Properties

Label 2888.1.bb.a.2395.1
Level $2888$
Weight $1$
Character 2888.2395
Analytic conductor $1.441$
Analytic rank $0$
Dimension $18$
Projective image $D_{19}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2888,1,Mod(115,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(38))
 
chi = DirichletCharacter(H, H._module([19, 19, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2888.115");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.bb (of order \(38\), degree \(18\), 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: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\Q(\zeta_{38})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{19}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{19} - \cdots)\)

Embedding invariants

Embedding label 2395.1
Root \(0.677282 - 0.735724i\) of defining polynomial
Character \(\chi\) \(=\) 2888.2395
Dual form 2888.1.bb.a.1939.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.789141 + 0.614213i) q^{2} +(0.917421 + 0.996584i) q^{3} +(0.245485 + 0.969400i) q^{4} +(0.111859 + 1.34994i) q^{6} +(-0.401695 + 0.915773i) q^{8} +(-0.0689406 + 0.831990i) q^{9} +(-0.0405441 - 0.489294i) q^{11} +(-0.740876 + 1.13399i) q^{12} +(-0.879474 + 0.475947i) q^{16} +(-0.431796 + 1.70512i) q^{17} +(-0.565423 + 0.614213i) q^{18} +(0.789141 - 0.614213i) q^{19} +(0.268536 - 0.411024i) q^{22} +(-1.28117 + 0.439826i) q^{24} +(0.546948 - 0.837166i) q^{25} +(0.176545 - 0.137410i) q^{27} +(-0.986361 - 0.164595i) q^{32} +(0.450427 - 0.489294i) q^{33} +(-1.38806 + 1.08037i) q^{34} +(-0.823455 + 0.137410i) q^{36} +1.00000 q^{38} +(-1.28117 - 0.439826i) q^{41} +(-1.55676 + 0.259777i) q^{43} +(0.464369 - 0.159418i) q^{44} +(-1.28117 - 0.439826i) q^{48} +(-0.879474 - 0.475947i) q^{49} +(0.945817 - 0.324699i) q^{50} +(-2.09544 + 1.13399i) q^{51} +0.223718 q^{54} +(1.33609 + 0.222954i) q^{57} +(1.49277 + 0.512467i) q^{59} +(-0.677282 - 0.735724i) q^{64} +(0.655981 - 0.109464i) q^{66} +(0.0663435 - 0.151248i) q^{67} -1.75895 q^{68} +(-0.734221 - 0.397341i) q^{72} +(0.464369 - 1.83375i) q^{73} +(1.33609 - 0.222954i) q^{75} +(0.789141 + 0.614213i) q^{76} +(1.12236 + 0.187289i) q^{81} +(-0.740876 - 1.13399i) q^{82} +(1.54695 + 0.837166i) q^{83} +(-1.38806 - 0.751179i) q^{86} +(0.464369 + 0.159418i) q^{88} +(-0.431796 - 1.70512i) q^{89} +(-0.740876 - 1.13399i) q^{96} +(-0.803391 - 1.83155i) q^{97} +(-0.401695 - 0.915773i) q^{98} +0.409883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} + 17 q^{3} - q^{4} - 2 q^{6} - q^{8} + 16 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} - q^{19} - 2 q^{22} - 2 q^{24} - q^{25} + 15 q^{27} - q^{32} - 4 q^{33} - 2 q^{34}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{4}{19}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.789141 + 0.614213i 0.789141 + 0.614213i
\(3\) 0.917421 + 0.996584i 0.917421 + 0.996584i 1.00000 \(0\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(4\) 0.245485 + 0.969400i 0.245485 + 0.969400i
\(5\) 0 0 0.879474 0.475947i \(-0.157895\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(6\) 0.111859 + 1.34994i 0.111859 + 1.34994i
\(7\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(8\) −0.401695 + 0.915773i −0.401695 + 0.915773i
\(9\) −0.0689406 + 0.831990i −0.0689406 + 0.831990i
\(10\) 0 0
\(11\) −0.0405441 0.489294i −0.0405441 0.489294i −0.986361 0.164595i \(-0.947368\pi\)
0.945817 0.324699i \(-0.105263\pi\)
\(12\) −0.740876 + 1.13399i −0.740876 + 1.13399i
\(13\) 0 0 −0.677282 0.735724i \(-0.736842\pi\)
0.677282 + 0.735724i \(0.263158\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.879474 + 0.475947i −0.879474 + 0.475947i
\(17\) −0.431796 + 1.70512i −0.431796 + 1.70512i 0.245485 + 0.969400i \(0.421053\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(18\) −0.565423 + 0.614213i −0.565423 + 0.614213i
\(19\) 0.789141 0.614213i 0.789141 0.614213i
\(20\) 0 0
\(21\) 0 0
\(22\) 0.268536 0.411024i 0.268536 0.411024i
\(23\) 0 0 0.677282 0.735724i \(-0.263158\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(24\) −1.28117 + 0.439826i −1.28117 + 0.439826i
\(25\) 0.546948 0.837166i 0.546948 0.837166i
\(26\) 0 0
\(27\) 0.176545 0.137410i 0.176545 0.137410i
\(28\) 0 0
\(29\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(30\) 0 0
\(31\) 0 0 0.789141 0.614213i \(-0.210526\pi\)
−0.789141 + 0.614213i \(0.789474\pi\)
\(32\) −0.986361 0.164595i −0.986361 0.164595i
\(33\) 0.450427 0.489294i 0.450427 0.489294i
\(34\) −1.38806 + 1.08037i −1.38806 + 1.08037i
\(35\) 0 0
\(36\) −0.823455 + 0.137410i −0.823455 + 0.137410i
\(37\) 0 0 −0.0825793 0.996584i \(-0.526316\pi\)
0.0825793 + 0.996584i \(0.473684\pi\)
\(38\) 1.00000 1.00000
\(39\) 0 0
\(40\) 0 0
\(41\) −1.28117 0.439826i −1.28117 0.439826i −0.401695 0.915773i \(-0.631579\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(42\) 0 0
\(43\) −1.55676 + 0.259777i −1.55676 + 0.259777i −0.879474 0.475947i \(-0.842105\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(44\) 0.464369 0.159418i 0.464369 0.159418i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.0825793 0.996584i \(-0.526316\pi\)
0.0825793 + 0.996584i \(0.473684\pi\)
\(48\) −1.28117 0.439826i −1.28117 0.439826i
\(49\) −0.879474 0.475947i −0.879474 0.475947i
\(50\) 0.945817 0.324699i 0.945817 0.324699i
\(51\) −2.09544 + 1.13399i −2.09544 + 1.13399i
\(52\) 0 0
\(53\) 0 0 −0.0825793 0.996584i \(-0.526316\pi\)
0.0825793 + 0.996584i \(0.473684\pi\)
\(54\) 0.223718 0.223718
\(55\) 0 0
\(56\) 0 0
\(57\) 1.33609 + 0.222954i 1.33609 + 0.222954i
\(58\) 0 0
\(59\) 1.49277 + 0.512467i 1.49277 + 0.512467i 0.945817 0.324699i \(-0.105263\pi\)
0.546948 + 0.837166i \(0.315789\pi\)
\(60\) 0 0
\(61\) 0 0 −0.401695 0.915773i \(-0.631579\pi\)
0.401695 + 0.915773i \(0.368421\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.677282 0.735724i −0.677282 0.735724i
\(65\) 0 0
\(66\) 0.655981 0.109464i 0.655981 0.109464i
\(67\) 0.0663435 0.151248i 0.0663435 0.151248i −0.879474 0.475947i \(-0.842105\pi\)
0.945817 + 0.324699i \(0.105263\pi\)
\(68\) −1.75895 −1.75895
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.401695 0.915773i \(-0.368421\pi\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(72\) −0.734221 0.397341i −0.734221 0.397341i
\(73\) 0.464369 1.83375i 0.464369 1.83375i −0.0825793 0.996584i \(-0.526316\pi\)
0.546948 0.837166i \(-0.315789\pi\)
\(74\) 0 0
\(75\) 1.33609 0.222954i 1.33609 0.222954i
\(76\) 0.789141 + 0.614213i 0.789141 + 0.614213i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.986361 0.164595i \(-0.0526316\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(80\) 0 0
\(81\) 1.12236 + 0.187289i 1.12236 + 0.187289i
\(82\) −0.740876 1.13399i −0.740876 1.13399i
\(83\) 1.54695 + 0.837166i 1.54695 + 0.837166i 1.00000 \(0\)
0.546948 + 0.837166i \(0.315789\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.38806 0.751179i −1.38806 0.751179i
\(87\) 0 0
\(88\) 0.464369 + 0.159418i 0.464369 + 0.159418i
\(89\) −0.431796 1.70512i −0.431796 1.70512i −0.677282 0.735724i \(-0.736842\pi\)
0.245485 0.969400i \(-0.421053\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −0.740876 1.13399i −0.740876 1.13399i
\(97\) −0.803391 1.83155i −0.803391 1.83155i −0.401695 0.915773i \(-0.631579\pi\)
−0.401695 0.915773i \(-0.631579\pi\)
\(98\) −0.401695 0.915773i −0.401695 0.915773i
\(99\) 0.409883 0.409883
\(100\) 0.945817 + 0.324699i 0.945817 + 0.324699i
\(101\) 0 0 0.0825793 0.996584i \(-0.473684\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(102\) −2.35011 0.392164i −2.35011 0.392164i
\(103\) 0 0 −0.245485 0.969400i \(-0.578947\pi\)
0.245485 + 0.969400i \(0.421053\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.49277 + 1.16187i 1.49277 + 1.16187i 0.945817 + 0.324699i \(0.105263\pi\)
0.546948 + 0.837166i \(0.315789\pi\)
\(108\) 0.176545 + 0.137410i 0.176545 + 0.137410i
\(109\) 0 0 0.0825793 0.996584i \(-0.473684\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.387445 + 1.52999i 0.387445 + 1.52999i 0.789141 + 0.614213i \(0.210526\pi\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(114\) 0.917421 + 0.996584i 0.917421 + 0.996584i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0.863238 + 1.32128i 0.863238 + 1.32128i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.748596 0.124919i 0.748596 0.124919i
\(122\) 0 0
\(123\) −0.737047 1.68030i −0.737047 1.68030i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −0.0825793 0.996584i −0.0825793 0.996584i
\(129\) −1.68709 1.31311i −1.68709 1.31311i
\(130\) 0 0
\(131\) −1.38806 + 1.08037i −1.38806 + 1.08037i −0.401695 + 0.915773i \(0.631579\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(132\) 0.584895 + 0.316529i 0.584895 + 0.316529i
\(133\) 0 0
\(134\) 0.145253 0.0786068i 0.145253 0.0786068i
\(135\) 0 0
\(136\) −1.38806 1.08037i −1.38806 1.08037i
\(137\) 0.792434 1.80657i 0.792434 1.80657i 0.245485 0.969400i \(-0.421053\pi\)
0.546948 0.837166i \(-0.315789\pi\)
\(138\) 0 0
\(139\) −1.07898 + 0.180049i −1.07898 + 0.180049i −0.677282 0.735724i \(-0.736842\pi\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.335352 0.764525i −0.335352 0.764525i
\(145\) 0 0
\(146\) 1.49277 1.16187i 1.49277 1.16187i
\(147\) −0.332526 1.31311i −0.332526 1.31311i
\(148\) 0 0
\(149\) 0 0 0.945817 0.324699i \(-0.105263\pi\)
−0.945817 + 0.324699i \(0.894737\pi\)
\(150\) 1.19130 + 0.644701i 1.19130 + 0.644701i
\(151\) 0 0 0.945817 0.324699i \(-0.105263\pi\)
−0.945817 + 0.324699i \(0.894737\pi\)
\(152\) 0.245485 + 0.969400i 0.245485 + 0.969400i
\(153\) −1.38888 0.476802i −1.38888 0.476802i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.677282 0.735724i \(-0.736842\pi\)
0.677282 + 0.735724i \(0.263158\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0.770666 + 0.837166i 0.770666 + 0.837166i
\(163\) 0.863238 + 0.671885i 0.863238 + 0.671885i 0.945817 0.324699i \(-0.105263\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(164\) 0.111859 1.34994i 0.111859 1.34994i
\(165\) 0 0
\(166\) 0.706561 + 1.61080i 0.706561 + 1.61080i
\(167\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(168\) 0 0
\(169\) −0.0825793 + 0.996584i −0.0825793 + 0.996584i
\(170\) 0 0
\(171\) 0.456615 + 0.698901i 0.456615 + 0.698901i
\(172\) −0.633988 1.44535i −0.633988 1.44535i
\(173\) 0 0 0.945817 0.324699i \(-0.105263\pi\)
−0.945817 + 0.324699i \(0.894737\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.268536 + 0.411024i 0.268536 + 0.411024i
\(177\) 0.858777 + 1.95781i 0.858777 + 1.95781i
\(178\) 0.706561 1.61080i 0.706561 1.61080i
\(179\) −1.55676 0.259777i −1.55676 0.259777i −0.677282 0.735724i \(-0.736842\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(180\) 0 0
\(181\) 0 0 0.789141 0.614213i \(-0.210526\pi\)
−0.789141 + 0.614213i \(0.789474\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.851814 + 0.142143i 0.851814 + 0.142143i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.401695 0.915773i \(-0.368421\pi\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(192\) 0.111859 1.34994i 0.111859 1.34994i
\(193\) 0.0136387 + 0.164595i 0.0136387 + 0.164595i 1.00000 \(0\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(194\) 0.490971 1.93880i 0.490971 1.93880i
\(195\) 0 0
\(196\) 0.245485 0.969400i 0.245485 0.969400i
\(197\) 0 0 0.0825793 0.996584i \(-0.473684\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(198\) 0.323455 + 0.251755i 0.323455 + 0.251755i
\(199\) 0 0 −0.677282 0.735724i \(-0.736842\pi\)
0.677282 + 0.735724i \(0.263158\pi\)
\(200\) 0.546948 + 0.837166i 0.546948 + 0.837166i
\(201\) 0.211596 0.0726411i 0.211596 0.0726411i
\(202\) 0 0
\(203\) 0 0
\(204\) −1.61369 1.75294i −1.61369 1.75294i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.332526 0.361219i −0.332526 0.361219i
\(210\) 0 0
\(211\) 0.706561 + 0.382372i 0.706561 + 0.382372i 0.789141 0.614213i \(-0.210526\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.464369 + 1.83375i 0.464369 + 1.83375i
\(215\) 0 0
\(216\) 0.0549195 + 0.216872i 0.0549195 + 0.216872i
\(217\) 0 0
\(218\) 0 0
\(219\) 2.25351 1.21954i 2.25351 1.21954i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.789141 0.614213i \(-0.789474\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(224\) 0 0
\(225\) 0.658807 + 0.512770i 0.658807 + 0.512770i
\(226\) −0.633988 + 1.44535i −0.633988 + 1.44535i
\(227\) 0.706561 0.382372i 0.706561 0.382372i −0.0825793 0.996584i \(-0.526316\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(228\) 0.111859 + 1.34994i 0.111859 + 1.34994i
\(229\) 0 0 −0.879474 0.475947i \(-0.842105\pi\)
0.879474 + 0.475947i \(0.157895\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.165159 1.99317i −0.165159 1.99317i −0.0825793 0.996584i \(-0.526316\pi\)
−0.0825793 0.996584i \(-0.526316\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.130333 + 1.57289i −0.130333 + 1.57289i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(240\) 0 0
\(241\) −1.28117 + 1.39172i −1.28117 + 1.39172i −0.401695 + 0.915773i \(0.631579\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(242\) 0.667474 + 0.361219i 0.667474 + 0.361219i
\(243\) 0.720667 + 1.10306i 0.720667 + 1.10306i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.450427 1.77870i 0.450427 1.77870i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.584895 + 2.30970i 0.584895 + 2.30970i
\(250\) 0 0
\(251\) −0.0405441 + 0.489294i −0.0405441 + 0.489294i 0.945817 + 0.324699i \(0.105263\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.546948 0.837166i 0.546948 0.837166i
\(257\) −0.156210 + 0.0536269i −0.156210 + 0.0536269i −0.401695 0.915773i \(-0.631579\pi\)
0.245485 + 0.969400i \(0.421053\pi\)
\(258\) −0.524819 2.07246i −0.524819 2.07246i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −1.75895 −1.75895
\(263\) 0 0 −0.401695 0.915773i \(-0.631579\pi\)
0.401695 + 0.915773i \(0.368421\pi\)
\(264\) 0.267148 + 0.609036i 0.267148 + 0.609036i
\(265\) 0 0
\(266\) 0 0
\(267\) 1.30316 1.99464i 1.30316 1.99464i
\(268\) 0.162906 + 0.0271842i 0.162906 + 0.0271842i
\(269\) 0 0 −0.879474 0.475947i \(-0.842105\pi\)
0.879474 + 0.475947i \(0.157895\pi\)
\(270\) 0 0
\(271\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(272\) −0.431796 1.70512i −0.431796 1.70512i
\(273\) 0 0
\(274\) 1.73496 0.938912i 1.73496 0.938912i
\(275\) −0.431796 0.233676i −0.431796 0.233676i
\(276\) 0 0
\(277\) 0 0 −0.677282 0.735724i \(-0.736842\pi\)
0.677282 + 0.735724i \(0.263158\pi\)
\(278\) −0.962053 0.520637i −0.962053 0.520637i
\(279\) 0 0
\(280\) 0 0
\(281\) −0.439413 + 0.672572i −0.439413 + 0.672572i −0.986361 0.164595i \(-0.947368\pi\)
0.546948 + 0.837166i \(0.315789\pi\)
\(282\) 0 0
\(283\) −1.35456 + 1.47145i −1.35456 + 1.47145i −0.677282 + 0.735724i \(0.736842\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.204941 0.809295i 0.204941 0.809295i
\(289\) −1.84153 0.996584i −1.84153 0.996584i
\(290\) 0 0
\(291\) 1.08824 2.48095i 1.08824 2.48095i
\(292\) 1.89163 1.89163
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0.544122 1.24047i 0.544122 1.24047i
\(295\) 0 0
\(296\) 0 0
\(297\) −0.0743919 0.0808112i −0.0743919 0.0808112i
\(298\) 0 0
\(299\) 0 0
\(300\) 0.544122 + 1.24047i 0.544122 + 1.24047i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.401695 + 0.915773i −0.401695 + 0.915773i
\(305\) 0 0
\(306\) −0.803162 1.22933i −0.803162 1.22933i
\(307\) −1.35456 −1.35456 −0.677282 0.735724i \(-0.736842\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.945817 0.324699i \(-0.105263\pi\)
−0.945817 + 0.324699i \(0.894737\pi\)
\(312\) 0 0
\(313\) −1.86584 0.640542i −1.86584 0.640542i −0.986361 0.164595i \(-0.947368\pi\)
−0.879474 0.475947i \(-0.842105\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.945817 0.324699i \(-0.105263\pi\)
−0.945817 + 0.324699i \(0.894737\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.211596 + 2.55359i 0.211596 + 2.55359i
\(322\) 0 0
\(323\) 0.706561 + 1.61080i 0.706561 + 1.61080i
\(324\) 0.0939655 + 1.13399i 0.0939655 + 1.13399i
\(325\) 0 0
\(326\) 0.268536 + 1.06042i 0.268536 + 1.06042i
\(327\) 0 0
\(328\) 0.917421 0.996584i 0.917421 0.996584i
\(329\) 0 0
\(330\) 0 0
\(331\) −0.439413 0.672572i −0.439413 0.672572i 0.546948 0.837166i \(-0.315789\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(332\) −0.431796 + 1.70512i −0.431796 + 1.70512i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.03463 0.355188i 1.03463 0.355188i 0.245485 0.969400i \(-0.421053\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(338\) −0.677282 + 0.735724i −0.677282 + 0.735724i
\(339\) −1.16931 + 1.78976i −1.16931 + 1.78976i
\(340\) 0 0
\(341\) 0 0
\(342\) −0.0689406 + 0.831990i −0.0689406 + 0.831990i
\(343\) 0 0
\(344\) 0.387445 1.52999i 0.387445 1.52999i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.387445 0.301561i 0.387445 0.301561i −0.401695 0.915773i \(-0.631579\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(348\) 0 0
\(349\) 0 0 0.546948 0.837166i \(-0.315789\pi\)
−0.546948 + 0.837166i \(0.684211\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.0405441 + 0.489294i −0.0405441 + 0.489294i
\(353\) −0.439413 + 1.00176i −0.439413 + 1.00176i 0.546948 + 0.837166i \(0.315789\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(354\) −0.524819 + 2.07246i −0.524819 + 2.07246i
\(355\) 0 0
\(356\) 1.54695 0.837166i 1.54695 0.837166i
\(357\) 0 0
\(358\) −1.06894 1.16118i −1.06894 1.16118i
\(359\) 0 0 −0.789141 0.614213i \(-0.789474\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(360\) 0 0
\(361\) 0.245485 0.969400i 0.245485 0.969400i
\(362\) 0 0
\(363\) 0.811270 + 0.631437i 0.811270 + 0.631437i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.0825793 0.996584i \(-0.526316\pi\)
0.0825793 + 0.996584i \(0.473684\pi\)
\(368\) 0 0
\(369\) 0.454255 1.03560i 0.454255 1.03560i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.546948 0.837166i \(-0.315789\pi\)
−0.546948 + 0.837166i \(0.684211\pi\)
\(374\) 0.584895 + 0.635365i 0.584895 + 0.635365i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.111859 0.121511i 0.111859 0.121511i −0.677282 0.735724i \(-0.736842\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.546948 0.837166i \(-0.315789\pi\)
−0.546948 + 0.837166i \(0.684211\pi\)
\(384\) 0.917421 0.996584i 0.917421 0.996584i
\(385\) 0 0
\(386\) −0.0903332 + 0.138265i −0.0903332 + 0.138265i
\(387\) −0.108808 1.31311i −0.108808 1.31311i
\(388\) 1.57828 1.22843i 1.57828 1.22843i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.789141 0.614213i 0.789141 0.614213i
\(393\) −2.35011 0.392164i −2.35011 0.392164i
\(394\) 0 0
\(395\) 0 0
\(396\) 0.100620 + 0.397341i 0.100620 + 0.397341i
\(397\) 0 0 0.986361 0.164595i \(-0.0526316\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.0825793 + 0.996584i −0.0825793 + 0.996584i
\(401\) 0.162906 + 1.96598i 0.162906 + 1.96598i 0.245485 + 0.969400i \(0.421053\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(402\) 0.211596 + 0.0726411i 0.211596 + 0.0726411i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −0.196754 2.37447i −0.196754 2.37447i
\(409\) −0.759861 0.260861i −0.759861 0.260861i −0.0825793 0.996584i \(-0.526316\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(410\) 0 0
\(411\) 2.52739 0.867655i 2.52739 0.867655i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.16931 0.910111i −1.16931 0.910111i
\(418\) −0.0405441 0.489294i −0.0405441 0.489294i
\(419\) 0.387445 0.301561i 0.387445 0.301561i −0.401695 0.915773i \(-0.631579\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(420\) 0 0
\(421\) 0 0 0.0825793 0.996584i \(-0.473684\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(422\) 0.322718 + 0.735724i 0.322718 + 0.735724i
\(423\) 0 0
\(424\) 0 0
\(425\) 1.19130 + 1.29410i 1.19130 + 1.29410i
\(426\) 0 0
\(427\) 0 0
\(428\) −0.759861 + 1.73231i −0.759861 + 1.73231i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.401695 0.915773i \(-0.368421\pi\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(432\) −0.0898664 + 0.204875i −0.0898664 + 0.204875i
\(433\) 1.19130 + 0.644701i 1.19130 + 0.644701i 0.945817 0.324699i \(-0.105263\pi\)
0.245485 + 0.969400i \(0.421053\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 2.52739 + 0.421747i 2.52739 + 0.421747i
\(439\) 0 0 0.677282 0.735724i \(-0.263158\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(440\) 0 0
\(441\) 0.456615 0.698901i 0.456615 0.698901i
\(442\) 0 0
\(443\) 0.863238 + 1.32128i 0.863238 + 1.32128i 0.945817 + 0.324699i \(0.105263\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.28117 0.439826i −1.28117 0.439826i −0.401695 0.915773i \(-0.631579\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(450\) 0.204941 + 0.809295i 0.204941 + 0.809295i
\(451\) −0.163260 + 0.644701i −0.163260 + 0.644701i
\(452\) −1.38806 + 0.751179i −1.38806 + 0.751179i
\(453\) 0 0
\(454\) 0.792434 + 0.132234i 0.792434 + 0.132234i
\(455\) 0 0
\(456\) −0.740876 + 1.13399i −0.740876 + 1.13399i
\(457\) −0.439413 0.672572i −0.439413 0.672572i 0.546948 0.837166i \(-0.315789\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(458\) 0 0
\(459\) 0.158070 + 0.360364i 0.158070 + 0.360364i
\(460\) 0 0
\(461\) 0 0 −0.945817 0.324699i \(-0.894737\pi\)
0.945817 + 0.324699i \(0.105263\pi\)
\(462\) 0 0
\(463\) 0 0 −0.986361 0.164595i \(-0.947368\pi\)
0.986361 + 0.164595i \(0.0526316\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.09390 1.67433i 1.09390 1.67433i
\(467\) 1.03463 1.58361i 1.03463 1.58361i 0.245485 0.969400i \(-0.421053\pi\)
0.789141 0.614213i \(-0.210526\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.06894 + 1.16118i −1.06894 + 1.16118i
\(473\) 0.190224 + 0.751179i 0.190224 + 0.751179i
\(474\) 0 0
\(475\) −0.0825793 0.996584i −0.0825793 0.996584i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.546948 0.837166i \(-0.684211\pi\)
0.546948 + 0.837166i \(0.315789\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1.86584 + 0.311353i −1.86584 + 0.311353i
\(483\) 0 0
\(484\) 0.304866 + 0.695024i 0.304866 + 0.695024i
\(485\) 0 0
\(486\) −0.108808 + 1.31311i −0.108808 + 1.31311i
\(487\) 0 0 −0.546948 0.837166i \(-0.684211\pi\)
0.546948 + 0.837166i \(0.315789\pi\)
\(488\) 0 0
\(489\) 0.122362 + 1.47669i 0.122362 + 1.47669i
\(490\) 0 0
\(491\) −1.07898 1.65150i −1.07898 1.65150i −0.677282 0.735724i \(-0.736842\pi\)
−0.401695 0.915773i \(-0.631579\pi\)
\(492\) 1.44795 1.12698i 1.44795 1.12698i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −0.957082 + 2.18193i −0.957082 + 2.18193i
\(499\) −0.633988 0.493453i −0.633988 0.493453i 0.245485 0.969400i \(-0.421053\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −0.332526 + 0.361219i −0.332526 + 0.361219i
\(503\) 0 0 0.879474 0.475947i \(-0.157895\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.06894 + 0.831990i −1.06894 + 0.831990i
\(508\) 0 0
\(509\) 0 0 0.945817 0.324699i \(-0.105263\pi\)
−0.945817 + 0.324699i \(0.894737\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.945817 0.324699i 0.945817 0.324699i
\(513\) 0.0549195 0.216872i 0.0549195 0.216872i
\(514\) −0.156210 0.0536269i −0.156210 0.0536269i
\(515\) 0 0
\(516\) 0.858777 1.95781i 0.858777 1.95781i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.464369 0.159418i 0.464369 0.159418i −0.0825793 0.996584i \(-0.526316\pi\)
0.546948 + 0.837166i \(0.315789\pi\)
\(522\) 0 0
\(523\) 1.19130 + 1.29410i 1.19130 + 1.29410i 0.945817 + 0.324699i \(0.105263\pi\)
0.245485 + 0.969400i \(0.421053\pi\)
\(524\) −1.38806 1.08037i −1.38806 1.08037i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −0.163260 + 0.644701i −0.163260 + 0.644701i
\(529\) −0.0825793 0.996584i −0.0825793 0.996584i
\(530\) 0 0
\(531\) −0.529280 + 1.20664i −0.529280 + 1.20664i
\(532\) 0 0
\(533\) 0 0
\(534\) 2.25351 0.773631i 2.25351 0.773631i
\(535\) 0 0
\(536\) 0.111859 + 0.121511i 0.111859 + 0.121511i
\(537\) −1.16931 1.78976i −1.16931 1.78976i
\(538\) 0 0
\(539\) −0.197221 + 0.449618i −0.197221 + 0.449618i
\(540\) 0 0
\(541\) 0 0 0.789141 0.614213i \(-0.210526\pi\)
−0.789141 + 0.614213i \(0.789474\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0.706561 1.61080i 0.706561 1.61080i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.111859 + 0.121511i 0.111859 + 0.121511i 0.789141 0.614213i \(-0.210526\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(548\) 1.94582 + 0.324699i 1.94582 + 0.324699i
\(549\) 0 0
\(550\) −0.197221 0.449618i −0.197221 0.449618i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.439413 1.00176i −0.439413 1.00176i
\(557\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0.639815 + 0.979309i 0.639815 + 0.979309i
\(562\) −0.759861 + 0.260861i −0.759861 + 0.260861i
\(563\) −1.28117 + 1.39172i −1.28117 + 1.39172i −0.401695 + 0.915773i \(0.631579\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.97272 + 0.329189i −1.97272 + 0.329189i
\(567\) 0 0
\(568\) 0 0
\(569\) −1.86584 0.640542i −1.86584 0.640542i −0.986361 0.164595i \(-0.947368\pi\)
−0.879474 0.475947i \(-0.842105\pi\)
\(570\) 0 0
\(571\) 1.78914 0.614213i 1.78914 0.614213i 0.789141 0.614213i \(-0.210526\pi\)
1.00000 \(0\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.658807 0.512770i 0.658807 0.512770i
\(577\) −0.332526 1.31311i −0.332526 1.31311i −0.879474 0.475947i \(-0.842105\pi\)
0.546948 0.837166i \(-0.315789\pi\)
\(578\) −0.841109 1.91753i −0.841109 1.91753i
\(579\) −0.151520 + 0.164595i −0.151520 + 0.164595i
\(580\) 0 0
\(581\) 0 0
\(582\) 2.38261 1.28940i 2.38261 1.28940i
\(583\) 0 0
\(584\) 1.49277 + 1.16187i 1.49277 + 1.16187i
\(585\) 0 0
\(586\) 0 0
\(587\) −0.197221 + 0.449618i −0.197221 + 0.449618i −0.986361 0.164595i \(-0.947368\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(588\) 1.19130 0.644701i 1.19130 0.644701i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.863238 + 0.671885i 0.863238 + 0.671885i 0.945817 0.324699i \(-0.105263\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(594\) −0.00907043 0.109464i −0.00907043 0.109464i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.401695 0.915773i \(-0.631579\pi\)
0.401695 + 0.915773i \(0.368421\pi\)
\(600\) −0.332526 + 1.31311i −0.332526 + 1.31311i
\(601\) −1.97272 + 0.329189i −1.97272 + 0.329189i −0.986361 + 0.164595i \(0.947368\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(602\) 0 0
\(603\) 0.121263 + 0.0656242i 0.121263 + 0.0656242i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(608\) −0.879474 + 0.475947i −0.879474 + 0.475947i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0.121263 1.46343i 0.121263 1.46343i
\(613\) 0 0 0.0825793 0.996584i \(-0.473684\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(614\) −1.06894 0.831990i −1.06894 0.831990i
\(615\) 0 0
\(616\) 0 0
\(617\) −0.962053 + 1.47253i −0.962053 + 1.47253i −0.0825793 + 0.996584i \(0.526316\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(618\) 0 0
\(619\) −0.197221 0.778807i −0.197221 0.778807i −0.986361 0.164595i \(-0.947368\pi\)
0.789141 0.614213i \(-0.210526\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.401695 0.915773i −0.401695 0.915773i
\(626\) −1.07898 1.65150i −1.07898 1.65150i
\(627\) 0.0549195 0.662780i 0.0549195 0.662780i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.879474 0.475947i \(-0.157895\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(632\) 0 0
\(633\) 0.267148 + 1.05494i 0.267148 + 1.05494i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.55676 0.259777i −1.55676 0.259777i −0.677282 0.735724i \(-0.736842\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(642\) −1.40147 + 2.14510i −1.40147 + 2.14510i
\(643\) −1.86584 + 0.311353i −1.86584 + 0.311353i −0.986361 0.164595i \(-0.947368\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.431796 + 1.70512i −0.431796 + 1.70512i
\(647\) 0 0 0.986361 0.164595i \(-0.0526316\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(648\) −0.622362 + 0.952596i −0.622362 + 0.952596i
\(649\) 0.190224 0.751179i 0.190224 0.751179i
\(650\) 0 0
\(651\) 0 0
\(652\) −0.439413 + 1.00176i −0.439413 + 1.00176i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.33609 0.222954i 1.33609 0.222954i
\(657\) 1.49365 + 0.512770i 1.49365 + 0.512770i
\(658\) 0 0
\(659\) 0.464369 + 0.159418i 0.464369 + 0.159418i 0.546948 0.837166i \(-0.315789\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(660\) 0 0
\(661\) 0 0 −0.401695 0.915773i \(-0.631579\pi\)
0.401695 + 0.915773i \(0.368421\pi\)
\(662\) 0.0663435 0.800647i 0.0663435 0.800647i
\(663\) 0 0
\(664\) −1.38806 + 1.08037i −1.38806 + 1.08037i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.19130 + 0.644701i 1.19130 + 0.644701i 0.945817 0.324699i \(-0.105263\pi\)
0.245485 + 0.969400i \(0.421053\pi\)
\(674\) 1.03463 + 0.355188i 1.03463 + 0.355188i
\(675\) −0.0184745 0.222954i −0.0184745 0.222954i
\(676\) −0.986361 + 0.164595i −0.986361 + 0.164595i
\(677\) 0 0 −0.401695 0.915773i \(-0.631579\pi\)
0.401695 + 0.915773i \(0.368421\pi\)
\(678\) −2.02204 + 0.694169i −2.02204 + 0.694169i
\(679\) 0 0
\(680\) 0 0
\(681\) 1.02928 + 0.353352i 1.02928 + 0.353352i
\(682\) 0 0
\(683\) 0.111859 1.34994i 0.111859 1.34994i −0.677282 0.735724i \(-0.736842\pi\)
0.789141 0.614213i \(-0.210526\pi\)
\(684\) −0.565423 + 0.614213i −0.565423 + 0.614213i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.24549 0.969400i 1.24549 0.969400i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.130333 + 0.101443i −0.130333 + 0.101443i −0.677282 0.735724i \(-0.736842\pi\)
0.546948 + 0.837166i \(0.315789\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0.490971 0.490971
\(695\) 0 0
\(696\) 0 0
\(697\) 1.30316 1.99464i 1.30316 1.99464i
\(698\) 0 0
\(699\) 1.83484 1.99317i 1.83484 1.99317i
\(700\) 0 0
\(701\) 0 0 0.879474 0.475947i \(-0.157895\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.332526 + 0.361219i −0.332526 + 0.361219i
\(705\) 0 0
\(706\) −0.962053 + 0.520637i −0.962053 + 0.520637i
\(707\) 0 0
\(708\) −1.68709 + 1.31311i −1.68709 + 1.31311i
\(709\) 0 0 −0.677282 0.735724i \(-0.736842\pi\)
0.677282 + 0.735724i \(0.263158\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.73496 + 0.289513i 1.73496 + 0.289513i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −0.130333 1.57289i −0.130333 1.57289i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.677282 0.735724i \(-0.736842\pi\)
0.677282 + 0.735724i \(0.263158\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.789141 0.614213i 0.789141 0.614213i
\(723\) −2.56234 −2.56234
\(724\) 0 0
\(725\) 0 0
\(726\) 0.252369 + 0.996584i 0.252369 + 0.996584i
\(727\) 0 0 0.879474 0.475947i \(-0.157895\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(728\) 0 0
\(729\) −0.158807 + 0.627115i −0.158807 + 0.627115i
\(730\) 0 0
\(731\) 0.229250 2.76663i 0.229250 2.76663i
\(732\) 0 0
\(733\) 0 0 −0.0825793 0.996584i \(-0.526316\pi\)
0.0825793 + 0.996584i \(0.473684\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.0766945 0.0263293i −0.0766945 0.0263293i
\(738\) 0.994549 0.538223i 0.994549 0.538223i
\(739\) 0.268536 1.06042i 0.268536 1.06042i −0.677282 0.735724i \(-0.736842\pi\)
0.945817 0.324699i \(-0.105263\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.879474 0.475947i \(-0.157895\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.803162 + 1.22933i −0.803162 + 1.22933i
\(748\) 0.0713149 + 0.860643i 0.0713149 + 0.860643i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(752\) 0 0
\(753\) −0.524819 + 0.408483i −0.524819 + 0.408483i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.245485 0.969400i \(-0.578947\pi\)
0.245485 + 0.969400i \(0.421053\pi\)
\(758\) 0.162906 0.0271842i 0.162906 0.0271842i
\(759\) 0 0
\(760\) 0 0
\(761\) −0.156210 + 1.88517i −0.156210 + 1.88517i 0.245485 + 0.969400i \(0.421053\pi\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.33609 0.222954i 1.33609 0.222954i
\(769\) 0.0136387 + 0.164595i 0.0136387 + 0.164595i 1.00000 \(0\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(770\) 0 0
\(771\) −0.196754 0.106478i −0.196754 0.106478i
\(772\) −0.156210 + 0.0536269i −0.156210 + 0.0536269i
\(773\) 0 0 0.879474 0.475947i \(-0.157895\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(774\) 0.720667 1.10306i 0.720667 1.10306i
\(775\) 0 0
\(776\) 2.00000 2.00000
\(777\) 0 0
\(778\) 0 0
\(779\) −1.28117 + 0.439826i −1.28117 + 0.439826i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) −1.61369 1.75294i −1.61369 1.75294i
\(787\) −0.156210 0.0536269i −0.156210 0.0536269i 0.245485 0.969400i \(-0.421053\pi\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.164648 + 0.375360i −0.164648 + 0.375360i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.986361 0.164595i \(-0.0526316\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.677282 + 0.735724i −0.677282 + 0.735724i
\(801\) 1.44841 0.241698i 1.44841 0.241698i
\(802\) −1.07898 + 1.65150i −1.07898 + 1.65150i
\(803\) −0.916071 0.152865i −0.916071 0.152865i
\(804\) 0.122362 + 0.187289i 0.122362 + 0.187289i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.19130 0.644701i 1.19130 0.644701i 0.245485 0.969400i \(-0.421053\pi\)
0.945817 + 0.324699i \(0.105263\pi\)
\(810\) 0 0
\(811\) 0.464369 + 1.83375i 0.464369 + 1.83375i 0.546948 + 0.837166i \(0.315789\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 1.30316 1.99464i 1.30316 1.99464i
\(817\) −1.06894 + 1.16118i −1.06894 + 1.16118i
\(818\) −0.439413 0.672572i −0.439413 0.672572i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 2.52739 + 0.867655i 2.52739 + 0.867655i
\(823\) 0 0 0.0825793 0.996584i \(-0.473684\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(824\) 0 0
\(825\) −0.163260 0.644701i −0.163260 0.644701i
\(826\) 0 0
\(827\) −0.740876 + 1.13399i −0.740876 + 1.13399i 0.245485 + 0.969400i \(0.421053\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(828\) 0 0
\(829\) 0 0 −0.789141 0.614213i \(-0.789474\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.19130 1.29410i 1.19130 1.29410i
\(834\) −0.363749 1.43641i −0.363749 1.43641i
\(835\) 0 0
\(836\) 0.268536 0.411024i 0.268536 0.411024i
\(837\) 0 0
\(838\) 0.490971 0.490971
\(839\) 0 0 0.789141 0.614213i \(-0.210526\pi\)
−0.789141 + 0.614213i \(0.789474\pi\)
\(840\) 0 0
\(841\) −0.879474 0.475947i −0.879474 0.475947i
\(842\) 0 0
\(843\) −1.07340 + 0.179119i −1.07340 + 0.179119i
\(844\) −0.197221 + 0.778807i −0.197221 + 0.778807i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −2.70913 −2.70913
\(850\) 0.145253 + 1.75294i 0.145253 + 1.75294i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.789141 0.614213i \(-0.210526\pi\)
−0.789141 + 0.614213i \(0.789474\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.66364 + 0.900319i −1.66364 + 0.900319i
\(857\) 0.322718 0.735724i 0.322718 0.735724i −0.677282 0.735724i \(-0.736842\pi\)
1.00000 \(0\)
\(858\) 0 0
\(859\) 0.706561 1.61080i 0.706561 1.61080i −0.0825793 0.996584i \(-0.526316\pi\)
0.789141 0.614213i \(-0.210526\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.677282 0.735724i \(-0.263158\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(864\) −0.196754 + 0.106478i −0.196754 + 0.106478i
\(865\) 0 0
\(866\) 0.544122 + 1.24047i 0.544122 + 1.24047i
\(867\) −0.696274 2.74952i −0.696274 2.74952i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.57921 0.542145i 1.57921 0.542145i
\(874\) 0 0
\(875\) 0 0
\(876\) 1.73542 + 1.88517i 1.73542 + 1.88517i
\(877\) 0 0 0.401695 0.915773i \(-0.368421\pi\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.06894 + 1.16118i −1.06894 + 1.16118i −0.0825793 + 0.996584i \(0.526316\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(882\) 0.789607 0.271073i 0.789607 0.271073i
\(883\) −0.962053 1.47253i −0.962053 1.47253i −0.879474 0.475947i \(-0.842105\pi\)
−0.0825793 0.996584i \(-0.526316\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.130333 + 1.57289i −0.130333 + 1.57289i
\(887\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.0461343 0.556759i 0.0461343 0.556759i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.740876 1.13399i −0.740876 1.13399i
\(899\) 0 0
\(900\) −0.335352 + 0.764525i −0.335352 + 0.764525i
\(901\) 0 0
\(902\) −0.524819 + 0.408483i −0.524819 + 0.408483i
\(903\) 0 0
\(904\) −1.55676 0.259777i −1.55676 0.259777i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.598305 + 0.915773i 0.598305 + 0.915773i 1.00000 \(0\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(908\) 0.544122 + 0.591074i 0.544122 + 0.591074i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.401695 0.915773i \(-0.631579\pi\)
0.401695 + 0.915773i \(0.368421\pi\)
\(912\) −1.28117 + 0.439826i −1.28117 + 0.439826i
\(913\) 0.346901 0.790855i 0.346901 0.790855i
\(914\) 0.0663435 0.800647i 0.0663435 0.800647i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −0.0966005 + 0.381467i −0.0966005 + 0.381467i
\(919\) 0 0 0.0825793 0.996584i \(-0.473684\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(920\) 0 0
\(921\) −1.24270 1.34994i −1.24270 1.34994i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.33609 + 1.45138i 1.33609 + 1.45138i 0.789141 + 0.614213i \(0.210526\pi\)
0.546948 + 0.837166i \(0.315789\pi\)
\(930\) 0 0
\(931\) −0.986361 + 0.164595i −0.986361 + 0.164595i
\(932\) 1.89163 0.649399i 1.89163 0.649399i
\(933\) 0 0
\(934\) 1.78914 0.614213i 1.78914 0.614213i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.24549 0.969400i 1.24549 0.969400i 0.245485 0.969400i \(-0.421053\pi\)
1.00000 \(0\)
\(938\) 0 0
\(939\) −1.07340 2.44711i −1.07340 2.44711i
\(940\) 0 0
\(941\) 0 0 0.879474 0.475947i \(-0.157895\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.55676 + 0.259777i −1.55676 + 0.259777i
\(945\) 0 0
\(946\) −0.311270 + 0.709624i −0.311270 + 0.709624i
\(947\) 1.24549 + 0.969400i 1.24549 + 0.969400i 1.00000 \(0\)
0.245485 + 0.969400i \(0.421053\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.546948 0.837166i 0.546948 0.837166i
\(951\) 0 0
\(952\) 0 0
\(953\) 0.598305 + 0.915773i 0.598305 + 0.915773i 1.00000 \(0\)
−0.401695 + 0.915773i \(0.631579\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.245485 0.969400i 0.245485 0.969400i
\(962\) 0 0
\(963\) −1.06957 + 1.16187i −1.06957 + 1.16187i
\(964\) −1.66364 0.900319i −1.66364 0.900319i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −0.186311 + 0.735724i −0.186311 + 0.735724i
\(969\) −0.957082 + 2.18193i −0.957082 + 2.18193i
\(970\) 0 0
\(971\) 0.490971 + 1.93880i 0.490971 + 1.93880i 0.245485 + 0.969400i \(0.421053\pi\)
0.245485 + 0.969400i \(0.421053\pi\)
\(972\) −0.892396 + 0.969400i −0.892396 + 0.969400i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.07898 + 1.65150i −1.07898 + 1.65150i −0.401695 + 0.915773i \(0.631579\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(978\) −0.810441 + 1.24047i −0.810441 + 1.24047i
\(979\) −0.816800 + 0.280408i −0.816800 + 0.280408i
\(980\) 0 0
\(981\) 0 0
\(982\) 0.162906 1.96598i 0.162906 1.96598i
\(983\) 0 0 −0.945817 0.324699i \(-0.894737\pi\)
0.945817 + 0.324699i \(0.105263\pi\)
\(984\) 1.83484 1.83484
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.879474 0.475947i \(-0.842105\pi\)
0.879474 + 0.475947i \(0.157895\pi\)
\(992\) 0 0
\(993\) 0.267148 1.05494i 0.267148 1.05494i
\(994\) 0 0
\(995\) 0 0
\(996\) −2.09544 + 1.13399i −2.09544 + 1.13399i
\(997\) 0 0 −0.879474 0.475947i \(-0.842105\pi\)
0.879474 + 0.475947i \(0.157895\pi\)
\(998\) −0.197221 0.778807i −0.197221 0.778807i
\(999\) 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 2888.1.bb.a.2395.1 yes 18
8.3 odd 2 CM 2888.1.bb.a.2395.1 yes 18
361.134 even 19 inner 2888.1.bb.a.1939.1 18
2888.1939 odd 38 inner 2888.1.bb.a.1939.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2888.1.bb.a.1939.1 18 361.134 even 19 inner
2888.1.bb.a.1939.1 18 2888.1939 odd 38 inner
2888.1.bb.a.2395.1 yes 18 1.1 even 1 trivial
2888.1.bb.a.2395.1 yes 18 8.3 odd 2 CM