Properties

Label 2548.1.de.a.779.1
Level $2548$
Weight $1$
Character 2548.779
Analytic conductor $1.272$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -52
Inner twists $4$

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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 779.1
Root \(-0.733052 + 0.680173i\) of defining polynomial
Character \(\chi\) \(=\) 2548.779
Dual form 2548.1.de.a.2391.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.988831 - 0.149042i) q^{2} +(0.955573 - 0.294755i) q^{4} +(-0.955573 - 0.294755i) q^{7} +(0.900969 - 0.433884i) q^{8} +(0.365341 + 0.930874i) q^{9} +O(q^{10})\) \(q+(0.988831 - 0.149042i) q^{2} +(0.955573 - 0.294755i) q^{4} +(-0.955573 - 0.294755i) q^{7} +(0.900969 - 0.433884i) q^{8} +(0.365341 + 0.930874i) q^{9} +(-0.603718 + 1.53825i) q^{11} +(0.623490 + 0.781831i) q^{13} +(-0.988831 - 0.149042i) q^{14} +(0.826239 - 0.563320i) q^{16} +(1.44973 - 1.34515i) q^{17} +(0.500000 + 0.866025i) q^{18} +(0.365341 - 0.632789i) q^{19} +(-0.367711 + 1.61105i) q^{22} +(-0.988831 - 0.149042i) q^{25} +(0.733052 + 0.680173i) q^{26} -1.00000 q^{28} +(-0.0332580 - 0.145713i) q^{29} +(0.623490 + 1.07992i) q^{31} +(0.733052 - 0.680173i) q^{32} +(1.23305 - 1.54620i) q^{34} +(0.623490 + 0.781831i) q^{36} +(0.266948 - 0.680173i) q^{38} +(-0.123490 + 1.64786i) q^{44} +(0.147791 - 0.0222759i) q^{47} +(0.826239 + 0.563320i) q^{49} -1.00000 q^{50} +(0.826239 + 0.563320i) q^{52} +(-0.955573 + 0.294755i) q^{53} +(-0.988831 + 0.149042i) q^{56} +(-0.0546039 - 0.139129i) q^{58} +(-0.142820 - 1.90580i) q^{59} +(-0.955573 - 0.294755i) q^{61} +(0.777479 + 0.974928i) q^{62} +(-0.0747301 - 0.997204i) q^{63} +(0.623490 - 0.781831i) q^{64} +(-0.733052 - 1.26968i) q^{67} +(0.988831 - 1.71271i) q^{68} +(-0.440071 + 1.92808i) q^{71} +(0.733052 + 0.680173i) q^{72} +(0.162592 - 0.712362i) q^{76} +(1.03030 - 1.29196i) q^{77} +(-0.733052 + 0.680173i) q^{81} +(0.277479 - 0.347948i) q^{83} +(0.123490 + 1.64786i) q^{88} +(-0.365341 - 0.930874i) q^{91} +(0.142820 - 0.0440542i) q^{94} +(0.900969 + 0.433884i) q^{98} -1.65248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + q^{4} - q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\) \(1275\)
\(\chi(n)\) \(-1\) \(e\left(\frac{4}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.988831 0.149042i 0.988831 0.149042i
\(3\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(4\) 0.955573 0.294755i 0.955573 0.294755i
\(5\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(6\) 0 0
\(7\) −0.955573 0.294755i −0.955573 0.294755i
\(8\) 0.900969 0.433884i 0.900969 0.433884i
\(9\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(10\) 0 0
\(11\) −0.603718 + 1.53825i −0.603718 + 1.53825i 0.222521 + 0.974928i \(0.428571\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(12\) 0 0
\(13\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(14\) −0.988831 0.149042i −0.988831 0.149042i
\(15\) 0 0
\(16\) 0.826239 0.563320i 0.826239 0.563320i
\(17\) 1.44973 1.34515i 1.44973 1.34515i 0.623490 0.781831i \(-0.285714\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(18\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(19\) 0.365341 0.632789i 0.365341 0.632789i −0.623490 0.781831i \(-0.714286\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.367711 + 1.61105i −0.367711 + 1.61105i
\(23\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(24\) 0 0
\(25\) −0.988831 0.149042i −0.988831 0.149042i
\(26\) 0.733052 + 0.680173i 0.733052 + 0.680173i
\(27\) 0 0
\(28\) −1.00000 −1.00000
\(29\) −0.0332580 0.145713i −0.0332580 0.145713i 0.955573 0.294755i \(-0.0952381\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(30\) 0 0
\(31\) 0.623490 + 1.07992i 0.623490 + 1.07992i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(32\) 0.733052 0.680173i 0.733052 0.680173i
\(33\) 0 0
\(34\) 1.23305 1.54620i 1.23305 1.54620i
\(35\) 0 0
\(36\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(37\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(38\) 0.266948 0.680173i 0.266948 0.680173i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(42\) 0 0
\(43\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(44\) −0.123490 + 1.64786i −0.123490 + 1.64786i
\(45\) 0 0
\(46\) 0 0
\(47\) 0.147791 0.0222759i 0.147791 0.0222759i −0.0747301 0.997204i \(-0.523810\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(48\) 0 0
\(49\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(53\) −0.955573 + 0.294755i −0.955573 + 0.294755i −0.733052 0.680173i \(-0.761905\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(57\) 0 0
\(58\) −0.0546039 0.139129i −0.0546039 0.139129i
\(59\) −0.142820 1.90580i −0.142820 1.90580i −0.365341 0.930874i \(-0.619048\pi\)
0.222521 0.974928i \(-0.428571\pi\)
\(60\) 0 0
\(61\) −0.955573 0.294755i −0.955573 0.294755i −0.222521 0.974928i \(-0.571429\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(62\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(63\) −0.0747301 0.997204i −0.0747301 0.997204i
\(64\) 0.623490 0.781831i 0.623490 0.781831i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.733052 1.26968i −0.733052 1.26968i −0.955573 0.294755i \(-0.904762\pi\)
0.222521 0.974928i \(-0.428571\pi\)
\(68\) 0.988831 1.71271i 0.988831 1.71271i
\(69\) 0 0
\(70\) 0 0
\(71\) −0.440071 + 1.92808i −0.440071 + 1.92808i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(72\) 0.733052 + 0.680173i 0.733052 + 0.680173i
\(73\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.162592 0.712362i 0.162592 0.712362i
\(77\) 1.03030 1.29196i 1.03030 1.29196i
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(82\) 0 0
\(83\) 0.277479 0.347948i 0.277479 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.123490 + 1.64786i 0.123490 + 1.64786i
\(89\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(90\) 0 0
\(91\) −0.365341 0.930874i −0.365341 0.930874i
\(92\) 0 0
\(93\) 0 0
\(94\) 0.142820 0.0440542i 0.142820 0.0440542i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(99\) −1.65248 −1.65248
\(100\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(101\) −1.48883 1.01507i −1.48883 1.01507i −0.988831 0.149042i \(-0.952381\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(102\) 0 0
\(103\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(104\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(105\) 0 0
\(106\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(107\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(108\) 0 0
\(109\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(113\) 1.19158 1.49419i 1.19158 1.49419i 0.365341 0.930874i \(-0.380952\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.0747301 0.129436i −0.0747301 0.129436i
\(117\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(118\) −0.425270 1.86323i −0.425270 1.86323i
\(119\) −1.78181 + 0.858075i −1.78181 + 0.858075i
\(120\) 0 0
\(121\) −1.26868 1.17716i −1.26868 1.17716i
\(122\) −0.988831 0.149042i −0.988831 0.149042i
\(123\) 0 0
\(124\) 0.914101 + 0.848162i 0.914101 + 0.848162i
\(125\) 0 0
\(126\) −0.222521 0.974928i −0.222521 0.974928i
\(127\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(128\) 0.500000 0.866025i 0.500000 0.866025i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(132\) 0 0
\(133\) −0.535628 + 0.496990i −0.535628 + 0.496990i
\(134\) −0.914101 1.14625i −0.914101 1.14625i
\(135\) 0 0
\(136\) 0.722521 1.84095i 0.722521 1.84095i
\(137\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(138\) 0 0
\(139\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.147791 + 1.97213i −0.147791 + 1.97213i
\(143\) −1.57906 + 0.487076i −1.57906 + 0.487076i
\(144\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(150\) 0 0
\(151\) −0.698220 + 0.215372i −0.698220 + 0.215372i −0.623490 0.781831i \(-0.714286\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(152\) 0.0546039 0.728639i 0.0546039 0.728639i
\(153\) 1.78181 + 0.858075i 1.78181 + 0.858075i
\(154\) 0.826239 1.43109i 0.826239 1.43109i
\(155\) 0 0
\(156\) 0 0
\(157\) 0.123490 + 1.64786i 0.123490 + 1.64786i 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
\(163\) −1.57906 + 1.07659i −1.57906 + 1.07659i −0.623490 + 0.781831i \(0.714286\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.222521 0.385418i 0.222521 0.385418i
\(167\) −0.326239 1.42935i −0.326239 1.42935i −0.826239 0.563320i \(-0.809524\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(168\) 0 0
\(169\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(170\) 0 0
\(171\) 0.722521 + 0.108903i 0.722521 + 0.108903i
\(172\) 0 0
\(173\) 1.07473 + 0.997204i 1.07473 + 0.997204i 1.00000 \(0\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(174\) 0 0
\(175\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(176\) 0.367711 + 1.61105i 0.367711 + 1.61105i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(180\) 0 0
\(181\) −0.914101 + 1.14625i −0.914101 + 1.14625i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(182\) −0.500000 0.866025i −0.500000 0.866025i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.19395 + 3.04213i 1.19395 + 3.04213i
\(188\) 0.134659 0.0648483i 0.134659 0.0648483i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(192\) 0 0
\(193\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −1.63402 + 0.246289i −1.63402 + 0.246289i
\(199\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(200\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(201\) 0 0
\(202\) −1.62349 0.781831i −1.62349 0.781831i
\(203\) −0.0111692 + 0.149042i −0.0111692 + 0.149042i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(209\) 0.752824 + 0.944011i 0.752824 + 0.944011i
\(210\) 0 0
\(211\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(212\) −0.826239 + 0.563320i −0.826239 + 0.563320i
\(213\) 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
\(221\) 1.95557 + 0.294755i 1.95557 + 0.294755i
\(222\) 0 0
\(223\) 0.367711 1.61105i 0.367711 1.61105i −0.365341 0.930874i \(-0.619048\pi\)
0.733052 0.680173i \(-0.238095\pi\)
\(224\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(225\) −0.222521 0.974928i −0.222521 0.974928i
\(226\) 0.955573 1.65510i 0.955573 1.65510i
\(227\) −0.900969 1.56052i −0.900969 1.56052i −0.826239 0.563320i \(-0.809524\pi\)
−0.0747301 0.997204i \(-0.523810\pi\)
\(228\) 0 0
\(229\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.0931869 0.116853i −0.0931869 0.116853i
\(233\) −1.40097 0.432142i −1.40097 0.432142i −0.500000 0.866025i \(-0.666667\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(234\) −0.365341 + 0.930874i −0.365341 + 0.930874i
\(235\) 0 0
\(236\) −0.698220 1.77904i −0.698220 1.77904i
\(237\) 0 0
\(238\) −1.63402 + 1.11406i −1.63402 + 1.11406i
\(239\) 1.72188 + 0.829215i 1.72188 + 0.829215i 0.988831 + 0.149042i \(0.0476190\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(240\) 0 0
\(241\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(242\) −1.42996 0.974928i −1.42996 0.974928i
\(243\) 0 0
\(244\) −1.00000 −1.00000
\(245\) 0 0
\(246\) 0 0
\(247\) 0.722521 0.108903i 0.722521 0.108903i
\(248\) 1.03030 + 0.702449i 1.03030 + 0.702449i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(252\) −0.365341 0.930874i −0.365341 0.930874i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.365341 0.930874i 0.365341 0.930874i
\(257\) −0.425270 0.131178i −0.425270 0.131178i 0.0747301 0.997204i \(-0.476190\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.123490 0.0841939i 0.123490 0.0841939i
\(262\) 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
\(266\) −0.455573 + 0.571270i −0.455573 + 0.571270i
\(267\) 0 0
\(268\) −1.07473 0.997204i −1.07473 0.997204i
\(269\) −1.88980 0.284841i −1.88980 0.284841i −0.900969 0.433884i \(-0.857143\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(270\) 0 0
\(271\) −0.733052 0.680173i −0.733052 0.680173i 0.222521 0.974928i \(-0.428571\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(272\) 0.440071 1.92808i 0.440071 1.92808i
\(273\) 0 0
\(274\) 0 0
\(275\) 0.826239 1.43109i 0.826239 1.43109i
\(276\) 0 0
\(277\) −0.535628 + 0.496990i −0.535628 + 0.496990i −0.900969 0.433884i \(-0.857143\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(278\) 0 0
\(279\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(280\) 0 0
\(281\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(282\) 0 0
\(283\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(284\) 0.147791 + 1.97213i 0.147791 + 1.97213i
\(285\) 0 0
\(286\) −1.48883 + 0.716983i −1.48883 + 0.716983i
\(287\) 0 0
\(288\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(289\) 0.217550 2.90301i 0.217550 2.90301i
\(290\) 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
\(302\) −0.658322 + 0.317031i −0.658322 + 0.317031i
\(303\) 0 0
\(304\) −0.0546039 0.728639i −0.0546039 0.728639i
\(305\) 0 0
\(306\) 1.88980 + 0.582926i 1.88980 + 0.582926i
\(307\) 0.623490 + 0.781831i 0.623490 + 0.781831i 0.988831 0.149042i \(-0.0476190\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(308\) 0.603718 1.53825i 0.603718 1.53825i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(312\) 0 0
\(313\) 0.900969 1.56052i 0.900969 1.56052i 0.0747301 0.997204i \(-0.476190\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(314\) 0.367711 + 1.61105i 0.367711 + 1.61105i
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(318\) 0 0
\(319\) 0.244221 + 0.0368104i 0.244221 + 0.0368104i
\(320\) 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 + 1.29991i −1.40097 + 1.29991i
\(327\) 0 0
\(328\) 0 0
\(329\) −0.147791 0.0222759i −0.147791 0.0222759i
\(330\) 0 0
\(331\) 1.72188 + 0.531130i 1.72188 + 0.531130i 0.988831 0.149042i \(-0.0476190\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(332\) 0.162592 0.414278i 0.162592 0.414278i
\(333\) 0 0
\(334\) −0.535628 1.36476i −0.535628 1.36476i
\(335\) 0 0
\(336\) 0 0
\(337\) 1.78181 + 0.858075i 1.78181 + 0.858075i 0.955573 + 0.294755i \(0.0952381\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(338\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i
\(339\) 0 0
\(340\) 0 0
\(341\) −2.03759 + 0.307117i −2.03759 + 0.307117i
\(342\) 0.730682 0.730682
\(343\) −0.623490 0.781831i −0.623490 0.781831i
\(344\) 0 0
\(345\) 0 0
\(346\) 1.21135 + 0.825886i 1.21135 + 0.825886i
\(347\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(348\) 0 0
\(349\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(350\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(351\) 0 0
\(352\) 0.603718 + 1.53825i 0.603718 + 1.53825i
\(353\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.48883 1.01507i 1.48883 1.01507i 0.500000 0.866025i \(-0.333333\pi\)
0.988831 0.149042i \(-0.0476190\pi\)
\(360\) 0 0
\(361\) 0.233052 + 0.403658i 0.233052 + 0.403658i
\(362\) −0.733052 + 1.26968i −0.733052 + 1.26968i
\(363\) 0 0
\(364\) −0.623490 0.781831i −0.623490 0.781831i
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.00000 1.00000
\(372\) 0 0
\(373\) 0.733052 1.26968i 0.733052 1.26968i −0.222521 0.974928i \(-0.571429\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(374\) 1.63402 + 2.83021i 1.63402 + 2.83021i
\(375\) 0 0
\(376\) 0.123490 0.0841939i 0.123490 0.0841939i
\(377\) 0.0931869 0.116853i 0.0931869 0.116853i
\(378\) 0 0
\(379\) 0.277479 + 0.347948i 0.277479 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.162592 + 0.414278i 0.162592 + 0.414278i 0.988831 0.149042i \(-0.0476190\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.63402 1.11406i −1.63402 1.11406i −0.900969 0.433884i \(-0.857143\pi\)
−0.733052 0.680173i \(-0.761905\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.988831 + 0.149042i 0.988831 + 0.149042i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.57906 + 0.487076i −1.57906 + 0.487076i
\(397\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(401\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(402\) 0 0
\(403\) −0.455573 + 1.16078i −0.455573 + 1.16078i
\(404\) −1.72188 0.531130i −1.72188 0.531130i
\(405\) 0 0
\(406\) 0.0111692 + 0.149042i 0.0111692 + 0.149042i
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.425270 + 1.86323i −0.425270 + 1.86323i
\(414\) 0 0
\(415\) 0 0
\(416\) 0.988831 + 0.149042i 0.988831 + 0.149042i
\(417\) 0 0
\(418\) 0.885113 + 0.821265i 0.885113 + 0.821265i
\(419\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(420\) 0 0
\(421\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(422\) 0 0
\(423\) 0.0747301 + 0.129436i 0.0747301 + 0.129436i
\(424\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(425\) −1.63402 + 1.11406i −1.63402 + 1.11406i
\(426\) 0 0
\(427\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.0931869 1.24349i −0.0931869 1.24349i −0.826239 0.563320i \(-0.809524\pi\)
0.733052 0.680173i \(-0.238095\pi\)
\(432\) 0 0
\(433\) −1.72188 + 0.829215i −1.72188 + 0.829215i −0.733052 + 0.680173i \(0.761905\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(434\) −0.455573 1.16078i −0.455573 1.16078i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(440\) 0 0
\(441\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(442\) 1.97766 1.97766
\(443\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.123490 1.64786i 0.123490 1.64786i
\(447\) 0 0
\(448\) −0.826239 + 0.563320i −0.826239 + 0.563320i
\(449\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(450\) −0.365341 0.930874i −0.365341 0.930874i
\(451\) 0 0
\(452\) 0.698220 1.77904i 0.698220 1.77904i
\(453\) 0 0
\(454\) −1.12349 1.40881i −1.12349 1.40881i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(462\) 0 0
\(463\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i 0.900969 + 0.433884i \(0.142857\pi\)
−1.00000 \(\pi\)
\(464\) −0.109562 0.101659i −0.109562 0.101659i
\(465\) 0 0
\(466\) −1.44973 0.218511i −1.44973 0.218511i
\(467\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(468\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(469\) 0.326239 + 1.42935i 0.326239 + 1.42935i
\(470\) 0 0
\(471\) 0 0
\(472\) −0.955573 1.65510i −0.955573 1.65510i
\(473\) 0 0
\(474\) 0 0
\(475\) −0.455573 + 0.571270i −0.455573 + 0.571270i
\(476\) −1.44973 + 1.34515i −1.44973 + 1.34515i
\(477\) −0.623490 0.781831i −0.623490 0.781831i
\(478\) 1.82624 + 0.563320i 1.82624 + 0.563320i
\(479\) 0.365341 0.930874i 0.365341 0.930874i −0.623490 0.781831i \(-0.714286\pi\)
0.988831 0.149042i \(-0.0476190\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.55929 0.750915i −1.55929 0.750915i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.63402 + 1.11406i 1.63402 + 1.11406i 0.900969 + 0.433884i \(0.142857\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(488\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −0.244221 0.166507i −0.244221 0.166507i
\(494\) 0.698220 0.215372i 0.698220 0.215372i
\(495\) 0 0
\(496\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(497\) 0.988831 1.71271i 0.988831 1.71271i
\(498\) 0 0
\(499\) 0.658322 + 1.67738i 0.658322 + 1.67738i 0.733052 + 0.680173i \(0.238095\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(504\) −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
\(514\) −0.440071 0.0663300i −0.440071 0.0663300i
\(515\) 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 \(-0.380952\pi\)
\(522\) 0.109562 0.101659i 0.109562 0.101659i
\(523\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.35654 + 0.726897i 2.35654 + 0.726897i
\(528\) 0 0
\(529\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(530\) 0 0
\(531\) 1.72188 0.829215i 1.72188 0.829215i
\(532\) −0.365341 + 0.632789i −0.365341 + 0.632789i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −1.21135 0.825886i −1.21135 0.825886i
\(537\) 0 0
\(538\) −1.91115 −1.91115
\(539\) −1.36534 + 0.930874i −1.36534 + 0.930874i
\(540\) 0 0
\(541\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(542\) −0.826239 0.563320i −0.826239 0.563320i
\(543\) 0 0
\(544\) 0.147791 1.97213i 0.147791 1.97213i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(548\) 0 0
\(549\) −0.0747301 0.997204i −0.0747301 0.997204i
\(550\) 0.603718 1.53825i 0.603718 1.53825i
\(551\) −0.104356 0.0321896i −0.104356 0.0321896i
\(552\) 0 0
\(553\) 0 0
\(554\) −0.455573 + 0.571270i −0.455573 + 0.571270i
\(555\) 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 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.900969 0.433884i 0.900969 0.433884i
\(568\) 0.440071 + 1.92808i 0.440071 + 1.92808i
\(569\) −0.365341 + 0.632789i −0.365341 + 0.632789i −0.988831 0.149042i \(-0.952381\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(570\) 0 0
\(571\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(572\) −1.36534 + 0.930874i −1.36534 + 0.930874i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(577\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(578\) −0.217550 2.90301i −0.217550 2.90301i
\(579\) 0 0
\(580\) 0 0
\(581\) −0.367711 + 0.250701i −0.367711 + 0.250701i
\(582\) 0 0
\(583\) 0.123490 1.64786i 0.123490 1.64786i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) 0 0
\(589\) 0.911146 0.911146
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(600\) 0 0
\(601\) 0.0931869 + 0.116853i 0.0931869 + 0.116853i 0.826239 0.563320i \(-0.190476\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(602\) 0 0
\(603\) 0.914101 1.14625i 0.914101 1.14625i
\(604\) −0.603718 + 0.411608i −0.603718 + 0.411608i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) −0.162592 0.712362i −0.162592 0.712362i
\(609\) 0 0
\(610\) 0 0
\(611\) 0.109562 + 0.101659i 0.109562 + 0.101659i
\(612\) 1.95557 + 0.294755i 1.95557 + 0.294755i
\(613\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(614\) 0.733052 + 0.680173i 0.733052 + 0.680173i
\(615\) 0 0
\(616\) 0.367711 1.61105i 0.367711 1.61105i
\(617\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(618\) 0 0
\(619\) −0.222521 0.385418i −0.222521 0.385418i 0.733052 0.680173i \(-0.238095\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(626\) 0.658322 1.67738i 0.658322 1.67738i
\(627\) 0 0
\(628\) 0.603718 + 1.53825i 0.603718 + 1.53825i
\(629\) 0 0
\(630\) 0 0
\(631\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(638\) 0.246980 0.246980
\(639\) −1.95557 + 0.294755i −1.95557 + 0.294755i
\(640\) 0 0
\(641\) −1.72188 + 0.531130i −1.72188 + 0.531130i −0.988831 0.149042i \(-0.952381\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(642\) 0 0
\(643\) 1.72188 + 0.829215i 1.72188 + 0.829215i 0.988831 + 0.149042i \(0.0476190\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.527933 1.34515i −0.527933 1.34515i
\(647\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(648\) −0.365341 + 0.930874i −0.365341 + 0.930874i
\(649\) 3.01782 + 0.930874i 3.01782 + 0.930874i
\(650\) −0.623490 0.781831i −0.623490 0.781831i
\(651\) 0 0
\(652\) −1.19158 + 1.49419i −1.19158 + 1.49419i
\(653\) −0.367711 + 0.250701i −0.367711 + 0.250701i −0.733052 0.680173i \(-0.761905\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) −0.149460 −0.149460
\(659\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(660\) 0 0
\(661\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(662\) 1.78181 + 0.268565i 1.78181 + 0.268565i
\(663\) 0 0
\(664\) 0.0990311 0.433884i 0.0990311 0.433884i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.733052 1.26968i −0.733052 1.26968i
\(669\) 0 0
\(670\) 0 0
\(671\) 1.03030 1.29196i 1.03030 1.29196i
\(672\) 0 0
\(673\) 0.777479 + 0.974928i 0.777479 + 0.974928i 1.00000 \(0\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(674\) 1.88980 + 0.582926i 1.88980 + 0.582926i
\(675\) 0 0
\(676\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(677\) 0.603718 + 1.53825i 0.603718 + 1.53825i 0.826239 + 0.563320i \(0.190476\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −1.96906 + 0.607374i −1.96906 + 0.607374i
\(683\) −1.03030 0.702449i −1.03030 0.702449i −0.0747301 0.997204i \(-0.523810\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(684\) 0.722521 0.108903i 0.722521 0.108903i
\(685\) 0 0
\(686\) −0.733052 0.680173i −0.733052 0.680173i
\(687\) 0 0
\(688\) 0 0
\(689\) −0.826239 0.563320i −0.826239 0.563320i
\(690\) 0 0
\(691\) −0.123490 + 1.64786i −0.123490 + 1.64786i 0.500000 + 0.866025i \(0.333333\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(692\) 1.32091 + 0.636119i 1.32091 + 0.636119i
\(693\) 1.57906 + 0.487076i 1.57906 + 0.487076i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.988831 + 0.149042i 0.988831 + 0.149042i
\(701\) 0.777479 0.974928i 0.777479 0.974928i −0.222521 0.974928i \(-0.571429\pi\)
1.00000 \(0\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.826239 + 1.43109i 0.826239 + 1.43109i
\(705\) 0 0
\(706\) 0 0
\(707\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(708\) 0 0
\(709\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 1.32091 1.22563i 1.32091 1.22563i
\(719\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.290611 + 0.364415i 0.290611 + 0.364415i
\(723\) 0 0
\(724\) −0.535628 + 1.36476i −0.535628 + 1.36476i
\(725\) 0.0111692 + 0.149042i 0.0111692 + 0.149042i
\(726\) 0 0
\(727\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(728\) −0.733052 0.680173i −0.733052 0.680173i
\(729\) −0.900969 0.433884i −0.900969 0.433884i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.39564 0.361085i 2.39564 0.361085i
\(738\) 0 0
\(739\) −1.19158 + 0.367554i −1.19158 + 0.367554i −0.826239 0.563320i \(-0.809524\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.988831 0.149042i 0.988831 0.149042i
\(743\) 0.134659 0.0648483i 0.134659 0.0648483i −0.365341 0.930874i \(-0.619048\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.535628 1.36476i 0.535628 1.36476i
\(747\) 0.425270 + 0.131178i 0.425270 + 0.131178i
\(748\) 2.03759 + 2.55506i 2.03759 + 2.55506i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(752\) 0.109562 0.101659i 0.109562 0.101659i
\(753\) 0 0
\(754\) 0.0747301 0.129436i 0.0747301 0.129436i
\(755\) 0 0
\(756\) 0 0
\(757\) 0.326239 1.42935i 0.326239 1.42935i −0.500000 0.866025i \(-0.666667\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(758\) 0.326239 + 0.302705i 0.326239 + 0.302705i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(767\) 1.40097 1.29991i 1.40097 1.29991i
\(768\) 0 0
\(769\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(774\) 0 0
\(775\) −0.455573 1.16078i −0.455573 1.16078i
\(776\) 0 0
\(777\) 0 0
\(778\) −1.78181 0.858075i −1.78181 0.858075i
\(779\) 0 0
\(780\) 0 0
\(781\) −2.70018 1.84095i −2.70018 1.84095i
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −0.123490 0.0841939i −0.123490 0.0841939i 0.500000 0.866025i \(-0.333333\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.57906 + 1.07659i −1.57906 + 1.07659i
\(792\) −1.48883 + 0.716983i −1.48883 + 0.716983i
\(793\) −0.365341 0.930874i −0.365341 0.930874i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.12349 1.40881i −1.12349 1.40881i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(798\) 0 0
\(799\) 0.184292 0.231095i 0.184292 0.231095i
\(800\) −0.826239 + 0.563320i −0.826239 + 0.563320i
\(801\) 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
\(808\) −1.78181 0.268565i −1.78181 0.268565i
\(809\) 0.988831 + 0.149042i 0.988831 + 0.149042i 0.623490 0.781831i \(-0.285714\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(810\) 0 0
\(811\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i 0.900969 + 0.433884i \(0.142857\pi\)
−1.00000 \(\pi\)
\(812\) 0.0332580 + 0.145713i 0.0332580 + 0.145713i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0.733052 0.680173i 0.733052 0.680173i
\(820\) 0 0
\(821\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(822\) 0 0
\(823\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −0.142820 + 1.90580i −0.142820 + 1.90580i
\(827\) 0.658322 + 0.317031i 0.658322 + 0.317031i 0.733052 0.680173i \(-0.238095\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(828\) 0 0
\(829\) 0.698220 0.215372i 0.698220 0.215372i 0.0747301 0.997204i \(-0.476190\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000 1.00000
\(833\) 1.95557 0.294755i 1.95557 0.294755i
\(834\) 0 0
\(835\) 0 0
\(836\) 0.997630 + 0.680173i 0.997630 + 0.680173i
\(837\) 0 0
\(838\) 0 0
\(839\) −1.32091 0.636119i −1.32091 0.636119i −0.365341 0.930874i \(-0.619048\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(840\) 0 0
\(841\) 0.880843 0.424191i 0.880843 0.424191i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0.0931869 + 0.116853i 0.0931869 + 0.116853i
\(847\) 0.865341 + 1.49881i 0.865341 + 1.49881i
\(848\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(849\) 0 0
\(850\) −1.44973 + 1.34515i −1.44973 + 1.34515i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(854\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(855\) 0 0
\(856\) 0 0
\(857\) 1.44973 + 0.218511i 1.44973 + 0.218511i 0.826239 0.563320i \(-0.190476\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(858\) 0 0
\(859\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(860\) 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 \(0.523810\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.57906 + 1.07659i −1.57906 + 1.07659i
\(867\) 0 0
\(868\) −0.623490 1.07992i −0.623490 1.07992i
\(869\) 0 0
\(870\) 0 0
\(871\) 0.535628 1.36476i 0.535628 1.36476i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(882\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 1.95557 0.294755i 1.95557 0.294755i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.603718 1.53825i −0.603718 1.53825i
\(892\) −0.123490 1.64786i −0.123490 1.64786i
\(893\) 0.0398981 0.101659i 0.0398981 0.101659i
\(894\) 0 0
\(895\) 0 0
\(896\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(897\) 0 0
\(898\) 0 0
\(899\) 0.136622 0.126766i 0.136622 0.126766i
\(900\) −0.500000 0.866025i −0.500000 0.866025i
\(901\) −0.988831 + 1.71271i −0.988831 + 1.71271i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.425270 1.86323i 0.425270 1.86323i
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(908\) −1.32091 1.22563i −1.32091 1.22563i
\(909\) 0.400969 1.75676i 0.400969 1.75676i
\(910\) 0 0
\(911\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(912\) 0 0
\(913\) 0.367711 + 0.636894i 0.367711 + 0.636894i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.78181 + 0.858075i −1.78181 + 0.858075i
\(924\) 0 0
\(925\) 0 0
\(926\) −0.0332580 + 0.443797i −0.0332580 + 0.443797i
\(927\) 0 0
\(928\) −0.123490 0.0841939i −0.123490 0.0841939i
\(929\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(930\) 0 0
\(931\) 0.658322 0.317031i 0.658322 0.317031i
\(932\) −1.46610 −1.46610
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −0.0747301 + 0.997204i −0.0747301 + 0.997204i
\(937\) 1.32091 + 0.636119i 1.32091 + 0.636119i 0.955573 0.294755i \(-0.0952381\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(938\) 0.535628 + 1.36476i 0.535628 + 1.36476i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.19158 1.49419i −1.19158 1.49419i
\(945\) 0 0
\(946\) 0 0
\(947\) 0.826239 0.563320i 0.826239 0.563320i −0.0747301 0.997204i \(-0.523810\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −0.365341 + 0.632789i −0.365341 + 0.632789i
\(951\) 0 0
\(952\) −1.23305 + 1.54620i −1.23305 + 1.54620i
\(953\) −0.162592 + 0.712362i −0.162592 + 0.712362i 0.826239 + 0.563320i \(0.190476\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(954\) −0.733052 0.680173i −0.733052 0.680173i
\(955\) 0 0
\(956\) 1.88980 + 0.284841i 1.88980 + 0.284841i
\(957\) 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 \(-0.0476190\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(968\) −1.65379 0.510127i −1.65379 0.510127i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.78181 + 0.858075i 1.78181 + 0.858075i
\(975\) 0 0
\(976\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(977\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.63402 + 1.11406i 1.63402 + 1.11406i 0.900969 + 0.433884i \(0.142857\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.266310 0.128248i −0.266310 0.128248i
\(987\) 0 0
\(988\) 0.658322 0.317031i 0.658322 0.317031i
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(992\) 1.19158 + 0.367554i 1.19158 + 0.367554i
\(993\) 0 0
\(994\) 0.722521 1.84095i 0.722521 1.84095i
\(995\) 0 0
\(996\) 0 0
\(997\) −1.21135 + 1.12397i −1.21135 + 1.12397i −0.222521 + 0.974928i \(0.571429\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(998\) 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.a.779.1 12
4.3 odd 2 2548.1.de.b.779.1 yes 12
13.12 even 2 2548.1.de.b.779.1 yes 12
49.39 even 21 inner 2548.1.de.a.2391.1 yes 12
52.51 odd 2 CM 2548.1.de.a.779.1 12
196.39 odd 42 2548.1.de.b.2391.1 yes 12
637.480 even 42 2548.1.de.b.2391.1 yes 12
2548.2391 odd 42 inner 2548.1.de.a.2391.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2548.1.de.a.779.1 12 1.1 even 1 trivial
2548.1.de.a.779.1 12 52.51 odd 2 CM
2548.1.de.a.2391.1 yes 12 49.39 even 21 inner
2548.1.de.a.2391.1 yes 12 2548.2391 odd 42 inner
2548.1.de.b.779.1 yes 12 4.3 odd 2
2548.1.de.b.779.1 yes 12 13.12 even 2
2548.1.de.b.2391.1 yes 12 196.39 odd 42
2548.1.de.b.2391.1 yes 12 637.480 even 42