Properties

Label 371.1.n.a.97.1
Level $371$
Weight $1$
Character 371.97
Analytic conductor $0.185$
Analytic rank $0$
Dimension $12$
Projective image $D_{13}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [371,1,Mod(13,371)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(371, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 12]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("371.13");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 371.n (of order \(26\), degree \(12\), 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: \(0.185153119687\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{13}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\)

Embedding invariants

Embedding label 97.1
Root \(0.354605 - 0.935016i\) of defining polynomial
Character \(\chi\) \(=\) 371.97
Dual form 371.1.n.a.153.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.213460 + 0.112032i) q^{2} +(-0.535051 - 0.775155i) q^{4} +(0.885456 + 0.464723i) q^{7} +(-0.0564276 - 0.464723i) q^{8} +(-0.748511 - 0.663123i) q^{9} +O(q^{10})\) \(q+(0.213460 + 0.112032i) q^{2} +(-0.535051 - 0.775155i) q^{4} +(0.885456 + 0.464723i) q^{7} +(-0.0564276 - 0.464723i) q^{8} +(-0.748511 - 0.663123i) q^{9} +(1.45352 - 0.358261i) q^{11} +(0.136945 + 0.198399i) q^{14} +(-0.293978 + 0.775155i) q^{16} +(-0.0854858 - 0.225408i) q^{18} +(0.350405 + 0.0863671i) q^{22} -1.94188 q^{23} +(0.120537 + 0.992709i) q^{25} +(-0.113532 - 0.935016i) q^{28} +(-0.234068 - 0.0576926i) q^{29} +(-0.500000 + 0.442961i) q^{32} +(-0.113532 + 0.935016i) q^{36} +(-0.402877 + 1.06230i) q^{37} +(0.251489 + 0.663123i) q^{43} +(-1.05542 - 0.935016i) q^{44} +(-0.414514 - 0.217554i) q^{46} +(0.568065 + 0.822984i) q^{49} +(-0.0854858 + 0.225408i) q^{50} +(-0.748511 + 0.663123i) q^{53} +(0.166003 - 0.437715i) q^{56} +(-0.0435007 - 0.0385383i) q^{58} +(-0.354605 - 0.935016i) q^{63} +(0.648582 - 0.159861i) q^{64} +(-0.850405 - 1.23202i) q^{67} +(-0.627974 - 1.65583i) q^{71} +(-0.265932 + 0.385269i) q^{72} +(-0.205010 + 0.181623i) q^{74} +(1.45352 + 0.358261i) q^{77} +(1.56806 - 0.822984i) q^{79} +(0.120537 + 0.992709i) q^{81} +(-0.0206083 + 0.169725i) q^{86} +(-0.248511 - 0.655269i) q^{88} +(1.03901 + 1.50526i) q^{92} +(0.0290582 + 0.239316i) q^{98} +(-1.32555 - 0.695701i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 3 q^{4} - q^{7} + 9 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 3 q^{4} - q^{7} + 9 q^{8} - q^{9} - 2 q^{11} - 2 q^{14} - 5 q^{16} - 2 q^{18} - 4 q^{22} - 2 q^{23} - q^{25} - 3 q^{28} - 2 q^{29} - 6 q^{32} - 3 q^{36} - 2 q^{37} + 11 q^{43} + 7 q^{44} - 4 q^{46} - q^{49} - 2 q^{50} - q^{53} + 9 q^{56} - 4 q^{58} - q^{63} + 6 q^{64} - 2 q^{67} - 2 q^{71} - 4 q^{72} + 9 q^{74} - 2 q^{77} + 11 q^{79} - q^{81} - 4 q^{86} + 5 q^{88} - 6 q^{92} + 11 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).

\(n\) \(213\) \(267\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{13}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.213460 + 0.112032i 0.213460 + 0.112032i 0.568065 0.822984i \(-0.307692\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(3\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(4\) −0.535051 0.775155i −0.535051 0.775155i
\(5\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(6\) 0 0
\(7\) 0.885456 + 0.464723i 0.885456 + 0.464723i
\(8\) −0.0564276 0.464723i −0.0564276 0.464723i
\(9\) −0.748511 0.663123i −0.748511 0.663123i
\(10\) 0 0
\(11\) 1.45352 0.358261i 1.45352 0.358261i 0.568065 0.822984i \(-0.307692\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(12\) 0 0
\(13\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(14\) 0.136945 + 0.198399i 0.136945 + 0.198399i
\(15\) 0 0
\(16\) −0.293978 + 0.775155i −0.293978 + 0.775155i
\(17\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(18\) −0.0854858 0.225408i −0.0854858 0.225408i
\(19\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.350405 + 0.0863671i 0.350405 + 0.0863671i
\(23\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(24\) 0 0
\(25\) 0.120537 + 0.992709i 0.120537 + 0.992709i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.113532 0.935016i −0.113532 0.935016i
\(29\) −0.234068 0.0576926i −0.234068 0.0576926i 0.120537 0.992709i \(-0.461538\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(30\) 0 0
\(31\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(32\) −0.500000 + 0.442961i −0.500000 + 0.442961i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.113532 + 0.935016i −0.113532 + 0.935016i
\(37\) −0.402877 + 1.06230i −0.402877 + 1.06230i 0.568065 + 0.822984i \(0.307692\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(42\) 0 0
\(43\) 0.251489 + 0.663123i 0.251489 + 0.663123i 1.00000 \(0\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(44\) −1.05542 0.935016i −1.05542 0.935016i
\(45\) 0 0
\(46\) −0.414514 0.217554i −0.414514 0.217554i
\(47\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(48\) 0 0
\(49\) 0.568065 + 0.822984i 0.568065 + 0.822984i
\(50\) −0.0854858 + 0.225408i −0.0854858 + 0.225408i
\(51\) 0 0
\(52\) 0 0
\(53\) −0.748511 + 0.663123i −0.748511 + 0.663123i
\(54\) 0 0
\(55\) 0 0
\(56\) 0.166003 0.437715i 0.166003 0.437715i
\(57\) 0 0
\(58\) −0.0435007 0.0385383i −0.0435007 0.0385383i
\(59\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(60\) 0 0
\(61\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(62\) 0 0
\(63\) −0.354605 0.935016i −0.354605 0.935016i
\(64\) 0.648582 0.159861i 0.648582 0.159861i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.850405 1.23202i −0.850405 1.23202i −0.970942 0.239316i \(-0.923077\pi\)
0.120537 0.992709i \(-0.461538\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.627974 1.65583i −0.627974 1.65583i −0.748511 0.663123i \(-0.769231\pi\)
0.120537 0.992709i \(-0.461538\pi\)
\(72\) −0.265932 + 0.385269i −0.265932 + 0.385269i
\(73\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(74\) −0.205010 + 0.181623i −0.205010 + 0.181623i
\(75\) 0 0
\(76\) 0 0
\(77\) 1.45352 + 0.358261i 1.45352 + 0.358261i
\(78\) 0 0
\(79\) 1.56806 0.822984i 1.56806 0.822984i 0.568065 0.822984i \(-0.307692\pi\)
1.00000 \(0\)
\(80\) 0 0
\(81\) 0.120537 + 0.992709i 0.120537 + 0.992709i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.0206083 + 0.169725i −0.0206083 + 0.169725i
\(87\) 0 0
\(88\) −0.248511 0.655269i −0.248511 0.655269i
\(89\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.03901 + 1.50526i 1.03901 + 1.50526i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(98\) 0.0290582 + 0.239316i 0.0290582 + 0.239316i
\(99\) −1.32555 0.695701i −1.32555 0.695701i
\(100\) 0.705010 0.624584i 0.705010 0.624584i
\(101\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(102\) 0 0
\(103\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −0.234068 + 0.0576926i −0.234068 + 0.0576926i
\(107\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(108\) 0 0
\(109\) −0.709210 + 1.87003i −0.709210 + 1.87003i −0.354605 + 0.935016i \(0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.620537 + 0.549748i −0.620537 + 0.549748i
\(113\) 1.00599 + 0.527986i 1.00599 + 0.527986i 0.885456 0.464723i \(-0.153846\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.0805177 + 0.212308i 0.0805177 + 0.212308i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.09892 0.576756i 1.09892 0.576756i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0.0290582 0.239316i 0.0290582 0.239316i
\(127\) 1.45352 1.28771i 1.45352 1.28771i 0.568065 0.822984i \(-0.307692\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(128\) 0.804938 + 0.198399i 0.804938 + 0.198399i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.0435007 0.358261i −0.0435007 0.358261i
\(135\) 0 0
\(136\) 0 0
\(137\) 0.688601 + 0.169725i 0.688601 + 0.169725i 0.568065 0.822984i \(-0.307692\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(138\) 0 0
\(139\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.0514595 0.423807i 0.0514595 0.423807i
\(143\) 0 0
\(144\) 0.734068 0.385269i 0.734068 0.385269i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.03901 0.256092i 1.03901 0.256092i
\(149\) −0.627974 1.65583i −0.627974 1.65583i −0.748511 0.663123i \(-0.769231\pi\)
0.120537 0.992709i \(-0.461538\pi\)
\(150\) 0 0
\(151\) −0.0854858 0.704039i −0.0854858 0.704039i −0.970942 0.239316i \(-0.923077\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.270132 + 0.239316i 0.270132 + 0.239316i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(158\) 0.426920 0.426920
\(159\) 0 0
\(160\) 0 0
\(161\) −1.71945 0.902438i −1.71945 0.902438i
\(162\) −0.0854858 + 0.225408i −0.0854858 + 0.225408i
\(163\) 1.00599 + 1.45743i 1.00599 + 1.45743i 0.885456 + 0.464723i \(0.153846\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(168\) 0 0
\(169\) −0.354605 0.935016i −0.354605 0.935016i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.379463 0.549748i 0.379463 0.549748i
\(173\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(174\) 0 0
\(175\) −0.354605 + 0.935016i −0.354605 + 0.935016i
\(176\) −0.149595 + 1.23202i −0.149595 + 1.23202i
\(177\) 0 0
\(178\) 0 0
\(179\) 0.136945 1.12785i 0.136945 1.12785i −0.748511 0.663123i \(-0.769231\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(180\) 0 0
\(181\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.109576 + 0.902438i 0.109576 + 0.902438i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.49702 + 1.32625i −1.49702 + 1.32625i −0.748511 + 0.663123i \(0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(192\) 0 0
\(193\) 0.645395 0.935016i 0.645395 0.935016i −0.354605 0.935016i \(-0.615385\pi\)
1.00000 \(0\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.333997 0.880676i 0.333997 0.880676i
\(197\) −1.71945 + 0.902438i −1.71945 + 0.902438i −0.748511 + 0.663123i \(0.769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(198\) −0.205010 0.297008i −0.205010 0.297008i
\(199\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(200\) 0.454533 0.112032i 0.454533 0.112032i
\(201\) 0 0
\(202\) 0 0
\(203\) −0.180446 0.159861i −0.180446 0.159861i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.45352 + 1.28771i 1.45352 + 1.28771i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(212\) 0.914514 + 0.225408i 0.914514 + 0.225408i
\(213\) 0 0
\(214\) −0.151388 0.0794545i −0.151388 0.0794545i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.360892 + 0.319722i −0.360892 + 0.319722i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(224\) −0.648582 + 0.159861i −0.648582 + 0.159861i
\(225\) 0.568065 0.822984i 0.568065 0.822984i
\(226\) 0.155588 + 0.225408i 0.155588 + 0.225408i
\(227\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(228\) 0 0
\(229\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.0136032 + 0.112032i −0.0136032 + 0.112032i
\(233\) 1.12054 0.992709i 1.12054 0.992709i 0.120537 0.992709i \(-0.461538\pi\)
1.00000 \(0\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.00599 0.527986i 1.00599 0.527986i 0.120537 0.992709i \(-0.461538\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(240\) 0 0
\(241\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(242\) 0.299190 0.299190
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(252\) −0.535051 + 0.775155i −0.535051 + 0.775155i
\(253\) −2.82257 + 0.695701i −2.82257 + 0.695701i
\(254\) 0.454533 0.112032i 0.454533 0.112032i
\(255\) 0 0
\(256\) −0.350405 0.310432i −0.350405 0.310432i
\(257\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(258\) 0 0
\(259\) −0.850405 + 0.753393i −0.850405 + 0.753393i
\(260\) 0 0
\(261\) 0.136945 + 0.198399i 0.136945 + 0.198399i
\(262\) 0 0
\(263\) −1.32555 0.695701i −1.32555 0.695701i −0.354605 0.935016i \(-0.615385\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.500000 + 1.31839i −0.500000 + 1.31839i
\(269\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(270\) 0 0
\(271\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.127974 + 0.113375i 0.127974 + 0.113375i
\(275\) 0.530851 + 1.39974i 0.530851 + 1.39974i
\(276\) 0 0
\(277\) −1.71945 + 0.423807i −1.71945 + 0.423807i −0.970942 0.239316i \(-0.923077\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.251489 0.663123i 0.251489 0.663123i −0.748511 0.663123i \(-0.769231\pi\)
1.00000 \(0\)
\(282\) 0 0
\(283\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(284\) −0.947528 + 1.37273i −0.947528 + 1.37273i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.667993 0.667993
\(289\) −0.970942 0.239316i −0.970942 0.239316i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.516409 + 0.127283i 0.516409 + 0.127283i
\(297\) 0 0
\(298\) 0.0514595 0.423807i 0.0514595 0.423807i
\(299\) 0 0
\(300\) 0 0
\(301\) −0.0854858 + 0.704039i −0.0854858 + 0.704039i
\(302\) 0.0606274 0.159861i 0.0606274 0.159861i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(308\) −0.500000 1.31839i −0.500000 1.31839i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(312\) 0 0
\(313\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.47693 0.775155i −1.47693 0.775155i
\(317\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(318\) 0 0
\(319\) −0.360892 −0.360892
\(320\) 0 0
\(321\) 0 0
\(322\) −0.265932 0.385269i −0.265932 0.385269i
\(323\) 0 0
\(324\) 0.705010 0.624584i 0.705010 0.624584i
\(325\) 0 0
\(326\) 0.0514595 + 0.423807i 0.0514595 + 0.423807i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.00599 1.45743i 1.00599 1.45743i 0.120537 0.992709i \(-0.461538\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(332\) 0 0
\(333\) 1.00599 0.527986i 1.00599 0.527986i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.850405 + 1.23202i −0.850405 + 1.23202i 0.120537 + 0.992709i \(0.461538\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(338\) 0.0290582 0.239316i 0.0290582 0.239316i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.120537 + 0.992709i 0.120537 + 0.992709i
\(344\) 0.293978 0.154291i 0.293978 0.154291i
\(345\) 0 0
\(346\) 0 0
\(347\) 1.45352 + 0.358261i 1.45352 + 0.358261i 0.885456 0.464723i \(-0.153846\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(348\) 0 0
\(349\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(350\) −0.180446 + 0.159861i −0.180446 + 0.159861i
\(351\) 0 0
\(352\) −0.568065 + 0.822984i −0.568065 + 0.822984i
\(353\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0.155588 0.225408i 0.155588 0.225408i
\(359\) −0.234068 + 0.0576926i −0.234068 + 0.0576926i −0.354605 0.935016i \(-0.615385\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(360\) 0 0
\(361\) −0.354605 0.935016i −0.354605 0.935016i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(368\) 0.570870 1.50526i 0.570870 1.50526i
\(369\) 0 0
\(370\) 0 0
\(371\) −0.970942 + 0.239316i −0.970942 + 0.239316i
\(372\) 0 0
\(373\) 0.213460 + 0.112032i 0.213460 + 0.112032i 0.568065 0.822984i \(-0.307692\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.180446 1.48611i −0.180446 1.48611i −0.748511 0.663123i \(-0.769231\pi\)
0.568065 0.822984i \(-0.307692\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.468136 + 0.115385i −0.468136 + 0.115385i
\(383\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.242518 0.127283i 0.242518 0.127283i
\(387\) 0.251489 0.663123i 0.251489 0.663123i
\(388\) 0 0
\(389\) 0.530851 + 1.39974i 0.530851 + 1.39974i 0.885456 + 0.464723i \(0.153846\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.350405 0.310432i 0.350405 0.310432i
\(393\) 0 0
\(394\) −0.468136 −0.468136
\(395\) 0 0
\(396\) 0.169959 + 1.39974i 0.169959 + 1.39974i
\(397\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.804938 0.198399i −0.804938 0.198399i
\(401\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −0.0206083 0.0543397i −0.0206083 0.0543397i
\(407\) −0.205010 + 1.68841i −0.205010 + 1.68841i
\(408\) 0 0
\(409\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0.166003 + 0.437715i 0.166003 + 0.437715i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(420\) 0 0
\(421\) −0.402877 + 1.06230i −0.402877 + 1.06230i 0.568065 + 0.822984i \(0.307692\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(422\) −0.414514 0.217554i −0.414514 0.217554i
\(423\) 0 0
\(424\) 0.350405 + 0.310432i 0.350405 + 0.310432i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0.379463 + 0.549748i 0.379463 + 0.549748i
\(429\) 0 0
\(430\) 0 0
\(431\) 1.56806 + 0.822984i 1.56806 + 0.822984i 1.00000 \(0\)
0.568065 + 0.822984i \(0.307692\pi\)
\(432\) 0 0
\(433\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.82903 0.450815i 1.82903 0.450815i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(440\) 0 0
\(441\) 0.120537 0.992709i 0.120537 0.992709i
\(442\) 0 0
\(443\) 0.645395 0.935016i 0.645395 0.935016i −0.354605 0.935016i \(-0.615385\pi\)
1.00000 \(0\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.648582 + 0.159861i 0.648582 + 0.159861i
\(449\) −0.234068 1.92773i −0.234068 1.92773i −0.354605 0.935016i \(-0.615385\pi\)
0.120537 0.992709i \(-0.461538\pi\)
\(450\) 0.213460 0.112032i 0.213460 0.112032i
\(451\) 0 0
\(452\) −0.128987 1.06230i −0.128987 1.06230i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.0290582 0.239316i 0.0290582 0.239316i −0.970942 0.239316i \(-0.923077\pi\)
1.00000 \(0\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(462\) 0 0
\(463\) 0.645395 + 0.935016i 0.645395 + 0.935016i 1.00000 \(0\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(464\) 0.113532 0.164479i 0.113532 0.164479i
\(465\) 0 0
\(466\) 0.350405 0.0863671i 0.350405 0.0863671i
\(467\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(468\) 0 0
\(469\) −0.180446 1.48611i −0.180446 1.48611i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.603116 + 0.873764i 0.603116 + 0.873764i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.00000 1.00000
\(478\) 0.273891 0.273891
\(479\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.03505 0.543237i −1.03505 0.543237i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.627974 1.65583i −0.627974 1.65583i −0.748511 0.663123i \(-0.769231\pi\)
0.120537 0.992709i \(-0.461538\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.13613 + 1.64597i 1.13613 + 1.64597i 0.568065 + 0.822984i \(0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.213460 1.75800i 0.213460 1.75800i
\(498\) 0 0
\(499\) −0.234068 0.0576926i −0.234068 0.0576926i 0.120537 0.992709i \(-0.461538\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(504\) −0.414514 + 0.217554i −0.414514 + 0.217554i
\(505\) 0 0
\(506\) −0.680446 0.167715i −0.680446 0.167715i
\(507\) 0 0
\(508\) −1.77588 0.437715i −1.77588 0.437715i
\(509\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.333997 0.880676i −0.333997 0.880676i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −0.265932 + 0.0655463i −0.265932 + 0.0655463i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(522\) 0.00700515 + 0.0576926i 0.00700515 + 0.0576926i
\(523\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.205010 0.297008i −0.205010 0.297008i
\(527\) 0 0
\(528\) 0 0
\(529\) 2.77091 2.77091
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −0.524564 + 0.464723i −0.524564 + 0.464723i
\(537\) 0 0
\(538\) 0 0
\(539\) 1.12054 + 0.992709i 1.12054 + 0.992709i
\(540\) 0 0
\(541\) 0.688601 0.169725i 0.688601 0.169725i 0.120537 0.992709i \(-0.461538\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.136945 1.12785i 0.136945 1.12785i −0.748511 0.663123i \(-0.769231\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(548\) −0.236874 0.624584i −0.236874 0.624584i
\(549\) 0 0
\(550\) −0.0435007 + 0.358261i −0.0435007 + 0.358261i
\(551\) 0 0
\(552\) 0 0
\(553\) 1.77091 1.77091
\(554\) −0.414514 0.102169i −0.414514 0.102169i
\(555\) 0 0
\(556\) 0 0
\(557\) −1.71945 + 0.902438i −1.71945 + 0.902438i −0.748511 + 0.663123i \(0.769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.127974 0.113375i 0.127974 0.113375i
\(563\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.354605 + 0.935016i −0.354605 + 0.935016i
\(568\) −0.734068 + 0.385269i −0.734068 + 0.385269i
\(569\) 1.00599 + 1.45743i 1.00599 + 1.45743i 0.885456 + 0.464723i \(0.153846\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(570\) 0 0
\(571\) 0.688601 0.169725i 0.688601 0.169725i 0.120537 0.992709i \(-0.461538\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.234068 1.92773i −0.234068 1.92773i
\(576\) −0.591478 0.310432i −0.591478 0.310432i
\(577\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(578\) −0.180446 0.159861i −0.180446 0.159861i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.850405 + 1.23202i −0.850405 + 1.23202i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.705010 0.624584i −0.705010 0.624584i
\(593\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.947528 + 1.37273i −0.947528 + 1.37273i
\(597\) 0 0
\(598\) 0 0
\(599\) −0.402877 + 1.06230i −0.402877 + 1.06230i 0.568065 + 0.822984i \(0.307692\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(600\) 0 0
\(601\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(602\) −0.0971229 + 0.140707i −0.0971229 + 0.140707i
\(603\) −0.180446 + 1.48611i −0.180446 + 1.48611i
\(604\) −0.500000 + 0.442961i −0.500000 + 0.442961i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.0844733 0.695701i 0.0844733 0.695701i
\(617\) −0.402877 + 0.583668i −0.402877 + 0.583668i −0.970942 0.239316i \(-0.923077\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(618\) 0 0
\(619\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.970942 + 0.239316i −0.970942 + 0.239316i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.530851 + 0.470293i 0.530851 + 0.470293i 0.885456 0.464723i \(-0.153846\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(632\) −0.470942 0.682277i −0.470942 0.682277i
\(633\) 0 0
\(634\) 0.378019 + 0.198399i 0.378019 + 0.198399i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −0.0770360 0.0404316i −0.0770360 0.0404316i
\(639\) −0.627974 + 1.65583i −0.627974 + 1.65583i
\(640\) 0 0
\(641\) −1.32555 1.17433i −1.32555 1.17433i −0.970942 0.239316i \(-0.923077\pi\)
−0.354605 0.935016i \(-0.615385\pi\)
\(642\) 0 0
\(643\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(644\) 0.220465 + 1.81569i 0.220465 + 1.81569i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(648\) 0.454533 0.112032i 0.454533 0.112032i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.591478 1.55960i 0.591478 1.55960i
\(653\) 0.0290582 0.239316i 0.0290582 0.239316i −0.970942 0.239316i \(-0.923077\pi\)
1.00000 \(0\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(660\) 0 0
\(661\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(662\) 0.378019 0.198399i 0.378019 0.198399i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.273891 0.273891
\(667\) 0.454533 + 0.112032i 0.454533 + 0.112032i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.627974 + 1.65583i −0.627974 + 1.65583i 0.120537 + 0.992709i \(0.461538\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(674\) −0.319554 + 0.167715i −0.319554 + 0.167715i
\(675\) 0 0
\(676\) −0.535051 + 0.775155i −0.535051 + 0.775155i
\(677\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.180446 + 0.159861i −0.180446 + 0.159861i −0.748511 0.663123i \(-0.769231\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.0854858 + 0.225408i −0.0854858 + 0.225408i
\(687\) 0 0
\(688\) −0.587955 −0.587955
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(692\) 0 0
\(693\) −0.850405 1.23202i −0.850405 1.23202i
\(694\) 0.270132 + 0.239316i 0.270132 + 0.239316i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.914514 0.225408i 0.914514 0.225408i
\(701\) 1.45352 0.358261i 1.45352 0.358261i 0.568065 0.822984i \(-0.307692\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.885456 0.464723i 0.885456 0.464723i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.0854858 + 0.704039i −0.0854858 + 0.704039i 0.885456 + 0.464723i \(0.153846\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(710\) 0 0
\(711\) −1.71945 0.423807i −1.71945 0.423807i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −0.947528 + 0.497301i −0.947528 + 0.497301i
\(717\) 0 0
\(718\) −0.0564276 0.0139082i −0.0564276 0.0139082i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.0290582 0.239316i 0.0290582 0.239316i
\(723\) 0 0
\(724\) 0 0
\(725\) 0.0290582 0.239316i 0.0290582 0.239316i
\(726\) 0 0
\(727\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(728\) 0 0
\(729\) 0.568065 0.822984i 0.568065 0.822984i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0.970942 0.860179i 0.970942 0.860179i
\(737\) −1.67747 1.48611i −1.67747 1.48611i
\(738\) 0 0
\(739\) −0.402877 + 1.06230i −0.402877 + 1.06230i 0.568065 + 0.822984i \(0.307692\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.234068 0.0576926i −0.234068 0.0576926i
\(743\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.0330139 + 0.0478288i 0.0330139 + 0.0478288i
\(747\) 0 0
\(748\) 0 0
\(749\) −0.627974 0.329586i −0.627974 0.329586i
\(750\) 0 0
\(751\) −0.180446 0.159861i −0.180446 0.159861i 0.568065 0.822984i \(-0.307692\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00599 0.527986i 1.00599 0.527986i 0.120537 0.992709i \(-0.461538\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(758\) 0.127974 0.337440i 0.127974 0.337440i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(762\) 0 0
\(763\) −1.49702 + 1.32625i −1.49702 + 1.32625i
\(764\) 1.82903 + 0.450815i 1.82903 + 0.450815i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.07010 −1.07010
\(773\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(774\) 0.127974 0.113375i 0.127974 0.113375i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −0.0435007 + 0.358261i −0.0435007 + 0.358261i
\(779\) 0 0
\(780\) 0 0
\(781\) −1.50599 2.18181i −1.50599 2.18181i
\(782\) 0 0
\(783\) 0 0
\(784\) −0.804938 + 0.198399i −0.804938 + 0.198399i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(788\) 1.61952 + 0.849992i 1.61952 + 0.849992i
\(789\) 0 0
\(790\) 0 0
\(791\) 0.645395 + 0.935016i 0.645395 + 0.935016i
\(792\) −0.248511 + 0.655269i −0.248511 + 0.655269i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.500000 0.442961i −0.500000 0.442961i
\(801\) 0 0
\(802\) 0.242518 + 0.127283i 0.242518 + 0.127283i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.645395 + 0.935016i 0.645395 + 0.935016i 1.00000 \(0\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(810\) 0 0
\(811\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(812\) −0.0273694 + 0.225408i −0.0273694 + 0.225408i
\(813\) 0 0
\(814\) −0.232918 + 0.337440i −0.232918 + 0.337440i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.32555 + 0.695701i −1.32555 + 0.695701i −0.970942 0.239316i \(-0.923077\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(822\) 0 0
\(823\) −0.0854858 0.704039i −0.0854858 0.704039i −0.970942 0.239316i \(-0.923077\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.530851 0.470293i 0.530851 0.470293i −0.354605 0.935016i \(-0.615385\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(828\) 0.220465 1.81569i 0.220465 1.81569i
\(829\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(840\) 0 0
\(841\) −0.833997 0.437715i −0.833997 0.437715i
\(842\) −0.205010 + 0.181623i −0.205010 + 0.181623i
\(843\) 0 0
\(844\) 1.03901 + 1.50526i 1.03901 + 1.50526i
\(845\) 0 0
\(846\) 0 0
\(847\) 1.24107 1.24107
\(848\) −0.293978 0.775155i −0.293978 0.775155i
\(849\) 0 0
\(850\) 0 0
\(851\) 0.782340 2.06286i 0.782340 2.06286i
\(852\) 0 0
\(853\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0.0400190 + 0.329586i 0.0400190 + 0.329586i
\(857\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(858\) 0 0
\(859\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0.242518 + 0.351348i 0.242518 + 0.351348i
\(863\) −0.627974 + 0.329586i −0.627974 + 0.329586i −0.748511 0.663123i \(-0.769231\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.98437 1.75800i 1.98437 1.75800i
\(870\) 0 0
\(871\) 0 0
\(872\) 0.909066 + 0.224065i 0.909066 + 0.224065i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.10312 0.271894i −1.10312 0.271894i −0.354605 0.935016i \(-0.615385\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(882\) 0.136945 0.198399i 0.136945 0.198399i
\(883\) 0.688601 + 1.81569i 0.688601 + 1.81569i 0.568065 + 0.822984i \(0.307692\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.242518 0.127283i 0.242518 0.127283i
\(887\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(888\) 0 0
\(889\) 1.88546 0.464723i 1.88546 0.464723i
\(890\) 0 0
\(891\) 0.530851 + 1.39974i 0.530851 + 1.39974i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0.620537 + 0.549748i 0.620537 + 0.549748i
\(897\) 0 0
\(898\) 0.166003 0.437715i 0.166003 0.437715i
\(899\) 0 0
\(900\) −0.941884 −0.941884
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.188601 0.497301i 0.188601 0.497301i
\(905\) 0 0
\(906\) 0 0
\(907\) 1.12054 0.992709i 1.12054 0.992709i 0.120537 0.992709i \(-0.461538\pi\)
1.00000 \(0\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.688601 + 1.81569i 0.688601 + 1.81569i 0.568065 + 0.822984i \(0.307692\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.0330139 0.0478288i 0.0330139 0.0478288i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −0.402877 1.06230i −0.402877 1.06230i −0.970942 0.239316i \(-0.923077\pi\)
0.568065 0.822984i \(-0.307692\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.10312 0.271894i −1.10312 0.271894i
\(926\) 0.0330139 + 0.271894i 0.0330139 + 0.271894i
\(927\) 0 0
\(928\) 0.142590 0.0748369i 0.142590 0.0748369i
\(929\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.36905 0.337440i −1.36905 0.337440i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(938\) 0.127974 0.337440i 0.127974 0.337440i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0.0308511 + 0.254082i 0.0308511 + 0.254082i
\(947\) 1.00599 + 0.527986i 1.00599 + 0.527986i 0.885456 0.464723i \(-0.153846\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(954\) 0.213460 + 0.112032i 0.213460 + 0.112032i
\(955\) 0 0
\(956\) −0.947528 0.497301i −0.947528 0.497301i
\(957\) 0 0
\(958\) 0 0
\(959\) 0.530851 + 0.470293i 0.530851 + 0.470293i
\(960\) 0 0
\(961\) 0.885456 + 0.464723i 0.885456 + 0.464723i
\(962\) 0 0
\(963\) 0.530851 + 0.470293i 0.530851 + 0.470293i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0.136945 0.198399i 0.136945 0.198399i −0.748511 0.663123i \(-0.769231\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(968\) −0.330041 0.478147i −0.330041 0.478147i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.0514595 0.423807i 0.0514595 0.423807i
\(975\) 0 0
\(976\) 0 0
\(977\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.77091 0.929446i 1.77091 0.929446i
\(982\) 0.0581164 + 0.478631i 0.0581164 + 0.478631i
\(983\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.488363 1.28771i −0.488363 1.28771i
\(990\) 0 0
\(991\) 0.530851 1.39974i 0.530851 1.39974i −0.354605 0.935016i \(-0.615385\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0.242518 0.351348i 0.242518 0.351348i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(998\) −0.0435007 0.0385383i −0.0435007 0.0385383i
\(999\) 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 371.1.n.a.97.1 12
3.2 odd 2 3339.1.ck.a.2323.1 12
7.2 even 3 2597.1.bf.a.521.1 24
7.3 odd 6 2597.1.bf.a.362.1 24
7.4 even 3 2597.1.bf.a.362.1 24
7.5 odd 6 2597.1.bf.a.521.1 24
7.6 odd 2 CM 371.1.n.a.97.1 12
21.20 even 2 3339.1.ck.a.2323.1 12
53.47 even 13 inner 371.1.n.a.153.1 yes 12
159.47 odd 26 3339.1.ck.a.2008.1 12
371.47 odd 78 2597.1.bf.a.2432.1 24
371.100 even 39 2597.1.bf.a.2432.1 24
371.153 odd 26 inner 371.1.n.a.153.1 yes 12
371.206 odd 78 2597.1.bf.a.2273.1 24
371.312 even 39 2597.1.bf.a.2273.1 24
1113.524 even 26 3339.1.ck.a.2008.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
371.1.n.a.97.1 12 1.1 even 1 trivial
371.1.n.a.97.1 12 7.6 odd 2 CM
371.1.n.a.153.1 yes 12 53.47 even 13 inner
371.1.n.a.153.1 yes 12 371.153 odd 26 inner
2597.1.bf.a.362.1 24 7.3 odd 6
2597.1.bf.a.362.1 24 7.4 even 3
2597.1.bf.a.521.1 24 7.2 even 3
2597.1.bf.a.521.1 24 7.5 odd 6
2597.1.bf.a.2273.1 24 371.206 odd 78
2597.1.bf.a.2273.1 24 371.312 even 39
2597.1.bf.a.2432.1 24 371.47 odd 78
2597.1.bf.a.2432.1 24 371.100 even 39
3339.1.ck.a.2008.1 12 159.47 odd 26
3339.1.ck.a.2008.1 12 1113.524 even 26
3339.1.ck.a.2323.1 12 3.2 odd 2
3339.1.ck.a.2323.1 12 21.20 even 2