Properties

Label 6525.2.a.cb.1.4
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $9$
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] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, 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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
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: \( x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 \) 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 \(1.23642\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.23642 q^{2} -0.471260 q^{4} -3.27673 q^{7} +3.05552 q^{8} +O(q^{10})\) \(q-1.23642 q^{2} -0.471260 q^{4} -3.27673 q^{7} +3.05552 q^{8} -2.30259 q^{11} +5.57932 q^{13} +4.05142 q^{14} -2.83539 q^{16} +1.94031 q^{17} -3.59158 q^{19} +2.84698 q^{22} +1.66263 q^{23} -6.89840 q^{26} +1.54419 q^{28} -1.00000 q^{29} +1.70877 q^{31} -2.60530 q^{32} -2.39904 q^{34} -9.16633 q^{37} +4.44071 q^{38} +8.54038 q^{41} -3.56165 q^{43} +1.08512 q^{44} -2.05571 q^{46} -11.5640 q^{47} +3.73698 q^{49} -2.62931 q^{52} -9.66312 q^{53} -10.0121 q^{56} +1.23642 q^{58} +9.83385 q^{59} +5.42714 q^{61} -2.11276 q^{62} +8.89204 q^{64} +5.20114 q^{67} -0.914391 q^{68} +6.02596 q^{71} +15.5143 q^{73} +11.3334 q^{74} +1.69257 q^{76} +7.54498 q^{77} +12.2465 q^{79} -10.5595 q^{82} -10.7179 q^{83} +4.40370 q^{86} -7.03562 q^{88} +2.53756 q^{89} -18.2820 q^{91} -0.783531 q^{92} +14.2980 q^{94} -5.89998 q^{97} -4.62048 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 2 q^{2} + 10 q^{4} + q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 2 q^{2} + 10 q^{4} + q^{7} - 9 q^{8} - 2 q^{11} + q^{13} + 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} + 3 q^{22} - 16 q^{23} - 6 q^{26} - 4 q^{28} - 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38} + 10 q^{41} + 3 q^{43} + 13 q^{44} + 4 q^{46} - 26 q^{47} - 8 q^{49} - 9 q^{52} - 22 q^{53} - 22 q^{56} + 2 q^{58} - 4 q^{59} + 7 q^{61} - 28 q^{62} + 9 q^{64} + 5 q^{67} - 39 q^{68} - 10 q^{73} + 34 q^{74} - 2 q^{76} - 34 q^{77} + 10 q^{79} - 8 q^{82} - 46 q^{83} - 28 q^{86} + 2 q^{88} - 4 q^{89} - 21 q^{91} - 20 q^{92} + 5 q^{94} + 7 q^{97} - 51 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23642 −0.874282 −0.437141 0.899393i \(-0.644009\pi\)
−0.437141 + 0.899393i \(0.644009\pi\)
\(3\) 0 0
\(4\) −0.471260 −0.235630
\(5\) 0 0
\(6\) 0 0
\(7\) −3.27673 −1.23849 −0.619244 0.785198i \(-0.712561\pi\)
−0.619244 + 0.785198i \(0.712561\pi\)
\(8\) 3.05552 1.08029
\(9\) 0 0
\(10\) 0 0
\(11\) −2.30259 −0.694258 −0.347129 0.937817i \(-0.612843\pi\)
−0.347129 + 0.937817i \(0.612843\pi\)
\(12\) 0 0
\(13\) 5.57932 1.54743 0.773713 0.633536i \(-0.218397\pi\)
0.773713 + 0.633536i \(0.218397\pi\)
\(14\) 4.05142 1.08279
\(15\) 0 0
\(16\) −2.83539 −0.708848
\(17\) 1.94031 0.470594 0.235297 0.971923i \(-0.424394\pi\)
0.235297 + 0.971923i \(0.424394\pi\)
\(18\) 0 0
\(19\) −3.59158 −0.823965 −0.411982 0.911192i \(-0.635164\pi\)
−0.411982 + 0.911192i \(0.635164\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.84698 0.606977
\(23\) 1.66263 0.346682 0.173341 0.984862i \(-0.444544\pi\)
0.173341 + 0.984862i \(0.444544\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.89840 −1.35289
\(27\) 0 0
\(28\) 1.54419 0.291825
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 1.70877 0.306903 0.153452 0.988156i \(-0.450961\pi\)
0.153452 + 0.988156i \(0.450961\pi\)
\(32\) −2.60530 −0.460556
\(33\) 0 0
\(34\) −2.39904 −0.411432
\(35\) 0 0
\(36\) 0 0
\(37\) −9.16633 −1.50693 −0.753467 0.657485i \(-0.771620\pi\)
−0.753467 + 0.657485i \(0.771620\pi\)
\(38\) 4.44071 0.720378
\(39\) 0 0
\(40\) 0 0
\(41\) 8.54038 1.33378 0.666892 0.745154i \(-0.267624\pi\)
0.666892 + 0.745154i \(0.267624\pi\)
\(42\) 0 0
\(43\) −3.56165 −0.543146 −0.271573 0.962418i \(-0.587544\pi\)
−0.271573 + 0.962418i \(0.587544\pi\)
\(44\) 1.08512 0.163588
\(45\) 0 0
\(46\) −2.05571 −0.303098
\(47\) −11.5640 −1.68678 −0.843391 0.537300i \(-0.819444\pi\)
−0.843391 + 0.537300i \(0.819444\pi\)
\(48\) 0 0
\(49\) 3.73698 0.533854
\(50\) 0 0
\(51\) 0 0
\(52\) −2.62931 −0.364620
\(53\) −9.66312 −1.32733 −0.663666 0.748029i \(-0.731000\pi\)
−0.663666 + 0.748029i \(0.731000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −10.0121 −1.33793
\(57\) 0 0
\(58\) 1.23642 0.162350
\(59\) 9.83385 1.28026 0.640129 0.768268i \(-0.278881\pi\)
0.640129 + 0.768268i \(0.278881\pi\)
\(60\) 0 0
\(61\) 5.42714 0.694874 0.347437 0.937703i \(-0.387052\pi\)
0.347437 + 0.937703i \(0.387052\pi\)
\(62\) −2.11276 −0.268320
\(63\) 0 0
\(64\) 8.89204 1.11150
\(65\) 0 0
\(66\) 0 0
\(67\) 5.20114 0.635421 0.317710 0.948188i \(-0.397086\pi\)
0.317710 + 0.948188i \(0.397086\pi\)
\(68\) −0.914391 −0.110886
\(69\) 0 0
\(70\) 0 0
\(71\) 6.02596 0.715150 0.357575 0.933884i \(-0.383604\pi\)
0.357575 + 0.933884i \(0.383604\pi\)
\(72\) 0 0
\(73\) 15.5143 1.81581 0.907905 0.419177i \(-0.137681\pi\)
0.907905 + 0.419177i \(0.137681\pi\)
\(74\) 11.3334 1.31749
\(75\) 0 0
\(76\) 1.69257 0.194151
\(77\) 7.54498 0.859830
\(78\) 0 0
\(79\) 12.2465 1.37784 0.688922 0.724835i \(-0.258084\pi\)
0.688922 + 0.724835i \(0.258084\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −10.5595 −1.16610
\(83\) −10.7179 −1.17644 −0.588221 0.808700i \(-0.700172\pi\)
−0.588221 + 0.808700i \(0.700172\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.40370 0.474863
\(87\) 0 0
\(88\) −7.03562 −0.749999
\(89\) 2.53756 0.268981 0.134491 0.990915i \(-0.457060\pi\)
0.134491 + 0.990915i \(0.457060\pi\)
\(90\) 0 0
\(91\) −18.2820 −1.91647
\(92\) −0.783531 −0.0816887
\(93\) 0 0
\(94\) 14.2980 1.47472
\(95\) 0 0
\(96\) 0 0
\(97\) −5.89998 −0.599053 −0.299526 0.954088i \(-0.596829\pi\)
−0.299526 + 0.954088i \(0.596829\pi\)
\(98\) −4.62048 −0.466739
\(99\) 0 0
\(100\) 0 0
\(101\) 4.33789 0.431636 0.215818 0.976434i \(-0.430758\pi\)
0.215818 + 0.976434i \(0.430758\pi\)
\(102\) 0 0
\(103\) 7.62183 0.751001 0.375501 0.926822i \(-0.377471\pi\)
0.375501 + 0.926822i \(0.377471\pi\)
\(104\) 17.0477 1.67167
\(105\) 0 0
\(106\) 11.9477 1.16046
\(107\) −4.19703 −0.405742 −0.202871 0.979205i \(-0.565027\pi\)
−0.202871 + 0.979205i \(0.565027\pi\)
\(108\) 0 0
\(109\) 4.02677 0.385695 0.192847 0.981229i \(-0.438228\pi\)
0.192847 + 0.981229i \(0.438228\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 9.29082 0.877900
\(113\) −7.40527 −0.696629 −0.348314 0.937378i \(-0.613246\pi\)
−0.348314 + 0.937378i \(0.613246\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.471260 0.0437554
\(117\) 0 0
\(118\) −12.1588 −1.11931
\(119\) −6.35788 −0.582826
\(120\) 0 0
\(121\) −5.69807 −0.518006
\(122\) −6.71024 −0.607516
\(123\) 0 0
\(124\) −0.805274 −0.0723157
\(125\) 0 0
\(126\) 0 0
\(127\) 12.9643 1.15040 0.575199 0.818014i \(-0.304925\pi\)
0.575199 + 0.818014i \(0.304925\pi\)
\(128\) −5.78371 −0.511213
\(129\) 0 0
\(130\) 0 0
\(131\) −17.6941 −1.54594 −0.772971 0.634442i \(-0.781230\pi\)
−0.772971 + 0.634442i \(0.781230\pi\)
\(132\) 0 0
\(133\) 11.7686 1.02047
\(134\) −6.43081 −0.555537
\(135\) 0 0
\(136\) 5.92866 0.508378
\(137\) 22.5543 1.92694 0.963471 0.267813i \(-0.0863010\pi\)
0.963471 + 0.267813i \(0.0863010\pi\)
\(138\) 0 0
\(139\) −3.20075 −0.271484 −0.135742 0.990744i \(-0.543342\pi\)
−0.135742 + 0.990744i \(0.543342\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −7.45064 −0.625243
\(143\) −12.8469 −1.07431
\(144\) 0 0
\(145\) 0 0
\(146\) −19.1822 −1.58753
\(147\) 0 0
\(148\) 4.31973 0.355079
\(149\) 0.612365 0.0501669 0.0250835 0.999685i \(-0.492015\pi\)
0.0250835 + 0.999685i \(0.492015\pi\)
\(150\) 0 0
\(151\) 12.4363 1.01205 0.506026 0.862518i \(-0.331114\pi\)
0.506026 + 0.862518i \(0.331114\pi\)
\(152\) −10.9741 −0.890121
\(153\) 0 0
\(154\) −9.32878 −0.751734
\(155\) 0 0
\(156\) 0 0
\(157\) −0.378638 −0.0302186 −0.0151093 0.999886i \(-0.504810\pi\)
−0.0151093 + 0.999886i \(0.504810\pi\)
\(158\) −15.1419 −1.20463
\(159\) 0 0
\(160\) 0 0
\(161\) −5.44799 −0.429362
\(162\) 0 0
\(163\) 2.54446 0.199297 0.0996487 0.995023i \(-0.468228\pi\)
0.0996487 + 0.995023i \(0.468228\pi\)
\(164\) −4.02474 −0.314280
\(165\) 0 0
\(166\) 13.2519 1.02854
\(167\) 10.7799 0.834175 0.417088 0.908866i \(-0.363051\pi\)
0.417088 + 0.908866i \(0.363051\pi\)
\(168\) 0 0
\(169\) 18.1289 1.39453
\(170\) 0 0
\(171\) 0 0
\(172\) 1.67846 0.127982
\(173\) −23.1221 −1.75794 −0.878970 0.476877i \(-0.841769\pi\)
−0.878970 + 0.476877i \(0.841769\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.52875 0.492123
\(177\) 0 0
\(178\) −3.13750 −0.235165
\(179\) −23.7081 −1.77203 −0.886014 0.463659i \(-0.846536\pi\)
−0.886014 + 0.463659i \(0.846536\pi\)
\(180\) 0 0
\(181\) 11.6809 0.868233 0.434116 0.900857i \(-0.357061\pi\)
0.434116 + 0.900857i \(0.357061\pi\)
\(182\) 22.6042 1.67554
\(183\) 0 0
\(184\) 5.08020 0.374517
\(185\) 0 0
\(186\) 0 0
\(187\) −4.46774 −0.326714
\(188\) 5.44965 0.397457
\(189\) 0 0
\(190\) 0 0
\(191\) −8.68267 −0.628256 −0.314128 0.949381i \(-0.601712\pi\)
−0.314128 + 0.949381i \(0.601712\pi\)
\(192\) 0 0
\(193\) −2.73727 −0.197033 −0.0985164 0.995135i \(-0.531410\pi\)
−0.0985164 + 0.995135i \(0.531410\pi\)
\(194\) 7.29487 0.523741
\(195\) 0 0
\(196\) −1.76109 −0.125792
\(197\) −26.2457 −1.86993 −0.934964 0.354741i \(-0.884569\pi\)
−0.934964 + 0.354741i \(0.884569\pi\)
\(198\) 0 0
\(199\) 9.94800 0.705195 0.352597 0.935775i \(-0.385299\pi\)
0.352597 + 0.935775i \(0.385299\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −5.36347 −0.377372
\(203\) 3.27673 0.229982
\(204\) 0 0
\(205\) 0 0
\(206\) −9.42380 −0.656587
\(207\) 0 0
\(208\) −15.8196 −1.09689
\(209\) 8.26994 0.572044
\(210\) 0 0
\(211\) 7.76180 0.534344 0.267172 0.963649i \(-0.413911\pi\)
0.267172 + 0.963649i \(0.413911\pi\)
\(212\) 4.55384 0.312759
\(213\) 0 0
\(214\) 5.18930 0.354733
\(215\) 0 0
\(216\) 0 0
\(217\) −5.59917 −0.380096
\(218\) −4.97879 −0.337206
\(219\) 0 0
\(220\) 0 0
\(221\) 10.8256 0.728210
\(222\) 0 0
\(223\) −9.63976 −0.645526 −0.322763 0.946480i \(-0.604612\pi\)
−0.322763 + 0.946480i \(0.604612\pi\)
\(224\) 8.53687 0.570394
\(225\) 0 0
\(226\) 9.15603 0.609050
\(227\) −9.21807 −0.611825 −0.305913 0.952060i \(-0.598962\pi\)
−0.305913 + 0.952060i \(0.598962\pi\)
\(228\) 0 0
\(229\) −6.89586 −0.455691 −0.227846 0.973697i \(-0.573168\pi\)
−0.227846 + 0.973697i \(0.573168\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.05552 −0.200605
\(233\) 13.6229 0.892466 0.446233 0.894917i \(-0.352765\pi\)
0.446233 + 0.894917i \(0.352765\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.63430 −0.301667
\(237\) 0 0
\(238\) 7.86102 0.509554
\(239\) 0.749144 0.0484581 0.0242290 0.999706i \(-0.492287\pi\)
0.0242290 + 0.999706i \(0.492287\pi\)
\(240\) 0 0
\(241\) −12.5380 −0.807645 −0.403822 0.914837i \(-0.632319\pi\)
−0.403822 + 0.914837i \(0.632319\pi\)
\(242\) 7.04522 0.452884
\(243\) 0 0
\(244\) −2.55760 −0.163733
\(245\) 0 0
\(246\) 0 0
\(247\) −20.0386 −1.27503
\(248\) 5.22117 0.331545
\(249\) 0 0
\(250\) 0 0
\(251\) −11.7441 −0.741281 −0.370640 0.928776i \(-0.620862\pi\)
−0.370640 + 0.928776i \(0.620862\pi\)
\(252\) 0 0
\(253\) −3.82836 −0.240687
\(254\) −16.0294 −1.00577
\(255\) 0 0
\(256\) −10.6330 −0.664560
\(257\) −13.9304 −0.868955 −0.434477 0.900683i \(-0.643067\pi\)
−0.434477 + 0.900683i \(0.643067\pi\)
\(258\) 0 0
\(259\) 30.0356 1.86632
\(260\) 0 0
\(261\) 0 0
\(262\) 21.8774 1.35159
\(263\) −25.8059 −1.59126 −0.795631 0.605781i \(-0.792861\pi\)
−0.795631 + 0.605781i \(0.792861\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −14.5510 −0.892180
\(267\) 0 0
\(268\) −2.45109 −0.149724
\(269\) 11.4590 0.698667 0.349333 0.936998i \(-0.386408\pi\)
0.349333 + 0.936998i \(0.386408\pi\)
\(270\) 0 0
\(271\) −21.4783 −1.30472 −0.652358 0.757911i \(-0.726220\pi\)
−0.652358 + 0.757911i \(0.726220\pi\)
\(272\) −5.50154 −0.333580
\(273\) 0 0
\(274\) −27.8866 −1.68469
\(275\) 0 0
\(276\) 0 0
\(277\) −7.16907 −0.430747 −0.215374 0.976532i \(-0.569097\pi\)
−0.215374 + 0.976532i \(0.569097\pi\)
\(278\) 3.95748 0.237354
\(279\) 0 0
\(280\) 0 0
\(281\) −14.0042 −0.835420 −0.417710 0.908580i \(-0.637167\pi\)
−0.417710 + 0.908580i \(0.637167\pi\)
\(282\) 0 0
\(283\) −10.3513 −0.615324 −0.307662 0.951496i \(-0.599547\pi\)
−0.307662 + 0.951496i \(0.599547\pi\)
\(284\) −2.83980 −0.168511
\(285\) 0 0
\(286\) 15.8842 0.939253
\(287\) −27.9846 −1.65188
\(288\) 0 0
\(289\) −13.2352 −0.778541
\(290\) 0 0
\(291\) 0 0
\(292\) −7.31127 −0.427859
\(293\) −26.6063 −1.55435 −0.777177 0.629282i \(-0.783349\pi\)
−0.777177 + 0.629282i \(0.783349\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −28.0079 −1.62793
\(297\) 0 0
\(298\) −0.757142 −0.0438601
\(299\) 9.27634 0.536465
\(300\) 0 0
\(301\) 11.6706 0.672680
\(302\) −15.3765 −0.884819
\(303\) 0 0
\(304\) 10.1835 0.584066
\(305\) 0 0
\(306\) 0 0
\(307\) −29.2808 −1.67114 −0.835571 0.549383i \(-0.814863\pi\)
−0.835571 + 0.549383i \(0.814863\pi\)
\(308\) −3.55565 −0.202602
\(309\) 0 0
\(310\) 0 0
\(311\) 5.03104 0.285284 0.142642 0.989774i \(-0.454440\pi\)
0.142642 + 0.989774i \(0.454440\pi\)
\(312\) 0 0
\(313\) 27.5875 1.55934 0.779670 0.626191i \(-0.215387\pi\)
0.779670 + 0.626191i \(0.215387\pi\)
\(314\) 0.468157 0.0264196
\(315\) 0 0
\(316\) −5.77131 −0.324662
\(317\) 7.70663 0.432848 0.216424 0.976300i \(-0.430561\pi\)
0.216424 + 0.976300i \(0.430561\pi\)
\(318\) 0 0
\(319\) 2.30259 0.128920
\(320\) 0 0
\(321\) 0 0
\(322\) 6.73601 0.375383
\(323\) −6.96878 −0.387753
\(324\) 0 0
\(325\) 0 0
\(326\) −3.14602 −0.174242
\(327\) 0 0
\(328\) 26.0953 1.44087
\(329\) 37.8921 2.08906
\(330\) 0 0
\(331\) 18.9943 1.04402 0.522010 0.852940i \(-0.325182\pi\)
0.522010 + 0.852940i \(0.325182\pi\)
\(332\) 5.05092 0.277205
\(333\) 0 0
\(334\) −13.3285 −0.729305
\(335\) 0 0
\(336\) 0 0
\(337\) 20.1983 1.10027 0.550136 0.835075i \(-0.314576\pi\)
0.550136 + 0.835075i \(0.314576\pi\)
\(338\) −22.4149 −1.21921
\(339\) 0 0
\(340\) 0 0
\(341\) −3.93459 −0.213070
\(342\) 0 0
\(343\) 10.6921 0.577317
\(344\) −10.8827 −0.586755
\(345\) 0 0
\(346\) 28.5887 1.53694
\(347\) −23.7995 −1.27763 −0.638813 0.769362i \(-0.720574\pi\)
−0.638813 + 0.769362i \(0.720574\pi\)
\(348\) 0 0
\(349\) −22.8554 −1.22342 −0.611710 0.791082i \(-0.709518\pi\)
−0.611710 + 0.791082i \(0.709518\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.99894 0.319745
\(353\) 17.5338 0.933231 0.466615 0.884460i \(-0.345473\pi\)
0.466615 + 0.884460i \(0.345473\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.19585 −0.0633801
\(357\) 0 0
\(358\) 29.3132 1.54925
\(359\) −26.6594 −1.40703 −0.703513 0.710682i \(-0.748387\pi\)
−0.703513 + 0.710682i \(0.748387\pi\)
\(360\) 0 0
\(361\) −6.10055 −0.321082
\(362\) −14.4425 −0.759081
\(363\) 0 0
\(364\) 8.61556 0.451578
\(365\) 0 0
\(366\) 0 0
\(367\) 22.5869 1.17902 0.589512 0.807760i \(-0.299320\pi\)
0.589512 + 0.807760i \(0.299320\pi\)
\(368\) −4.71420 −0.245745
\(369\) 0 0
\(370\) 0 0
\(371\) 31.6634 1.64388
\(372\) 0 0
\(373\) −11.0978 −0.574623 −0.287311 0.957837i \(-0.592761\pi\)
−0.287311 + 0.957837i \(0.592761\pi\)
\(374\) 5.52402 0.285640
\(375\) 0 0
\(376\) −35.3340 −1.82221
\(377\) −5.57932 −0.287350
\(378\) 0 0
\(379\) −28.2474 −1.45097 −0.725485 0.688238i \(-0.758384\pi\)
−0.725485 + 0.688238i \(0.758384\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 10.7354 0.549273
\(383\) −7.24537 −0.370221 −0.185110 0.982718i \(-0.559264\pi\)
−0.185110 + 0.982718i \(0.559264\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.38442 0.172262
\(387\) 0 0
\(388\) 2.78043 0.141155
\(389\) 34.4482 1.74659 0.873296 0.487190i \(-0.161978\pi\)
0.873296 + 0.487190i \(0.161978\pi\)
\(390\) 0 0
\(391\) 3.22601 0.163147
\(392\) 11.4184 0.576717
\(393\) 0 0
\(394\) 32.4508 1.63485
\(395\) 0 0
\(396\) 0 0
\(397\) 13.6412 0.684632 0.342316 0.939585i \(-0.388789\pi\)
0.342316 + 0.939585i \(0.388789\pi\)
\(398\) −12.2999 −0.616540
\(399\) 0 0
\(400\) 0 0
\(401\) 10.3741 0.518059 0.259029 0.965869i \(-0.416597\pi\)
0.259029 + 0.965869i \(0.416597\pi\)
\(402\) 0 0
\(403\) 9.53376 0.474910
\(404\) −2.04428 −0.101707
\(405\) 0 0
\(406\) −4.05142 −0.201069
\(407\) 21.1063 1.04620
\(408\) 0 0
\(409\) 38.5394 1.90565 0.952825 0.303520i \(-0.0981620\pi\)
0.952825 + 0.303520i \(0.0981620\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.59187 −0.176959
\(413\) −32.2229 −1.58558
\(414\) 0 0
\(415\) 0 0
\(416\) −14.5358 −0.712677
\(417\) 0 0
\(418\) −10.2251 −0.500128
\(419\) 9.87578 0.482464 0.241232 0.970468i \(-0.422449\pi\)
0.241232 + 0.970468i \(0.422449\pi\)
\(420\) 0 0
\(421\) −10.2857 −0.501292 −0.250646 0.968079i \(-0.580643\pi\)
−0.250646 + 0.968079i \(0.580643\pi\)
\(422\) −9.59686 −0.467168
\(423\) 0 0
\(424\) −29.5259 −1.43390
\(425\) 0 0
\(426\) 0 0
\(427\) −17.7833 −0.860594
\(428\) 1.97789 0.0956051
\(429\) 0 0
\(430\) 0 0
\(431\) −15.2705 −0.735553 −0.367776 0.929914i \(-0.619881\pi\)
−0.367776 + 0.929914i \(0.619881\pi\)
\(432\) 0 0
\(433\) 13.8370 0.664966 0.332483 0.943109i \(-0.392114\pi\)
0.332483 + 0.943109i \(0.392114\pi\)
\(434\) 6.92294 0.332312
\(435\) 0 0
\(436\) −1.89766 −0.0908813
\(437\) −5.97146 −0.285654
\(438\) 0 0
\(439\) −6.48836 −0.309673 −0.154836 0.987940i \(-0.549485\pi\)
−0.154836 + 0.987940i \(0.549485\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −13.3850 −0.636661
\(443\) −22.1026 −1.05012 −0.525062 0.851064i \(-0.675958\pi\)
−0.525062 + 0.851064i \(0.675958\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 11.9188 0.564372
\(447\) 0 0
\(448\) −29.1368 −1.37659
\(449\) 0.247203 0.0116662 0.00583311 0.999983i \(-0.498143\pi\)
0.00583311 + 0.999983i \(0.498143\pi\)
\(450\) 0 0
\(451\) −19.6650 −0.925990
\(452\) 3.48981 0.164147
\(453\) 0 0
\(454\) 11.3974 0.534908
\(455\) 0 0
\(456\) 0 0
\(457\) −35.4153 −1.65666 −0.828328 0.560243i \(-0.810708\pi\)
−0.828328 + 0.560243i \(0.810708\pi\)
\(458\) 8.52619 0.398403
\(459\) 0 0
\(460\) 0 0
\(461\) 13.5690 0.631969 0.315985 0.948764i \(-0.397665\pi\)
0.315985 + 0.948764i \(0.397665\pi\)
\(462\) 0 0
\(463\) −7.88217 −0.366316 −0.183158 0.983084i \(-0.558632\pi\)
−0.183158 + 0.983084i \(0.558632\pi\)
\(464\) 2.83539 0.131630
\(465\) 0 0
\(466\) −16.8437 −0.780268
\(467\) −20.8411 −0.964411 −0.482206 0.876058i \(-0.660164\pi\)
−0.482206 + 0.876058i \(0.660164\pi\)
\(468\) 0 0
\(469\) −17.0428 −0.786961
\(470\) 0 0
\(471\) 0 0
\(472\) 30.0475 1.38305
\(473\) 8.20102 0.377083
\(474\) 0 0
\(475\) 0 0
\(476\) 2.99622 0.137331
\(477\) 0 0
\(478\) −0.926258 −0.0423661
\(479\) 28.9598 1.32321 0.661604 0.749853i \(-0.269876\pi\)
0.661604 + 0.749853i \(0.269876\pi\)
\(480\) 0 0
\(481\) −51.1419 −2.33187
\(482\) 15.5023 0.706110
\(483\) 0 0
\(484\) 2.68527 0.122058
\(485\) 0 0
\(486\) 0 0
\(487\) −23.3354 −1.05743 −0.528714 0.848800i \(-0.677326\pi\)
−0.528714 + 0.848800i \(0.677326\pi\)
\(488\) 16.5827 0.750665
\(489\) 0 0
\(490\) 0 0
\(491\) −25.9951 −1.17314 −0.586570 0.809898i \(-0.699522\pi\)
−0.586570 + 0.809898i \(0.699522\pi\)
\(492\) 0 0
\(493\) −1.94031 −0.0873872
\(494\) 24.7762 1.11473
\(495\) 0 0
\(496\) −4.84502 −0.217548
\(497\) −19.7455 −0.885706
\(498\) 0 0
\(499\) −31.9575 −1.43061 −0.715307 0.698810i \(-0.753713\pi\)
−0.715307 + 0.698810i \(0.753713\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 14.5207 0.648089
\(503\) −4.75172 −0.211869 −0.105934 0.994373i \(-0.533783\pi\)
−0.105934 + 0.994373i \(0.533783\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 4.73346 0.210428
\(507\) 0 0
\(508\) −6.10957 −0.271068
\(509\) 10.8427 0.480595 0.240297 0.970699i \(-0.422755\pi\)
0.240297 + 0.970699i \(0.422755\pi\)
\(510\) 0 0
\(511\) −50.8361 −2.24886
\(512\) 24.7143 1.09223
\(513\) 0 0
\(514\) 17.2239 0.759712
\(515\) 0 0
\(516\) 0 0
\(517\) 26.6272 1.17106
\(518\) −37.1367 −1.63169
\(519\) 0 0
\(520\) 0 0
\(521\) 33.8929 1.48487 0.742437 0.669916i \(-0.233670\pi\)
0.742437 + 0.669916i \(0.233670\pi\)
\(522\) 0 0
\(523\) −25.7652 −1.12664 −0.563318 0.826240i \(-0.690475\pi\)
−0.563318 + 0.826240i \(0.690475\pi\)
\(524\) 8.33853 0.364270
\(525\) 0 0
\(526\) 31.9070 1.39121
\(527\) 3.31554 0.144427
\(528\) 0 0
\(529\) −20.2357 −0.879812
\(530\) 0 0
\(531\) 0 0
\(532\) −5.54610 −0.240454
\(533\) 47.6496 2.06393
\(534\) 0 0
\(535\) 0 0
\(536\) 15.8922 0.686438
\(537\) 0 0
\(538\) −14.1681 −0.610832
\(539\) −8.60473 −0.370632
\(540\) 0 0
\(541\) 4.96888 0.213629 0.106814 0.994279i \(-0.465935\pi\)
0.106814 + 0.994279i \(0.465935\pi\)
\(542\) 26.5563 1.14069
\(543\) 0 0
\(544\) −5.05509 −0.216735
\(545\) 0 0
\(546\) 0 0
\(547\) 9.03353 0.386246 0.193123 0.981175i \(-0.438138\pi\)
0.193123 + 0.981175i \(0.438138\pi\)
\(548\) −10.6289 −0.454046
\(549\) 0 0
\(550\) 0 0
\(551\) 3.59158 0.153006
\(552\) 0 0
\(553\) −40.1287 −1.70644
\(554\) 8.86399 0.376595
\(555\) 0 0
\(556\) 1.50839 0.0639699
\(557\) −22.5265 −0.954477 −0.477238 0.878774i \(-0.658362\pi\)
−0.477238 + 0.878774i \(0.658362\pi\)
\(558\) 0 0
\(559\) −19.8716 −0.840479
\(560\) 0 0
\(561\) 0 0
\(562\) 17.3151 0.730393
\(563\) −14.0601 −0.592561 −0.296280 0.955101i \(-0.595746\pi\)
−0.296280 + 0.955101i \(0.595746\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 12.7986 0.537967
\(567\) 0 0
\(568\) 18.4125 0.772570
\(569\) −5.59236 −0.234444 −0.117222 0.993106i \(-0.537399\pi\)
−0.117222 + 0.993106i \(0.537399\pi\)
\(570\) 0 0
\(571\) −20.9909 −0.878443 −0.439221 0.898379i \(-0.644746\pi\)
−0.439221 + 0.898379i \(0.644746\pi\)
\(572\) 6.05424 0.253140
\(573\) 0 0
\(574\) 34.6007 1.44421
\(575\) 0 0
\(576\) 0 0
\(577\) −25.7139 −1.07048 −0.535242 0.844699i \(-0.679779\pi\)
−0.535242 + 0.844699i \(0.679779\pi\)
\(578\) 16.3643 0.680665
\(579\) 0 0
\(580\) 0 0
\(581\) 35.1197 1.45701
\(582\) 0 0
\(583\) 22.2502 0.921510
\(584\) 47.4042 1.96160
\(585\) 0 0
\(586\) 32.8966 1.35894
\(587\) 9.26970 0.382602 0.191301 0.981531i \(-0.438729\pi\)
0.191301 + 0.981531i \(0.438729\pi\)
\(588\) 0 0
\(589\) −6.13717 −0.252878
\(590\) 0 0
\(591\) 0 0
\(592\) 25.9901 1.06819
\(593\) 31.5912 1.29729 0.648647 0.761089i \(-0.275335\pi\)
0.648647 + 0.761089i \(0.275335\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.288584 −0.0118208
\(597\) 0 0
\(598\) −11.4695 −0.469022
\(599\) −13.1545 −0.537480 −0.268740 0.963213i \(-0.586607\pi\)
−0.268740 + 0.963213i \(0.586607\pi\)
\(600\) 0 0
\(601\) 34.2150 1.39566 0.697829 0.716265i \(-0.254150\pi\)
0.697829 + 0.716265i \(0.254150\pi\)
\(602\) −14.4297 −0.588113
\(603\) 0 0
\(604\) −5.86073 −0.238470
\(605\) 0 0
\(606\) 0 0
\(607\) 3.51939 0.142848 0.0714238 0.997446i \(-0.477246\pi\)
0.0714238 + 0.997446i \(0.477246\pi\)
\(608\) 9.35714 0.379482
\(609\) 0 0
\(610\) 0 0
\(611\) −64.5193 −2.61017
\(612\) 0 0
\(613\) −5.90592 −0.238538 −0.119269 0.992862i \(-0.538055\pi\)
−0.119269 + 0.992862i \(0.538055\pi\)
\(614\) 36.2034 1.46105
\(615\) 0 0
\(616\) 23.0538 0.928866
\(617\) −30.3173 −1.22053 −0.610264 0.792198i \(-0.708937\pi\)
−0.610264 + 0.792198i \(0.708937\pi\)
\(618\) 0 0
\(619\) −23.1044 −0.928646 −0.464323 0.885666i \(-0.653702\pi\)
−0.464323 + 0.885666i \(0.653702\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.22049 −0.249419
\(623\) −8.31491 −0.333130
\(624\) 0 0
\(625\) 0 0
\(626\) −34.1098 −1.36330
\(627\) 0 0
\(628\) 0.178437 0.00712042
\(629\) −17.7855 −0.709155
\(630\) 0 0
\(631\) −27.6228 −1.09965 −0.549824 0.835280i \(-0.685305\pi\)
−0.549824 + 0.835280i \(0.685305\pi\)
\(632\) 37.4196 1.48847
\(633\) 0 0
\(634\) −9.52865 −0.378431
\(635\) 0 0
\(636\) 0 0
\(637\) 20.8498 0.826099
\(638\) −2.84698 −0.112713
\(639\) 0 0
\(640\) 0 0
\(641\) 10.7286 0.423756 0.211878 0.977296i \(-0.432042\pi\)
0.211878 + 0.977296i \(0.432042\pi\)
\(642\) 0 0
\(643\) −18.0482 −0.711751 −0.355876 0.934533i \(-0.615817\pi\)
−0.355876 + 0.934533i \(0.615817\pi\)
\(644\) 2.56742 0.101171
\(645\) 0 0
\(646\) 8.61635 0.339006
\(647\) −26.9395 −1.05910 −0.529551 0.848278i \(-0.677640\pi\)
−0.529551 + 0.848278i \(0.677640\pi\)
\(648\) 0 0
\(649\) −22.6433 −0.888829
\(650\) 0 0
\(651\) 0 0
\(652\) −1.19910 −0.0469605
\(653\) −13.4857 −0.527737 −0.263868 0.964559i \(-0.584998\pi\)
−0.263868 + 0.964559i \(0.584998\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −24.2153 −0.945450
\(657\) 0 0
\(658\) −46.8507 −1.82643
\(659\) −27.5560 −1.07343 −0.536716 0.843763i \(-0.680335\pi\)
−0.536716 + 0.843763i \(0.680335\pi\)
\(660\) 0 0
\(661\) 18.1148 0.704583 0.352292 0.935890i \(-0.385403\pi\)
0.352292 + 0.935890i \(0.385403\pi\)
\(662\) −23.4849 −0.912768
\(663\) 0 0
\(664\) −32.7488 −1.27090
\(665\) 0 0
\(666\) 0 0
\(667\) −1.66263 −0.0643772
\(668\) −5.08015 −0.196557
\(669\) 0 0
\(670\) 0 0
\(671\) −12.4965 −0.482422
\(672\) 0 0
\(673\) −45.5751 −1.75679 −0.878396 0.477933i \(-0.841386\pi\)
−0.878396 + 0.477933i \(0.841386\pi\)
\(674\) −24.9736 −0.961948
\(675\) 0 0
\(676\) −8.54342 −0.328593
\(677\) −10.2385 −0.393497 −0.196748 0.980454i \(-0.563038\pi\)
−0.196748 + 0.980454i \(0.563038\pi\)
\(678\) 0 0
\(679\) 19.3327 0.741920
\(680\) 0 0
\(681\) 0 0
\(682\) 4.86482 0.186283
\(683\) 41.9562 1.60541 0.802705 0.596376i \(-0.203393\pi\)
0.802705 + 0.596376i \(0.203393\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.2199 −0.504738
\(687\) 0 0
\(688\) 10.0987 0.385008
\(689\) −53.9137 −2.05395
\(690\) 0 0
\(691\) 50.6633 1.92732 0.963662 0.267126i \(-0.0860739\pi\)
0.963662 + 0.267126i \(0.0860739\pi\)
\(692\) 10.8965 0.414224
\(693\) 0 0
\(694\) 29.4263 1.11701
\(695\) 0 0
\(696\) 0 0
\(697\) 16.5710 0.627671
\(698\) 28.2589 1.06961
\(699\) 0 0
\(700\) 0 0
\(701\) 17.2666 0.652151 0.326076 0.945344i \(-0.394274\pi\)
0.326076 + 0.945344i \(0.394274\pi\)
\(702\) 0 0
\(703\) 32.9216 1.24166
\(704\) −20.4747 −0.771671
\(705\) 0 0
\(706\) −21.6792 −0.815907
\(707\) −14.2141 −0.534577
\(708\) 0 0
\(709\) −18.3307 −0.688425 −0.344213 0.938892i \(-0.611854\pi\)
−0.344213 + 0.938892i \(0.611854\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 7.75357 0.290577
\(713\) 2.84104 0.106398
\(714\) 0 0
\(715\) 0 0
\(716\) 11.1727 0.417543
\(717\) 0 0
\(718\) 32.9622 1.23014
\(719\) −10.6931 −0.398785 −0.199392 0.979920i \(-0.563897\pi\)
−0.199392 + 0.979920i \(0.563897\pi\)
\(720\) 0 0
\(721\) −24.9747 −0.930107
\(722\) 7.54286 0.280716
\(723\) 0 0
\(724\) −5.50474 −0.204582
\(725\) 0 0
\(726\) 0 0
\(727\) −5.74209 −0.212962 −0.106481 0.994315i \(-0.533958\pi\)
−0.106481 + 0.994315i \(0.533958\pi\)
\(728\) −55.8609 −2.07034
\(729\) 0 0
\(730\) 0 0
\(731\) −6.91070 −0.255601
\(732\) 0 0
\(733\) 6.03079 0.222752 0.111376 0.993778i \(-0.464474\pi\)
0.111376 + 0.993778i \(0.464474\pi\)
\(734\) −27.9269 −1.03080
\(735\) 0 0
\(736\) −4.33164 −0.159667
\(737\) −11.9761 −0.441146
\(738\) 0 0
\(739\) 23.7939 0.875273 0.437637 0.899152i \(-0.355816\pi\)
0.437637 + 0.899152i \(0.355816\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −39.1494 −1.43722
\(743\) −24.4388 −0.896574 −0.448287 0.893890i \(-0.647966\pi\)
−0.448287 + 0.893890i \(0.647966\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 13.7216 0.502383
\(747\) 0 0
\(748\) 2.10547 0.0769836
\(749\) 13.7525 0.502507
\(750\) 0 0
\(751\) 38.9811 1.42244 0.711220 0.702970i \(-0.248143\pi\)
0.711220 + 0.702970i \(0.248143\pi\)
\(752\) 32.7885 1.19567
\(753\) 0 0
\(754\) 6.89840 0.251225
\(755\) 0 0
\(756\) 0 0
\(757\) −6.78690 −0.246674 −0.123337 0.992365i \(-0.539360\pi\)
−0.123337 + 0.992365i \(0.539360\pi\)
\(758\) 34.9257 1.26856
\(759\) 0 0
\(760\) 0 0
\(761\) 8.25388 0.299203 0.149601 0.988746i \(-0.452201\pi\)
0.149601 + 0.988746i \(0.452201\pi\)
\(762\) 0 0
\(763\) −13.1946 −0.477678
\(764\) 4.09180 0.148036
\(765\) 0 0
\(766\) 8.95833 0.323678
\(767\) 54.8662 1.98110
\(768\) 0 0
\(769\) −12.3938 −0.446930 −0.223465 0.974712i \(-0.571737\pi\)
−0.223465 + 0.974712i \(0.571737\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.28997 0.0464269
\(773\) 38.1311 1.37148 0.685740 0.727846i \(-0.259479\pi\)
0.685740 + 0.727846i \(0.259479\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −18.0275 −0.647150
\(777\) 0 0
\(778\) −42.5925 −1.52701
\(779\) −30.6735 −1.09899
\(780\) 0 0
\(781\) −13.8753 −0.496499
\(782\) −3.98872 −0.142636
\(783\) 0 0
\(784\) −10.5958 −0.378421
\(785\) 0 0
\(786\) 0 0
\(787\) 37.5823 1.33967 0.669833 0.742512i \(-0.266366\pi\)
0.669833 + 0.742512i \(0.266366\pi\)
\(788\) 12.3686 0.440612
\(789\) 0 0
\(790\) 0 0
\(791\) 24.2651 0.862767
\(792\) 0 0
\(793\) 30.2798 1.07527
\(794\) −16.8663 −0.598561
\(795\) 0 0
\(796\) −4.68810 −0.166165
\(797\) 14.9045 0.527945 0.263972 0.964530i \(-0.414967\pi\)
0.263972 + 0.964530i \(0.414967\pi\)
\(798\) 0 0
\(799\) −22.4377 −0.793790
\(800\) 0 0
\(801\) 0 0
\(802\) −12.8268 −0.452930
\(803\) −35.7231 −1.26064
\(804\) 0 0
\(805\) 0 0
\(806\) −11.7878 −0.415206
\(807\) 0 0
\(808\) 13.2545 0.466292
\(809\) 12.5838 0.442424 0.221212 0.975226i \(-0.428999\pi\)
0.221212 + 0.975226i \(0.428999\pi\)
\(810\) 0 0
\(811\) −27.9894 −0.982841 −0.491420 0.870922i \(-0.663522\pi\)
−0.491420 + 0.870922i \(0.663522\pi\)
\(812\) −1.54419 −0.0541906
\(813\) 0 0
\(814\) −26.0963 −0.914675
\(815\) 0 0
\(816\) 0 0
\(817\) 12.7919 0.447533
\(818\) −47.6509 −1.66608
\(819\) 0 0
\(820\) 0 0
\(821\) 1.20609 0.0420929 0.0210464 0.999778i \(-0.493300\pi\)
0.0210464 + 0.999778i \(0.493300\pi\)
\(822\) 0 0
\(823\) −25.7664 −0.898162 −0.449081 0.893491i \(-0.648249\pi\)
−0.449081 + 0.893491i \(0.648249\pi\)
\(824\) 23.2887 0.811299
\(825\) 0 0
\(826\) 39.8411 1.38625
\(827\) −56.1443 −1.95233 −0.976165 0.217029i \(-0.930363\pi\)
−0.976165 + 0.217029i \(0.930363\pi\)
\(828\) 0 0
\(829\) −14.6534 −0.508935 −0.254467 0.967081i \(-0.581900\pi\)
−0.254467 + 0.967081i \(0.581900\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 49.6116 1.71997
\(833\) 7.25089 0.251229
\(834\) 0 0
\(835\) 0 0
\(836\) −3.89730 −0.134791
\(837\) 0 0
\(838\) −12.2106 −0.421809
\(839\) −8.72873 −0.301349 −0.150675 0.988583i \(-0.548145\pi\)
−0.150675 + 0.988583i \(0.548145\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 12.7174 0.438271
\(843\) 0 0
\(844\) −3.65783 −0.125908
\(845\) 0 0
\(846\) 0 0
\(847\) 18.6710 0.641545
\(848\) 27.3987 0.940876
\(849\) 0 0
\(850\) 0 0
\(851\) −15.2402 −0.522427
\(852\) 0 0
\(853\) −42.3533 −1.45015 −0.725075 0.688670i \(-0.758195\pi\)
−0.725075 + 0.688670i \(0.758195\pi\)
\(854\) 21.9876 0.752402
\(855\) 0 0
\(856\) −12.8241 −0.438319
\(857\) −49.8924 −1.70429 −0.852145 0.523305i \(-0.824699\pi\)
−0.852145 + 0.523305i \(0.824699\pi\)
\(858\) 0 0
\(859\) −33.4585 −1.14159 −0.570795 0.821092i \(-0.693365\pi\)
−0.570795 + 0.821092i \(0.693365\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 18.8807 0.643081
\(863\) −36.8016 −1.25274 −0.626371 0.779525i \(-0.715460\pi\)
−0.626371 + 0.779525i \(0.715460\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −17.1084 −0.581368
\(867\) 0 0
\(868\) 2.63867 0.0895622
\(869\) −28.1988 −0.956579
\(870\) 0 0
\(871\) 29.0189 0.983267
\(872\) 12.3039 0.416662
\(873\) 0 0
\(874\) 7.38325 0.249742
\(875\) 0 0
\(876\) 0 0
\(877\) −19.7336 −0.666358 −0.333179 0.942864i \(-0.608121\pi\)
−0.333179 + 0.942864i \(0.608121\pi\)
\(878\) 8.02235 0.270741
\(879\) 0 0
\(880\) 0 0
\(881\) −38.2999 −1.29036 −0.645179 0.764032i \(-0.723217\pi\)
−0.645179 + 0.764032i \(0.723217\pi\)
\(882\) 0 0
\(883\) 32.9689 1.10949 0.554745 0.832020i \(-0.312816\pi\)
0.554745 + 0.832020i \(0.312816\pi\)
\(884\) −5.10169 −0.171588
\(885\) 0 0
\(886\) 27.3281 0.918105
\(887\) 15.0022 0.503724 0.251862 0.967763i \(-0.418957\pi\)
0.251862 + 0.967763i \(0.418957\pi\)
\(888\) 0 0
\(889\) −42.4806 −1.42475
\(890\) 0 0
\(891\) 0 0
\(892\) 4.54284 0.152105
\(893\) 41.5330 1.38985
\(894\) 0 0
\(895\) 0 0
\(896\) 18.9517 0.633131
\(897\) 0 0
\(898\) −0.305647 −0.0101996
\(899\) −1.70877 −0.0569905
\(900\) 0 0
\(901\) −18.7494 −0.624634
\(902\) 24.3143 0.809577
\(903\) 0 0
\(904\) −22.6269 −0.752561
\(905\) 0 0
\(906\) 0 0
\(907\) −45.4276 −1.50840 −0.754199 0.656646i \(-0.771974\pi\)
−0.754199 + 0.656646i \(0.771974\pi\)
\(908\) 4.34411 0.144164
\(909\) 0 0
\(910\) 0 0
\(911\) −59.7103 −1.97829 −0.989146 0.146937i \(-0.953059\pi\)
−0.989146 + 0.146937i \(0.953059\pi\)
\(912\) 0 0
\(913\) 24.6790 0.816755
\(914\) 43.7882 1.44839
\(915\) 0 0
\(916\) 3.24975 0.107375
\(917\) 57.9789 1.91463
\(918\) 0 0
\(919\) −1.59828 −0.0527224 −0.0263612 0.999652i \(-0.508392\pi\)
−0.0263612 + 0.999652i \(0.508392\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −16.7770 −0.552520
\(923\) 33.6208 1.10664
\(924\) 0 0
\(925\) 0 0
\(926\) 9.74569 0.320263
\(927\) 0 0
\(928\) 2.60530 0.0855231
\(929\) −35.5116 −1.16510 −0.582549 0.812795i \(-0.697945\pi\)
−0.582549 + 0.812795i \(0.697945\pi\)
\(930\) 0 0
\(931\) −13.4216 −0.439877
\(932\) −6.41994 −0.210292
\(933\) 0 0
\(934\) 25.7684 0.843168
\(935\) 0 0
\(936\) 0 0
\(937\) 52.7492 1.72324 0.861621 0.507552i \(-0.169449\pi\)
0.861621 + 0.507552i \(0.169449\pi\)
\(938\) 21.0720 0.688026
\(939\) 0 0
\(940\) 0 0
\(941\) 27.2935 0.889743 0.444872 0.895594i \(-0.353249\pi\)
0.444872 + 0.895594i \(0.353249\pi\)
\(942\) 0 0
\(943\) 14.1995 0.462399
\(944\) −27.8828 −0.907508
\(945\) 0 0
\(946\) −10.1399 −0.329677
\(947\) −21.1187 −0.686266 −0.343133 0.939287i \(-0.611488\pi\)
−0.343133 + 0.939287i \(0.611488\pi\)
\(948\) 0 0
\(949\) 86.5592 2.80983
\(950\) 0 0
\(951\) 0 0
\(952\) −19.4266 −0.629621
\(953\) −3.73049 −0.120842 −0.0604211 0.998173i \(-0.519244\pi\)
−0.0604211 + 0.998173i \(0.519244\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −0.353042 −0.0114182
\(957\) 0 0
\(958\) −35.8066 −1.15686
\(959\) −73.9043 −2.38650
\(960\) 0 0
\(961\) −28.0801 −0.905810
\(962\) 63.2330 2.03871
\(963\) 0 0
\(964\) 5.90867 0.190305
\(965\) 0 0
\(966\) 0 0
\(967\) 22.8453 0.734654 0.367327 0.930092i \(-0.380273\pi\)
0.367327 + 0.930092i \(0.380273\pi\)
\(968\) −17.4106 −0.559597
\(969\) 0 0
\(970\) 0 0
\(971\) 42.3917 1.36042 0.680208 0.733019i \(-0.261890\pi\)
0.680208 + 0.733019i \(0.261890\pi\)
\(972\) 0 0
\(973\) 10.4880 0.336230
\(974\) 28.8524 0.924492
\(975\) 0 0
\(976\) −15.3881 −0.492560
\(977\) 0.0842590 0.00269568 0.00134784 0.999999i \(-0.499571\pi\)
0.00134784 + 0.999999i \(0.499571\pi\)
\(978\) 0 0
\(979\) −5.84297 −0.186742
\(980\) 0 0
\(981\) 0 0
\(982\) 32.1409 1.02566
\(983\) −15.0122 −0.478813 −0.239407 0.970919i \(-0.576953\pi\)
−0.239407 + 0.970919i \(0.576953\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.39904 0.0764011
\(987\) 0 0
\(988\) 9.44339 0.300434
\(989\) −5.92170 −0.188299
\(990\) 0 0
\(991\) 3.36536 0.106904 0.0534521 0.998570i \(-0.482978\pi\)
0.0534521 + 0.998570i \(0.482978\pi\)
\(992\) −4.45185 −0.141346
\(993\) 0 0
\(994\) 24.4137 0.774357
\(995\) 0 0
\(996\) 0 0
\(997\) −1.94573 −0.0616220 −0.0308110 0.999525i \(-0.509809\pi\)
−0.0308110 + 0.999525i \(0.509809\pi\)
\(998\) 39.5130 1.25076
\(999\) 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 6525.2.a.cb.1.4 yes 9
3.2 odd 2 6525.2.a.cd.1.6 yes 9
5.4 even 2 6525.2.a.cc.1.6 yes 9
15.14 odd 2 6525.2.a.ca.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6525.2.a.ca.1.4 9 15.14 odd 2
6525.2.a.cb.1.4 yes 9 1.1 even 1 trivial
6525.2.a.cc.1.6 yes 9 5.4 even 2
6525.2.a.cd.1.6 yes 9 3.2 odd 2