Properties

Label 3528.1.fb.a.2413.1
Level $3528$
Weight $1$
Character 3528.2413
Analytic conductor $1.761$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,1,Mod(397,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 21, 0, 29]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.397");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.fb (of order \(42\), 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: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{42})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{42}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

Embedding invariants

Embedding label 2413.1
Root \(0.930874 - 0.365341i\) of defining polynomial
Character \(\chi\) \(=\) 3528.2413
Dual form 3528.1.fb.a.829.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.563320 - 0.826239i) q^{2} +(-0.365341 + 0.930874i) q^{4} +(-1.90580 - 0.587862i) q^{5} +(0.955573 - 0.294755i) q^{7} +(0.974928 - 0.222521i) q^{8} +O(q^{10})\) \(q+(-0.563320 - 0.826239i) q^{2} +(-0.365341 + 0.930874i) q^{4} +(-1.90580 - 0.587862i) q^{5} +(0.955573 - 0.294755i) q^{7} +(0.974928 - 0.222521i) q^{8} +(0.587862 + 1.90580i) q^{10} +(0.728639 - 0.0546039i) q^{11} +(-0.781831 - 0.623490i) q^{14} +(-0.733052 - 0.680173i) q^{16} +(1.24349 - 1.55929i) q^{20} +(-0.455573 - 0.571270i) q^{22} +(2.46026 + 1.67738i) q^{25} +(-0.0747301 + 0.997204i) q^{28} +(1.29196 + 1.03030i) q^{29} +(-1.17809 + 0.680173i) q^{31} +(-0.149042 + 0.988831i) q^{32} -1.99441 q^{35} +(-1.98883 - 0.149042i) q^{40} +(-0.215372 + 0.698220i) q^{44} +(0.826239 - 0.563320i) q^{49} -2.97766i q^{50} +(-0.930874 - 0.365341i) q^{53} +(-1.42074 - 0.324275i) q^{55} +(0.866025 - 0.500000i) q^{56} +(0.123490 - 1.64786i) q^{58} +(1.65510 - 0.510531i) q^{59} +(1.22563 + 0.590232i) q^{62} +(0.900969 - 0.433884i) q^{64} +(1.12349 + 1.64786i) q^{70} +(1.09839 - 1.61105i) q^{73} +(0.680173 - 0.266948i) q^{77} +(0.955573 - 1.65510i) q^{79} +(0.997204 + 1.72721i) q^{80} +(-1.01507 + 0.488831i) q^{83} +(0.698220 - 0.215372i) q^{88} +0.298085i q^{97} +(-0.930874 - 0.365341i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{4} + 2 q^{7} + 6 q^{10} + 2 q^{16} + 10 q^{22} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 22 q^{40} + 2 q^{49} - 14 q^{55} - 16 q^{58} + 4 q^{64} + 8 q^{70} + 2 q^{79} - 2 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{42}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.563320 0.826239i −0.563320 0.826239i
\(3\) 0 0
\(4\) −0.365341 + 0.930874i −0.365341 + 0.930874i
\(5\) −1.90580 0.587862i −1.90580 0.587862i −0.974928 0.222521i \(-0.928571\pi\)
−0.930874 0.365341i \(-0.880952\pi\)
\(6\) 0 0
\(7\) 0.955573 0.294755i 0.955573 0.294755i
\(8\) 0.974928 0.222521i 0.974928 0.222521i
\(9\) 0 0
\(10\) 0.587862 + 1.90580i 0.587862 + 1.90580i
\(11\) 0.728639 0.0546039i 0.728639 0.0546039i 0.294755 0.955573i \(-0.404762\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(12\) 0 0
\(13\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(14\) −0.781831 0.623490i −0.781831 0.623490i
\(15\) 0 0
\(16\) −0.733052 0.680173i −0.733052 0.680173i
\(17\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 1.24349 1.55929i 1.24349 1.55929i
\(21\) 0 0
\(22\) −0.455573 0.571270i −0.455573 0.571270i
\(23\) 0 0 −0.149042 0.988831i \(-0.547619\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(24\) 0 0
\(25\) 2.46026 + 1.67738i 2.46026 + 1.67738i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i
\(29\) 1.29196 + 1.03030i 1.29196 + 1.03030i 0.997204 + 0.0747301i \(0.0238095\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(30\) 0 0
\(31\) −1.17809 + 0.680173i −1.17809 + 0.680173i −0.955573 0.294755i \(-0.904762\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(32\) −0.149042 + 0.988831i −0.149042 + 0.988831i
\(33\) 0 0
\(34\) 0 0
\(35\) −1.99441 −1.99441
\(36\) 0 0
\(37\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.98883 0.149042i −1.98883 0.149042i
\(41\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(42\) 0 0
\(43\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(44\) −0.215372 + 0.698220i −0.215372 + 0.698220i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(48\) 0 0
\(49\) 0.826239 0.563320i 0.826239 0.563320i
\(50\) 2.97766i 2.97766i
\(51\) 0 0
\(52\) 0 0
\(53\) −0.930874 0.365341i −0.930874 0.365341i −0.149042 0.988831i \(-0.547619\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(54\) 0 0
\(55\) −1.42074 0.324275i −1.42074 0.324275i
\(56\) 0.866025 0.500000i 0.866025 0.500000i
\(57\) 0 0
\(58\) 0.123490 1.64786i 0.123490 1.64786i
\(59\) 1.65510 0.510531i 1.65510 0.510531i 0.680173 0.733052i \(-0.261905\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(60\) 0 0
\(61\) 0 0 0.930874 0.365341i \(-0.119048\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(62\) 1.22563 + 0.590232i 1.22563 + 0.590232i
\(63\) 0 0
\(64\) 0.900969 0.433884i 0.900969 0.433884i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 1.12349 + 1.64786i 1.12349 + 1.64786i
\(71\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(72\) 0 0
\(73\) 1.09839 1.61105i 1.09839 1.61105i 0.365341 0.930874i \(-0.380952\pi\)
0.733052 0.680173i \(-0.238095\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.680173 0.266948i 0.680173 0.266948i
\(78\) 0 0
\(79\) 0.955573 1.65510i 0.955573 1.65510i 0.222521 0.974928i \(-0.428571\pi\)
0.733052 0.680173i \(-0.238095\pi\)
\(80\) 0.997204 + 1.72721i 0.997204 + 1.72721i
\(81\) 0 0
\(82\) 0 0
\(83\) −1.01507 + 0.488831i −1.01507 + 0.488831i −0.866025 0.500000i \(-0.833333\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.698220 0.215372i 0.698220 0.215372i
\(89\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.298085i 0.298085i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(98\) −0.930874 0.365341i −0.930874 0.365341i
\(99\) 0 0
\(100\) −2.46026 + 1.67738i −2.46026 + 1.67738i
\(101\) 1.14625 1.06356i 1.14625 1.06356i 0.149042 0.988831i \(-0.452381\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(102\) 0 0
\(103\) 0.460898 1.49419i 0.460898 1.49419i −0.365341 0.930874i \(-0.619048\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(107\) 1.90580 + 0.142820i 1.90580 + 0.142820i 0.974928 0.222521i \(-0.0714286\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(108\) 0 0
\(109\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(110\) 0.532403 + 1.35654i 0.532403 + 1.35654i
\(111\) 0 0
\(112\) −0.900969 0.433884i −0.900969 0.433884i
\(113\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.43109 + 0.826239i −1.43109 + 0.826239i
\(117\) 0 0
\(118\) −1.35417 1.07992i −1.35417 1.07992i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.460898 + 0.0694692i −0.460898 + 0.0694692i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.202749 1.34515i −0.202749 1.34515i
\(125\) −2.45921 3.08375i −2.45921 3.08375i
\(126\) 0 0
\(127\) −0.914101 + 1.14625i −0.914101 + 1.14625i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(128\) −0.866025 0.500000i −0.866025 0.500000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.432142 + 0.400969i 0.432142 + 0.400969i 0.866025 0.500000i \(-0.166667\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.294755 0.955573i \(-0.595238\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(138\) 0 0
\(139\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(140\) 0.728639 1.85654i 0.728639 1.85654i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.85654 2.72305i −1.85654 2.72305i
\(146\) −1.94986 −1.94986
\(147\) 0 0
\(148\) 0 0
\(149\) 0.250701 + 0.367711i 0.250701 + 0.367711i 0.930874 0.365341i \(-0.119048\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(150\) 0 0
\(151\) 0.365341 0.930874i 0.365341 0.930874i −0.623490 0.781831i \(-0.714286\pi\)
0.988831 0.149042i \(-0.0476190\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.603718 0.411608i −0.603718 0.411608i
\(155\) 2.64506 0.603718i 2.64506 0.603718i
\(156\) 0 0
\(157\) 0 0 −0.294755 0.955573i \(-0.595238\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(158\) −1.90580 + 0.142820i −1.90580 + 0.142820i
\(159\) 0 0
\(160\) 0.865341 1.79690i 0.865341 1.79690i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.975699 + 0.563320i 0.975699 + 0.563320i
\(167\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(168\) 0 0
\(169\) −0.623490 0.781831i −0.623490 0.781831i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.858075 + 0.129334i −0.858075 + 0.129334i −0.563320 0.826239i \(-0.690476\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(174\) 0 0
\(175\) 2.84537 + 0.877681i 2.84537 + 0.877681i
\(176\) −0.571270 0.455573i −0.571270 0.455573i
\(177\) 0 0
\(178\) 0 0
\(179\) 0.268565 1.78181i 0.268565 1.78181i −0.294755 0.955573i \(-0.595238\pi\)
0.563320 0.826239i \(-0.309524\pi\)
\(180\) 0 0
\(181\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.294755 0.955573i \(-0.404762\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(192\) 0 0
\(193\) −1.40097 + 1.29991i −1.40097 + 1.29991i −0.500000 + 0.866025i \(0.666667\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(194\) 0.246289 0.167917i 0.246289 0.167917i
\(195\) 0 0
\(196\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(197\) 1.24698i 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(198\) 0 0
\(199\) 1.32624 + 1.42935i 1.32624 + 1.42935i 0.826239 + 0.563320i \(0.190476\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 2.77183 + 1.08786i 2.77183 + 1.08786i
\(201\) 0 0
\(202\) −1.52446 0.347948i −1.52446 0.347948i
\(203\) 1.53825 + 0.603718i 1.53825 + 0.603718i
\(204\) 0 0
\(205\) 0 0
\(206\) −1.49419 + 0.460898i −1.49419 + 0.460898i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(212\) 0.680173 0.733052i 0.680173 0.733052i
\(213\) 0 0
\(214\) −0.955573 1.65510i −0.955573 1.65510i
\(215\) 0 0
\(216\) 0 0
\(217\) −0.925270 + 0.997204i −0.925270 + 0.997204i
\(218\) 0 0
\(219\) 0 0
\(220\) 0.820914 1.20406i 0.820914 1.20406i
\(221\) 0 0
\(222\) 0 0
\(223\) 0.233052 0.185853i 0.233052 0.185853i −0.500000 0.866025i \(-0.666667\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(224\) 0.149042 + 0.988831i 0.149042 + 0.988831i
\(225\) 0 0
\(226\) 0 0
\(227\) 0.930874 + 1.61232i 0.930874 + 1.61232i 0.781831 + 0.623490i \(0.214286\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(228\) 0 0
\(229\) 0 0 0.680173 0.733052i \(-0.261905\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.48883 + 0.716983i 1.48883 + 0.716983i
\(233\) 0 0 0.930874 0.365341i \(-0.119048\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.129436 + 1.72721i −0.129436 + 1.72721i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(240\) 0 0
\(241\) −1.04876 0.411608i −1.04876 0.411608i −0.222521 0.974928i \(-0.571429\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(242\) 0.317031 + 0.341678i 0.317031 + 0.341678i
\(243\) 0 0
\(244\) 0 0
\(245\) −1.90580 + 0.587862i −1.90580 + 0.587862i
\(246\) 0 0
\(247\) 0 0
\(248\) −0.997204 + 0.925270i −0.997204 + 0.925270i
\(249\) 0 0
\(250\) −1.16259 + 3.76903i −1.16259 + 3.76903i
\(251\) −0.302705 + 1.32624i −0.302705 + 1.32624i 0.563320 + 0.826239i \(0.309524\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.46200 + 0.109562i 1.46200 + 0.109562i
\(255\) 0 0
\(256\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(257\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0.0878620 0.582926i 0.0878620 0.582926i
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 1.55929 + 1.24349i 1.55929 + 1.24349i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.930874 + 0.634659i 0.930874 + 0.634659i 0.930874 0.365341i \(-0.119048\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0.0444272 + 0.294755i 0.0444272 + 0.294755i 1.00000 \(0\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.88423 + 1.08786i 1.88423 + 1.08786i
\(276\) 0 0
\(277\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −1.94440 + 0.443797i −1.94440 + 0.443797i
\(281\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(282\) 0 0
\(283\) 0 0 0.997204 0.0747301i \(-0.0238095\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(290\) −1.20406 + 3.06789i −1.20406 + 3.06789i
\(291\) 0 0
\(292\) 1.09839 + 1.61105i 1.09839 + 1.61105i
\(293\) 0.298085 0.298085 0.149042 0.988831i \(-0.452381\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(294\) 0 0
\(295\) −3.45442 −3.45442
\(296\) 0 0
\(297\) 0 0
\(298\) 0.162592 0.414278i 0.162592 0.414278i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −0.974928 + 0.222521i −0.974928 + 0.222521i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(308\) 0.730682i 0.730682i
\(309\) 0 0
\(310\) −1.98883 1.84537i −1.98883 1.84537i
\(311\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(312\) 0 0
\(313\) −0.510531 0.294755i −0.510531 0.294755i 0.222521 0.974928i \(-0.428571\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.19158 + 1.49419i 1.19158 + 1.49419i
\(317\) −0.294755 1.95557i −0.294755 1.95557i −0.294755 0.955573i \(-0.595238\pi\)
1.00000i \(-0.5\pi\)
\(318\) 0 0
\(319\) 0.997630 + 0.680173i 0.997630 + 0.680173i
\(320\) −1.97213 + 0.297251i −1.97213 + 0.297251i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(332\) −0.0841939 1.12349i −0.0841939 1.12349i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.222521 0.974928i 0.222521 0.974928i −0.733052 0.680173i \(-0.761905\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(338\) −0.294755 + 0.955573i −0.294755 + 0.955573i
\(339\) 0 0
\(340\) 0 0
\(341\) −0.821265 + 0.559929i −0.821265 + 0.559929i
\(342\) 0 0
\(343\) 0.623490 0.781831i 0.623490 0.781831i
\(344\) 0 0
\(345\) 0 0
\(346\) 0.590232 + 0.636119i 0.590232 + 0.636119i
\(347\) −1.67738 0.658322i −1.67738 0.658322i −0.680173 0.733052i \(-0.738095\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(348\) 0 0
\(349\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(350\) −0.877681 2.84537i −0.877681 2.84537i
\(351\) 0 0
\(352\) −0.0546039 + 0.728639i −0.0546039 + 0.728639i
\(353\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(359\) 0 0 0.680173 0.733052i \(-0.261905\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(360\) 0 0
\(361\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.04039 + 2.42463i −3.04039 + 2.42463i
\(366\) 0 0
\(367\) 0.975699 1.43109i 0.975699 1.43109i 0.0747301 0.997204i \(-0.476190\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.997204 0.0747301i −0.997204 0.0747301i
\(372\) 0 0
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(384\) 0 0
\(385\) −1.45320 + 0.108903i −1.45320 + 0.108903i
\(386\) 1.86323 + 0.425270i 1.86323 + 0.425270i
\(387\) 0 0
\(388\) −0.277479 0.108903i −0.277479 0.108903i
\(389\) 1.22563 + 1.32091i 1.22563 + 1.32091i 0.930874 + 0.365341i \(0.119048\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.680173 0.733052i 0.680173 0.733052i
\(393\) 0 0
\(394\) −1.03030 + 0.702449i −1.03030 + 0.702449i
\(395\) −2.79410 + 2.59255i −2.79410 + 2.59255i
\(396\) 0 0
\(397\) 0 0 0.294755 0.955573i \(-0.404762\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(398\) 0.433884 1.90097i 0.433884 1.90097i
\(399\) 0 0
\(400\) −0.662592 2.90301i −0.662592 2.90301i
\(401\) 0 0 −0.997204 0.0747301i \(-0.976190\pi\)
0.997204 + 0.0747301i \(0.0238095\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.571270 + 1.45557i 0.571270 + 1.45557i
\(405\) 0 0
\(406\) −0.367711 1.61105i −0.367711 1.61105i
\(407\) 0 0
\(408\) 0 0
\(409\) 0.258149 1.71271i 0.258149 1.71271i −0.365341 0.930874i \(-0.619048\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.22252 + 0.974928i 1.22252 + 0.974928i
\(413\) 1.43109 0.975699i 1.43109 0.975699i
\(414\) 0 0
\(415\) 2.22188 0.334895i 2.22188 0.334895i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.541044 0.678448i −0.541044 0.678448i 0.433884 0.900969i \(-0.357143\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(420\) 0 0
\(421\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.988831 0.149042i −0.988831 0.149042i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.829215 + 1.72188i −0.829215 + 1.72188i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.294755 0.955573i \(-0.595238\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(432\) 0 0
\(433\) 0.846011 0.193096i 0.846011 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(434\) 1.34515 + 0.202749i 1.34515 + 0.202749i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.04876 + 1.53825i 1.04876 + 1.53825i 0.826239 + 0.563320i \(0.190476\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(440\) −1.45728 −1.45728
\(441\) 0 0
\(442\) 0 0
\(443\) 1.11406 + 1.63402i 1.11406 + 1.63402i 0.680173 + 0.733052i \(0.261905\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.284841 0.0878620i −0.284841 0.0878620i
\(447\) 0 0
\(448\) 0.733052 0.680173i 0.733052 0.680173i
\(449\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0.807782 1.67738i 0.807782 1.67738i
\(455\) 0 0
\(456\) 0 0
\(457\) −1.21135 1.12397i −1.21135 1.12397i −0.988831 0.149042i \(-0.952381\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.21572 + 1.52446i −1.21572 + 1.52446i −0.433884 + 0.900969i \(0.642857\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(462\) 0 0
\(463\) 1.12349 + 1.40881i 1.12349 + 1.40881i 0.900969 + 0.433884i \(0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(464\) −0.246289 1.63402i −0.246289 1.63402i
\(465\) 0 0
\(466\) 0 0
\(467\) −1.54620 + 0.233052i −1.54620 + 0.233052i −0.866025 0.500000i \(-0.833333\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.50000 0.866025i 1.50000 0.866025i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0.250701 + 1.09839i 0.250701 + 1.09839i
\(483\) 0 0
\(484\) 0.103718 0.454418i 0.103718 0.454418i
\(485\) 0.175233 0.568090i 0.175233 0.568090i
\(486\) 0 0
\(487\) −1.07473 + 0.997204i −1.07473 + 0.997204i −0.0747301 + 0.997204i \(0.523810\pi\)
−1.00000 \(1.00000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.55929 + 1.24349i 1.55929 + 1.24349i
\(491\) 0.149460i 0.149460i −0.997204 0.0747301i \(-0.976190\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.32624 + 0.302705i 1.32624 + 0.302705i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(500\) 3.76903 1.16259i 3.76903 1.16259i
\(501\) 0 0
\(502\) 1.26631 0.496990i 1.26631 0.496990i
\(503\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(504\) 0 0
\(505\) −2.80974 + 1.35310i −2.80974 + 1.35310i
\(506\) 0 0
\(507\) 0 0
\(508\) −0.733052 1.26968i −0.733052 1.26968i
\(509\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0.574730 1.86323i 0.574730 1.86323i
\(512\) 0.781831 0.623490i 0.781831 0.623490i
\(513\) 0 0
\(514\) 0 0
\(515\) −1.75676 + 2.57669i −1.75676 + 2.57669i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 0 0 0.680173 0.733052i \(-0.261905\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(524\) −0.531130 + 0.255779i −0.531130 + 0.255779i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(530\) 0.149042 1.98883i 0.149042 1.98883i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −3.54812 1.39254i −3.54812 1.39254i
\(536\) 0 0
\(537\) 0 0
\(538\) 1.12664i 1.12664i
\(539\) 0.571270 0.455573i 0.571270 0.455573i
\(540\) 0 0
\(541\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(542\) 0.218511 0.202749i 0.218511 0.202749i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −0.162592 2.16964i −0.162592 2.16964i
\(551\) 0 0
\(552\) 0 0
\(553\) 0.425270 1.86323i 0.425270 1.86323i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.129436 0.0747301i 0.129436 0.0747301i −0.433884 0.900969i \(-0.642857\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.46200 + 1.35654i 1.46200 + 1.35654i
\(561\) 0 0
\(562\) 0 0
\(563\) −0.487076 0.332083i −0.487076 0.332083i 0.294755 0.955573i \(-0.404762\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.35654 + 0.101659i −1.35654 + 0.101659i −0.733052 0.680173i \(-0.761905\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(578\) −0.294755 0.955573i −0.294755 0.955573i
\(579\) 0 0
\(580\) 3.21308 0.733365i 3.21308 0.733365i
\(581\) −0.825886 + 0.766310i −0.825886 + 0.766310i
\(582\) 0 0
\(583\) −0.698220 0.215372i −0.698220 0.215372i
\(584\) 0.712362 1.81507i 0.712362 1.81507i
\(585\) 0 0
\(586\) −0.167917 0.246289i −0.167917 0.246289i
\(587\) 1.86175 1.86175 0.930874 0.365341i \(-0.119048\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 1.94594 + 2.85417i 1.94594 + 2.85417i
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.433884 + 0.0990311i −0.433884 + 0.0990311i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.997204 0.0747301i \(-0.0238095\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(600\) 0 0
\(601\) −0.590232 + 1.22563i −0.590232 + 1.22563i 0.365341 + 0.930874i \(0.380952\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0.733052 + 0.680173i 0.733052 + 0.680173i
\(605\) 0.919218 + 0.138550i 0.919218 + 0.138550i
\(606\) 0 0
\(607\) 1.61232 + 0.930874i 1.61232 + 0.930874i 0.988831 + 0.149042i \(0.0476190\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.603718 0.411608i 0.603718 0.411608i
\(617\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(618\) 0 0
\(619\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) −0.404364 + 2.68278i −0.404364 + 2.68278i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.78608 + 4.55086i 1.78608 + 4.55086i
\(626\) 0.0440542 + 0.587862i 0.0440542 + 0.587862i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.440071 + 1.92808i −0.440071 + 1.92808i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(632\) 0.563320 1.82624i 0.563320 1.82624i
\(633\) 0 0
\(634\) −1.44973 + 1.34515i −1.44973 + 1.34515i
\(635\) 2.41593 1.64715i 2.41593 1.64715i
\(636\) 0 0
\(637\) 0 0
\(638\) 1.20744i 1.20744i
\(639\) 0 0
\(640\) 1.35654 + 1.46200i 1.35654 + 1.46200i
\(641\) 0 0 −0.930874 0.365341i \(-0.880952\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(642\) 0 0
\(643\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(648\) 0 0
\(649\) 1.17809 0.462368i 1.17809 0.462368i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.101659 0.109562i 0.101659 0.109562i −0.680173 0.733052i \(-0.738095\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(654\) 0 0
\(655\) −0.587862 1.01821i −0.587862 1.01821i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.974928 0.777479i 0.974928 0.777479i 1.00000i \(-0.5\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(660\) 0 0
\(661\) 0 0 0.563320 0.826239i \(-0.309524\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −0.880843 + 0.702449i −0.880843 + 0.702449i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.658322 + 0.317031i 0.658322 + 0.317031i 0.733052 0.680173i \(-0.238095\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(674\) −0.930874 + 0.365341i −0.930874 + 0.365341i
\(675\) 0 0
\(676\) 0.955573 0.294755i 0.955573 0.294755i
\(677\) 0.139129 1.85654i 0.139129 1.85654i −0.294755 0.955573i \(-0.595238\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(678\) 0 0
\(679\) 0.0878620 + 0.284841i 0.0878620 + 0.284841i
\(680\) 0 0
\(681\) 0 0
\(682\) 0.925270 + 0.363142i 0.925270 + 0.363142i
\(683\) 0.101659 + 0.109562i 0.101659 + 0.109562i 0.781831 0.623490i \(-0.214286\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.997204 0.0747301i −0.997204 0.0747301i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.294755 0.955573i \(-0.404762\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(692\) 0.193096 0.846011i 0.193096 0.846011i
\(693\) 0 0
\(694\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.85654 + 2.32803i −1.85654 + 2.32803i
\(701\) −0.636119 1.32091i −0.636119 1.32091i −0.930874 0.365341i \(-0.880952\pi\)
0.294755 0.955573i \(-0.404762\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.632789 0.365341i 0.632789 0.365341i
\(705\) 0 0
\(706\) 0 0
\(707\) 0.781831 1.35417i 0.781831 1.35417i
\(708\) 0 0
\(709\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.56052 + 0.900969i 1.56052 + 0.900969i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(720\) 0 0
\(721\) 1.56366i 1.56366i
\(722\) 0.433884 0.900969i 0.433884 0.900969i
\(723\) 0 0
\(724\) 0 0
\(725\) 1.45035 + 4.70191i 1.45035 + 4.70191i
\(726\) 0 0
\(727\) −1.09839 + 0.250701i −1.09839 + 0.250701i −0.733052 0.680173i \(-0.761905\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 3.71604 + 1.14625i 3.71604 + 1.14625i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.563320 0.826239i \(-0.690476\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(734\) −1.73205 −1.73205
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(743\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(744\) 0 0
\(745\) −0.261623 0.848162i −0.261623 0.848162i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.86323 0.425270i 1.86323 0.425270i
\(750\) 0 0
\(751\) 0.535628 + 0.496990i 0.535628 + 0.496990i 0.900969 0.433884i \(-0.142857\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.24349 + 1.55929i −1.24349 + 1.55929i
\(756\) 0 0
\(757\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.255779 + 0.531130i 0.255779 + 0.531130i 0.988831 0.149042i \(-0.0476190\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(770\) 0.908598 + 1.13935i 0.908598 + 1.13935i
\(771\) 0 0
\(772\) −0.698220 1.77904i −0.698220 1.77904i
\(773\) −0.116853 1.55929i −0.116853 1.55929i −0.680173 0.733052i \(-0.738095\pi\)
0.563320 0.826239i \(-0.309524\pi\)
\(774\) 0 0
\(775\) −4.03932 0.302705i −4.03932 0.302705i
\(776\) 0.0663300 + 0.290611i 0.0663300 + 0.290611i
\(777\) 0 0
\(778\) 0.400969 1.75676i 0.400969 1.75676i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.988831 0.149042i −0.988831 0.149042i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.680173 0.733052i \(-0.738095\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(788\) 1.16078 + 0.455573i 1.16078 + 0.455573i
\(789\) 0 0
\(790\) 3.71604 + 0.848162i 3.71604 + 0.848162i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −1.81507 + 0.712362i −1.81507 + 0.712362i
\(797\) 0.268565 + 0.129334i 0.268565 + 0.129334i 0.563320 0.826239i \(-0.309524\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.02532 + 2.18278i −2.02532 + 2.18278i
\(801\) 0 0
\(802\) 0 0
\(803\) 0.712362 1.23385i 0.712362 1.23385i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.880843 1.29196i 0.880843 1.29196i
\(809\) 0 0 0.563320 0.826239i \(-0.309524\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(810\) 0 0
\(811\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(812\) −1.12397 + 1.21135i −1.12397 + 1.21135i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.56052 + 0.751509i −1.56052 + 0.751509i
\(819\) 0 0
\(820\) 0 0
\(821\) −1.77904 + 0.698220i −1.77904 + 0.698220i −0.781831 + 0.623490i \(0.785714\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(822\) 0 0
\(823\) 0.425270 0.131178i 0.425270 0.131178i −0.0747301 0.997204i \(-0.523810\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0.116853 1.55929i 0.116853 1.55929i
\(825\) 0 0
\(826\) −1.61232 0.632789i −1.61232 0.632789i
\(827\) −0.145713 0.0332580i −0.145713 0.0332580i 0.149042 0.988831i \(-0.452381\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(828\) 0 0
\(829\) 0 0 −0.930874 0.365341i \(-0.880952\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(830\) −1.52833 1.64715i −1.52833 1.64715i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −0.255779 + 0.829215i −0.255779 + 0.829215i
\(839\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(840\) 0 0
\(841\) 0.385113 + 1.68729i 0.385113 + 1.68729i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.728639 + 1.85654i 0.728639 + 1.85654i
\(846\) 0 0
\(847\) −0.419945 + 0.202235i −0.419945 + 0.202235i
\(848\) 0.433884 + 0.900969i 0.433884 + 0.900969i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.88980 0.284841i 1.88980 0.284841i
\(857\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(858\) 0 0
\(859\) 0 0 −0.149042 0.988831i \(-0.547619\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(864\) 0 0
\(865\) 1.71135 + 0.257945i 1.71135 + 0.257945i
\(866\) −0.636119 0.590232i −0.636119 0.590232i
\(867\) 0 0
\(868\) −0.590232 1.22563i −0.590232 1.22563i
\(869\) 0.605893 1.25815i 0.605893 1.25815i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.25890 2.22188i −3.25890 2.22188i
\(876\) 0 0
\(877\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(878\) 0.680173 1.73305i 0.680173 1.73305i
\(879\) 0 0
\(880\) 0.820914 + 1.20406i 0.820914 + 1.20406i
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.722521 1.84095i 0.722521 1.84095i
\(887\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(888\) 0 0
\(889\) −0.535628 + 1.36476i −0.535628 + 1.36476i
\(890\) 0 0
\(891\) 0 0
\(892\) 0.0878620 + 0.284841i 0.0878620 + 0.284841i
\(893\) 0 0
\(894\) 0 0
\(895\) −1.55929 + 3.23790i −1.55929 + 3.23790i
\(896\) −0.974928 0.222521i −0.974928 0.222521i
\(897\) 0 0
\(898\) 0 0
\(899\) −2.22283 0.335038i −2.22283 0.335038i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(908\) −1.84095 + 0.277479i −1.84095 + 0.277479i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(912\) 0 0
\(913\) −0.712926 + 0.411608i −0.712926 + 0.411608i
\(914\) −0.246289 + 1.63402i −0.246289 + 1.63402i
\(915\) 0 0
\(916\) 0 0
\(917\) 0.531130 + 0.255779i 0.531130 + 0.255779i
\(918\) 0 0
\(919\) 0.455573 + 1.16078i 0.455573 + 1.16078i 0.955573 + 0.294755i \(0.0952381\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.94440 + 0.145713i 1.94440 + 0.145713i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0.531130 1.72188i 0.531130 1.72188i
\(927\) 0 0
\(928\) −1.21135 + 1.12397i −1.21135 + 1.12397i
\(929\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 1.06356 + 1.14625i 1.06356 + 1.14625i
\(935\) 0 0
\(936\) 0 0
\(937\) −1.32624 0.302705i −1.32624 0.302705i −0.500000 0.866025i \(-0.666667\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.284841 0.0878620i 0.284841 0.0878620i −0.149042 0.988831i \(-0.547619\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.56052 0.751509i −1.56052 0.751509i
\(945\) 0 0
\(946\) 0 0
\(947\) 0.302705 0.326239i 0.302705 0.326239i −0.563320 0.826239i \(-0.690476\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.425270 0.736589i 0.425270 0.736589i
\(962\) 0 0
\(963\) 0 0
\(964\) 0.766310 0.825886i 0.766310 0.825886i
\(965\) 3.43414 1.65379i 3.43414 1.65379i
\(966\) 0 0
\(967\) −1.72188 0.829215i −1.72188 0.829215i −0.988831 0.149042i \(-0.952381\pi\)
−0.733052 0.680173i \(-0.761905\pi\)
\(968\) −0.433884 + 0.170287i −0.433884 + 0.170287i
\(969\) 0 0
\(970\) −0.568090 + 0.175233i −0.568090 + 0.175233i
\(971\) −0.101659 + 1.35654i −0.101659 + 1.35654i 0.680173 + 0.733052i \(0.261905\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.42935 + 0.326239i 1.42935 + 0.326239i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.680173 0.733052i \(-0.738095\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.149042 1.98883i 0.149042 1.98883i
\(981\) 0 0
\(982\) −0.123490 + 0.0841939i −0.123490 + 0.0841939i
\(983\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(984\) 0 0
\(985\) −0.733052 + 2.37650i −0.733052 + 2.37650i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.0546039 + 0.728639i 0.0546039 + 0.728639i 0.955573 + 0.294755i \(0.0952381\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(992\) −0.496990 1.26631i −0.496990 1.26631i
\(993\) 0 0
\(994\) 0 0
\(995\) −1.68729 3.50369i −1.68729 3.50369i
\(996\) 0 0
\(997\) 0 0 0.149042 0.988831i \(-0.452381\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(998\) 0 0
\(999\) 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 3528.1.fb.a.2413.1 yes 24
3.2 odd 2 inner 3528.1.fb.a.2413.2 yes 24
8.5 even 2 inner 3528.1.fb.a.2413.2 yes 24
24.5 odd 2 CM 3528.1.fb.a.2413.1 yes 24
49.45 odd 42 inner 3528.1.fb.a.829.2 yes 24
147.143 even 42 inner 3528.1.fb.a.829.1 24
392.45 odd 42 inner 3528.1.fb.a.829.1 24
1176.437 even 42 inner 3528.1.fb.a.829.2 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3528.1.fb.a.829.1 24 147.143 even 42 inner
3528.1.fb.a.829.1 24 392.45 odd 42 inner
3528.1.fb.a.829.2 yes 24 49.45 odd 42 inner
3528.1.fb.a.829.2 yes 24 1176.437 even 42 inner
3528.1.fb.a.2413.1 yes 24 1.1 even 1 trivial
3528.1.fb.a.2413.1 yes 24 24.5 odd 2 CM
3528.1.fb.a.2413.2 yes 24 3.2 odd 2 inner
3528.1.fb.a.2413.2 yes 24 8.5 even 2 inner