Properties

Label 2548.1.de.b.1143.1
Level $2548$
Weight $1$
Character 2548.1143
Analytic conductor $1.272$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -52
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,1,Mod(51,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 26, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.51");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2548.de (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: \(1.27161765219\)
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 1143.1
Root \(0.955573 - 0.294755i\) of defining polynomial
Character \(\chi\) \(=\) 2548.1143
Dual form 2548.1.de.b.1663.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.365341 + 0.930874i) q^{2} +(-0.733052 + 0.680173i) q^{4} +(-0.733052 - 0.680173i) q^{7} +(-0.900969 - 0.433884i) q^{8} +(-0.988831 + 0.149042i) q^{9} +O(q^{10})\) \(q+(0.365341 + 0.930874i) q^{2} +(-0.733052 + 0.680173i) q^{4} +(-0.733052 - 0.680173i) q^{7} +(-0.900969 - 0.433884i) q^{8} +(-0.988831 + 0.149042i) q^{9} +(-0.147791 - 0.0222759i) q^{11} +(0.623490 - 0.781831i) q^{13} +(0.365341 - 0.930874i) q^{14} +(0.0747301 - 0.997204i) q^{16} +(0.698220 - 0.215372i) q^{17} +(-0.500000 - 0.866025i) q^{18} +(0.988831 - 1.71271i) q^{19} +(-0.0332580 - 0.145713i) q^{22} +(0.365341 - 0.930874i) q^{25} +(0.955573 + 0.294755i) q^{26} +1.00000 q^{28} +(-0.367711 + 1.61105i) q^{29} +(-0.623490 - 1.07992i) q^{31} +(0.955573 - 0.294755i) q^{32} +(0.455573 + 0.571270i) q^{34} +(0.623490 - 0.781831i) q^{36} +(1.95557 + 0.294755i) q^{38} +(0.123490 - 0.0841939i) q^{44} +(0.603718 + 1.53825i) q^{47} +(0.0747301 + 0.997204i) q^{49} +1.00000 q^{50} +(0.0747301 + 0.997204i) q^{52} +(0.733052 - 0.680173i) q^{53} +(0.365341 + 0.930874i) q^{56} +(-1.63402 + 0.246289i) q^{58} +(-1.21135 - 0.825886i) q^{59} +(0.733052 + 0.680173i) q^{61} +(0.777479 - 0.974928i) q^{62} +(0.826239 + 0.563320i) q^{63} +(0.623490 + 0.781831i) q^{64} +(-0.955573 - 1.65510i) q^{67} +(-0.365341 + 0.632789i) q^{68} +(-0.162592 - 0.712362i) q^{71} +(0.955573 + 0.294755i) q^{72} +(0.440071 + 1.92808i) q^{76} +(0.0931869 + 0.116853i) q^{77} +(0.955573 - 0.294755i) q^{81} +(-0.277479 - 0.347948i) q^{83} +(0.123490 + 0.0841939i) q^{88} +(-0.988831 + 0.149042i) q^{91} +(-1.21135 + 1.12397i) q^{94} +(-0.900969 + 0.433884i) q^{98} +0.149460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} + q^{4} + q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{2} + q^{4} + q^{7} - 2 q^{8} + q^{9} - q^{11} - 2 q^{13} + q^{14} + q^{16} - q^{17} - 6 q^{18} - q^{19} + 2 q^{22} + q^{25} + q^{26} + 12 q^{28} + 2 q^{29} + 2 q^{31} + q^{32} - 5 q^{34} - 2 q^{36} + 13 q^{38} - 8 q^{44} - q^{47} + q^{49} + 12 q^{50} + q^{52} - q^{53} + q^{56} - q^{58} - q^{59} - q^{61} + 10 q^{62} + q^{63} - 2 q^{64} - q^{67} - q^{68} + 2 q^{71} + q^{72} + 2 q^{76} + 2 q^{77} + q^{81} - 4 q^{83} - 8 q^{88} + q^{91} - q^{94} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\) \(1275\)
\(\chi(n)\) \(-1\) \(e\left(\frac{10}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(3\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(4\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(5\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(6\) 0 0
\(7\) −0.733052 0.680173i −0.733052 0.680173i
\(8\) −0.900969 0.433884i −0.900969 0.433884i
\(9\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(10\) 0 0
\(11\) −0.147791 0.0222759i −0.147791 0.0222759i 0.0747301 0.997204i \(-0.476190\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(12\) 0 0
\(13\) 0.623490 0.781831i 0.623490 0.781831i
\(14\) 0.365341 0.930874i 0.365341 0.930874i
\(15\) 0 0
\(16\) 0.0747301 0.997204i 0.0747301 0.997204i
\(17\) 0.698220 0.215372i 0.698220 0.215372i 0.0747301 0.997204i \(-0.476190\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(18\) −0.500000 0.866025i −0.500000 0.866025i
\(19\) 0.988831 1.71271i 0.988831 1.71271i 0.365341 0.930874i \(-0.380952\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.0332580 0.145713i −0.0332580 0.145713i
\(23\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(24\) 0 0
\(25\) 0.365341 0.930874i 0.365341 0.930874i
\(26\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(27\) 0 0
\(28\) 1.00000 1.00000
\(29\) −0.367711 + 1.61105i −0.367711 + 1.61105i 0.365341 + 0.930874i \(0.380952\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(30\) 0 0
\(31\) −0.623490 1.07992i −0.623490 1.07992i −0.988831 0.149042i \(-0.952381\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(32\) 0.955573 0.294755i 0.955573 0.294755i
\(33\) 0 0
\(34\) 0.455573 + 0.571270i 0.455573 + 0.571270i
\(35\) 0 0
\(36\) 0.623490 0.781831i 0.623490 0.781831i
\(37\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(38\) 1.95557 + 0.294755i 1.95557 + 0.294755i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(42\) 0 0
\(43\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(44\) 0.123490 0.0841939i 0.123490 0.0841939i
\(45\) 0 0
\(46\) 0 0
\(47\) 0.603718 + 1.53825i 0.603718 + 1.53825i 0.826239 + 0.563320i \(0.190476\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(48\) 0 0
\(49\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(50\) 1.00000 1.00000
\(51\) 0 0
\(52\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(53\) 0.733052 0.680173i 0.733052 0.680173i −0.222521 0.974928i \(-0.571429\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(57\) 0 0
\(58\) −1.63402 + 0.246289i −1.63402 + 0.246289i
\(59\) −1.21135 0.825886i −1.21135 0.825886i −0.222521 0.974928i \(-0.571429\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(60\) 0 0
\(61\) 0.733052 + 0.680173i 0.733052 + 0.680173i 0.955573 0.294755i \(-0.0952381\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(62\) 0.777479 0.974928i 0.777479 0.974928i
\(63\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(64\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.955573 1.65510i −0.955573 1.65510i −0.733052 0.680173i \(-0.761905\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(68\) −0.365341 + 0.632789i −0.365341 + 0.632789i
\(69\) 0 0
\(70\) 0 0
\(71\) −0.162592 0.712362i −0.162592 0.712362i −0.988831 0.149042i \(-0.952381\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(72\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(73\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.440071 + 1.92808i 0.440071 + 1.92808i
\(77\) 0.0931869 + 0.116853i 0.0931869 + 0.116853i
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 0.955573 0.294755i 0.955573 0.294755i
\(82\) 0 0
\(83\) −0.277479 0.347948i −0.277479 0.347948i 0.623490 0.781831i \(-0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.123490 + 0.0841939i 0.123490 + 0.0841939i
\(89\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(90\) 0 0
\(91\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(92\) 0 0
\(93\) 0 0
\(94\) −1.21135 + 1.12397i −1.21135 + 1.12397i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(99\) 0.149460 0.149460
\(100\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(101\) −0.134659 1.79690i −0.134659 1.79690i −0.500000 0.866025i \(-0.666667\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(102\) 0 0
\(103\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(104\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(105\) 0 0
\(106\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(107\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(108\) 0 0
\(109\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(113\) −0.914101 1.14625i −0.914101 1.14625i −0.988831 0.149042i \(-0.952381\pi\)
0.0747301 0.997204i \(-0.476190\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.826239 1.43109i −0.826239 1.43109i
\(117\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(118\) 0.326239 1.42935i 0.326239 1.42935i
\(119\) −0.658322 0.317031i −0.658322 0.317031i
\(120\) 0 0
\(121\) −0.934227 0.288171i −0.934227 0.288171i
\(122\) −0.365341 + 0.930874i −0.365341 + 0.930874i
\(123\) 0 0
\(124\) 1.19158 + 0.367554i 1.19158 + 0.367554i
\(125\) 0 0
\(126\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(127\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(128\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(132\) 0 0
\(133\) −1.88980 + 0.582926i −1.88980 + 0.582926i
\(134\) 1.19158 1.49419i 1.19158 1.49419i
\(135\) 0 0
\(136\) −0.722521 0.108903i −0.722521 0.108903i
\(137\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(138\) 0 0
\(139\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.603718 0.411608i 0.603718 0.411608i
\(143\) −0.109562 + 0.101659i −0.109562 + 0.101659i
\(144\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(150\) 0 0
\(151\) 1.44973 1.34515i 1.44973 1.34515i 0.623490 0.781831i \(-0.285714\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(152\) −1.63402 + 1.11406i −1.63402 + 1.11406i
\(153\) −0.658322 + 0.317031i −0.658322 + 0.317031i
\(154\) −0.0747301 + 0.129436i −0.0747301 + 0.129436i
\(155\) 0 0
\(156\) 0 0
\(157\) 0.123490 + 0.0841939i 0.123490 + 0.0841939i 0.623490 0.781831i \(-0.285714\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(163\) −0.109562 + 1.46200i −0.109562 + 1.46200i 0.623490 + 0.781831i \(0.285714\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.222521 0.385418i 0.222521 0.385418i
\(167\) −0.425270 + 1.86323i −0.425270 + 1.86323i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) −0.222521 0.974928i −0.222521 0.974928i
\(170\) 0 0
\(171\) −0.722521 + 1.84095i −0.722521 + 1.84095i
\(172\) 0 0
\(173\) 1.82624 + 0.563320i 1.82624 + 0.563320i 1.00000 \(0\)
0.826239 + 0.563320i \(0.190476\pi\)
\(174\) 0 0
\(175\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(176\) −0.0332580 + 0.145713i −0.0332580 + 0.145713i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(180\) 0 0
\(181\) 1.19158 + 1.49419i 1.19158 + 1.49419i 0.826239 + 0.563320i \(0.190476\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(182\) −0.500000 0.866025i −0.500000 0.866025i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.107988 + 0.0162766i −0.107988 + 0.0162766i
\(188\) −1.48883 0.716983i −1.48883 0.716983i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(192\) 0 0
\(193\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.733052 0.680173i −0.733052 0.680173i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0.0546039 + 0.139129i 0.0546039 + 0.139129i
\(199\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(200\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(201\) 0 0
\(202\) 1.62349 0.781831i 1.62349 0.781831i
\(203\) 1.36534 0.930874i 1.36534 0.930874i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −0.733052 0.680173i −0.733052 0.680173i
\(209\) −0.184292 + 0.231095i −0.184292 + 0.231095i
\(210\) 0 0
\(211\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(212\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.266948 0.680173i 0.266948 0.680173i
\(222\) 0 0
\(223\) −0.0332580 0.145713i −0.0332580 0.145713i 0.955573 0.294755i \(-0.0952381\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(224\) −0.900969 0.433884i −0.900969 0.433884i
\(225\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(226\) 0.733052 1.26968i 0.733052 1.26968i
\(227\) 0.900969 + 1.56052i 0.900969 + 1.56052i 0.826239 + 0.563320i \(0.190476\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(228\) 0 0
\(229\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.03030 1.29196i 1.03030 1.29196i
\(233\) −1.40097 1.29991i −1.40097 1.29991i −0.900969 0.433884i \(-0.857143\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(234\) −0.988831 0.149042i −0.988831 0.149042i
\(235\) 0 0
\(236\) 1.44973 0.218511i 1.44973 0.218511i
\(237\) 0 0
\(238\) 0.0546039 0.728639i 0.0546039 0.728639i
\(239\) 1.32091 0.636119i 1.32091 0.636119i 0.365341 0.930874i \(-0.380952\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(240\) 0 0
\(241\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(242\) −0.0730607 0.974928i −0.0730607 0.974928i
\(243\) 0 0
\(244\) −1.00000 −1.00000
\(245\) 0 0
\(246\) 0 0
\(247\) −0.722521 1.84095i −0.722521 1.84095i
\(248\) 0.0931869 + 1.24349i 0.0931869 + 1.24349i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(252\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.988831 0.149042i −0.988831 0.149042i
\(257\) 0.326239 + 0.302705i 0.326239 + 0.302705i 0.826239 0.563320i \(-0.190476\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.123490 1.64786i 0.123490 1.64786i
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.23305 1.54620i −1.23305 1.54620i
\(267\) 0 0
\(268\) 1.82624 + 0.563320i 1.82624 + 0.563320i
\(269\) −0.535628 + 1.36476i −0.535628 + 1.36476i 0.365341 + 0.930874i \(0.380952\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(270\) 0 0
\(271\) −0.955573 0.294755i −0.955573 0.294755i −0.222521 0.974928i \(-0.571429\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(272\) −0.162592 0.712362i −0.162592 0.712362i
\(273\) 0 0
\(274\) 0 0
\(275\) −0.0747301 + 0.129436i −0.0747301 + 0.129436i
\(276\) 0 0
\(277\) −1.88980 + 0.582926i −1.88980 + 0.582926i −0.900969 + 0.433884i \(0.857143\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(278\) 0 0
\(279\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(280\) 0 0
\(281\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(282\) 0 0
\(283\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(284\) 0.603718 + 0.411608i 0.603718 + 0.411608i
\(285\) 0 0
\(286\) −0.134659 0.0648483i −0.134659 0.0648483i
\(287\) 0 0
\(288\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(289\) −0.385113 + 0.262566i −0.385113 + 0.262566i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 1.78181 + 0.858075i 1.78181 + 0.858075i
\(303\) 0 0
\(304\) −1.63402 1.11406i −1.63402 1.11406i
\(305\) 0 0
\(306\) −0.535628 0.496990i −0.535628 0.496990i
\(307\) −0.623490 + 0.781831i −0.623490 + 0.781831i −0.988831 0.149042i \(-0.952381\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(308\) −0.147791 0.0222759i −0.147791 0.0222759i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(312\) 0 0
\(313\) 0.900969 1.56052i 0.900969 1.56052i 0.0747301 0.997204i \(-0.476190\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(314\) −0.0332580 + 0.145713i −0.0332580 + 0.145713i
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(318\) 0 0
\(319\) 0.0902318 0.229907i 0.0902318 0.229907i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.321552 1.40881i 0.321552 1.40881i
\(324\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(325\) −0.500000 0.866025i −0.500000 0.866025i
\(326\) −1.40097 + 0.432142i −1.40097 + 0.432142i
\(327\) 0 0
\(328\) 0 0
\(329\) 0.603718 1.53825i 0.603718 1.53825i
\(330\) 0 0
\(331\) 1.32091 + 1.22563i 1.32091 + 1.22563i 0.955573 + 0.294755i \(0.0952381\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(332\) 0.440071 + 0.0663300i 0.440071 + 0.0663300i
\(333\) 0 0
\(334\) −1.88980 + 0.284841i −1.88980 + 0.284841i
\(335\) 0 0
\(336\) 0 0
\(337\) −0.658322 + 0.317031i −0.658322 + 0.317031i −0.733052 0.680173i \(-0.761905\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(338\) 0.826239 0.563320i 0.826239 0.563320i
\(339\) 0 0
\(340\) 0 0
\(341\) 0.0680900 + 0.173490i 0.0680900 + 0.173490i
\(342\) −1.97766 −1.97766
\(343\) 0.623490 0.781831i 0.623490 0.781831i
\(344\) 0 0
\(345\) 0 0
\(346\) 0.142820 + 1.90580i 0.142820 + 1.90580i
\(347\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(348\) 0 0
\(349\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(350\) −0.733052 0.680173i −0.733052 0.680173i
\(351\) 0 0
\(352\) −0.147791 + 0.0222759i −0.147791 + 0.0222759i
\(353\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.134659 + 1.79690i −0.134659 + 1.79690i 0.365341 + 0.930874i \(0.380952\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) −1.45557 2.52113i −1.45557 2.52113i
\(362\) −0.955573 + 1.65510i −0.955573 + 1.65510i
\(363\) 0 0
\(364\) 0.623490 0.781831i 0.623490 0.781831i
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.00000 −1.00000
\(372\) 0 0
\(373\) −0.955573 + 1.65510i −0.955573 + 1.65510i −0.222521 + 0.974928i \(0.571429\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(374\) −0.0546039 0.0945768i −0.0546039 0.0945768i
\(375\) 0 0
\(376\) 0.123490 1.64786i 0.123490 1.64786i
\(377\) 1.03030 + 1.29196i 1.03030 + 1.29196i
\(378\) 0 0
\(379\) −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.440071 0.0663300i 0.440071 0.0663300i 0.0747301 0.997204i \(-0.476190\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.0546039 + 0.728639i 0.0546039 + 0.728639i 0.955573 + 0.294755i \(0.0952381\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.365341 0.930874i 0.365341 0.930874i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −0.109562 + 0.101659i −0.109562 + 0.101659i
\(397\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.900969 0.433884i −0.900969 0.433884i
\(401\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(402\) 0 0
\(403\) −1.23305 0.185853i −1.23305 0.185853i
\(404\) 1.32091 + 1.22563i 1.32091 + 1.22563i
\(405\) 0 0
\(406\) 1.36534 + 0.930874i 1.36534 + 0.930874i
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.326239 + 1.42935i 0.326239 + 1.42935i
\(414\) 0 0
\(415\) 0 0
\(416\) 0.365341 0.930874i 0.365341 0.930874i
\(417\) 0 0
\(418\) −0.282450 0.0871242i −0.282450 0.0871242i
\(419\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(420\) 0 0
\(421\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(422\) 0 0
\(423\) −0.826239 1.43109i −0.826239 1.43109i
\(424\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(425\) 0.0546039 0.728639i 0.0546039 0.728639i
\(426\) 0 0
\(427\) −0.0747301 0.997204i −0.0747301 0.997204i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.03030 + 0.702449i 1.03030 + 0.702449i 0.955573 0.294755i \(-0.0952381\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(432\) 0 0
\(433\) 1.32091 + 0.636119i 1.32091 + 0.636119i 0.955573 0.294755i \(-0.0952381\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(434\) −1.23305 + 0.185853i −1.23305 + 0.185853i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(440\) 0 0
\(441\) −0.222521 0.974928i −0.222521 0.974928i
\(442\) 0.730682 0.730682
\(443\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.123490 0.0841939i 0.123490 0.0841939i
\(447\) 0 0
\(448\) 0.0747301 0.997204i 0.0747301 0.997204i
\(449\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(450\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(451\) 0 0
\(452\) 1.44973 + 0.218511i 1.44973 + 0.218511i
\(453\) 0 0
\(454\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(462\) 0 0
\(463\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(464\) 1.57906 + 0.487076i 1.57906 + 0.487076i
\(465\) 0 0
\(466\) 0.698220 1.77904i 0.698220 1.77904i
\(467\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(468\) −0.222521 0.974928i −0.222521 0.974928i
\(469\) −0.425270 + 1.86323i −0.425270 + 1.86323i
\(470\) 0 0
\(471\) 0 0
\(472\) 0.733052 + 1.26968i 0.733052 + 1.26968i
\(473\) 0 0
\(474\) 0 0
\(475\) −1.23305 1.54620i −1.23305 1.54620i
\(476\) 0.698220 0.215372i 0.698220 0.215372i
\(477\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(478\) 1.07473 + 0.997204i 1.07473 + 0.997204i
\(479\) 0.988831 + 0.149042i 0.988831 + 0.149042i 0.623490 0.781831i \(-0.285714\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.880843 0.424191i 0.880843 0.424191i
\(485\) 0 0
\(486\) 0 0
\(487\) 0.0546039 + 0.728639i 0.0546039 + 0.728639i 0.955573 + 0.294755i \(0.0952381\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(488\) −0.365341 0.930874i −0.365341 0.930874i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0.0902318 + 1.20406i 0.0902318 + 1.20406i
\(494\) 1.44973 1.34515i 1.44973 1.34515i
\(495\) 0 0
\(496\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(497\) −0.365341 + 0.632789i −0.365341 + 0.632789i
\(498\) 0 0
\(499\) 1.78181 0.268565i 1.78181 0.268565i 0.826239 0.563320i \(-0.190476\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(504\) −0.500000 0.866025i −0.500000 0.866025i
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.222521 0.974928i −0.222521 0.974928i
\(513\) 0 0
\(514\) −0.162592 + 0.414278i −0.162592 + 0.414278i
\(515\) 0 0
\(516\) 0 0
\(517\) −0.0549581 0.240787i −0.0549581 0.240787i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.623490 1.07992i −0.623490 1.07992i −0.988831 0.149042i \(-0.952381\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(522\) 1.57906 0.487076i 1.57906 0.487076i
\(523\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.667917 0.619736i −0.667917 0.619736i
\(528\) 0 0
\(529\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(530\) 0 0
\(531\) 1.32091 + 0.636119i 1.32091 + 0.636119i
\(532\) 0.988831 1.71271i 0.988831 1.71271i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.142820 + 1.90580i 0.142820 + 1.90580i
\(537\) 0 0
\(538\) −1.46610 −1.46610
\(539\) 0.0111692 0.149042i 0.0111692 0.149042i
\(540\) 0 0
\(541\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(542\) −0.0747301 0.997204i −0.0747301 0.997204i
\(543\) 0 0
\(544\) 0.603718 0.411608i 0.603718 0.411608i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(548\) 0 0
\(549\) −0.826239 0.563320i −0.826239 0.563320i
\(550\) −0.147791 0.0222759i −0.147791 0.0222759i
\(551\) 2.39564 + 2.22283i 2.39564 + 2.22283i
\(552\) 0 0
\(553\) 0 0
\(554\) −1.23305 1.54620i −1.23305 1.54620i
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) −0.623490 + 1.07992i −0.623490 + 1.07992i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.900969 0.433884i −0.900969 0.433884i
\(568\) −0.162592 + 0.712362i −0.162592 + 0.712362i
\(569\) 0.988831 1.71271i 0.988831 1.71271i 0.365341 0.930874i \(-0.380952\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(570\) 0 0
\(571\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(572\) 0.0111692 0.149042i 0.0111692 0.149042i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.733052 0.680173i −0.733052 0.680173i
\(577\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(578\) −0.385113 0.262566i −0.385113 0.262566i
\(579\) 0 0
\(580\) 0 0
\(581\) −0.0332580 + 0.443797i −0.0332580 + 0.443797i
\(582\) 0 0
\(583\) −0.123490 + 0.0841939i −0.123490 + 0.0841939i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(588\) 0 0
\(589\) −2.46610 −2.46610
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(600\) 0 0
\(601\) 1.03030 1.29196i 1.03030 1.29196i 0.0747301 0.997204i \(-0.476190\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(602\) 0 0
\(603\) 1.19158 + 1.49419i 1.19158 + 1.49419i
\(604\) −0.147791 + 1.97213i −0.147791 + 1.97213i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0.440071 1.92808i 0.440071 1.92808i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.57906 + 0.487076i 1.57906 + 0.487076i
\(612\) 0.266948 0.680173i 0.266948 0.680173i
\(613\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(614\) −0.955573 0.294755i −0.955573 0.294755i
\(615\) 0 0
\(616\) −0.0332580 0.145713i −0.0332580 0.145713i
\(617\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(618\) 0 0
\(619\) 0.222521 + 0.385418i 0.222521 + 0.385418i 0.955573 0.294755i \(-0.0952381\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.733052 0.680173i −0.733052 0.680173i
\(626\) 1.78181 + 0.268565i 1.78181 + 0.268565i
\(627\) 0 0
\(628\) −0.147791 + 0.0222759i −0.147791 + 0.0222759i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.400969 0.193096i 0.400969 0.193096i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(638\) 0.246980 0.246980
\(639\) 0.266948 + 0.680173i 0.266948 + 0.680173i
\(640\) 0 0
\(641\) 1.32091 1.22563i 1.32091 1.22563i 0.365341 0.930874i \(-0.380952\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(642\) 0 0
\(643\) 1.32091 0.636119i 1.32091 0.636119i 0.365341 0.930874i \(-0.380952\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.42890 0.215372i 1.42890 0.215372i
\(647\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(648\) −0.988831 0.149042i −0.988831 0.149042i
\(649\) 0.160629 + 0.149042i 0.160629 + 0.149042i
\(650\) 0.623490 0.781831i 0.623490 0.781831i
\(651\) 0 0
\(652\) −0.914101 1.14625i −0.914101 1.14625i
\(653\) −0.0332580 + 0.443797i −0.0332580 + 0.443797i 0.955573 + 0.294755i \(0.0952381\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 1.65248 1.65248
\(659\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(660\) 0 0
\(661\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(662\) −0.658322 + 1.67738i −0.658322 + 1.67738i
\(663\) 0 0
\(664\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.955573 1.65510i −0.955573 1.65510i
\(669\) 0 0
\(670\) 0 0
\(671\) −0.0931869 0.116853i −0.0931869 0.116853i
\(672\) 0 0
\(673\) 0.777479 0.974928i 0.777479 0.974928i −0.222521 0.974928i \(-0.571429\pi\)
1.00000 \(0\)
\(674\) −0.535628 0.496990i −0.535628 0.496990i
\(675\) 0 0
\(676\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(677\) −0.147791 + 0.0222759i −0.147791 + 0.0222759i −0.222521 0.974928i \(-0.571429\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −0.136622 + 0.126766i −0.136622 + 0.126766i
\(683\) 0.0931869 + 1.24349i 0.0931869 + 1.24349i 0.826239 + 0.563320i \(0.190476\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(684\) −0.722521 1.84095i −0.722521 1.84095i
\(685\) 0 0
\(686\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(687\) 0 0
\(688\) 0 0
\(689\) −0.0747301 0.997204i −0.0747301 0.997204i
\(690\) 0 0
\(691\) 0.123490 0.0841939i 0.123490 0.0841939i −0.500000 0.866025i \(-0.666667\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(692\) −1.72188 + 0.829215i −1.72188 + 0.829215i
\(693\) −0.109562 0.101659i −0.109562 0.101659i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.365341 0.930874i 0.365341 0.930874i
\(701\) 0.777479 + 0.974928i 0.777479 + 0.974928i 1.00000 \(0\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.0747301 0.129436i −0.0747301 0.129436i
\(705\) 0 0
\(706\) 0 0
\(707\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(708\) 0 0
\(709\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −1.72188 + 0.531130i −1.72188 + 0.531130i
\(719\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.81507 2.27603i 1.81507 2.27603i
\(723\) 0 0
\(724\) −1.88980 0.284841i −1.88980 0.284841i
\(725\) 1.36534 + 0.930874i 1.36534 + 0.930874i
\(726\) 0 0
\(727\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(728\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(729\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.104356 + 0.265895i 0.104356 + 0.265895i
\(738\) 0 0
\(739\) −0.914101 + 0.848162i −0.914101 + 0.848162i −0.988831 0.149042i \(-0.952381\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.365341 0.930874i −0.365341 0.930874i
\(743\) −1.48883 0.716983i −1.48883 0.716983i −0.500000 0.866025i \(-0.666667\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.88980 0.284841i −1.88980 0.284841i
\(747\) 0.326239 + 0.302705i 0.326239 + 0.302705i
\(748\) 0.0680900 0.0853822i 0.0680900 0.0853822i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(752\) 1.57906 0.487076i 1.57906 0.487076i
\(753\) 0 0
\(754\) −0.826239 + 1.43109i −0.826239 + 1.43109i
\(755\) 0 0
\(756\) 0 0
\(757\) −0.425270 1.86323i −0.425270 1.86323i −0.500000 0.866025i \(-0.666667\pi\)
0.0747301 0.997204i \(-0.476190\pi\)
\(758\) −0.425270 0.131178i −0.425270 0.131178i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(767\) −1.40097 + 0.432142i −1.40097 + 0.432142i
\(768\) 0 0
\(769\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(774\) 0 0
\(775\) −1.23305 + 0.185853i −1.23305 + 0.185853i
\(776\) 0 0
\(777\) 0 0
\(778\) −0.658322 + 0.317031i −0.658322 + 0.317031i
\(779\) 0 0
\(780\) 0 0
\(781\) 0.00816111 + 0.108903i 0.00816111 + 0.108903i
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 0.123490 + 1.64786i 0.123490 + 1.64786i 0.623490 + 0.781831i \(0.285714\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.109562 + 1.46200i −0.109562 + 1.46200i
\(792\) −0.134659 0.0648483i −0.134659 0.0648483i
\(793\) 0.988831 0.149042i 0.988831 0.149042i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.12349 + 1.40881i −1.12349 + 1.40881i −0.222521 + 0.974928i \(0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(798\) 0 0
\(799\) 0.752824 + 0.944011i 0.752824 + 0.944011i
\(800\) 0.0747301 0.997204i 0.0747301 0.997204i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −0.277479 1.21572i −0.277479 1.21572i
\(807\) 0 0
\(808\) −0.658322 + 1.67738i −0.658322 + 1.67738i
\(809\) −0.365341 + 0.930874i −0.365341 + 0.930874i 0.623490 + 0.781831i \(0.285714\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(810\) 0 0
\(811\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(812\) −0.367711 + 1.61105i −0.367711 + 1.61105i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0.955573 0.294755i 0.955573 0.294755i
\(820\) 0 0
\(821\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(822\) 0 0
\(823\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −1.21135 + 0.825886i −1.21135 + 0.825886i
\(827\) 1.78181 0.858075i 1.78181 0.858075i 0.826239 0.563320i \(-0.190476\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(828\) 0 0
\(829\) 1.44973 1.34515i 1.44973 1.34515i 0.623490 0.781831i \(-0.285714\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000 1.00000
\(833\) 0.266948 + 0.680173i 0.266948 + 0.680173i
\(834\) 0 0
\(835\) 0 0
\(836\) −0.0220888 0.294755i −0.0220888 0.294755i
\(837\) 0 0
\(838\) 0 0
\(839\) −1.72188 + 0.829215i −1.72188 + 0.829215i −0.733052 + 0.680173i \(0.761905\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(840\) 0 0
\(841\) −1.55929 0.750915i −1.55929 0.750915i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 1.03030 1.29196i 1.03030 1.29196i
\(847\) 0.488831 + 0.846680i 0.488831 + 0.846680i
\(848\) −0.623490 0.781831i −0.623490 0.781831i
\(849\) 0 0
\(850\) 0.698220 0.215372i 0.698220 0.215372i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(854\) 0.900969 0.433884i 0.900969 0.433884i
\(855\) 0 0
\(856\) 0 0
\(857\) 0.698220 1.77904i 0.698220 1.77904i 0.0747301 0.997204i \(-0.476190\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(858\) 0 0
\(859\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(863\) 0.900969 1.56052i 0.900969 1.56052i 0.0747301 0.997204i \(-0.476190\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −0.109562 + 1.46200i −0.109562 + 1.46200i
\(867\) 0 0
\(868\) −0.623490 1.07992i −0.623490 1.07992i
\(869\) 0 0
\(870\) 0 0
\(871\) −1.88980 0.284841i −1.88980 0.284841i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(882\) 0.826239 0.563320i 0.826239 0.563320i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0.266948 + 0.680173i 0.266948 + 0.680173i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.147791 + 0.0222759i −0.147791 + 0.0222759i
\(892\) 0.123490 + 0.0841939i 0.123490 + 0.0841939i
\(893\) 3.23154 + 0.487076i 3.23154 + 0.487076i
\(894\) 0 0
\(895\) 0 0
\(896\) 0.955573 0.294755i 0.955573 0.294755i
\(897\) 0 0
\(898\) 0 0
\(899\) 1.96906 0.607374i 1.96906 0.607374i
\(900\) −0.500000 0.866025i −0.500000 0.866025i
\(901\) 0.365341 0.632789i 0.365341 0.632789i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.326239 + 1.42935i 0.326239 + 1.42935i
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(908\) −1.72188 0.531130i −1.72188 0.531130i
\(909\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(910\) 0 0
\(911\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(912\) 0 0
\(913\) 0.0332580 + 0.0576046i 0.0332580 + 0.0576046i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.658322 0.317031i −0.658322 0.317031i
\(924\) 0 0
\(925\) 0 0
\(926\) −0.367711 + 0.250701i −0.367711 + 0.250701i
\(927\) 0 0
\(928\) 0.123490 + 1.64786i 0.123490 + 1.64786i
\(929\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(930\) 0 0
\(931\) 1.78181 + 0.858075i 1.78181 + 0.858075i
\(932\) 1.91115 1.91115
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.826239 0.563320i 0.826239 0.563320i
\(937\) −1.72188 + 0.829215i −1.72188 + 0.829215i −0.733052 + 0.680173i \(0.761905\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(938\) −1.88980 + 0.284841i −1.88980 + 0.284841i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −0.914101 + 1.14625i −0.914101 + 1.14625i
\(945\) 0 0
\(946\) 0 0
\(947\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i 0.826239 + 0.563320i \(0.190476\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.988831 1.71271i 0.988831 1.71271i
\(951\) 0 0
\(952\) 0.455573 + 0.571270i 0.455573 + 0.571270i
\(953\) 0.440071 + 1.92808i 0.440071 + 1.92808i 0.365341 + 0.930874i \(0.380952\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(954\) −0.955573 0.294755i −0.955573 0.294755i
\(955\) 0 0
\(956\) −0.535628 + 1.36476i −0.535628 + 1.36476i
\(957\) 0 0
\(958\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(959\) 0 0
\(960\) 0 0
\(961\) −0.277479 + 0.480608i −0.277479 + 0.480608i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.623490 + 0.781831i −0.623490 + 0.781831i −0.988831 0.149042i \(-0.952381\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(968\) 0.716677 + 0.664979i 0.716677 + 0.664979i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −0.658322 + 0.317031i −0.658322 + 0.317031i
\(975\) 0 0
\(976\) 0.733052 0.680173i 0.733052 0.680173i
\(977\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0.0546039 + 0.728639i 0.0546039 + 0.728639i 0.955573 + 0.294755i \(0.0952381\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.08786 + 0.523887i −1.08786 + 0.523887i
\(987\) 0 0
\(988\) 1.78181 + 0.858075i 1.78181 + 0.858075i
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(992\) −0.914101 0.848162i −0.914101 0.848162i
\(993\) 0 0
\(994\) −0.722521 0.108903i −0.722521 0.108903i
\(995\) 0 0
\(996\) 0 0
\(997\) 0.142820 0.0440542i 0.142820 0.0440542i −0.222521 0.974928i \(-0.571429\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(998\) 0.900969 + 1.56052i 0.900969 + 1.56052i
\(999\) 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 2548.1.de.b.1143.1 yes 12
4.3 odd 2 2548.1.de.a.1143.1 12
13.12 even 2 2548.1.de.a.1143.1 12
49.46 even 21 inner 2548.1.de.b.1663.1 yes 12
52.51 odd 2 CM 2548.1.de.b.1143.1 yes 12
196.95 odd 42 2548.1.de.a.1663.1 yes 12
637.389 even 42 2548.1.de.a.1663.1 yes 12
2548.1663 odd 42 inner 2548.1.de.b.1663.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2548.1.de.a.1143.1 12 4.3 odd 2
2548.1.de.a.1143.1 12 13.12 even 2
2548.1.de.a.1663.1 yes 12 196.95 odd 42
2548.1.de.a.1663.1 yes 12 637.389 even 42
2548.1.de.b.1143.1 yes 12 1.1 even 1 trivial
2548.1.de.b.1143.1 yes 12 52.51 odd 2 CM
2548.1.de.b.1663.1 yes 12 49.46 even 21 inner
2548.1.de.b.1663.1 yes 12 2548.1663 odd 42 inner