Properties

Label 1407.1.cb.a.1319.1
Level $1407$
Weight $1$
Character 1407.1319
Analytic conductor $0.702$
Analytic rank $0$
Dimension $20$
Projective image $D_{66}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1407,1,Mod(101,1407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 11, 65]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1407.101");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1407 = 3 \cdot 7 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1407.cb (of order \(66\), degree \(20\), minimal)

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: no
Analytic conductor: \(0.702184472775\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{66}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{66} - \cdots)\)

Embedding invariants

Embedding label 1319.1
Root \(0.928368 + 0.371662i\) of defining polynomial
Character \(\chi\) \(=\) 1407.1319
Dual form 1407.1.cb.a.1391.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.580057 + 0.814576i) q^{3} +(0.142315 - 0.989821i) q^{4} +(-0.0475819 - 0.998867i) q^{7} +(-0.327068 + 0.945001i) q^{9} +O(q^{10})\) \(q+(0.580057 + 0.814576i) q^{3} +(0.142315 - 0.989821i) q^{4} +(-0.0475819 - 0.998867i) q^{7} +(-0.327068 + 0.945001i) q^{9} +(0.888835 - 0.458227i) q^{12} +(1.16413 - 0.600149i) q^{13} +(-0.959493 - 0.281733i) q^{16} +(0.143677 + 0.124497i) q^{19} +(0.786053 - 0.618159i) q^{21} +(-0.888835 + 0.458227i) q^{25} +(-0.959493 + 0.281733i) q^{27} +(-0.995472 - 0.0950560i) q^{28} +(1.61435 - 1.03748i) q^{31} +(0.888835 + 0.458227i) q^{36} +0.0951638 q^{37} +(1.16413 + 0.600149i) q^{39} +(1.07028 - 0.153882i) q^{43} +(-0.327068 - 0.945001i) q^{48} +(-0.995472 + 0.0950560i) q^{49} +(-0.428368 - 1.23769i) q^{52} +(-0.0180713 + 0.189251i) q^{57} +(-1.91030 + 0.560914i) q^{61} +(0.959493 + 0.281733i) q^{63} +(-0.415415 + 0.909632i) q^{64} +(-0.0475819 + 0.998867i) q^{67} +(-1.34125 + 1.40667i) q^{73} +(-0.888835 - 0.458227i) q^{75} +(0.143677 - 0.124497i) q^{76} +(0.915415 + 1.77566i) q^{79} +(-0.786053 - 0.618159i) q^{81} +(-0.500000 - 0.866025i) q^{84} +(-0.654861 - 1.13425i) q^{91} +(1.78153 + 0.713215i) q^{93} +(0.580057 - 1.00469i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{3} + 2 q^{4} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{3} + 2 q^{4} - q^{7} + q^{9} - q^{12} + 2 q^{13} - 2 q^{16} - q^{21} + q^{25} - 2 q^{27} + q^{28} + 4 q^{31} - q^{36} + 2 q^{37} + 2 q^{39} + q^{48} + q^{49} + 9 q^{52} - 19 q^{57} - 2 q^{61} + 2 q^{63} + 2 q^{64} - q^{67} - 8 q^{73} + q^{75} + 8 q^{79} + q^{81} - 10 q^{84} - 2 q^{91} - 2 q^{93} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1407\mathbb{Z}\right)^\times\).

\(n\) \(337\) \(470\) \(1207\)
\(\chi(n)\) \(e\left(\frac{29}{66}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(3\) 0.580057 + 0.814576i 0.580057 + 0.814576i
\(4\) 0.142315 0.989821i 0.142315 0.989821i
\(5\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(6\) 0 0
\(7\) −0.0475819 0.998867i −0.0475819 0.998867i
\(8\) 0 0
\(9\) −0.327068 + 0.945001i −0.327068 + 0.945001i
\(10\) 0 0
\(11\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(12\) 0.888835 0.458227i 0.888835 0.458227i
\(13\) 1.16413 0.600149i 1.16413 0.600149i 0.235759 0.971812i \(-0.424242\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.959493 0.281733i −0.959493 0.281733i
\(17\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(18\) 0 0
\(19\) 0.143677 + 0.124497i 0.143677 + 0.124497i 0.723734 0.690079i \(-0.242424\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(20\) 0 0
\(21\) 0.786053 0.618159i 0.786053 0.618159i
\(22\) 0 0
\(23\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(24\) 0 0
\(25\) −0.888835 + 0.458227i −0.888835 + 0.458227i
\(26\) 0 0
\(27\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(28\) −0.995472 0.0950560i −0.995472 0.0950560i
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 1.61435 1.03748i 1.61435 1.03748i 0.654861 0.755750i \(-0.272727\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.888835 + 0.458227i 0.888835 + 0.458227i
\(37\) 0.0951638 0.0951638 0.0475819 0.998867i \(-0.484848\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(38\) 0 0
\(39\) 1.16413 + 0.600149i 1.16413 + 0.600149i
\(40\) 0 0
\(41\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(42\) 0 0
\(43\) 1.07028 0.153882i 1.07028 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(48\) −0.327068 0.945001i −0.327068 0.945001i
\(49\) −0.995472 + 0.0950560i −0.995472 + 0.0950560i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.428368 1.23769i −0.428368 1.23769i
\(53\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.0180713 + 0.189251i −0.0180713 + 0.189251i
\(58\) 0 0
\(59\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(60\) 0 0
\(61\) −1.91030 + 0.560914i −1.91030 + 0.560914i −0.928368 + 0.371662i \(0.878788\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(62\) 0 0
\(63\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(64\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.0475819 + 0.998867i −0.0475819 + 0.998867i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(72\) 0 0
\(73\) −1.34125 + 1.40667i −1.34125 + 1.40667i −0.500000 + 0.866025i \(0.666667\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(74\) 0 0
\(75\) −0.888835 0.458227i −0.888835 0.458227i
\(76\) 0.143677 0.124497i 0.143677 0.124497i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.915415 + 1.77566i 0.915415 + 1.77566i 0.500000 + 0.866025i \(0.333333\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(80\) 0 0
\(81\) −0.786053 0.618159i −0.786053 0.618159i
\(82\) 0 0
\(83\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(84\) −0.500000 0.866025i −0.500000 0.866025i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(90\) 0 0
\(91\) −0.654861 1.13425i −0.654861 1.13425i
\(92\) 0 0
\(93\) 1.78153 + 0.713215i 1.78153 + 0.713215i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.580057 1.00469i 0.580057 1.00469i −0.415415 0.909632i \(-0.636364\pi\)
0.995472 0.0950560i \(-0.0303030\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.327068 + 0.945001i 0.327068 + 0.945001i
\(101\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(102\) 0 0
\(103\) −0.378074 + 0.0180099i −0.378074 + 0.0180099i −0.235759 0.971812i \(-0.575758\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(108\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(109\) 0.258195 + 0.500828i 0.258195 + 0.500828i 0.981929 0.189251i \(-0.0606061\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(110\) 0 0
\(111\) 0.0552004 + 0.0775182i 0.0552004 + 0.0775182i
\(112\) −0.235759 + 0.971812i −0.235759 + 0.971812i
\(113\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.186393 + 1.29639i 0.186393 + 1.29639i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.841254 0.540641i −0.841254 0.540641i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.797176 1.74557i −0.797176 1.74557i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.581419 + 1.67990i −0.581419 + 1.67990i 0.142315 + 0.989821i \(0.454545\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(128\) 0 0
\(129\) 0.746170 + 0.782560i 0.746170 + 0.782560i
\(130\) 0 0
\(131\) 0 0 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(132\) 0 0
\(133\) 0.117519 0.149438i 0.117519 0.149438i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(138\) 0 0
\(139\) −0.797176 0.234072i −0.797176 0.234072i −0.142315 0.989821i \(-0.545455\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.580057 0.814576i 0.580057 0.814576i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.654861 0.755750i −0.654861 0.755750i
\(148\) 0.0135432 0.0941952i 0.0135432 0.0941952i
\(149\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(150\) 0 0
\(151\) 0.370638 0.291473i 0.370638 0.291473i −0.415415 0.909632i \(-0.636364\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0.759713 1.06687i 0.759713 1.06687i
\(157\) −0.143677 1.50465i −0.143677 1.50465i −0.723734 0.690079i \(-0.757576\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.928368 + 1.60798i −0.928368 + 1.60798i −0.142315 + 0.989821i \(0.545455\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(168\) 0 0
\(169\) 0.414955 0.582723i 0.414955 0.582723i
\(170\) 0 0
\(171\) −0.164642 + 0.0950560i −0.164642 + 0.0950560i
\(172\) 1.08128i 1.08128i
\(173\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(174\) 0 0
\(175\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(180\) 0 0
\(181\) −0.172932 1.81103i −0.172932 1.81103i −0.500000 0.866025i \(-0.666667\pi\)
0.327068 0.945001i \(-0.393939\pi\)
\(182\) 0 0
\(183\) −1.56499 1.23072i −1.56499 1.23072i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.327068 + 0.945001i 0.327068 + 0.945001i
\(190\) 0 0
\(191\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(192\) −0.981929 + 0.189251i −0.981929 + 0.189251i
\(193\) 0.0311250 + 0.653395i 0.0311250 + 0.653395i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.0475819 + 0.998867i −0.0475819 + 0.998867i
\(197\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(198\) 0 0
\(199\) 1.16832 + 0.404360i 1.16832 + 0.404360i 0.841254 0.540641i \(-0.181818\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(200\) 0 0
\(201\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.28605 + 0.247866i −1.28605 + 0.247866i
\(209\) 0 0
\(210\) 0 0
\(211\) −1.44091 0.137591i −1.44091 0.137591i −0.654861 0.755750i \(-0.727273\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.11312 1.56316i −1.11312 1.56316i
\(218\) 0 0
\(219\) −1.92384 0.276606i −1.92384 0.276606i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.48193 0.676774i 1.48193 0.676774i 0.500000 0.866025i \(-0.333333\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(224\) 0 0
\(225\) −0.142315 0.989821i −0.142315 0.989821i
\(226\) 0 0
\(227\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(228\) 0.184753 + 0.0448206i 0.184753 + 0.0448206i
\(229\) −1.41542 + 0.909632i −1.41542 + 0.909632i −0.415415 + 0.909632i \(0.636364\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.915415 + 1.77566i −0.915415 + 1.77566i
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) −0.890620 0.216062i −0.890620 0.216062i −0.235759 0.971812i \(-0.575758\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(242\) 0 0
\(243\) 0.0475819 0.998867i 0.0475819 0.998867i
\(244\) 0.283341 + 1.97068i 0.283341 + 1.97068i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.241975 + 0.0587025i 0.241975 + 0.0587025i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(252\) 0.415415 0.909632i 0.415415 0.909632i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(257\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(258\) 0 0
\(259\) −0.00452808 0.0950560i −0.00452808 0.0950560i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.981929 + 0.189251i 0.981929 + 0.189251i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) −1.95496 + 0.186677i −1.95496 + 0.186677i −0.995472 0.0950560i \(-0.969697\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(272\) 0 0
\(273\) 0.544078 1.19136i 0.544078 1.19136i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.54370 0.297523i 1.54370 0.297523i 0.654861 0.755750i \(-0.272727\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(278\) 0 0
\(279\) 0.452418 + 1.86489i 0.452418 + 1.86489i
\(280\) 0 0
\(281\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(282\) 0 0
\(283\) 1.30425 0.451405i 1.30425 0.451405i 0.415415 0.909632i \(-0.363636\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.235759 + 0.971812i −0.235759 + 0.971812i
\(290\) 0 0
\(291\) 1.15486 0.110276i 1.15486 0.110276i
\(292\) 1.20147 + 1.52779i 1.20147 + 1.52779i
\(293\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −0.580057 + 0.814576i −0.580057 + 0.814576i
\(301\) −0.204634 1.06174i −0.204634 1.06174i
\(302\) 0 0
\(303\) 0 0
\(304\) −0.102782 0.159932i −0.102782 0.159932i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.97740 0.0941952i 1.97740 0.0941952i 0.981929 0.189251i \(-0.0606061\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(308\) 0 0
\(309\) −0.233975 0.297523i −0.233975 0.297523i
\(310\) 0 0
\(311\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(312\) 0 0
\(313\) 0.379436 0.532843i 0.379436 0.532843i −0.580057 0.814576i \(-0.696970\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.88786 0.653395i 1.88786 0.653395i
\(317\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.723734 + 0.690079i −0.723734 + 0.690079i
\(325\) −0.759713 + 1.06687i −0.759713 + 1.06687i
\(326\) 0 0
\(327\) −0.258195 + 0.500828i −0.258195 + 0.500828i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.12459 1.43004i 1.12459 1.43004i 0.235759 0.971812i \(-0.424242\pi\)
0.888835 0.458227i \(-0.151515\pi\)
\(332\) 0 0
\(333\) −0.0311250 + 0.0899299i −0.0311250 + 0.0899299i
\(334\) 0 0
\(335\) 0 0
\(336\) −0.928368 + 0.371662i −0.928368 + 0.371662i
\(337\) −0.173440 0.899892i −0.173440 0.899892i −0.959493 0.281733i \(-0.909091\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(348\) 0 0
\(349\) −1.87076 0.268975i −1.87076 0.268975i −0.888835 0.458227i \(-0.848485\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(350\) 0 0
\(351\) −0.947890 + 0.903811i −0.947890 + 0.903811i
\(352\) 0 0
\(353\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(360\) 0 0
\(361\) −0.137171 0.954047i −0.137171 0.954047i
\(362\) 0 0
\(363\) −0.0475819 0.998867i −0.0475819 0.998867i
\(364\) −1.21590 + 0.486774i −1.21590 + 0.486774i
\(365\) 0 0
\(366\) 0 0
\(367\) 0.601300 + 1.31666i 0.601300 + 1.31666i 0.928368 + 0.371662i \(0.121212\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.959493 1.66189i 0.959493 1.66189i
\(373\) −1.30900 0.755750i −1.30900 0.755750i −0.327068 0.945001i \(-0.606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.53955 + 1.09631i −1.53955 + 1.09631i −0.580057 + 0.814576i \(0.696970\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(380\) 0 0
\(381\) −1.70566 + 0.500828i −1.70566 + 0.500828i
\(382\) 0 0
\(383\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.204634 + 1.06174i −0.204634 + 1.06174i
\(388\) −0.911911 0.717135i −0.911911 0.717135i
\(389\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.722372 0.175245i 0.722372 0.175245i 0.142315 0.989821i \(-0.454545\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(398\) 0 0
\(399\) 0.189897 + 0.00904590i 0.189897 + 0.00904590i
\(400\) 0.981929 0.189251i 0.981929 0.189251i
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) 1.25667 2.17661i 1.25667 2.17661i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.308779 + 0.356349i 0.308779 + 0.356349i 0.888835 0.458227i \(-0.151515\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.0359789 + 0.376789i −0.0359789 + 0.376789i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.271738 0.785135i −0.271738 0.785135i
\(418\) 0 0
\(419\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(420\) 0 0
\(421\) 0.550294 + 1.58997i 0.550294 + 1.58997i 0.786053 + 0.618159i \(0.212121\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.651174 + 1.88144i 0.651174 + 1.88144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.00000 1.00000
\(433\) 1.58006 + 0.814576i 1.58006 + 0.814576i 1.00000 \(0\)
0.580057 + 0.814576i \(0.303030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.532475 0.184291i 0.532475 0.184291i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.743325i 0.743325i −0.928368 0.371662i \(-0.878788\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(440\) 0 0
\(441\) 0.235759 0.971812i 0.235759 0.971812i
\(442\) 0 0
\(443\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(444\) 0.0845850 0.0436066i 0.0845850 0.0436066i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.928368 + 0.371662i 0.928368 + 0.371662i
\(449\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.452418 + 0.132842i 0.452418 + 0.132842i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.74555 0.899892i 1.74555 0.899892i 0.786053 0.618159i \(-0.212121\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(462\) 0 0
\(463\) 1.83673 + 0.445585i 1.83673 + 0.445585i 0.995472 0.0950560i \(-0.0303030\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(468\) 1.30972 1.30972
\(469\) 1.00000 1.00000
\(470\) 0 0
\(471\) 1.14231 0.989821i 1.14231 0.989821i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.184753 0.0448206i −0.184753 0.0448206i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(480\) 0 0
\(481\) 0.110783 0.0571125i 0.110783 0.0571125i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.735759 1.83784i −0.735759 1.83784i −0.500000 0.866025i \(-0.666667\pi\)
−0.235759 0.971812i \(-0.575758\pi\)
\(488\) 0 0
\(489\) −1.84833 + 0.176494i −1.84833 + 0.176494i
\(490\) 0 0
\(491\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.84125 + 0.540641i −1.84125 + 0.540641i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.916453i 0.916453i 0.888835 + 0.458227i \(0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.715370 0.715370
\(508\) 1.58006 + 0.814576i 1.58006 + 0.814576i
\(509\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(510\) 0 0
\(511\) 1.46889 + 1.27280i 1.46889 + 1.27280i
\(512\) 0 0
\(513\) −0.172932 0.0789754i −0.172932 0.0789754i
\(514\) 0 0
\(515\) 0 0
\(516\) 0.880786 0.627205i 0.880786 0.627205i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(522\) 0 0
\(523\) 1.02181 1.58997i 1.02181 1.58997i 0.235759 0.971812i \(-0.424242\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(524\) 0 0
\(525\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.654861 0.755750i −0.654861 0.755750i
\(530\) 0 0
\(531\) 0 0
\(532\) −0.131192 0.137591i −0.131192 0.137591i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.20147 + 0.291473i −1.20147 + 0.291473i −0.786053 0.618159i \(-0.787879\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(542\) 0 0
\(543\) 1.37491 1.19136i 1.37491 1.19136i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.722372 0.175245i 0.722372 0.175245i 0.142315 0.989821i \(-0.454545\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(548\) 0 0
\(549\) 0.0947329 1.98869i 0.0947329 1.98869i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.73009 0.998867i 1.73009 0.998867i
\(554\) 0 0
\(555\) 0 0
\(556\) −0.345139 + 0.755750i −0.345139 + 0.755750i
\(557\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(558\) 0 0
\(559\) 1.15358 0.821464i 1.15358 0.821464i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.580057 + 0.814576i −0.580057 + 0.814576i
\(568\) 0 0
\(569\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(570\) 0 0
\(571\) 0.481929 + 1.05528i 0.481929 + 1.05528i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.723734 0.690079i −0.723734 0.690079i
\(577\) −0.0671040 0.466718i −0.0671040 0.466718i −0.995472 0.0950560i \(-0.969697\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(578\) 0 0
\(579\) −0.514186 + 0.404360i −0.514186 + 0.404360i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(588\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(589\) 0.361109 + 0.0519196i 0.361109 + 0.0519196i
\(590\) 0 0
\(591\) 0 0
\(592\) −0.0913090 0.0268107i −0.0913090 0.0268107i
\(593\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.348311 + 1.18624i 0.348311 + 1.18624i
\(598\) 0 0
\(599\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(600\) 0 0
\(601\) −0.261197 1.35522i −0.261197 1.35522i −0.841254 0.540641i \(-0.818182\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(602\) 0 0
\(603\) −0.928368 0.371662i −0.928368 0.371662i
\(604\) −0.235759 0.408346i −0.235759 0.408346i
\(605\) 0 0
\(606\) 0 0
\(607\) 1.16832 1.48564i 1.16832 1.48564i 0.327068 0.945001i \(-0.393939\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.911911 + 1.28060i −0.911911 + 1.28060i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(618\) 0 0
\(619\) −0.547582 + 0.132842i −0.547582 + 0.132842i −0.500000 0.866025i \(-0.666667\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −0.947890 0.903811i −0.947890 0.903811i
\(625\) 0.580057 0.814576i 0.580057 0.814576i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.50979 0.0719200i −1.50979 0.0719200i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.81720 0.0865641i 1.81720 0.0865641i 0.888835 0.458227i \(-0.151515\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(632\) 0 0
\(633\) −0.723734 1.25354i −0.723734 1.25354i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.10181 + 0.708089i −1.10181 + 0.708089i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −0.209419 0.713215i −0.209419 0.713215i −0.995472 0.0950560i \(-0.969697\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.627639 1.81344i 0.627639 1.81344i
\(652\) 1.45949 + 1.14776i 1.45949 + 1.14776i
\(653\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −0.890620 1.72756i −0.890620 1.72756i
\(658\) 0 0
\(659\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(660\) 0 0
\(661\) −0.279486 + 0.0538665i −0.279486 + 0.0538665i −0.327068 0.945001i \(-0.606061\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.41089 + 0.814576i 1.41089 + 0.814576i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.25544 0.573338i −1.25544 0.573338i −0.327068 0.945001i \(-0.606061\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(674\) 0 0
\(675\) 0.723734 0.690079i 0.723734 0.690079i
\(676\) −0.517738 0.493662i −0.517738 0.493662i
\(677\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(678\) 0 0
\(679\) −1.03115 0.531595i −1.03115 0.531595i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(684\) 0.0706575 + 0.176494i 0.0706575 + 0.176494i
\(685\) 0 0
\(686\) 0 0
\(687\) −1.56199 0.625325i −1.56199 0.625325i
\(688\) −1.07028 0.153882i −1.07028 0.153882i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.890620 + 0.216062i 0.890620 + 0.216062i 0.654861 0.755750i \(-0.272727\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.928368 0.371662i 0.928368 0.371662i
\(701\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(702\) 0 0
\(703\) 0.0136729 + 0.0118476i 0.0136729 + 0.0118476i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.698939 0.449181i 0.698939 0.449181i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(710\) 0 0
\(711\) −1.97740 + 0.284307i −1.97740 + 0.284307i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(720\) 0 0
\(721\) 0.0359789 + 0.376789i 0.0359789 + 0.376789i
\(722\) 0 0
\(723\) −0.340611 0.850806i −0.340611 0.850806i
\(724\) −1.81720 0.0865641i −1.81720 0.0865641i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.84833 + 0.176494i 1.84833 + 0.176494i 0.959493 0.281733i \(-0.0909091\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(728\) 0 0
\(729\) 0.841254 0.540641i 0.841254 0.540641i
\(730\) 0 0
\(731\) 0 0
\(732\) −1.44091 + 1.37391i −1.44091 + 1.37391i
\(733\) 1.54370 + 1.21398i 1.54370 + 1.21398i 0.888835 + 0.458227i \(0.151515\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.87076 0.647478i −1.87076 0.647478i −0.981929 0.189251i \(-0.939394\pi\)
−0.888835 0.458227i \(-0.848485\pi\)
\(740\) 0 0
\(741\) 0.0925417 + 0.231158i 0.0925417 + 0.231158i
\(742\) 0 0
\(743\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.45949 0.584293i 1.45949 0.584293i 0.500000 0.866025i \(-0.333333\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.981929 0.189251i 0.981929 0.189251i
\(757\) 0.184753 + 1.93482i 0.184753 + 1.93482i 0.327068 + 0.945001i \(0.393939\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(762\) 0 0
\(763\) 0.487975 0.281733i 0.487975 0.281733i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.0475819 + 0.998867i 0.0475819 + 0.998867i
\(769\) 0.975950 1.37053i 0.975950 1.37053i 0.0475819 0.998867i \(-0.484848\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.651174 + 0.0621796i 0.651174 + 0.0621796i
\(773\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(774\) 0 0
\(775\) −0.959493 + 1.66189i −0.959493 + 1.66189i
\(776\) 0 0
\(777\) 0.0748038 0.0588264i 0.0748038 0.0588264i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.981929 + 0.189251i 0.981929 + 0.189251i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.32254 1.04006i 1.32254 1.04006i 0.327068 0.945001i \(-0.393939\pi\)
0.995472 0.0950560i \(-0.0303030\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.88720 + 1.79944i −1.88720 + 1.79944i
\(794\) 0 0
\(795\) 0 0
\(796\) 0.566514 1.09888i 0.566514 1.09888i
\(797\) 0 0 −0.945001 0.327068i \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(810\) 0 0
\(811\) −0.581419 + 1.67990i −0.581419 + 1.67990i 0.142315 + 0.989821i \(0.454545\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(812\) 0 0
\(813\) −1.28605 1.48418i −1.28605 1.48418i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.172932 + 0.111137i 0.172932 + 0.111137i
\(818\) 0 0
\(819\) 1.28605 0.247866i 1.28605 0.247866i
\(820\) 0 0
\(821\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(822\) 0 0
\(823\) −1.92837 0.371662i −1.92837 0.371662i −0.928368 0.371662i \(-0.878788\pi\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(828\) 0 0
\(829\) 0.566514 + 1.09888i 0.566514 + 1.09888i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(830\) 0 0
\(831\) 1.13779 + 1.08488i 1.13779 + 1.08488i
\(832\) 0.0623191 + 1.30824i 0.0623191 + 1.30824i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.25667 + 1.45027i −1.25667 + 1.45027i
\(838\) 0 0
\(839\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(840\) 0 0
\(841\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) −0.341254 + 1.40667i −0.341254 + 1.40667i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(848\) 0 0
\(849\) 1.12424 + 0.800570i 1.12424 + 0.800570i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.566514 + 0.720381i −0.566514 + 0.720381i −0.981929 0.189251i \(-0.939394\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(858\) 0 0
\(859\) −0.915415 1.77566i −0.915415 1.77566i −0.500000 0.866025i \(-0.666667\pi\)
−0.415415 0.909632i \(-0.636364\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.928368 + 0.371662i −0.928368 + 0.371662i
\(868\) −1.70566 + 0.879330i −1.70566 + 0.879330i
\(869\) 0 0
\(870\) 0 0
\(871\) 0.544078 + 1.19136i 0.544078 + 1.19136i
\(872\) 0 0
\(873\) 0.759713 + 0.876756i 0.759713 + 0.876756i
\(874\) 0 0
\(875\) 0 0
\(876\) −0.547582 + 1.86489i −0.547582 + 1.86489i
\(877\) −0.627639 + 0.184291i −0.627639 + 0.184291i −0.580057 0.814576i \(-0.696970\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(882\) 0 0
\(883\) −1.62731 0.0775182i −1.62731 0.0775182i −0.786053 0.618159i \(-0.787879\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(888\) 0 0
\(889\) 1.70566 + 0.500828i 1.70566 + 0.500828i
\(890\) 0 0
\(891\) 0 0
\(892\) −0.458985 1.56316i −0.458985 1.56316i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.00000 −1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 0.746170 0.782560i 0.746170 0.782560i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.21769 + 0.782560i −1.21769 + 0.782560i −0.981929 0.189251i \(-0.939394\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(912\) 0.0706575 0.176494i 0.0706575 0.176494i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.698939 + 1.53046i 0.698939 + 1.53046i
\(917\) 0 0
\(918\) 0 0
\(919\) −1.46889 1.27280i −1.46889 1.27280i −0.888835 0.458227i \(-0.848485\pi\)
−0.580057 0.814576i \(-0.696970\pi\)
\(920\) 0 0
\(921\) 1.22373 + 1.55610i 1.22373 + 1.55610i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.0845850 + 0.0436066i −0.0845850 + 0.0436066i
\(926\) 0 0
\(927\) 0.106636 0.363170i 0.106636 0.363170i
\(928\) 0 0
\(929\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(930\) 0 0
\(931\) −0.154861 0.110276i −0.154861 0.110276i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.85674 −1.85674 −0.928368 0.371662i \(-0.878788\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(938\) 0 0
\(939\) 0.654136 0.654136
\(940\) 0 0
\(941\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(948\) 1.62731 + 1.15880i 1.62731 + 1.15880i
\(949\) −0.717180 + 2.44249i −0.717180 + 2.44249i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.11435 2.44009i 1.11435 2.44009i
\(962\) 0 0
\(963\) 0 0
\(964\) −0.340611 + 0.850806i −0.340611 + 0.850806i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.995472 1.72421i −0.995472 1.72421i −0.580057 0.814576i \(-0.696970\pi\)
−0.415415 0.909632i \(-0.636364\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(972\) −0.981929 0.189251i −0.981929 0.189251i
\(973\) −0.195876 + 0.807410i −0.195876 + 0.807410i
\(974\) 0 0
\(975\) −1.30972 −1.30972
\(976\) 1.99094 1.99094
\(977\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.557730 + 0.0801894i −0.557730 + 0.0801894i
\(982\) 0 0
\(983\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0.0925417 0.231158i 0.0925417 0.231158i
\(989\) 0 0
\(990\) 0 0
\(991\) 0.853157 1.08488i 0.853157 1.08488i −0.142315 0.989821i \(-0.545455\pi\)
0.995472 0.0950560i \(-0.0303030\pi\)
\(992\) 0 0
\(993\) 1.81720 + 0.0865641i 1.81720 + 0.0865641i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.62731 + 0.0775182i −1.62731 + 0.0775182i −0.841254 0.540641i \(-0.818182\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(998\) 0 0
\(999\) −0.0913090 + 0.0268107i −0.0913090 + 0.0268107i
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 1407.1.cb.a.1319.1 20
3.2 odd 2 CM 1407.1.cb.a.1319.1 20
7.5 odd 6 1407.1.ct.a.1118.1 yes 20
21.5 even 6 1407.1.ct.a.1118.1 yes 20
67.51 odd 66 1407.1.ct.a.185.1 yes 20
201.185 even 66 1407.1.ct.a.185.1 yes 20
469.453 even 66 inner 1407.1.cb.a.1391.1 yes 20
1407.1391 odd 66 inner 1407.1.cb.a.1391.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1407.1.cb.a.1319.1 20 1.1 even 1 trivial
1407.1.cb.a.1319.1 20 3.2 odd 2 CM
1407.1.cb.a.1391.1 yes 20 469.453 even 66 inner
1407.1.cb.a.1391.1 yes 20 1407.1391 odd 66 inner
1407.1.ct.a.185.1 yes 20 67.51 odd 66
1407.1.ct.a.185.1 yes 20 201.185 even 66
1407.1.ct.a.1118.1 yes 20 7.5 odd 6
1407.1.ct.a.1118.1 yes 20 21.5 even 6