Properties

Label 624.2.d.g.287.3
Level $624$
Weight $2$
Character 624.287
Analytic conductor $4.983$
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] = [624,2,Mod(287,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.d (of order \(2\), degree \(1\), 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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 287.3
Root \(-1.18614 + 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 624.287
Dual form 624.2.d.g.287.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.18614 - 1.26217i) q^{3} +0.939764i q^{5} +2.52434i q^{7} +(-0.186141 - 2.99422i) q^{9} +O(q^{10})\) \(q+(1.18614 - 1.26217i) q^{3} +0.939764i q^{5} +2.52434i q^{7} +(-0.186141 - 2.99422i) q^{9} +4.00000 q^{11} +1.00000 q^{13} +(1.18614 + 1.11469i) q^{15} -4.10891i q^{17} +3.46410i q^{19} +(3.18614 + 2.99422i) q^{21} +4.74456 q^{23} +4.11684 q^{25} +(-4.00000 - 3.31662i) q^{27} +6.63325i q^{31} +(4.74456 - 5.04868i) q^{33} -2.37228 q^{35} +0.372281 q^{37} +(1.18614 - 1.26217i) q^{39} -5.04868i q^{41} -0.644810i q^{43} +(2.81386 - 0.174928i) q^{45} -6.37228 q^{47} +0.627719 q^{49} +(-5.18614 - 4.87375i) q^{51} -11.9769i q^{53} +3.75906i q^{55} +(4.37228 + 4.10891i) q^{57} -12.0000 q^{59} -11.4891 q^{61} +(7.55842 - 0.469882i) q^{63} +0.939764i q^{65} +3.46410i q^{67} +(5.62772 - 5.98844i) q^{69} +11.1168 q^{71} -10.7446 q^{73} +(4.88316 - 5.19615i) q^{75} +10.0974i q^{77} -1.58457i q^{79} +(-8.93070 + 1.11469i) q^{81} +12.0000 q^{83} +3.86141 q^{85} +5.04868i q^{89} +2.52434i q^{91} +(8.37228 + 7.86797i) q^{93} -3.25544 q^{95} -2.74456 q^{97} +(-0.744563 - 11.9769i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 5 q^{9} + 16 q^{11} + 4 q^{13} - q^{15} + 7 q^{21} - 4 q^{23} - 18 q^{25} - 16 q^{27} - 4 q^{33} + 2 q^{35} - 10 q^{37} - q^{39} + 17 q^{45} - 14 q^{47} + 14 q^{49} - 15 q^{51} + 6 q^{57} - 48 q^{59} + 13 q^{63} + 34 q^{69} + 10 q^{71} - 20 q^{73} + 54 q^{75} - 7 q^{81} + 48 q^{83} - 42 q^{85} + 22 q^{93} - 36 q^{95} + 12 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.18614 1.26217i 0.684819 0.728714i
\(4\) 0 0
\(5\) 0.939764i 0.420275i 0.977672 + 0.210138i \(0.0673912\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 0 0
\(7\) 2.52434i 0.954110i 0.878873 + 0.477055i \(0.158296\pi\)
−0.878873 + 0.477055i \(0.841704\pi\)
\(8\) 0 0
\(9\) −0.186141 2.99422i −0.0620469 0.998073i
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.18614 + 1.11469i 0.306260 + 0.287812i
\(16\) 0 0
\(17\) 4.10891i 0.996557i −0.867017 0.498279i \(-0.833966\pi\)
0.867017 0.498279i \(-0.166034\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 3.18614 + 2.99422i 0.695273 + 0.653392i
\(22\) 0 0
\(23\) 4.74456 0.989310 0.494655 0.869090i \(-0.335294\pi\)
0.494655 + 0.869090i \(0.335294\pi\)
\(24\) 0 0
\(25\) 4.11684 0.823369
\(26\) 0 0
\(27\) −4.00000 3.31662i −0.769800 0.638285i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 6.63325i 1.19137i 0.803219 + 0.595683i \(0.203119\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 0 0
\(33\) 4.74456 5.04868i 0.825922 0.878862i
\(34\) 0 0
\(35\) −2.37228 −0.400989
\(36\) 0 0
\(37\) 0.372281 0.0612027 0.0306013 0.999532i \(-0.490258\pi\)
0.0306013 + 0.999532i \(0.490258\pi\)
\(38\) 0 0
\(39\) 1.18614 1.26217i 0.189935 0.202109i
\(40\) 0 0
\(41\) 5.04868i 0.788471i −0.919010 0.394235i \(-0.871009\pi\)
0.919010 0.394235i \(-0.128991\pi\)
\(42\) 0 0
\(43\) 0.644810i 0.0983326i −0.998791 0.0491663i \(-0.984344\pi\)
0.998791 0.0491663i \(-0.0156564\pi\)
\(44\) 0 0
\(45\) 2.81386 0.174928i 0.419465 0.0260768i
\(46\) 0 0
\(47\) −6.37228 −0.929493 −0.464746 0.885444i \(-0.653854\pi\)
−0.464746 + 0.885444i \(0.653854\pi\)
\(48\) 0 0
\(49\) 0.627719 0.0896741
\(50\) 0 0
\(51\) −5.18614 4.87375i −0.726205 0.682461i
\(52\) 0 0
\(53\) 11.9769i 1.64515i −0.568656 0.822575i \(-0.692536\pi\)
0.568656 0.822575i \(-0.307464\pi\)
\(54\) 0 0
\(55\) 3.75906i 0.506871i
\(56\) 0 0
\(57\) 4.37228 + 4.10891i 0.579123 + 0.544239i
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −11.4891 −1.47103 −0.735516 0.677507i \(-0.763060\pi\)
−0.735516 + 0.677507i \(0.763060\pi\)
\(62\) 0 0
\(63\) 7.55842 0.469882i 0.952272 0.0591996i
\(64\) 0 0
\(65\) 0.939764i 0.116563i
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 5.62772 5.98844i 0.677498 0.720923i
\(70\) 0 0
\(71\) 11.1168 1.31933 0.659663 0.751561i \(-0.270699\pi\)
0.659663 + 0.751561i \(0.270699\pi\)
\(72\) 0 0
\(73\) −10.7446 −1.25756 −0.628778 0.777585i \(-0.716445\pi\)
−0.628778 + 0.777585i \(0.716445\pi\)
\(74\) 0 0
\(75\) 4.88316 5.19615i 0.563858 0.600000i
\(76\) 0 0
\(77\) 10.0974i 1.15070i
\(78\) 0 0
\(79\) 1.58457i 0.178279i −0.996019 0.0891393i \(-0.971588\pi\)
0.996019 0.0891393i \(-0.0284116\pi\)
\(80\) 0 0
\(81\) −8.93070 + 1.11469i −0.992300 + 0.123855i
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 3.86141 0.418828
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.04868i 0.535159i 0.963536 + 0.267579i \(0.0862237\pi\)
−0.963536 + 0.267579i \(0.913776\pi\)
\(90\) 0 0
\(91\) 2.52434i 0.264623i
\(92\) 0 0
\(93\) 8.37228 + 7.86797i 0.868165 + 0.815870i
\(94\) 0 0
\(95\) −3.25544 −0.334001
\(96\) 0 0
\(97\) −2.74456 −0.278668 −0.139334 0.990245i \(-0.544496\pi\)
−0.139334 + 0.990245i \(0.544496\pi\)
\(98\) 0 0
\(99\) −0.744563 11.9769i −0.0748314 1.20372i
\(100\) 0 0
\(101\) 10.0974i 1.00472i 0.864657 + 0.502362i \(0.167536\pi\)
−0.864657 + 0.502362i \(0.832464\pi\)
\(102\) 0 0
\(103\) 8.51278i 0.838789i 0.907804 + 0.419394i \(0.137758\pi\)
−0.907804 + 0.419394i \(0.862242\pi\)
\(104\) 0 0
\(105\) −2.81386 + 2.99422i −0.274605 + 0.292206i
\(106\) 0 0
\(107\) −17.4891 −1.69074 −0.845369 0.534183i \(-0.820619\pi\)
−0.845369 + 0.534183i \(0.820619\pi\)
\(108\) 0 0
\(109\) −12.3723 −1.18505 −0.592525 0.805552i \(-0.701869\pi\)
−0.592525 + 0.805552i \(0.701869\pi\)
\(110\) 0 0
\(111\) 0.441578 0.469882i 0.0419127 0.0445992i
\(112\) 0 0
\(113\) 20.1947i 1.89976i 0.312618 + 0.949879i \(0.398794\pi\)
−0.312618 + 0.949879i \(0.601206\pi\)
\(114\) 0 0
\(115\) 4.45877i 0.415782i
\(116\) 0 0
\(117\) −0.186141 2.99422i −0.0172087 0.276816i
\(118\) 0 0
\(119\) 10.3723 0.950825
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −6.37228 5.98844i −0.574569 0.539959i
\(124\) 0 0
\(125\) 8.56768i 0.766317i
\(126\) 0 0
\(127\) 6.63325i 0.588606i 0.955712 + 0.294303i \(0.0950874\pi\)
−0.955712 + 0.294303i \(0.904913\pi\)
\(128\) 0 0
\(129\) −0.813859 0.764836i −0.0716563 0.0673400i
\(130\) 0 0
\(131\) −7.11684 −0.621802 −0.310901 0.950442i \(-0.600631\pi\)
−0.310901 + 0.950442i \(0.600631\pi\)
\(132\) 0 0
\(133\) −8.74456 −0.758250
\(134\) 0 0
\(135\) 3.11684 3.75906i 0.268255 0.323528i
\(136\) 0 0
\(137\) 15.1460i 1.29401i −0.762485 0.647006i \(-0.776021\pi\)
0.762485 0.647006i \(-0.223979\pi\)
\(138\) 0 0
\(139\) 20.8395i 1.76758i −0.467880 0.883792i \(-0.654982\pi\)
0.467880 0.883792i \(-0.345018\pi\)
\(140\) 0 0
\(141\) −7.55842 + 8.04290i −0.636534 + 0.677334i
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.744563 0.792287i 0.0614105 0.0653467i
\(148\) 0 0
\(149\) 17.0256i 1.39479i −0.716688 0.697394i \(-0.754343\pi\)
0.716688 0.697394i \(-0.245657\pi\)
\(150\) 0 0
\(151\) 5.69349i 0.463329i −0.972796 0.231665i \(-0.925583\pi\)
0.972796 0.231665i \(-0.0744172\pi\)
\(152\) 0 0
\(153\) −12.3030 + 0.764836i −0.994637 + 0.0618333i
\(154\) 0 0
\(155\) −6.23369 −0.500702
\(156\) 0 0
\(157\) 2.74456 0.219040 0.109520 0.993985i \(-0.465069\pi\)
0.109520 + 0.993985i \(0.465069\pi\)
\(158\) 0 0
\(159\) −15.1168 14.2063i −1.19884 1.12663i
\(160\) 0 0
\(161\) 11.9769i 0.943910i
\(162\) 0 0
\(163\) 7.22316i 0.565761i 0.959155 + 0.282881i \(0.0912900\pi\)
−0.959155 + 0.282881i \(0.908710\pi\)
\(164\) 0 0
\(165\) 4.74456 + 4.45877i 0.369364 + 0.347115i
\(166\) 0 0
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 10.3723 0.644810i 0.793188 0.0493099i
\(172\) 0 0
\(173\) 6.33830i 0.481892i −0.970539 0.240946i \(-0.922542\pi\)
0.970539 0.240946i \(-0.0774576\pi\)
\(174\) 0 0
\(175\) 10.3923i 0.785584i
\(176\) 0 0
\(177\) −14.2337 + 15.1460i −1.06987 + 1.13845i
\(178\) 0 0
\(179\) −5.62772 −0.420635 −0.210318 0.977633i \(-0.567450\pi\)
−0.210318 + 0.977633i \(0.567450\pi\)
\(180\) 0 0
\(181\) −24.2337 −1.80128 −0.900638 0.434570i \(-0.856900\pi\)
−0.900638 + 0.434570i \(0.856900\pi\)
\(182\) 0 0
\(183\) −13.6277 + 14.5012i −1.00739 + 1.07196i
\(184\) 0 0
\(185\) 0.349857i 0.0257220i
\(186\) 0 0
\(187\) 16.4356i 1.20189i
\(188\) 0 0
\(189\) 8.37228 10.0974i 0.608994 0.734474i
\(190\) 0 0
\(191\) −1.48913 −0.107749 −0.0538747 0.998548i \(-0.517157\pi\)
−0.0538747 + 0.998548i \(0.517157\pi\)
\(192\) 0 0
\(193\) 22.7446 1.63719 0.818595 0.574372i \(-0.194754\pi\)
0.818595 + 0.574372i \(0.194754\pi\)
\(194\) 0 0
\(195\) 1.18614 + 1.11469i 0.0849413 + 0.0798248i
\(196\) 0 0
\(197\) 0.939764i 0.0669554i 0.999439 + 0.0334777i \(0.0106583\pi\)
−0.999439 + 0.0334777i \(0.989342\pi\)
\(198\) 0 0
\(199\) 16.7306i 1.18600i 0.805202 + 0.593000i \(0.202057\pi\)
−0.805202 + 0.593000i \(0.797943\pi\)
\(200\) 0 0
\(201\) 4.37228 + 4.10891i 0.308397 + 0.289820i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.74456 0.331375
\(206\) 0 0
\(207\) −0.883156 14.2063i −0.0613836 0.987404i
\(208\) 0 0
\(209\) 13.8564i 0.958468i
\(210\) 0 0
\(211\) 6.28339i 0.432567i −0.976331 0.216283i \(-0.930607\pi\)
0.976331 0.216283i \(-0.0693935\pi\)
\(212\) 0 0
\(213\) 13.1861 14.0313i 0.903499 0.961411i
\(214\) 0 0
\(215\) 0.605969 0.0413268
\(216\) 0 0
\(217\) −16.7446 −1.13669
\(218\) 0 0
\(219\) −12.7446 + 13.5615i −0.861198 + 0.916398i
\(220\) 0 0
\(221\) 4.10891i 0.276395i
\(222\) 0 0
\(223\) 3.11425i 0.208545i −0.994549 0.104273i \(-0.966749\pi\)
0.994549 0.104273i \(-0.0332515\pi\)
\(224\) 0 0
\(225\) −0.766312 12.3267i −0.0510875 0.821782i
\(226\) 0 0
\(227\) −8.74456 −0.580397 −0.290199 0.956966i \(-0.593721\pi\)
−0.290199 + 0.956966i \(0.593721\pi\)
\(228\) 0 0
\(229\) −29.8614 −1.97330 −0.986649 0.162863i \(-0.947927\pi\)
−0.986649 + 0.162863i \(0.947927\pi\)
\(230\) 0 0
\(231\) 12.7446 + 11.9769i 0.838531 + 0.788021i
\(232\) 0 0
\(233\) 16.0858i 1.05382i −0.849923 0.526908i \(-0.823351\pi\)
0.849923 0.526908i \(-0.176649\pi\)
\(234\) 0 0
\(235\) 5.98844i 0.390643i
\(236\) 0 0
\(237\) −2.00000 1.87953i −0.129914 0.122088i
\(238\) 0 0
\(239\) −7.86141 −0.508512 −0.254256 0.967137i \(-0.581831\pi\)
−0.254256 + 0.967137i \(0.581831\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) −9.18614 + 12.5942i −0.589291 + 0.807921i
\(244\) 0 0
\(245\) 0.589907i 0.0376878i
\(246\) 0 0
\(247\) 3.46410i 0.220416i
\(248\) 0 0
\(249\) 14.2337 15.1460i 0.902023 0.959840i
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 18.9783 1.19315
\(254\) 0 0
\(255\) 4.58017 4.87375i 0.286821 0.305206i
\(256\) 0 0
\(257\) 0.349857i 0.0218234i 0.999940 + 0.0109117i \(0.00347338\pi\)
−0.999940 + 0.0109117i \(0.996527\pi\)
\(258\) 0 0
\(259\) 0.939764i 0.0583941i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.7446 −0.785863 −0.392932 0.919568i \(-0.628539\pi\)
−0.392932 + 0.919568i \(0.628539\pi\)
\(264\) 0 0
\(265\) 11.2554 0.691416
\(266\) 0 0
\(267\) 6.37228 + 5.98844i 0.389977 + 0.366487i
\(268\) 0 0
\(269\) 11.9769i 0.730243i 0.930960 + 0.365122i \(0.118973\pi\)
−0.930960 + 0.365122i \(0.881027\pi\)
\(270\) 0 0
\(271\) 15.7908i 0.959225i −0.877480 0.479613i \(-0.840777\pi\)
0.877480 0.479613i \(-0.159223\pi\)
\(272\) 0 0
\(273\) 3.18614 + 2.99422i 0.192834 + 0.181218i
\(274\) 0 0
\(275\) 16.4674 0.993020
\(276\) 0 0
\(277\) 24.9783 1.50080 0.750399 0.660985i \(-0.229861\pi\)
0.750399 + 0.660985i \(0.229861\pi\)
\(278\) 0 0
\(279\) 19.8614 1.23472i 1.18907 0.0739206i
\(280\) 0 0
\(281\) 21.4843i 1.28165i 0.767688 + 0.640824i \(0.221407\pi\)
−0.767688 + 0.640824i \(0.778593\pi\)
\(282\) 0 0
\(283\) 14.1514i 0.841211i −0.907244 0.420606i \(-0.861818\pi\)
0.907244 0.420606i \(-0.138182\pi\)
\(284\) 0 0
\(285\) −3.86141 + 4.10891i −0.228730 + 0.243391i
\(286\) 0 0
\(287\) 12.7446 0.752288
\(288\) 0 0
\(289\) 0.116844 0.00687317
\(290\) 0 0
\(291\) −3.25544 + 3.46410i −0.190837 + 0.203069i
\(292\) 0 0
\(293\) 25.5932i 1.49517i −0.664165 0.747586i \(-0.731213\pi\)
0.664165 0.747586i \(-0.268787\pi\)
\(294\) 0 0
\(295\) 11.2772i 0.656582i
\(296\) 0 0
\(297\) −16.0000 13.2665i −0.928414 0.769800i
\(298\) 0 0
\(299\) 4.74456 0.274385
\(300\) 0 0
\(301\) 1.62772 0.0938201
\(302\) 0 0
\(303\) 12.7446 + 11.9769i 0.732156 + 0.688054i
\(304\) 0 0
\(305\) 10.7971i 0.618238i
\(306\) 0 0
\(307\) 19.2000i 1.09580i 0.836543 + 0.547902i \(0.184573\pi\)
−0.836543 + 0.547902i \(0.815427\pi\)
\(308\) 0 0
\(309\) 10.7446 + 10.0974i 0.611237 + 0.574418i
\(310\) 0 0
\(311\) 20.7446 1.17632 0.588158 0.808746i \(-0.299853\pi\)
0.588158 + 0.808746i \(0.299853\pi\)
\(312\) 0 0
\(313\) 20.3723 1.15151 0.575755 0.817622i \(-0.304708\pi\)
0.575755 + 0.817622i \(0.304708\pi\)
\(314\) 0 0
\(315\) 0.441578 + 7.10313i 0.0248801 + 0.400216i
\(316\) 0 0
\(317\) 17.0256i 0.956250i −0.878292 0.478125i \(-0.841317\pi\)
0.878292 0.478125i \(-0.158683\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −20.7446 + 22.0742i −1.15785 + 1.23206i
\(322\) 0 0
\(323\) 14.2337 0.791984
\(324\) 0 0
\(325\) 4.11684 0.228361
\(326\) 0 0
\(327\) −14.6753 + 15.6159i −0.811544 + 0.863562i
\(328\) 0 0
\(329\) 16.0858i 0.886838i
\(330\) 0 0
\(331\) 2.17448i 0.119520i −0.998213 0.0597602i \(-0.980966\pi\)
0.998213 0.0597602i \(-0.0190336\pi\)
\(332\) 0 0
\(333\) −0.0692967 1.11469i −0.00379744 0.0610847i
\(334\) 0 0
\(335\) −3.25544 −0.177864
\(336\) 0 0
\(337\) 2.88316 0.157056 0.0785278 0.996912i \(-0.474978\pi\)
0.0785278 + 0.996912i \(0.474978\pi\)
\(338\) 0 0
\(339\) 25.4891 + 23.9538i 1.38438 + 1.30099i
\(340\) 0 0
\(341\) 26.5330i 1.43684i
\(342\) 0 0
\(343\) 19.2549i 1.03967i
\(344\) 0 0
\(345\) 5.62772 + 5.28873i 0.302986 + 0.284735i
\(346\) 0 0
\(347\) −16.6060 −0.891455 −0.445728 0.895169i \(-0.647055\pi\)
−0.445728 + 0.895169i \(0.647055\pi\)
\(348\) 0 0
\(349\) 14.8832 0.796677 0.398339 0.917238i \(-0.369587\pi\)
0.398339 + 0.917238i \(0.369587\pi\)
\(350\) 0 0
\(351\) −4.00000 3.31662i −0.213504 0.177028i
\(352\) 0 0
\(353\) 33.4612i 1.78096i −0.455022 0.890480i \(-0.650369\pi\)
0.455022 0.890480i \(-0.349631\pi\)
\(354\) 0 0
\(355\) 10.4472i 0.554480i
\(356\) 0 0
\(357\) 12.3030 13.0916i 0.651143 0.692879i
\(358\) 0 0
\(359\) −21.4891 −1.13415 −0.567076 0.823665i \(-0.691926\pi\)
−0.567076 + 0.823665i \(0.691926\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) 5.93070 6.31084i 0.311281 0.331233i
\(364\) 0 0
\(365\) 10.0974i 0.528520i
\(366\) 0 0
\(367\) 33.7562i 1.76206i −0.473063 0.881028i \(-0.656852\pi\)
0.473063 0.881028i \(-0.343148\pi\)
\(368\) 0 0
\(369\) −15.1168 + 0.939764i −0.786951 + 0.0489222i
\(370\) 0 0
\(371\) 30.2337 1.56965
\(372\) 0 0
\(373\) 1.25544 0.0650041 0.0325020 0.999472i \(-0.489652\pi\)
0.0325020 + 0.999472i \(0.489652\pi\)
\(374\) 0 0
\(375\) 10.8139 + 10.1625i 0.558425 + 0.524788i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 17.3205i 0.889695i 0.895606 + 0.444847i \(0.146742\pi\)
−0.895606 + 0.444847i \(0.853258\pi\)
\(380\) 0 0
\(381\) 8.37228 + 7.86797i 0.428925 + 0.403088i
\(382\) 0 0
\(383\) −4.88316 −0.249518 −0.124759 0.992187i \(-0.539816\pi\)
−0.124759 + 0.992187i \(0.539816\pi\)
\(384\) 0 0
\(385\) −9.48913 −0.483611
\(386\) 0 0
\(387\) −1.93070 + 0.120025i −0.0981432 + 0.00610123i
\(388\) 0 0
\(389\) 37.8102i 1.91705i 0.285006 + 0.958526i \(0.408004\pi\)
−0.285006 + 0.958526i \(0.591996\pi\)
\(390\) 0 0
\(391\) 19.4950i 0.985904i
\(392\) 0 0
\(393\) −8.44158 + 8.98266i −0.425821 + 0.453115i
\(394\) 0 0
\(395\) 1.48913 0.0749260
\(396\) 0 0
\(397\) 4.51087 0.226394 0.113197 0.993573i \(-0.463891\pi\)
0.113197 + 0.993573i \(0.463891\pi\)
\(398\) 0 0
\(399\) −10.3723 + 11.0371i −0.519264 + 0.552547i
\(400\) 0 0
\(401\) 18.9051i 0.944075i −0.881579 0.472037i \(-0.843519\pi\)
0.881579 0.472037i \(-0.156481\pi\)
\(402\) 0 0
\(403\) 6.63325i 0.330426i
\(404\) 0 0
\(405\) −1.04755 8.39275i −0.0520530 0.417039i
\(406\) 0 0
\(407\) 1.48913 0.0738132
\(408\) 0 0
\(409\) 12.9783 0.641733 0.320867 0.947124i \(-0.396026\pi\)
0.320867 + 0.947124i \(0.396026\pi\)
\(410\) 0 0
\(411\) −19.1168 17.9653i −0.942964 0.886164i
\(412\) 0 0
\(413\) 30.2921i 1.49057i
\(414\) 0 0
\(415\) 11.2772i 0.553574i
\(416\) 0 0
\(417\) −26.3030 24.7186i −1.28806 1.21047i
\(418\) 0 0
\(419\) 27.8614 1.36112 0.680559 0.732693i \(-0.261737\pi\)
0.680559 + 0.732693i \(0.261737\pi\)
\(420\) 0 0
\(421\) −10.8832 −0.530413 −0.265206 0.964192i \(-0.585440\pi\)
−0.265206 + 0.964192i \(0.585440\pi\)
\(422\) 0 0
\(423\) 1.18614 + 19.0800i 0.0576721 + 0.927702i
\(424\) 0 0
\(425\) 16.9157i 0.820534i
\(426\) 0 0
\(427\) 29.0024i 1.40353i
\(428\) 0 0
\(429\) 4.74456 5.04868i 0.229070 0.243752i
\(430\) 0 0
\(431\) −33.3505 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(432\) 0 0
\(433\) 31.3505 1.50661 0.753305 0.657671i \(-0.228458\pi\)
0.753305 + 0.657671i \(0.228458\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.4356i 0.786224i
\(438\) 0 0
\(439\) 28.1176i 1.34198i −0.741467 0.670989i \(-0.765870\pi\)
0.741467 0.670989i \(-0.234130\pi\)
\(440\) 0 0
\(441\) −0.116844 1.87953i −0.00556400 0.0895013i
\(442\) 0 0
\(443\) 2.37228 0.112710 0.0563552 0.998411i \(-0.482052\pi\)
0.0563552 + 0.998411i \(0.482052\pi\)
\(444\) 0 0
\(445\) −4.74456 −0.224914
\(446\) 0 0
\(447\) −21.4891 20.1947i −1.01640 0.955177i
\(448\) 0 0
\(449\) 6.92820i 0.326962i −0.986546 0.163481i \(-0.947728\pi\)
0.986546 0.163481i \(-0.0522723\pi\)
\(450\) 0 0
\(451\) 20.1947i 0.950931i
\(452\) 0 0
\(453\) −7.18614 6.75327i −0.337634 0.317297i
\(454\) 0 0
\(455\) −2.37228 −0.111214
\(456\) 0 0
\(457\) 28.9783 1.35555 0.677773 0.735271i \(-0.262945\pi\)
0.677773 + 0.735271i \(0.262945\pi\)
\(458\) 0 0
\(459\) −13.6277 + 16.4356i −0.636087 + 0.767150i
\(460\) 0 0
\(461\) 17.3754i 0.809254i 0.914482 + 0.404627i \(0.132599\pi\)
−0.914482 + 0.404627i \(0.867401\pi\)
\(462\) 0 0
\(463\) 37.5152i 1.74348i −0.489969 0.871740i \(-0.662992\pi\)
0.489969 0.871740i \(-0.337008\pi\)
\(464\) 0 0
\(465\) −7.39403 + 7.86797i −0.342890 + 0.364868i
\(466\) 0 0
\(467\) −25.4891 −1.17950 −0.589748 0.807587i \(-0.700773\pi\)
−0.589748 + 0.807587i \(0.700773\pi\)
\(468\) 0 0
\(469\) −8.74456 −0.403786
\(470\) 0 0
\(471\) 3.25544 3.46410i 0.150003 0.159617i
\(472\) 0 0
\(473\) 2.57924i 0.118594i
\(474\) 0 0
\(475\) 14.2612i 0.654347i
\(476\) 0 0
\(477\) −35.8614 + 2.22938i −1.64198 + 0.102076i
\(478\) 0 0
\(479\) −27.1168 −1.23900 −0.619500 0.784997i \(-0.712665\pi\)
−0.619500 + 0.784997i \(0.712665\pi\)
\(480\) 0 0
\(481\) 0.372281 0.0169746
\(482\) 0 0
\(483\) 15.1168 + 14.2063i 0.687840 + 0.646407i
\(484\) 0 0
\(485\) 2.57924i 0.117117i
\(486\) 0 0
\(487\) 40.6844i 1.84358i 0.387684 + 0.921792i \(0.373275\pi\)
−0.387684 + 0.921792i \(0.626725\pi\)
\(488\) 0 0
\(489\) 9.11684 + 8.56768i 0.412278 + 0.387444i
\(490\) 0 0
\(491\) −29.3505 −1.32457 −0.662285 0.749252i \(-0.730413\pi\)
−0.662285 + 0.749252i \(0.730413\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 11.2554 0.699713i 0.505894 0.0314498i
\(496\) 0 0
\(497\) 28.0627i 1.25878i
\(498\) 0 0
\(499\) 23.6588i 1.05911i 0.848275 + 0.529557i \(0.177642\pi\)
−0.848275 + 0.529557i \(0.822358\pi\)
\(500\) 0 0
\(501\) 4.74456 5.04868i 0.211971 0.225558i
\(502\) 0 0
\(503\) −6.23369 −0.277946 −0.138973 0.990296i \(-0.544380\pi\)
−0.138973 + 0.990296i \(0.544380\pi\)
\(504\) 0 0
\(505\) −9.48913 −0.422261
\(506\) 0 0
\(507\) 1.18614 1.26217i 0.0526784 0.0560549i
\(508\) 0 0
\(509\) 30.8820i 1.36882i 0.729098 + 0.684409i \(0.239940\pi\)
−0.729098 + 0.684409i \(0.760060\pi\)
\(510\) 0 0
\(511\) 27.1229i 1.19985i
\(512\) 0 0
\(513\) 11.4891 13.8564i 0.507257 0.611775i
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) −25.4891 −1.12101
\(518\) 0 0
\(519\) −8.00000 7.51811i −0.351161 0.330008i
\(520\) 0 0
\(521\) 33.7013i 1.47648i 0.674539 + 0.738239i \(0.264342\pi\)
−0.674539 + 0.738239i \(0.735658\pi\)
\(522\) 0 0
\(523\) 29.9971i 1.31168i 0.754899 + 0.655841i \(0.227686\pi\)
−0.754899 + 0.655841i \(0.772314\pi\)
\(524\) 0 0
\(525\) 13.1168 + 12.3267i 0.572466 + 0.537983i
\(526\) 0 0
\(527\) 27.2554 1.18727
\(528\) 0 0
\(529\) −0.489125 −0.0212663
\(530\) 0 0
\(531\) 2.23369 + 35.9306i 0.0969338 + 1.55926i
\(532\) 0 0
\(533\) 5.04868i 0.218682i
\(534\) 0 0
\(535\) 16.4356i 0.710575i
\(536\) 0 0
\(537\) −6.67527 + 7.10313i −0.288059 + 0.306523i
\(538\) 0 0
\(539\) 2.51087 0.108151
\(540\) 0 0
\(541\) −28.3723 −1.21982 −0.609910 0.792471i \(-0.708794\pi\)
−0.609910 + 0.792471i \(0.708794\pi\)
\(542\) 0 0
\(543\) −28.7446 + 30.5870i −1.23355 + 1.31261i
\(544\) 0 0
\(545\) 11.6270i 0.498047i
\(546\) 0 0
\(547\) 33.9962i 1.45357i −0.686864 0.726786i \(-0.741013\pi\)
0.686864 0.726786i \(-0.258987\pi\)
\(548\) 0 0
\(549\) 2.13859 + 34.4010i 0.0912729 + 1.46820i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 0.441578 + 0.414979i 0.0187439 + 0.0176149i
\(556\) 0 0
\(557\) 20.4348i 0.865848i −0.901430 0.432924i \(-0.857482\pi\)
0.901430 0.432924i \(-0.142518\pi\)
\(558\) 0 0
\(559\) 0.644810i 0.0272726i
\(560\) 0 0
\(561\) −20.7446 19.4950i −0.875836 0.823079i
\(562\) 0 0
\(563\) 34.0951 1.43694 0.718468 0.695560i \(-0.244843\pi\)
0.718468 + 0.695560i \(0.244843\pi\)
\(564\) 0 0
\(565\) −18.9783 −0.798421
\(566\) 0 0
\(567\) −2.81386 22.5441i −0.118171 0.946764i
\(568\) 0 0
\(569\) 0.349857i 0.0146667i 0.999973 + 0.00733337i \(0.00233431\pi\)
−0.999973 + 0.00733337i \(0.997666\pi\)
\(570\) 0 0
\(571\) 20.8395i 0.872106i −0.899921 0.436053i \(-0.856376\pi\)
0.899921 0.436053i \(-0.143624\pi\)
\(572\) 0 0
\(573\) −1.76631 + 1.87953i −0.0737887 + 0.0785184i
\(574\) 0 0
\(575\) 19.5326 0.814567
\(576\) 0 0
\(577\) −2.74456 −0.114258 −0.0571288 0.998367i \(-0.518195\pi\)
−0.0571288 + 0.998367i \(0.518195\pi\)
\(578\) 0 0
\(579\) 26.9783 28.7075i 1.12118 1.19304i
\(580\) 0 0
\(581\) 30.2921i 1.25673i
\(582\) 0 0
\(583\) 47.9075i 1.98413i
\(584\) 0 0
\(585\) 2.81386 0.174928i 0.116339 0.00723239i
\(586\) 0 0
\(587\) 0.744563 0.0307314 0.0153657 0.999882i \(-0.495109\pi\)
0.0153657 + 0.999882i \(0.495109\pi\)
\(588\) 0 0
\(589\) −22.9783 −0.946802
\(590\) 0 0
\(591\) 1.18614 + 1.11469i 0.0487913 + 0.0458523i
\(592\) 0 0
\(593\) 37.2203i 1.52845i 0.644948 + 0.764226i \(0.276879\pi\)
−0.644948 + 0.764226i \(0.723121\pi\)
\(594\) 0 0
\(595\) 9.74749i 0.399608i
\(596\) 0 0
\(597\) 21.1168 + 19.8448i 0.864255 + 0.812195i
\(598\) 0 0
\(599\) −1.48913 −0.0608440 −0.0304220 0.999537i \(-0.509685\pi\)
−0.0304220 + 0.999537i \(0.509685\pi\)
\(600\) 0 0
\(601\) −44.8397 −1.82905 −0.914524 0.404532i \(-0.867434\pi\)
−0.914524 + 0.404532i \(0.867434\pi\)
\(602\) 0 0
\(603\) 10.3723 0.644810i 0.422392 0.0262587i
\(604\) 0 0
\(605\) 4.69882i 0.191034i
\(606\) 0 0
\(607\) 25.5383i 1.03657i −0.855208 0.518284i \(-0.826571\pi\)
0.855208 0.518284i \(-0.173429\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6.37228 −0.257795
\(612\) 0 0
\(613\) 15.4891 0.625600 0.312800 0.949819i \(-0.398733\pi\)
0.312800 + 0.949819i \(0.398733\pi\)
\(614\) 0 0
\(615\) 5.62772 5.98844i 0.226932 0.241477i
\(616\) 0 0
\(617\) 30.8820i 1.24326i 0.783311 + 0.621630i \(0.213529\pi\)
−0.783311 + 0.621630i \(0.786471\pi\)
\(618\) 0 0
\(619\) 33.7562i 1.35677i 0.734705 + 0.678387i \(0.237321\pi\)
−0.734705 + 0.678387i \(0.762679\pi\)
\(620\) 0 0
\(621\) −18.9783 15.7359i −0.761571 0.631461i
\(622\) 0 0
\(623\) −12.7446 −0.510600
\(624\) 0 0
\(625\) 12.5326 0.501305
\(626\) 0 0
\(627\) 17.4891 + 16.4356i 0.698448 + 0.656377i
\(628\) 0 0
\(629\) 1.52967i 0.0609920i
\(630\) 0 0
\(631\) 10.7422i 0.427639i 0.976873 + 0.213819i \(0.0685904\pi\)
−0.976873 + 0.213819i \(0.931410\pi\)
\(632\) 0 0
\(633\) −7.93070 7.45299i −0.315217 0.296230i
\(634\) 0 0
\(635\) −6.23369 −0.247376
\(636\) 0 0
\(637\) 0.627719 0.0248711
\(638\) 0 0
\(639\) −2.06930 33.2863i −0.0818601 1.31678i
\(640\) 0 0
\(641\) 37.8102i 1.49341i −0.665154 0.746706i \(-0.731634\pi\)
0.665154 0.746706i \(-0.268366\pi\)
\(642\) 0 0
\(643\) 9.21249i 0.363305i −0.983363 0.181653i \(-0.941855\pi\)
0.983363 0.181653i \(-0.0581446\pi\)
\(644\) 0 0
\(645\) 0.718765 0.764836i 0.0283013 0.0301154i
\(646\) 0 0
\(647\) 34.9783 1.37514 0.687568 0.726120i \(-0.258678\pi\)
0.687568 + 0.726120i \(0.258678\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) −19.8614 + 21.1345i −0.778430 + 0.828325i
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 6.68815i 0.261328i
\(656\) 0 0
\(657\) 2.00000 + 32.1716i 0.0780274 + 1.25513i
\(658\) 0 0
\(659\) 17.4891 0.681280 0.340640 0.940194i \(-0.389356\pi\)
0.340640 + 0.940194i \(0.389356\pi\)
\(660\) 0 0
\(661\) −0.510875 −0.0198707 −0.00993536 0.999951i \(-0.503163\pi\)
−0.00993536 + 0.999951i \(0.503163\pi\)
\(662\) 0 0
\(663\) −5.18614 4.87375i −0.201413 0.189281i
\(664\) 0 0
\(665\) 8.21782i 0.318674i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.93070 3.69393i −0.151970 0.142816i
\(670\) 0 0
\(671\) −45.9565 −1.77413
\(672\) 0 0
\(673\) 33.1168 1.27656 0.638280 0.769804i \(-0.279646\pi\)
0.638280 + 0.769804i \(0.279646\pi\)
\(674\) 0 0
\(675\) −16.4674 13.6540i −0.633830 0.525544i
\(676\) 0 0
\(677\) 7.51811i 0.288944i −0.989509 0.144472i \(-0.953852\pi\)
0.989509 0.144472i \(-0.0461484\pi\)
\(678\) 0 0
\(679\) 6.92820i 0.265880i
\(680\) 0 0
\(681\) −10.3723 + 11.0371i −0.397467 + 0.422943i
\(682\) 0 0
\(683\) 22.9783 0.879238 0.439619 0.898184i \(-0.355113\pi\)
0.439619 + 0.898184i \(0.355113\pi\)
\(684\) 0 0
\(685\) 14.2337 0.543841
\(686\) 0 0
\(687\) −35.4198 + 37.6901i −1.35135 + 1.43797i
\(688\) 0 0
\(689\) 11.9769i 0.456283i
\(690\) 0 0
\(691\) 15.4410i 0.587403i 0.955897 + 0.293701i \(0.0948871\pi\)
−0.955897 + 0.293701i \(0.905113\pi\)
\(692\) 0 0
\(693\) 30.2337 1.87953i 1.14848 0.0713974i
\(694\) 0 0
\(695\) 19.5842 0.742872
\(696\) 0 0
\(697\) −20.7446 −0.785756
\(698\) 0 0
\(699\) −20.3030 19.0800i −0.767929 0.721672i
\(700\) 0 0
\(701\) 9.39764i 0.354944i −0.984126 0.177472i \(-0.943208\pi\)
0.984126 0.177472i \(-0.0567919\pi\)
\(702\) 0 0
\(703\) 1.28962i 0.0486390i
\(704\) 0 0
\(705\) −7.55842 7.10313i −0.284667 0.267519i
\(706\) 0 0
\(707\) −25.4891 −0.958617
\(708\) 0 0
\(709\) 24.9783 0.938078 0.469039 0.883177i \(-0.344600\pi\)
0.469039 + 0.883177i \(0.344600\pi\)
\(710\) 0 0
\(711\) −4.74456 + 0.294954i −0.177935 + 0.0110616i
\(712\) 0 0
\(713\) 31.4719i 1.17863i
\(714\) 0 0
\(715\) 3.75906i 0.140581i
\(716\) 0 0
\(717\) −9.32473 + 9.92242i −0.348239 + 0.370560i
\(718\) 0 0
\(719\) −7.72281 −0.288012 −0.144006 0.989577i \(-0.545999\pi\)
−0.144006 + 0.989577i \(0.545999\pi\)
\(720\) 0 0
\(721\) −21.4891 −0.800297
\(722\) 0 0
\(723\) 21.3505 22.7190i 0.794035 0.844930i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 41.3841i 1.53485i 0.641139 + 0.767425i \(0.278462\pi\)
−0.641139 + 0.767425i \(0.721538\pi\)
\(728\) 0 0
\(729\) 5.00000 + 26.5330i 0.185185 + 0.982704i
\(730\) 0 0
\(731\) −2.64947 −0.0979941
\(732\) 0 0
\(733\) 11.3505 0.419241 0.209621 0.977783i \(-0.432777\pi\)
0.209621 + 0.977783i \(0.432777\pi\)
\(734\) 0 0
\(735\) 0.744563 + 0.699713i 0.0274636 + 0.0258093i
\(736\) 0 0
\(737\) 13.8564i 0.510407i
\(738\) 0 0
\(739\) 19.8997i 0.732024i 0.930610 + 0.366012i \(0.119277\pi\)
−0.930610 + 0.366012i \(0.880723\pi\)
\(740\) 0 0
\(741\) 4.37228 + 4.10891i 0.160620 + 0.150945i
\(742\) 0 0
\(743\) 1.62772 0.0597152 0.0298576 0.999554i \(-0.490495\pi\)
0.0298576 + 0.999554i \(0.490495\pi\)
\(744\) 0 0
\(745\) 16.0000 0.586195
\(746\) 0 0
\(747\) −2.23369 35.9306i −0.0817264 1.31463i
\(748\) 0 0
\(749\) 44.1485i 1.61315i
\(750\) 0 0
\(751\) 0.884861i 0.0322890i −0.999870 0.0161445i \(-0.994861\pi\)
0.999870 0.0161445i \(-0.00513918\pi\)
\(752\) 0 0
\(753\) 18.9783 20.1947i 0.691606 0.735936i
\(754\) 0 0
\(755\) 5.35053 0.194726
\(756\) 0 0
\(757\) −28.9783 −1.05323 −0.526616 0.850103i \(-0.676539\pi\)
−0.526616 + 0.850103i \(0.676539\pi\)
\(758\) 0 0
\(759\) 22.5109 23.9538i 0.817093 0.869466i
\(760\) 0 0
\(761\) 40.9793i 1.48550i 0.669569 + 0.742749i \(0.266479\pi\)
−0.669569 + 0.742749i \(0.733521\pi\)
\(762\) 0 0
\(763\) 31.2318i 1.13067i
\(764\) 0 0
\(765\) −0.718765 11.5619i −0.0259870 0.418021i
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 9.72281 0.350614 0.175307 0.984514i \(-0.443908\pi\)
0.175307 + 0.984514i \(0.443908\pi\)
\(770\) 0 0
\(771\) 0.441578 + 0.414979i 0.0159030 + 0.0149451i
\(772\) 0 0
\(773\) 31.2318i 1.12333i −0.827365 0.561665i \(-0.810161\pi\)
0.827365 0.561665i \(-0.189839\pi\)
\(774\) 0 0
\(775\) 27.3081i 0.980934i
\(776\) 0 0
\(777\) 1.18614 + 1.11469i 0.0425526 + 0.0399894i
\(778\) 0 0
\(779\) 17.4891 0.626613
\(780\) 0 0
\(781\) 44.4674 1.59117
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.57924i 0.0920570i
\(786\) 0 0
\(787\) 27.4179i 0.977341i 0.872468 + 0.488671i \(0.162518\pi\)
−0.872468 + 0.488671i \(0.837482\pi\)
\(788\) 0 0
\(789\) −15.1168 + 16.0858i −0.538174 + 0.572669i
\(790\) 0 0
\(791\) −50.9783 −1.81258
\(792\) 0 0
\(793\) −11.4891 −0.407991
\(794\) 0 0
\(795\) 13.3505 14.2063i 0.473495 0.503844i
\(796\) 0 0
\(797\) 25.8333i 0.915062i 0.889194 + 0.457531i \(0.151266\pi\)
−0.889194 + 0.457531i \(0.848734\pi\)
\(798\) 0 0
\(799\) 26.1831i 0.926293i
\(800\) 0 0
\(801\) 15.1168 0.939764i 0.534127 0.0332049i
\(802\) 0 0
\(803\) −42.9783 −1.51667
\(804\) 0 0
\(805\) −11.2554 −0.396702
\(806\) 0 0
\(807\) 15.1168 + 14.2063i 0.532138 + 0.500084i
\(808\) 0 0
\(809\) 26.1831i 0.920550i −0.887776 0.460275i \(-0.847751\pi\)
0.887776 0.460275i \(-0.152249\pi\)
\(810\) 0 0
\(811\) 33.0564i 1.16077i 0.814343 + 0.580384i \(0.197098\pi\)
−0.814343 + 0.580384i \(0.802902\pi\)
\(812\) 0 0
\(813\) −19.9307 18.7302i −0.699000 0.656895i
\(814\) 0 0
\(815\) −6.78806 −0.237775
\(816\) 0 0
\(817\) 2.23369 0.0781468
\(818\) 0 0
\(819\) 7.55842 0.469882i 0.264113 0.0164190i
\(820\) 0 0
\(821\) 50.7268i 1.77038i −0.465232 0.885189i \(-0.654029\pi\)
0.465232 0.885189i \(-0.345971\pi\)
\(822\) 0 0
\(823\) 12.2718i 0.427769i 0.976859 + 0.213885i \(0.0686117\pi\)
−0.976859 + 0.213885i \(0.931388\pi\)
\(824\) 0 0
\(825\) 19.5326 20.7846i 0.680039 0.723627i
\(826\) 0 0
\(827\) 40.7446 1.41683 0.708414 0.705798i \(-0.249411\pi\)
0.708414 + 0.705798i \(0.249411\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 29.6277 31.5268i 1.02777 1.09365i
\(832\) 0 0
\(833\) 2.57924i 0.0893654i
\(834\) 0 0
\(835\) 3.75906i 0.130087i
\(836\) 0 0
\(837\) 22.0000 26.5330i 0.760431 0.917115i
\(838\) 0 0
\(839\) 1.02175 0.0352747 0.0176374 0.999844i \(-0.494386\pi\)
0.0176374 + 0.999844i \(0.494386\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 27.1168 + 25.4834i 0.933954 + 0.877696i
\(844\) 0 0
\(845\) 0.939764i 0.0323289i
\(846\) 0 0
\(847\) 12.6217i 0.433686i
\(848\) 0 0
\(849\) −17.8614 16.7855i −0.613002 0.576077i
\(850\) 0 0
\(851\) 1.76631 0.0605484
\(852\) 0 0
\(853\) 14.6060 0.500099 0.250049 0.968233i \(-0.419553\pi\)
0.250049 + 0.968233i \(0.419553\pi\)
\(854\) 0 0
\(855\) 0.605969 + 9.74749i 0.0207237 + 0.333357i
\(856\) 0 0
\(857\) 10.0974i 0.344919i 0.985017 + 0.172459i \(0.0551714\pi\)
−0.985017 + 0.172459i \(0.944829\pi\)
\(858\) 0 0
\(859\) 26.8280i 0.915358i −0.889118 0.457679i \(-0.848681\pi\)
0.889118 0.457679i \(-0.151319\pi\)
\(860\) 0 0
\(861\) 15.1168 16.0858i 0.515181 0.548202i
\(862\) 0 0
\(863\) 9.35053 0.318296 0.159148 0.987255i \(-0.449125\pi\)
0.159148 + 0.987255i \(0.449125\pi\)
\(864\) 0 0
\(865\) 5.95650 0.202527
\(866\) 0 0
\(867\) 0.138593 0.147477i 0.00470688 0.00500858i
\(868\) 0 0
\(869\) 6.33830i 0.215012i
\(870\) 0 0
\(871\) 3.46410i 0.117377i
\(872\) 0 0
\(873\) 0.510875 + 8.21782i 0.0172905 + 0.278131i
\(874\) 0 0
\(875\) −21.6277 −0.731150
\(876\) 0 0
\(877\) −1.11684 −0.0377131 −0.0188566 0.999822i \(-0.506003\pi\)
−0.0188566 + 0.999822i \(0.506003\pi\)
\(878\) 0 0
\(879\) −32.3030 30.3572i −1.08955 1.02392i
\(880\) 0 0
\(881\) 11.6270i 0.391724i 0.980631 + 0.195862i \(0.0627505\pi\)
−0.980631 + 0.195862i \(0.937249\pi\)
\(882\) 0 0
\(883\) 21.4294i 0.721157i 0.932729 + 0.360579i \(0.117421\pi\)
−0.932729 + 0.360579i \(0.882579\pi\)
\(884\) 0 0
\(885\) −14.2337 13.3763i −0.478460 0.449640i
\(886\) 0 0
\(887\) −38.2337 −1.28376 −0.641881 0.766804i \(-0.721846\pi\)
−0.641881 + 0.766804i \(0.721846\pi\)
\(888\) 0 0
\(889\) −16.7446 −0.561595
\(890\) 0 0
\(891\) −35.7228 + 4.45877i −1.19676 + 0.149374i
\(892\) 0 0
\(893\) 22.0742i 0.738686i
\(894\) 0 0
\(895\) 5.28873i 0.176783i
\(896\) 0 0
\(897\) 5.62772 5.98844i 0.187904 0.199948i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −49.2119 −1.63949
\(902\) 0 0
\(903\) 1.93070 2.05446i 0.0642498 0.0683680i
\(904\) 0 0
\(905\) 22.7739i 0.757031i
\(906\) 0 0
\(907\) 16.3807i 0.543914i −0.962309 0.271957i \(-0.912329\pi\)
0.962309 0.271957i \(-0.0876708\pi\)
\(908\) 0 0
\(909\) 30.2337 1.87953i 1.00279 0.0623400i
\(910\) 0 0
\(911\) 10.9783 0.363726 0.181863 0.983324i \(-0.441787\pi\)
0.181863 + 0.983324i \(0.441787\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 0 0
\(915\) −13.6277 12.8068i −0.450518 0.423381i
\(916\) 0 0
\(917\) 17.9653i 0.593267i
\(918\) 0 0
\(919\) 56.4203i 1.86113i 0.366122 + 0.930567i \(0.380685\pi\)
−0.366122 + 0.930567i \(0.619315\pi\)
\(920\) 0 0
\(921\) 24.2337 + 22.7739i 0.798527 + 0.750427i
\(922\) 0 0
\(923\) 11.1168 0.365915
\(924\) 0 0
\(925\) 1.53262 0.0503924
\(926\) 0 0
\(927\) 25.4891 1.58457i 0.837173 0.0520442i
\(928\) 0 0
\(929\) 40.9793i 1.34449i −0.740330 0.672244i \(-0.765331\pi\)
0.740330 0.672244i \(-0.234669\pi\)
\(930\) 0 0
\(931\) 2.17448i 0.0712657i
\(932\) 0 0
\(933\) 24.6060 26.1831i 0.805563 0.857198i
\(934\) 0 0
\(935\) 15.4456 0.505126
\(936\) 0 0
\(937\) −12.5109 −0.408712 −0.204356 0.978897i \(-0.565510\pi\)
−0.204356 + 0.978897i \(0.565510\pi\)
\(938\) 0 0
\(939\) 24.1644 25.7133i 0.788575 0.839121i
\(940\) 0 0
\(941\) 7.27806i 0.237258i −0.992939 0.118629i \(-0.962150\pi\)
0.992939 0.118629i \(-0.0378499\pi\)
\(942\) 0 0
\(943\) 23.9538i 0.780042i
\(944\) 0 0
\(945\) 9.48913 + 7.86797i 0.308681 + 0.255945i
\(946\) 0 0
\(947\) 22.9783 0.746693 0.373346 0.927692i \(-0.378210\pi\)
0.373346 + 0.927692i \(0.378210\pi\)
\(948\) 0 0
\(949\) −10.7446 −0.348783
\(950\) 0 0
\(951\) −21.4891 20.1947i −0.696833 0.654858i
\(952\) 0 0
\(953\) 3.40920i 0.110435i 0.998474 + 0.0552174i \(0.0175852\pi\)
−0.998474 + 0.0552174i \(0.982415\pi\)
\(954\) 0 0
\(955\) 1.39943i 0.0452844i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 38.2337 1.23463
\(960\) 0 0
\(961\) −13.0000 −0.419355
\(962\) 0 0
\(963\) 3.25544 + 52.3663i 0.104905 + 1.68748i
\(964\) 0 0
\(965\) 21.3745i 0.688070i
\(966\) 0 0
\(967\) 14.5012i 0.466328i 0.972437 + 0.233164i \(0.0749078\pi\)
−0.972437 + 0.233164i \(0.925092\pi\)
\(968\) 0 0
\(969\) 16.8832 17.9653i 0.542365 0.577129i
\(970\) 0 0
\(971\) −11.8614 −0.380651 −0.190325 0.981721i \(-0.560954\pi\)
−0.190325 + 0.981721i \(0.560954\pi\)
\(972\) 0 0
\(973\) 52.6060 1.68647
\(974\) 0 0
\(975\) 4.88316 5.19615i 0.156386 0.166410i
\(976\) 0 0
\(977\) 22.6641i 0.725090i 0.931966 + 0.362545i \(0.118092\pi\)
−0.931966 + 0.362545i \(0.881908\pi\)
\(978\) 0 0
\(979\) 20.1947i 0.645425i
\(980\) 0 0
\(981\) 2.30298 + 37.0453i 0.0735286 + 1.18277i
\(982\) 0 0
\(983\) 51.1168 1.63037 0.815187 0.579198i \(-0.196634\pi\)
0.815187 + 0.579198i \(0.196634\pi\)
\(984\) 0 0
\(985\) −0.883156 −0.0281397
\(986\) 0 0
\(987\) −20.3030 19.0800i −0.646251 0.607323i
\(988\) 0 0
\(989\) 3.05934i 0.0972814i
\(990\) 0 0
\(991\) 6.04334i 0.191973i −0.995383 0.0959865i \(-0.969399\pi\)
0.995383 0.0959865i \(-0.0306006\pi\)
\(992\) 0 0
\(993\) −2.74456 2.57924i −0.0870961 0.0818497i
\(994\) 0 0
\(995\) −15.7228 −0.498447
\(996\) 0 0
\(997\) 42.7446 1.35373 0.676867 0.736105i \(-0.263337\pi\)
0.676867 + 0.736105i \(0.263337\pi\)
\(998\) 0 0
\(999\) −1.48913 1.23472i −0.0471138 0.0390647i
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 624.2.d.g.287.3 4
3.2 odd 2 624.2.d.i.287.1 yes 4
4.3 odd 2 624.2.d.i.287.2 yes 4
8.3 odd 2 2496.2.d.i.1535.3 4
8.5 even 2 2496.2.d.l.1535.2 4
12.11 even 2 inner 624.2.d.g.287.4 yes 4
24.5 odd 2 2496.2.d.i.1535.4 4
24.11 even 2 2496.2.d.l.1535.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
624.2.d.g.287.3 4 1.1 even 1 trivial
624.2.d.g.287.4 yes 4 12.11 even 2 inner
624.2.d.i.287.1 yes 4 3.2 odd 2
624.2.d.i.287.2 yes 4 4.3 odd 2
2496.2.d.i.1535.3 4 8.3 odd 2
2496.2.d.i.1535.4 4 24.5 odd 2
2496.2.d.l.1535.1 4 24.11 even 2
2496.2.d.l.1535.2 4 8.5 even 2