Properties

Label 224.2.i.a.65.2
Level $224$
Weight $2$
Character 224.65
Analytic conductor $1.789$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,2,Mod(65,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.i (of order \(3\), degree \(2\), 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: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 65.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 224.65
Dual form 224.2.i.a.193.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.207107 + 0.358719i) q^{3} +(0.914214 - 1.58346i) q^{5} +(-1.00000 - 2.44949i) q^{7} +(1.41421 - 2.44949i) q^{9} +O(q^{10})\) \(q+(0.207107 + 0.358719i) q^{3} +(0.914214 - 1.58346i) q^{5} +(-1.00000 - 2.44949i) q^{7} +(1.41421 - 2.44949i) q^{9} +(1.20711 + 2.09077i) q^{11} +2.82843 q^{13} +0.757359 q^{15} +(0.0857864 + 0.148586i) q^{17} +(-3.20711 + 5.55487i) q^{19} +(0.671573 - 0.866025i) q^{21} +(2.62132 - 4.54026i) q^{23} +(0.828427 + 1.43488i) q^{25} +2.41421 q^{27} -2.82843 q^{29} +(-2.79289 - 4.83743i) q^{31} +(-0.500000 + 0.866025i) q^{33} +(-4.79289 - 0.655892i) q^{35} +(-4.32843 + 7.49706i) q^{37} +(0.585786 + 1.01461i) q^{39} -6.82843 q^{41} +9.65685 q^{43} +(-2.58579 - 4.47871i) q^{45} +(-5.20711 + 9.01897i) q^{47} +(-5.00000 + 4.89898i) q^{49} +(-0.0355339 + 0.0615465i) q^{51} +(0.500000 + 0.866025i) q^{53} +4.41421 q^{55} -2.65685 q^{57} +(-5.44975 - 9.43924i) q^{59} +(-4.32843 + 7.49706i) q^{61} +(-7.41421 - 1.01461i) q^{63} +(2.58579 - 4.47871i) q^{65} +(1.37868 + 2.38794i) q^{67} +2.17157 q^{69} +13.6569 q^{71} +(7.32843 + 12.6932i) q^{73} +(-0.343146 + 0.594346i) q^{75} +(3.91421 - 5.04757i) q^{77} +(-3.03553 + 5.25770i) q^{79} +(-3.74264 - 6.48244i) q^{81} -7.31371 q^{83} +0.313708 q^{85} +(-0.585786 - 1.01461i) q^{87} +(4.50000 - 7.79423i) q^{89} +(-2.82843 - 6.92820i) q^{91} +(1.15685 - 2.00373i) q^{93} +(5.86396 + 10.1567i) q^{95} -1.17157 q^{97} +6.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{11} + 20 q^{15} + 6 q^{17} - 10 q^{19} + 14 q^{21} + 2 q^{23} - 8 q^{25} + 4 q^{27} - 14 q^{31} - 2 q^{33} - 22 q^{35} - 6 q^{37} + 8 q^{39} - 16 q^{41} + 16 q^{43} - 16 q^{45} - 18 q^{47} - 20 q^{49} + 14 q^{51} + 2 q^{53} + 12 q^{55} + 12 q^{57} - 2 q^{59} - 6 q^{61} - 24 q^{63} + 16 q^{65} + 14 q^{67} + 20 q^{69} + 32 q^{71} + 18 q^{73} - 24 q^{75} + 10 q^{77} + 2 q^{79} + 2 q^{81} + 16 q^{83} - 44 q^{85} - 8 q^{87} + 18 q^{89} - 18 q^{93} - 2 q^{95} - 16 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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 0
\(3\) 0.207107 + 0.358719i 0.119573 + 0.207107i 0.919599 0.392859i \(-0.128514\pi\)
−0.800025 + 0.599966i \(0.795181\pi\)
\(4\) 0 0
\(5\) 0.914214 1.58346i 0.408849 0.708147i −0.585912 0.810374i \(-0.699264\pi\)
0.994761 + 0.102228i \(0.0325970\pi\)
\(6\) 0 0
\(7\) −1.00000 2.44949i −0.377964 0.925820i
\(8\) 0 0
\(9\) 1.41421 2.44949i 0.471405 0.816497i
\(10\) 0 0
\(11\) 1.20711 + 2.09077i 0.363956 + 0.630391i 0.988608 0.150513i \(-0.0480924\pi\)
−0.624652 + 0.780903i \(0.714759\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0.757359 0.195549
\(16\) 0 0
\(17\) 0.0857864 + 0.148586i 0.0208063 + 0.0360375i 0.876241 0.481873i \(-0.160043\pi\)
−0.855435 + 0.517911i \(0.826710\pi\)
\(18\) 0 0
\(19\) −3.20711 + 5.55487i −0.735761 + 1.27438i 0.218628 + 0.975808i \(0.429842\pi\)
−0.954389 + 0.298567i \(0.903491\pi\)
\(20\) 0 0
\(21\) 0.671573 0.866025i 0.146549 0.188982i
\(22\) 0 0
\(23\) 2.62132 4.54026i 0.546583 0.946710i −0.451922 0.892057i \(-0.649262\pi\)
0.998505 0.0546524i \(-0.0174051\pi\)
\(24\) 0 0
\(25\) 0.828427 + 1.43488i 0.165685 + 0.286976i
\(26\) 0 0
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) −2.79289 4.83743i −0.501618 0.868829i −0.999998 0.00186981i \(-0.999405\pi\)
0.498380 0.866959i \(-0.333929\pi\)
\(32\) 0 0
\(33\) −0.500000 + 0.866025i −0.0870388 + 0.150756i
\(34\) 0 0
\(35\) −4.79289 0.655892i −0.810147 0.110866i
\(36\) 0 0
\(37\) −4.32843 + 7.49706i −0.711589 + 1.23251i 0.252671 + 0.967552i \(0.418691\pi\)
−0.964260 + 0.264956i \(0.914642\pi\)
\(38\) 0 0
\(39\) 0.585786 + 1.01461i 0.0938009 + 0.162468i
\(40\) 0 0
\(41\) −6.82843 −1.06642 −0.533211 0.845983i \(-0.679015\pi\)
−0.533211 + 0.845983i \(0.679015\pi\)
\(42\) 0 0
\(43\) 9.65685 1.47266 0.736328 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) 0 0
\(45\) −2.58579 4.47871i −0.385466 0.667647i
\(46\) 0 0
\(47\) −5.20711 + 9.01897i −0.759535 + 1.31555i 0.183554 + 0.983010i \(0.441240\pi\)
−0.943088 + 0.332543i \(0.892093\pi\)
\(48\) 0 0
\(49\) −5.00000 + 4.89898i −0.714286 + 0.699854i
\(50\) 0 0
\(51\) −0.0355339 + 0.0615465i −0.00497574 + 0.00861824i
\(52\) 0 0
\(53\) 0.500000 + 0.866025i 0.0686803 + 0.118958i 0.898321 0.439340i \(-0.144788\pi\)
−0.829640 + 0.558298i \(0.811454\pi\)
\(54\) 0 0
\(55\) 4.41421 0.595212
\(56\) 0 0
\(57\) −2.65685 −0.351909
\(58\) 0 0
\(59\) −5.44975 9.43924i −0.709497 1.22888i −0.965044 0.262088i \(-0.915589\pi\)
0.255547 0.966797i \(-0.417744\pi\)
\(60\) 0 0
\(61\) −4.32843 + 7.49706i −0.554198 + 0.959900i 0.443767 + 0.896142i \(0.353642\pi\)
−0.997965 + 0.0637575i \(0.979692\pi\)
\(62\) 0 0
\(63\) −7.41421 1.01461i −0.934103 0.127829i
\(64\) 0 0
\(65\) 2.58579 4.47871i 0.320727 0.555516i
\(66\) 0 0
\(67\) 1.37868 + 2.38794i 0.168433 + 0.291734i 0.937869 0.346990i \(-0.112796\pi\)
−0.769436 + 0.638723i \(0.779463\pi\)
\(68\) 0 0
\(69\) 2.17157 0.261427
\(70\) 0 0
\(71\) 13.6569 1.62077 0.810385 0.585897i \(-0.199258\pi\)
0.810385 + 0.585897i \(0.199258\pi\)
\(72\) 0 0
\(73\) 7.32843 + 12.6932i 0.857728 + 1.48563i 0.874091 + 0.485762i \(0.161458\pi\)
−0.0163639 + 0.999866i \(0.505209\pi\)
\(74\) 0 0
\(75\) −0.343146 + 0.594346i −0.0396231 + 0.0686292i
\(76\) 0 0
\(77\) 3.91421 5.04757i 0.446066 0.575224i
\(78\) 0 0
\(79\) −3.03553 + 5.25770i −0.341524 + 0.591537i −0.984716 0.174168i \(-0.944276\pi\)
0.643192 + 0.765705i \(0.277610\pi\)
\(80\) 0 0
\(81\) −3.74264 6.48244i −0.415849 0.720272i
\(82\) 0 0
\(83\) −7.31371 −0.802784 −0.401392 0.915906i \(-0.631473\pi\)
−0.401392 + 0.915906i \(0.631473\pi\)
\(84\) 0 0
\(85\) 0.313708 0.0340265
\(86\) 0 0
\(87\) −0.585786 1.01461i −0.0628029 0.108778i
\(88\) 0 0
\(89\) 4.50000 7.79423i 0.476999 0.826187i −0.522654 0.852545i \(-0.675058\pi\)
0.999653 + 0.0263586i \(0.00839118\pi\)
\(90\) 0 0
\(91\) −2.82843 6.92820i −0.296500 0.726273i
\(92\) 0 0
\(93\) 1.15685 2.00373i 0.119960 0.207777i
\(94\) 0 0
\(95\) 5.86396 + 10.1567i 0.601630 + 1.04205i
\(96\) 0 0
\(97\) −1.17157 −0.118955 −0.0594776 0.998230i \(-0.518943\pi\)
−0.0594776 + 0.998230i \(0.518943\pi\)
\(98\) 0 0
\(99\) 6.82843 0.686283
\(100\) 0 0
\(101\) −6.74264 11.6786i −0.670918 1.16206i −0.977644 0.210266i \(-0.932567\pi\)
0.306726 0.951798i \(-0.400766\pi\)
\(102\) 0 0
\(103\) −4.20711 + 7.28692i −0.414539 + 0.718002i −0.995380 0.0960150i \(-0.969390\pi\)
0.580841 + 0.814017i \(0.302724\pi\)
\(104\) 0 0
\(105\) −0.757359 1.85514i −0.0739107 0.181044i
\(106\) 0 0
\(107\) 6.44975 11.1713i 0.623521 1.07997i −0.365304 0.930888i \(-0.619035\pi\)
0.988825 0.149081i \(-0.0476316\pi\)
\(108\) 0 0
\(109\) 4.08579 + 7.07679i 0.391347 + 0.677834i 0.992628 0.121205i \(-0.0386758\pi\)
−0.601280 + 0.799038i \(0.705342\pi\)
\(110\) 0 0
\(111\) −3.58579 −0.340348
\(112\) 0 0
\(113\) 18.1421 1.70667 0.853334 0.521364i \(-0.174577\pi\)
0.853334 + 0.521364i \(0.174577\pi\)
\(114\) 0 0
\(115\) −4.79289 8.30153i −0.446940 0.774122i
\(116\) 0 0
\(117\) 4.00000 6.92820i 0.369800 0.640513i
\(118\) 0 0
\(119\) 0.278175 0.358719i 0.0255002 0.0328838i
\(120\) 0 0
\(121\) 2.58579 4.47871i 0.235071 0.407156i
\(122\) 0 0
\(123\) −1.41421 2.44949i −0.127515 0.220863i
\(124\) 0 0
\(125\) 12.1716 1.08866
\(126\) 0 0
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) 0 0
\(129\) 2.00000 + 3.46410i 0.176090 + 0.304997i
\(130\) 0 0
\(131\) −3.86396 + 6.69258i −0.337596 + 0.584733i −0.983980 0.178279i \(-0.942947\pi\)
0.646384 + 0.763012i \(0.276280\pi\)
\(132\) 0 0
\(133\) 16.8137 + 2.30090i 1.45793 + 0.199514i
\(134\) 0 0
\(135\) 2.20711 3.82282i 0.189958 0.329016i
\(136\) 0 0
\(137\) −5.57107 9.64937i −0.475968 0.824402i 0.523653 0.851932i \(-0.324569\pi\)
−0.999621 + 0.0275304i \(0.991236\pi\)
\(138\) 0 0
\(139\) 15.3137 1.29889 0.649446 0.760408i \(-0.275001\pi\)
0.649446 + 0.760408i \(0.275001\pi\)
\(140\) 0 0
\(141\) −4.31371 −0.363280
\(142\) 0 0
\(143\) 3.41421 + 5.91359i 0.285511 + 0.494519i
\(144\) 0 0
\(145\) −2.58579 + 4.47871i −0.214738 + 0.371937i
\(146\) 0 0
\(147\) −2.79289 0.778985i −0.230354 0.0642496i
\(148\) 0 0
\(149\) −3.08579 + 5.34474i −0.252797 + 0.437858i −0.964295 0.264831i \(-0.914684\pi\)
0.711497 + 0.702689i \(0.248017\pi\)
\(150\) 0 0
\(151\) −0.449747 0.778985i −0.0365999 0.0633929i 0.847145 0.531361i \(-0.178319\pi\)
−0.883745 + 0.467969i \(0.844986\pi\)
\(152\) 0 0
\(153\) 0.485281 0.0392327
\(154\) 0 0
\(155\) −10.2132 −0.820344
\(156\) 0 0
\(157\) −4.67157 8.09140i −0.372832 0.645764i 0.617168 0.786831i \(-0.288280\pi\)
−0.990000 + 0.141067i \(0.954947\pi\)
\(158\) 0 0
\(159\) −0.207107 + 0.358719i −0.0164246 + 0.0284483i
\(160\) 0 0
\(161\) −13.7426 1.88064i −1.08307 0.148215i
\(162\) 0 0
\(163\) −3.37868 + 5.85204i −0.264639 + 0.458368i −0.967469 0.252990i \(-0.918586\pi\)
0.702830 + 0.711358i \(0.251919\pi\)
\(164\) 0 0
\(165\) 0.914214 + 1.58346i 0.0711714 + 0.123273i
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 9.07107 + 15.7116i 0.693682 + 1.20149i
\(172\) 0 0
\(173\) −5.50000 + 9.52628i −0.418157 + 0.724270i −0.995754 0.0920525i \(-0.970657\pi\)
0.577597 + 0.816322i \(0.303991\pi\)
\(174\) 0 0
\(175\) 2.68629 3.46410i 0.203065 0.261861i
\(176\) 0 0
\(177\) 2.25736 3.90986i 0.169674 0.293883i
\(178\) 0 0
\(179\) −2.79289 4.83743i −0.208751 0.361567i 0.742571 0.669768i \(-0.233606\pi\)
−0.951321 + 0.308201i \(0.900273\pi\)
\(180\) 0 0
\(181\) −9.31371 −0.692283 −0.346141 0.938182i \(-0.612508\pi\)
−0.346141 + 0.938182i \(0.612508\pi\)
\(182\) 0 0
\(183\) −3.58579 −0.265069
\(184\) 0 0
\(185\) 7.91421 + 13.7078i 0.581865 + 1.00782i
\(186\) 0 0
\(187\) −0.207107 + 0.358719i −0.0151451 + 0.0262322i
\(188\) 0 0
\(189\) −2.41421 5.91359i −0.175608 0.430150i
\(190\) 0 0
\(191\) 4.44975 7.70719i 0.321972 0.557673i −0.658923 0.752211i \(-0.728987\pi\)
0.980895 + 0.194538i \(0.0623208\pi\)
\(192\) 0 0
\(193\) 4.57107 + 7.91732i 0.329033 + 0.569901i 0.982320 0.187209i \(-0.0599440\pi\)
−0.653288 + 0.757110i \(0.726611\pi\)
\(194\) 0 0
\(195\) 2.14214 0.153402
\(196\) 0 0
\(197\) −18.8284 −1.34147 −0.670735 0.741697i \(-0.734021\pi\)
−0.670735 + 0.741697i \(0.734021\pi\)
\(198\) 0 0
\(199\) −5.62132 9.73641i −0.398485 0.690196i 0.595054 0.803685i \(-0.297131\pi\)
−0.993539 + 0.113489i \(0.963797\pi\)
\(200\) 0 0
\(201\) −0.571068 + 0.989118i −0.0402800 + 0.0697670i
\(202\) 0 0
\(203\) 2.82843 + 6.92820i 0.198517 + 0.486265i
\(204\) 0 0
\(205\) −6.24264 + 10.8126i −0.436005 + 0.755183i
\(206\) 0 0
\(207\) −7.41421 12.8418i −0.515323 0.892566i
\(208\) 0 0
\(209\) −15.4853 −1.07114
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 2.82843 + 4.89898i 0.193801 + 0.335673i
\(214\) 0 0
\(215\) 8.82843 15.2913i 0.602094 1.04286i
\(216\) 0 0
\(217\) −9.05635 + 11.6786i −0.614785 + 0.792795i
\(218\) 0 0
\(219\) −3.03553 + 5.25770i −0.205122 + 0.355282i
\(220\) 0 0
\(221\) 0.242641 + 0.420266i 0.0163218 + 0.0282702i
\(222\) 0 0
\(223\) 2.34315 0.156909 0.0784543 0.996918i \(-0.475002\pi\)
0.0784543 + 0.996918i \(0.475002\pi\)
\(224\) 0 0
\(225\) 4.68629 0.312419
\(226\) 0 0
\(227\) 8.03553 + 13.9180i 0.533337 + 0.923767i 0.999242 + 0.0389321i \(0.0123956\pi\)
−0.465905 + 0.884835i \(0.654271\pi\)
\(228\) 0 0
\(229\) 2.08579 3.61269i 0.137833 0.238733i −0.788843 0.614594i \(-0.789320\pi\)
0.926676 + 0.375861i \(0.122653\pi\)
\(230\) 0 0
\(231\) 2.62132 + 0.358719i 0.172470 + 0.0236020i
\(232\) 0 0
\(233\) −1.08579 + 1.88064i −0.0711322 + 0.123205i −0.899398 0.437131i \(-0.855995\pi\)
0.828266 + 0.560336i \(0.189328\pi\)
\(234\) 0 0
\(235\) 9.52082 + 16.4905i 0.621070 + 1.07572i
\(236\) 0 0
\(237\) −2.51472 −0.163349
\(238\) 0 0
\(239\) 1.31371 0.0849767 0.0424884 0.999097i \(-0.486471\pi\)
0.0424884 + 0.999097i \(0.486471\pi\)
\(240\) 0 0
\(241\) −5.15685 8.93193i −0.332182 0.575356i 0.650757 0.759286i \(-0.274452\pi\)
−0.982939 + 0.183929i \(0.941118\pi\)
\(242\) 0 0
\(243\) 5.17157 8.95743i 0.331757 0.574619i
\(244\) 0 0
\(245\) 3.18629 + 12.3960i 0.203565 + 0.791954i
\(246\) 0 0
\(247\) −9.07107 + 15.7116i −0.577178 + 0.999702i
\(248\) 0 0
\(249\) −1.51472 2.62357i −0.0959914 0.166262i
\(250\) 0 0
\(251\) −30.9706 −1.95484 −0.977422 0.211295i \(-0.932232\pi\)
−0.977422 + 0.211295i \(0.932232\pi\)
\(252\) 0 0
\(253\) 12.6569 0.795730
\(254\) 0 0
\(255\) 0.0649712 + 0.112533i 0.00406865 + 0.00704711i
\(256\) 0 0
\(257\) −2.25736 + 3.90986i −0.140810 + 0.243890i −0.927802 0.373073i \(-0.878304\pi\)
0.786992 + 0.616964i \(0.211637\pi\)
\(258\) 0 0
\(259\) 22.6924 + 3.10538i 1.41004 + 0.192959i
\(260\) 0 0
\(261\) −4.00000 + 6.92820i −0.247594 + 0.428845i
\(262\) 0 0
\(263\) −5.44975 9.43924i −0.336046 0.582048i 0.647639 0.761947i \(-0.275756\pi\)
−0.983685 + 0.179899i \(0.942423\pi\)
\(264\) 0 0
\(265\) 1.82843 0.112319
\(266\) 0 0
\(267\) 3.72792 0.228145
\(268\) 0 0
\(269\) −6.32843 10.9612i −0.385851 0.668314i 0.606036 0.795437i \(-0.292759\pi\)
−0.991887 + 0.127124i \(0.959425\pi\)
\(270\) 0 0
\(271\) 15.1066 26.1654i 0.917661 1.58943i 0.114702 0.993400i \(-0.463409\pi\)
0.802958 0.596035i \(-0.203258\pi\)
\(272\) 0 0
\(273\) 1.89949 2.44949i 0.114963 0.148250i
\(274\) 0 0
\(275\) −2.00000 + 3.46410i −0.120605 + 0.208893i
\(276\) 0 0
\(277\) −5.15685 8.93193i −0.309845 0.536668i 0.668483 0.743727i \(-0.266944\pi\)
−0.978328 + 0.207060i \(0.933611\pi\)
\(278\) 0 0
\(279\) −15.7990 −0.945861
\(280\) 0 0
\(281\) −12.4853 −0.744809 −0.372405 0.928070i \(-0.621467\pi\)
−0.372405 + 0.928070i \(0.621467\pi\)
\(282\) 0 0
\(283\) −2.79289 4.83743i −0.166020 0.287556i 0.770997 0.636839i \(-0.219758\pi\)
−0.937017 + 0.349283i \(0.886425\pi\)
\(284\) 0 0
\(285\) −2.42893 + 4.20703i −0.143878 + 0.249203i
\(286\) 0 0
\(287\) 6.82843 + 16.7262i 0.403069 + 0.987314i
\(288\) 0 0
\(289\) 8.48528 14.6969i 0.499134 0.864526i
\(290\) 0 0
\(291\) −0.242641 0.420266i −0.0142238 0.0246364i
\(292\) 0 0
\(293\) 28.6274 1.67243 0.836216 0.548401i \(-0.184763\pi\)
0.836216 + 0.548401i \(0.184763\pi\)
\(294\) 0 0
\(295\) −19.9289 −1.16031
\(296\) 0 0
\(297\) 2.91421 + 5.04757i 0.169100 + 0.292889i
\(298\) 0 0
\(299\) 7.41421 12.8418i 0.428775 0.742660i
\(300\) 0 0
\(301\) −9.65685 23.6544i −0.556612 1.36341i
\(302\) 0 0
\(303\) 2.79289 4.83743i 0.160448 0.277903i
\(304\) 0 0
\(305\) 7.91421 + 13.7078i 0.453167 + 0.784907i
\(306\) 0 0
\(307\) 14.3431 0.818607 0.409303 0.912398i \(-0.365772\pi\)
0.409303 + 0.912398i \(0.365772\pi\)
\(308\) 0 0
\(309\) −3.48528 −0.198271
\(310\) 0 0
\(311\) 9.37868 + 16.2443i 0.531816 + 0.921133i 0.999310 + 0.0371364i \(0.0118236\pi\)
−0.467494 + 0.883996i \(0.654843\pi\)
\(312\) 0 0
\(313\) 14.6421 25.3609i 0.827622 1.43348i −0.0722760 0.997385i \(-0.523026\pi\)
0.899898 0.436099i \(-0.143640\pi\)
\(314\) 0 0
\(315\) −8.38478 + 10.8126i −0.472429 + 0.609219i
\(316\) 0 0
\(317\) 4.84315 8.38857i 0.272018 0.471149i −0.697360 0.716721i \(-0.745642\pi\)
0.969379 + 0.245571i \(0.0789756\pi\)
\(318\) 0 0
\(319\) −3.41421 5.91359i −0.191159 0.331098i
\(320\) 0 0
\(321\) 5.34315 0.298225
\(322\) 0 0
\(323\) −1.10051 −0.0612337
\(324\) 0 0
\(325\) 2.34315 + 4.05845i 0.129974 + 0.225122i
\(326\) 0 0
\(327\) −1.69239 + 2.93130i −0.0935893 + 0.162101i
\(328\) 0 0
\(329\) 27.2990 + 3.73578i 1.50504 + 0.205960i
\(330\) 0 0
\(331\) −14.2071 + 24.6074i −0.780893 + 1.35255i 0.150529 + 0.988606i \(0.451902\pi\)
−0.931422 + 0.363941i \(0.881431\pi\)
\(332\) 0 0
\(333\) 12.2426 + 21.2049i 0.670893 + 1.16202i
\(334\) 0 0
\(335\) 5.04163 0.275454
\(336\) 0 0
\(337\) 9.17157 0.499607 0.249804 0.968296i \(-0.419634\pi\)
0.249804 + 0.968296i \(0.419634\pi\)
\(338\) 0 0
\(339\) 3.75736 + 6.50794i 0.204072 + 0.353463i
\(340\) 0 0
\(341\) 6.74264 11.6786i 0.365134 0.632431i
\(342\) 0 0
\(343\) 17.0000 + 7.34847i 0.917914 + 0.396780i
\(344\) 0 0
\(345\) 1.98528 3.43861i 0.106884 0.185128i
\(346\) 0 0
\(347\) −17.4497 30.2238i −0.936752 1.62250i −0.771480 0.636253i \(-0.780483\pi\)
−0.165271 0.986248i \(-0.552850\pi\)
\(348\) 0 0
\(349\) 5.17157 0.276828 0.138414 0.990374i \(-0.455800\pi\)
0.138414 + 0.990374i \(0.455800\pi\)
\(350\) 0 0
\(351\) 6.82843 0.364474
\(352\) 0 0
\(353\) −1.08579 1.88064i −0.0577906 0.100096i 0.835683 0.549212i \(-0.185072\pi\)
−0.893473 + 0.449116i \(0.851739\pi\)
\(354\) 0 0
\(355\) 12.4853 21.6251i 0.662650 1.14774i
\(356\) 0 0
\(357\) 0.186292 + 0.0254934i 0.00985959 + 0.00134925i
\(358\) 0 0
\(359\) −5.69239 + 9.85951i −0.300433 + 0.520365i −0.976234 0.216719i \(-0.930464\pi\)
0.675801 + 0.737084i \(0.263798\pi\)
\(360\) 0 0
\(361\) −11.0711 19.1757i −0.582688 1.00924i
\(362\) 0 0
\(363\) 2.14214 0.112433
\(364\) 0 0
\(365\) 26.7990 1.40272
\(366\) 0 0
\(367\) −2.13604 3.69973i −0.111500 0.193124i 0.804875 0.593444i \(-0.202232\pi\)
−0.916375 + 0.400320i \(0.868899\pi\)
\(368\) 0 0
\(369\) −9.65685 + 16.7262i −0.502716 + 0.870729i
\(370\) 0 0
\(371\) 1.62132 2.09077i 0.0841748 0.108547i
\(372\) 0 0
\(373\) 4.15685 7.19988i 0.215234 0.372796i −0.738111 0.674679i \(-0.764282\pi\)
0.953345 + 0.301883i \(0.0976153\pi\)
\(374\) 0 0
\(375\) 2.52082 + 4.36618i 0.130174 + 0.225469i
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −34.9706 −1.79632 −0.898159 0.439672i \(-0.855095\pi\)
−0.898159 + 0.439672i \(0.855095\pi\)
\(380\) 0 0
\(381\) −1.17157 2.02922i −0.0600215 0.103960i
\(382\) 0 0
\(383\) 5.62132 9.73641i 0.287236 0.497507i −0.685913 0.727684i \(-0.740597\pi\)
0.973149 + 0.230176i \(0.0739303\pi\)
\(384\) 0 0
\(385\) −4.41421 10.8126i −0.224969 0.551060i
\(386\) 0 0
\(387\) 13.6569 23.6544i 0.694217 1.20242i
\(388\) 0 0
\(389\) −9.08579 15.7370i −0.460668 0.797900i 0.538327 0.842736i \(-0.319057\pi\)
−0.998994 + 0.0448365i \(0.985723\pi\)
\(390\) 0 0
\(391\) 0.899495 0.0454894
\(392\) 0 0
\(393\) −3.20101 −0.161470
\(394\) 0 0
\(395\) 5.55025 + 9.61332i 0.279264 + 0.483699i
\(396\) 0 0
\(397\) 2.08579 3.61269i 0.104683 0.181316i −0.808926 0.587911i \(-0.799951\pi\)
0.913608 + 0.406595i \(0.133284\pi\)
\(398\) 0 0
\(399\) 2.65685 + 6.50794i 0.133009 + 0.325804i
\(400\) 0 0
\(401\) −3.91421 + 6.77962i −0.195466 + 0.338558i −0.947053 0.321076i \(-0.895955\pi\)
0.751587 + 0.659634i \(0.229289\pi\)
\(402\) 0 0
\(403\) −7.89949 13.6823i −0.393502 0.681565i
\(404\) 0 0
\(405\) −13.6863 −0.680077
\(406\) 0 0
\(407\) −20.8995 −1.03595
\(408\) 0 0
\(409\) −7.22792 12.5191i −0.357398 0.619031i 0.630128 0.776492i \(-0.283003\pi\)
−0.987525 + 0.157461i \(0.949669\pi\)
\(410\) 0 0
\(411\) 2.30761 3.99690i 0.113826 0.197153i
\(412\) 0 0
\(413\) −17.6716 + 22.7883i −0.869561 + 1.12134i
\(414\) 0 0
\(415\) −6.68629 + 11.5810i −0.328217 + 0.568489i
\(416\) 0 0
\(417\) 3.17157 + 5.49333i 0.155313 + 0.269009i
\(418\) 0 0
\(419\) 9.65685 0.471768 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(420\) 0 0
\(421\) −16.4853 −0.803443 −0.401722 0.915762i \(-0.631588\pi\)
−0.401722 + 0.915762i \(0.631588\pi\)
\(422\) 0 0
\(423\) 14.7279 + 25.5095i 0.716096 + 1.24031i
\(424\) 0 0
\(425\) −0.142136 + 0.246186i −0.00689459 + 0.0119418i
\(426\) 0 0
\(427\) 22.6924 + 3.10538i 1.09816 + 0.150280i
\(428\) 0 0
\(429\) −1.41421 + 2.44949i −0.0682789 + 0.118262i
\(430\) 0 0
\(431\) 6.37868 + 11.0482i 0.307250 + 0.532173i 0.977760 0.209728i \(-0.0672578\pi\)
−0.670510 + 0.741901i \(0.733924\pi\)
\(432\) 0 0
\(433\) 11.5147 0.553362 0.276681 0.960962i \(-0.410765\pi\)
0.276681 + 0.960962i \(0.410765\pi\)
\(434\) 0 0
\(435\) −2.14214 −0.102708
\(436\) 0 0
\(437\) 16.8137 + 29.1222i 0.804309 + 1.39310i
\(438\) 0 0
\(439\) −11.3492 + 19.6575i −0.541670 + 0.938200i 0.457139 + 0.889395i \(0.348874\pi\)
−0.998808 + 0.0488041i \(0.984459\pi\)
\(440\) 0 0
\(441\) 4.92893 + 19.1757i 0.234711 + 0.913126i
\(442\) 0 0
\(443\) 5.96447 10.3308i 0.283380 0.490829i −0.688835 0.724918i \(-0.741877\pi\)
0.972215 + 0.234089i \(0.0752108\pi\)
\(444\) 0 0
\(445\) −8.22792 14.2512i −0.390041 0.675571i
\(446\) 0 0
\(447\) −2.55635 −0.120911
\(448\) 0 0
\(449\) 1.17157 0.0552899 0.0276450 0.999618i \(-0.491199\pi\)
0.0276450 + 0.999618i \(0.491199\pi\)
\(450\) 0 0
\(451\) −8.24264 14.2767i −0.388131 0.672262i
\(452\) 0 0
\(453\) 0.186292 0.322666i 0.00875274 0.0151602i
\(454\) 0 0
\(455\) −13.5563 1.85514i −0.635531 0.0869705i
\(456\) 0 0
\(457\) −9.64214 + 16.7007i −0.451040 + 0.781224i −0.998451 0.0556397i \(-0.982280\pi\)
0.547411 + 0.836864i \(0.315614\pi\)
\(458\) 0 0
\(459\) 0.207107 + 0.358719i 0.00966692 + 0.0167436i
\(460\) 0 0
\(461\) −25.4558 −1.18560 −0.592798 0.805351i \(-0.701977\pi\)
−0.592798 + 0.805351i \(0.701977\pi\)
\(462\) 0 0
\(463\) −11.3137 −0.525793 −0.262896 0.964824i \(-0.584678\pi\)
−0.262896 + 0.964824i \(0.584678\pi\)
\(464\) 0 0
\(465\) −2.11522 3.66367i −0.0980911 0.169899i
\(466\) 0 0
\(467\) 6.27817 10.8741i 0.290519 0.503194i −0.683413 0.730032i \(-0.739505\pi\)
0.973933 + 0.226837i \(0.0728386\pi\)
\(468\) 0 0
\(469\) 4.47056 5.76500i 0.206431 0.266203i
\(470\) 0 0
\(471\) 1.93503 3.35157i 0.0891614 0.154432i
\(472\) 0 0
\(473\) 11.6569 + 20.1903i 0.535983 + 0.928349i
\(474\) 0 0
\(475\) −10.6274 −0.487619
\(476\) 0 0
\(477\) 2.82843 0.129505
\(478\) 0 0
\(479\) 14.3492 + 24.8536i 0.655634 + 1.13559i 0.981735 + 0.190256i \(0.0609317\pi\)
−0.326101 + 0.945335i \(0.605735\pi\)
\(480\) 0 0
\(481\) −12.2426 + 21.2049i −0.558216 + 0.966859i
\(482\) 0 0
\(483\) −2.17157 5.31925i −0.0988100 0.242034i
\(484\) 0 0
\(485\) −1.07107 + 1.85514i −0.0486347 + 0.0842377i
\(486\) 0 0
\(487\) 5.86396 + 10.1567i 0.265721 + 0.460243i 0.967752 0.251903i \(-0.0810565\pi\)
−0.702031 + 0.712146i \(0.747723\pi\)
\(488\) 0 0
\(489\) −2.79899 −0.126575
\(490\) 0 0
\(491\) 3.65685 0.165032 0.0825158 0.996590i \(-0.473705\pi\)
0.0825158 + 0.996590i \(0.473705\pi\)
\(492\) 0 0
\(493\) −0.242641 0.420266i −0.0109280 0.0189278i
\(494\) 0 0
\(495\) 6.24264 10.8126i 0.280586 0.485989i
\(496\) 0 0
\(497\) −13.6569 33.4523i −0.612594 1.50054i
\(498\) 0 0
\(499\) 2.27817 3.94591i 0.101985 0.176643i −0.810517 0.585715i \(-0.800814\pi\)
0.912502 + 0.409071i \(0.134147\pi\)
\(500\) 0 0
\(501\) 0.414214 + 0.717439i 0.0185057 + 0.0320528i
\(502\) 0 0
\(503\) −10.3431 −0.461178 −0.230589 0.973051i \(-0.574065\pi\)
−0.230589 + 0.973051i \(0.574065\pi\)
\(504\) 0 0
\(505\) −24.6569 −1.09722
\(506\) 0 0
\(507\) −1.03553 1.79360i −0.0459897 0.0796565i
\(508\) 0 0
\(509\) −20.7426 + 35.9273i −0.919401 + 1.59245i −0.119075 + 0.992885i \(0.537993\pi\)
−0.800326 + 0.599565i \(0.795340\pi\)
\(510\) 0 0
\(511\) 23.7635 30.6441i 1.05123 1.35562i
\(512\) 0 0
\(513\) −7.74264 + 13.4106i −0.341846 + 0.592095i
\(514\) 0 0
\(515\) 7.69239 + 13.3236i 0.338967 + 0.587108i
\(516\) 0 0
\(517\) −25.1421 −1.10575
\(518\) 0 0
\(519\) −4.55635 −0.200002
\(520\) 0 0
\(521\) −3.50000 6.06218i −0.153338 0.265589i 0.779115 0.626881i \(-0.215669\pi\)
−0.932453 + 0.361293i \(0.882336\pi\)
\(522\) 0 0
\(523\) −8.86396 + 15.3528i −0.387594 + 0.671332i −0.992125 0.125249i \(-0.960027\pi\)
0.604531 + 0.796581i \(0.293360\pi\)
\(524\) 0 0
\(525\) 1.79899 + 0.246186i 0.0785144 + 0.0107444i
\(526\) 0 0
\(527\) 0.479185 0.829972i 0.0208736 0.0361542i
\(528\) 0 0
\(529\) −2.24264 3.88437i −0.0975061 0.168886i
\(530\) 0 0
\(531\) −30.8284 −1.33784
\(532\) 0 0
\(533\) −19.3137 −0.836570
\(534\) 0 0
\(535\) −11.7929 20.4259i −0.509851 0.883088i
\(536\) 0 0
\(537\) 1.15685 2.00373i 0.0499219 0.0864673i
\(538\) 0 0
\(539\) −16.2782 4.54026i −0.701151 0.195563i
\(540\) 0 0
\(541\) 12.9142 22.3681i 0.555225 0.961679i −0.442661 0.896689i \(-0.645965\pi\)
0.997886 0.0649894i \(-0.0207013\pi\)
\(542\) 0 0
\(543\) −1.92893 3.34101i −0.0827784 0.143376i
\(544\) 0 0
\(545\) 14.9411 0.640007
\(546\) 0 0
\(547\) 10.9706 0.469067 0.234534 0.972108i \(-0.424644\pi\)
0.234534 + 0.972108i \(0.424644\pi\)
\(548\) 0 0
\(549\) 12.2426 + 21.2049i 0.522503 + 0.905002i
\(550\) 0 0
\(551\) 9.07107 15.7116i 0.386440 0.669335i
\(552\) 0 0
\(553\) 15.9142 + 2.17781i 0.676741 + 0.0926099i
\(554\) 0 0
\(555\) −3.27817 + 5.67796i −0.139151 + 0.241016i
\(556\) 0 0
\(557\) 0.0857864 + 0.148586i 0.00363489 + 0.00629581i 0.867837 0.496849i \(-0.165510\pi\)
−0.864202 + 0.503145i \(0.832176\pi\)
\(558\) 0 0
\(559\) 27.3137 1.15525
\(560\) 0 0
\(561\) −0.171573 −0.00724381
\(562\) 0 0
\(563\) 7.03553 + 12.1859i 0.296512 + 0.513575i 0.975336 0.220727i \(-0.0708431\pi\)
−0.678823 + 0.734302i \(0.737510\pi\)
\(564\) 0 0
\(565\) 16.5858 28.7274i 0.697769 1.20857i
\(566\) 0 0
\(567\) −12.1360 + 15.6500i −0.509666 + 0.657238i
\(568\) 0 0
\(569\) −18.3284 + 31.7458i −0.768368 + 1.33085i 0.170080 + 0.985430i \(0.445597\pi\)
−0.938448 + 0.345422i \(0.887736\pi\)
\(570\) 0 0
\(571\) 9.20711 + 15.9472i 0.385305 + 0.667369i 0.991812 0.127710i \(-0.0407628\pi\)
−0.606506 + 0.795079i \(0.707429\pi\)
\(572\) 0 0
\(573\) 3.68629 0.153997
\(574\) 0 0
\(575\) 8.68629 0.362243
\(576\) 0 0
\(577\) −19.5000 33.7750i −0.811796 1.40607i −0.911606 0.411065i \(-0.865157\pi\)
0.0998105 0.995006i \(-0.468176\pi\)
\(578\) 0 0
\(579\) −1.89340 + 3.27946i −0.0786869 + 0.136290i
\(580\) 0 0
\(581\) 7.31371 + 17.9149i 0.303424 + 0.743233i
\(582\) 0 0
\(583\) −1.20711 + 2.09077i −0.0499933 + 0.0865909i
\(584\) 0 0
\(585\) −7.31371 12.6677i −0.302385 0.523746i
\(586\) 0 0
\(587\) 28.9706 1.19574 0.597872 0.801592i \(-0.296013\pi\)
0.597872 + 0.801592i \(0.296013\pi\)
\(588\) 0 0
\(589\) 35.8284 1.47628
\(590\) 0 0
\(591\) −3.89949 6.75412i −0.160404 0.277828i
\(592\) 0 0
\(593\) −2.74264 + 4.75039i −0.112627 + 0.195075i −0.916829 0.399281i \(-0.869260\pi\)
0.804202 + 0.594356i \(0.202593\pi\)
\(594\) 0 0
\(595\) −0.313708 0.768426i −0.0128608 0.0315024i
\(596\) 0 0
\(597\) 2.32843 4.03295i 0.0952962 0.165058i
\(598\) 0 0
\(599\) −17.9350 31.0644i −0.732805 1.26926i −0.955680 0.294409i \(-0.904877\pi\)
0.222874 0.974847i \(-0.428456\pi\)
\(600\) 0 0
\(601\) 26.1421 1.06636 0.533180 0.846002i \(-0.320997\pi\)
0.533180 + 0.846002i \(0.320997\pi\)
\(602\) 0 0
\(603\) 7.79899 0.317599
\(604\) 0 0
\(605\) −4.72792 8.18900i −0.192217 0.332930i
\(606\) 0 0
\(607\) −17.6924 + 30.6441i −0.718112 + 1.24381i 0.243635 + 0.969867i \(0.421660\pi\)
−0.961747 + 0.273939i \(0.911673\pi\)
\(608\) 0 0
\(609\) −1.89949 + 2.44949i −0.0769714 + 0.0992583i
\(610\) 0 0
\(611\) −14.7279 + 25.5095i −0.595828 + 1.03200i
\(612\) 0 0
\(613\) −2.25736 3.90986i −0.0911739 0.157918i 0.816831 0.576876i \(-0.195729\pi\)
−0.908005 + 0.418959i \(0.862395\pi\)
\(614\) 0 0
\(615\) −5.17157 −0.208538
\(616\) 0 0
\(617\) 19.1127 0.769448 0.384724 0.923032i \(-0.374297\pi\)
0.384724 + 0.923032i \(0.374297\pi\)
\(618\) 0 0
\(619\) −8.96447 15.5269i −0.360312 0.624079i 0.627700 0.778455i \(-0.283997\pi\)
−0.988012 + 0.154376i \(0.950663\pi\)
\(620\) 0 0
\(621\) 6.32843 10.9612i 0.253951 0.439856i
\(622\) 0 0
\(623\) −23.5919 3.22848i −0.945189 0.129346i
\(624\) 0 0
\(625\) 6.98528 12.0989i 0.279411 0.483954i
\(626\) 0 0
\(627\) −3.20711 5.55487i −0.128080 0.221840i
\(628\) 0 0
\(629\) −1.48528 −0.0592220
\(630\) 0 0
\(631\) 29.6569 1.18062 0.590310 0.807176i \(-0.299005\pi\)
0.590310 + 0.807176i \(0.299005\pi\)
\(632\) 0 0
\(633\) −2.48528 4.30463i −0.0987811 0.171094i
\(634\) 0 0
\(635\) −5.17157 + 8.95743i −0.205228 + 0.355465i
\(636\) 0 0
\(637\) −14.1421 + 13.8564i −0.560332 + 0.549011i
\(638\) 0 0
\(639\) 19.3137 33.4523i 0.764038 1.32335i
\(640\) 0 0
\(641\) 8.50000 + 14.7224i 0.335730 + 0.581501i 0.983625 0.180229i \(-0.0576838\pi\)
−0.647895 + 0.761730i \(0.724350\pi\)
\(642\) 0 0
\(643\) 11.0294 0.434959 0.217479 0.976065i \(-0.430217\pi\)
0.217479 + 0.976065i \(0.430217\pi\)
\(644\) 0 0
\(645\) 7.31371 0.287977
\(646\) 0 0
\(647\) −13.9350 24.1362i −0.547843 0.948891i −0.998422 0.0561548i \(-0.982116\pi\)
0.450580 0.892736i \(-0.351217\pi\)
\(648\) 0 0
\(649\) 13.1569 22.7883i 0.516452 0.894521i
\(650\) 0 0
\(651\) −6.06497 0.829972i −0.237705 0.0325292i
\(652\) 0 0
\(653\) 19.4706 33.7240i 0.761942 1.31972i −0.179906 0.983684i \(-0.557579\pi\)
0.941848 0.336038i \(-0.109087\pi\)
\(654\) 0 0
\(655\) 7.06497 + 12.2369i 0.276051 + 0.478135i
\(656\) 0 0
\(657\) 41.4558 1.61735
\(658\) 0 0
\(659\) 34.6274 1.34889 0.674446 0.738324i \(-0.264382\pi\)
0.674446 + 0.738324i \(0.264382\pi\)
\(660\) 0 0
\(661\) 15.3284 + 26.5496i 0.596207 + 1.03266i 0.993375 + 0.114915i \(0.0366595\pi\)
−0.397169 + 0.917746i \(0.630007\pi\)
\(662\) 0 0
\(663\) −0.100505 + 0.174080i −0.00390329 + 0.00676070i
\(664\) 0 0
\(665\) 19.0147 24.5204i 0.737359 0.950860i
\(666\) 0 0
\(667\) −7.41421 + 12.8418i −0.287079 + 0.497236i
\(668\) 0 0
\(669\) 0.485281 + 0.840532i 0.0187621 + 0.0324968i
\(670\) 0 0
\(671\) −20.8995 −0.806816
\(672\) 0 0
\(673\) −2.14214 −0.0825733 −0.0412866 0.999147i \(-0.513146\pi\)
−0.0412866 + 0.999147i \(0.513146\pi\)
\(674\) 0 0
\(675\) 2.00000 + 3.46410i 0.0769800 + 0.133333i
\(676\) 0 0
\(677\) 9.39949 16.2804i 0.361252 0.625707i −0.626915 0.779087i \(-0.715683\pi\)
0.988167 + 0.153381i \(0.0490161\pi\)
\(678\) 0 0
\(679\) 1.17157 + 2.86976i 0.0449608 + 0.110131i
\(680\) 0 0
\(681\) −3.32843 + 5.76500i −0.127546 + 0.220915i
\(682\) 0 0
\(683\) 8.72183 + 15.1066i 0.333731 + 0.578040i 0.983240 0.182314i \(-0.0583588\pi\)
−0.649509 + 0.760354i \(0.725025\pi\)
\(684\) 0 0
\(685\) −20.3726 −0.778396
\(686\) 0 0
\(687\) 1.72792 0.0659243
\(688\) 0 0
\(689\) 1.41421 + 2.44949i 0.0538772 + 0.0933181i
\(690\) 0 0
\(691\) −13.5208 + 23.4187i −0.514356 + 0.890891i 0.485505 + 0.874234i \(0.338636\pi\)
−0.999861 + 0.0166570i \(0.994698\pi\)
\(692\) 0 0
\(693\) −6.82843 16.7262i −0.259390 0.635374i
\(694\) 0 0
\(695\) 14.0000 24.2487i 0.531050 0.919806i
\(696\) 0 0
\(697\) −0.585786 1.01461i −0.0221882 0.0384312i
\(698\) 0 0
\(699\) −0.899495 −0.0340220
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 0 0
\(703\) −27.7635 48.0877i −1.04712 1.81366i
\(704\) 0 0
\(705\) −3.94365 + 6.83060i −0.148526 + 0.257255i
\(706\) 0 0
\(707\) −21.8640 + 28.1946i −0.822279 + 1.06037i
\(708\) 0 0
\(709\) 18.2990 31.6948i 0.687233 1.19032i −0.285496 0.958380i \(-0.592158\pi\)
0.972729 0.231943i \(-0.0745082\pi\)
\(710\) 0 0
\(711\) 8.58579 + 14.8710i 0.321992 + 0.557707i
\(712\) 0 0
\(713\) −29.2843 −1.09670
\(714\) 0 0
\(715\) 12.4853 0.466923
\(716\) 0 0
\(717\) 0.272078 + 0.471253i 0.0101609 + 0.0175993i
\(718\) 0 0
\(719\) −15.5208 + 26.8828i −0.578829 + 1.00256i 0.416785 + 0.909005i \(0.363157\pi\)
−0.995614 + 0.0935558i \(0.970177\pi\)
\(720\) 0 0
\(721\) 22.0563 + 3.01834i 0.821421 + 0.112409i
\(722\) 0 0
\(723\) 2.13604 3.69973i 0.0794401 0.137594i
\(724\) 0 0
\(725\) −2.34315 4.05845i −0.0870222 0.150727i
\(726\) 0 0
\(727\) 34.3431 1.27372 0.636858 0.770981i \(-0.280234\pi\)
0.636858 + 0.770981i \(0.280234\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 0.828427 + 1.43488i 0.0306405 + 0.0530709i
\(732\) 0 0
\(733\) −11.8431 + 20.5129i −0.437437 + 0.757662i −0.997491 0.0707935i \(-0.977447\pi\)
0.560054 + 0.828456i \(0.310780\pi\)
\(734\) 0 0
\(735\) −3.78680 + 3.71029i −0.139678 + 0.136856i
\(736\) 0 0
\(737\) −3.32843 + 5.76500i −0.122604 + 0.212357i
\(738\) 0 0
\(739\) 3.34924 + 5.80106i 0.123204 + 0.213395i 0.921029 0.389493i \(-0.127350\pi\)
−0.797826 + 0.602888i \(0.794016\pi\)
\(740\) 0 0
\(741\) −7.51472 −0.276060
\(742\) 0 0
\(743\) 26.3431 0.966436 0.483218 0.875500i \(-0.339468\pi\)
0.483218 + 0.875500i \(0.339468\pi\)
\(744\) 0 0
\(745\) 5.64214 + 9.77247i 0.206712 + 0.358035i
\(746\) 0 0
\(747\) −10.3431 + 17.9149i −0.378436 + 0.655470i
\(748\) 0 0
\(749\) −33.8137 4.62730i −1.23553 0.169078i
\(750\) 0 0
\(751\) −10.8640 + 18.8169i −0.396432 + 0.686640i −0.993283 0.115712i \(-0.963085\pi\)
0.596851 + 0.802352i \(0.296418\pi\)
\(752\) 0 0
\(753\) −6.41421 11.1097i −0.233747 0.404862i
\(754\) 0 0
\(755\) −1.64466 −0.0598553
\(756\) 0 0
\(757\) −6.14214 −0.223240 −0.111620 0.993751i \(-0.535604\pi\)
−0.111620 + 0.993751i \(0.535604\pi\)
\(758\) 0 0
\(759\) 2.62132 + 4.54026i 0.0951479 + 0.164801i
\(760\) 0 0
\(761\) 21.0563 36.4707i 0.763292 1.32206i −0.177853 0.984057i \(-0.556915\pi\)
0.941145 0.338004i \(-0.109752\pi\)
\(762\) 0 0
\(763\) 13.2487 17.0849i 0.479636 0.618514i
\(764\) 0 0
\(765\) 0.443651 0.768426i 0.0160402 0.0277825i
\(766\) 0 0
\(767\) −15.4142 26.6982i −0.556575 0.964016i
\(768\) 0 0
\(769\) 13.8579 0.499727 0.249864 0.968281i \(-0.419614\pi\)
0.249864 + 0.968281i \(0.419614\pi\)
\(770\) 0 0
\(771\) −1.87006 −0.0673485
\(772\) 0 0
\(773\) 4.57107 + 7.91732i 0.164410 + 0.284766i 0.936446 0.350813i \(-0.114095\pi\)
−0.772036 + 0.635579i \(0.780761\pi\)
\(774\) 0 0
\(775\) 4.62742 8.01492i 0.166222 0.287904i
\(776\) 0 0
\(777\) 3.58579 + 8.78335i 0.128639 + 0.315101i
\(778\) 0 0
\(779\) 21.8995 37.9310i 0.784631 1.35902i
\(780\) 0 0
\(781\) 16.4853 + 28.5533i 0.589890 + 1.02172i
\(782\) 0 0
\(783\) −6.82843 −0.244028
\(784\) 0 0
\(785\) −17.0833 −0.609728
\(786\) 0 0
\(787\) −23.1066 40.0218i −0.823661 1.42662i −0.902938 0.429771i \(-0.858594\pi\)
0.0792766 0.996853i \(-0.474739\pi\)
\(788\) 0 0
\(789\) 2.25736 3.90986i 0.0803641 0.139195i
\(790\) 0 0
\(791\) −18.1421 44.4390i −0.645060 1.58007i
\(792\) 0 0
\(793\) −12.2426 + 21.2049i −0.434749 + 0.753007i
\(794\) 0 0
\(795\) 0.378680 + 0.655892i 0.0134304 + 0.0232621i
\(796\) 0 0
\(797\) 47.1127 1.66882 0.834409 0.551146i \(-0.185809\pi\)
0.834409 + 0.551146i \(0.185809\pi\)
\(798\) 0 0
\(799\) −1.78680 −0.0632123
\(800\) 0 0
\(801\) −12.7279 22.0454i −0.449719 0.778936i
\(802\) 0 0
\(803\) −17.6924 + 30.6441i −0.624351 + 1.08141i
\(804\) 0 0
\(805\) −15.5416 + 20.0417i −0.547771 + 0.706376i
\(806\) 0 0
\(807\) 2.62132 4.54026i 0.0922748 0.159825i
\(808\) 0 0
\(809\) 8.98528 + 15.5630i 0.315906 + 0.547165i 0.979630 0.200813i \(-0.0643584\pi\)
−0.663724 + 0.747978i \(0.731025\pi\)
\(810\) 0 0
\(811\) −17.6569 −0.620016 −0.310008 0.950734i \(-0.600332\pi\)
−0.310008 + 0.950734i \(0.600332\pi\)
\(812\) 0 0
\(813\) 12.5147 0.438910
\(814\) 0 0
\(815\) 6.17767 + 10.7000i 0.216394 + 0.374806i
\(816\) 0 0
\(817\) −30.9706 + 53.6426i −1.08352 + 1.87672i
\(818\) 0 0
\(819\) −20.9706 2.86976i −0.732771 0.100277i
\(820\) 0 0
\(821\) −7.57107 + 13.1135i −0.264232 + 0.457663i −0.967362 0.253398i \(-0.918452\pi\)
0.703130 + 0.711061i \(0.251785\pi\)
\(822\) 0 0
\(823\) 24.0061 + 41.5798i 0.836800 + 1.44938i 0.892556 + 0.450936i \(0.148910\pi\)
−0.0557566 + 0.998444i \(0.517757\pi\)
\(824\) 0 0
\(825\) −1.65685 −0.0576843
\(826\) 0 0
\(827\) 41.6569 1.44855 0.724275 0.689511i \(-0.242174\pi\)
0.724275 + 0.689511i \(0.242174\pi\)
\(828\) 0 0
\(829\) −0.600505 1.04011i −0.0208564 0.0361243i 0.855409 0.517953i \(-0.173306\pi\)
−0.876265 + 0.481829i \(0.839973\pi\)
\(830\) 0 0
\(831\) 2.13604 3.69973i 0.0740984 0.128342i
\(832\) 0 0
\(833\) −1.15685 0.322666i −0.0400826 0.0111797i
\(834\) 0 0
\(835\) 1.82843 3.16693i 0.0632753 0.109596i
\(836\) 0 0
\(837\) −6.74264 11.6786i −0.233060 0.403671i
\(838\) 0 0
\(839\) −3.31371 −0.114402 −0.0572010 0.998363i \(-0.518218\pi\)
−0.0572010 + 0.998363i \(0.518218\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) −2.58579 4.47871i −0.0890592 0.154255i
\(844\) 0 0
\(845\) −4.57107 + 7.91732i −0.157250 + 0.272364i
\(846\) 0 0
\(847\) −13.5563 1.85514i −0.465802 0.0637435i
\(848\) 0 0
\(849\) 1.15685 2.00373i 0.0397031 0.0687678i
\(850\) 0 0
\(851\) 22.6924 + 39.3044i 0.777885 + 1.34734i
\(852\) 0 0
\(853\) −40.4853 −1.38619 −0.693095 0.720846i \(-0.743753\pi\)
−0.693095 + 0.720846i \(0.743753\pi\)
\(854\) 0 0
\(855\) 33.1716 1.13444
\(856\) 0 0
\(857\) −12.6716 21.9478i −0.432853 0.749723i 0.564265 0.825594i \(-0.309160\pi\)
−0.997118 + 0.0758709i \(0.975826\pi\)
\(858\) 0 0
\(859\) −2.72183 + 4.71434i −0.0928675 + 0.160851i −0.908717 0.417414i \(-0.862937\pi\)
0.815849 + 0.578265i \(0.196270\pi\)
\(860\) 0 0
\(861\) −4.58579 + 5.91359i −0.156283 + 0.201535i
\(862\) 0 0
\(863\) −16.6924 + 28.9121i −0.568216 + 0.984178i 0.428527 + 0.903529i \(0.359033\pi\)
−0.996743 + 0.0806492i \(0.974301\pi\)
\(864\) 0 0
\(865\) 10.0563 + 17.4181i 0.341926 + 0.592233i
\(866\) 0 0
\(867\) 7.02944 0.238732
\(868\) 0 0
\(869\) −14.6569 −0.497200
\(870\) 0 0
\(871\) 3.89949 + 6.75412i 0.132129 + 0.228855i
\(872\) 0 0
\(873\) −1.65685 + 2.86976i −0.0560760 + 0.0971265i
\(874\) 0 0
\(875\) −12.1716 29.8141i −0.411474 1.00790i
\(876\) 0 0
\(877\) −25.7132 + 44.5366i −0.868273 + 1.50389i −0.00451320 + 0.999990i \(0.501437\pi\)
−0.863760 + 0.503903i \(0.831897\pi\)
\(878\) 0 0
\(879\) 5.92893 + 10.2692i 0.199978 + 0.346372i
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) −41.2548 −1.38834 −0.694168 0.719813i \(-0.744227\pi\)
−0.694168 + 0.719813i \(0.744227\pi\)
\(884\) 0 0
\(885\) −4.12742 7.14890i −0.138742 0.240308i
\(886\) 0 0
\(887\) −15.5503 + 26.9338i −0.522126 + 0.904349i 0.477542 + 0.878609i \(0.341528\pi\)
−0.999669 + 0.0257406i \(0.991806\pi\)
\(888\) 0 0
\(889\) 5.65685 + 13.8564i 0.189725 + 0.464729i
\(890\) 0 0
\(891\) 9.03553 15.6500i 0.302702 0.524295i
\(892\) 0 0
\(893\) −33.3995 57.8496i −1.11767 1.93586i
\(894\) 0 0
\(895\) −10.2132 −0.341390
\(896\) 0 0
\(897\) 6.14214 0.205080
\(898\) 0 0
\(899\) 7.89949 + 13.6823i 0.263463 + 0.456331i
\(900\) 0 0
\(901\) −0.0857864 + 0.148586i −0.00285796 + 0.00495013i
\(902\) 0 0
\(903\) 6.48528 8.36308i 0.215817 0.278306i
\(904\) 0 0
\(905\) −8.51472 + 14.7479i −0.283039 + 0.490238i
\(906\) 0 0
\(907\) −12.7635 22.1070i −0.423804 0.734049i 0.572504 0.819902i \(-0.305972\pi\)
−0.996308 + 0.0858524i \(0.972639\pi\)
\(908\) 0 0
\(909\) −38.1421 −1.26509
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −8.82843 15.2913i −0.292178 0.506068i
\(914\) 0 0
\(915\) −3.27817 + 5.67796i −0.108373 + 0.187708i
\(916\) 0 0
\(917\) 20.2574 + 2.77216i 0.668957 + 0.0915447i
\(918\) 0 0
\(919\) 3.27817 5.67796i 0.108137 0.187299i −0.806879 0.590717i \(-0.798845\pi\)
0.915016 + 0.403419i \(0.132178\pi\)
\(920\) 0 0
\(921\) 2.97056 + 5.14517i 0.0978834 + 0.169539i
\(922\) 0 0
\(923\) 38.6274 1.27144
\(924\) 0 0
\(925\) −14.3431 −0.471600
\(926\) 0 0
\(927\) 11.8995 + 20.6105i 0.390831 + 0.676939i
\(928\) 0 0
\(929\) 9.67157 16.7517i 0.317314 0.549604i −0.662613 0.748962i \(-0.730552\pi\)
0.979927 + 0.199358i \(0.0638857\pi\)
\(930\) 0 0
\(931\) −11.1777 43.4859i −0.366333 1.42519i
\(932\) 0 0
\(933\) −3.88478 + 6.72863i −0.127182 + 0.220285i
\(934\) 0 0
\(935\) 0.378680 + 0.655892i 0.0123841 + 0.0214500i
\(936\) 0 0
\(937\) 16.6274 0.543194 0.271597 0.962411i \(-0.412448\pi\)
0.271597 + 0.962411i \(0.412448\pi\)
\(938\) 0 0
\(939\) 12.1299 0.395846
\(940\) 0 0
\(941\) 19.1274 + 33.1297i 0.623536 + 1.08000i 0.988822 + 0.149101i \(0.0476378\pi\)
−0.365286 + 0.930895i \(0.619029\pi\)
\(942\) 0 0
\(943\) −17.8995 + 31.0028i −0.582888 + 1.00959i
\(944\) 0 0
\(945\) −11.5711 1.58346i −0.376407 0.0515101i
\(946\) 0 0
\(947\) 9.10660 15.7731i 0.295925 0.512557i −0.679275 0.733884i \(-0.737706\pi\)
0.975200 + 0.221327i \(0.0710389\pi\)
\(948\) 0 0
\(949\) 20.7279 + 35.9018i 0.672857 + 1.16542i
\(950\) 0 0
\(951\) 4.01219 0.130104
\(952\) 0 0
\(953\) −27.1127 −0.878266 −0.439133 0.898422i \(-0.644714\pi\)
−0.439133 + 0.898422i \(0.644714\pi\)
\(954\) 0 0
\(955\) −8.13604 14.0920i −0.263276 0.456007i
\(956\) 0 0
\(957\) 1.41421 2.44949i 0.0457150 0.0791808i
\(958\) 0 0
\(959\) −18.0650 + 23.2956i −0.583348 + 0.752256i
\(960\) 0 0
\(961\) −0.100505 + 0.174080i −0.00324210 + 0.00561548i
\(962\) 0 0
\(963\) −18.2426 31.5972i −0.587861 1.01820i
\(964\) 0 0
\(965\) 16.7157 0.538098
\(966\) 0 0
\(967\) −18.3431 −0.589876 −0.294938 0.955516i \(-0.595299\pi\)
−0.294938 + 0.955516i \(0.595299\pi\)
\(968\) 0 0
\(969\) −0.227922 0.394773i −0.00732191 0.0126819i
\(970\) 0 0
\(971\) 17.7929 30.8182i 0.571001 0.989003i −0.425462 0.904976i \(-0.639889\pi\)
0.996463 0.0840268i \(-0.0267781\pi\)
\(972\) 0 0
\(973\) −15.3137 37.5108i −0.490935 1.20254i
\(974\) 0 0
\(975\) −0.970563 + 1.68106i −0.0310829 + 0.0538371i
\(976\) 0 0
\(977\) 17.2574 + 29.8906i 0.552112 + 0.956286i 0.998122 + 0.0612578i \(0.0195112\pi\)
−0.446010 + 0.895028i \(0.647155\pi\)
\(978\) 0 0
\(979\) 21.7279 0.694427
\(980\) 0 0
\(981\) 23.1127 0.737932
\(982\) 0 0
\(983\) −13.4203 23.2447i −0.428041 0.741389i 0.568658 0.822574i \(-0.307463\pi\)
−0.996699 + 0.0811848i \(0.974130\pi\)
\(984\) 0 0
\(985\) −17.2132 + 29.8141i −0.548458 + 0.949958i
\(986\) 0 0
\(987\) 4.31371 + 10.5664i 0.137307 + 0.336332i
\(988\) 0 0
\(989\) 25.3137 43.8446i 0.804929 1.39418i
\(990\) 0 0
\(991\) 22.3492 + 38.7100i 0.709947 + 1.22966i 0.964876 + 0.262705i \(0.0846145\pi\)
−0.254929 + 0.966960i \(0.582052\pi\)
\(992\) 0 0
\(993\) −11.7696 −0.373495
\(994\) 0 0
\(995\) −20.5563 −0.651680
\(996\) 0 0
\(997\) 22.6421 + 39.2173i 0.717084 + 1.24203i 0.962150 + 0.272519i \(0.0878568\pi\)
−0.245067 + 0.969506i \(0.578810\pi\)
\(998\) 0 0
\(999\) −10.4497 + 18.0995i −0.330615 + 0.572643i
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 224.2.i.a.65.2 4
3.2 odd 2 2016.2.s.q.289.1 4
4.3 odd 2 224.2.i.d.65.1 yes 4
7.2 even 3 1568.2.a.w.1.1 2
7.3 odd 6 1568.2.i.x.1537.1 4
7.4 even 3 inner 224.2.i.a.193.2 yes 4
7.5 odd 6 1568.2.a.j.1.2 2
7.6 odd 2 1568.2.i.x.961.1 4
8.3 odd 2 448.2.i.g.65.2 4
8.5 even 2 448.2.i.j.65.1 4
12.11 even 2 2016.2.s.s.289.1 4
21.11 odd 6 2016.2.s.q.865.1 4
28.3 even 6 1568.2.i.o.1537.2 4
28.11 odd 6 224.2.i.d.193.1 yes 4
28.19 even 6 1568.2.a.u.1.1 2
28.23 odd 6 1568.2.a.l.1.2 2
28.27 even 2 1568.2.i.o.961.2 4
56.5 odd 6 3136.2.a.bx.1.1 2
56.11 odd 6 448.2.i.g.193.2 4
56.19 even 6 3136.2.a.be.1.2 2
56.37 even 6 3136.2.a.bd.1.2 2
56.51 odd 6 3136.2.a.bw.1.1 2
56.53 even 6 448.2.i.j.193.1 4
84.11 even 6 2016.2.s.s.865.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.i.a.65.2 4 1.1 even 1 trivial
224.2.i.a.193.2 yes 4 7.4 even 3 inner
224.2.i.d.65.1 yes 4 4.3 odd 2
224.2.i.d.193.1 yes 4 28.11 odd 6
448.2.i.g.65.2 4 8.3 odd 2
448.2.i.g.193.2 4 56.11 odd 6
448.2.i.j.65.1 4 8.5 even 2
448.2.i.j.193.1 4 56.53 even 6
1568.2.a.j.1.2 2 7.5 odd 6
1568.2.a.l.1.2 2 28.23 odd 6
1568.2.a.u.1.1 2 28.19 even 6
1568.2.a.w.1.1 2 7.2 even 3
1568.2.i.o.961.2 4 28.27 even 2
1568.2.i.o.1537.2 4 28.3 even 6
1568.2.i.x.961.1 4 7.6 odd 2
1568.2.i.x.1537.1 4 7.3 odd 6
2016.2.s.q.289.1 4 3.2 odd 2
2016.2.s.q.865.1 4 21.11 odd 6
2016.2.s.s.289.1 4 12.11 even 2
2016.2.s.s.865.1 4 84.11 even 6
3136.2.a.bd.1.2 2 56.37 even 6
3136.2.a.be.1.2 2 56.19 even 6
3136.2.a.bw.1.1 2 56.51 odd 6
3136.2.a.bx.1.1 2 56.5 odd 6