Properties

Label 2028.2.i.e.529.1
Level $2028$
Weight $2$
Character 2028.529
Analytic conductor $16.194$
Analytic rank $1$
Dimension $2$
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] = [2028,2,Mod(529,2028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2028.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2028.i (of order \(3\), degree \(2\), not 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: \(16.1936615299\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2028.529
Dual form 2028.2.i.e.2005.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} -4.00000 q^{5} +(1.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} -4.00000 q^{5} +(1.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(2.00000 - 3.46410i) q^{11} +(-2.00000 + 3.46410i) q^{15} +(-1.00000 - 1.73205i) q^{17} +(1.00000 + 1.73205i) q^{19} +2.00000 q^{21} +11.0000 q^{25} -1.00000 q^{27} +(3.00000 - 5.19615i) q^{29} -10.0000 q^{31} +(-2.00000 - 3.46410i) q^{33} +(-4.00000 - 6.92820i) q^{35} +(-5.00000 + 8.66025i) q^{37} +(-4.00000 + 6.92820i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(2.00000 + 3.46410i) q^{45} -4.00000 q^{47} +(1.50000 - 2.59808i) q^{49} -2.00000 q^{51} -10.0000 q^{53} +(-8.00000 + 13.8564i) q^{55} +2.00000 q^{57} +(4.00000 + 6.92820i) q^{59} +(7.00000 + 12.1244i) q^{61} +(1.00000 - 1.73205i) q^{63} +(-1.00000 + 1.73205i) q^{67} +(-8.00000 - 13.8564i) q^{71} -10.0000 q^{73} +(5.50000 - 9.52628i) q^{75} +8.00000 q^{77} -16.0000 q^{79} +(-0.500000 + 0.866025i) q^{81} +(4.00000 + 6.92820i) q^{85} +(-3.00000 - 5.19615i) q^{87} +(2.00000 - 3.46410i) q^{89} +(-5.00000 + 8.66025i) q^{93} +(-4.00000 - 6.92820i) q^{95} +(1.00000 + 1.73205i) q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 8 q^{5} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 8 q^{5} + 2 q^{7} - q^{9} + 4 q^{11} - 4 q^{15} - 2 q^{17} + 2 q^{19} + 4 q^{21} + 22 q^{25} - 2 q^{27} + 6 q^{29} - 20 q^{31} - 4 q^{33} - 8 q^{35} - 10 q^{37} - 8 q^{41} - 4 q^{43} + 4 q^{45} - 8 q^{47} + 3 q^{49} - 4 q^{51} - 20 q^{53} - 16 q^{55} + 4 q^{57} + 8 q^{59} + 14 q^{61} + 2 q^{63} - 2 q^{67} - 16 q^{71} - 20 q^{73} + 11 q^{75} + 16 q^{77} - 32 q^{79} - q^{81} + 8 q^{85} - 6 q^{87} + 4 q^{89} - 10 q^{93} - 8 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 0
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 1.00000 + 1.73205i 0.377964 + 0.654654i 0.990766 0.135583i \(-0.0432908\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 2.00000 3.46410i 0.603023 1.04447i −0.389338 0.921095i \(-0.627296\pi\)
0.992361 0.123371i \(-0.0393705\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −2.00000 + 3.46410i −0.516398 + 0.894427i
\(16\) 0 0
\(17\) −1.00000 1.73205i −0.242536 0.420084i 0.718900 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) 1.00000 + 1.73205i 0.229416 + 0.397360i 0.957635 0.287984i \(-0.0929851\pi\)
−0.728219 + 0.685344i \(0.759652\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.00000 5.19615i 0.557086 0.964901i −0.440652 0.897678i \(-0.645253\pi\)
0.997738 0.0672232i \(-0.0214140\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) −2.00000 3.46410i −0.348155 0.603023i
\(34\) 0 0
\(35\) −4.00000 6.92820i −0.676123 1.17108i
\(36\) 0 0
\(37\) −5.00000 + 8.66025i −0.821995 + 1.42374i 0.0821995 + 0.996616i \(0.473806\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 + 6.92820i −0.624695 + 1.08200i 0.363905 + 0.931436i \(0.381443\pi\)
−0.988600 + 0.150567i \(0.951890\pi\)
\(42\) 0 0
\(43\) −2.00000 3.46410i −0.304997 0.528271i 0.672264 0.740312i \(-0.265322\pi\)
−0.977261 + 0.212041i \(0.931989\pi\)
\(44\) 0 0
\(45\) 2.00000 + 3.46410i 0.298142 + 0.516398i
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −8.00000 + 13.8564i −1.07872 + 1.86840i
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 4.00000 + 6.92820i 0.520756 + 0.901975i 0.999709 + 0.0241347i \(0.00768307\pi\)
−0.478953 + 0.877841i \(0.658984\pi\)
\(60\) 0 0
\(61\) 7.00000 + 12.1244i 0.896258 + 1.55236i 0.832240 + 0.554416i \(0.187058\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 1.00000 1.73205i 0.125988 0.218218i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 13.8564i −0.949425 1.64445i −0.746639 0.665230i \(-0.768333\pi\)
−0.202787 0.979223i \(-0.565000\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 5.50000 9.52628i 0.635085 1.10000i
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 4.00000 + 6.92820i 0.433861 + 0.751469i
\(86\) 0 0
\(87\) −3.00000 5.19615i −0.321634 0.557086i
\(88\) 0 0
\(89\) 2.00000 3.46410i 0.212000 0.367194i −0.740341 0.672232i \(-0.765336\pi\)
0.952340 + 0.305038i \(0.0986691\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.00000 + 8.66025i −0.518476 + 0.898027i
\(94\) 0 0
\(95\) −4.00000 6.92820i −0.410391 0.710819i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −5.00000 + 8.66025i −0.497519 + 0.861727i −0.999996 0.00286291i \(-0.999089\pi\)
0.502477 + 0.864590i \(0.332422\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) −6.00000 + 10.3923i −0.580042 + 1.00466i 0.415432 + 0.909624i \(0.363630\pi\)
−0.995474 + 0.0950377i \(0.969703\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 5.00000 + 8.66025i 0.474579 + 0.821995i
\(112\) 0 0
\(113\) −3.00000 5.19615i −0.282216 0.488813i 0.689714 0.724082i \(-0.257736\pi\)
−0.971930 + 0.235269i \(0.924403\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.00000 3.46410i 0.183340 0.317554i
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 4.00000 + 6.92820i 0.360668 + 0.624695i
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) −6.00000 + 10.3923i −0.532414 + 0.922168i 0.466870 + 0.884326i \(0.345382\pi\)
−0.999284 + 0.0378419i \(0.987952\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −2.00000 + 3.46410i −0.173422 + 0.300376i
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −4.00000 6.92820i −0.341743 0.591916i 0.643013 0.765855i \(-0.277684\pi\)
−0.984757 + 0.173939i \(0.944351\pi\)
\(138\) 0 0
\(139\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(140\) 0 0
\(141\) −2.00000 + 3.46410i −0.168430 + 0.291730i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −12.0000 + 20.7846i −0.996546 + 1.72607i
\(146\) 0 0
\(147\) −1.50000 2.59808i −0.123718 0.214286i
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 0 0
\(153\) −1.00000 + 1.73205i −0.0808452 + 0.140028i
\(154\) 0 0
\(155\) 40.0000 3.21288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −5.00000 + 8.66025i −0.396526 + 0.686803i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00000 1.73205i −0.0783260 0.135665i 0.824202 0.566296i \(-0.191624\pi\)
−0.902528 + 0.430632i \(0.858291\pi\)
\(164\) 0 0
\(165\) 8.00000 + 13.8564i 0.622799 + 1.07872i
\(166\) 0 0
\(167\) 6.00000 10.3923i 0.464294 0.804181i −0.534875 0.844931i \(-0.679641\pi\)
0.999169 + 0.0407502i \(0.0129748\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 1.00000 1.73205i 0.0764719 0.132453i
\(172\) 0 0
\(173\) −1.00000 1.73205i −0.0760286 0.131685i 0.825505 0.564396i \(-0.190891\pi\)
−0.901533 + 0.432710i \(0.857557\pi\)
\(174\) 0 0
\(175\) 11.0000 + 19.0526i 0.831522 + 1.44024i
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) −6.00000 + 10.3923i −0.448461 + 0.776757i −0.998286 0.0585225i \(-0.981361\pi\)
0.549825 + 0.835280i \(0.314694\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 0 0
\(185\) 20.0000 34.6410i 1.47043 2.54686i
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) −1.00000 1.73205i −0.0727393 0.125988i
\(190\) 0 0
\(191\) 4.00000 + 6.92820i 0.289430 + 0.501307i 0.973674 0.227946i \(-0.0732010\pi\)
−0.684244 + 0.729253i \(0.739868\pi\)
\(192\) 0 0
\(193\) 7.00000 12.1244i 0.503871 0.872730i −0.496119 0.868255i \(-0.665242\pi\)
0.999990 0.00447566i \(-0.00142465\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 10.3923i 0.427482 0.740421i −0.569166 0.822222i \(-0.692734\pi\)
0.996649 + 0.0818013i \(0.0260673\pi\)
\(198\) 0 0
\(199\) −2.00000 3.46410i −0.141776 0.245564i 0.786389 0.617731i \(-0.211948\pi\)
−0.928166 + 0.372168i \(0.878615\pi\)
\(200\) 0 0
\(201\) 1.00000 + 1.73205i 0.0705346 + 0.122169i
\(202\) 0 0
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 16.0000 27.7128i 1.11749 1.93555i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 6.00000 10.3923i 0.413057 0.715436i −0.582165 0.813070i \(-0.697794\pi\)
0.995222 + 0.0976347i \(0.0311277\pi\)
\(212\) 0 0
\(213\) −16.0000 −1.09630
\(214\) 0 0
\(215\) 8.00000 + 13.8564i 0.545595 + 0.944999i
\(216\) 0 0
\(217\) −10.0000 17.3205i −0.678844 1.17579i
\(218\) 0 0
\(219\) −5.00000 + 8.66025i −0.337869 + 0.585206i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.00000 + 12.1244i −0.468755 + 0.811907i −0.999362 0.0357107i \(-0.988630\pi\)
0.530607 + 0.847618i \(0.321964\pi\)
\(224\) 0 0
\(225\) −5.50000 9.52628i −0.366667 0.635085i
\(226\) 0 0
\(227\) 2.00000 + 3.46410i 0.132745 + 0.229920i 0.924734 0.380615i \(-0.124288\pi\)
−0.791989 + 0.610535i \(0.790954\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 4.00000 6.92820i 0.263181 0.455842i
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) −8.00000 + 13.8564i −0.519656 + 0.900070i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 1.00000 + 1.73205i 0.0644157 + 0.111571i 0.896435 0.443176i \(-0.146148\pi\)
−0.832019 + 0.554747i \(0.812815\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) −6.00000 + 10.3923i −0.383326 + 0.663940i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.0000 + 17.3205i 0.631194 + 1.09326i 0.987308 + 0.158818i \(0.0507683\pi\)
−0.356113 + 0.934443i \(0.615898\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) −15.0000 + 25.9808i −0.935674 + 1.62064i −0.162247 + 0.986750i \(0.551874\pi\)
−0.773427 + 0.633885i \(0.781459\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 8.00000 13.8564i 0.493301 0.854423i −0.506669 0.862141i \(-0.669123\pi\)
0.999970 + 0.00771799i \(0.00245674\pi\)
\(264\) 0 0
\(265\) 40.0000 2.45718
\(266\) 0 0
\(267\) −2.00000 3.46410i −0.122398 0.212000i
\(268\) 0 0
\(269\) 7.00000 + 12.1244i 0.426798 + 0.739235i 0.996586 0.0825561i \(-0.0263084\pi\)
−0.569789 + 0.821791i \(0.692975\pi\)
\(270\) 0 0
\(271\) 5.00000 8.66025i 0.303728 0.526073i −0.673249 0.739416i \(-0.735102\pi\)
0.976977 + 0.213343i \(0.0684351\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.0000 38.1051i 1.32665 2.29783i
\(276\) 0 0
\(277\) −11.0000 19.0526i −0.660926 1.14476i −0.980373 0.197153i \(-0.936830\pi\)
0.319447 0.947604i \(-0.396503\pi\)
\(278\) 0 0
\(279\) 5.00000 + 8.66025i 0.299342 + 0.518476i
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) 8.00000 13.8564i 0.475551 0.823678i −0.524057 0.851683i \(-0.675582\pi\)
0.999608 + 0.0280052i \(0.00891551\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −16.0000 −0.944450
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) −4.00000 6.92820i −0.233682 0.404750i 0.725206 0.688531i \(-0.241744\pi\)
−0.958889 + 0.283782i \(0.908411\pi\)
\(294\) 0 0
\(295\) −16.0000 27.7128i −0.931556 1.61350i
\(296\) 0 0
\(297\) −2.00000 + 3.46410i −0.116052 + 0.201008i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000 6.92820i 0.230556 0.399335i
\(302\) 0 0
\(303\) 5.00000 + 8.66025i 0.287242 + 0.497519i
\(304\) 0 0
\(305\) −28.0000 48.4974i −1.60328 2.77695i
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) −4.00000 + 6.92820i −0.227552 + 0.394132i
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −4.00000 + 6.92820i −0.225374 + 0.390360i
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) −12.0000 20.7846i −0.671871 1.16371i
\(320\) 0 0
\(321\) 6.00000 + 10.3923i 0.334887 + 0.580042i
\(322\) 0 0
\(323\) 2.00000 3.46410i 0.111283 0.192748i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −1.00000 + 1.73205i −0.0553001 + 0.0957826i
\(328\) 0 0
\(329\) −4.00000 6.92820i −0.220527 0.381964i
\(330\) 0 0
\(331\) −1.00000 1.73205i −0.0549650 0.0952021i 0.837234 0.546845i \(-0.184171\pi\)
−0.892199 + 0.451643i \(0.850838\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 4.00000 6.92820i 0.218543 0.378528i
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −20.0000 + 34.6410i −1.08306 + 1.87592i
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 10.3923i −0.322097 0.557888i 0.658824 0.752297i \(-0.271054\pi\)
−0.980921 + 0.194409i \(0.937721\pi\)
\(348\) 0 0
\(349\) −9.00000 + 15.5885i −0.481759 + 0.834431i −0.999781 0.0209364i \(-0.993335\pi\)
0.518022 + 0.855367i \(0.326669\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.00000 13.8564i 0.425797 0.737502i −0.570697 0.821160i \(-0.693327\pi\)
0.996495 + 0.0836583i \(0.0266604\pi\)
\(354\) 0 0
\(355\) 32.0000 + 55.4256i 1.69838 + 2.94169i
\(356\) 0 0
\(357\) −2.00000 3.46410i −0.105851 0.183340i
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 7.50000 12.9904i 0.394737 0.683704i
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 40.0000 2.09370
\(366\) 0 0
\(367\) −2.00000 + 3.46410i −0.104399 + 0.180825i −0.913493 0.406855i \(-0.866625\pi\)
0.809093 + 0.587680i \(0.199959\pi\)
\(368\) 0 0
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) −10.0000 17.3205i −0.519174 0.899236i
\(372\) 0 0
\(373\) 9.00000 + 15.5885i 0.466002 + 0.807140i 0.999246 0.0388219i \(-0.0123605\pi\)
−0.533244 + 0.845962i \(0.679027\pi\)
\(374\) 0 0
\(375\) −12.0000 + 20.7846i −0.619677 + 1.07331i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.00000 + 5.19615i −0.154100 + 0.266908i −0.932731 0.360573i \(-0.882581\pi\)
0.778631 + 0.627482i \(0.215914\pi\)
\(380\) 0 0
\(381\) 6.00000 + 10.3923i 0.307389 + 0.532414i
\(382\) 0 0
\(383\) −6.00000 10.3923i −0.306586 0.531022i 0.671027 0.741433i \(-0.265853\pi\)
−0.977613 + 0.210411i \(0.932520\pi\)
\(384\) 0 0
\(385\) −32.0000 −1.63087
\(386\) 0 0
\(387\) −2.00000 + 3.46410i −0.101666 + 0.176090i
\(388\) 0 0
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 2.00000 3.46410i 0.100887 0.174741i
\(394\) 0 0
\(395\) 64.0000 3.22019
\(396\) 0 0
\(397\) −9.00000 15.5885i −0.451697 0.782362i 0.546795 0.837267i \(-0.315848\pi\)
−0.998492 + 0.0549046i \(0.982515\pi\)
\(398\) 0 0
\(399\) 2.00000 + 3.46410i 0.100125 + 0.173422i
\(400\) 0 0
\(401\) 6.00000 10.3923i 0.299626 0.518967i −0.676425 0.736512i \(-0.736472\pi\)
0.976050 + 0.217545i \(0.0698049\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.00000 3.46410i 0.0993808 0.172133i
\(406\) 0 0
\(407\) 20.0000 + 34.6410i 0.991363 + 1.71709i
\(408\) 0 0
\(409\) −7.00000 12.1244i −0.346128 0.599511i 0.639430 0.768849i \(-0.279170\pi\)
−0.985558 + 0.169338i \(0.945837\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) −8.00000 + 13.8564i −0.393654 + 0.681829i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.0000 17.3205i 0.488532 0.846162i −0.511381 0.859354i \(-0.670866\pi\)
0.999913 + 0.0131919i \(0.00419923\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 2.00000 + 3.46410i 0.0972433 + 0.168430i
\(424\) 0 0
\(425\) −11.0000 19.0526i −0.533578 0.924185i
\(426\) 0 0
\(427\) −14.0000 + 24.2487i −0.677507 + 1.17348i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.0000 17.3205i 0.481683 0.834300i −0.518096 0.855323i \(-0.673359\pi\)
0.999779 + 0.0210230i \(0.00669232\pi\)
\(432\) 0 0
\(433\) 9.00000 + 15.5885i 0.432512 + 0.749133i 0.997089 0.0762473i \(-0.0242938\pi\)
−0.564577 + 0.825381i \(0.690961\pi\)
\(434\) 0 0
\(435\) 12.0000 + 20.7846i 0.575356 + 0.996546i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −8.00000 + 13.8564i −0.381819 + 0.661330i −0.991322 0.131453i \(-0.958036\pi\)
0.609503 + 0.792784i \(0.291369\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −8.00000 + 13.8564i −0.379236 + 0.656857i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.00000 10.3923i −0.283158 0.490443i 0.689003 0.724758i \(-0.258049\pi\)
−0.972161 + 0.234315i \(0.924715\pi\)
\(450\) 0 0
\(451\) 16.0000 + 27.7128i 0.753411 + 1.30495i
\(452\) 0 0
\(453\) 9.00000 15.5885i 0.422857 0.732410i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.00000 15.5885i 0.421002 0.729197i −0.575036 0.818128i \(-0.695012\pi\)
0.996038 + 0.0889312i \(0.0283451\pi\)
\(458\) 0 0
\(459\) 1.00000 + 1.73205i 0.0466760 + 0.0808452i
\(460\) 0 0
\(461\) −6.00000 10.3923i −0.279448 0.484018i 0.691800 0.722089i \(-0.256818\pi\)
−0.971248 + 0.238071i \(0.923485\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) 0 0
\(465\) 20.0000 34.6410i 0.927478 1.60644i
\(466\) 0 0
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 1.00000 1.73205i 0.0460776 0.0798087i
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 11.0000 + 19.0526i 0.504715 + 0.874191i
\(476\) 0 0
\(477\) 5.00000 + 8.66025i 0.228934 + 0.396526i
\(478\) 0 0
\(479\) 8.00000 13.8564i 0.365529 0.633115i −0.623332 0.781958i \(-0.714221\pi\)
0.988861 + 0.148842i \(0.0475547\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.00000 6.92820i −0.181631 0.314594i
\(486\) 0 0
\(487\) 13.0000 + 22.5167i 0.589086 + 1.02033i 0.994352 + 0.106129i \(0.0338455\pi\)
−0.405266 + 0.914199i \(0.632821\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) −6.00000 + 10.3923i −0.270776 + 0.468998i −0.969061 0.246822i \(-0.920614\pi\)
0.698285 + 0.715820i \(0.253947\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 16.0000 0.719147
\(496\) 0 0
\(497\) 16.0000 27.7128i 0.717698 1.24309i
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) −6.00000 10.3923i −0.268060 0.464294i
\(502\) 0 0
\(503\) −8.00000 13.8564i −0.356702 0.617827i 0.630705 0.776022i \(-0.282766\pi\)
−0.987408 + 0.158196i \(0.949432\pi\)
\(504\) 0 0
\(505\) 20.0000 34.6410i 0.889988 1.54150i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.00000 + 10.3923i −0.265945 + 0.460631i −0.967811 0.251679i \(-0.919017\pi\)
0.701866 + 0.712309i \(0.252351\pi\)
\(510\) 0 0
\(511\) −10.0000 17.3205i −0.442374 0.766214i
\(512\) 0 0
\(513\) −1.00000 1.73205i −0.0441511 0.0764719i
\(514\) 0 0
\(515\) 32.0000 1.41009
\(516\) 0 0
\(517\) −8.00000 + 13.8564i −0.351840 + 0.609404i
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 0 0
\(523\) −8.00000 + 13.8564i −0.349816 + 0.605898i −0.986216 0.165460i \(-0.947089\pi\)
0.636401 + 0.771358i \(0.280422\pi\)
\(524\) 0 0
\(525\) 22.0000 0.960159
\(526\) 0 0
\(527\) 10.0000 + 17.3205i 0.435607 + 0.754493i
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 4.00000 6.92820i 0.173585 0.300658i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 24.0000 41.5692i 1.03761 1.79719i
\(536\) 0 0
\(537\) 6.00000 + 10.3923i 0.258919 + 0.448461i
\(538\) 0 0
\(539\) −6.00000 10.3923i −0.258438 0.447628i
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 1.00000 1.73205i 0.0429141 0.0743294i
\(544\) 0 0
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 0 0
\(549\) 7.00000 12.1244i 0.298753 0.517455i
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −16.0000 27.7128i −0.680389 1.17847i
\(554\) 0 0
\(555\) −20.0000 34.6410i −0.848953 1.47043i
\(556\) 0 0
\(557\) −12.0000 + 20.7846i −0.508456 + 0.880672i 0.491496 + 0.870880i \(0.336450\pi\)
−0.999952 + 0.00979220i \(0.996883\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −4.00000 + 6.92820i −0.168880 + 0.292509i
\(562\) 0 0
\(563\) −14.0000 24.2487i −0.590030 1.02196i −0.994228 0.107290i \(-0.965783\pi\)
0.404198 0.914671i \(-0.367551\pi\)
\(564\) 0 0
\(565\) 12.0000 + 20.7846i 0.504844 + 0.874415i
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −3.00000 + 5.19615i −0.125767 + 0.217834i −0.922032 0.387113i \(-0.873472\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 0 0
\(579\) −7.00000 12.1244i −0.290910 0.503871i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −20.0000 + 34.6410i −0.828315 + 1.43468i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.00000 6.92820i 0.165098 0.285958i −0.771592 0.636117i \(-0.780539\pi\)
0.936690 + 0.350160i \(0.113873\pi\)
\(588\) 0 0
\(589\) −10.0000 17.3205i −0.412043 0.713679i
\(590\) 0 0
\(591\) −6.00000 10.3923i −0.246807 0.427482i
\(592\) 0 0
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) −8.00000 + 13.8564i −0.327968 + 0.568057i
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 11.0000 19.0526i 0.448699 0.777170i −0.549602 0.835426i \(-0.685221\pi\)
0.998302 + 0.0582563i \(0.0185541\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) 10.0000 + 17.3205i 0.406558 + 0.704179i
\(606\) 0 0
\(607\) 2.00000 + 3.46410i 0.0811775 + 0.140604i 0.903756 0.428048i \(-0.140799\pi\)
−0.822578 + 0.568652i \(0.807465\pi\)
\(608\) 0 0
\(609\) 6.00000 10.3923i 0.243132 0.421117i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 7.00000 12.1244i 0.282727 0.489698i −0.689328 0.724449i \(-0.742094\pi\)
0.972056 + 0.234751i \(0.0754275\pi\)
\(614\) 0 0
\(615\) −16.0000 27.7128i −0.645182 1.11749i
\(616\) 0 0
\(617\) 24.0000 + 41.5692i 0.966204 + 1.67351i 0.706346 + 0.707867i \(0.250342\pi\)
0.259858 + 0.965647i \(0.416324\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 4.00000 6.92820i 0.159745 0.276686i
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 15.0000 + 25.9808i 0.597141 + 1.03428i 0.993241 + 0.116071i \(0.0370299\pi\)
−0.396100 + 0.918207i \(0.629637\pi\)
\(632\) 0 0
\(633\) −6.00000 10.3923i −0.238479 0.413057i
\(634\) 0 0
\(635\) 24.0000 41.5692i 0.952411 1.64962i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −8.00000 + 13.8564i −0.316475 + 0.548151i
\(640\) 0 0
\(641\) −13.0000 22.5167i −0.513469 0.889355i −0.999878 0.0156233i \(-0.995027\pi\)
0.486409 0.873731i \(-0.338307\pi\)
\(642\) 0 0
\(643\) −13.0000 22.5167i −0.512670 0.887970i −0.999892 0.0146923i \(-0.995323\pi\)
0.487222 0.873278i \(-0.338010\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) 12.0000 20.7846i 0.471769 0.817127i −0.527710 0.849425i \(-0.676949\pi\)
0.999478 + 0.0322975i \(0.0102824\pi\)
\(648\) 0 0
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) 0 0
\(653\) 23.0000 39.8372i 0.900060 1.55895i 0.0726446 0.997358i \(-0.476856\pi\)
0.827415 0.561591i \(-0.189811\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) 5.00000 + 8.66025i 0.195069 + 0.337869i
\(658\) 0 0
\(659\) 6.00000 + 10.3923i 0.233727 + 0.404827i 0.958902 0.283738i \(-0.0915745\pi\)
−0.725175 + 0.688565i \(0.758241\pi\)
\(660\) 0 0
\(661\) −5.00000 + 8.66025i −0.194477 + 0.336845i −0.946729 0.322031i \(-0.895634\pi\)
0.752252 + 0.658876i \(0.228968\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 13.8564i 0.310227 0.537328i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 7.00000 + 12.1244i 0.270636 + 0.468755i
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) −11.0000 + 19.0526i −0.424019 + 0.734422i −0.996328 0.0856156i \(-0.972714\pi\)
0.572309 + 0.820038i \(0.306048\pi\)
\(674\) 0 0
\(675\) −11.0000 −0.423390
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) −2.00000 + 3.46410i −0.0767530 + 0.132940i
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) −10.0000 17.3205i −0.382639 0.662751i 0.608799 0.793324i \(-0.291651\pi\)
−0.991439 + 0.130573i \(0.958318\pi\)
\(684\) 0 0
\(685\) 16.0000 + 27.7128i 0.611329 + 1.05885i
\(686\) 0 0
\(687\) −1.00000 + 1.73205i −0.0381524 + 0.0660819i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −5.00000 + 8.66025i −0.190209 + 0.329452i −0.945319 0.326146i \(-0.894250\pi\)
0.755110 + 0.655598i \(0.227583\pi\)
\(692\) 0 0
\(693\) −4.00000 6.92820i −0.151947 0.263181i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 0 0
\(699\) −9.00000 + 15.5885i −0.340411 + 0.589610i
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 8.00000 13.8564i 0.301297 0.521862i
\(706\) 0 0
\(707\) −20.0000 −0.752177
\(708\) 0 0
\(709\) −3.00000 5.19615i −0.112667 0.195146i 0.804178 0.594389i \(-0.202606\pi\)
−0.916845 + 0.399244i \(0.869273\pi\)
\(710\) 0 0
\(711\) 8.00000 + 13.8564i 0.300023 + 0.519656i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 + 20.7846i −0.448148 + 0.776215i
\(718\) 0 0
\(719\) 24.0000 + 41.5692i 0.895049 + 1.55027i 0.833744 + 0.552151i \(0.186193\pi\)
0.0613050 + 0.998119i \(0.480474\pi\)
\(720\) 0 0
\(721\) −8.00000 13.8564i −0.297936 0.516040i
\(722\) 0 0
\(723\) 2.00000 0.0743808
\(724\) 0 0
\(725\) 33.0000 57.1577i 1.22559 2.12278i
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 + 6.92820i −0.147945 + 0.256249i
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) 6.00000 + 10.3923i 0.221313 + 0.383326i
\(736\) 0 0
\(737\) 4.00000 + 6.92820i 0.147342 + 0.255204i
\(738\) 0 0
\(739\) −11.0000 + 19.0526i −0.404642 + 0.700860i −0.994280 0.106808i \(-0.965937\pi\)
0.589638 + 0.807668i \(0.299270\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 + 41.5692i −0.880475 + 1.52503i −0.0296605 + 0.999560i \(0.509443\pi\)
−0.850814 + 0.525467i \(0.823891\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −4.00000 + 6.92820i −0.145962 + 0.252814i −0.929731 0.368238i \(-0.879961\pi\)
0.783769 + 0.621052i \(0.213294\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) −72.0000 −2.62035
\(756\) 0 0
\(757\) −9.00000 + 15.5885i −0.327111 + 0.566572i −0.981937 0.189207i \(-0.939408\pi\)
0.654827 + 0.755779i \(0.272742\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20.0000 34.6410i −0.724999 1.25574i −0.958974 0.283493i \(-0.908507\pi\)
0.233975 0.972243i \(-0.424827\pi\)
\(762\) 0 0
\(763\) −2.00000 3.46410i −0.0724049 0.125409i
\(764\) 0 0
\(765\) 4.00000 6.92820i 0.144620 0.250490i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 11.0000 19.0526i 0.396670 0.687053i −0.596643 0.802507i \(-0.703499\pi\)
0.993313 + 0.115454i \(0.0368323\pi\)
\(770\) 0 0
\(771\) 15.0000 + 25.9808i 0.540212 + 0.935674i
\(772\) 0 0
\(773\) 24.0000 + 41.5692i 0.863220 + 1.49514i 0.868804 + 0.495156i \(0.164889\pi\)
−0.00558380 + 0.999984i \(0.501777\pi\)
\(774\) 0 0
\(775\) −110.000 −3.95132
\(776\) 0 0
\(777\) −10.0000 + 17.3205i −0.358748 + 0.621370i
\(778\) 0 0
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) 0 0
\(783\) −3.00000 + 5.19615i −0.107211 + 0.185695i
\(784\) 0 0
\(785\) −8.00000 −0.285532
\(786\) 0 0
\(787\) 25.0000 + 43.3013i 0.891154 + 1.54352i 0.838494 + 0.544911i \(0.183437\pi\)
0.0526599 + 0.998613i \(0.483230\pi\)
\(788\) 0 0
\(789\) −8.00000 13.8564i −0.284808 0.493301i
\(790\) 0 0
\(791\) 6.00000 10.3923i 0.213335 0.369508i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 20.0000 34.6410i 0.709327 1.22859i
\(796\) 0 0
\(797\) 13.0000 + 22.5167i 0.460484 + 0.797581i 0.998985 0.0450436i \(-0.0143427\pi\)
−0.538501 + 0.842625i \(0.681009\pi\)
\(798\) 0 0
\(799\) 4.00000 + 6.92820i 0.141510 + 0.245102i
\(800\) 0 0
\(801\) −4.00000 −0.141333
\(802\) 0 0
\(803\) −20.0000 + 34.6410i −0.705785 + 1.22245i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000 0.492823
\(808\) 0 0
\(809\) −19.0000 + 32.9090i −0.668004 + 1.15702i 0.310457 + 0.950587i \(0.399518\pi\)
−0.978461 + 0.206430i \(0.933815\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 0 0
\(813\) −5.00000 8.66025i −0.175358 0.303728i
\(814\) 0 0
\(815\) 4.00000 + 6.92820i 0.140114 + 0.242684i
\(816\) 0 0
\(817\) 4.00000 6.92820i 0.139942 0.242387i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0000 + 20.7846i −0.418803 + 0.725388i −0.995819 0.0913446i \(-0.970884\pi\)
0.577016 + 0.816733i \(0.304217\pi\)
\(822\) 0 0
\(823\) −16.0000 27.7128i −0.557725 0.966008i −0.997686 0.0679910i \(-0.978341\pi\)
0.439961 0.898017i \(-0.354992\pi\)
\(824\) 0 0
\(825\) −22.0000 38.1051i −0.765942 1.32665i
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) 7.00000 12.1244i 0.243120 0.421096i −0.718481 0.695546i \(-0.755162\pi\)
0.961601 + 0.274450i \(0.0884958\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −24.0000 + 41.5692i −0.830554 + 1.43856i
\(836\) 0 0
\(837\) 10.0000 0.345651
\(838\) 0 0
\(839\) 8.00000 + 13.8564i 0.276191 + 0.478376i 0.970435 0.241363i \(-0.0775945\pi\)
−0.694244 + 0.719740i \(0.744261\pi\)
\(840\) 0 0
\(841\) −3.50000 6.06218i −0.120690 0.209041i
\(842\) 0 0
\(843\) −4.00000 + 6.92820i −0.137767 + 0.238620i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.00000 8.66025i 0.171802 0.297570i
\(848\) 0 0
\(849\) −8.00000 13.8564i −0.274559 0.475551i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) −4.00000 + 6.92820i −0.136797 + 0.236940i
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) 0 0
\(861\) −8.00000 + 13.8564i −0.272639 + 0.472225i
\(862\) 0 0
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 4.00000 + 6.92820i 0.136004 + 0.235566i
\(866\) 0 0
\(867\) −6.50000 11.2583i −0.220752 0.382353i
\(868\) 0 0
\(869\) −32.0000 + 55.4256i −1.08553 + 1.88019i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.00000 1.73205i 0.0338449 0.0586210i
\(874\) 0 0
\(875\) −24.0000 41.5692i −0.811348 1.40530i
\(876\) 0 0
\(877\) −5.00000 8.66025i −0.168838 0.292436i 0.769174 0.639040i \(-0.220668\pi\)
−0.938012 + 0.346604i \(0.887335\pi\)
\(878\) 0 0
\(879\) −8.00000 −0.269833
\(880\) 0 0
\(881\) −7.00000 + 12.1244i −0.235836 + 0.408480i −0.959515 0.281656i \(-0.909116\pi\)
0.723679 + 0.690136i \(0.242449\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 0 0
\(885\) −32.0000 −1.07567
\(886\) 0 0
\(887\) −24.0000 + 41.5692i −0.805841 + 1.39576i 0.109881 + 0.993945i \(0.464953\pi\)
−0.915722 + 0.401813i \(0.868380\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 2.00000 + 3.46410i 0.0670025 + 0.116052i
\(892\) 0 0
\(893\) −4.00000 6.92820i −0.133855 0.231843i
\(894\) 0 0
\(895\) 24.0000 41.5692i 0.802232 1.38951i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −30.0000 + 51.9615i −1.00056 + 1.73301i
\(900\) 0 0
\(901\) 10.0000 + 17.3205i 0.333148 + 0.577030i
\(902\) 0 0
\(903\) −4.00000 6.92820i −0.133112 0.230556i
\(904\) 0 0
\(905\) −8.00000 −0.265929
\(906\) 0 0
\(907\) 28.0000 48.4974i 0.929725 1.61033i 0.145944 0.989293i \(-0.453378\pi\)
0.783781 0.621038i \(-0.213289\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −56.0000 −1.85130
\(916\) 0 0
\(917\) 4.00000 + 6.92820i 0.132092 + 0.228789i
\(918\) 0 0
\(919\) −28.0000 48.4974i −0.923635 1.59978i −0.793742 0.608254i \(-0.791870\pi\)
−0.129893 0.991528i \(-0.541463\pi\)
\(920\) 0 0
\(921\) 11.0000 19.0526i 0.362462 0.627803i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −55.0000 + 95.2628i −1.80839 + 3.13222i
\(926\) 0 0
\(927\) 4.00000 + 6.92820i 0.131377 + 0.227552i
\(928\) 0 0
\(929\) −8.00000 13.8564i −0.262471 0.454614i 0.704427 0.709777i \(-0.251204\pi\)
−0.966898 + 0.255163i \(0.917871\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 12.0000 20.7846i 0.392862 0.680458i
\(934\) 0 0
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) −3.00000 + 5.19615i −0.0979013 + 0.169570i
\(940\) 0 0
\(941\) 8.00000 0.260793 0.130396 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 4.00000 + 6.92820i 0.130120 + 0.225374i
\(946\) 0 0
\(947\) −4.00000 + 6.92820i −0.129983 + 0.225136i −0.923670 0.383190i \(-0.874825\pi\)
0.793687 + 0.608326i \(0.208159\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 + 10.3923i −0.194563 + 0.336994i
\(952\) 0 0
\(953\) 21.0000 + 36.3731i 0.680257 + 1.17824i 0.974902 + 0.222633i \(0.0714650\pi\)
−0.294646 + 0.955607i \(0.595202\pi\)
\(954\) 0 0
\(955\) −16.0000 27.7128i −0.517748 0.896766i
\(956\) 0 0
\(957\) −24.0000 −0.775810
\(958\) 0 0
\(959\) 8.00000 13.8564i 0.258333 0.447447i
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) −28.0000 + 48.4974i −0.901352 + 1.56119i
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) −2.00000 3.46410i −0.0642493 0.111283i
\(970\) 0 0
\(971\) 14.0000 + 24.2487i 0.449281 + 0.778178i 0.998339 0.0576061i \(-0.0183467\pi\)
−0.549058 + 0.835784i \(0.685013\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.00000 10.3923i 0.191957 0.332479i −0.753942 0.656941i \(-0.771850\pi\)
0.945899 + 0.324462i \(0.105183\pi\)
\(978\) 0 0
\(979\) −8.00000 13.8564i −0.255681 0.442853i
\(980\) 0 0
\(981\) 1.00000 + 1.73205i 0.0319275 + 0.0553001i
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −24.0000 + 41.5692i −0.764704 + 1.32451i
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 18.0000 31.1769i 0.571789 0.990367i −0.424594 0.905384i \(-0.639583\pi\)
0.996382 0.0849833i \(-0.0270837\pi\)
\(992\) 0 0
\(993\) −2.00000 −0.0634681
\(994\) 0 0
\(995\) 8.00000 + 13.8564i 0.253617 + 0.439278i
\(996\) 0 0
\(997\) −11.0000 19.0526i −0.348373 0.603401i 0.637587 0.770378i \(-0.279933\pi\)
−0.985961 + 0.166978i \(0.946599\pi\)
\(998\) 0 0
\(999\) 5.00000 8.66025i 0.158193 0.273998i
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 2028.2.i.e.529.1 2
13.2 odd 12 2028.2.q.h.361.2 4
13.3 even 3 inner 2028.2.i.e.2005.1 2
13.4 even 6 2028.2.a.c.1.1 1
13.5 odd 4 2028.2.q.h.1837.2 4
13.6 odd 12 2028.2.b.a.337.2 2
13.7 odd 12 2028.2.b.a.337.1 2
13.8 odd 4 2028.2.q.h.1837.1 4
13.9 even 3 156.2.a.a.1.1 1
13.10 even 6 2028.2.i.g.2005.1 2
13.11 odd 12 2028.2.q.h.361.1 4
13.12 even 2 2028.2.i.g.529.1 2
39.17 odd 6 6084.2.a.b.1.1 1
39.20 even 12 6084.2.b.j.4393.2 2
39.32 even 12 6084.2.b.j.4393.1 2
39.35 odd 6 468.2.a.d.1.1 1
52.35 odd 6 624.2.a.e.1.1 1
52.43 odd 6 8112.2.a.bi.1.1 1
65.9 even 6 3900.2.a.m.1.1 1
65.22 odd 12 3900.2.h.b.1249.2 2
65.48 odd 12 3900.2.h.b.1249.1 2
91.48 odd 6 7644.2.a.k.1.1 1
104.35 odd 6 2496.2.a.o.1.1 1
104.61 even 6 2496.2.a.bc.1.1 1
117.22 even 3 4212.2.i.l.2809.1 2
117.61 even 3 4212.2.i.l.1405.1 2
117.74 odd 6 4212.2.i.b.1405.1 2
117.113 odd 6 4212.2.i.b.2809.1 2
156.35 even 6 1872.2.a.s.1.1 1
312.35 even 6 7488.2.a.d.1.1 1
312.269 odd 6 7488.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.a.a.1.1 1 13.9 even 3
468.2.a.d.1.1 1 39.35 odd 6
624.2.a.e.1.1 1 52.35 odd 6
1872.2.a.s.1.1 1 156.35 even 6
2028.2.a.c.1.1 1 13.4 even 6
2028.2.b.a.337.1 2 13.7 odd 12
2028.2.b.a.337.2 2 13.6 odd 12
2028.2.i.e.529.1 2 1.1 even 1 trivial
2028.2.i.e.2005.1 2 13.3 even 3 inner
2028.2.i.g.529.1 2 13.12 even 2
2028.2.i.g.2005.1 2 13.10 even 6
2028.2.q.h.361.1 4 13.11 odd 12
2028.2.q.h.361.2 4 13.2 odd 12
2028.2.q.h.1837.1 4 13.8 odd 4
2028.2.q.h.1837.2 4 13.5 odd 4
2496.2.a.o.1.1 1 104.35 odd 6
2496.2.a.bc.1.1 1 104.61 even 6
3900.2.a.m.1.1 1 65.9 even 6
3900.2.h.b.1249.1 2 65.48 odd 12
3900.2.h.b.1249.2 2 65.22 odd 12
4212.2.i.b.1405.1 2 117.74 odd 6
4212.2.i.b.2809.1 2 117.113 odd 6
4212.2.i.l.1405.1 2 117.61 even 3
4212.2.i.l.2809.1 2 117.22 even 3
6084.2.a.b.1.1 1 39.17 odd 6
6084.2.b.j.4393.1 2 39.32 even 12
6084.2.b.j.4393.2 2 39.20 even 12
7488.2.a.c.1.1 1 312.269 odd 6
7488.2.a.d.1.1 1 312.35 even 6
7644.2.a.k.1.1 1 91.48 odd 6
8112.2.a.bi.1.1 1 52.43 odd 6