Properties

Label 6016.2.a.o.1.8
Level $6016$
Weight $2$
Character 6016.1
Self dual yes
Analytic conductor $48.038$
Analytic rank $0$
Dimension $13$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, 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("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} - 6230 x^{4} + 1488 x^{3} + 4720 x^{2} - 1440 x - 832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.18016\) of defining polynomial
Character \(\chi\) \(=\) 6016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.18016 q^{3} +2.06312 q^{5} +0.279748 q^{7} -1.60723 q^{9} +O(q^{10})\) \(q+1.18016 q^{3} +2.06312 q^{5} +0.279748 q^{7} -1.60723 q^{9} +3.49808 q^{11} +3.50765 q^{13} +2.43481 q^{15} -2.93033 q^{17} +2.42881 q^{19} +0.330146 q^{21} -2.39220 q^{23} -0.743524 q^{25} -5.43726 q^{27} +3.15996 q^{29} -4.46282 q^{31} +4.12829 q^{33} +0.577153 q^{35} +6.97509 q^{37} +4.13959 q^{39} +8.78599 q^{41} +12.9727 q^{43} -3.31590 q^{45} -1.00000 q^{47} -6.92174 q^{49} -3.45825 q^{51} +4.29816 q^{53} +7.21697 q^{55} +2.86638 q^{57} +1.39738 q^{59} -3.74584 q^{61} -0.449617 q^{63} +7.23672 q^{65} +7.66539 q^{67} -2.82317 q^{69} -1.67409 q^{71} +7.21968 q^{73} -0.877477 q^{75} +0.978579 q^{77} +2.44688 q^{79} -1.59515 q^{81} +14.4732 q^{83} -6.04563 q^{85} +3.72926 q^{87} -4.63757 q^{89} +0.981257 q^{91} -5.26683 q^{93} +5.01092 q^{95} +3.16215 q^{97} -5.62220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9} + 10 q^{11} - 4 q^{13} + 14 q^{15} + 10 q^{17} + 8 q^{19} - 10 q^{21} + 18 q^{23} + 23 q^{25} + 16 q^{27} - 14 q^{29} + 4 q^{31} + 14 q^{33} + 14 q^{35} - 16 q^{37} + 12 q^{39} + 10 q^{41} + 12 q^{43} - 10 q^{45} - 13 q^{47} + 9 q^{49} + 22 q^{51} - 26 q^{53} - 2 q^{55} + 20 q^{57} + 30 q^{59} - 18 q^{61} + 12 q^{63} - 4 q^{65} + 4 q^{67} - 2 q^{69} + 36 q^{71} + 10 q^{73} + 38 q^{75} - 42 q^{77} + 21 q^{81} + 12 q^{83} - 4 q^{85} + 6 q^{87} + 50 q^{89} - 4 q^{91} - 52 q^{93} + 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) \) 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
\(3\) 1.18016 0.681365 0.340682 0.940178i \(-0.389342\pi\)
0.340682 + 0.940178i \(0.389342\pi\)
\(4\) 0 0
\(5\) 2.06312 0.922657 0.461328 0.887230i \(-0.347373\pi\)
0.461328 + 0.887230i \(0.347373\pi\)
\(6\) 0 0
\(7\) 0.279748 0.105735 0.0528673 0.998602i \(-0.483164\pi\)
0.0528673 + 0.998602i \(0.483164\pi\)
\(8\) 0 0
\(9\) −1.60723 −0.535742
\(10\) 0 0
\(11\) 3.49808 1.05471 0.527356 0.849645i \(-0.323184\pi\)
0.527356 + 0.849645i \(0.323184\pi\)
\(12\) 0 0
\(13\) 3.50765 0.972847 0.486424 0.873723i \(-0.338301\pi\)
0.486424 + 0.873723i \(0.338301\pi\)
\(14\) 0 0
\(15\) 2.43481 0.628666
\(16\) 0 0
\(17\) −2.93033 −0.710709 −0.355355 0.934732i \(-0.615640\pi\)
−0.355355 + 0.934732i \(0.615640\pi\)
\(18\) 0 0
\(19\) 2.42881 0.557206 0.278603 0.960406i \(-0.410129\pi\)
0.278603 + 0.960406i \(0.410129\pi\)
\(20\) 0 0
\(21\) 0.330146 0.0720439
\(22\) 0 0
\(23\) −2.39220 −0.498808 −0.249404 0.968400i \(-0.580235\pi\)
−0.249404 + 0.968400i \(0.580235\pi\)
\(24\) 0 0
\(25\) −0.743524 −0.148705
\(26\) 0 0
\(27\) −5.43726 −1.04640
\(28\) 0 0
\(29\) 3.15996 0.586790 0.293395 0.955991i \(-0.405215\pi\)
0.293395 + 0.955991i \(0.405215\pi\)
\(30\) 0 0
\(31\) −4.46282 −0.801545 −0.400773 0.916178i \(-0.631258\pi\)
−0.400773 + 0.916178i \(0.631258\pi\)
\(32\) 0 0
\(33\) 4.12829 0.718643
\(34\) 0 0
\(35\) 0.577153 0.0975567
\(36\) 0 0
\(37\) 6.97509 1.14670 0.573348 0.819312i \(-0.305644\pi\)
0.573348 + 0.819312i \(0.305644\pi\)
\(38\) 0 0
\(39\) 4.13959 0.662864
\(40\) 0 0
\(41\) 8.78599 1.37214 0.686071 0.727535i \(-0.259334\pi\)
0.686071 + 0.727535i \(0.259334\pi\)
\(42\) 0 0
\(43\) 12.9727 1.97833 0.989163 0.146822i \(-0.0469045\pi\)
0.989163 + 0.146822i \(0.0469045\pi\)
\(44\) 0 0
\(45\) −3.31590 −0.494306
\(46\) 0 0
\(47\) −1.00000 −0.145865
\(48\) 0 0
\(49\) −6.92174 −0.988820
\(50\) 0 0
\(51\) −3.45825 −0.484252
\(52\) 0 0
\(53\) 4.29816 0.590398 0.295199 0.955436i \(-0.404614\pi\)
0.295199 + 0.955436i \(0.404614\pi\)
\(54\) 0 0
\(55\) 7.21697 0.973136
\(56\) 0 0
\(57\) 2.86638 0.379661
\(58\) 0 0
\(59\) 1.39738 0.181923 0.0909614 0.995854i \(-0.471006\pi\)
0.0909614 + 0.995854i \(0.471006\pi\)
\(60\) 0 0
\(61\) −3.74584 −0.479606 −0.239803 0.970822i \(-0.577083\pi\)
−0.239803 + 0.970822i \(0.577083\pi\)
\(62\) 0 0
\(63\) −0.449617 −0.0566465
\(64\) 0 0
\(65\) 7.23672 0.897604
\(66\) 0 0
\(67\) 7.66539 0.936476 0.468238 0.883602i \(-0.344889\pi\)
0.468238 + 0.883602i \(0.344889\pi\)
\(68\) 0 0
\(69\) −2.82317 −0.339870
\(70\) 0 0
\(71\) −1.67409 −0.198677 −0.0993387 0.995054i \(-0.531673\pi\)
−0.0993387 + 0.995054i \(0.531673\pi\)
\(72\) 0 0
\(73\) 7.21968 0.844999 0.422500 0.906363i \(-0.361153\pi\)
0.422500 + 0.906363i \(0.361153\pi\)
\(74\) 0 0
\(75\) −0.877477 −0.101322
\(76\) 0 0
\(77\) 0.978579 0.111519
\(78\) 0 0
\(79\) 2.44688 0.275296 0.137648 0.990481i \(-0.456046\pi\)
0.137648 + 0.990481i \(0.456046\pi\)
\(80\) 0 0
\(81\) −1.59515 −0.177239
\(82\) 0 0
\(83\) 14.4732 1.58864 0.794321 0.607499i \(-0.207827\pi\)
0.794321 + 0.607499i \(0.207827\pi\)
\(84\) 0 0
\(85\) −6.04563 −0.655740
\(86\) 0 0
\(87\) 3.72926 0.399818
\(88\) 0 0
\(89\) −4.63757 −0.491581 −0.245791 0.969323i \(-0.579048\pi\)
−0.245791 + 0.969323i \(0.579048\pi\)
\(90\) 0 0
\(91\) 0.981257 0.102864
\(92\) 0 0
\(93\) −5.26683 −0.546145
\(94\) 0 0
\(95\) 5.01092 0.514110
\(96\) 0 0
\(97\) 3.16215 0.321068 0.160534 0.987030i \(-0.448678\pi\)
0.160534 + 0.987030i \(0.448678\pi\)
\(98\) 0 0
\(99\) −5.62220 −0.565053
\(100\) 0 0
\(101\) 7.46764 0.743058 0.371529 0.928421i \(-0.378834\pi\)
0.371529 + 0.928421i \(0.378834\pi\)
\(102\) 0 0
\(103\) 4.99761 0.492429 0.246215 0.969215i \(-0.420813\pi\)
0.246215 + 0.969215i \(0.420813\pi\)
\(104\) 0 0
\(105\) 0.681133 0.0664717
\(106\) 0 0
\(107\) 4.44704 0.429912 0.214956 0.976624i \(-0.431039\pi\)
0.214956 + 0.976624i \(0.431039\pi\)
\(108\) 0 0
\(109\) −20.5089 −1.96439 −0.982197 0.187852i \(-0.939847\pi\)
−0.982197 + 0.187852i \(0.939847\pi\)
\(110\) 0 0
\(111\) 8.23171 0.781319
\(112\) 0 0
\(113\) −7.31188 −0.687844 −0.343922 0.938998i \(-0.611756\pi\)
−0.343922 + 0.938998i \(0.611756\pi\)
\(114\) 0 0
\(115\) −4.93540 −0.460228
\(116\) 0 0
\(117\) −5.63759 −0.521195
\(118\) 0 0
\(119\) −0.819752 −0.0751466
\(120\) 0 0
\(121\) 1.23657 0.112415
\(122\) 0 0
\(123\) 10.3689 0.934929
\(124\) 0 0
\(125\) −11.8496 −1.05986
\(126\) 0 0
\(127\) −4.55329 −0.404039 −0.202020 0.979381i \(-0.564750\pi\)
−0.202020 + 0.979381i \(0.564750\pi\)
\(128\) 0 0
\(129\) 15.3099 1.34796
\(130\) 0 0
\(131\) 18.1327 1.58426 0.792132 0.610350i \(-0.208971\pi\)
0.792132 + 0.610350i \(0.208971\pi\)
\(132\) 0 0
\(133\) 0.679452 0.0589160
\(134\) 0 0
\(135\) −11.2177 −0.965468
\(136\) 0 0
\(137\) −7.06635 −0.603719 −0.301860 0.953352i \(-0.597607\pi\)
−0.301860 + 0.953352i \(0.597607\pi\)
\(138\) 0 0
\(139\) −3.88583 −0.329592 −0.164796 0.986328i \(-0.552697\pi\)
−0.164796 + 0.986328i \(0.552697\pi\)
\(140\) 0 0
\(141\) −1.18016 −0.0993873
\(142\) 0 0
\(143\) 12.2700 1.02607
\(144\) 0 0
\(145\) 6.51939 0.541406
\(146\) 0 0
\(147\) −8.16875 −0.673747
\(148\) 0 0
\(149\) −4.46359 −0.365672 −0.182836 0.983143i \(-0.558528\pi\)
−0.182836 + 0.983143i \(0.558528\pi\)
\(150\) 0 0
\(151\) −5.81966 −0.473598 −0.236799 0.971559i \(-0.576098\pi\)
−0.236799 + 0.971559i \(0.576098\pi\)
\(152\) 0 0
\(153\) 4.70970 0.380757
\(154\) 0 0
\(155\) −9.20734 −0.739551
\(156\) 0 0
\(157\) −0.881406 −0.0703438 −0.0351719 0.999381i \(-0.511198\pi\)
−0.0351719 + 0.999381i \(0.511198\pi\)
\(158\) 0 0
\(159\) 5.07251 0.402277
\(160\) 0 0
\(161\) −0.669211 −0.0527412
\(162\) 0 0
\(163\) 4.22097 0.330612 0.165306 0.986242i \(-0.447139\pi\)
0.165306 + 0.986242i \(0.447139\pi\)
\(164\) 0 0
\(165\) 8.51717 0.663061
\(166\) 0 0
\(167\) 9.27379 0.717628 0.358814 0.933409i \(-0.383181\pi\)
0.358814 + 0.933409i \(0.383181\pi\)
\(168\) 0 0
\(169\) −0.696382 −0.0535679
\(170\) 0 0
\(171\) −3.90364 −0.298519
\(172\) 0 0
\(173\) −25.3560 −1.92778 −0.963892 0.266295i \(-0.914200\pi\)
−0.963892 + 0.266295i \(0.914200\pi\)
\(174\) 0 0
\(175\) −0.207999 −0.0157233
\(176\) 0 0
\(177\) 1.64912 0.123956
\(178\) 0 0
\(179\) −7.59962 −0.568022 −0.284011 0.958821i \(-0.591665\pi\)
−0.284011 + 0.958821i \(0.591665\pi\)
\(180\) 0 0
\(181\) 16.6388 1.23675 0.618374 0.785884i \(-0.287792\pi\)
0.618374 + 0.785884i \(0.287792\pi\)
\(182\) 0 0
\(183\) −4.42069 −0.326786
\(184\) 0 0
\(185\) 14.3905 1.05801
\(186\) 0 0
\(187\) −10.2505 −0.749593
\(188\) 0 0
\(189\) −1.52106 −0.110641
\(190\) 0 0
\(191\) −10.9636 −0.793299 −0.396650 0.917970i \(-0.629827\pi\)
−0.396650 + 0.917970i \(0.629827\pi\)
\(192\) 0 0
\(193\) 7.68560 0.553221 0.276611 0.960982i \(-0.410789\pi\)
0.276611 + 0.960982i \(0.410789\pi\)
\(194\) 0 0
\(195\) 8.54047 0.611596
\(196\) 0 0
\(197\) 13.1032 0.933567 0.466783 0.884372i \(-0.345413\pi\)
0.466783 + 0.884372i \(0.345413\pi\)
\(198\) 0 0
\(199\) 1.51169 0.107161 0.0535806 0.998564i \(-0.482937\pi\)
0.0535806 + 0.998564i \(0.482937\pi\)
\(200\) 0 0
\(201\) 9.04638 0.638082
\(202\) 0 0
\(203\) 0.883991 0.0620440
\(204\) 0 0
\(205\) 18.1266 1.26601
\(206\) 0 0
\(207\) 3.84480 0.267232
\(208\) 0 0
\(209\) 8.49616 0.587692
\(210\) 0 0
\(211\) 17.3271 1.19285 0.596424 0.802669i \(-0.296588\pi\)
0.596424 + 0.802669i \(0.296588\pi\)
\(212\) 0 0
\(213\) −1.97569 −0.135372
\(214\) 0 0
\(215\) 26.7644 1.82532
\(216\) 0 0
\(217\) −1.24846 −0.0847511
\(218\) 0 0
\(219\) 8.52037 0.575753
\(220\) 0 0
\(221\) −10.2786 −0.691411
\(222\) 0 0
\(223\) −19.7548 −1.32288 −0.661439 0.749999i \(-0.730054\pi\)
−0.661439 + 0.749999i \(0.730054\pi\)
\(224\) 0 0
\(225\) 1.19501 0.0796674
\(226\) 0 0
\(227\) 28.0461 1.86149 0.930743 0.365673i \(-0.119161\pi\)
0.930743 + 0.365673i \(0.119161\pi\)
\(228\) 0 0
\(229\) −1.24499 −0.0822714 −0.0411357 0.999154i \(-0.513098\pi\)
−0.0411357 + 0.999154i \(0.513098\pi\)
\(230\) 0 0
\(231\) 1.15488 0.0759855
\(232\) 0 0
\(233\) 25.9085 1.69732 0.848660 0.528938i \(-0.177410\pi\)
0.848660 + 0.528938i \(0.177410\pi\)
\(234\) 0 0
\(235\) −2.06312 −0.134583
\(236\) 0 0
\(237\) 2.88771 0.187577
\(238\) 0 0
\(239\) −9.89745 −0.640213 −0.320106 0.947382i \(-0.603719\pi\)
−0.320106 + 0.947382i \(0.603719\pi\)
\(240\) 0 0
\(241\) 2.78140 0.179166 0.0895829 0.995979i \(-0.471447\pi\)
0.0895829 + 0.995979i \(0.471447\pi\)
\(242\) 0 0
\(243\) 14.4292 0.925636
\(244\) 0 0
\(245\) −14.2804 −0.912341
\(246\) 0 0
\(247\) 8.51940 0.542077
\(248\) 0 0
\(249\) 17.0807 1.08244
\(250\) 0 0
\(251\) −8.89038 −0.561156 −0.280578 0.959831i \(-0.590526\pi\)
−0.280578 + 0.959831i \(0.590526\pi\)
\(252\) 0 0
\(253\) −8.36810 −0.526098
\(254\) 0 0
\(255\) −7.13480 −0.446799
\(256\) 0 0
\(257\) −22.1651 −1.38262 −0.691310 0.722558i \(-0.742966\pi\)
−0.691310 + 0.722558i \(0.742966\pi\)
\(258\) 0 0
\(259\) 1.95126 0.121246
\(260\) 0 0
\(261\) −5.07877 −0.314368
\(262\) 0 0
\(263\) 31.6693 1.95282 0.976408 0.215936i \(-0.0692802\pi\)
0.976408 + 0.215936i \(0.0692802\pi\)
\(264\) 0 0
\(265\) 8.86764 0.544735
\(266\) 0 0
\(267\) −5.47307 −0.334946
\(268\) 0 0
\(269\) −26.5301 −1.61757 −0.808785 0.588105i \(-0.799874\pi\)
−0.808785 + 0.588105i \(0.799874\pi\)
\(270\) 0 0
\(271\) −3.48366 −0.211617 −0.105809 0.994387i \(-0.533743\pi\)
−0.105809 + 0.994387i \(0.533743\pi\)
\(272\) 0 0
\(273\) 1.15804 0.0700877
\(274\) 0 0
\(275\) −2.60091 −0.156841
\(276\) 0 0
\(277\) 16.2766 0.977965 0.488983 0.872294i \(-0.337368\pi\)
0.488983 + 0.872294i \(0.337368\pi\)
\(278\) 0 0
\(279\) 7.17275 0.429421
\(280\) 0 0
\(281\) 26.8688 1.60286 0.801429 0.598090i \(-0.204073\pi\)
0.801429 + 0.598090i \(0.204073\pi\)
\(282\) 0 0
\(283\) 6.99091 0.415567 0.207783 0.978175i \(-0.433375\pi\)
0.207783 + 0.978175i \(0.433375\pi\)
\(284\) 0 0
\(285\) 5.91369 0.350297
\(286\) 0 0
\(287\) 2.45786 0.145083
\(288\) 0 0
\(289\) −8.41317 −0.494893
\(290\) 0 0
\(291\) 3.73184 0.218764
\(292\) 0 0
\(293\) −24.9302 −1.45644 −0.728220 0.685343i \(-0.759652\pi\)
−0.728220 + 0.685343i \(0.759652\pi\)
\(294\) 0 0
\(295\) 2.88296 0.167852
\(296\) 0 0
\(297\) −19.0200 −1.10365
\(298\) 0 0
\(299\) −8.39099 −0.485264
\(300\) 0 0
\(301\) 3.62909 0.209178
\(302\) 0 0
\(303\) 8.81301 0.506294
\(304\) 0 0
\(305\) −7.72813 −0.442511
\(306\) 0 0
\(307\) −9.38854 −0.535832 −0.267916 0.963442i \(-0.586335\pi\)
−0.267916 + 0.963442i \(0.586335\pi\)
\(308\) 0 0
\(309\) 5.89797 0.335524
\(310\) 0 0
\(311\) 20.0444 1.13661 0.568306 0.822817i \(-0.307599\pi\)
0.568306 + 0.822817i \(0.307599\pi\)
\(312\) 0 0
\(313\) −28.1356 −1.59032 −0.795160 0.606400i \(-0.792613\pi\)
−0.795160 + 0.606400i \(0.792613\pi\)
\(314\) 0 0
\(315\) −0.927616 −0.0522652
\(316\) 0 0
\(317\) −19.4954 −1.09497 −0.547484 0.836816i \(-0.684414\pi\)
−0.547484 + 0.836816i \(0.684414\pi\)
\(318\) 0 0
\(319\) 11.0538 0.618894
\(320\) 0 0
\(321\) 5.24822 0.292927
\(322\) 0 0
\(323\) −7.11720 −0.396012
\(324\) 0 0
\(325\) −2.60802 −0.144667
\(326\) 0 0
\(327\) −24.2037 −1.33847
\(328\) 0 0
\(329\) −0.279748 −0.0154230
\(330\) 0 0
\(331\) 23.1069 1.27007 0.635035 0.772484i \(-0.280986\pi\)
0.635035 + 0.772484i \(0.280986\pi\)
\(332\) 0 0
\(333\) −11.2105 −0.614333
\(334\) 0 0
\(335\) 15.8146 0.864046
\(336\) 0 0
\(337\) 20.4810 1.11567 0.557835 0.829952i \(-0.311632\pi\)
0.557835 + 0.829952i \(0.311632\pi\)
\(338\) 0 0
\(339\) −8.62918 −0.468673
\(340\) 0 0
\(341\) −15.6113 −0.845399
\(342\) 0 0
\(343\) −3.89457 −0.210287
\(344\) 0 0
\(345\) −5.82455 −0.313583
\(346\) 0 0
\(347\) −20.0707 −1.07745 −0.538726 0.842481i \(-0.681094\pi\)
−0.538726 + 0.842481i \(0.681094\pi\)
\(348\) 0 0
\(349\) −4.97410 −0.266258 −0.133129 0.991099i \(-0.542502\pi\)
−0.133129 + 0.991099i \(0.542502\pi\)
\(350\) 0 0
\(351\) −19.0720 −1.01799
\(352\) 0 0
\(353\) 1.00102 0.0532788 0.0266394 0.999645i \(-0.491519\pi\)
0.0266394 + 0.999645i \(0.491519\pi\)
\(354\) 0 0
\(355\) −3.45384 −0.183311
\(356\) 0 0
\(357\) −0.967438 −0.0512022
\(358\) 0 0
\(359\) −16.7311 −0.883036 −0.441518 0.897252i \(-0.645560\pi\)
−0.441518 + 0.897252i \(0.645560\pi\)
\(360\) 0 0
\(361\) −13.1009 −0.689521
\(362\) 0 0
\(363\) 1.45935 0.0765959
\(364\) 0 0
\(365\) 14.8951 0.779644
\(366\) 0 0
\(367\) 0.525038 0.0274067 0.0137034 0.999906i \(-0.495638\pi\)
0.0137034 + 0.999906i \(0.495638\pi\)
\(368\) 0 0
\(369\) −14.1211 −0.735113
\(370\) 0 0
\(371\) 1.20240 0.0624255
\(372\) 0 0
\(373\) −1.43021 −0.0740534 −0.0370267 0.999314i \(-0.511789\pi\)
−0.0370267 + 0.999314i \(0.511789\pi\)
\(374\) 0 0
\(375\) −13.9844 −0.722152
\(376\) 0 0
\(377\) 11.0840 0.570857
\(378\) 0 0
\(379\) 19.0852 0.980342 0.490171 0.871626i \(-0.336934\pi\)
0.490171 + 0.871626i \(0.336934\pi\)
\(380\) 0 0
\(381\) −5.37360 −0.275298
\(382\) 0 0
\(383\) −18.6095 −0.950904 −0.475452 0.879742i \(-0.657715\pi\)
−0.475452 + 0.879742i \(0.657715\pi\)
\(384\) 0 0
\(385\) 2.01893 0.102894
\(386\) 0 0
\(387\) −20.8501 −1.05987
\(388\) 0 0
\(389\) −8.96907 −0.454750 −0.227375 0.973807i \(-0.573014\pi\)
−0.227375 + 0.973807i \(0.573014\pi\)
\(390\) 0 0
\(391\) 7.00992 0.354507
\(392\) 0 0
\(393\) 21.3995 1.07946
\(394\) 0 0
\(395\) 5.04822 0.254004
\(396\) 0 0
\(397\) −6.64572 −0.333539 −0.166770 0.985996i \(-0.553334\pi\)
−0.166770 + 0.985996i \(0.553334\pi\)
\(398\) 0 0
\(399\) 0.801862 0.0401433
\(400\) 0 0
\(401\) 1.85295 0.0925320 0.0462660 0.998929i \(-0.485268\pi\)
0.0462660 + 0.998929i \(0.485268\pi\)
\(402\) 0 0
\(403\) −15.6540 −0.779781
\(404\) 0 0
\(405\) −3.29099 −0.163531
\(406\) 0 0
\(407\) 24.3994 1.20943
\(408\) 0 0
\(409\) 30.0196 1.48437 0.742187 0.670192i \(-0.233788\pi\)
0.742187 + 0.670192i \(0.233788\pi\)
\(410\) 0 0
\(411\) −8.33942 −0.411353
\(412\) 0 0
\(413\) 0.390912 0.0192355
\(414\) 0 0
\(415\) 29.8600 1.46577
\(416\) 0 0
\(417\) −4.58590 −0.224572
\(418\) 0 0
\(419\) −2.42035 −0.118242 −0.0591209 0.998251i \(-0.518830\pi\)
−0.0591209 + 0.998251i \(0.518830\pi\)
\(420\) 0 0
\(421\) −20.7969 −1.01358 −0.506789 0.862070i \(-0.669168\pi\)
−0.506789 + 0.862070i \(0.669168\pi\)
\(422\) 0 0
\(423\) 1.60723 0.0781460
\(424\) 0 0
\(425\) 2.17877 0.105686
\(426\) 0 0
\(427\) −1.04789 −0.0507109
\(428\) 0 0
\(429\) 14.4806 0.699130
\(430\) 0 0
\(431\) 5.73175 0.276089 0.138044 0.990426i \(-0.455918\pi\)
0.138044 + 0.990426i \(0.455918\pi\)
\(432\) 0 0
\(433\) −33.2943 −1.60002 −0.800012 0.599984i \(-0.795174\pi\)
−0.800012 + 0.599984i \(0.795174\pi\)
\(434\) 0 0
\(435\) 7.69391 0.368895
\(436\) 0 0
\(437\) −5.81018 −0.277939
\(438\) 0 0
\(439\) −25.2471 −1.20498 −0.602489 0.798127i \(-0.705824\pi\)
−0.602489 + 0.798127i \(0.705824\pi\)
\(440\) 0 0
\(441\) 11.1248 0.529752
\(442\) 0 0
\(443\) 15.8718 0.754094 0.377047 0.926194i \(-0.376940\pi\)
0.377047 + 0.926194i \(0.376940\pi\)
\(444\) 0 0
\(445\) −9.56787 −0.453561
\(446\) 0 0
\(447\) −5.26775 −0.249156
\(448\) 0 0
\(449\) 5.61052 0.264777 0.132388 0.991198i \(-0.457735\pi\)
0.132388 + 0.991198i \(0.457735\pi\)
\(450\) 0 0
\(451\) 30.7341 1.44721
\(452\) 0 0
\(453\) −6.86813 −0.322693
\(454\) 0 0
\(455\) 2.02445 0.0949078
\(456\) 0 0
\(457\) −23.6225 −1.10502 −0.552508 0.833508i \(-0.686329\pi\)
−0.552508 + 0.833508i \(0.686329\pi\)
\(458\) 0 0
\(459\) 15.9330 0.743686
\(460\) 0 0
\(461\) −6.96173 −0.324240 −0.162120 0.986771i \(-0.551833\pi\)
−0.162120 + 0.986771i \(0.551833\pi\)
\(462\) 0 0
\(463\) 3.69193 0.171578 0.0857892 0.996313i \(-0.472659\pi\)
0.0857892 + 0.996313i \(0.472659\pi\)
\(464\) 0 0
\(465\) −10.8661 −0.503904
\(466\) 0 0
\(467\) −17.7477 −0.821267 −0.410634 0.911800i \(-0.634692\pi\)
−0.410634 + 0.911800i \(0.634692\pi\)
\(468\) 0 0
\(469\) 2.14437 0.0990180
\(470\) 0 0
\(471\) −1.04020 −0.0479298
\(472\) 0 0
\(473\) 45.3797 2.08656
\(474\) 0 0
\(475\) −1.80588 −0.0828593
\(476\) 0 0
\(477\) −6.90812 −0.316301
\(478\) 0 0
\(479\) 27.3638 1.25028 0.625142 0.780511i \(-0.285041\pi\)
0.625142 + 0.780511i \(0.285041\pi\)
\(480\) 0 0
\(481\) 24.4662 1.11556
\(482\) 0 0
\(483\) −0.789775 −0.0359360
\(484\) 0 0
\(485\) 6.52390 0.296235
\(486\) 0 0
\(487\) 43.9223 1.99031 0.995155 0.0983196i \(-0.0313467\pi\)
0.995155 + 0.0983196i \(0.0313467\pi\)
\(488\) 0 0
\(489\) 4.98142 0.225268
\(490\) 0 0
\(491\) 14.0950 0.636097 0.318048 0.948074i \(-0.396973\pi\)
0.318048 + 0.948074i \(0.396973\pi\)
\(492\) 0 0
\(493\) −9.25973 −0.417037
\(494\) 0 0
\(495\) −11.5993 −0.521350
\(496\) 0 0
\(497\) −0.468321 −0.0210071
\(498\) 0 0
\(499\) −17.6023 −0.787987 −0.393993 0.919113i \(-0.628907\pi\)
−0.393993 + 0.919113i \(0.628907\pi\)
\(500\) 0 0
\(501\) 10.9445 0.488966
\(502\) 0 0
\(503\) −8.94532 −0.398852 −0.199426 0.979913i \(-0.563908\pi\)
−0.199426 + 0.979913i \(0.563908\pi\)
\(504\) 0 0
\(505\) 15.4067 0.685588
\(506\) 0 0
\(507\) −0.821842 −0.0364993
\(508\) 0 0
\(509\) 19.4504 0.862123 0.431061 0.902323i \(-0.358139\pi\)
0.431061 + 0.902323i \(0.358139\pi\)
\(510\) 0 0
\(511\) 2.01969 0.0893457
\(512\) 0 0
\(513\) −13.2060 −0.583061
\(514\) 0 0
\(515\) 10.3107 0.454343
\(516\) 0 0
\(517\) −3.49808 −0.153845
\(518\) 0 0
\(519\) −29.9241 −1.31352
\(520\) 0 0
\(521\) −21.8725 −0.958253 −0.479126 0.877746i \(-0.659046\pi\)
−0.479126 + 0.877746i \(0.659046\pi\)
\(522\) 0 0
\(523\) −16.5279 −0.722714 −0.361357 0.932427i \(-0.617687\pi\)
−0.361357 + 0.932427i \(0.617687\pi\)
\(524\) 0 0
\(525\) −0.245472 −0.0107133
\(526\) 0 0
\(527\) 13.0775 0.569666
\(528\) 0 0
\(529\) −17.2774 −0.751191
\(530\) 0 0
\(531\) −2.24590 −0.0974636
\(532\) 0 0
\(533\) 30.8182 1.33488
\(534\) 0 0
\(535\) 9.17480 0.396661
\(536\) 0 0
\(537\) −8.96876 −0.387031
\(538\) 0 0
\(539\) −24.2128 −1.04292
\(540\) 0 0
\(541\) 7.56928 0.325429 0.162714 0.986673i \(-0.447975\pi\)
0.162714 + 0.986673i \(0.447975\pi\)
\(542\) 0 0
\(543\) 19.6364 0.842677
\(544\) 0 0
\(545\) −42.3123 −1.81246
\(546\) 0 0
\(547\) −27.2004 −1.16301 −0.581503 0.813544i \(-0.697535\pi\)
−0.581503 + 0.813544i \(0.697535\pi\)
\(548\) 0 0
\(549\) 6.02041 0.256945
\(550\) 0 0
\(551\) 7.67493 0.326963
\(552\) 0 0
\(553\) 0.684510 0.0291083
\(554\) 0 0
\(555\) 16.9830 0.720889
\(556\) 0 0
\(557\) −27.7806 −1.17710 −0.588551 0.808460i \(-0.700301\pi\)
−0.588551 + 0.808460i \(0.700301\pi\)
\(558\) 0 0
\(559\) 45.5039 1.92461
\(560\) 0 0
\(561\) −12.0972 −0.510746
\(562\) 0 0
\(563\) −0.709879 −0.0299178 −0.0149589 0.999888i \(-0.504762\pi\)
−0.0149589 + 0.999888i \(0.504762\pi\)
\(564\) 0 0
\(565\) −15.0853 −0.634644
\(566\) 0 0
\(567\) −0.446240 −0.0187403
\(568\) 0 0
\(569\) 33.1083 1.38797 0.693986 0.719988i \(-0.255853\pi\)
0.693986 + 0.719988i \(0.255853\pi\)
\(570\) 0 0
\(571\) −0.640242 −0.0267933 −0.0133966 0.999910i \(-0.504264\pi\)
−0.0133966 + 0.999910i \(0.504264\pi\)
\(572\) 0 0
\(573\) −12.9388 −0.540526
\(574\) 0 0
\(575\) 1.77866 0.0741751
\(576\) 0 0
\(577\) −15.5999 −0.649431 −0.324715 0.945812i \(-0.605269\pi\)
−0.324715 + 0.945812i \(0.605269\pi\)
\(578\) 0 0
\(579\) 9.07022 0.376946
\(580\) 0 0
\(581\) 4.04884 0.167974
\(582\) 0 0
\(583\) 15.0353 0.622699
\(584\) 0 0
\(585\) −11.6310 −0.480884
\(586\) 0 0
\(587\) −0.178942 −0.00738572 −0.00369286 0.999993i \(-0.501175\pi\)
−0.00369286 + 0.999993i \(0.501175\pi\)
\(588\) 0 0
\(589\) −10.8393 −0.446626
\(590\) 0 0
\(591\) 15.4639 0.636100
\(592\) 0 0
\(593\) −47.7952 −1.96272 −0.981358 0.192190i \(-0.938441\pi\)
−0.981358 + 0.192190i \(0.938441\pi\)
\(594\) 0 0
\(595\) −1.69125 −0.0693345
\(596\) 0 0
\(597\) 1.78404 0.0730159
\(598\) 0 0
\(599\) −11.5227 −0.470806 −0.235403 0.971898i \(-0.575641\pi\)
−0.235403 + 0.971898i \(0.575641\pi\)
\(600\) 0 0
\(601\) −2.30751 −0.0941253 −0.0470627 0.998892i \(-0.514986\pi\)
−0.0470627 + 0.998892i \(0.514986\pi\)
\(602\) 0 0
\(603\) −12.3200 −0.501710
\(604\) 0 0
\(605\) 2.55119 0.103721
\(606\) 0 0
\(607\) −30.4093 −1.23427 −0.617137 0.786855i \(-0.711708\pi\)
−0.617137 + 0.786855i \(0.711708\pi\)
\(608\) 0 0
\(609\) 1.04325 0.0422746
\(610\) 0 0
\(611\) −3.50765 −0.141904
\(612\) 0 0
\(613\) 27.1057 1.09479 0.547394 0.836875i \(-0.315620\pi\)
0.547394 + 0.836875i \(0.315620\pi\)
\(614\) 0 0
\(615\) 21.3922 0.862618
\(616\) 0 0
\(617\) 15.8852 0.639513 0.319756 0.947500i \(-0.396399\pi\)
0.319756 + 0.947500i \(0.396399\pi\)
\(618\) 0 0
\(619\) 23.2692 0.935269 0.467635 0.883922i \(-0.345106\pi\)
0.467635 + 0.883922i \(0.345106\pi\)
\(620\) 0 0
\(621\) 13.0070 0.521952
\(622\) 0 0
\(623\) −1.29735 −0.0519772
\(624\) 0 0
\(625\) −20.7296 −0.829182
\(626\) 0 0
\(627\) 10.0268 0.400432
\(628\) 0 0
\(629\) −20.4393 −0.814968
\(630\) 0 0
\(631\) −13.4704 −0.536248 −0.268124 0.963384i \(-0.586404\pi\)
−0.268124 + 0.963384i \(0.586404\pi\)
\(632\) 0 0
\(633\) 20.4488 0.812765
\(634\) 0 0
\(635\) −9.39399 −0.372789
\(636\) 0 0
\(637\) −24.2791 −0.961971
\(638\) 0 0
\(639\) 2.69063 0.106440
\(640\) 0 0
\(641\) −22.2199 −0.877634 −0.438817 0.898577i \(-0.644602\pi\)
−0.438817 + 0.898577i \(0.644602\pi\)
\(642\) 0 0
\(643\) −8.57458 −0.338148 −0.169074 0.985603i \(-0.554078\pi\)
−0.169074 + 0.985603i \(0.554078\pi\)
\(644\) 0 0
\(645\) 31.5862 1.24371
\(646\) 0 0
\(647\) 12.0914 0.475360 0.237680 0.971343i \(-0.423613\pi\)
0.237680 + 0.971343i \(0.423613\pi\)
\(648\) 0 0
\(649\) 4.88813 0.191876
\(650\) 0 0
\(651\) −1.47338 −0.0577464
\(652\) 0 0
\(653\) 27.0993 1.06048 0.530239 0.847848i \(-0.322102\pi\)
0.530239 + 0.847848i \(0.322102\pi\)
\(654\) 0 0
\(655\) 37.4100 1.46173
\(656\) 0 0
\(657\) −11.6037 −0.452702
\(658\) 0 0
\(659\) 11.2447 0.438032 0.219016 0.975721i \(-0.429715\pi\)
0.219016 + 0.975721i \(0.429715\pi\)
\(660\) 0 0
\(661\) 40.6805 1.58229 0.791143 0.611631i \(-0.209486\pi\)
0.791143 + 0.611631i \(0.209486\pi\)
\(662\) 0 0
\(663\) −12.1303 −0.471104
\(664\) 0 0
\(665\) 1.40179 0.0543592
\(666\) 0 0
\(667\) −7.55925 −0.292695
\(668\) 0 0
\(669\) −23.3138 −0.901363
\(670\) 0 0
\(671\) −13.1032 −0.505845
\(672\) 0 0
\(673\) −3.11190 −0.119955 −0.0599774 0.998200i \(-0.519103\pi\)
−0.0599774 + 0.998200i \(0.519103\pi\)
\(674\) 0 0
\(675\) 4.04273 0.155605
\(676\) 0 0
\(677\) −38.3779 −1.47498 −0.737491 0.675357i \(-0.763989\pi\)
−0.737491 + 0.675357i \(0.763989\pi\)
\(678\) 0 0
\(679\) 0.884603 0.0339480
\(680\) 0 0
\(681\) 33.0989 1.26835
\(682\) 0 0
\(683\) 26.3628 1.00874 0.504372 0.863487i \(-0.331724\pi\)
0.504372 + 0.863487i \(0.331724\pi\)
\(684\) 0 0
\(685\) −14.5788 −0.557025
\(686\) 0 0
\(687\) −1.46929 −0.0560568
\(688\) 0 0
\(689\) 15.0765 0.574367
\(690\) 0 0
\(691\) 33.2289 1.26409 0.632044 0.774932i \(-0.282216\pi\)
0.632044 + 0.774932i \(0.282216\pi\)
\(692\) 0 0
\(693\) −1.57280 −0.0597456
\(694\) 0 0
\(695\) −8.01695 −0.304100
\(696\) 0 0
\(697\) −25.7458 −0.975193
\(698\) 0 0
\(699\) 30.5761 1.15650
\(700\) 0 0
\(701\) −7.59061 −0.286693 −0.143347 0.989673i \(-0.545786\pi\)
−0.143347 + 0.989673i \(0.545786\pi\)
\(702\) 0 0
\(703\) 16.9411 0.638947
\(704\) 0 0
\(705\) −2.43481 −0.0917003
\(706\) 0 0
\(707\) 2.08906 0.0785670
\(708\) 0 0
\(709\) 15.5138 0.582633 0.291316 0.956627i \(-0.405907\pi\)
0.291316 + 0.956627i \(0.405907\pi\)
\(710\) 0 0
\(711\) −3.93269 −0.147488
\(712\) 0 0
\(713\) 10.6759 0.399817
\(714\) 0 0
\(715\) 25.3146 0.946713
\(716\) 0 0
\(717\) −11.6806 −0.436218
\(718\) 0 0
\(719\) −27.6361 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(720\) 0 0
\(721\) 1.39807 0.0520668
\(722\) 0 0
\(723\) 3.28249 0.122077
\(724\) 0 0
\(725\) −2.34951 −0.0872585
\(726\) 0 0
\(727\) −24.6442 −0.914003 −0.457002 0.889466i \(-0.651077\pi\)
−0.457002 + 0.889466i \(0.651077\pi\)
\(728\) 0 0
\(729\) 21.8142 0.807935
\(730\) 0 0
\(731\) −38.0144 −1.40601
\(732\) 0 0
\(733\) 52.3115 1.93217 0.966084 0.258226i \(-0.0831380\pi\)
0.966084 + 0.258226i \(0.0831380\pi\)
\(734\) 0 0
\(735\) −16.8531 −0.621638
\(736\) 0 0
\(737\) 26.8142 0.987712
\(738\) 0 0
\(739\) −6.43918 −0.236869 −0.118435 0.992962i \(-0.537788\pi\)
−0.118435 + 0.992962i \(0.537788\pi\)
\(740\) 0 0
\(741\) 10.0542 0.369352
\(742\) 0 0
\(743\) 25.2625 0.926792 0.463396 0.886151i \(-0.346631\pi\)
0.463396 + 0.886151i \(0.346631\pi\)
\(744\) 0 0
\(745\) −9.20893 −0.337389
\(746\) 0 0
\(747\) −23.2617 −0.851101
\(748\) 0 0
\(749\) 1.24405 0.0454566
\(750\) 0 0
\(751\) 36.3072 1.32487 0.662435 0.749120i \(-0.269523\pi\)
0.662435 + 0.749120i \(0.269523\pi\)
\(752\) 0 0
\(753\) −10.4921 −0.382352
\(754\) 0 0
\(755\) −12.0067 −0.436968
\(756\) 0 0
\(757\) 2.29994 0.0835929 0.0417964 0.999126i \(-0.486692\pi\)
0.0417964 + 0.999126i \(0.486692\pi\)
\(758\) 0 0
\(759\) −9.87568 −0.358465
\(760\) 0 0
\(761\) 3.44599 0.124917 0.0624584 0.998048i \(-0.480106\pi\)
0.0624584 + 0.998048i \(0.480106\pi\)
\(762\) 0 0
\(763\) −5.73731 −0.207705
\(764\) 0 0
\(765\) 9.71669 0.351308
\(766\) 0 0
\(767\) 4.90151 0.176983
\(768\) 0 0
\(769\) −5.47883 −0.197572 −0.0987858 0.995109i \(-0.531496\pi\)
−0.0987858 + 0.995109i \(0.531496\pi\)
\(770\) 0 0
\(771\) −26.1583 −0.942069
\(772\) 0 0
\(773\) 10.2329 0.368052 0.184026 0.982921i \(-0.441087\pi\)
0.184026 + 0.982921i \(0.441087\pi\)
\(774\) 0 0
\(775\) 3.31821 0.119194
\(776\) 0 0
\(777\) 2.30280 0.0826125
\(778\) 0 0
\(779\) 21.3395 0.764566
\(780\) 0 0
\(781\) −5.85608 −0.209547
\(782\) 0 0
\(783\) −17.1815 −0.614018
\(784\) 0 0
\(785\) −1.81845 −0.0649032
\(786\) 0 0
\(787\) −5.82336 −0.207580 −0.103790 0.994599i \(-0.533097\pi\)
−0.103790 + 0.994599i \(0.533097\pi\)
\(788\) 0 0
\(789\) 37.3748 1.33058
\(790\) 0 0
\(791\) −2.04548 −0.0727289
\(792\) 0 0
\(793\) −13.1391 −0.466583
\(794\) 0 0
\(795\) 10.4652 0.371163
\(796\) 0 0
\(797\) 38.9127 1.37836 0.689179 0.724591i \(-0.257971\pi\)
0.689179 + 0.724591i \(0.257971\pi\)
\(798\) 0 0
\(799\) 2.93033 0.103668
\(800\) 0 0
\(801\) 7.45362 0.263361
\(802\) 0 0
\(803\) 25.2550 0.891230
\(804\) 0 0
\(805\) −1.38066 −0.0486620
\(806\) 0 0
\(807\) −31.3097 −1.10216
\(808\) 0 0
\(809\) −13.6415 −0.479608 −0.239804 0.970821i \(-0.577083\pi\)
−0.239804 + 0.970821i \(0.577083\pi\)
\(810\) 0 0
\(811\) 31.3721 1.10162 0.550812 0.834629i \(-0.314318\pi\)
0.550812 + 0.834629i \(0.314318\pi\)
\(812\) 0 0
\(813\) −4.11127 −0.144189
\(814\) 0 0
\(815\) 8.70839 0.305042
\(816\) 0 0
\(817\) 31.5083 1.10234
\(818\) 0 0
\(819\) −1.57710 −0.0551084
\(820\) 0 0
\(821\) −31.7420 −1.10781 −0.553903 0.832581i \(-0.686862\pi\)
−0.553903 + 0.832581i \(0.686862\pi\)
\(822\) 0 0
\(823\) −13.0929 −0.456388 −0.228194 0.973616i \(-0.573282\pi\)
−0.228194 + 0.973616i \(0.573282\pi\)
\(824\) 0 0
\(825\) −3.06948 −0.106866
\(826\) 0 0
\(827\) 40.7565 1.41724 0.708621 0.705589i \(-0.249317\pi\)
0.708621 + 0.705589i \(0.249317\pi\)
\(828\) 0 0
\(829\) −12.8485 −0.446246 −0.223123 0.974790i \(-0.571625\pi\)
−0.223123 + 0.974790i \(0.571625\pi\)
\(830\) 0 0
\(831\) 19.2090 0.666351
\(832\) 0 0
\(833\) 20.2830 0.702763
\(834\) 0 0
\(835\) 19.1330 0.662124
\(836\) 0 0
\(837\) 24.2655 0.838738
\(838\) 0 0
\(839\) 15.8858 0.548440 0.274220 0.961667i \(-0.411580\pi\)
0.274220 + 0.961667i \(0.411580\pi\)
\(840\) 0 0
\(841\) −19.0146 −0.655677
\(842\) 0 0
\(843\) 31.7094 1.09213
\(844\) 0 0
\(845\) −1.43672 −0.0494247
\(846\) 0 0
\(847\) 0.345927 0.0118862
\(848\) 0 0
\(849\) 8.25039 0.283153
\(850\) 0 0
\(851\) −16.6858 −0.571981
\(852\) 0 0
\(853\) 29.8272 1.02126 0.510631 0.859800i \(-0.329412\pi\)
0.510631 + 0.859800i \(0.329412\pi\)
\(854\) 0 0
\(855\) −8.05368 −0.275430
\(856\) 0 0
\(857\) −29.1512 −0.995786 −0.497893 0.867238i \(-0.665893\pi\)
−0.497893 + 0.867238i \(0.665893\pi\)
\(858\) 0 0
\(859\) −13.3351 −0.454987 −0.227494 0.973780i \(-0.573053\pi\)
−0.227494 + 0.973780i \(0.573053\pi\)
\(860\) 0 0
\(861\) 2.90066 0.0988543
\(862\) 0 0
\(863\) −15.2707 −0.519821 −0.259910 0.965633i \(-0.583693\pi\)
−0.259910 + 0.965633i \(0.583693\pi\)
\(864\) 0 0
\(865\) −52.3126 −1.77868
\(866\) 0 0
\(867\) −9.92888 −0.337203
\(868\) 0 0
\(869\) 8.55940 0.290358
\(870\) 0 0
\(871\) 26.8875 0.911049
\(872\) 0 0
\(873\) −5.08229 −0.172009
\(874\) 0 0
\(875\) −3.31489 −0.112064
\(876\) 0 0
\(877\) −19.7918 −0.668322 −0.334161 0.942516i \(-0.608453\pi\)
−0.334161 + 0.942516i \(0.608453\pi\)
\(878\) 0 0
\(879\) −29.4216 −0.992367
\(880\) 0 0
\(881\) −26.9189 −0.906920 −0.453460 0.891277i \(-0.649811\pi\)
−0.453460 + 0.891277i \(0.649811\pi\)
\(882\) 0 0
\(883\) −16.6624 −0.560733 −0.280367 0.959893i \(-0.590456\pi\)
−0.280367 + 0.959893i \(0.590456\pi\)
\(884\) 0 0
\(885\) 3.40235 0.114369
\(886\) 0 0
\(887\) −44.7724 −1.50331 −0.751654 0.659557i \(-0.770744\pi\)
−0.751654 + 0.659557i \(0.770744\pi\)
\(888\) 0 0
\(889\) −1.27377 −0.0427209
\(890\) 0 0
\(891\) −5.57997 −0.186936
\(892\) 0 0
\(893\) −2.42881 −0.0812769
\(894\) 0 0
\(895\) −15.6790 −0.524090
\(896\) 0 0
\(897\) −9.90270 −0.330642
\(898\) 0 0
\(899\) −14.1023 −0.470339
\(900\) 0 0
\(901\) −12.5950 −0.419601
\(902\) 0 0
\(903\) 4.28291 0.142526
\(904\) 0 0
\(905\) 34.3278 1.14109
\(906\) 0 0
\(907\) −30.9720 −1.02841 −0.514205 0.857667i \(-0.671913\pi\)
−0.514205 + 0.857667i \(0.671913\pi\)
\(908\) 0 0
\(909\) −12.0022 −0.398087
\(910\) 0 0
\(911\) −53.4237 −1.77000 −0.885002 0.465586i \(-0.845843\pi\)
−0.885002 + 0.465586i \(0.845843\pi\)
\(912\) 0 0
\(913\) 50.6285 1.67556
\(914\) 0 0
\(915\) −9.12042 −0.301512
\(916\) 0 0
\(917\) 5.07259 0.167512
\(918\) 0 0
\(919\) −28.0113 −0.924009 −0.462004 0.886878i \(-0.652870\pi\)
−0.462004 + 0.886878i \(0.652870\pi\)
\(920\) 0 0
\(921\) −11.0800 −0.365097
\(922\) 0 0
\(923\) −5.87211 −0.193283
\(924\) 0 0
\(925\) −5.18614 −0.170519
\(926\) 0 0
\(927\) −8.03228 −0.263815
\(928\) 0 0
\(929\) 10.0405 0.329419 0.164709 0.986342i \(-0.447331\pi\)
0.164709 + 0.986342i \(0.447331\pi\)
\(930\) 0 0
\(931\) −16.8116 −0.550977
\(932\) 0 0
\(933\) 23.6555 0.774447
\(934\) 0 0
\(935\) −21.1481 −0.691617
\(936\) 0 0
\(937\) −31.2383 −1.02051 −0.510255 0.860023i \(-0.670449\pi\)
−0.510255 + 0.860023i \(0.670449\pi\)
\(938\) 0 0
\(939\) −33.2045 −1.08359
\(940\) 0 0
\(941\) −28.8715 −0.941184 −0.470592 0.882351i \(-0.655960\pi\)
−0.470592 + 0.882351i \(0.655960\pi\)
\(942\) 0 0
\(943\) −21.0178 −0.684434
\(944\) 0 0
\(945\) −3.13813 −0.102083
\(946\) 0 0
\(947\) 35.1273 1.14148 0.570742 0.821130i \(-0.306656\pi\)
0.570742 + 0.821130i \(0.306656\pi\)
\(948\) 0 0
\(949\) 25.3241 0.822056
\(950\) 0 0
\(951\) −23.0076 −0.746073
\(952\) 0 0
\(953\) 8.72292 0.282563 0.141282 0.989969i \(-0.454878\pi\)
0.141282 + 0.989969i \(0.454878\pi\)
\(954\) 0 0
\(955\) −22.6193 −0.731943
\(956\) 0 0
\(957\) 13.0452 0.421693
\(958\) 0 0
\(959\) −1.97679 −0.0638340
\(960\) 0 0
\(961\) −11.0833 −0.357525
\(962\) 0 0
\(963\) −7.14740 −0.230322
\(964\) 0 0
\(965\) 15.8563 0.510433
\(966\) 0 0
\(967\) −47.5185 −1.52809 −0.764047 0.645161i \(-0.776790\pi\)
−0.764047 + 0.645161i \(0.776790\pi\)
\(968\) 0 0
\(969\) −8.39942 −0.269828
\(970\) 0 0
\(971\) 15.6662 0.502754 0.251377 0.967889i \(-0.419117\pi\)
0.251377 + 0.967889i \(0.419117\pi\)
\(972\) 0 0
\(973\) −1.08705 −0.0348493
\(974\) 0 0
\(975\) −3.07788 −0.0985711
\(976\) 0 0
\(977\) 19.0680 0.610041 0.305020 0.952346i \(-0.401337\pi\)
0.305020 + 0.952346i \(0.401337\pi\)
\(978\) 0 0
\(979\) −16.2226 −0.518476
\(980\) 0 0
\(981\) 32.9624 1.05241
\(982\) 0 0
\(983\) 6.59597 0.210379 0.105189 0.994452i \(-0.466455\pi\)
0.105189 + 0.994452i \(0.466455\pi\)
\(984\) 0 0
\(985\) 27.0336 0.861361
\(986\) 0 0
\(987\) −0.330146 −0.0105087
\(988\) 0 0
\(989\) −31.0334 −0.986804
\(990\) 0 0
\(991\) −54.7341 −1.73868 −0.869342 0.494210i \(-0.835457\pi\)
−0.869342 + 0.494210i \(0.835457\pi\)
\(992\) 0 0
\(993\) 27.2698 0.865381
\(994\) 0 0
\(995\) 3.11881 0.0988730
\(996\) 0 0
\(997\) 46.7121 1.47939 0.739694 0.672943i \(-0.234970\pi\)
0.739694 + 0.672943i \(0.234970\pi\)
\(998\) 0 0
\(999\) −37.9253 −1.19990
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 6016.2.a.o.1.8 yes 13
4.3 odd 2 6016.2.a.m.1.6 13
8.3 odd 2 6016.2.a.p.1.8 yes 13
8.5 even 2 6016.2.a.n.1.6 yes 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6016.2.a.m.1.6 13 4.3 odd 2
6016.2.a.n.1.6 yes 13 8.5 even 2
6016.2.a.o.1.8 yes 13 1.1 even 1 trivial
6016.2.a.p.1.8 yes 13 8.3 odd 2