
\(732\) 0 0
\(733\) 13.0319 0.481345 0.240672 0.970606i \(-0.422632\pi\)
0.240672 + 0.970606i \(0.422632\pi\)
\(734\) 0 0
\(735\) 0.838579 0.0309314
\(736\) 0 0
\(737\) −20.8198 −0.766906
\(738\) 0 0
\(739\) −2.84386 −0.104613 −0.0523065 0.998631i \(-0.516657\pi\)
−0.0523065 + 0.998631i \(0.516657\pi\)
\(740\) 0 0
\(741\) −4.41458 −0.162173
\(742\) 0 0
\(743\) 30.3233 1.11246 0.556228 0.831030i \(-0.312248\pi\)
0.556228 + 0.831030i \(0.312248\pi\)
\(744\) 0 0
\(745\) −4.46923 −0.163740
\(746\) 0 0
\(747\) −13.5665 −0.496372
\(748\) 0 0
\(749\) 23.0576 0.842506
\(750\) 0 0
\(751\) 19.3929 0.707656 0.353828 0.935311i \(-0.384880\pi\)
0.353828 + 0.935311i \(0.384880\pi\)
\(752\) 0 0
\(753\) −7.53054 −0.274428
\(754\) 0 0
\(755\) 6.38244 0.232281
\(756\) 0 0
\(757\) −27.7720 −1.00939 −0.504696 0.863297i \(-0.668395\pi\)
−0.504696 + 0.863297i \(0.668395\pi\)
\(758\) 0 0
\(759\) 0.686351 0.0249130
\(760\) 0 0
\(761\) −30.4853 −1.10509 −0.552545 0.833483i \(-0.686343\pi\)
−0.552545 + 0.833483i \(0.686343\pi\)
\(762\) 0 0
\(763\) 21.5928 0.781711
\(764\) 0 0
\(765\) −10.6505 −0.385070
\(766\) 0 0
\(767\) −40.0691 −1.44681
\(768\) 0 0
\(769\) 45.3891 1.63677 0.818386 0.574669i \(-0.194869\pi\)
0.818386 + 0.574669i \(0.194869\pi\)
\(770\) 0 0
\(771\) −4.87233 −0.175473
\(772\) 0 0
\(773\) 2.17984 0.0784035 0.0392017 0.999231i \(-0.487519\pi\)
0.0392017 + 0.999231i \(0.487519\pi\)
\(774\) 0 0
\(775\) 35.5490 1.27696
\(776\) 0 0
\(777\) −7.28997 −0.261526
\(778\) 0 0
\(779\) 11.7814 0.422111
\(780\) 0 0
\(781\) 50.8336 1.81897
\(782\) 0 0
\(783\) −8.14351 −0.291025
\(784\) 0 0
\(785\) 4.69321 0.167508
\(786\) 0 0
\(787\) −34.5630 −1.23204 −0.616019 0.787731i \(-0.711255\pi\)
−0.616019 + 0.787731i \(0.711255\pi\)
\(788\) 0 0
\(789\) −6.01732 −0.214222
\(790\) 0 0
\(791\) 4.82767 0.171652
\(792\) 0 0
\(793\) 23.1586 0.822388
\(794\) 0 0
\(795\) 1.16407 0.0412851
\(796\) 0 0
\(797\) 46.3705 1.64253 0.821264 0.570549i \(-0.193270\pi\)
0.821264 + 0.570549i \(0.193270\pi\)
\(798\) 0 0
\(799\) −22.6267 −0.800473
\(800\) 0 0
\(801\) 32.4409 1.14624
\(802\) 0 0
\(803\) −38.1518 −1.34635
\(804\) 0 0
\(805\) 0.328559 0.0115802
\(806\) 0 0
\(807\) 4.61062 0.162302
\(808\) 0 0
\(809\) −2.71862 −0.0955815 −0.0477908 0.998857i \(-0.515218\pi\)
−0.0477908 + 0.998857i \(0.515218\pi\)
\(810\) 0 0
\(811\) −21.8647 −0.767773 −0.383887 0.923380i \(-0.625415\pi\)
−0.383887 + 0.923380i \(0.625415\pi\)
\(812\) 0 0
\(813\) −4.37150 −0.153315
\(814\) 0 0
\(815\) −2.27090 −0.0795463
\(816\) 0 0
\(817\) −8.15189 −0.285198
\(818\) 0 0
\(819\) −24.5671 −0.858443
\(820\) 0 0
\(821\) −47.1656 −1.64609 −0.823046 0.567975i \(-0.807727\pi\)
−0.823046 + 0.567975i \(0.807727\pi\)
\(822\) 0 0
\(823\) 10.0300 0.349624 0.174812 0.984602i \(-0.444068\pi\)
0.174812 + 0.984602i \(0.444068\pi\)
\(824\) 0 0
\(825\) 9.83019 0.342243
\(826\) 0 0
\(827\) −34.3947 −1.19602 −0.598010 0.801488i \(-0.704042\pi\)
−0.598010 + 0.801488i \(0.704042\pi\)
\(828\) 0 0
\(829\) −26.0071 −0.903265 −0.451633 0.892204i \(-0.649158\pi\)
−0.451633 + 0.892204i \(0.649158\pi\)
\(830\) 0 0
\(831\) −0.833995 −0.0289310
\(832\) 0 0
\(833\) −24.8888 −0.862346
\(834\) 0 0
\(835\) 0.528623 0.0182938
\(836\) 0 0
\(837\) −20.0772 −0.693969
\(838\) 0 0
\(839\) −27.0861 −0.935117 −0.467559 0.883962i \(-0.654866\pi\)
−0.467559 + 0.883962i \(0.654866\pi\)
\(840\) 0 0
\(841\) −19.6699 −0.678271
\(842\) 0 0
\(843\) −11.4204 −0.393341
\(844\) 0 0
\(845\) 4.67158 0.160707
\(846\) 0 0
\(847\) −17.7973 −0.611523
\(848\) 0 0
\(849\) −4.45493 −0.152893
\(850\) 0 0
\(851\) 2.76605 0.0948191
\(852\) 0 0
\(853\) −50.3832 −1.72509 −0.862544 0.505982i \(-0.831130\pi\)
−0.862544 + 0.505982i \(0.831130\pi\)
\(854\) 0 0
\(855\) 3.02237 0.103363
\(856\) 0 0
\(857\) 38.9044 1.32895 0.664474 0.747312i \(-0.268656\pi\)
0.664474 + 0.747312i \(0.268656\pi\)
\(858\) 0 0
\(859\) −5.08700 −0.173566 −0.0867831 0.996227i \(-0.527659\pi\)
−0.0867831 + 0.996227i \(0.527659\pi\)
\(860\) 0 0
\(861\) −4.99001 −0.170059
\(862\) 0 0
\(863\) 35.1898 1.19787 0.598937 0.800796i \(-0.295590\pi\)
0.598937 + 0.800796i \(0.295590\pi\)
\(864\) 0 0
\(865\) 0.311435 0.0105891
\(866\) 0 0
\(867\) −16.2280 −0.551131
\(868\) 0 0
\(869\) −46.6375 −1.58207
\(870\) 0 0
\(871\) 21.5209 0.729206
\(872\) 0 0
\(873\) −4.99751 −0.169140
\(874\) 0 0
\(875\) 9.69008 0.327585
\(876\) 0 0
\(877\) −51.1941 −1.72870 −0.864351 0.502889i \(-0.832270\pi\)
−0.864351 + 0.502889i \(0.832270\pi\)
\(878\) 0 0
\(879\) −9.25895 −0.312297
\(880\) 0 0
\(881\) −20.6653 −0.696231 −0.348115 0.937452i \(-0.613178\pi\)
−0.348115 + 0.937452i \(0.613178\pi\)
\(882\) 0 0
\(883\) −30.9195 −1.04052 −0.520262 0.854007i \(-0.674166\pi\)
−0.520262 + 0.854007i \(0.674166\pi\)
\(884\) 0 0
\(885\) −2.08790 −0.0701840
\(886\) 0 0
\(887\) 16.0054 0.537408 0.268704 0.963223i \(-0.413405\pi\)
0.268704 + 0.963223i \(0.413405\pi\)
\(888\) 0 0
\(889\) −16.9797 −0.569483
\(890\) 0 0
\(891\) 32.2577 1.08067
\(892\) 0 0
\(893\) 6.42092 0.214868
\(894\) 0 0
\(895\) −9.87523 −0.330093
\(896\) 0 0
\(897\) −0.709463 −0.0236883
\(898\) 0 0
\(899\) 23.0027 0.767182
\(900\) 0 0
\(901\) −34.5492 −1.15100
\(902\) 0 0
\(903\) 3.45274 0.114900
\(904\) 0 0
\(905\) 0.821128 0.0272952
\(906\) 0 0
\(907\) 40.8755 1.35725 0.678624 0.734486i \(-0.262577\pi\)
0.678624 + 0.734486i \(0.262577\pi\)
\(908\) 0 0
\(909\) −8.05108 −0.267037
\(910\) 0 0
\(911\) −28.5013 −0.944291 −0.472146 0.881521i \(-0.656520\pi\)
−0.472146 + 0.881521i \(0.656520\pi\)
\(912\) 0 0
\(913\) 21.9998 0.728086
\(914\) 0 0
\(915\) 1.20674 0.0398936
\(916\) 0 0
\(917\) 9.58321 0.316466
\(918\) 0 0
\(919\) −43.0660 −1.42062 −0.710309 0.703890i \(-0.751445\pi\)
−0.710309 + 0.703890i \(0.751445\pi\)
\(920\) 0 0
\(921\) 8.88865 0.292891
\(922\) 0 0
\(923\) −52.5454 −1.72955
\(924\) 0 0
\(925\) 39.6165 1.30258
\(926\) 0 0
\(927\) 6.10706 0.200582
\(928\) 0 0
\(929\) 1.94430 0.0637904 0.0318952 0.999491i \(-0.489846\pi\)
0.0318952 + 0.999491i \(0.489846\pi\)
\(930\) 0 0
\(931\) 7.06287 0.231476
\(932\) 0 0
\(933\) 11.0309 0.361136
\(934\) 0 0
\(935\) 17.2712 0.564827
\(936\) 0 0
\(937\) 31.9326 1.04319 0.521596 0.853192i \(-0.325337\pi\)
0.521596 + 0.853192i \(0.325337\pi\)
\(938\) 0 0
\(939\) −10.9371 −0.356920
\(940\) 0 0
\(941\) 10.9641 0.357419 0.178709 0.983902i \(-0.442808\pi\)
0.178709 + 0.983902i \(0.442808\pi\)
\(942\) 0 0
\(943\) 1.89337 0.0616567
\(944\) 0 0
\(945\) −2.65769 −0.0864547
\(946\) 0 0
\(947\) 21.7390 0.706421 0.353211 0.935544i \(-0.385090\pi\)
0.353211 + 0.935544i \(0.385090\pi\)
\(948\) 0 0
\(949\) 39.4365 1.28016
\(950\) 0 0
\(951\) −0.0233604 −0.000757513 0
\(952\) 0 0
\(953\) −8.91288 −0.288717 −0.144358 0.989525i \(-0.546112\pi\)
−0.144358 + 0.989525i \(0.546112\pi\)
\(954\) 0 0
\(955\) −0.139945 −0.00452852
\(956\) 0 0
\(957\) 6.36081 0.205616
\(958\) 0 0
\(959\) −32.0802 −1.03592
\(960\) 0 0
\(961\) 25.7113 0.829396
\(962\) 0 0
\(963\) 34.0870 1.09844
\(964\) 0 0
\(965\) −6.67870 −0.214995
\(966\) 0 0
\(967\) 42.8796 1.37891 0.689457 0.724326i \(-0.257849\pi\)
0.689457 + 0.724326i \(0.257849\pi\)
\(968\) 0 0
\(969\) 6.82731 0.219325
\(970\) 0 0
\(971\) 34.9981 1.12314 0.561571 0.827429i \(-0.310197\pi\)
0.561571 + 0.827429i \(0.310197\pi\)
\(972\) 0 0
\(973\) −8.65243 −0.277384
\(974\) 0 0
\(975\) −10.1612 −0.325419
\(976\) 0 0
\(977\) −29.2187 −0.934790 −0.467395 0.884049i \(-0.654807\pi\)
−0.467395 + 0.884049i \(0.654807\pi\)
\(978\) 0 0
\(979\) −52.6071 −1.68133
\(980\) 0 0
\(981\) 31.9215 1.01918
\(982\) 0 0
\(983\) −50.0259 −1.59558 −0.797789 0.602937i \(-0.793997\pi\)
−0.797789 + 0.602937i \(0.793997\pi\)
\(984\) 0 0
\(985\) 9.69986 0.309063
\(986\) 0 0
\(987\) −2.71959 −0.0865655
\(988\) 0 0
\(989\) −1.31008 −0.0416582
\(990\) 0 0
\(991\) 22.6908 0.720797 0.360398 0.932798i \(-0.382641\pi\)
0.360398 + 0.932798i \(0.382641\pi\)
\(992\) 0 0
\(993\) 15.1091 0.479471
\(994\) 0 0
\(995\) 3.98948 0.126475
\(996\) 0 0
\(997\) −34.7647 −1.10101 −0.550505 0.834832i \(-0.685565\pi\)
−0.550505 + 0.834832i \(0.685565\pi\)
\(998\) 0 0
\(999\) −22.3744 −0.707894
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 1336.2.a.c.1.4 9
4.3 odd 2 2672.2.a.m.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.c.1.4 9 1.1 even 1 trivial
2672.2.a.m.1.6 9 4.3 odd 2