Properties

Label 1960.1.cz.b.1579.1
Level $1960$
Weight $1$
Character 1960.1579
Analytic conductor $0.978$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1960,1,Mod(179,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 21, 21, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.179");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1960.cz (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: \(0.978167424761\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} + \cdots)\)

Embedding invariants

Embedding label 1579.1
Root \(-0.988831 - 0.149042i\) of defining polynomial
Character \(\chi\) \(=\) 1960.1579
Dual form 1960.1.cz.b.499.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.733052 + 0.680173i) q^{2} +(0.0747301 - 0.997204i) q^{4} +(0.365341 - 0.930874i) q^{5} +(0.623490 - 0.781831i) q^{7} +(0.623490 + 0.781831i) q^{8} +(0.955573 - 0.294755i) q^{9} +O(q^{10})\) \(q+(-0.733052 + 0.680173i) q^{2} +(0.0747301 - 0.997204i) q^{4} +(0.365341 - 0.930874i) q^{5} +(0.623490 - 0.781831i) q^{7} +(0.623490 + 0.781831i) q^{8} +(0.955573 - 0.294755i) q^{9} +(0.365341 + 0.930874i) q^{10} +(0.698220 + 0.215372i) q^{11} +(-0.0332580 - 0.145713i) q^{13} +(0.0747301 + 0.997204i) q^{14} +(-0.988831 - 0.149042i) q^{16} +(-0.500000 + 0.866025i) q^{18} +(0.988831 + 1.71271i) q^{19} +(-0.900969 - 0.433884i) q^{20} +(-0.658322 + 0.317031i) q^{22} +(-0.826239 - 0.563320i) q^{23} +(-0.733052 - 0.680173i) q^{25} +(0.123490 + 0.0841939i) q^{26} +(-0.733052 - 0.680173i) q^{28} +(0.826239 - 0.563320i) q^{32} +(-0.500000 - 0.866025i) q^{35} +(-0.222521 - 0.974928i) q^{36} +(0.142820 + 1.90580i) q^{37} +(-1.88980 - 0.582926i) q^{38} +(0.955573 - 0.294755i) q^{40} +(-0.914101 - 1.14625i) q^{41} +(0.266948 - 0.680173i) q^{44} +(0.0747301 - 0.997204i) q^{45} +(0.988831 - 0.149042i) q^{46} +(-1.21135 + 1.12397i) q^{47} +(-0.222521 - 0.974928i) q^{49} +1.00000 q^{50} +(-0.147791 + 0.0222759i) q^{52} +(0.123490 - 1.64786i) q^{53} +(0.455573 - 0.571270i) q^{55} +1.00000 q^{56} +(0.455573 + 1.16078i) q^{59} +(0.365341 - 0.930874i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(-0.147791 - 0.0222759i) q^{65} +(0.955573 + 0.294755i) q^{70} +(0.826239 + 0.563320i) q^{72} +(-1.40097 - 1.29991i) q^{74} +(1.78181 - 0.858075i) q^{76} +(0.603718 - 0.411608i) q^{77} +(-0.500000 + 0.866025i) q^{80} +(0.826239 - 0.563320i) q^{81} +(1.44973 + 0.218511i) q^{82} +(0.266948 + 0.680173i) q^{88} +(-1.72188 + 0.531130i) q^{89} +(0.623490 + 0.781831i) q^{90} +(-0.134659 - 0.0648483i) q^{91} +(-0.623490 + 0.781831i) q^{92} +(0.123490 - 1.64786i) q^{94} +(1.95557 - 0.294755i) q^{95} +(0.826239 + 0.563320i) q^{98} +0.730682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} + q^{4} + q^{5} - 2 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{2} + q^{4} + q^{5} - 2 q^{7} - 2 q^{8} + q^{9} + q^{10} - q^{11} + 2 q^{13} + q^{14} + q^{16} - 6 q^{18} - q^{19} - 2 q^{20} + 2 q^{22} - q^{23} + q^{25} - 8 q^{26} + q^{28} + q^{32} - 6 q^{35} - 2 q^{36} - q^{37} - q^{38} + q^{40} + 2 q^{41} + 13 q^{44} + q^{45} - q^{46} - q^{47} - 2 q^{49} + 12 q^{50} - q^{52} - 8 q^{53} - 5 q^{55} + 12 q^{56} - 5 q^{59} + q^{63} - 2 q^{64} - q^{65} + q^{70} + q^{72} - 8 q^{74} + 2 q^{76} - q^{77} - 6 q^{80} + q^{81} - q^{82} + 13 q^{88} + 2 q^{89} - 2 q^{90} - 5 q^{91} + 2 q^{92} - 8 q^{94} + 13 q^{95} + q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(-1\) \(e\left(\frac{20}{21}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(3\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(4\) 0.0747301 0.997204i 0.0747301 0.997204i
\(5\) 0.365341 0.930874i 0.365341 0.930874i
\(6\) 0 0
\(7\) 0.623490 0.781831i 0.623490 0.781831i
\(8\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(9\) 0.955573 0.294755i 0.955573 0.294755i
\(10\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(11\) 0.698220 + 0.215372i 0.698220 + 0.215372i 0.623490 0.781831i \(-0.285714\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(12\) 0 0
\(13\) −0.0332580 0.145713i −0.0332580 0.145713i 0.955573 0.294755i \(-0.0952381\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(14\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(15\) 0 0
\(16\) −0.988831 0.149042i −0.988831 0.149042i
\(17\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(18\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(19\) 0.988831 + 1.71271i 0.988831 + 1.71271i 0.623490 + 0.781831i \(0.285714\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(20\) −0.900969 0.433884i −0.900969 0.433884i
\(21\) 0 0
\(22\) −0.658322 + 0.317031i −0.658322 + 0.317031i
\(23\) −0.826239 0.563320i −0.826239 0.563320i 0.0747301 0.997204i \(-0.476190\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(24\) 0 0
\(25\) −0.733052 0.680173i −0.733052 0.680173i
\(26\) 0.123490 + 0.0841939i 0.123490 + 0.0841939i
\(27\) 0 0
\(28\) −0.733052 0.680173i −0.733052 0.680173i
\(29\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0.826239 0.563320i 0.826239 0.563320i
\(33\) 0 0
\(34\) 0 0
\(35\) −0.500000 0.866025i −0.500000 0.866025i
\(36\) −0.222521 0.974928i −0.222521 0.974928i
\(37\) 0.142820 + 1.90580i 0.142820 + 1.90580i 0.365341 + 0.930874i \(0.380952\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(38\) −1.88980 0.582926i −1.88980 0.582926i
\(39\) 0 0
\(40\) 0.955573 0.294755i 0.955573 0.294755i
\(41\) −0.914101 1.14625i −0.914101 1.14625i −0.988831 0.149042i \(-0.952381\pi\)
0.0747301 0.997204i \(-0.476190\pi\)
\(42\) 0 0
\(43\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(44\) 0.266948 0.680173i 0.266948 0.680173i
\(45\) 0.0747301 0.997204i 0.0747301 0.997204i
\(46\) 0.988831 0.149042i 0.988831 0.149042i
\(47\) −1.21135 + 1.12397i −1.21135 + 1.12397i −0.222521 + 0.974928i \(0.571429\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(48\) 0 0
\(49\) −0.222521 0.974928i −0.222521 0.974928i
\(50\) 1.00000 1.00000
\(51\) 0 0
\(52\) −0.147791 + 0.0222759i −0.147791 + 0.0222759i
\(53\) 0.123490 1.64786i 0.123490 1.64786i −0.500000 0.866025i \(-0.666667\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(54\) 0 0
\(55\) 0.455573 0.571270i 0.455573 0.571270i
\(56\) 1.00000 1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) 0.455573 + 1.16078i 0.455573 + 1.16078i 0.955573 + 0.294755i \(0.0952381\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(62\) 0 0
\(63\) 0.365341 0.930874i 0.365341 0.930874i
\(64\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(65\) −0.147791 0.0222759i −0.147791 0.0222759i
\(66\) 0 0
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(71\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(72\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(73\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(74\) −1.40097 1.29991i −1.40097 1.29991i
\(75\) 0 0
\(76\) 1.78181 0.858075i 1.78181 0.858075i
\(77\) 0.603718 0.411608i 0.603718 0.411608i
\(78\) 0 0
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(81\) 0.826239 0.563320i 0.826239 0.563320i
\(82\) 1.44973 + 0.218511i 1.44973 + 0.218511i
\(83\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.266948 + 0.680173i 0.266948 + 0.680173i
\(89\) −1.72188 + 0.531130i −1.72188 + 0.531130i −0.988831 0.149042i \(-0.952381\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(90\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(91\) −0.134659 0.0648483i −0.134659 0.0648483i
\(92\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(93\) 0 0
\(94\) 0.123490 1.64786i 0.123490 1.64786i
\(95\) 1.95557 0.294755i 1.95557 0.294755i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(99\) 0.730682 0.730682
\(100\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(101\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(102\) 0 0
\(103\) −0.162592 + 0.414278i −0.162592 + 0.414278i −0.988831 0.149042i \(-0.952381\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(104\) 0.0931869 0.116853i 0.0931869 0.116853i
\(105\) 0 0
\(106\) 1.03030 + 1.29196i 1.03030 + 1.29196i
\(107\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(108\) 0 0
\(109\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(110\) 0.0546039 + 0.728639i 0.0546039 + 0.728639i
\(111\) 0 0
\(112\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(113\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(114\) 0 0
\(115\) −0.826239 + 0.563320i −0.826239 + 0.563320i
\(116\) 0 0
\(117\) −0.0747301 0.129436i −0.0747301 0.129436i
\(118\) −1.12349 0.541044i −1.12349 0.541044i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.385113 0.262566i −0.385113 0.262566i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(126\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(127\) 1.32091 + 0.636119i 1.32091 + 0.636119i 0.955573 0.294755i \(-0.0952381\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(128\) −0.500000 0.866025i −0.500000 0.866025i
\(129\) 0 0
\(130\) 0.123490 0.0841939i 0.123490 0.0841939i
\(131\) −1.88980 0.284841i −1.88980 0.284841i −0.900969 0.433884i \(-0.857143\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(132\) 0 0
\(133\) 1.95557 + 0.294755i 1.95557 + 0.294755i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(138\) 0 0
\(139\) −0.277479 0.347948i −0.277479 0.347948i 0.623490 0.781831i \(-0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(140\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(141\) 0 0
\(142\) 0 0
\(143\) 0.00816111 0.108903i 0.00816111 0.108903i
\(144\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.91115 1.91115
\(149\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(150\) 0 0
\(151\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(152\) −0.722521 + 1.84095i −0.722521 + 1.84095i
\(153\) 0 0
\(154\) −0.162592 + 0.712362i −0.162592 + 0.712362i
\(155\) 0 0
\(156\) 0 0
\(157\) −0.365341 0.930874i −0.365341 0.930874i −0.988831 0.149042i \(-0.952381\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −0.222521 0.974928i −0.222521 0.974928i
\(161\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(162\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(163\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(164\) −1.21135 + 0.825886i −1.21135 + 0.825886i
\(165\) 0 0
\(166\) 0 0
\(167\) 0.900969 + 0.433884i 0.900969 + 0.433884i 0.826239 0.563320i \(-0.190476\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(168\) 0 0
\(169\) 0.880843 0.424191i 0.880843 0.424191i
\(170\) 0 0
\(171\) 1.44973 + 1.34515i 1.44973 + 1.34515i
\(172\) 0 0
\(173\) 1.57906 + 1.07659i 1.57906 + 1.07659i 0.955573 + 0.294755i \(0.0952381\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(174\) 0 0
\(175\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(176\) −0.658322 0.317031i −0.658322 0.317031i
\(177\) 0 0
\(178\) 0.900969 1.56052i 0.900969 1.56052i
\(179\) −1.63402 + 1.11406i −1.63402 + 1.11406i −0.733052 + 0.680173i \(0.761905\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(180\) −0.988831 0.149042i −0.988831 0.149042i
\(181\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(182\) 0.142820 0.0440542i 0.142820 0.0440542i
\(183\) 0 0
\(184\) −0.0747301 0.997204i −0.0747301 0.997204i
\(185\) 1.82624 + 0.563320i 1.82624 + 0.563320i
\(186\) 0 0
\(187\) 0 0
\(188\) 1.03030 + 1.29196i 1.03030 + 1.29196i
\(189\) 0 0
\(190\) −1.23305 + 1.54620i −1.23305 + 1.54620i
\(191\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(192\) 0 0
\(193\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(197\) 0.149460 0.149460 0.0747301 0.997204i \(-0.476190\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(198\) −0.535628 + 0.496990i −0.535628 + 0.496990i
\(199\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(200\) 0.0747301 0.997204i 0.0747301 0.997204i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.40097 + 0.432142i −1.40097 + 0.432142i
\(206\) −0.162592 0.414278i −0.162592 0.414278i
\(207\) −0.955573 0.294755i −0.955573 0.294755i
\(208\) 0.0111692 + 0.149042i 0.0111692 + 0.149042i
\(209\) 0.321552 + 1.40881i 0.321552 + 1.40881i
\(210\) 0 0
\(211\) 0.326239 1.42935i 0.326239 1.42935i −0.500000 0.866025i \(-0.666667\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(212\) −1.63402 0.246289i −1.63402 0.246289i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −0.535628 0.496990i −0.535628 0.496990i
\(221\) 0 0
\(222\) 0 0
\(223\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(224\) 0.0747301 0.997204i 0.0747301 0.997204i
\(225\) −0.900969 0.433884i −0.900969 0.433884i
\(226\) 0 0
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 0 0
\(229\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(230\) 0.222521 0.974928i 0.222521 0.974928i
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(234\) 0.142820 + 0.0440542i 0.142820 + 0.0440542i
\(235\) 0.603718 + 1.53825i 0.603718 + 1.53825i
\(236\) 1.19158 0.367554i 1.19158 0.367554i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(240\) 0 0
\(241\) 0.0546039 0.728639i 0.0546039 0.728639i −0.900969 0.433884i \(-0.857143\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(242\) 0.460898 0.0694692i 0.460898 0.0694692i
\(243\) 0 0
\(244\) 0 0
\(245\) −0.988831 0.149042i −0.988831 0.149042i
\(246\) 0 0
\(247\) 0.216677 0.201047i 0.216677 0.201047i
\(248\) 0 0
\(249\) 0 0
\(250\) 0.365341 0.930874i 0.365341 0.930874i
\(251\) 0.455573 0.571270i 0.455573 0.571270i −0.500000 0.866025i \(-0.666667\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(252\) −0.900969 0.433884i −0.900969 0.433884i
\(253\) −0.455573 0.571270i −0.455573 0.571270i
\(254\) −1.40097 + 0.432142i −1.40097 + 0.432142i
\(255\) 0 0
\(256\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(257\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(258\) 0 0
\(259\) 1.57906 + 1.07659i 1.57906 + 1.07659i
\(260\) −0.0332580 + 0.145713i −0.0332580 + 0.145713i
\(261\) 0 0
\(262\) 1.57906 1.07659i 1.57906 1.07659i
\(263\) 0.900969 1.56052i 0.900969 1.56052i 0.0747301 0.997204i \(-0.476190\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(264\) 0 0
\(265\) −1.48883 0.716983i −1.48883 0.716983i
\(266\) −1.63402 + 1.11406i −1.63402 + 1.11406i
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(270\) 0 0
\(271\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.365341 0.632789i −0.365341 0.632789i
\(276\) 0 0
\(277\) −1.48883 + 1.01507i −1.48883 + 1.01507i −0.500000 + 0.866025i \(0.666667\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(278\) 0.440071 + 0.0663300i 0.440071 + 0.0663300i
\(279\) 0 0
\(280\) 0.365341 0.930874i 0.365341 0.930874i
\(281\) −0.367711 1.61105i −0.367711 1.61105i −0.733052 0.680173i \(-0.761905\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(282\) 0 0
\(283\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0.0680900 + 0.0853822i 0.0680900 + 0.0853822i
\(287\) −1.46610 −1.46610
\(288\) 0.623490 0.781831i 0.623490 0.781831i
\(289\) 0.365341 0.930874i 0.365341 0.930874i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.46610 −1.46610 −0.733052 0.680173i \(-0.761905\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(294\) 0 0
\(295\) 1.24698 1.24698
\(296\) −1.40097 + 1.29991i −1.40097 + 1.29991i
\(297\) 0 0
\(298\) 0 0
\(299\) −0.0546039 + 0.139129i −0.0546039 + 0.139129i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.722521 1.84095i −0.722521 1.84095i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(308\) −0.365341 0.632789i −0.365341 0.632789i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(312\) 0 0
\(313\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(314\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(315\) −0.733052 0.680173i −0.733052 0.680173i
\(316\) 0 0
\(317\) −0.367711 0.250701i −0.367711 0.250701i 0.365341 0.930874i \(-0.380952\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(321\) 0 0
\(322\) 0.500000 0.866025i 0.500000 0.866025i
\(323\) 0 0
\(324\) −0.500000 0.866025i −0.500000 0.866025i
\(325\) −0.0747301 + 0.129436i −0.0747301 + 0.129436i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.326239 1.42935i 0.326239 1.42935i
\(329\) 0.123490 + 1.64786i 0.123490 + 1.64786i
\(330\) 0 0
\(331\) 0.142820 + 1.90580i 0.142820 + 1.90580i 0.365341 + 0.930874i \(0.380952\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(332\) 0 0
\(333\) 0.698220 + 1.77904i 0.698220 + 1.77904i
\(334\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(338\) −0.357180 + 0.910080i −0.357180 + 0.910080i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −1.97766 −1.97766
\(343\) −0.900969 0.433884i −0.900969 0.433884i
\(344\) 0 0
\(345\) 0 0
\(346\) −1.88980 + 0.284841i −1.88980 + 0.284841i
\(347\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(348\) 0 0
\(349\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(350\) 0.623490 0.781831i 0.623490 0.781831i
\(351\) 0 0
\(352\) 0.698220 0.215372i 0.698220 0.215372i
\(353\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(357\) 0 0
\(358\) 0.440071 1.92808i 0.440071 1.92808i
\(359\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(360\) 0.826239 0.563320i 0.826239 0.563320i
\(361\) −1.45557 + 2.52113i −1.45557 + 2.52113i
\(362\) 0 0
\(363\) 0 0
\(364\) −0.0747301 + 0.129436i −0.0747301 + 0.129436i
\(365\) 0 0
\(366\) 0 0
\(367\) 1.07473 + 0.997204i 1.07473 + 0.997204i 1.00000 \(0\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(368\) 0.733052 + 0.680173i 0.733052 + 0.680173i
\(369\) −1.21135 0.825886i −1.21135 0.825886i
\(370\) −1.72188 + 0.829215i −1.72188 + 0.829215i
\(371\) −1.21135 1.12397i −1.21135 1.12397i
\(372\) 0 0
\(373\) 0.222521 + 0.385418i 0.222521 + 0.385418i 0.955573 0.294755i \(-0.0952381\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.63402 0.246289i −1.63402 0.246289i
\(377\) 0 0
\(378\) 0 0
\(379\) −0.0332580 0.145713i −0.0332580 0.145713i 0.955573 0.294755i \(-0.0952381\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(380\) −0.147791 1.97213i −0.147791 1.97213i
\(381\) 0 0
\(382\) 0 0
\(383\) −1.40097 + 0.432142i −1.40097 + 0.432142i −0.900969 0.433884i \(-0.857143\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 0 0
\(385\) −0.162592 0.712362i −0.162592 0.712362i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.623490 0.781831i 0.623490 0.781831i
\(393\) 0 0
\(394\) −0.109562 + 0.101659i −0.109562 + 0.101659i
\(395\) 0 0
\(396\) 0.0546039 0.728639i 0.0546039 0.728639i
\(397\) −0.658322 + 1.67738i −0.658322 + 1.67738i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(401\) 1.57906 0.487076i 1.57906 0.487076i 0.623490 0.781831i \(-0.285714\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.222521 0.974928i −0.222521 0.974928i
\(406\) 0 0
\(407\) −0.310737 + 1.36143i −0.310737 + 1.36143i
\(408\) 0 0
\(409\) 1.03030 0.702449i 1.03030 0.702449i 0.0747301 0.997204i \(-0.476190\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(410\) 0.733052 1.26968i 0.733052 1.26968i
\(411\) 0 0
\(412\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(413\) 1.19158 + 0.367554i 1.19158 + 0.367554i
\(414\) 0.900969 0.433884i 0.900969 0.433884i
\(415\) 0 0
\(416\) −0.109562 0.101659i −0.109562 0.101659i
\(417\) 0 0
\(418\) −1.19395 0.814021i −1.19395 0.814021i
\(419\) −1.48883 + 0.716983i −1.48883 + 0.716983i −0.988831 0.149042i \(-0.952381\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(422\) 0.733052 + 1.26968i 0.733052 + 1.26968i
\(423\) −0.826239 + 1.43109i −0.826239 + 1.43109i
\(424\) 1.36534 0.930874i 1.36534 0.930874i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(432\) 0 0
\(433\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.147791 1.97213i 0.147791 1.97213i
\(438\) 0 0
\(439\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(440\) 0.730682 0.730682
\(441\) −0.500000 0.866025i −0.500000 0.866025i
\(442\) 0 0
\(443\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(444\) 0 0
\(445\) −0.134659 + 1.79690i −0.134659 + 1.79690i
\(446\) 0.455573 1.16078i 0.455573 1.16078i
\(447\) 0 0
\(448\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(449\) 1.19158 + 1.49419i 1.19158 + 1.49419i 0.826239 + 0.563320i \(0.190476\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(450\) 0.955573 0.294755i 0.955573 0.294755i
\(451\) −0.391374 0.997204i −0.391374 0.997204i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.109562 + 0.101659i −0.109562 + 0.101659i
\(456\) 0 0
\(457\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(461\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(462\) 0 0
\(463\) −1.48883 + 0.716983i −1.48883 + 0.716983i −0.988831 0.149042i \(-0.952381\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(468\) −0.134659 + 0.0648483i −0.134659 + 0.0648483i
\(469\) 0 0
\(470\) −1.48883 0.716983i −1.48883 0.716983i
\(471\) 0 0
\(472\) −0.623490 + 1.07992i −0.623490 + 1.07992i
\(473\) 0 0
\(474\) 0 0
\(475\) 0.440071 1.92808i 0.440071 1.92808i
\(476\) 0 0
\(477\) −0.367711 1.61105i −0.367711 1.61105i
\(478\) 0 0
\(479\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(480\) 0 0
\(481\) 0.272950 0.0841939i 0.272950 0.0841939i
\(482\) 0.455573 + 0.571270i 0.455573 + 0.571270i
\(483\) 0 0
\(484\) −0.290611 + 0.364415i −0.290611 + 0.364415i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.78181 0.268565i 1.78181 0.268565i 0.826239 0.563320i \(-0.190476\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0.826239 0.563320i 0.826239 0.563320i
\(491\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −0.0220888 + 0.294755i −0.0220888 + 0.294755i
\(495\) 0.266948 0.680173i 0.266948 0.680173i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.19158 0.367554i 1.19158 0.367554i 0.365341 0.930874i \(-0.380952\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(500\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(501\) 0 0
\(502\) 0.0546039 + 0.728639i 0.0546039 + 0.728639i
\(503\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(504\) 0.955573 0.294755i 0.955573 0.294755i
\(505\) 0 0
\(506\) 0.722521 + 0.108903i 0.722521 + 0.108903i
\(507\) 0 0
\(508\) 0.733052 1.26968i 0.733052 1.26968i
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(513\) 0 0
\(514\) 0 0
\(515\) 0.326239 + 0.302705i 0.326239 + 0.302705i
\(516\) 0 0
\(517\) −1.08786 + 0.523887i −1.08786 + 0.523887i
\(518\) −1.88980 + 0.284841i −1.88980 + 0.284841i
\(519\) 0 0
\(520\) −0.0747301 0.129436i −0.0747301 0.129436i
\(521\) −0.826239 + 1.43109i −0.826239 + 1.43109i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(522\) 0 0
\(523\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(524\) −0.425270 + 1.86323i −0.425270 + 1.86323i
\(525\) 0 0
\(526\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 1.57906 0.487076i 1.57906 0.487076i
\(531\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(532\) 0.440071 1.92808i 0.440071 1.92808i
\(533\) −0.136622 + 0.171318i −0.136622 + 0.171318i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.0546039 0.728639i 0.0546039 0.728639i
\(540\) 0 0
\(541\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.698220 + 0.215372i 0.698220 + 0.215372i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.400969 1.75676i 0.400969 1.75676i
\(555\) 0 0
\(556\) −0.367711 + 0.250701i −0.367711 + 0.250701i
\(557\) −0.0747301 + 0.129436i −0.0747301 + 0.129436i −0.900969 0.433884i \(-0.857143\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(561\) 0 0
\(562\) 1.36534 + 0.930874i 1.36534 + 0.930874i
\(563\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.0747301 0.997204i 0.0747301 0.997204i
\(568\) 0 0
\(569\) 0.988831 + 1.71271i 0.988831 + 1.71271i 0.623490 + 0.781831i \(0.285714\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(570\) 0 0
\(571\) −0.367711 + 0.250701i −0.367711 + 0.250701i −0.733052 0.680173i \(-0.761905\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(572\) −0.107988 0.0162766i −0.107988 0.0162766i
\(573\) 0 0
\(574\) 1.07473 0.997204i 1.07473 0.997204i
\(575\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(576\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(577\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(578\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.441126 1.12397i 0.441126 1.12397i
\(584\) 0 0
\(585\) −0.147791 + 0.0222759i −0.147791 + 0.0222759i
\(586\) 1.07473 0.997204i 1.07473 0.997204i
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −0.914101 + 0.848162i −0.914101 + 0.848162i
\(591\) 0 0
\(592\) 0.142820 1.90580i 0.142820 1.90580i
\(593\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −0.0546039 0.139129i −0.0546039 0.139129i
\(599\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(600\) 0 0
\(601\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.385113 + 0.262566i −0.385113 + 0.262566i
\(606\) 0 0
\(607\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 1.78181 + 0.858075i 1.78181 + 0.858075i
\(609\) 0 0
\(610\) 0 0
\(611\) 0.204064 + 0.139129i 0.204064 + 0.139129i
\(612\) 0 0
\(613\) 1.44973 + 1.34515i 1.44973 + 1.34515i 0.826239 + 0.563320i \(0.190476\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.698220 + 0.215372i 0.698220 + 0.215372i
\(617\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(618\) 0 0
\(619\) 0.733052 1.26968i 0.733052 1.26968i −0.222521 0.974928i \(-0.571429\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.658322 + 1.67738i −0.658322 + 1.67738i
\(624\) 0 0
\(625\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(626\) 0 0
\(627\) 0 0
\(628\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(629\) 0 0
\(630\) 1.00000 1.00000
\(631\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.440071 0.0663300i 0.440071 0.0663300i
\(635\) 1.07473 0.997204i 1.07473 0.997204i
\(636\) 0 0
\(637\) −0.134659 + 0.0648483i −0.134659 + 0.0648483i
\(638\) 0 0
\(639\) 0 0
\(640\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(641\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i 0.826239 + 0.563320i \(0.190476\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(642\) 0 0
\(643\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(644\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(645\) 0 0
\(646\) 0 0
\(647\) −0.722521 1.84095i −0.722521 1.84095i −0.500000 0.866025i \(-0.666667\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(648\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(649\) 0.0680900 + 0.908598i 0.0680900 + 0.908598i
\(650\) −0.0332580 0.145713i −0.0332580 0.145713i
\(651\) 0 0
\(652\) 0 0
\(653\) 1.95557 + 0.294755i 1.95557 + 0.294755i 1.00000 \(0\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(654\) 0 0
\(655\) −0.955573 + 1.65510i −0.955573 + 1.65510i
\(656\) 0.733052 + 1.26968i 0.733052 + 1.26968i
\(657\) 0 0
\(658\) −1.21135 1.12397i −1.21135 1.12397i
\(659\) 1.62349 0.781831i 1.62349 0.781831i 0.623490 0.781831i \(-0.285714\pi\)
1.00000 \(0\)
\(660\) 0 0
\(661\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(662\) −1.40097 1.29991i −1.40097 1.29991i
\(663\) 0 0
\(664\) 0 0
\(665\) 0.988831 1.71271i 0.988831 1.71271i
\(666\) −1.72188 0.829215i −1.72188 0.829215i
\(667\) 0 0
\(668\) 0.500000 0.866025i 0.500000 0.866025i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −0.357180 0.910080i −0.357180 0.910080i
\(677\) 0.698220 0.215372i 0.698220 0.215372i 0.0747301 0.997204i \(-0.476190\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(684\) 1.44973 1.34515i 1.44973 1.34515i
\(685\) 0 0
\(686\) 0.955573 0.294755i 0.955573 0.294755i
\(687\) 0 0
\(688\) 0 0
\(689\) −0.244221 + 0.0368104i −0.244221 + 0.0368104i
\(690\) 0 0
\(691\) −0.162592 + 0.414278i −0.162592 + 0.414278i −0.988831 0.149042i \(-0.952381\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(692\) 1.19158 1.49419i 1.19158 1.49419i
\(693\) 0.455573 0.571270i 0.455573 0.571270i
\(694\) 0 0
\(695\) −0.425270 + 0.131178i −0.425270 + 0.131178i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(701\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(702\) 0 0
\(703\) −3.12285 + 2.12912i −3.12285 + 2.12912i
\(704\) −0.365341 + 0.632789i −0.365341 + 0.632789i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.48883 1.01507i −1.48883 1.01507i
\(713\) 0 0
\(714\) 0 0
\(715\) −0.0983929 0.0473835i −0.0983929 0.0473835i
\(716\) 0.988831 + 1.71271i 0.988831 + 1.71271i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(720\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(721\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(722\) −0.647791 2.83816i −0.647791 2.83816i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.19158 + 1.49419i 1.19158 + 1.49419i 0.826239 + 0.563320i \(0.190476\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(728\) −0.0332580 0.145713i −0.0332580 0.145713i
\(729\) 0.623490 0.781831i 0.623490 0.781831i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.44973 1.34515i 1.44973 1.34515i 0.623490 0.781831i \(-0.285714\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(734\) −1.46610 −1.46610
\(735\) 0 0
\(736\) −1.00000 −1.00000
\(737\) 0 0
\(738\) 1.44973 0.218511i 1.44973 0.218511i
\(739\) 0.0111692 0.149042i 0.0111692 0.149042i −0.988831 0.149042i \(-0.952381\pi\)
1.00000 \(0\)
\(740\) 0.698220 1.77904i 0.698220 1.77904i
\(741\) 0 0
\(742\) 1.65248 1.65248
\(743\) 1.19158 + 1.49419i 1.19158 + 1.49419i 0.826239 + 0.563320i \(0.190476\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.425270 0.131178i −0.425270 0.131178i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(752\) 1.36534 0.930874i 1.36534 0.930874i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(758\) 0.123490 + 0.0841939i 0.123490 + 0.0841939i
\(759\) 0 0
\(760\) 1.44973 + 1.34515i 1.44973 + 1.34515i
\(761\) 1.36534 + 0.930874i 1.36534 + 0.930874i 1.00000 \(0\)
0.365341 + 0.930874i \(0.380952\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0.733052 1.26968i 0.733052 1.26968i
\(767\) 0.153989 0.104988i 0.153989 0.104988i
\(768\) 0 0
\(769\) −0.425270 + 1.86323i −0.425270 + 1.86323i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0.603718 + 0.411608i 0.603718 + 0.411608i
\(771\) 0 0
\(772\) 0 0
\(773\) 1.57906 + 0.487076i 1.57906 + 0.487076i 0.955573 0.294755i \(-0.0952381\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.05929 2.69903i 1.05929 2.69903i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(785\) −1.00000 −1.00000
\(786\) 0 0
\(787\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(788\) 0.0111692 0.149042i 0.0111692 0.149042i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.455573 + 0.571270i 0.455573 + 0.571270i
\(793\) 0 0
\(794\) −0.658322 1.67738i −0.658322 1.67738i
\(795\) 0 0
\(796\) 0 0
\(797\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.988831 0.149042i −0.988831 0.149042i
\(801\) −1.48883 + 1.01507i −1.48883 + 1.01507i
\(802\) −0.826239 + 1.43109i −0.826239 + 1.43109i
\(803\) 0 0
\(804\) 0 0
\(805\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.07473 + 0.997204i 1.07473 + 0.997204i 1.00000 \(0\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(810\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(811\) −1.72188 + 0.829215i −1.72188 + 0.829215i −0.733052 + 0.680173i \(0.761905\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −0.698220 1.20935i −0.698220 1.20935i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(819\) −0.147791 0.0222759i −0.147791 0.0222759i
\(820\) 0.326239 + 1.42935i 0.326239 + 1.42935i
\(821\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(822\) 0 0
\(823\) −0.658322 1.67738i −0.658322 1.67738i −0.733052 0.680173i \(-0.761905\pi\)
0.0747301 0.997204i \(-0.476190\pi\)
\(824\) −0.425270 + 0.131178i −0.425270 + 0.131178i
\(825\) 0 0
\(826\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(827\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(828\) −0.365341 + 0.930874i −0.365341 + 0.930874i
\(829\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.149460 0.149460
\(833\) 0 0
\(834\) 0 0
\(835\) 0.733052 0.680173i 0.733052 0.680173i
\(836\) 1.42890 0.215372i 1.42890 0.215372i
\(837\) 0 0
\(838\) 0.603718 1.53825i 0.603718 1.53825i
\(839\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(840\) 0 0
\(841\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(842\) 0 0
\(843\) 0 0
\(844\) −1.40097 0.432142i −1.40097 0.432142i
\(845\) −0.0730607 0.974928i −0.0730607 0.974928i
\(846\) −0.367711 1.61105i −0.367711 1.61105i
\(847\) −0.445396 + 0.137386i −0.445396 + 0.137386i
\(848\) −0.367711 + 1.61105i −0.367711 + 1.61105i
\(849\) 0 0
\(850\) 0 0
\(851\) 0.955573 1.65510i 0.955573 1.65510i
\(852\) 0 0
\(853\) −1.72188 0.829215i −1.72188 0.829215i −0.988831 0.149042i \(-0.952381\pi\)
−0.733052 0.680173i \(-0.761905\pi\)
\(854\) 0 0
\(855\) 1.78181 0.858075i 1.78181 0.858075i
\(856\) 0 0
\(857\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(858\) 0 0
\(859\) −1.48883 1.01507i −1.48883 1.01507i −0.988831 0.149042i \(-0.952381\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.955573 1.65510i −0.955573 1.65510i −0.733052 0.680173i \(-0.761905\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(864\) 0 0
\(865\) 1.57906 1.07659i 1.57906 1.07659i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 1.23305 + 1.54620i 1.23305 + 1.54620i
\(875\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(876\) 0 0
\(877\) 0.0546039 0.139129i 0.0546039 0.139129i −0.900969 0.433884i \(-0.857143\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −0.535628 + 0.496990i −0.535628 + 0.496990i
\(881\) 0.730682 0.730682 0.365341 0.930874i \(-0.380952\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(882\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.455573 1.16078i 0.455573 1.16078i −0.500000 0.866025i \(-0.666667\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(888\) 0 0
\(889\) 1.32091 0.636119i 1.32091 0.636119i
\(890\) −1.12349 1.40881i −1.12349 1.40881i
\(891\) 0.698220 0.215372i 0.698220 0.215372i
\(892\) 0.455573 + 1.16078i 0.455573 + 1.16078i
\(893\) −3.12285 0.963272i −3.12285 0.963272i
\(894\) 0 0
\(895\) 0.440071 + 1.92808i 0.440071 + 1.92808i
\(896\) −0.988831 0.149042i −0.988831 0.149042i
\(897\) 0 0
\(898\) −1.88980 0.284841i −1.88980 0.284841i
\(899\) 0 0
\(900\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(901\) 0 0
\(902\) 0.965168 + 0.464800i 0.965168 + 0.464800i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0.0111692 0.149042i 0.0111692 0.149042i
\(911\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.40097 + 1.29991i −1.40097 + 1.29991i
\(918\) 0 0
\(919\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(920\) −0.955573 0.294755i −0.955573 0.294755i
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.19158 1.49419i 1.19158 1.49419i
\(926\) 0.603718 1.53825i 0.603718 1.53825i
\(927\) −0.0332580 + 0.443797i −0.0332580 + 0.443797i
\(928\) 0 0
\(929\) −0.109562 + 0.101659i −0.109562 + 0.101659i −0.733052 0.680173i \(-0.761905\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(930\) 0 0
\(931\) 1.44973 1.34515i 1.44973 1.34515i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.0546039 0.139129i 0.0546039 0.139129i
\(937\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.57906 0.487076i 1.57906 0.487076i
\(941\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(942\) 0 0
\(943\) 0.109562 + 1.46200i 0.109562 + 1.46200i
\(944\) −0.277479 1.21572i −0.277479 1.21572i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.988831 + 1.71271i 0.988831 + 1.71271i
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(954\) 1.36534 + 0.930874i 1.36534 + 0.930874i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.500000 0.866025i −0.500000 0.866025i
\(962\) −0.142820 + 0.247372i −0.142820 + 0.247372i
\(963\) 0 0
\(964\) −0.722521 0.108903i −0.722521 0.108903i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(968\) −0.0348320 0.464800i −0.0348320 0.464800i
\(969\) 0 0
\(970\) 0 0
\(971\) −0.955573 + 0.294755i −0.955573 + 0.294755i −0.733052 0.680173i \(-0.761905\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(972\) 0 0
\(973\) −0.445042 −0.445042
\(974\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(978\) 0 0
\(979\) −1.31664 −1.31664
\(980\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(981\) 0 0
\(982\) 1.32091 1.22563i 1.32091 1.22563i
\(983\) 0.988831 0.149042i 0.988831 0.149042i 0.365341 0.930874i \(-0.380952\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(984\) 0 0
\(985\) 0.0546039 0.139129i 0.0546039 0.139129i
\(986\) 0 0
\(987\) 0 0
\(988\) −0.184292 0.231095i −0.184292 0.231095i
\(989\) 0 0
\(990\) 0.266948 + 0.680173i 0.266948 + 0.680173i
\(991\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.03030 0.702449i 1.03030 0.702449i 0.0747301 0.997204i \(-0.476190\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(998\) −0.623490 + 1.07992i −0.623490 + 1.07992i
\(999\) 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 1960.1.cz.b.1579.1 yes 12
5.4 even 2 1960.1.cz.a.1579.1 yes 12
8.3 odd 2 1960.1.cz.a.1579.1 yes 12
40.19 odd 2 CM 1960.1.cz.b.1579.1 yes 12
49.9 even 21 inner 1960.1.cz.b.499.1 yes 12
245.9 even 42 1960.1.cz.a.499.1 12
392.107 odd 42 1960.1.cz.a.499.1 12
1960.499 odd 42 inner 1960.1.cz.b.499.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1960.1.cz.a.499.1 12 245.9 even 42
1960.1.cz.a.499.1 12 392.107 odd 42
1960.1.cz.a.1579.1 yes 12 5.4 even 2
1960.1.cz.a.1579.1 yes 12 8.3 odd 2
1960.1.cz.b.499.1 yes 12 49.9 even 21 inner
1960.1.cz.b.499.1 yes 12 1960.499 odd 42 inner
1960.1.cz.b.1579.1 yes 12 1.1 even 1 trivial
1960.1.cz.b.1579.1 yes 12 40.19 odd 2 CM