Properties

Label 4012.2.a.h.1.13
Level $4012$
Weight $2$
Character 4012.1
Self dual yes
Analytic conductor $32.036$
Analytic rank $1$
Dimension $15$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} - 249 x^{6} + 2736 x^{5} - 801 x^{4} - 900 x^{3} + 429 x^{2} - 36 x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(1.66914\) of defining polynomial
Character \(\chi\) \(=\) 4012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.17000 q^{3} -3.20913 q^{5} +1.70252 q^{7} +1.70890 q^{9} +O(q^{10})\) \(q+2.17000 q^{3} -3.20913 q^{5} +1.70252 q^{7} +1.70890 q^{9} +1.56324 q^{11} -1.34519 q^{13} -6.96381 q^{15} +1.00000 q^{17} -8.30535 q^{19} +3.69448 q^{21} -0.270390 q^{23} +5.29850 q^{25} -2.80168 q^{27} +3.47669 q^{29} -7.97386 q^{31} +3.39224 q^{33} -5.46362 q^{35} +7.41140 q^{37} -2.91906 q^{39} -2.68101 q^{41} +3.08191 q^{43} -5.48409 q^{45} -2.60815 q^{47} -4.10141 q^{49} +2.17000 q^{51} -3.06637 q^{53} -5.01664 q^{55} -18.0226 q^{57} -1.00000 q^{59} -0.954765 q^{61} +2.90945 q^{63} +4.31688 q^{65} -4.38979 q^{67} -0.586746 q^{69} -10.9061 q^{71} -1.18272 q^{73} +11.4978 q^{75} +2.66146 q^{77} -0.0741392 q^{79} -11.2064 q^{81} -8.26055 q^{83} -3.20913 q^{85} +7.54442 q^{87} -5.83589 q^{89} -2.29022 q^{91} -17.3033 q^{93} +26.6529 q^{95} -5.57873 q^{97} +2.67143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 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
\(3\) 2.17000 1.25285 0.626425 0.779482i \(-0.284517\pi\)
0.626425 + 0.779482i \(0.284517\pi\)
\(4\) 0 0
\(5\) −3.20913 −1.43517 −0.717583 0.696473i \(-0.754752\pi\)
−0.717583 + 0.696473i \(0.754752\pi\)
\(6\) 0 0
\(7\) 1.70252 0.643494 0.321747 0.946826i \(-0.395730\pi\)
0.321747 + 0.946826i \(0.395730\pi\)
\(8\) 0 0
\(9\) 1.70890 0.569634
\(10\) 0 0
\(11\) 1.56324 0.471335 0.235668 0.971834i \(-0.424272\pi\)
0.235668 + 0.971834i \(0.424272\pi\)
\(12\) 0 0
\(13\) −1.34519 −0.373088 −0.186544 0.982447i \(-0.559729\pi\)
−0.186544 + 0.982447i \(0.559729\pi\)
\(14\) 0 0
\(15\) −6.96381 −1.79805
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −8.30535 −1.90538 −0.952689 0.303947i \(-0.901695\pi\)
−0.952689 + 0.303947i \(0.901695\pi\)
\(20\) 0 0
\(21\) 3.69448 0.806201
\(22\) 0 0
\(23\) −0.270390 −0.0563801 −0.0281901 0.999603i \(-0.508974\pi\)
−0.0281901 + 0.999603i \(0.508974\pi\)
\(24\) 0 0
\(25\) 5.29850 1.05970
\(26\) 0 0
\(27\) −2.80168 −0.539184
\(28\) 0 0
\(29\) 3.47669 0.645605 0.322803 0.946466i \(-0.395375\pi\)
0.322803 + 0.946466i \(0.395375\pi\)
\(30\) 0 0
\(31\) −7.97386 −1.43215 −0.716073 0.698025i \(-0.754062\pi\)
−0.716073 + 0.698025i \(0.754062\pi\)
\(32\) 0 0
\(33\) 3.39224 0.590512
\(34\) 0 0
\(35\) −5.46362 −0.923520
\(36\) 0 0
\(37\) 7.41140 1.21843 0.609214 0.793006i \(-0.291485\pi\)
0.609214 + 0.793006i \(0.291485\pi\)
\(38\) 0 0
\(39\) −2.91906 −0.467424
\(40\) 0 0
\(41\) −2.68101 −0.418704 −0.209352 0.977840i \(-0.567135\pi\)
−0.209352 + 0.977840i \(0.567135\pi\)
\(42\) 0 0
\(43\) 3.08191 0.469987 0.234993 0.971997i \(-0.424493\pi\)
0.234993 + 0.971997i \(0.424493\pi\)
\(44\) 0 0
\(45\) −5.48409 −0.817519
\(46\) 0 0
\(47\) −2.60815 −0.380438 −0.190219 0.981742i \(-0.560920\pi\)
−0.190219 + 0.981742i \(0.560920\pi\)
\(48\) 0 0
\(49\) −4.10141 −0.585916
\(50\) 0 0
\(51\) 2.17000 0.303861
\(52\) 0 0
\(53\) −3.06637 −0.421198 −0.210599 0.977573i \(-0.567541\pi\)
−0.210599 + 0.977573i \(0.567541\pi\)
\(54\) 0 0
\(55\) −5.01664 −0.676444
\(56\) 0 0
\(57\) −18.0226 −2.38715
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) −0.954765 −0.122245 −0.0611226 0.998130i \(-0.519468\pi\)
−0.0611226 + 0.998130i \(0.519468\pi\)
\(62\) 0 0
\(63\) 2.90945 0.366556
\(64\) 0 0
\(65\) 4.31688 0.535444
\(66\) 0 0
\(67\) −4.38979 −0.536298 −0.268149 0.963377i \(-0.586412\pi\)
−0.268149 + 0.963377i \(0.586412\pi\)
\(68\) 0 0
\(69\) −0.586746 −0.0706359
\(70\) 0 0
\(71\) −10.9061 −1.29431 −0.647155 0.762358i \(-0.724041\pi\)
−0.647155 + 0.762358i \(0.724041\pi\)
\(72\) 0 0
\(73\) −1.18272 −0.138426 −0.0692132 0.997602i \(-0.522049\pi\)
−0.0692132 + 0.997602i \(0.522049\pi\)
\(74\) 0 0
\(75\) 11.4978 1.32765
\(76\) 0 0
\(77\) 2.66146 0.303301
\(78\) 0 0
\(79\) −0.0741392 −0.00834132 −0.00417066 0.999991i \(-0.501328\pi\)
−0.00417066 + 0.999991i \(0.501328\pi\)
\(80\) 0 0
\(81\) −11.2064 −1.24515
\(82\) 0 0
\(83\) −8.26055 −0.906713 −0.453357 0.891329i \(-0.649774\pi\)
−0.453357 + 0.891329i \(0.649774\pi\)
\(84\) 0 0
\(85\) −3.20913 −0.348079
\(86\) 0 0
\(87\) 7.54442 0.808847
\(88\) 0 0
\(89\) −5.83589 −0.618603 −0.309301 0.950964i \(-0.600095\pi\)
−0.309301 + 0.950964i \(0.600095\pi\)
\(90\) 0 0
\(91\) −2.29022 −0.240080
\(92\) 0 0
\(93\) −17.3033 −1.79427
\(94\) 0 0
\(95\) 26.6529 2.73453
\(96\) 0 0
\(97\) −5.57873 −0.566434 −0.283217 0.959056i \(-0.591402\pi\)
−0.283217 + 0.959056i \(0.591402\pi\)
\(98\) 0 0
\(99\) 2.67143 0.268488
\(100\) 0 0
\(101\) −6.24181 −0.621083 −0.310542 0.950560i \(-0.600510\pi\)
−0.310542 + 0.950560i \(0.600510\pi\)
\(102\) 0 0
\(103\) 15.9810 1.57466 0.787329 0.616534i \(-0.211464\pi\)
0.787329 + 0.616534i \(0.211464\pi\)
\(104\) 0 0
\(105\) −11.8561 −1.15703
\(106\) 0 0
\(107\) −19.5421 −1.88920 −0.944602 0.328218i \(-0.893552\pi\)
−0.944602 + 0.328218i \(0.893552\pi\)
\(108\) 0 0
\(109\) 1.92754 0.184625 0.0923123 0.995730i \(-0.470574\pi\)
0.0923123 + 0.995730i \(0.470574\pi\)
\(110\) 0 0
\(111\) 16.0827 1.52651
\(112\) 0 0
\(113\) 11.6973 1.10038 0.550192 0.835038i \(-0.314554\pi\)
0.550192 + 0.835038i \(0.314554\pi\)
\(114\) 0 0
\(115\) 0.867715 0.0809149
\(116\) 0 0
\(117\) −2.29880 −0.212524
\(118\) 0 0
\(119\) 1.70252 0.156070
\(120\) 0 0
\(121\) −8.55628 −0.777843
\(122\) 0 0
\(123\) −5.81780 −0.524573
\(124\) 0 0
\(125\) −0.957936 −0.0856804
\(126\) 0 0
\(127\) 19.9148 1.76716 0.883578 0.468285i \(-0.155128\pi\)
0.883578 + 0.468285i \(0.155128\pi\)
\(128\) 0 0
\(129\) 6.68775 0.588823
\(130\) 0 0
\(131\) 1.79773 0.157069 0.0785344 0.996911i \(-0.474976\pi\)
0.0785344 + 0.996911i \(0.474976\pi\)
\(132\) 0 0
\(133\) −14.1401 −1.22610
\(134\) 0 0
\(135\) 8.99096 0.773819
\(136\) 0 0
\(137\) −6.05801 −0.517571 −0.258785 0.965935i \(-0.583322\pi\)
−0.258785 + 0.965935i \(0.583322\pi\)
\(138\) 0 0
\(139\) 10.2402 0.868562 0.434281 0.900777i \(-0.357002\pi\)
0.434281 + 0.900777i \(0.357002\pi\)
\(140\) 0 0
\(141\) −5.65969 −0.476632
\(142\) 0 0
\(143\) −2.10286 −0.175850
\(144\) 0 0
\(145\) −11.1571 −0.926550
\(146\) 0 0
\(147\) −8.90006 −0.734065
\(148\) 0 0
\(149\) −13.1741 −1.07927 −0.539634 0.841900i \(-0.681437\pi\)
−0.539634 + 0.841900i \(0.681437\pi\)
\(150\) 0 0
\(151\) −16.7238 −1.36096 −0.680481 0.732766i \(-0.738229\pi\)
−0.680481 + 0.732766i \(0.738229\pi\)
\(152\) 0 0
\(153\) 1.70890 0.138157
\(154\) 0 0
\(155\) 25.5891 2.05537
\(156\) 0 0
\(157\) 17.2040 1.37303 0.686513 0.727118i \(-0.259141\pi\)
0.686513 + 0.727118i \(0.259141\pi\)
\(158\) 0 0
\(159\) −6.65402 −0.527698
\(160\) 0 0
\(161\) −0.460345 −0.0362803
\(162\) 0 0
\(163\) −0.297541 −0.0233052 −0.0116526 0.999932i \(-0.503709\pi\)
−0.0116526 + 0.999932i \(0.503709\pi\)
\(164\) 0 0
\(165\) −10.8861 −0.847483
\(166\) 0 0
\(167\) −8.11071 −0.627625 −0.313813 0.949485i \(-0.601606\pi\)
−0.313813 + 0.949485i \(0.601606\pi\)
\(168\) 0 0
\(169\) −11.1905 −0.860805
\(170\) 0 0
\(171\) −14.1930 −1.08537
\(172\) 0 0
\(173\) 4.30029 0.326945 0.163472 0.986548i \(-0.447731\pi\)
0.163472 + 0.986548i \(0.447731\pi\)
\(174\) 0 0
\(175\) 9.02083 0.681911
\(176\) 0 0
\(177\) −2.17000 −0.163107
\(178\) 0 0
\(179\) −6.34978 −0.474605 −0.237302 0.971436i \(-0.576263\pi\)
−0.237302 + 0.971436i \(0.576263\pi\)
\(180\) 0 0
\(181\) 2.66110 0.197798 0.0988988 0.995097i \(-0.468468\pi\)
0.0988988 + 0.995097i \(0.468468\pi\)
\(182\) 0 0
\(183\) −2.07184 −0.153155
\(184\) 0 0
\(185\) −23.7841 −1.74865
\(186\) 0 0
\(187\) 1.56324 0.114316
\(188\) 0 0
\(189\) −4.76993 −0.346962
\(190\) 0 0
\(191\) −13.6624 −0.988579 −0.494289 0.869297i \(-0.664572\pi\)
−0.494289 + 0.869297i \(0.664572\pi\)
\(192\) 0 0
\(193\) −19.7543 −1.42195 −0.710973 0.703220i \(-0.751745\pi\)
−0.710973 + 0.703220i \(0.751745\pi\)
\(194\) 0 0
\(195\) 9.36764 0.670831
\(196\) 0 0
\(197\) 18.6912 1.33170 0.665848 0.746088i \(-0.268070\pi\)
0.665848 + 0.746088i \(0.268070\pi\)
\(198\) 0 0
\(199\) 10.1404 0.718834 0.359417 0.933177i \(-0.382976\pi\)
0.359417 + 0.933177i \(0.382976\pi\)
\(200\) 0 0
\(201\) −9.52584 −0.671901
\(202\) 0 0
\(203\) 5.91915 0.415443
\(204\) 0 0
\(205\) 8.60371 0.600909
\(206\) 0 0
\(207\) −0.462069 −0.0321160
\(208\) 0 0
\(209\) −12.9833 −0.898071
\(210\) 0 0
\(211\) 15.6987 1.08074 0.540372 0.841426i \(-0.318284\pi\)
0.540372 + 0.841426i \(0.318284\pi\)
\(212\) 0 0
\(213\) −23.6661 −1.62158
\(214\) 0 0
\(215\) −9.89024 −0.674509
\(216\) 0 0
\(217\) −13.5757 −0.921578
\(218\) 0 0
\(219\) −2.56649 −0.173428
\(220\) 0 0
\(221\) −1.34519 −0.0904872
\(222\) 0 0
\(223\) 9.20369 0.616325 0.308162 0.951334i \(-0.400286\pi\)
0.308162 + 0.951334i \(0.400286\pi\)
\(224\) 0 0
\(225\) 9.05462 0.603642
\(226\) 0 0
\(227\) −19.1557 −1.27141 −0.635706 0.771931i \(-0.719291\pi\)
−0.635706 + 0.771931i \(0.719291\pi\)
\(228\) 0 0
\(229\) −11.3955 −0.753036 −0.376518 0.926409i \(-0.622879\pi\)
−0.376518 + 0.926409i \(0.622879\pi\)
\(230\) 0 0
\(231\) 5.77536 0.379991
\(232\) 0 0
\(233\) 19.4525 1.27438 0.637188 0.770708i \(-0.280097\pi\)
0.637188 + 0.770708i \(0.280097\pi\)
\(234\) 0 0
\(235\) 8.36989 0.545991
\(236\) 0 0
\(237\) −0.160882 −0.0104504
\(238\) 0 0
\(239\) 8.34105 0.539538 0.269769 0.962925i \(-0.413053\pi\)
0.269769 + 0.962925i \(0.413053\pi\)
\(240\) 0 0
\(241\) −25.8279 −1.66372 −0.831861 0.554984i \(-0.812724\pi\)
−0.831861 + 0.554984i \(0.812724\pi\)
\(242\) 0 0
\(243\) −15.9128 −1.02080
\(244\) 0 0
\(245\) 13.1620 0.840886
\(246\) 0 0
\(247\) 11.1723 0.710874
\(248\) 0 0
\(249\) −17.9254 −1.13598
\(250\) 0 0
\(251\) −22.5608 −1.42403 −0.712014 0.702166i \(-0.752217\pi\)
−0.712014 + 0.702166i \(0.752217\pi\)
\(252\) 0 0
\(253\) −0.422684 −0.0265739
\(254\) 0 0
\(255\) −6.96381 −0.436091
\(256\) 0 0
\(257\) −13.0467 −0.813829 −0.406915 0.913466i \(-0.633395\pi\)
−0.406915 + 0.913466i \(0.633395\pi\)
\(258\) 0 0
\(259\) 12.6181 0.784050
\(260\) 0 0
\(261\) 5.94132 0.367759
\(262\) 0 0
\(263\) −20.6859 −1.27555 −0.637775 0.770223i \(-0.720145\pi\)
−0.637775 + 0.770223i \(0.720145\pi\)
\(264\) 0 0
\(265\) 9.84036 0.604489
\(266\) 0 0
\(267\) −12.6639 −0.775017
\(268\) 0 0
\(269\) −5.82717 −0.355289 −0.177644 0.984095i \(-0.556848\pi\)
−0.177644 + 0.984095i \(0.556848\pi\)
\(270\) 0 0
\(271\) −2.98666 −0.181427 −0.0907134 0.995877i \(-0.528915\pi\)
−0.0907134 + 0.995877i \(0.528915\pi\)
\(272\) 0 0
\(273\) −4.96977 −0.300784
\(274\) 0 0
\(275\) 8.28284 0.499474
\(276\) 0 0
\(277\) 2.62682 0.157830 0.0789151 0.996881i \(-0.474854\pi\)
0.0789151 + 0.996881i \(0.474854\pi\)
\(278\) 0 0
\(279\) −13.6265 −0.815800
\(280\) 0 0
\(281\) −7.19113 −0.428987 −0.214494 0.976725i \(-0.568810\pi\)
−0.214494 + 0.976725i \(0.568810\pi\)
\(282\) 0 0
\(283\) −13.7265 −0.815956 −0.407978 0.912992i \(-0.633766\pi\)
−0.407978 + 0.912992i \(0.633766\pi\)
\(284\) 0 0
\(285\) 57.8369 3.42596
\(286\) 0 0
\(287\) −4.56449 −0.269433
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −12.1058 −0.709657
\(292\) 0 0
\(293\) 2.54175 0.148490 0.0742452 0.997240i \(-0.476345\pi\)
0.0742452 + 0.997240i \(0.476345\pi\)
\(294\) 0 0
\(295\) 3.20913 0.186843
\(296\) 0 0
\(297\) −4.37971 −0.254136
\(298\) 0 0
\(299\) 0.363725 0.0210348
\(300\) 0 0
\(301\) 5.24703 0.302434
\(302\) 0 0
\(303\) −13.5447 −0.778125
\(304\) 0 0
\(305\) 3.06396 0.175442
\(306\) 0 0
\(307\) 5.42090 0.309387 0.154694 0.987962i \(-0.450561\pi\)
0.154694 + 0.987962i \(0.450561\pi\)
\(308\) 0 0
\(309\) 34.6788 1.97281
\(310\) 0 0
\(311\) −21.5270 −1.22068 −0.610342 0.792138i \(-0.708968\pi\)
−0.610342 + 0.792138i \(0.708968\pi\)
\(312\) 0 0
\(313\) 25.0437 1.41555 0.707776 0.706437i \(-0.249699\pi\)
0.707776 + 0.706437i \(0.249699\pi\)
\(314\) 0 0
\(315\) −9.33679 −0.526069
\(316\) 0 0
\(317\) 16.7274 0.939504 0.469752 0.882798i \(-0.344343\pi\)
0.469752 + 0.882798i \(0.344343\pi\)
\(318\) 0 0
\(319\) 5.43491 0.304296
\(320\) 0 0
\(321\) −42.4063 −2.36689
\(322\) 0 0
\(323\) −8.30535 −0.462122
\(324\) 0 0
\(325\) −7.12749 −0.395362
\(326\) 0 0
\(327\) 4.18276 0.231307
\(328\) 0 0
\(329\) −4.44044 −0.244809
\(330\) 0 0
\(331\) 20.8107 1.14386 0.571931 0.820302i \(-0.306194\pi\)
0.571931 + 0.820302i \(0.306194\pi\)
\(332\) 0 0
\(333\) 12.6654 0.694058
\(334\) 0 0
\(335\) 14.0874 0.769677
\(336\) 0 0
\(337\) 32.3757 1.76362 0.881809 0.471606i \(-0.156326\pi\)
0.881809 + 0.471606i \(0.156326\pi\)
\(338\) 0 0
\(339\) 25.3830 1.37862
\(340\) 0 0
\(341\) −12.4651 −0.675021
\(342\) 0 0
\(343\) −18.9004 −1.02053
\(344\) 0 0
\(345\) 1.88294 0.101374
\(346\) 0 0
\(347\) 31.4754 1.68969 0.844844 0.535013i \(-0.179693\pi\)
0.844844 + 0.535013i \(0.179693\pi\)
\(348\) 0 0
\(349\) 0.0830546 0.00444581 0.00222291 0.999998i \(-0.499292\pi\)
0.00222291 + 0.999998i \(0.499292\pi\)
\(350\) 0 0
\(351\) 3.76879 0.201163
\(352\) 0 0
\(353\) 20.3492 1.08308 0.541539 0.840676i \(-0.317842\pi\)
0.541539 + 0.840676i \(0.317842\pi\)
\(354\) 0 0
\(355\) 34.9989 1.85755
\(356\) 0 0
\(357\) 3.69448 0.195533
\(358\) 0 0
\(359\) −8.57604 −0.452626 −0.226313 0.974055i \(-0.572667\pi\)
−0.226313 + 0.974055i \(0.572667\pi\)
\(360\) 0 0
\(361\) 49.9788 2.63046
\(362\) 0 0
\(363\) −18.5671 −0.974521
\(364\) 0 0
\(365\) 3.79549 0.198665
\(366\) 0 0
\(367\) −34.7526 −1.81407 −0.907036 0.421054i \(-0.861660\pi\)
−0.907036 + 0.421054i \(0.861660\pi\)
\(368\) 0 0
\(369\) −4.58159 −0.238508
\(370\) 0 0
\(371\) −5.22056 −0.271038
\(372\) 0 0
\(373\) 3.82563 0.198084 0.0990419 0.995083i \(-0.468422\pi\)
0.0990419 + 0.995083i \(0.468422\pi\)
\(374\) 0 0
\(375\) −2.07872 −0.107345
\(376\) 0 0
\(377\) −4.67680 −0.240868
\(378\) 0 0
\(379\) 31.1141 1.59823 0.799113 0.601181i \(-0.205303\pi\)
0.799113 + 0.601181i \(0.205303\pi\)
\(380\) 0 0
\(381\) 43.2152 2.21398
\(382\) 0 0
\(383\) 20.3739 1.04106 0.520530 0.853844i \(-0.325735\pi\)
0.520530 + 0.853844i \(0.325735\pi\)
\(384\) 0 0
\(385\) −8.54096 −0.435287
\(386\) 0 0
\(387\) 5.26668 0.267721
\(388\) 0 0
\(389\) −18.3645 −0.931118 −0.465559 0.885017i \(-0.654147\pi\)
−0.465559 + 0.885017i \(0.654147\pi\)
\(390\) 0 0
\(391\) −0.270390 −0.0136742
\(392\) 0 0
\(393\) 3.90109 0.196784
\(394\) 0 0
\(395\) 0.237922 0.0119712
\(396\) 0 0
\(397\) 31.3639 1.57411 0.787055 0.616882i \(-0.211605\pi\)
0.787055 + 0.616882i \(0.211605\pi\)
\(398\) 0 0
\(399\) −30.6839 −1.53612
\(400\) 0 0
\(401\) −12.0286 −0.600682 −0.300341 0.953832i \(-0.597100\pi\)
−0.300341 + 0.953832i \(0.597100\pi\)
\(402\) 0 0
\(403\) 10.7263 0.534317
\(404\) 0 0
\(405\) 35.9626 1.78700
\(406\) 0 0
\(407\) 11.5858 0.574288
\(408\) 0 0
\(409\) −1.27287 −0.0629394 −0.0314697 0.999505i \(-0.510019\pi\)
−0.0314697 + 0.999505i \(0.510019\pi\)
\(410\) 0 0
\(411\) −13.1459 −0.648438
\(412\) 0 0
\(413\) −1.70252 −0.0837758
\(414\) 0 0
\(415\) 26.5092 1.30128
\(416\) 0 0
\(417\) 22.2212 1.08818
\(418\) 0 0
\(419\) 22.4485 1.09668 0.548340 0.836256i \(-0.315260\pi\)
0.548340 + 0.836256i \(0.315260\pi\)
\(420\) 0 0
\(421\) 32.3270 1.57552 0.787761 0.615981i \(-0.211240\pi\)
0.787761 + 0.615981i \(0.211240\pi\)
\(422\) 0 0
\(423\) −4.45707 −0.216710
\(424\) 0 0
\(425\) 5.29850 0.257015
\(426\) 0 0
\(427\) −1.62551 −0.0786640
\(428\) 0 0
\(429\) −4.56320 −0.220313
\(430\) 0 0
\(431\) −19.2314 −0.926345 −0.463173 0.886268i \(-0.653289\pi\)
−0.463173 + 0.886268i \(0.653289\pi\)
\(432\) 0 0
\(433\) −24.4820 −1.17653 −0.588265 0.808668i \(-0.700189\pi\)
−0.588265 + 0.808668i \(0.700189\pi\)
\(434\) 0 0
\(435\) −24.2110 −1.16083
\(436\) 0 0
\(437\) 2.24568 0.107425
\(438\) 0 0
\(439\) 12.1768 0.581168 0.290584 0.956849i \(-0.406150\pi\)
0.290584 + 0.956849i \(0.406150\pi\)
\(440\) 0 0
\(441\) −7.00891 −0.333758
\(442\) 0 0
\(443\) −36.8064 −1.74872 −0.874362 0.485274i \(-0.838720\pi\)
−0.874362 + 0.485274i \(0.838720\pi\)
\(444\) 0 0
\(445\) 18.7281 0.887797
\(446\) 0 0
\(447\) −28.5879 −1.35216
\(448\) 0 0
\(449\) −10.3308 −0.487539 −0.243770 0.969833i \(-0.578384\pi\)
−0.243770 + 0.969833i \(0.578384\pi\)
\(450\) 0 0
\(451\) −4.19107 −0.197350
\(452\) 0 0
\(453\) −36.2906 −1.70508
\(454\) 0 0
\(455\) 7.34960 0.344555
\(456\) 0 0
\(457\) −17.9065 −0.837630 −0.418815 0.908072i \(-0.637554\pi\)
−0.418815 + 0.908072i \(0.637554\pi\)
\(458\) 0 0
\(459\) −2.80168 −0.130771
\(460\) 0 0
\(461\) −16.9047 −0.787330 −0.393665 0.919254i \(-0.628793\pi\)
−0.393665 + 0.919254i \(0.628793\pi\)
\(462\) 0 0
\(463\) 42.4949 1.97490 0.987452 0.157922i \(-0.0504795\pi\)
0.987452 + 0.157922i \(0.0504795\pi\)
\(464\) 0 0
\(465\) 55.5284 2.57507
\(466\) 0 0
\(467\) −8.64917 −0.400236 −0.200118 0.979772i \(-0.564133\pi\)
−0.200118 + 0.979772i \(0.564133\pi\)
\(468\) 0 0
\(469\) −7.47372 −0.345104
\(470\) 0 0
\(471\) 37.3326 1.72020
\(472\) 0 0
\(473\) 4.81777 0.221521
\(474\) 0 0
\(475\) −44.0059 −2.01913
\(476\) 0 0
\(477\) −5.24012 −0.239929
\(478\) 0 0
\(479\) 16.3819 0.748510 0.374255 0.927326i \(-0.377898\pi\)
0.374255 + 0.927326i \(0.377898\pi\)
\(480\) 0 0
\(481\) −9.96974 −0.454581
\(482\) 0 0
\(483\) −0.998949 −0.0454538
\(484\) 0 0
\(485\) 17.9029 0.812927
\(486\) 0 0
\(487\) 39.7254 1.80013 0.900064 0.435757i \(-0.143519\pi\)
0.900064 + 0.435757i \(0.143519\pi\)
\(488\) 0 0
\(489\) −0.645664 −0.0291979
\(490\) 0 0
\(491\) 18.7675 0.846967 0.423484 0.905904i \(-0.360807\pi\)
0.423484 + 0.905904i \(0.360807\pi\)
\(492\) 0 0
\(493\) 3.47669 0.156582
\(494\) 0 0
\(495\) −8.57295 −0.385325
\(496\) 0 0
\(497\) −18.5678 −0.832880
\(498\) 0 0
\(499\) 25.2522 1.13045 0.565223 0.824938i \(-0.308790\pi\)
0.565223 + 0.824938i \(0.308790\pi\)
\(500\) 0 0
\(501\) −17.6002 −0.786321
\(502\) 0 0
\(503\) −13.8627 −0.618107 −0.309053 0.951045i \(-0.600012\pi\)
−0.309053 + 0.951045i \(0.600012\pi\)
\(504\) 0 0
\(505\) 20.0308 0.891358
\(506\) 0 0
\(507\) −24.2833 −1.07846
\(508\) 0 0
\(509\) −14.0478 −0.622656 −0.311328 0.950302i \(-0.600774\pi\)
−0.311328 + 0.950302i \(0.600774\pi\)
\(510\) 0 0
\(511\) −2.01360 −0.0890765
\(512\) 0 0
\(513\) 23.2690 1.02735
\(514\) 0 0
\(515\) −51.2852 −2.25989
\(516\) 0 0
\(517\) −4.07717 −0.179314
\(518\) 0 0
\(519\) 9.33162 0.409613
\(520\) 0 0
\(521\) −5.73417 −0.251219 −0.125609 0.992080i \(-0.540089\pi\)
−0.125609 + 0.992080i \(0.540089\pi\)
\(522\) 0 0
\(523\) 8.07052 0.352899 0.176450 0.984310i \(-0.443539\pi\)
0.176450 + 0.984310i \(0.443539\pi\)
\(524\) 0 0
\(525\) 19.5752 0.854332
\(526\) 0 0
\(527\) −7.97386 −0.347347
\(528\) 0 0
\(529\) −22.9269 −0.996821
\(530\) 0 0
\(531\) −1.70890 −0.0741600
\(532\) 0 0
\(533\) 3.60647 0.156214
\(534\) 0 0
\(535\) 62.7130 2.71132
\(536\) 0 0
\(537\) −13.7790 −0.594609
\(538\) 0 0
\(539\) −6.41150 −0.276163
\(540\) 0 0
\(541\) 6.49022 0.279037 0.139518 0.990219i \(-0.455445\pi\)
0.139518 + 0.990219i \(0.455445\pi\)
\(542\) 0 0
\(543\) 5.77458 0.247811
\(544\) 0 0
\(545\) −6.18572 −0.264967
\(546\) 0 0
\(547\) 0.491942 0.0210339 0.0105170 0.999945i \(-0.496652\pi\)
0.0105170 + 0.999945i \(0.496652\pi\)
\(548\) 0 0
\(549\) −1.63160 −0.0696350
\(550\) 0 0
\(551\) −28.8751 −1.23012
\(552\) 0 0
\(553\) −0.126224 −0.00536758
\(554\) 0 0
\(555\) −51.6116 −2.19079
\(556\) 0 0
\(557\) 19.0867 0.808727 0.404364 0.914598i \(-0.367493\pi\)
0.404364 + 0.914598i \(0.367493\pi\)
\(558\) 0 0
\(559\) −4.14575 −0.175347
\(560\) 0 0
\(561\) 3.39224 0.143220
\(562\) 0 0
\(563\) 1.43278 0.0603843 0.0301922 0.999544i \(-0.490388\pi\)
0.0301922 + 0.999544i \(0.490388\pi\)
\(564\) 0 0
\(565\) −37.5380 −1.57923
\(566\) 0 0
\(567\) −19.0791 −0.801247
\(568\) 0 0
\(569\) 26.6751 1.11828 0.559139 0.829074i \(-0.311132\pi\)
0.559139 + 0.829074i \(0.311132\pi\)
\(570\) 0 0
\(571\) 15.3926 0.644158 0.322079 0.946713i \(-0.395618\pi\)
0.322079 + 0.946713i \(0.395618\pi\)
\(572\) 0 0
\(573\) −29.6475 −1.23854
\(574\) 0 0
\(575\) −1.43266 −0.0597461
\(576\) 0 0
\(577\) −37.5558 −1.56347 −0.781733 0.623613i \(-0.785664\pi\)
−0.781733 + 0.623613i \(0.785664\pi\)
\(578\) 0 0
\(579\) −42.8668 −1.78148
\(580\) 0 0
\(581\) −14.0638 −0.583464
\(582\) 0 0
\(583\) −4.79347 −0.198525
\(584\) 0 0
\(585\) 7.37713 0.305007
\(586\) 0 0
\(587\) −22.3466 −0.922342 −0.461171 0.887311i \(-0.652571\pi\)
−0.461171 + 0.887311i \(0.652571\pi\)
\(588\) 0 0
\(589\) 66.2257 2.72878
\(590\) 0 0
\(591\) 40.5600 1.66841
\(592\) 0 0
\(593\) 5.76185 0.236611 0.118305 0.992977i \(-0.462254\pi\)
0.118305 + 0.992977i \(0.462254\pi\)
\(594\) 0 0
\(595\) −5.46362 −0.223987
\(596\) 0 0
\(597\) 22.0047 0.900592
\(598\) 0 0
\(599\) −13.7931 −0.563572 −0.281786 0.959477i \(-0.590927\pi\)
−0.281786 + 0.959477i \(0.590927\pi\)
\(600\) 0 0
\(601\) −13.9342 −0.568389 −0.284194 0.958767i \(-0.591726\pi\)
−0.284194 + 0.958767i \(0.591726\pi\)
\(602\) 0 0
\(603\) −7.50172 −0.305494
\(604\) 0 0
\(605\) 27.4582 1.11633
\(606\) 0 0
\(607\) 27.8102 1.12878 0.564391 0.825508i \(-0.309111\pi\)
0.564391 + 0.825508i \(0.309111\pi\)
\(608\) 0 0
\(609\) 12.8446 0.520488
\(610\) 0 0
\(611\) 3.50845 0.141937
\(612\) 0 0
\(613\) −0.181586 −0.00733417 −0.00366709 0.999993i \(-0.501167\pi\)
−0.00366709 + 0.999993i \(0.501167\pi\)
\(614\) 0 0
\(615\) 18.6701 0.752850
\(616\) 0 0
\(617\) 3.99176 0.160702 0.0803510 0.996767i \(-0.474396\pi\)
0.0803510 + 0.996767i \(0.474396\pi\)
\(618\) 0 0
\(619\) −13.2337 −0.531907 −0.265953 0.963986i \(-0.585687\pi\)
−0.265953 + 0.963986i \(0.585687\pi\)
\(620\) 0 0
\(621\) 0.757546 0.0303993
\(622\) 0 0
\(623\) −9.93574 −0.398067
\(624\) 0 0
\(625\) −23.4184 −0.936735
\(626\) 0 0
\(627\) −28.1737 −1.12515
\(628\) 0 0
\(629\) 7.41140 0.295512
\(630\) 0 0
\(631\) 23.4711 0.934368 0.467184 0.884160i \(-0.345269\pi\)
0.467184 + 0.884160i \(0.345269\pi\)
\(632\) 0 0
\(633\) 34.0662 1.35401
\(634\) 0 0
\(635\) −63.9092 −2.53616
\(636\) 0 0
\(637\) 5.51717 0.218598
\(638\) 0 0
\(639\) −18.6374 −0.737283
\(640\) 0 0
\(641\) −6.22616 −0.245918 −0.122959 0.992412i \(-0.539238\pi\)
−0.122959 + 0.992412i \(0.539238\pi\)
\(642\) 0 0
\(643\) −3.44042 −0.135677 −0.0678385 0.997696i \(-0.521610\pi\)
−0.0678385 + 0.997696i \(0.521610\pi\)
\(644\) 0 0
\(645\) −21.4618 −0.845059
\(646\) 0 0
\(647\) 27.6489 1.08699 0.543495 0.839413i \(-0.317101\pi\)
0.543495 + 0.839413i \(0.317101\pi\)
\(648\) 0 0
\(649\) −1.56324 −0.0613626
\(650\) 0 0
\(651\) −29.4592 −1.15460
\(652\) 0 0
\(653\) 12.0889 0.473076 0.236538 0.971622i \(-0.423987\pi\)
0.236538 + 0.971622i \(0.423987\pi\)
\(654\) 0 0
\(655\) −5.76916 −0.225420
\(656\) 0 0
\(657\) −2.02114 −0.0788524
\(658\) 0 0
\(659\) −7.48788 −0.291687 −0.145843 0.989308i \(-0.546590\pi\)
−0.145843 + 0.989308i \(0.546590\pi\)
\(660\) 0 0
\(661\) 26.4560 1.02902 0.514509 0.857485i \(-0.327974\pi\)
0.514509 + 0.857485i \(0.327974\pi\)
\(662\) 0 0
\(663\) −2.91906 −0.113367
\(664\) 0 0
\(665\) 45.3773 1.75965
\(666\) 0 0
\(667\) −0.940061 −0.0363993
\(668\) 0 0
\(669\) 19.9720 0.772163
\(670\) 0 0
\(671\) −1.49253 −0.0576184
\(672\) 0 0
\(673\) 8.39833 0.323732 0.161866 0.986813i \(-0.448249\pi\)
0.161866 + 0.986813i \(0.448249\pi\)
\(674\) 0 0
\(675\) −14.8447 −0.571374
\(676\) 0 0
\(677\) −18.2587 −0.701740 −0.350870 0.936424i \(-0.614114\pi\)
−0.350870 + 0.936424i \(0.614114\pi\)
\(678\) 0 0
\(679\) −9.49792 −0.364497
\(680\) 0 0
\(681\) −41.5680 −1.59289
\(682\) 0 0
\(683\) 25.5791 0.978758 0.489379 0.872071i \(-0.337224\pi\)
0.489379 + 0.872071i \(0.337224\pi\)
\(684\) 0 0
\(685\) 19.4409 0.742800
\(686\) 0 0
\(687\) −24.7282 −0.943441
\(688\) 0 0
\(689\) 4.12484 0.157144
\(690\) 0 0
\(691\) −17.2637 −0.656741 −0.328370 0.944549i \(-0.606499\pi\)
−0.328370 + 0.944549i \(0.606499\pi\)
\(692\) 0 0
\(693\) 4.54817 0.172771
\(694\) 0 0
\(695\) −32.8621 −1.24653
\(696\) 0 0
\(697\) −2.68101 −0.101551
\(698\) 0 0
\(699\) 42.2119 1.59660
\(700\) 0 0
\(701\) 18.3280 0.692240 0.346120 0.938190i \(-0.387499\pi\)
0.346120 + 0.938190i \(0.387499\pi\)
\(702\) 0 0
\(703\) −61.5543 −2.32156
\(704\) 0 0
\(705\) 18.1627 0.684045
\(706\) 0 0
\(707\) −10.6268 −0.399663
\(708\) 0 0
\(709\) −2.74511 −0.103095 −0.0515474 0.998671i \(-0.516415\pi\)
−0.0515474 + 0.998671i \(0.516415\pi\)
\(710\) 0 0
\(711\) −0.126697 −0.00475150
\(712\) 0 0
\(713\) 2.15605 0.0807447
\(714\) 0 0
\(715\) 6.74833 0.252373
\(716\) 0 0
\(717\) 18.1001 0.675960
\(718\) 0 0
\(719\) −6.11780 −0.228155 −0.114078 0.993472i \(-0.536391\pi\)
−0.114078 + 0.993472i \(0.536391\pi\)
\(720\) 0 0
\(721\) 27.2081 1.01328
\(722\) 0 0
\(723\) −56.0466 −2.08439
\(724\) 0 0
\(725\) 18.4213 0.684148
\(726\) 0 0
\(727\) −8.88321 −0.329460 −0.164730 0.986339i \(-0.552675\pi\)
−0.164730 + 0.986339i \(0.552675\pi\)
\(728\) 0 0
\(729\) −0.911609 −0.0337633
\(730\) 0 0
\(731\) 3.08191 0.113989
\(732\) 0 0
\(733\) 26.1133 0.964517 0.482259 0.876029i \(-0.339817\pi\)
0.482259 + 0.876029i \(0.339817\pi\)
\(734\) 0 0
\(735\) 28.5614 1.05350
\(736\) 0 0
\(737\) −6.86230 −0.252776
\(738\) 0 0
\(739\) 21.6675 0.797050 0.398525 0.917157i \(-0.369522\pi\)
0.398525 + 0.917157i \(0.369522\pi\)
\(740\) 0 0
\(741\) 24.2438 0.890619
\(742\) 0 0
\(743\) −17.1589 −0.629499 −0.314749 0.949175i \(-0.601920\pi\)
−0.314749 + 0.949175i \(0.601920\pi\)
\(744\) 0 0
\(745\) 42.2775 1.54893
\(746\) 0 0
\(747\) −14.1165 −0.516495
\(748\) 0 0
\(749\) −33.2709 −1.21569
\(750\) 0 0
\(751\) −51.5615 −1.88151 −0.940753 0.339091i \(-0.889880\pi\)
−0.940753 + 0.339091i \(0.889880\pi\)
\(752\) 0 0
\(753\) −48.9570 −1.78409
\(754\) 0 0
\(755\) 53.6687 1.95321
\(756\) 0 0
\(757\) −0.725707 −0.0263763 −0.0131881 0.999913i \(-0.504198\pi\)
−0.0131881 + 0.999913i \(0.504198\pi\)
\(758\) 0 0
\(759\) −0.917225 −0.0332932
\(760\) 0 0
\(761\) −18.5554 −0.672633 −0.336317 0.941749i \(-0.609181\pi\)
−0.336317 + 0.941749i \(0.609181\pi\)
\(762\) 0 0
\(763\) 3.28168 0.118805
\(764\) 0 0
\(765\) −5.48409 −0.198278
\(766\) 0 0
\(767\) 1.34519 0.0485720
\(768\) 0 0
\(769\) 45.1388 1.62775 0.813874 0.581041i \(-0.197355\pi\)
0.813874 + 0.581041i \(0.197355\pi\)
\(770\) 0 0
\(771\) −28.3113 −1.01961
\(772\) 0 0
\(773\) 25.4320 0.914724 0.457362 0.889281i \(-0.348794\pi\)
0.457362 + 0.889281i \(0.348794\pi\)
\(774\) 0 0
\(775\) −42.2495 −1.51765
\(776\) 0 0
\(777\) 27.3813 0.982298
\(778\) 0 0
\(779\) 22.2668 0.797789
\(780\) 0 0
\(781\) −17.0488 −0.610054
\(782\) 0 0
\(783\) −9.74058 −0.348100
\(784\) 0 0
\(785\) −55.2097 −1.97052
\(786\) 0 0
\(787\) −29.7429 −1.06022 −0.530110 0.847929i \(-0.677849\pi\)
−0.530110 + 0.847929i \(0.677849\pi\)
\(788\) 0 0
\(789\) −44.8885 −1.59807
\(790\) 0 0
\(791\) 19.9149 0.708091
\(792\) 0 0
\(793\) 1.28434 0.0456082
\(794\) 0 0
\(795\) 21.3536 0.757334
\(796\) 0 0
\(797\) 43.5190 1.54152 0.770761 0.637124i \(-0.219876\pi\)
0.770761 + 0.637124i \(0.219876\pi\)
\(798\) 0 0
\(799\) −2.60815 −0.0922697
\(800\) 0 0
\(801\) −9.97296 −0.352377
\(802\) 0 0
\(803\) −1.84887 −0.0652452
\(804\) 0 0
\(805\) 1.47731 0.0520682
\(806\) 0 0
\(807\) −12.6450 −0.445124
\(808\) 0 0
\(809\) −3.43994 −0.120942 −0.0604709 0.998170i \(-0.519260\pi\)
−0.0604709 + 0.998170i \(0.519260\pi\)
\(810\) 0 0
\(811\) −35.4132 −1.24353 −0.621763 0.783205i \(-0.713583\pi\)
−0.621763 + 0.783205i \(0.713583\pi\)
\(812\) 0 0
\(813\) −6.48106 −0.227301
\(814\) 0 0
\(815\) 0.954847 0.0334468
\(816\) 0 0
\(817\) −25.5963 −0.895503
\(818\) 0 0
\(819\) −3.91376 −0.136758
\(820\) 0 0
\(821\) −10.3915 −0.362666 −0.181333 0.983422i \(-0.558041\pi\)
−0.181333 + 0.983422i \(0.558041\pi\)
\(822\) 0 0
\(823\) 21.6806 0.755740 0.377870 0.925859i \(-0.376657\pi\)
0.377870 + 0.925859i \(0.376657\pi\)
\(824\) 0 0
\(825\) 17.9738 0.625766
\(826\) 0 0
\(827\) 49.7237 1.72906 0.864531 0.502579i \(-0.167615\pi\)
0.864531 + 0.502579i \(0.167615\pi\)
\(828\) 0 0
\(829\) −4.31747 −0.149952 −0.0749760 0.997185i \(-0.523888\pi\)
−0.0749760 + 0.997185i \(0.523888\pi\)
\(830\) 0 0
\(831\) 5.70020 0.197738
\(832\) 0 0
\(833\) −4.10141 −0.142105
\(834\) 0 0
\(835\) 26.0283 0.900746
\(836\) 0 0
\(837\) 22.3402 0.772191
\(838\) 0 0
\(839\) −25.4729 −0.879422 −0.439711 0.898139i \(-0.644919\pi\)
−0.439711 + 0.898139i \(0.644919\pi\)
\(840\) 0 0
\(841\) −16.9126 −0.583194
\(842\) 0 0
\(843\) −15.6048 −0.537457
\(844\) 0 0
\(845\) 35.9116 1.23540
\(846\) 0 0
\(847\) −14.5673 −0.500537
\(848\) 0 0
\(849\) −29.7865 −1.02227
\(850\) 0 0
\(851\) −2.00397 −0.0686951
\(852\) 0 0
\(853\) 4.39660 0.150537 0.0752683 0.997163i \(-0.476019\pi\)
0.0752683 + 0.997163i \(0.476019\pi\)
\(854\) 0 0
\(855\) 45.5472 1.55768
\(856\) 0 0
\(857\) −47.0133 −1.60595 −0.802973 0.596016i \(-0.796749\pi\)
−0.802973 + 0.596016i \(0.796749\pi\)
\(858\) 0 0
\(859\) 11.4665 0.391233 0.195616 0.980681i \(-0.437329\pi\)
0.195616 + 0.980681i \(0.437329\pi\)
\(860\) 0 0
\(861\) −9.90495 −0.337560
\(862\) 0 0
\(863\) 41.0470 1.39726 0.698629 0.715485i \(-0.253794\pi\)
0.698629 + 0.715485i \(0.253794\pi\)
\(864\) 0 0
\(865\) −13.8002 −0.469220
\(866\) 0 0
\(867\) 2.17000 0.0736971
\(868\) 0 0
\(869\) −0.115898 −0.00393155
\(870\) 0 0
\(871\) 5.90510 0.200087
\(872\) 0 0
\(873\) −9.53350 −0.322660
\(874\) 0 0
\(875\) −1.63091 −0.0551348
\(876\) 0 0
\(877\) −32.7545 −1.10604 −0.553021 0.833167i \(-0.686525\pi\)
−0.553021 + 0.833167i \(0.686525\pi\)
\(878\) 0 0
\(879\) 5.51559 0.186036
\(880\) 0 0
\(881\) −18.4053 −0.620090 −0.310045 0.950722i \(-0.600344\pi\)
−0.310045 + 0.950722i \(0.600344\pi\)
\(882\) 0 0
\(883\) 39.3984 1.32586 0.662931 0.748680i \(-0.269312\pi\)
0.662931 + 0.748680i \(0.269312\pi\)
\(884\) 0 0
\(885\) 6.96381 0.234086
\(886\) 0 0
\(887\) −7.03073 −0.236069 −0.118034 0.993009i \(-0.537659\pi\)
−0.118034 + 0.993009i \(0.537659\pi\)
\(888\) 0 0
\(889\) 33.9055 1.13715
\(890\) 0 0
\(891\) −17.5182 −0.586883
\(892\) 0 0
\(893\) 21.6616 0.724878
\(894\) 0 0
\(895\) 20.3773 0.681137
\(896\) 0 0
\(897\) 0.789284 0.0263534
\(898\) 0 0
\(899\) −27.7226 −0.924601
\(900\) 0 0
\(901\) −3.06637 −0.102155
\(902\) 0 0
\(903\) 11.3861 0.378904
\(904\) 0 0
\(905\) −8.53980 −0.283872
\(906\) 0 0
\(907\) −33.3275 −1.10662 −0.553311 0.832975i \(-0.686636\pi\)
−0.553311 + 0.832975i \(0.686636\pi\)
\(908\) 0 0
\(909\) −10.6666 −0.353790
\(910\) 0 0
\(911\) −27.5422 −0.912515 −0.456258 0.889848i \(-0.650810\pi\)
−0.456258 + 0.889848i \(0.650810\pi\)
\(912\) 0 0
\(913\) −12.9132 −0.427366
\(914\) 0 0
\(915\) 6.64880 0.219803
\(916\) 0 0
\(917\) 3.06069 0.101073
\(918\) 0 0
\(919\) −25.0702 −0.826988 −0.413494 0.910507i \(-0.635692\pi\)
−0.413494 + 0.910507i \(0.635692\pi\)
\(920\) 0 0
\(921\) 11.7634 0.387616
\(922\) 0 0
\(923\) 14.6707 0.482892
\(924\) 0 0
\(925\) 39.2693 1.29117
\(926\) 0 0
\(927\) 27.3100 0.896978
\(928\) 0 0
\(929\) −19.1222 −0.627380 −0.313690 0.949525i \(-0.601565\pi\)
−0.313690 + 0.949525i \(0.601565\pi\)
\(930\) 0 0
\(931\) 34.0636 1.11639
\(932\) 0 0
\(933\) −46.7136 −1.52933
\(934\) 0 0
\(935\) −5.01664 −0.164062
\(936\) 0 0
\(937\) 32.7035 1.06838 0.534188 0.845366i \(-0.320617\pi\)
0.534188 + 0.845366i \(0.320617\pi\)
\(938\) 0 0
\(939\) 54.3448 1.77347
\(940\) 0 0
\(941\) 20.4221 0.665742 0.332871 0.942972i \(-0.391983\pi\)
0.332871 + 0.942972i \(0.391983\pi\)
\(942\) 0 0
\(943\) 0.724918 0.0236066
\(944\) 0 0
\(945\) 15.3073 0.497947
\(946\) 0 0
\(947\) −15.5386 −0.504935 −0.252468 0.967605i \(-0.581242\pi\)
−0.252468 + 0.967605i \(0.581242\pi\)
\(948\) 0 0
\(949\) 1.59098 0.0516453
\(950\) 0 0
\(951\) 36.2984 1.17706
\(952\) 0 0
\(953\) 25.9031 0.839085 0.419543 0.907736i \(-0.362191\pi\)
0.419543 + 0.907736i \(0.362191\pi\)
\(954\) 0 0
\(955\) 43.8445 1.41877
\(956\) 0 0
\(957\) 11.7937 0.381238
\(958\) 0 0
\(959\) −10.3139 −0.333053
\(960\) 0 0
\(961\) 32.5824 1.05104
\(962\) 0 0
\(963\) −33.3955 −1.07616
\(964\) 0 0
\(965\) 63.3941 2.04073
\(966\) 0 0
\(967\) −29.7864 −0.957865 −0.478933 0.877852i \(-0.658976\pi\)
−0.478933 + 0.877852i \(0.658976\pi\)
\(968\) 0 0
\(969\) −18.0226 −0.578970
\(970\) 0 0
\(971\) 3.28341 0.105369 0.0526847 0.998611i \(-0.483222\pi\)
0.0526847 + 0.998611i \(0.483222\pi\)
\(972\) 0 0
\(973\) 17.4342 0.558914
\(974\) 0 0
\(975\) −15.4667 −0.495329
\(976\) 0 0
\(977\) −9.89636 −0.316613 −0.158306 0.987390i \(-0.550603\pi\)
−0.158306 + 0.987390i \(0.550603\pi\)
\(978\) 0 0
\(979\) −9.12290 −0.291569
\(980\) 0 0
\(981\) 3.29397 0.105168
\(982\) 0 0
\(983\) 35.9896 1.14789 0.573945 0.818894i \(-0.305412\pi\)
0.573945 + 0.818894i \(0.305412\pi\)
\(984\) 0 0
\(985\) −59.9826 −1.91120
\(986\) 0 0
\(987\) −9.63575 −0.306709
\(988\) 0 0
\(989\) −0.833317 −0.0264979
\(990\) 0 0
\(991\) −60.7949 −1.93121 −0.965607 0.260005i \(-0.916276\pi\)
−0.965607 + 0.260005i \(0.916276\pi\)
\(992\) 0 0
\(993\) 45.1593 1.43309
\(994\) 0 0
\(995\) −32.5419 −1.03165
\(996\) 0 0
\(997\) 18.4164 0.583252 0.291626 0.956532i \(-0.405804\pi\)
0.291626 + 0.956532i \(0.405804\pi\)
\(998\) 0 0
\(999\) −20.7644 −0.656957
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 4012.2.a.h.1.13 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.a.h.1.13 15 1.1 even 1 trivial