Properties

Label 1352.1.p.c.147.6
Level $1352$
Weight $1$
Character 1352.147
Analytic conductor $0.675$
Analytic rank $0$
Dimension $12$
Projective image $D_{7}$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1352,1,Mod(147,1352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1352.147");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1352.p (of order \(6\), degree \(2\), not 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: \(0.674735897080\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.2471326208.1

Embedding invariants

Embedding label 147.6
Root \(1.07992 + 0.623490i\) of defining polynomial
Character \(\chi\) \(=\) 1352.147
Dual form 1352.1.p.c.699.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.900969 - 1.56052i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.56052 - 0.900969i) q^{6} +1.00000i q^{8} +(-1.12349 - 1.94594i) q^{9} +O(q^{10})\) \(q+(0.866025 + 0.500000i) q^{2} +(0.900969 - 1.56052i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.56052 - 0.900969i) q^{6} +1.00000i q^{8} +(-1.12349 - 1.94594i) q^{9} +(0.385418 + 0.222521i) q^{11} +1.80194 q^{12} +(-0.500000 + 0.866025i) q^{16} +(-0.222521 - 0.385418i) q^{17} -2.24698i q^{18} +(-1.07992 + 0.623490i) q^{19} +(0.222521 + 0.385418i) q^{22} +(1.56052 + 0.900969i) q^{24} -1.00000 q^{25} -2.24698 q^{27} +(-0.866025 + 0.500000i) q^{32} +(0.694498 - 0.400969i) q^{33} -0.445042i q^{34} +(1.12349 - 1.94594i) q^{36} -1.24698 q^{38} +(1.07992 + 0.623490i) q^{41} +(0.623490 + 1.07992i) q^{43} +0.445042i q^{44} +(0.900969 + 1.56052i) q^{48} +(0.500000 - 0.866025i) q^{49} +(-0.866025 - 0.500000i) q^{50} -0.801938 q^{51} +(-1.94594 - 1.12349i) q^{54} +2.24698i q^{57} +(-1.56052 + 0.900969i) q^{59} -1.00000 q^{64} +0.801938 q^{66} +(-1.56052 - 0.900969i) q^{67} +(0.222521 - 0.385418i) q^{68} +(1.94594 - 1.12349i) q^{72} -1.80194i q^{73} +(-0.900969 + 1.56052i) q^{75} +(-1.07992 - 0.623490i) q^{76} +(-0.900969 + 1.56052i) q^{81} +(0.623490 + 1.07992i) q^{82} -1.24698i q^{83} +1.24698i q^{86} +(-0.222521 + 0.385418i) q^{88} +(1.56052 + 0.900969i) q^{89} +1.80194i q^{96} +(0.385418 - 0.222521i) q^{97} +(0.866025 - 0.500000i) q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9} + 4 q^{12} - 6 q^{16} - 2 q^{17} + 2 q^{22} - 12 q^{25} - 8 q^{27} + 4 q^{36} + 4 q^{38} - 2 q^{43} + 2 q^{48} + 6 q^{49} + 8 q^{51} - 12 q^{64} - 8 q^{66} + 2 q^{68} - 2 q^{75} - 2 q^{81} - 2 q^{82} - 2 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(3\) 0.900969 1.56052i 0.900969 1.56052i 0.0747301 0.997204i \(-0.476190\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(4\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.56052 0.900969i 1.56052 0.900969i
\(7\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 1.00000i 1.00000i
\(9\) −1.12349 1.94594i −1.12349 1.94594i
\(10\) 0 0
\(11\) 0.385418 + 0.222521i 0.385418 + 0.222521i 0.680173 0.733052i \(-0.261905\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(12\) 1.80194 1.80194
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) −0.222521 0.385418i −0.222521 0.385418i 0.733052 0.680173i \(-0.238095\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(18\) 2.24698i 2.24698i
\(19\) −1.07992 + 0.623490i −1.07992 + 0.623490i −0.930874 0.365341i \(-0.880952\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 1.56052 + 0.900969i 1.56052 + 0.900969i
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) −2.24698 −2.24698
\(28\) 0 0
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(33\) 0.694498 0.400969i 0.694498 0.400969i
\(34\) 0.445042i 0.445042i
\(35\) 0 0
\(36\) 1.12349 1.94594i 1.12349 1.94594i
\(37\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(38\) −1.24698 −1.24698
\(39\) 0 0
\(40\) 0 0
\(41\) 1.07992 + 0.623490i 1.07992 + 0.623490i 0.930874 0.365341i \(-0.119048\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(42\) 0 0
\(43\) 0.623490 + 1.07992i 0.623490 + 1.07992i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(44\) 0.445042i 0.445042i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0.900969 + 1.56052i 0.900969 + 1.56052i
\(49\) 0.500000 0.866025i 0.500000 0.866025i
\(50\) −0.866025 0.500000i −0.866025 0.500000i
\(51\) −0.801938 −0.801938
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −1.94594 1.12349i −1.94594 1.12349i
\(55\) 0 0
\(56\) 0 0
\(57\) 2.24698i 2.24698i
\(58\) 0 0
\(59\) −1.56052 + 0.900969i −1.56052 + 0.900969i −0.563320 + 0.826239i \(0.690476\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0.801938 0.801938
\(67\) −1.56052 0.900969i −1.56052 0.900969i −0.997204 0.0747301i \(-0.976190\pi\)
−0.563320 0.826239i \(-0.690476\pi\)
\(68\) 0.222521 0.385418i 0.222521 0.385418i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 1.94594 1.12349i 1.94594 1.12349i
\(73\) 1.80194i 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(74\) 0 0
\(75\) −0.900969 + 1.56052i −0.900969 + 1.56052i
\(76\) −1.07992 0.623490i −1.07992 0.623490i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −0.900969 + 1.56052i −0.900969 + 1.56052i
\(82\) 0.623490 + 1.07992i 0.623490 + 1.07992i
\(83\) 1.24698i 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.24698i 1.24698i
\(87\) 0 0
\(88\) −0.222521 + 0.385418i −0.222521 + 0.385418i
\(89\) 1.56052 + 0.900969i 1.56052 + 0.900969i 0.997204 + 0.0747301i \(0.0238095\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.80194i 1.80194i
\(97\) 0.385418 0.222521i 0.385418 0.222521i −0.294755 0.955573i \(-0.595238\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(98\) 0.866025 0.500000i 0.866025 0.500000i
\(99\) 1.00000i 1.00000i
\(100\) −0.500000 0.866025i −0.500000 0.866025i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) −0.694498 0.400969i −0.694498 0.400969i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.623490 + 1.07992i −0.623490 + 1.07992i 0.365341 + 0.930874i \(0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(108\) −1.12349 1.94594i −1.12349 1.94594i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.900969 + 1.56052i 0.900969 + 1.56052i 0.826239 + 0.563320i \(0.190476\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(114\) −1.12349 + 1.94594i −1.12349 + 1.94594i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −1.80194 −1.80194
\(119\) 0 0
\(120\) 0 0
\(121\) −0.400969 0.694498i −0.400969 0.694498i
\(122\) 0 0
\(123\) 1.94594 1.12349i 1.94594 1.12349i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) −0.866025 0.500000i −0.866025 0.500000i
\(129\) 2.24698 2.24698
\(130\) 0 0
\(131\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(132\) 0.694498 + 0.400969i 0.694498 + 0.400969i
\(133\) 0 0
\(134\) −0.900969 1.56052i −0.900969 1.56052i
\(135\) 0 0
\(136\) 0.385418 0.222521i 0.385418 0.222521i
\(137\) −1.56052 + 0.900969i −1.56052 + 0.900969i −0.563320 + 0.826239i \(0.690476\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(138\) 0 0
\(139\) 0.222521 + 0.385418i 0.222521 + 0.385418i 0.955573 0.294755i \(-0.0952381\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.24698 2.24698
\(145\) 0 0
\(146\) 0.900969 1.56052i 0.900969 1.56052i
\(147\) −0.900969 1.56052i −0.900969 1.56052i
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) −1.56052 + 0.900969i −1.56052 + 0.900969i
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −0.623490 1.07992i −0.623490 1.07992i
\(153\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −1.56052 + 0.900969i −1.56052 + 0.900969i
\(163\) −0.385418 + 0.222521i −0.385418 + 0.222521i −0.680173 0.733052i \(-0.738095\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(164\) 1.24698i 1.24698i
\(165\) 0 0
\(166\) 0.623490 1.07992i 0.623490 1.07992i
\(167\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 2.42655 + 1.40097i 2.42655 + 1.40097i
\(172\) −0.623490 + 1.07992i −0.623490 + 1.07992i
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.385418 + 0.222521i −0.385418 + 0.222521i
\(177\) 3.24698i 3.24698i
\(178\) 0.900969 + 1.56052i 0.900969 + 1.56052i
\(179\) −0.222521 + 0.385418i −0.222521 + 0.385418i −0.955573 0.294755i \(-0.904762\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.198062i 0.198062i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) −0.900969 + 1.56052i −0.900969 + 1.56052i
\(193\) −1.07992 0.623490i −1.07992 0.623490i −0.149042 0.988831i \(-0.547619\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(194\) 0.445042 0.445042
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(198\) 0.500000 0.866025i 0.500000 0.866025i
\(199\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 1.00000i 1.00000i
\(201\) −2.81197 + 1.62349i −2.81197 + 1.62349i
\(202\) 0 0
\(203\) 0 0
\(204\) −0.400969 0.694498i −0.400969 0.694498i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.554958 −0.554958
\(210\) 0 0
\(211\) 0.222521 0.385418i 0.222521 0.385418i −0.733052 0.680173i \(-0.761905\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −1.07992 + 0.623490i −1.07992 + 0.623490i
\(215\) 0 0
\(216\) 2.24698i 2.24698i
\(217\) 0 0
\(218\) 0 0
\(219\) −2.81197 1.62349i −2.81197 1.62349i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(224\) 0 0
\(225\) 1.12349 + 1.94594i 1.12349 + 1.94594i
\(226\) 1.80194i 1.80194i
\(227\) 1.56052 0.900969i 1.56052 0.900969i 0.563320 0.826239i \(-0.309524\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(228\) −1.94594 + 1.12349i −1.94594 + 1.12349i
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.56052 0.900969i −1.56052 0.900969i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(242\) 0.801938i 0.801938i
\(243\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(244\) 0 0
\(245\) 0 0
\(246\) 2.24698 2.24698
\(247\) 0 0
\(248\) 0 0
\(249\) −1.94594 1.12349i −1.94594 1.12349i
\(250\) 0 0
\(251\) −0.222521 0.385418i −0.222521 0.385418i 0.733052 0.680173i \(-0.238095\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) 0.623490 1.07992i 0.623490 1.07992i −0.365341 0.930874i \(-0.619048\pi\)
0.988831 0.149042i \(-0.0476190\pi\)
\(258\) 1.94594 + 1.12349i 1.94594 + 1.12349i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 1.07992 + 0.623490i 1.07992 + 0.623490i
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0.400969 + 0.694498i 0.400969 + 0.694498i
\(265\) 0 0
\(266\) 0 0
\(267\) 2.81197 1.62349i 2.81197 1.62349i
\(268\) 1.80194i 1.80194i
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0.445042 0.445042
\(273\) 0 0
\(274\) −1.80194 −1.80194
\(275\) −0.385418 0.222521i −0.385418 0.222521i
\(276\) 0 0
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0.445042i 0.445042i
\(279\) 0 0
\(280\) 0 0
\(281\) 0.445042i 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(282\) 0 0
\(283\) 0.623490 1.07992i 0.623490 1.07992i −0.365341 0.930874i \(-0.619048\pi\)
0.988831 0.149042i \(-0.0476190\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.94594 + 1.12349i 1.94594 + 1.12349i
\(289\) 0.400969 0.694498i 0.400969 0.694498i
\(290\) 0 0
\(291\) 0.801938i 0.801938i
\(292\) 1.56052 0.900969i 1.56052 0.900969i
\(293\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(294\) 1.80194i 1.80194i
\(295\) 0 0
\(296\) 0 0
\(297\) −0.866025 0.500000i −0.866025 0.500000i
\(298\) 0 0
\(299\) 0 0
\(300\) −1.80194 −1.80194
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.24698i 1.24698i
\(305\) 0 0
\(306\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(307\) 0.445042i 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 1.12349 + 1.94594i 1.12349 + 1.94594i
\(322\) 0 0
\(323\) 0.480608 + 0.277479i 0.480608 + 0.277479i
\(324\) −1.80194 −1.80194
\(325\) 0 0
\(326\) −0.445042 −0.445042
\(327\) 0 0
\(328\) −0.623490 + 1.07992i −0.623490 + 1.07992i
\(329\) 0 0
\(330\) 0 0
\(331\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 1.07992 0.623490i 1.07992 0.623490i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(338\) 0 0
\(339\) 3.24698 3.24698
\(340\) 0 0
\(341\) 0 0
\(342\) 1.40097 + 2.42655i 1.40097 + 2.42655i
\(343\) 0 0
\(344\) −1.07992 + 0.623490i −1.07992 + 0.623490i
\(345\) 0 0
\(346\) 0 0
\(347\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(348\) 0 0
\(349\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.445042 −0.445042
\(353\) −0.385418 0.222521i −0.385418 0.222521i 0.294755 0.955573i \(-0.404762\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(354\) −1.62349 + 2.81197i −1.62349 + 2.81197i
\(355\) 0 0
\(356\) 1.80194i 1.80194i
\(357\) 0 0
\(358\) −0.385418 + 0.222521i −0.385418 + 0.222521i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 0.277479 0.480608i 0.277479 0.480608i
\(362\) 0 0
\(363\) −1.44504 −1.44504
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0 0
\(369\) 2.80194i 2.80194i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0.0990311 0.171527i 0.0990311 0.171527i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.56052 0.900969i −1.56052 0.900969i −0.997204 0.0747301i \(-0.976190\pi\)
−0.563320 0.826239i \(-0.690476\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) −1.56052 + 0.900969i −1.56052 + 0.900969i
\(385\) 0 0
\(386\) −0.623490 1.07992i −0.623490 1.07992i
\(387\) 1.40097 2.42655i 1.40097 2.42655i
\(388\) 0.385418 + 0.222521i 0.385418 + 0.222521i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(393\) 1.12349 1.94594i 1.12349 1.94594i
\(394\) 0 0
\(395\) 0 0
\(396\) 0.866025 0.500000i 0.866025 0.500000i
\(397\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.500000 0.866025i 0.500000 0.866025i
\(401\) 0.385418 + 0.222521i 0.385418 + 0.222521i 0.680173 0.733052i \(-0.261905\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(402\) −3.24698 −3.24698
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0.801938i 0.801938i
\(409\) 1.56052 0.900969i 1.56052 0.900969i 0.563320 0.826239i \(-0.309524\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(410\) 0 0
\(411\) 3.24698i 3.24698i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.801938 0.801938
\(418\) −0.480608 0.277479i −0.480608 0.277479i
\(419\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0.385418 0.222521i 0.385418 0.222521i
\(423\) 0 0
\(424\) 0 0
\(425\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(426\) 0 0
\(427\) 0 0
\(428\) −1.24698 −1.24698
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) 1.12349 1.94594i 1.12349 1.94594i
\(433\) −0.900969 1.56052i −0.900969 1.56052i −0.826239 0.563320i \(-0.809524\pi\)
−0.0747301 0.997204i \(-0.523810\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −1.62349 2.81197i −1.62349 2.81197i
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) −2.24698 −2.24698
\(442\) 0 0
\(443\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.07992 0.623490i 1.07992 0.623490i 0.149042 0.988831i \(-0.452381\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(450\) 2.24698i 2.24698i
\(451\) 0.277479 + 0.480608i 0.277479 + 0.480608i
\(452\) −0.900969 + 1.56052i −0.900969 + 1.56052i
\(453\) 0 0
\(454\) 1.80194 1.80194
\(455\) 0 0
\(456\) −2.24698 −2.24698
\(457\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(458\) 0 0
\(459\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(460\) 0 0
\(461\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.56052 + 0.900969i 1.56052 + 0.900969i
\(467\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −0.900969 1.56052i −0.900969 1.56052i
\(473\) 0.554958i 0.554958i
\(474\) 0 0
\(475\) 1.07992 0.623490i 1.07992 0.623490i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.00000 2.00000
\(483\) 0 0
\(484\) 0.400969 0.694498i 0.400969 0.694498i
\(485\) 0 0
\(486\) 1.00000i 1.00000i
\(487\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(488\) 0 0
\(489\) 0.801938i 0.801938i
\(490\) 0 0
\(491\) −0.900969 + 1.56052i −0.900969 + 1.56052i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(492\) 1.94594 + 1.12349i 1.94594 + 1.12349i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −1.12349 1.94594i −1.12349 1.94594i
\(499\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.445042i 0.445042i
\(503\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 2.42655 1.40097i 2.42655 1.40097i
\(514\) 1.07992 0.623490i 1.07992 0.623490i
\(515\) 0 0
\(516\) 1.12349 + 1.94594i 1.12349 + 1.94594i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(522\) 0 0
\(523\) 0.222521 0.385418i 0.222521 0.385418i −0.733052 0.680173i \(-0.761905\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(524\) 0.623490 + 1.07992i 0.623490 + 1.07992i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0.801938i 0.801938i
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) 0 0
\(531\) 3.50647 + 2.02446i 3.50647 + 2.02446i
\(532\) 0 0
\(533\) 0 0
\(534\) 3.24698 3.24698
\(535\) 0 0
\(536\) 0.900969 1.56052i 0.900969 1.56052i
\(537\) 0.400969 + 0.694498i 0.400969 + 0.694498i
\(538\) 0 0
\(539\) 0.385418 0.222521i 0.385418 0.222521i
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0.385418 + 0.222521i 0.385418 + 0.222521i
\(545\) 0 0
\(546\) 0 0
\(547\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(548\) −1.56052 0.900969i −1.56052 0.900969i
\(549\) 0 0
\(550\) −0.222521 0.385418i −0.222521 0.385418i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.222521 + 0.385418i −0.222521 + 0.385418i
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −0.309081 0.178448i −0.309081 0.178448i
\(562\) 0.222521 0.385418i 0.222521 0.385418i
\(563\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.07992 0.623490i 1.07992 0.623490i
\(567\) 0 0
\(568\) 0 0
\(569\) −0.900969 + 1.56052i −0.900969 + 1.56052i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(570\) 0 0
\(571\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.12349 + 1.94594i 1.12349 + 1.94594i
\(577\) 1.24698i 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(578\) 0.694498 0.400969i 0.694498 0.400969i
\(579\) −1.94594 + 1.12349i −1.94594 + 1.12349i
\(580\) 0 0
\(581\) 0 0
\(582\) 0.400969 0.694498i 0.400969 0.694498i
\(583\) 0 0
\(584\) 1.80194 1.80194
\(585\) 0 0
\(586\) 0 0
\(587\) 1.07992 + 0.623490i 1.07992 + 0.623490i 0.930874 0.365341i \(-0.119048\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(588\) 0.900969 1.56052i 0.900969 1.56052i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.445042i 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(594\) −0.500000 0.866025i −0.500000 0.866025i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −1.56052 0.900969i −1.56052 0.900969i
\(601\) −0.623490 + 1.07992i −0.623490 + 1.07992i 0.365341 + 0.930874i \(0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(602\) 0 0
\(603\) 4.04892i 4.04892i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 0.623490 1.07992i 0.623490 1.07992i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.00000 −1.00000
\(613\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(614\) 0.222521 0.385418i 0.222521 0.385418i
\(615\) 0 0
\(616\) 0 0
\(617\) 1.56052 0.900969i 1.56052 0.900969i 0.563320 0.826239i \(-0.309524\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(618\) 0 0
\(619\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 1.00000
\(626\) −0.385418 0.222521i −0.385418 0.222521i
\(627\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(632\) 0 0
\(633\) −0.400969 0.694498i −0.400969 0.694498i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0.623490 + 1.07992i 0.623490 + 1.07992i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(642\) 2.24698i 2.24698i
\(643\) −1.07992 + 0.623490i −1.07992 + 0.623490i −0.930874 0.365341i \(-0.880952\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.277479 + 0.480608i 0.277479 + 0.480608i
\(647\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) −1.56052 0.900969i −1.56052 0.900969i
\(649\) −0.801938 −0.801938
\(650\) 0 0
\(651\) 0 0
\(652\) −0.385418 0.222521i −0.385418 0.222521i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.07992 + 0.623490i −1.07992 + 0.623490i
\(657\) −3.50647 + 2.02446i −3.50647 + 2.02446i
\(658\) 0 0
\(659\) 0.900969 + 1.56052i 0.900969 + 1.56052i 0.826239 + 0.563320i \(0.190476\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) −2.00000 −2.00000
\(663\) 0 0
\(664\) 1.24698 1.24698
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.900969 + 1.56052i −0.900969 + 1.56052i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(674\) −1.07992 0.623490i −1.07992 0.623490i
\(675\) 2.24698 2.24698
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 2.81197 + 1.62349i 2.81197 + 1.62349i
\(679\) 0 0
\(680\) 0 0
\(681\) 3.24698i 3.24698i
\(682\) 0 0
\(683\) 1.07992 0.623490i 1.07992 0.623490i 0.149042 0.988831i \(-0.452381\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(684\) 2.80194i 2.80194i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −1.24698 −1.24698
\(689\) 0 0
\(690\) 0 0
\(691\) 1.07992 + 0.623490i 1.07992 + 0.623490i 0.930874 0.365341i \(-0.119048\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 2.00000i 2.00000i
\(695\) 0 0
\(696\) 0 0
\(697\) 0.554958i 0.554958i
\(698\) 0 0
\(699\) 1.62349 2.81197i 1.62349 2.81197i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.385418 0.222521i −0.385418 0.222521i
\(705\) 0 0
\(706\) −0.222521 0.385418i −0.222521 0.385418i
\(707\) 0 0
\(708\) −2.81197 + 1.62349i −2.81197 + 1.62349i
\(709\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.900969 + 1.56052i −0.900969 + 1.56052i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −0.445042 −0.445042
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.480608 0.277479i 0.480608 0.277479i
\(723\) 3.60388i 3.60388i
\(724\) 0 0
\(725\) 0 0
\(726\) −1.25144 0.722521i −1.25144 0.722521i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.277479 0.480608i 0.277479 0.480608i
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.400969 0.694498i −0.400969 0.694498i
\(738\) 1.40097 2.42655i 1.40097 2.42655i
\(739\) 1.56052 + 0.900969i 1.56052 + 0.900969i 0.997204 + 0.0747301i \(0.0238095\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −2.42655 + 1.40097i −2.42655 + 1.40097i
\(748\) 0.171527 0.0990311i 0.171527 0.0990311i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) −0.801938 −0.801938
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(758\) −0.900969 1.56052i −0.900969 1.56052i
\(759\) 0 0
\(760\) 0 0
\(761\) −0.385418 + 0.222521i −0.385418 + 0.222521i −0.680173 0.733052i \(-0.738095\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.80194 −1.80194
\(769\) −0.385418 0.222521i −0.385418 0.222521i 0.294755 0.955573i \(-0.404762\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(770\) 0 0
\(771\) −1.12349 1.94594i −1.12349 1.94594i
\(772\) 1.24698i 1.24698i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 2.42655 1.40097i 2.42655 1.40097i
\(775\) 0 0
\(776\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(777\) 0 0
\(778\) 0 0
\(779\) −1.55496 −1.55496
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(785\) 0 0
\(786\) 1.94594 1.12349i 1.94594 1.12349i
\(787\) −0.385418 + 0.222521i −0.385418 + 0.222521i −0.680173 0.733052i \(-0.738095\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 1.00000 1.00000
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.866025 0.500000i 0.866025 0.500000i
\(801\) 4.04892i 4.04892i
\(802\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(803\) 0.400969 0.694498i 0.400969 0.694498i
\(804\) −2.81197 1.62349i −2.81197 1.62349i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.623490 + 1.07992i −0.623490 + 1.07992i 0.365341 + 0.930874i \(0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(810\) 0 0
\(811\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0.400969 0.694498i 0.400969 0.694498i
\(817\) −1.34663 0.777479i −1.34663 0.777479i
\(818\) 1.80194 1.80194
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(822\) −1.62349 + 2.81197i −1.62349 + 2.81197i
\(823\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0 0
\(825\) −0.694498 + 0.400969i −0.694498 + 0.400969i
\(826\) 0 0
\(827\) 1.80194i 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(828\) 0 0
\(829\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.445042 −0.445042
\(834\) 0.694498 + 0.400969i 0.694498 + 0.400969i
\(835\) 0 0
\(836\) −0.277479 0.480608i −0.277479 0.480608i
\(837\) 0 0
\(838\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(839\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 0 0
\(841\) −0.500000 0.866025i −0.500000 0.866025i
\(842\) 0 0
\(843\) −0.694498 0.400969i −0.694498 0.400969i
\(844\) 0.445042 0.445042
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.12349 1.94594i −1.12349 1.94594i
\(850\) 0.445042i 0.445042i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.07992 0.623490i −1.07992 0.623490i
\(857\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(858\) 0 0
\(859\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 1.94594 1.12349i 1.94594 1.12349i
\(865\) 0 0
\(866\) 1.80194i 1.80194i
\(867\) −0.722521 1.25144i −0.722521 1.25144i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.866025 0.500000i −0.866025 0.500000i
\(874\) 0 0
\(875\) 0 0
\(876\) 3.24698i 3.24698i
\(877\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.222521 + 0.385418i −0.222521 + 0.385418i −0.955573 0.294755i \(-0.904762\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(882\) −1.94594 1.12349i −1.94594 1.12349i
\(883\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.385418 0.222521i −0.385418 0.222521i
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.694498 + 0.400969i −0.694498 + 0.400969i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.24698 1.24698
\(899\) 0 0
\(900\) −1.12349 + 1.94594i −1.12349 + 1.94594i
\(901\) 0 0
\(902\) 0.554958i 0.554958i
\(903\) 0 0
\(904\) −1.56052 + 0.900969i −1.56052 + 0.900969i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.900969 + 1.56052i −0.900969 + 1.56052i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(908\) 1.56052 + 0.900969i 1.56052 + 0.900969i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −1.94594 1.12349i −1.94594 1.12349i
\(913\) 0.277479 0.480608i 0.277479 0.480608i
\(914\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 1.00000i 1.00000i
\(919\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(920\) 0 0
\(921\) −0.694498 0.400969i −0.694498 0.400969i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.07992 + 0.623490i −1.07992 + 0.623490i −0.930874 0.365341i \(-0.880952\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(930\) 0 0
\(931\) 1.24698i 1.24698i
\(932\) 0.900969 + 1.56052i 0.900969 + 1.56052i
\(933\) 0 0
\(934\) 1.56052 + 0.900969i 1.56052 + 0.900969i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(938\) 0 0
\(939\) −0.400969 + 0.694498i −0.400969 + 0.694498i
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.80194i 1.80194i
\(945\) 0 0
\(946\) −0.277479 + 0.480608i −0.277479 + 0.480608i
\(947\) −1.07992 0.623490i −1.07992 0.623490i −0.149042 0.988831i \(-0.547619\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1.24698 1.24698
\(951\) 0 0
\(952\) 0 0
\(953\) 0.623490 + 1.07992i 0.623490 + 1.07992i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1.00000 −1.00000
\(962\) 0 0
\(963\) 2.80194 2.80194
\(964\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0.694498 0.400969i 0.694498 0.400969i
\(969\) 0.866025 0.500000i 0.866025 0.500000i
\(970\) 0 0
\(971\) −0.623490 1.07992i −0.623490 1.07992i −0.988831 0.149042i \(-0.952381\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(972\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.07992 + 0.623490i 1.07992 + 0.623490i 0.930874 0.365341i \(-0.119048\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(978\) −0.400969 + 0.694498i −0.400969 + 0.694498i
\(979\) 0.400969 + 0.694498i 0.400969 + 0.694498i
\(980\) 0 0
\(981\) 0 0
\(982\) −1.56052 + 0.900969i −1.56052 + 0.900969i
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 1.12349 + 1.94594i 1.12349 + 1.94594i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) 3.60388i 3.60388i
\(994\) 0 0
\(995\) 0 0
\(996\) 2.24698i 2.24698i
\(997\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) −0.222521 + 0.385418i −0.222521 + 0.385418i
\(999\) 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 1352.1.p.c.147.6 12
8.3 odd 2 CM 1352.1.p.c.147.6 12
13.2 odd 12 1352.1.n.c.315.3 6
13.3 even 3 inner 1352.1.p.c.699.3 12
13.4 even 6 1352.1.h.a.675.4 6
13.5 odd 4 1352.1.n.c.867.3 6
13.6 odd 12 1352.1.g.b.339.1 3
13.7 odd 12 1352.1.g.c.339.1 yes 3
13.8 odd 4 1352.1.n.b.867.3 6
13.9 even 3 1352.1.h.a.675.1 6
13.10 even 6 inner 1352.1.p.c.699.6 12
13.11 odd 12 1352.1.n.b.315.3 6
13.12 even 2 inner 1352.1.p.c.147.3 12
104.3 odd 6 inner 1352.1.p.c.699.3 12
104.11 even 12 1352.1.n.b.315.3 6
104.19 even 12 1352.1.g.b.339.1 3
104.35 odd 6 1352.1.h.a.675.1 6
104.43 odd 6 1352.1.h.a.675.4 6
104.51 odd 2 inner 1352.1.p.c.147.3 12
104.59 even 12 1352.1.g.c.339.1 yes 3
104.67 even 12 1352.1.n.c.315.3 6
104.75 odd 6 inner 1352.1.p.c.699.6 12
104.83 even 4 1352.1.n.c.867.3 6
104.99 even 4 1352.1.n.b.867.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1352.1.g.b.339.1 3 13.6 odd 12
1352.1.g.b.339.1 3 104.19 even 12
1352.1.g.c.339.1 yes 3 13.7 odd 12
1352.1.g.c.339.1 yes 3 104.59 even 12
1352.1.h.a.675.1 6 13.9 even 3
1352.1.h.a.675.1 6 104.35 odd 6
1352.1.h.a.675.4 6 13.4 even 6
1352.1.h.a.675.4 6 104.43 odd 6
1352.1.n.b.315.3 6 13.11 odd 12
1352.1.n.b.315.3 6 104.11 even 12
1352.1.n.b.867.3 6 13.8 odd 4
1352.1.n.b.867.3 6 104.99 even 4
1352.1.n.c.315.3 6 13.2 odd 12
1352.1.n.c.315.3 6 104.67 even 12
1352.1.n.c.867.3 6 13.5 odd 4
1352.1.n.c.867.3 6 104.83 even 4
1352.1.p.c.147.3 12 13.12 even 2 inner
1352.1.p.c.147.3 12 104.51 odd 2 inner
1352.1.p.c.147.6 12 1.1 even 1 trivial
1352.1.p.c.147.6 12 8.3 odd 2 CM
1352.1.p.c.699.3 12 13.3 even 3 inner
1352.1.p.c.699.3 12 104.3 odd 6 inner
1352.1.p.c.699.6 12 13.10 even 6 inner
1352.1.p.c.699.6 12 104.75 odd 6 inner