Properties

Label 1911.1.ea.a.1172.1
Level $1911$
Weight $1$
Character 1911.1172
Analytic conductor $0.954$
Analytic rank $0$
Dimension $24$
Projective image $D_{84}$
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] = [1911,1,Mod(110,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(84))
 
chi = DirichletCharacter(H, H._module([42, 22, 35]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1911.110");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1911.ea (of order \(84\), degree \(24\), 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: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{84})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{84}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{84} - \cdots)\)

Embedding invariants

Embedding label 1172.1
Root \(-0.149042 + 0.988831i\) of defining polynomial
Character \(\chi\) \(=\) 1911.1172
Dual form 1911.1.ea.a.1580.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.974928 - 0.222521i) q^{3} +(-0.149042 + 0.988831i) q^{4} +(0.0747301 - 0.997204i) q^{7} +(0.900969 + 0.433884i) q^{9} +O(q^{10})\) \(q+(-0.974928 - 0.222521i) q^{3} +(-0.149042 + 0.988831i) q^{4} +(0.0747301 - 0.997204i) q^{7} +(0.900969 + 0.433884i) q^{9} +(0.365341 - 0.930874i) q^{12} +(-0.680173 - 0.733052i) q^{13} +(-0.955573 - 0.294755i) q^{16} +(-1.38956 - 1.38956i) q^{19} +(-0.294755 + 0.955573i) q^{21} +(0.563320 + 0.826239i) q^{25} +(-0.781831 - 0.623490i) q^{27} +(0.974928 + 0.222521i) q^{28} +(1.91970 - 0.514383i) q^{31} +(-0.563320 + 0.826239i) q^{36} +(-0.955573 - 0.705245i) q^{37} +(0.500000 + 0.866025i) q^{39} +(0.332083 - 1.07659i) q^{43} +(0.866025 + 0.500000i) q^{48} +(-0.988831 - 0.149042i) q^{49} +(0.826239 - 0.563320i) q^{52} +(1.04551 + 1.66393i) q^{57} +(-0.571270 + 0.455573i) q^{61} +(0.500000 - 0.866025i) q^{63} +(0.433884 - 0.900969i) q^{64} +(1.19745 - 1.19745i) q^{67} +(0.340799 - 1.80117i) q^{73} +(-0.365341 - 0.930874i) q^{75} +(1.58114 - 1.16694i) q^{76} +(-0.781831 - 1.35417i) q^{79} +(0.623490 + 0.781831i) q^{81} +(-0.900969 - 0.433884i) q^{84} +(-0.781831 + 0.623490i) q^{91} +(-1.98603 + 0.0743122i) q^{93} +(-0.359154 + 0.0962349i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 2 q^{7} + 4 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{19} - 2 q^{31} - 2 q^{37} + 12 q^{39} + 6 q^{43} + 2 q^{49} + 2 q^{52} + 2 q^{57} + 12 q^{63} + 4 q^{67} - 4 q^{73} - 2 q^{75} + 4 q^{76} - 4 q^{81} - 4 q^{84} + 2 q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times\).

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{31}{42}\right)\)

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 −0.652287 0.757972i \(-0.726190\pi\)
0.652287 + 0.757972i \(0.273810\pi\)
\(3\) −0.974928 0.222521i −0.974928 0.222521i
\(4\) −0.149042 + 0.988831i −0.149042 + 0.988831i
\(5\) 0 0 −0.884115 0.467269i \(-0.845238\pi\)
0.884115 + 0.467269i \(0.154762\pi\)
\(6\) 0 0
\(7\) 0.0747301 0.997204i 0.0747301 0.997204i
\(8\) 0 0
\(9\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(10\) 0 0
\(11\) 0 0 −0.943883 0.330279i \(-0.892857\pi\)
0.943883 + 0.330279i \(0.107143\pi\)
\(12\) 0.365341 0.930874i 0.365341 0.930874i
\(13\) −0.680173 0.733052i −0.680173 0.733052i
\(14\) 0 0
\(15\) 0 0
\(16\) −0.955573 0.294755i −0.955573 0.294755i
\(17\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(18\) 0 0
\(19\) −1.38956 1.38956i −1.38956 1.38956i −0.826239 0.563320i \(-0.809524\pi\)
−0.563320 0.826239i \(-0.690476\pi\)
\(20\) 0 0
\(21\) −0.294755 + 0.955573i −0.294755 + 0.955573i
\(22\) 0 0
\(23\) 0 0 0.930874 0.365341i \(-0.119048\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(24\) 0 0
\(25\) 0.563320 + 0.826239i 0.563320 + 0.826239i
\(26\) 0 0
\(27\) −0.781831 0.623490i −0.781831 0.623490i
\(28\) 0.974928 + 0.222521i 0.974928 + 0.222521i
\(29\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(30\) 0 0
\(31\) 1.91970 0.514383i 1.91970 0.514383i 0.930874 0.365341i \(-0.119048\pi\)
0.988831 0.149042i \(-0.0476190\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.563320 + 0.826239i −0.563320 + 0.826239i
\(37\) −0.955573 0.705245i −0.955573 0.705245i 1.00000i \(-0.5\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(40\) 0 0
\(41\) 0 0 −0.884115 0.467269i \(-0.845238\pi\)
0.884115 + 0.467269i \(0.154762\pi\)
\(42\) 0 0
\(43\) 0.332083 1.07659i 0.332083 1.07659i −0.623490 0.781831i \(-0.714286\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.982566 0.185912i \(-0.0595238\pi\)
−0.982566 + 0.185912i \(0.940476\pi\)
\(48\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(49\) −0.988831 0.149042i −0.988831 0.149042i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.826239 0.563320i 0.826239 0.563320i
\(53\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.04551 + 1.66393i 1.04551 + 1.66393i
\(58\) 0 0
\(59\) 0 0 0.0373912 0.999301i \(-0.488095\pi\)
−0.0373912 + 0.999301i \(0.511905\pi\)
\(60\) 0 0
\(61\) −0.571270 + 0.455573i −0.571270 + 0.455573i −0.866025 0.500000i \(-0.833333\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(62\) 0 0
\(63\) 0.500000 0.866025i 0.500000 0.866025i
\(64\) 0.433884 0.900969i 0.433884 0.900969i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.19745 1.19745i 1.19745 1.19745i 0.222521 0.974928i \(-0.428571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.916562 0.399892i \(-0.869048\pi\)
0.916562 + 0.399892i \(0.130952\pi\)
\(72\) 0 0
\(73\) 0.340799 1.80117i 0.340799 1.80117i −0.222521 0.974928i \(-0.571429\pi\)
0.563320 0.826239i \(-0.309524\pi\)
\(74\) 0 0
\(75\) −0.365341 0.930874i −0.365341 0.930874i
\(76\) 1.58114 1.16694i 1.58114 1.16694i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.781831 1.35417i −0.781831 1.35417i −0.930874 0.365341i \(-0.880952\pi\)
0.149042 0.988831i \(-0.452381\pi\)
\(80\) 0 0
\(81\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(82\) 0 0
\(83\) 0 0 −0.330279 0.943883i \(-0.607143\pi\)
0.330279 + 0.943883i \(0.392857\pi\)
\(84\) −0.900969 0.433884i −0.900969 0.433884i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.185912 0.982566i \(-0.440476\pi\)
−0.185912 + 0.982566i \(0.559524\pi\)
\(90\) 0 0
\(91\) −0.781831 + 0.623490i −0.781831 + 0.623490i
\(92\) 0 0
\(93\) −1.98603 + 0.0743122i −1.98603 + 0.0743122i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.359154 + 0.0962349i −0.359154 + 0.0962349i −0.433884 0.900969i \(-0.642857\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(101\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(102\) 0 0
\(103\) 1.57906 0.487076i 1.57906 0.487076i 0.623490 0.781831i \(-0.285714\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(108\) 0.733052 0.680173i 0.733052 0.680173i
\(109\) 0.0487796 + 0.0566829i 0.0487796 + 0.0566829i 0.781831 0.623490i \(-0.214286\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(110\) 0 0
\(111\) 0.774683 + 0.900198i 0.774683 + 0.900198i
\(112\) −0.365341 + 0.930874i −0.365341 + 0.930874i
\(113\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.294755 0.955573i −0.294755 0.955573i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.781831 + 0.623490i 0.781831 + 0.623490i
\(122\) 0 0
\(123\) 0 0
\(124\) 0.222521 + 1.97493i 0.222521 + 1.97493i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.139129 0.0546039i −0.139129 0.0546039i 0.294755 0.955573i \(-0.404762\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(128\) 0 0
\(129\) −0.563320 + 0.975699i −0.563320 + 0.975699i
\(130\) 0 0
\(131\) 0 0 −0.680173 0.733052i \(-0.738095\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(132\) 0 0
\(133\) −1.48952 + 1.28183i −1.48952 + 1.28183i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.999301 0.0373912i \(-0.988095\pi\)
0.999301 + 0.0373912i \(0.0119048\pi\)
\(138\) 0 0
\(139\) −1.32624 + 1.42935i −1.32624 + 1.42935i −0.500000 + 0.866025i \(0.666667\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.733052 0.680173i −0.733052 0.680173i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.930874 + 0.365341i 0.930874 + 0.365341i
\(148\) 0.839789 0.839789i 0.839789 0.839789i
\(149\) 0 0 0.330279 0.943883i \(-0.392857\pi\)
−0.330279 + 0.943883i \(0.607143\pi\)
\(150\) 0 0
\(151\) −0.754903 + 1.73026i −0.754903 + 1.73026i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −0.930874 + 0.365341i −0.930874 + 0.365341i
\(157\) −0.173761 + 0.563320i −0.173761 + 0.563320i 0.826239 + 0.563320i \(0.190476\pi\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.497204 0.791295i 0.497204 0.791295i −0.500000 0.866025i \(-0.666667\pi\)
0.997204 + 0.0747301i \(0.0238095\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.399892 0.916562i \(-0.630952\pi\)
0.399892 + 0.916562i \(0.369048\pi\)
\(168\) 0 0
\(169\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i
\(170\) 0 0
\(171\) −0.649042 1.85486i −0.649042 1.85486i
\(172\) 1.01507 + 0.488831i 1.01507 + 0.488831i
\(173\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(174\) 0 0
\(175\) 0.866025 0.500000i 0.866025 0.500000i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(180\) 0 0
\(181\) 1.72188 + 0.829215i 1.72188 + 0.829215i 0.988831 + 0.149042i \(0.0476190\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(182\) 0 0
\(183\) 0.658322 0.317031i 0.658322 0.317031i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.680173 + 0.733052i −0.680173 + 0.733052i
\(190\) 0 0
\(191\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(192\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(193\) −1.49720 + 0.940755i −1.49720 + 0.940755i −0.500000 + 0.866025i \(0.666667\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.294755 0.955573i 0.294755 0.955573i
\(197\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(198\) 0 0
\(199\) −1.90580 + 0.587862i −1.90580 + 0.587862i −0.930874 + 0.365341i \(0.880952\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(200\) 0 0
\(201\) −1.43388 + 0.900969i −1.43388 + 0.900969i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.433884 + 0.900969i 0.433884 + 0.900969i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.0222759 + 0.297251i 0.0222759 + 0.297251i 0.997204 + 0.0747301i \(0.0238095\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.369485 1.95278i −0.369485 1.95278i
\(218\) 0 0
\(219\) −0.733052 + 1.68017i −0.733052 + 1.68017i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.132974 0.180173i −0.132974 0.180173i 0.733052 0.680173i \(-0.238095\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(224\) 0 0
\(225\) 0.149042 + 0.988831i 0.149042 + 0.988831i
\(226\) 0 0
\(227\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(228\) −1.80117 + 0.785841i −1.80117 + 0.785841i
\(229\) 0.0487796 1.30366i 0.0487796 1.30366i −0.733052 0.680173i \(-0.761905\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.930874 0.365341i \(-0.880952\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.460898 + 1.49419i 0.460898 + 1.49419i
\(238\) 0 0
\(239\) 0 0 −0.846724 0.532032i \(-0.821429\pi\)
0.846724 + 0.532032i \(0.178571\pi\)
\(240\) 0 0
\(241\) 0.856144 0.631863i 0.856144 0.631863i −0.0747301 0.997204i \(-0.523810\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(242\) 0 0
\(243\) −0.433884 0.900969i −0.433884 0.900969i
\(244\) −0.365341 0.632789i −0.365341 0.632789i
\(245\) 0 0
\(246\) 0 0
\(247\) −0.0734787 + 1.96376i −0.0734787 + 1.96376i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(252\) 0.781831 + 0.623490i 0.781831 + 0.623490i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(257\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(258\) 0 0
\(259\) −0.774683 + 0.900198i −0.774683 + 0.900198i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.00560 + 1.36254i 1.00560 + 1.36254i
\(269\) 0 0 0.997204 0.0747301i \(-0.0238095\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(270\) 0 0
\(271\) −1.83184 0.799225i −1.83184 0.799225i −0.930874 0.365341i \(-0.880952\pi\)
−0.900969 0.433884i \(-0.857143\pi\)
\(272\) 0 0
\(273\) 0.900969 0.433884i 0.900969 0.433884i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.26631 0.496990i −1.26631 0.496990i −0.365341 0.930874i \(-0.619048\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(278\) 0 0
\(279\) 1.95278 + 0.369485i 1.95278 + 0.369485i
\(280\) 0 0
\(281\) 0 0 −0.943883 0.330279i \(-0.892857\pi\)
0.943883 + 0.330279i \(0.107143\pi\)
\(282\) 0 0
\(283\) 1.67738 0.807782i 1.67738 0.807782i 0.680173 0.733052i \(-0.261905\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.733052 0.680173i −0.733052 0.680173i
\(290\) 0 0
\(291\) 0.371563 0.0139029i 0.371563 0.0139029i
\(292\) 1.73026 + 0.605443i 1.73026 + 0.605443i
\(293\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.974928 0.222521i 0.974928 0.222521i
\(301\) −1.04876 0.411608i −1.04876 0.411608i
\(302\) 0 0
\(303\) 0 0
\(304\) 0.918245 + 1.73740i 0.918245 + 1.73740i
\(305\) 0 0
\(306\) 0 0
\(307\) −0.218169 + 0.623490i −0.218169 + 0.623490i 0.781831 + 0.623490i \(0.214286\pi\)
−1.00000 \(1.00000\pi\)
\(308\) 0 0
\(309\) −1.64786 + 0.123490i −1.64786 + 0.123490i
\(310\) 0 0
\(311\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(312\) 0 0
\(313\) −1.71271 + 0.988831i −1.71271 + 0.988831i −0.781831 + 0.623490i \(0.785714\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.45557 0.571270i 1.45557 0.571270i
\(317\) 0 0 0.804598 0.593820i \(-0.202381\pi\)
−0.804598 + 0.593820i \(0.797619\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(325\) 0.222521 0.974928i 0.222521 0.974928i
\(326\) 0 0
\(327\) −0.0349435 0.0661163i −0.0349435 0.0661163i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.50641 + 0.169732i 1.50641 + 0.169732i 0.826239 0.563320i \(-0.190476\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(332\) 0 0
\(333\) −0.554947 1.05001i −0.554947 1.05001i
\(334\) 0 0
\(335\) 0 0
\(336\) 0.563320 0.826239i 0.563320 0.826239i
\(337\) 1.42935 0.326239i 1.42935 0.326239i 0.563320 0.826239i \(-0.309524\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(348\) 0 0
\(349\) −0.0566829 + 1.51488i −0.0566829 + 1.51488i 0.623490 + 0.781831i \(0.285714\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(350\) 0 0
\(351\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(352\) 0 0
\(353\) 0 0 0.532032 0.846724i \(-0.321429\pi\)
−0.532032 + 0.846724i \(0.678571\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.999301 0.0373912i \(-0.988095\pi\)
0.999301 + 0.0373912i \(0.0119048\pi\)
\(360\) 0 0
\(361\) 2.86175i 2.86175i
\(362\) 0 0
\(363\) −0.623490 0.781831i −0.623490 0.781831i
\(364\) −0.500000 0.866025i −0.500000 0.866025i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.590232 + 1.22563i −0.590232 + 1.22563i 0.365341 + 0.930874i \(0.380952\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.222521 1.97493i 0.222521 1.97493i
\(373\) 0.867767 0.867767 0.433884 0.900969i \(-0.357143\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.649042 0.122805i 0.649042 0.122805i 0.149042 0.988831i \(-0.452381\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(380\) 0 0
\(381\) 0.123490 + 0.0841939i 0.123490 + 0.0841939i
\(382\) 0 0
\(383\) 0 0 −0.330279 0.943883i \(-0.607143\pi\)
0.330279 + 0.943883i \(0.392857\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.766310 0.825886i 0.766310 0.825886i
\(388\) −0.0416310 0.369485i −0.0416310 0.369485i
\(389\) 0 0 0.680173 0.733052i \(-0.261905\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.425511 + 0.677197i 0.425511 + 0.677197i 0.988831 0.149042i \(-0.0476190\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(398\) 0 0
\(399\) 1.73740 0.918245i 1.73740 0.918245i
\(400\) −0.294755 0.955573i −0.294755 0.955573i
\(401\) 0 0 −0.982566 0.185912i \(-0.940476\pi\)
0.982566 + 0.185912i \(0.0595238\pi\)
\(402\) 0 0
\(403\) −1.68280 1.05737i −1.68280 1.05737i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.47493 0.643504i 1.47493 0.643504i 0.500000 0.866025i \(-0.333333\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.246289 + 1.63402i 0.246289 + 1.63402i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.61105 1.09839i 1.61105 1.09839i
\(418\) 0 0
\(419\) 0 0 0.149042 0.988831i \(-0.452381\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(420\) 0 0
\(421\) −1.05737 0.119137i −1.05737 0.119137i −0.433884 0.900969i \(-0.642857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.411608 + 0.603718i 0.411608 + 0.603718i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.846724 0.532032i \(-0.821429\pi\)
0.846724 + 0.532032i \(0.178571\pi\)
\(432\) 0.563320 + 0.826239i 0.563320 + 0.826239i
\(433\) 0.636119 + 0.590232i 0.636119 + 0.590232i 0.930874 0.365341i \(-0.119048\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.0633201 + 0.0397866i −0.0633201 + 0.0397866i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.109562 1.46200i 0.109562 1.46200i −0.623490 0.781831i \(-0.714286\pi\)
0.733052 0.680173i \(-0.238095\pi\)
\(440\) 0 0
\(441\) −0.826239 0.563320i −0.826239 0.563320i
\(442\) 0 0
\(443\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(444\) −1.00560 + 0.631863i −1.00560 + 0.631863i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.866025 0.500000i −0.866025 0.500000i
\(449\) 0 0 −0.0373912 0.999301i \(-0.511905\pi\)
0.0373912 + 0.999301i \(0.488095\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.12099 1.51889i 1.12099 1.51889i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.197979 0.104635i 0.197979 0.104635i −0.365341 0.930874i \(-0.619048\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.804598 0.593820i \(-0.797619\pi\)
0.804598 + 0.593820i \(0.202381\pi\)
\(462\) 0 0
\(463\) −1.98603 + 0.223772i −1.98603 + 0.223772i −0.988831 + 0.149042i \(0.952381\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(468\) 0.988831 0.149042i 0.988831 0.149042i
\(469\) −1.10462 1.28359i −1.10462 1.28359i
\(470\) 0 0
\(471\) 0.294755 0.510531i 0.294755 0.510531i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.365341 1.93087i 0.365341 1.93087i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.330279 0.943883i \(-0.392857\pi\)
−0.330279 + 0.943883i \(0.607143\pi\)
\(480\) 0 0
\(481\) 0.132974 + 1.18017i 0.132974 + 1.18017i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(485\) 0 0
\(486\) 0 0
\(487\) 0.751927 1.42271i 0.751927 1.42271i −0.149042 0.988831i \(-0.547619\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(488\) 0 0
\(489\) −0.660818 + 0.660818i −0.660818 + 0.660818i
\(490\) 0 0
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.98603 0.0743122i −1.98603 0.0743122i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.606214 + 0.521689i 0.606214 + 0.521689i 0.900969 0.433884i \(-0.142857\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.997204 0.0747301i \(-0.976190\pi\)
0.997204 + 0.0747301i \(0.0238095\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.294755 0.955573i 0.294755 0.955573i
\(508\) 0.0747301 0.129436i 0.0747301 0.129436i
\(509\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(510\) 0 0
\(511\) −1.77066 0.474448i −1.77066 0.474448i
\(512\) 0 0
\(513\) 0.220025 + 1.95278i 0.220025 + 1.95278i
\(514\) 0 0
\(515\) 0 0
\(516\) −0.880843 0.702449i −0.880843 0.702449i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) −0.0878620 + 0.284841i −0.0878620 + 0.284841i −0.988831 0.149042i \(-0.952381\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(524\) 0 0
\(525\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.733052 0.680173i 0.733052 0.680173i
\(530\) 0 0
\(531\) 0 0
\(532\) −1.04551 1.66393i −1.04551 1.66393i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.242536 + 1.28183i 0.242536 + 1.28183i 0.866025 + 0.500000i \(0.166667\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(542\) 0 0
\(543\) −1.49419 1.19158i −1.49419 1.19158i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.440071 1.92808i 0.440071 1.92808i 0.0747301 0.997204i \(-0.476190\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(548\) 0 0
\(549\) −0.712362 + 0.162592i −0.712362 + 0.162592i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.40881 + 0.678448i −1.40881 + 0.678448i
\(554\) 0 0
\(555\) 0 0
\(556\) −1.21572 1.52446i −1.21572 1.52446i
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) −1.01507 + 0.488831i −1.01507 + 0.488831i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.826239 0.563320i 0.826239 0.563320i
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) −0.294755 1.95557i −0.294755 1.95557i −0.294755 0.955573i \(-0.595238\pi\)
1.00000i \(-0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.781831 0.623490i 0.781831 0.623490i
\(577\) 1.34027 1.15339i 1.34027 1.15339i 0.365341 0.930874i \(-0.380952\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(578\) 0 0
\(579\) 1.66900 0.584010i 1.66900 0.584010i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(588\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(589\) −3.38231 1.95278i −3.38231 1.95278i
\(590\) 0 0
\(591\) 0 0
\(592\) 0.705245 + 0.955573i 0.705245 + 0.955573i
\(593\) 0 0 0.467269 0.884115i \(-0.345238\pi\)
−0.467269 + 0.884115i \(0.654762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.98883 0.149042i 1.98883 0.149042i
\(598\) 0 0
\(599\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(600\) 0 0
\(601\) −0.563320 + 0.826239i −0.563320 + 0.826239i −0.997204 0.0747301i \(-0.976190\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(602\) 0 0
\(603\) 1.59842 0.559311i 1.59842 0.559311i
\(604\) −1.59842 1.00435i −1.59842 1.00435i
\(605\) 0 0
\(606\) 0 0
\(607\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.754903 0.264152i 0.754903 0.264152i 0.0747301 0.997204i \(-0.476190\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.593820 0.804598i \(-0.297619\pi\)
−0.593820 + 0.804598i \(0.702381\pi\)
\(618\) 0 0
\(619\) 0.500000 + 1.86603i 0.500000 + 1.86603i 0.500000 + 0.866025i \(0.333333\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −0.222521 0.974928i −0.222521 0.974928i
\(625\) −0.365341 + 0.930874i −0.365341 + 0.930874i
\(626\) 0 0
\(627\) 0 0
\(628\) −0.531130 0.255779i −0.531130 0.255779i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.751927 + 1.42271i 0.751927 + 1.42271i 0.900969 + 0.433884i \(0.142857\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(632\) 0 0
\(633\) 0.0444272 0.294755i 0.0444272 0.294755i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.563320 + 0.826239i 0.563320 + 0.826239i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.149042 0.988831i \(-0.452381\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(642\) 0 0
\(643\) 0.436680 + 0.826239i 0.436680 + 0.826239i 1.00000 \(0\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.0743122 + 1.98603i −0.0743122 + 1.98603i
\(652\) 0.708353 + 0.609587i 0.708353 + 0.609587i
\(653\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.08855 1.47493i 1.08855 1.47493i
\(658\) 0 0
\(659\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(660\) 0 0
\(661\) 1.33485 0.467085i 1.33485 0.467085i 0.433884 0.900969i \(-0.357143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.0895474 + 0.205245i 0.0895474 + 0.205245i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.332083 + 0.487076i −0.332083 + 0.487076i −0.955573 0.294755i \(-0.904762\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(674\) 0 0
\(675\) 0.0747301 0.997204i 0.0747301 0.997204i
\(676\) −0.974928 0.222521i −0.974928 0.222521i
\(677\) 0 0 0.997204 0.0747301i \(-0.0238095\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(678\) 0 0
\(679\) 0.0691263 + 0.365341i 0.0691263 + 0.365341i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.884115 0.467269i \(-0.845238\pi\)
0.884115 + 0.467269i \(0.154762\pi\)
\(684\) 1.93087 0.365341i 1.93087 0.365341i
\(685\) 0 0
\(686\) 0 0
\(687\) −0.337649 + 1.26012i −0.337649 + 1.26012i
\(688\) −0.634659 + 0.930874i −0.634659 + 0.930874i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.0193551 0.517276i −0.0193551 0.517276i −0.974928 0.222521i \(-0.928571\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(701\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(702\) 0 0
\(703\) 0.347845 + 2.30780i 0.347845 + 2.30780i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.29637 0.146066i 1.29637 0.146066i 0.563320 0.826239i \(-0.309524\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(710\) 0 0
\(711\) −0.116853 1.55929i −0.116853 1.55929i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(720\) 0 0
\(721\) −0.367711 1.61105i −0.367711 1.61105i
\(722\) 0 0
\(723\) −0.975281 + 0.425511i −0.975281 + 0.425511i
\(724\) −1.07659 + 1.57906i −1.07659 + 1.57906i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(728\) 0 0
\(729\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(730\) 0 0
\(731\) 0 0
\(732\) 0.215372 + 0.698220i 0.215372 + 0.698220i
\(733\) 0.350958 + 1.85486i 0.350958 + 1.85486i 0.500000 + 0.866025i \(0.333333\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.98603 0.223772i 1.98603 0.223772i 0.988831 0.149042i \(-0.0476190\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(740\) 0 0
\(741\) 0.508614 1.89817i 0.508614 1.89817i
\(742\) 0 0
\(743\) 0 0 0.467269 0.884115i \(-0.345238\pi\)
−0.467269 + 0.884115i \(0.654762\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.587862 1.90580i 0.587862 1.90580i 0.222521 0.974928i \(-0.428571\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −0.623490 0.781831i −0.623490 0.781831i
\(757\) 1.92808 + 0.290611i 1.92808 + 0.290611i 0.997204 0.0747301i \(-0.0238095\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.111964 0.993712i \(-0.535714\pi\)
0.111964 + 0.993712i \(0.464286\pi\)
\(762\) 0 0
\(763\) 0.0601697 0.0444073i 0.0601697 0.0444073i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.680173 0.733052i −0.680173 0.733052i
\(769\) 0.337649 0.392355i 0.337649 0.392355i −0.563320 0.826239i \(-0.690476\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.707101 1.62069i −0.707101 1.62069i
\(773\) 0 0 −0.185912 0.982566i \(-0.559524\pi\)
0.185912 + 0.982566i \(0.440476\pi\)
\(774\) 0 0
\(775\) 1.50641 + 1.29637i 1.50641 + 1.29637i
\(776\) 0 0
\(777\) 0.955573 0.705245i 0.955573 0.705245i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(785\) 0 0
\(786\) 0 0
\(787\) −0.751927 + 1.42271i −0.751927 + 1.42271i 0.149042 + 0.988831i \(0.452381\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.722521 + 0.108903i 0.722521 + 0.108903i
\(794\) 0 0
\(795\) 0 0
\(796\) −0.297251 1.97213i −0.297251 1.97213i
\(797\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −0.677197 1.55215i −0.677197 1.55215i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(810\) 0 0
\(811\) 1.75711 0.197979i 1.75711 0.197979i 0.826239 0.563320i \(-0.190476\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(812\) 0 0
\(813\) 1.60807 + 1.18681i 1.60807 + 1.18681i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.95743 + 1.03453i −1.95743 + 1.03453i
\(818\) 0 0
\(819\) −0.974928 + 0.222521i −0.974928 + 0.222521i
\(820\) 0 0
\(821\) 0 0 0.593820 0.804598i \(-0.297619\pi\)
−0.593820 + 0.804598i \(0.702381\pi\)
\(822\) 0 0
\(823\) −0.590232 0.636119i −0.590232 0.636119i 0.365341 0.930874i \(-0.380952\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.532032 0.846724i \(-0.321429\pi\)
−0.532032 + 0.846724i \(0.678571\pi\)
\(828\) 0 0
\(829\) 0.185853 0.233052i 0.185853 0.233052i −0.680173 0.733052i \(-0.738095\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(830\) 0 0
\(831\) 1.12397 + 0.766310i 1.12397 + 0.766310i
\(832\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.82160 0.794755i −1.82160 0.794755i
\(838\) 0 0
\(839\) 0 0 0.884115 0.467269i \(-0.154762\pi\)
−0.884115 + 0.467269i \(0.845238\pi\)
\(840\) 0 0
\(841\) −0.733052 0.680173i −0.733052 0.680173i
\(842\) 0 0
\(843\) 0 0
\(844\) −0.297251 0.0222759i −0.297251 0.0222759i
\(845\) 0 0
\(846\) 0 0
\(847\) 0.680173 0.733052i 0.680173 0.733052i
\(848\) 0 0
\(849\) −1.81507 + 0.414278i −1.81507 + 0.414278i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.87590 0.211363i −1.87590 0.211363i −0.900969 0.433884i \(-0.857143\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(858\) 0 0
\(859\) −0.0222759 + 0.147791i −0.0222759 + 0.147791i −0.997204 0.0747301i \(-0.976190\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.563320 + 0.826239i 0.563320 + 0.826239i
\(868\) 1.98603 0.0743122i 1.98603 0.0743122i
\(869\) 0 0
\(870\) 0 0
\(871\) −1.69226 0.0633201i −1.69226 0.0633201i
\(872\) 0 0
\(873\) −0.365341 0.0691263i −0.365341 0.0691263i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.55215 0.975281i −1.55215 0.975281i
\(877\) −0.351438 0.559311i −0.351438 0.559311i 0.623490 0.781831i \(-0.285714\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(882\) 0 0
\(883\) 1.91115i 1.91115i 0.294755 + 0.955573i \(0.404762\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.680173 0.733052i \(-0.261905\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(888\) 0 0
\(889\) −0.0648483 + 0.134659i −0.0648483 + 0.134659i
\(890\) 0 0
\(891\) 0 0
\(892\) 0.197979 0.104635i 0.197979 0.104635i
\(893\) 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
\(903\) 0.930874 + 0.634659i 0.930874 + 0.634659i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.636119 + 1.32091i −0.636119 + 1.32091i 0.294755 + 0.955573i \(0.404762\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(912\) −0.508614 1.89817i −0.508614 1.89817i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.28183 + 0.242536i 1.28183 + 0.242536i
\(917\) 0 0
\(918\) 0 0
\(919\) 0.367554 + 0.460898i 0.367554 + 0.460898i 0.930874 0.365341i \(-0.119048\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(920\) 0 0
\(921\) 0.351438 0.559311i 0.351438 0.559311i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.0444073 1.18681i 0.0444073 1.18681i
\(926\) 0 0
\(927\) 1.63402 + 0.246289i 1.63402 + 0.246289i
\(928\) 0 0
\(929\) 0 0 −0.943883 0.330279i \(-0.892857\pi\)
0.943883 + 0.330279i \(0.107143\pi\)
\(930\) 0 0
\(931\) 1.16694 + 1.58114i 1.16694 + 1.58114i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.145713 + 0.0332580i −0.145713 + 0.0332580i −0.294755 0.955573i \(-0.595238\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(938\) 0 0
\(939\) 1.88980 0.582926i 1.88980 0.582926i
\(940\) 0 0
\(941\) 0 0 −0.467269 0.884115i \(-0.654762\pi\)
0.467269 + 0.884115i \(0.345238\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.467269 0.884115i \(-0.654762\pi\)
0.467269 + 0.884115i \(0.345238\pi\)
\(948\) −1.54620 + 0.233052i −1.54620 + 0.233052i
\(949\) −1.55215 + 0.975281i −1.55215 + 0.975281i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.930874 0.365341i \(-0.119048\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.55465 1.47493i 2.55465 1.47493i
\(962\) 0 0
\(963\) 0 0
\(964\) 0.497204 + 0.940755i 0.497204 + 0.940755i
\(965\) 0 0
\(966\) 0 0
\(967\) 0.500684 1.43087i 0.500684 1.43087i −0.365341 0.930874i \(-0.619048\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(972\) 0.955573 0.294755i 0.955573 0.294755i
\(973\) 1.32624 + 1.42935i 1.32624 + 1.42935i
\(974\) 0 0
\(975\) −0.433884 + 0.900969i −0.433884 + 0.900969i
\(976\) 0.680173 0.266948i 0.680173 0.266948i
\(977\) 0 0 −0.532032 0.846724i \(-0.678571\pi\)
0.532032 + 0.846724i \(0.321429\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.0193551 + 0.0722342i 0.0193551 + 0.0722342i
\(982\) 0 0
\(983\) 0 0 0.999301 0.0373912i \(-0.0119048\pi\)
−0.999301 + 0.0373912i \(0.988095\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −1.93087 0.365341i −1.93087 0.365341i
\(989\) 0 0
\(990\) 0 0
\(991\) 1.48883 0.716983i 1.48883 0.716983i 0.500000 0.866025i \(-0.333333\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(992\) 0 0
\(993\) −1.43087 0.500684i −1.43087 0.500684i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.61232 0.632789i −1.61232 0.632789i −0.623490 0.781831i \(-0.714286\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(998\) 0 0
\(999\) 0.307384 + 1.14717i 0.307384 + 1.14717i
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 1911.1.ea.a.1172.1 24
3.2 odd 2 CM 1911.1.ea.a.1172.1 24
13.7 odd 12 1911.1.ep.a.878.1 yes 24
39.20 even 12 1911.1.ep.a.878.1 yes 24
49.12 odd 42 1911.1.ep.a.1874.1 yes 24
147.110 even 42 1911.1.ep.a.1874.1 yes 24
637.306 even 84 inner 1911.1.ea.a.1580.1 yes 24
1911.1580 odd 84 inner 1911.1.ea.a.1580.1 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1911.1.ea.a.1172.1 24 1.1 even 1 trivial
1911.1.ea.a.1172.1 24 3.2 odd 2 CM
1911.1.ea.a.1580.1 yes 24 637.306 even 84 inner
1911.1.ea.a.1580.1 yes 24 1911.1580 odd 84 inner
1911.1.ep.a.878.1 yes 24 13.7 odd 12
1911.1.ep.a.878.1 yes 24 39.20 even 12
1911.1.ep.a.1874.1 yes 24 49.12 odd 42
1911.1.ep.a.1874.1 yes 24 147.110 even 42