Properties

Label 672.2.h.e.575.4
Level $672$
Weight $2$
Character 672.575
Analytic conductor $5.366$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(575,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, 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("672.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.h (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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.4
Root \(1.14412 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 672.575
Dual form 672.2.h.e.575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 1.58114i) q^{3} +1.41421i q^{5} +1.00000i q^{7} +(-2.00000 - 2.23607i) q^{9} +O(q^{10})\) \(q+(-0.707107 + 1.58114i) q^{3} +1.41421i q^{5} +1.00000i q^{7} +(-2.00000 - 2.23607i) q^{9} -4.47214 q^{11} -0.837722 q^{13} +(-2.23607 - 1.00000i) q^{15} -1.64371i q^{17} +7.16228i q^{19} +(-1.58114 - 0.707107i) q^{21} -5.65685 q^{23} +3.00000 q^{25} +(4.94975 - 1.58114i) q^{27} -7.30056i q^{29} -6.32456i q^{31} +(3.16228 - 7.07107i) q^{33} -1.41421 q^{35} -8.32456 q^{37} +(0.592359 - 1.32456i) q^{39} +1.18472i q^{41} +4.32456i q^{43} +(3.16228 - 2.82843i) q^{45} -8.94427 q^{47} -1.00000 q^{49} +(2.59893 + 1.16228i) q^{51} +10.1290i q^{53} -6.32456i q^{55} +(-11.3246 - 5.06450i) q^{57} +1.41421 q^{59} -3.16228 q^{61} +(2.23607 - 2.00000i) q^{63} -1.18472i q^{65} +2.00000i q^{67} +(4.00000 - 8.94427i) q^{69} -1.18472 q^{71} -8.32456 q^{73} +(-2.12132 + 4.74342i) q^{75} -4.47214i q^{77} +4.00000i q^{79} +(-1.00000 + 8.94427i) q^{81} +10.3585 q^{83} +2.32456 q^{85} +(11.5432 + 5.16228i) q^{87} +10.1290i q^{89} -0.837722i q^{91} +(10.0000 + 4.47214i) q^{93} -10.1290 q^{95} +14.6491 q^{97} +(8.94427 + 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{9} - 32 q^{13} + 24 q^{25} - 16 q^{37} - 8 q^{49} - 40 q^{57} + 32 q^{69} - 16 q^{73} - 8 q^{81} - 32 q^{85} + 80 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) −0.707107 + 1.58114i −0.408248 + 0.912871i
\(4\) 0 0
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.00000 2.23607i −0.666667 0.745356i
\(10\) 0 0
\(11\) −4.47214 −1.34840 −0.674200 0.738549i \(-0.735511\pi\)
−0.674200 + 0.738549i \(0.735511\pi\)
\(12\) 0 0
\(13\) −0.837722 −0.232342 −0.116171 0.993229i \(-0.537062\pi\)
−0.116171 + 0.993229i \(0.537062\pi\)
\(14\) 0 0
\(15\) −2.23607 1.00000i −0.577350 0.258199i
\(16\) 0 0
\(17\) 1.64371i 0.398658i −0.979933 0.199329i \(-0.936124\pi\)
0.979933 0.199329i \(-0.0638762\pi\)
\(18\) 0 0
\(19\) 7.16228i 1.64314i 0.570108 + 0.821570i \(0.306901\pi\)
−0.570108 + 0.821570i \(0.693099\pi\)
\(20\) 0 0
\(21\) −1.58114 0.707107i −0.345033 0.154303i
\(22\) 0 0
\(23\) −5.65685 −1.17954 −0.589768 0.807573i \(-0.700781\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 4.94975 1.58114i 0.952579 0.304290i
\(28\) 0 0
\(29\) 7.30056i 1.35568i −0.735209 0.677840i \(-0.762916\pi\)
0.735209 0.677840i \(-0.237084\pi\)
\(30\) 0 0
\(31\) 6.32456i 1.13592i −0.823055 0.567962i \(-0.807732\pi\)
0.823055 0.567962i \(-0.192268\pi\)
\(32\) 0 0
\(33\) 3.16228 7.07107i 0.550482 1.23091i
\(34\) 0 0
\(35\) −1.41421 −0.239046
\(36\) 0 0
\(37\) −8.32456 −1.36855 −0.684274 0.729225i \(-0.739881\pi\)
−0.684274 + 0.729225i \(0.739881\pi\)
\(38\) 0 0
\(39\) 0.592359 1.32456i 0.0948534 0.212099i
\(40\) 0 0
\(41\) 1.18472i 0.185022i 0.995712 + 0.0925110i \(0.0294893\pi\)
−0.995712 + 0.0925110i \(0.970511\pi\)
\(42\) 0 0
\(43\) 4.32456i 0.659489i 0.944070 + 0.329744i \(0.106963\pi\)
−0.944070 + 0.329744i \(0.893037\pi\)
\(44\) 0 0
\(45\) 3.16228 2.82843i 0.471405 0.421637i
\(46\) 0 0
\(47\) −8.94427 −1.30466 −0.652328 0.757937i \(-0.726208\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.59893 + 1.16228i 0.363923 + 0.162751i
\(52\) 0 0
\(53\) 10.1290i 1.39132i 0.718369 + 0.695662i \(0.244889\pi\)
−0.718369 + 0.695662i \(0.755111\pi\)
\(54\) 0 0
\(55\) 6.32456i 0.852803i
\(56\) 0 0
\(57\) −11.3246 5.06450i −1.49997 0.670809i
\(58\) 0 0
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) −3.16228 −0.404888 −0.202444 0.979294i \(-0.564888\pi\)
−0.202444 + 0.979294i \(0.564888\pi\)
\(62\) 0 0
\(63\) 2.23607 2.00000i 0.281718 0.251976i
\(64\) 0 0
\(65\) 1.18472i 0.146946i
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 0 0
\(69\) 4.00000 8.94427i 0.481543 1.07676i
\(70\) 0 0
\(71\) −1.18472 −0.140600 −0.0703001 0.997526i \(-0.522396\pi\)
−0.0703001 + 0.997526i \(0.522396\pi\)
\(72\) 0 0
\(73\) −8.32456 −0.974316 −0.487158 0.873314i \(-0.661966\pi\)
−0.487158 + 0.873314i \(0.661966\pi\)
\(74\) 0 0
\(75\) −2.12132 + 4.74342i −0.244949 + 0.547723i
\(76\) 0 0
\(77\) 4.47214i 0.509647i
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) −1.00000 + 8.94427i −0.111111 + 0.993808i
\(82\) 0 0
\(83\) 10.3585 1.13699 0.568496 0.822686i \(-0.307526\pi\)
0.568496 + 0.822686i \(0.307526\pi\)
\(84\) 0 0
\(85\) 2.32456 0.252133
\(86\) 0 0
\(87\) 11.5432 + 5.16228i 1.23756 + 0.553454i
\(88\) 0 0
\(89\) 10.1290i 1.07367i 0.843687 + 0.536835i \(0.180380\pi\)
−0.843687 + 0.536835i \(0.819620\pi\)
\(90\) 0 0
\(91\) 0.837722i 0.0878172i
\(92\) 0 0
\(93\) 10.0000 + 4.47214i 1.03695 + 0.463739i
\(94\) 0 0
\(95\) −10.1290 −1.03921
\(96\) 0 0
\(97\) 14.6491 1.48739 0.743696 0.668518i \(-0.233071\pi\)
0.743696 + 0.668518i \(0.233071\pi\)
\(98\) 0 0
\(99\) 8.94427 + 10.0000i 0.898933 + 1.00504i
\(100\) 0 0
\(101\) 1.41421i 0.140720i 0.997522 + 0.0703598i \(0.0224147\pi\)
−0.997522 + 0.0703598i \(0.977585\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 1.00000 2.23607i 0.0975900 0.218218i
\(106\) 0 0
\(107\) −18.6143 −1.79951 −0.899755 0.436396i \(-0.856255\pi\)
−0.899755 + 0.436396i \(0.856255\pi\)
\(108\) 0 0
\(109\) 8.32456 0.797348 0.398674 0.917093i \(-0.369471\pi\)
0.398674 + 0.917093i \(0.369471\pi\)
\(110\) 0 0
\(111\) 5.88635 13.1623i 0.558708 1.24931i
\(112\) 0 0
\(113\) 15.7858i 1.48501i 0.669842 + 0.742504i \(0.266362\pi\)
−0.669842 + 0.742504i \(0.733638\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) 1.67544 + 1.87320i 0.154895 + 0.173178i
\(118\) 0 0
\(119\) 1.64371 0.150679
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) −1.87320 0.837722i −0.168901 0.0755349i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 6.00000i 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) 0 0
\(129\) −6.83772 3.05792i −0.602028 0.269235i
\(130\) 0 0
\(131\) 13.1869 1.15215 0.576073 0.817398i \(-0.304584\pi\)
0.576073 + 0.817398i \(0.304584\pi\)
\(132\) 0 0
\(133\) −7.16228 −0.621048
\(134\) 0 0
\(135\) 2.23607 + 7.00000i 0.192450 + 0.602464i
\(136\) 0 0
\(137\) 17.8885i 1.52832i 0.645026 + 0.764161i \(0.276847\pi\)
−0.645026 + 0.764161i \(0.723153\pi\)
\(138\) 0 0
\(139\) 0.837722i 0.0710547i −0.999369 0.0355273i \(-0.988689\pi\)
0.999369 0.0355273i \(-0.0113111\pi\)
\(140\) 0 0
\(141\) 6.32456 14.1421i 0.532624 1.19098i
\(142\) 0 0
\(143\) 3.74641 0.313290
\(144\) 0 0
\(145\) 10.3246 0.857408
\(146\) 0 0
\(147\) 0.707107 1.58114i 0.0583212 0.130410i
\(148\) 0 0
\(149\) 4.47214i 0.366372i 0.983078 + 0.183186i \(0.0586410\pi\)
−0.983078 + 0.183186i \(0.941359\pi\)
\(150\) 0 0
\(151\) 10.0000i 0.813788i 0.913475 + 0.406894i \(0.133388\pi\)
−0.913475 + 0.406894i \(0.866612\pi\)
\(152\) 0 0
\(153\) −3.67544 + 3.28742i −0.297142 + 0.265772i
\(154\) 0 0
\(155\) 8.94427 0.718421
\(156\) 0 0
\(157\) 5.48683 0.437897 0.218948 0.975736i \(-0.429737\pi\)
0.218948 + 0.975736i \(0.429737\pi\)
\(158\) 0 0
\(159\) −16.0153 7.16228i −1.27010 0.568006i
\(160\) 0 0
\(161\) 5.65685i 0.445823i
\(162\) 0 0
\(163\) 8.32456i 0.652029i 0.945365 + 0.326015i \(0.105706\pi\)
−0.945365 + 0.326015i \(0.894294\pi\)
\(164\) 0 0
\(165\) 10.0000 + 4.47214i 0.778499 + 0.348155i
\(166\) 0 0
\(167\) 17.4296 1.34874 0.674370 0.738394i \(-0.264415\pi\)
0.674370 + 0.738394i \(0.264415\pi\)
\(168\) 0 0
\(169\) −12.2982 −0.946017
\(170\) 0 0
\(171\) 16.0153 14.3246i 1.22472 1.09543i
\(172\) 0 0
\(173\) 21.6722i 1.64771i 0.566803 + 0.823853i \(0.308180\pi\)
−0.566803 + 0.823853i \(0.691820\pi\)
\(174\) 0 0
\(175\) 3.00000i 0.226779i
\(176\) 0 0
\(177\) −1.00000 + 2.23607i −0.0751646 + 0.168073i
\(178\) 0 0
\(179\) −25.1891 −1.88272 −0.941361 0.337401i \(-0.890452\pi\)
−0.941361 + 0.337401i \(0.890452\pi\)
\(180\) 0 0
\(181\) −26.1359 −1.94267 −0.971335 0.237716i \(-0.923601\pi\)
−0.971335 + 0.237716i \(0.923601\pi\)
\(182\) 0 0
\(183\) 2.23607 5.00000i 0.165295 0.369611i
\(184\) 0 0
\(185\) 11.7727i 0.865546i
\(186\) 0 0
\(187\) 7.35089i 0.537550i
\(188\) 0 0
\(189\) 1.58114 + 4.94975i 0.115011 + 0.360041i
\(190\) 0 0
\(191\) 21.4427 1.55154 0.775770 0.631016i \(-0.217362\pi\)
0.775770 + 0.631016i \(0.217362\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) 1.87320 + 0.837722i 0.134143 + 0.0599905i
\(196\) 0 0
\(197\) 1.18472i 0.0844077i 0.999109 + 0.0422038i \(0.0134379\pi\)
−0.999109 + 0.0422038i \(0.986562\pi\)
\(198\) 0 0
\(199\) 0.649111i 0.0460142i 0.999735 + 0.0230071i \(0.00732404\pi\)
−0.999735 + 0.0230071i \(0.992676\pi\)
\(200\) 0 0
\(201\) −3.16228 1.41421i −0.223050 0.0997509i
\(202\) 0 0
\(203\) 7.30056 0.512399
\(204\) 0 0
\(205\) −1.67544 −0.117018
\(206\) 0 0
\(207\) 11.3137 + 12.6491i 0.786357 + 0.879174i
\(208\) 0 0
\(209\) 32.0307i 2.21561i
\(210\) 0 0
\(211\) 26.6491i 1.83460i −0.398196 0.917300i \(-0.630364\pi\)
0.398196 0.917300i \(-0.369636\pi\)
\(212\) 0 0
\(213\) 0.837722 1.87320i 0.0573998 0.128350i
\(214\) 0 0
\(215\) −6.11584 −0.417097
\(216\) 0 0
\(217\) 6.32456 0.429339
\(218\) 0 0
\(219\) 5.88635 13.1623i 0.397763 0.889424i
\(220\) 0 0
\(221\) 1.37697i 0.0926251i
\(222\) 0 0
\(223\) 10.3246i 0.691383i 0.938348 + 0.345692i \(0.112356\pi\)
−0.938348 + 0.345692i \(0.887644\pi\)
\(224\) 0 0
\(225\) −6.00000 6.70820i −0.400000 0.447214i
\(226\) 0 0
\(227\) 4.70163 0.312058 0.156029 0.987752i \(-0.450131\pi\)
0.156029 + 0.987752i \(0.450131\pi\)
\(228\) 0 0
\(229\) −27.8114 −1.83783 −0.918914 0.394458i \(-0.870932\pi\)
−0.918914 + 0.394458i \(0.870932\pi\)
\(230\) 0 0
\(231\) 7.07107 + 3.16228i 0.465242 + 0.208063i
\(232\) 0 0
\(233\) 5.65685i 0.370593i 0.982683 + 0.185296i \(0.0593245\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 0 0
\(235\) 12.6491i 0.825137i
\(236\) 0 0
\(237\) −6.32456 2.82843i −0.410824 0.183726i
\(238\) 0 0
\(239\) 8.02629 0.519178 0.259589 0.965719i \(-0.416413\pi\)
0.259589 + 0.965719i \(0.416413\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −13.4350 7.90569i −0.861858 0.507151i
\(244\) 0 0
\(245\) 1.41421i 0.0903508i
\(246\) 0 0
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) −7.32456 + 16.3782i −0.464175 + 1.03793i
\(250\) 0 0
\(251\) −0.955223 −0.0602931 −0.0301466 0.999545i \(-0.509597\pi\)
−0.0301466 + 0.999545i \(0.509597\pi\)
\(252\) 0 0
\(253\) 25.2982 1.59049
\(254\) 0 0
\(255\) −1.64371 + 3.67544i −0.102933 + 0.230165i
\(256\) 0 0
\(257\) 10.5880i 0.660460i 0.943900 + 0.330230i \(0.107126\pi\)
−0.943900 + 0.330230i \(0.892874\pi\)
\(258\) 0 0
\(259\) 8.32456i 0.517263i
\(260\) 0 0
\(261\) −16.3246 + 14.6011i −1.01046 + 0.903787i
\(262\) 0 0
\(263\) −24.7301 −1.52492 −0.762462 0.647033i \(-0.776010\pi\)
−0.762462 + 0.647033i \(0.776010\pi\)
\(264\) 0 0
\(265\) −14.3246 −0.879950
\(266\) 0 0
\(267\) −16.0153 7.16228i −0.980123 0.438324i
\(268\) 0 0
\(269\) 24.5006i 1.49383i −0.664920 0.746915i \(-0.731534\pi\)
0.664920 0.746915i \(-0.268466\pi\)
\(270\) 0 0
\(271\) 14.3246i 0.870155i 0.900393 + 0.435077i \(0.143279\pi\)
−0.900393 + 0.435077i \(0.856721\pi\)
\(272\) 0 0
\(273\) 1.32456 + 0.592359i 0.0801657 + 0.0358512i
\(274\) 0 0
\(275\) −13.4164 −0.809040
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −14.1421 + 12.6491i −0.846668 + 0.757282i
\(280\) 0 0
\(281\) 12.2317i 0.729681i −0.931070 0.364841i \(-0.881123\pi\)
0.931070 0.364841i \(-0.118877\pi\)
\(282\) 0 0
\(283\) 25.4868i 1.51503i 0.652815 + 0.757517i \(0.273588\pi\)
−0.652815 + 0.757517i \(0.726412\pi\)
\(284\) 0 0
\(285\) 7.16228 16.0153i 0.424257 0.948667i
\(286\) 0 0
\(287\) −1.18472 −0.0699317
\(288\) 0 0
\(289\) 14.2982 0.841072
\(290\) 0 0
\(291\) −10.3585 + 23.1623i −0.607225 + 1.35780i
\(292\) 0 0
\(293\) 13.6459i 0.797202i −0.917124 0.398601i \(-0.869496\pi\)
0.917124 0.398601i \(-0.130504\pi\)
\(294\) 0 0
\(295\) 2.00000i 0.116445i
\(296\) 0 0
\(297\) −22.1359 + 7.07107i −1.28446 + 0.410305i
\(298\) 0 0
\(299\) 4.73887 0.274056
\(300\) 0 0
\(301\) −4.32456 −0.249263
\(302\) 0 0
\(303\) −2.23607 1.00000i −0.128459 0.0574485i
\(304\) 0 0
\(305\) 4.47214i 0.256074i
\(306\) 0 0
\(307\) 21.4868i 1.22632i −0.789959 0.613159i \(-0.789898\pi\)
0.789959 0.613159i \(-0.210102\pi\)
\(308\) 0 0
\(309\) 12.6491 + 5.65685i 0.719583 + 0.321807i
\(310\) 0 0
\(311\) −0.458991 −0.0260270 −0.0130135 0.999915i \(-0.504142\pi\)
−0.0130135 + 0.999915i \(0.504142\pi\)
\(312\) 0 0
\(313\) 32.9737 1.86378 0.931891 0.362739i \(-0.118158\pi\)
0.931891 + 0.362739i \(0.118158\pi\)
\(314\) 0 0
\(315\) 2.82843 + 3.16228i 0.159364 + 0.178174i
\(316\) 0 0
\(317\) 19.0733i 1.07126i −0.844452 0.535631i \(-0.820074\pi\)
0.844452 0.535631i \(-0.179926\pi\)
\(318\) 0 0
\(319\) 32.6491i 1.82800i
\(320\) 0 0
\(321\) 13.1623 29.4317i 0.734647 1.64272i
\(322\) 0 0
\(323\) 11.7727 0.655050
\(324\) 0 0
\(325\) −2.51317 −0.139405
\(326\) 0 0
\(327\) −5.88635 + 13.1623i −0.325516 + 0.727876i
\(328\) 0 0
\(329\) 8.94427i 0.493114i
\(330\) 0 0
\(331\) 8.97367i 0.493237i −0.969113 0.246619i \(-0.920680\pi\)
0.969113 0.246619i \(-0.0793195\pi\)
\(332\) 0 0
\(333\) 16.6491 + 18.6143i 0.912366 + 1.02006i
\(334\) 0 0
\(335\) −2.82843 −0.154533
\(336\) 0 0
\(337\) −32.6491 −1.77851 −0.889255 0.457411i \(-0.848777\pi\)
−0.889255 + 0.457411i \(0.848777\pi\)
\(338\) 0 0
\(339\) −24.9596 11.1623i −1.35562 0.606252i
\(340\) 0 0
\(341\) 28.2843i 1.53168i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 12.6491 + 5.65685i 0.681005 + 0.304555i
\(346\) 0 0
\(347\) −12.4984 −0.670951 −0.335475 0.942049i \(-0.608897\pi\)
−0.335475 + 0.942049i \(0.608897\pi\)
\(348\) 0 0
\(349\) −23.1623 −1.23985 −0.619924 0.784662i \(-0.712837\pi\)
−0.619924 + 0.784662i \(0.712837\pi\)
\(350\) 0 0
\(351\) −4.14651 + 1.32456i −0.221325 + 0.0706995i
\(352\) 0 0
\(353\) 4.01315i 0.213598i −0.994281 0.106799i \(-0.965940\pi\)
0.994281 0.106799i \(-0.0340602\pi\)
\(354\) 0 0
\(355\) 1.67544i 0.0889234i
\(356\) 0 0
\(357\) −1.16228 + 2.59893i −0.0615143 + 0.137550i
\(358\) 0 0
\(359\) −23.5454 −1.24268 −0.621339 0.783542i \(-0.713411\pi\)
−0.621339 + 0.783542i \(0.713411\pi\)
\(360\) 0 0
\(361\) −32.2982 −1.69991
\(362\) 0 0
\(363\) −6.36396 + 14.2302i −0.334021 + 0.746894i
\(364\) 0 0
\(365\) 11.7727i 0.616211i
\(366\) 0 0
\(367\) 5.67544i 0.296256i −0.988968 0.148128i \(-0.952675\pi\)
0.988968 0.148128i \(-0.0473247\pi\)
\(368\) 0 0
\(369\) 2.64911 2.36944i 0.137907 0.123348i
\(370\) 0 0
\(371\) −10.1290 −0.525871
\(372\) 0 0
\(373\) 5.35089 0.277059 0.138529 0.990358i \(-0.455762\pi\)
0.138529 + 0.990358i \(0.455762\pi\)
\(374\) 0 0
\(375\) −17.8885 8.00000i −0.923760 0.413118i
\(376\) 0 0
\(377\) 6.11584i 0.314982i
\(378\) 0 0
\(379\) 11.6754i 0.599727i 0.953982 + 0.299864i \(0.0969412\pi\)
−0.953982 + 0.299864i \(0.903059\pi\)
\(380\) 0 0
\(381\) 9.48683 + 4.24264i 0.486025 + 0.217357i
\(382\) 0 0
\(383\) 22.6274 1.15621 0.578103 0.815963i \(-0.303793\pi\)
0.578103 + 0.815963i \(0.303793\pi\)
\(384\) 0 0
\(385\) 6.32456 0.322329
\(386\) 0 0
\(387\) 9.67000 8.64911i 0.491554 0.439659i
\(388\) 0 0
\(389\) 33.2154i 1.68409i −0.539409 0.842044i \(-0.681352\pi\)
0.539409 0.842044i \(-0.318648\pi\)
\(390\) 0 0
\(391\) 9.29822i 0.470231i
\(392\) 0 0
\(393\) −9.32456 + 20.8503i −0.470362 + 1.05176i
\(394\) 0 0
\(395\) −5.65685 −0.284627
\(396\) 0 0
\(397\) 16.8377 0.845061 0.422531 0.906349i \(-0.361142\pi\)
0.422531 + 0.906349i \(0.361142\pi\)
\(398\) 0 0
\(399\) 5.06450 11.3246i 0.253542 0.566937i
\(400\) 0 0
\(401\) 12.4984i 0.624142i −0.950059 0.312071i \(-0.898977\pi\)
0.950059 0.312071i \(-0.101023\pi\)
\(402\) 0 0
\(403\) 5.29822i 0.263923i
\(404\) 0 0
\(405\) −12.6491 1.41421i −0.628539 0.0702728i
\(406\) 0 0
\(407\) 37.2285 1.84535
\(408\) 0 0
\(409\) −24.3246 −1.20277 −0.601386 0.798959i \(-0.705385\pi\)
−0.601386 + 0.798959i \(0.705385\pi\)
\(410\) 0 0
\(411\) −28.2843 12.6491i −1.39516 0.623935i
\(412\) 0 0
\(413\) 1.41421i 0.0695889i
\(414\) 0 0
\(415\) 14.6491i 0.719097i
\(416\) 0 0
\(417\) 1.32456 + 0.592359i 0.0648638 + 0.0290080i
\(418\) 0 0
\(419\) −7.98905 −0.390291 −0.195145 0.980774i \(-0.562518\pi\)
−0.195145 + 0.980774i \(0.562518\pi\)
\(420\) 0 0
\(421\) 2.64911 0.129110 0.0645549 0.997914i \(-0.479437\pi\)
0.0645549 + 0.997914i \(0.479437\pi\)
\(422\) 0 0
\(423\) 17.8885 + 20.0000i 0.869771 + 0.972433i
\(424\) 0 0
\(425\) 4.93113i 0.239195i
\(426\) 0 0
\(427\) 3.16228i 0.153033i
\(428\) 0 0
\(429\) −2.64911 + 5.92359i −0.127900 + 0.285994i
\(430\) 0 0
\(431\) −3.28742 −0.158349 −0.0791747 0.996861i \(-0.525228\pi\)
−0.0791747 + 0.996861i \(0.525228\pi\)
\(432\) 0 0
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 0 0
\(435\) −7.30056 + 16.3246i −0.350035 + 0.782703i
\(436\) 0 0
\(437\) 40.5160i 1.93814i
\(438\) 0 0
\(439\) 24.6491i 1.17644i −0.808702 0.588219i \(-0.799829\pi\)
0.808702 0.588219i \(-0.200171\pi\)
\(440\) 0 0
\(441\) 2.00000 + 2.23607i 0.0952381 + 0.106479i
\(442\) 0 0
\(443\) 12.9574 0.615625 0.307813 0.951447i \(-0.400403\pi\)
0.307813 + 0.951447i \(0.400403\pi\)
\(444\) 0 0
\(445\) −14.3246 −0.679049
\(446\) 0 0
\(447\) −7.07107 3.16228i −0.334450 0.149571i
\(448\) 0 0
\(449\) 16.9706i 0.800890i 0.916321 + 0.400445i \(0.131145\pi\)
−0.916321 + 0.400445i \(0.868855\pi\)
\(450\) 0 0
\(451\) 5.29822i 0.249483i
\(452\) 0 0
\(453\) −15.8114 7.07107i −0.742884 0.332228i
\(454\) 0 0
\(455\) 1.18472 0.0555405
\(456\) 0 0
\(457\) −0.649111 −0.0303641 −0.0151821 0.999885i \(-0.504833\pi\)
−0.0151821 + 0.999885i \(0.504833\pi\)
\(458\) 0 0
\(459\) −2.59893 8.13594i −0.121308 0.379753i
\(460\) 0 0
\(461\) 15.0974i 0.703154i −0.936159 0.351577i \(-0.885646\pi\)
0.936159 0.351577i \(-0.114354\pi\)
\(462\) 0 0
\(463\) 15.3509i 0.713416i 0.934216 + 0.356708i \(0.116101\pi\)
−0.934216 + 0.356708i \(0.883899\pi\)
\(464\) 0 0
\(465\) −6.32456 + 14.1421i −0.293294 + 0.655826i
\(466\) 0 0
\(467\) 9.44050 0.436854 0.218427 0.975853i \(-0.429907\pi\)
0.218427 + 0.975853i \(0.429907\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 0 0
\(471\) −3.87978 + 8.67544i −0.178771 + 0.399743i
\(472\) 0 0
\(473\) 19.3400i 0.889254i
\(474\) 0 0
\(475\) 21.4868i 0.985884i
\(476\) 0 0
\(477\) 22.6491 20.2580i 1.03703 0.927549i
\(478\) 0 0
\(479\) 5.65685 0.258468 0.129234 0.991614i \(-0.458748\pi\)
0.129234 + 0.991614i \(0.458748\pi\)
\(480\) 0 0
\(481\) 6.97367 0.317972
\(482\) 0 0
\(483\) 8.94427 + 4.00000i 0.406978 + 0.182006i
\(484\) 0 0
\(485\) 20.7170i 0.940709i
\(486\) 0 0
\(487\) 27.2982i 1.23700i −0.785785 0.618500i \(-0.787741\pi\)
0.785785 0.618500i \(-0.212259\pi\)
\(488\) 0 0
\(489\) −13.1623 5.88635i −0.595219 0.266190i
\(490\) 0 0
\(491\) 20.9837 0.946981 0.473491 0.880799i \(-0.342994\pi\)
0.473491 + 0.880799i \(0.342994\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) −14.1421 + 12.6491i −0.635642 + 0.568535i
\(496\) 0 0
\(497\) 1.18472i 0.0531419i
\(498\) 0 0
\(499\) 2.00000i 0.0895323i −0.998997 0.0447661i \(-0.985746\pi\)
0.998997 0.0447661i \(-0.0142543\pi\)
\(500\) 0 0
\(501\) −12.3246 + 27.5585i −0.550621 + 1.23122i
\(502\) 0 0
\(503\) 6.11584 0.272692 0.136346 0.990661i \(-0.456464\pi\)
0.136346 + 0.990661i \(0.456464\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 8.69616 19.4452i 0.386210 0.863591i
\(508\) 0 0
\(509\) 13.1869i 0.584500i −0.956342 0.292250i \(-0.905596\pi\)
0.956342 0.292250i \(-0.0944039\pi\)
\(510\) 0 0
\(511\) 8.32456i 0.368257i
\(512\) 0 0
\(513\) 11.3246 + 35.4515i 0.499991 + 1.56522i
\(514\) 0 0
\(515\) 11.3137 0.498542
\(516\) 0 0
\(517\) 40.0000 1.75920
\(518\) 0 0
\(519\) −34.2667 15.3246i −1.50414 0.672673i
\(520\) 0 0
\(521\) 24.7301i 1.08345i 0.840557 + 0.541723i \(0.182228\pi\)
−0.840557 + 0.541723i \(0.817772\pi\)
\(522\) 0 0
\(523\) 8.18861i 0.358063i −0.983843 0.179031i \(-0.942704\pi\)
0.983843 0.179031i \(-0.0572964\pi\)
\(524\) 0 0
\(525\) −4.74342 2.12132i −0.207020 0.0925820i
\(526\) 0 0
\(527\) −10.3957 −0.452845
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) −2.82843 3.16228i −0.122743 0.137231i
\(532\) 0 0
\(533\) 0.992465i 0.0429884i
\(534\) 0 0
\(535\) 26.3246i 1.13811i
\(536\) 0 0
\(537\) 17.8114 39.8275i 0.768618 1.71868i
\(538\) 0 0
\(539\) 4.47214 0.192629
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 18.4809 41.3246i 0.793091 1.77341i
\(544\) 0 0
\(545\) 11.7727i 0.504287i
\(546\) 0 0
\(547\) 23.6754i 1.01229i −0.862449 0.506144i \(-0.831070\pi\)
0.862449 0.506144i \(-0.168930\pi\)
\(548\) 0 0
\(549\) 6.32456 + 7.07107i 0.269925 + 0.301786i
\(550\) 0 0
\(551\) 52.2887 2.22757
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) 18.6143 + 8.32456i 0.790132 + 0.353358i
\(556\) 0 0
\(557\) 23.8121i 1.00895i 0.863426 + 0.504476i \(0.168314\pi\)
−0.863426 + 0.504476i \(0.831686\pi\)
\(558\) 0 0
\(559\) 3.62278i 0.153227i
\(560\) 0 0
\(561\) −11.6228 5.19786i −0.490714 0.219454i
\(562\) 0 0
\(563\) 13.6459 0.575106 0.287553 0.957765i \(-0.407158\pi\)
0.287553 + 0.957765i \(0.407158\pi\)
\(564\) 0 0
\(565\) −22.3246 −0.939201
\(566\) 0 0
\(567\) −8.94427 1.00000i −0.375624 0.0419961i
\(568\) 0 0
\(569\) 38.4133i 1.61037i −0.593025 0.805184i \(-0.702067\pi\)
0.593025 0.805184i \(-0.297933\pi\)
\(570\) 0 0
\(571\) 0.324555i 0.0135822i −0.999977 0.00679111i \(-0.997838\pi\)
0.999977 0.00679111i \(-0.00216169\pi\)
\(572\) 0 0
\(573\) −15.1623 + 33.9039i −0.633413 + 1.41636i
\(574\) 0 0
\(575\) −16.9706 −0.707721
\(576\) 0 0
\(577\) −15.2982 −0.636873 −0.318437 0.947944i \(-0.603158\pi\)
−0.318437 + 0.947944i \(0.603158\pi\)
\(578\) 0 0
\(579\) −11.3137 + 25.2982i −0.470182 + 1.05136i
\(580\) 0 0
\(581\) 10.3585i 0.429742i
\(582\) 0 0
\(583\) 45.2982i 1.87606i
\(584\) 0 0
\(585\) −2.64911 + 2.36944i −0.109527 + 0.0979641i
\(586\) 0 0
\(587\) 27.3290 1.12799 0.563995 0.825778i \(-0.309264\pi\)
0.563995 + 0.825778i \(0.309264\pi\)
\(588\) 0 0
\(589\) 45.2982 1.86648
\(590\) 0 0
\(591\) −1.87320 0.837722i −0.0770533 0.0344593i
\(592\) 0 0
\(593\) 15.3269i 0.629398i 0.949191 + 0.314699i \(0.101904\pi\)
−0.949191 + 0.314699i \(0.898096\pi\)
\(594\) 0 0
\(595\) 2.32456i 0.0952975i
\(596\) 0 0
\(597\) −1.02633 0.458991i −0.0420051 0.0187852i
\(598\) 0 0
\(599\) 2.10270 0.0859140 0.0429570 0.999077i \(-0.486322\pi\)
0.0429570 + 0.999077i \(0.486322\pi\)
\(600\) 0 0
\(601\) 33.6228 1.37150 0.685751 0.727836i \(-0.259474\pi\)
0.685751 + 0.727836i \(0.259474\pi\)
\(602\) 0 0
\(603\) 4.47214 4.00000i 0.182119 0.162893i
\(604\) 0 0
\(605\) 12.7279i 0.517464i
\(606\) 0 0
\(607\) 30.3246i 1.23084i 0.788201 + 0.615418i \(0.211013\pi\)
−0.788201 + 0.615418i \(0.788987\pi\)
\(608\) 0 0
\(609\) −5.16228 + 11.5432i −0.209186 + 0.467754i
\(610\) 0 0
\(611\) 7.49282 0.303127
\(612\) 0 0
\(613\) 32.3246 1.30558 0.652788 0.757540i \(-0.273599\pi\)
0.652788 + 0.757540i \(0.273599\pi\)
\(614\) 0 0
\(615\) 1.18472 2.64911i 0.0477725 0.106822i
\(616\) 0 0
\(617\) 18.1553i 0.730904i 0.930830 + 0.365452i \(0.119086\pi\)
−0.930830 + 0.365452i \(0.880914\pi\)
\(618\) 0 0
\(619\) 39.8114i 1.60015i 0.599897 + 0.800077i \(0.295208\pi\)
−0.599897 + 0.800077i \(0.704792\pi\)
\(620\) 0 0
\(621\) −28.0000 + 8.94427i −1.12360 + 0.358921i
\(622\) 0 0
\(623\) −10.1290 −0.405809
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 50.6450 + 22.6491i 2.02256 + 0.904518i
\(628\) 0 0
\(629\) 13.6831i 0.545583i
\(630\) 0 0
\(631\) 37.9473i 1.51066i 0.655345 + 0.755330i \(0.272523\pi\)
−0.655345 + 0.755330i \(0.727477\pi\)
\(632\) 0 0
\(633\) 42.1359 + 18.8438i 1.67475 + 0.748972i
\(634\) 0 0
\(635\) 8.48528 0.336728
\(636\) 0 0
\(637\) 0.837722 0.0331918
\(638\) 0 0
\(639\) 2.36944 + 2.64911i 0.0937335 + 0.104797i
\(640\) 0 0
\(641\) 21.4427i 0.846936i −0.905911 0.423468i \(-0.860813\pi\)
0.905911 0.423468i \(-0.139187\pi\)
\(642\) 0 0
\(643\) 11.1623i 0.440197i −0.975478 0.220099i \(-0.929362\pi\)
0.975478 0.220099i \(-0.0706379\pi\)
\(644\) 0 0
\(645\) 4.32456 9.67000i 0.170279 0.380756i
\(646\) 0 0
\(647\) −49.0012 −1.92644 −0.963219 0.268718i \(-0.913400\pi\)
−0.963219 + 0.268718i \(0.913400\pi\)
\(648\) 0 0
\(649\) −6.32456 −0.248261
\(650\) 0 0
\(651\) −4.47214 + 10.0000i −0.175277 + 0.391931i
\(652\) 0 0
\(653\) 3.09516i 0.121123i −0.998164 0.0605616i \(-0.980711\pi\)
0.998164 0.0605616i \(-0.0192891\pi\)
\(654\) 0 0
\(655\) 18.6491i 0.728681i
\(656\) 0 0
\(657\) 16.6491 + 18.6143i 0.649544 + 0.726212i
\(658\) 0 0
\(659\) −28.0175 −1.09141 −0.545704 0.837978i \(-0.683738\pi\)
−0.545704 + 0.837978i \(0.683738\pi\)
\(660\) 0 0
\(661\) 41.4868 1.61365 0.806825 0.590790i \(-0.201184\pi\)
0.806825 + 0.590790i \(0.201184\pi\)
\(662\) 0 0
\(663\) −2.17718 0.973666i −0.0845548 0.0378141i
\(664\) 0 0
\(665\) 10.1290i 0.392785i
\(666\) 0 0
\(667\) 41.2982i 1.59907i
\(668\) 0 0
\(669\) −16.3246 7.30056i −0.631144 0.282256i
\(670\) 0 0
\(671\) 14.1421 0.545951
\(672\) 0 0
\(673\) −23.2982 −0.898080 −0.449040 0.893512i \(-0.648234\pi\)
−0.449040 + 0.893512i \(0.648234\pi\)
\(674\) 0 0
\(675\) 14.8492 4.74342i 0.571548 0.182574i
\(676\) 0 0
\(677\) 45.2176i 1.73785i 0.494941 + 0.868927i \(0.335190\pi\)
−0.494941 + 0.868927i \(0.664810\pi\)
\(678\) 0 0
\(679\) 14.6491i 0.562181i
\(680\) 0 0
\(681\) −3.32456 + 7.43393i −0.127397 + 0.284869i
\(682\) 0 0
\(683\) 11.5060 0.440263 0.220132 0.975470i \(-0.429351\pi\)
0.220132 + 0.975470i \(0.429351\pi\)
\(684\) 0 0
\(685\) −25.2982 −0.966595
\(686\) 0 0
\(687\) 19.6656 43.9737i 0.750290 1.67770i
\(688\) 0 0
\(689\) 8.48528i 0.323263i
\(690\) 0 0
\(691\) 8.83772i 0.336203i −0.985770 0.168101i \(-0.946236\pi\)
0.985770 0.168101i \(-0.0537636\pi\)
\(692\) 0 0
\(693\) −10.0000 + 8.94427i −0.379869 + 0.339765i
\(694\) 0 0
\(695\) 1.18472 0.0449389
\(696\) 0 0
\(697\) 1.94733 0.0737605
\(698\) 0 0
\(699\) −8.94427 4.00000i −0.338303 0.151294i
\(700\) 0 0
\(701\) 35.5848i 1.34402i 0.740542 + 0.672010i \(0.234569\pi\)
−0.740542 + 0.672010i \(0.765431\pi\)
\(702\) 0 0
\(703\) 59.6228i 2.24872i
\(704\) 0 0
\(705\) 20.0000 + 8.94427i 0.753244 + 0.336861i
\(706\) 0 0
\(707\) −1.41421 −0.0531870
\(708\) 0 0
\(709\) −40.9737 −1.53880 −0.769399 0.638768i \(-0.779444\pi\)
−0.769399 + 0.638768i \(0.779444\pi\)
\(710\) 0 0
\(711\) 8.94427 8.00000i 0.335436 0.300023i
\(712\) 0 0
\(713\) 35.7771i 1.33986i
\(714\) 0 0
\(715\) 5.29822i 0.198142i
\(716\) 0 0
\(717\) −5.67544 + 12.6907i −0.211953 + 0.473942i
\(718\) 0 0
\(719\) 23.5454 0.878095 0.439048 0.898464i \(-0.355316\pi\)
0.439048 + 0.898464i \(0.355316\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) −9.89949 + 22.1359i −0.368166 + 0.823245i
\(724\) 0 0
\(725\) 21.9017i 0.813408i
\(726\) 0 0
\(727\) 8.64911i 0.320778i 0.987054 + 0.160389i \(0.0512748\pi\)
−0.987054 + 0.160389i \(0.948725\pi\)
\(728\) 0 0
\(729\) 22.0000 15.6525i 0.814815 0.579721i
\(730\) 0 0
\(731\) 7.10831 0.262910
\(732\) 0 0
\(733\) 1.86406 0.0688505 0.0344252 0.999407i \(-0.489040\pi\)
0.0344252 + 0.999407i \(0.489040\pi\)
\(734\) 0 0
\(735\) 2.23607 + 1.00000i 0.0824786 + 0.0368856i
\(736\) 0 0
\(737\) 8.94427i 0.329466i
\(738\) 0 0
\(739\) 4.97367i 0.182959i 0.995807 + 0.0914796i \(0.0291596\pi\)
−0.995807 + 0.0914796i \(0.970840\pi\)
\(740\) 0 0
\(741\) 9.48683 + 4.24264i 0.348508 + 0.155857i
\(742\) 0 0
\(743\) 12.2317 0.448737 0.224369 0.974504i \(-0.427968\pi\)
0.224369 + 0.974504i \(0.427968\pi\)
\(744\) 0 0
\(745\) −6.32456 −0.231714
\(746\) 0 0
\(747\) −20.7170 23.1623i −0.757994 0.847463i
\(748\) 0 0
\(749\) 18.6143i 0.680151i
\(750\) 0 0
\(751\) 42.6491i 1.55629i −0.628086 0.778144i \(-0.716161\pi\)
0.628086 0.778144i \(-0.283839\pi\)
\(752\) 0 0
\(753\) 0.675445 1.51034i 0.0246146 0.0550399i
\(754\) 0 0
\(755\) −14.1421 −0.514685
\(756\) 0 0
\(757\) −7.67544 −0.278969 −0.139484 0.990224i \(-0.544545\pi\)
−0.139484 + 0.990224i \(0.544545\pi\)
\(758\) 0 0
\(759\) −17.8885 + 40.0000i −0.649313 + 1.45191i
\(760\) 0 0
\(761\) 4.47214i 0.162115i 0.996709 + 0.0810574i \(0.0258297\pi\)
−0.996709 + 0.0810574i \(0.974170\pi\)
\(762\) 0 0
\(763\) 8.32456i 0.301369i
\(764\) 0 0
\(765\) −4.64911 5.19786i −0.168089 0.187929i
\(766\) 0 0
\(767\) −1.18472 −0.0427777
\(768\) 0 0
\(769\) 23.2982 0.840155 0.420078 0.907488i \(-0.362003\pi\)
0.420078 + 0.907488i \(0.362003\pi\)
\(770\) 0 0
\(771\) −16.7411 7.48683i −0.602915 0.269632i
\(772\) 0 0
\(773\) 25.8776i 0.930752i 0.885113 + 0.465376i \(0.154081\pi\)
−0.885113 + 0.465376i \(0.845919\pi\)
\(774\) 0 0
\(775\) 18.9737i 0.681554i
\(776\) 0 0
\(777\) 13.1623 + 5.88635i 0.472194 + 0.211172i
\(778\) 0 0
\(779\) −8.48528 −0.304017
\(780\) 0 0
\(781\) 5.29822 0.189585
\(782\) 0 0
\(783\) −11.5432 36.1359i −0.412520 1.29139i
\(784\) 0 0
\(785\) 7.75955i 0.276950i
\(786\) 0 0
\(787\) 32.8377i 1.17054i 0.810839 + 0.585269i \(0.199011\pi\)
−0.810839 + 0.585269i \(0.800989\pi\)
\(788\) 0 0
\(789\) 17.4868 39.1017i 0.622548 1.39206i
\(790\) 0 0
\(791\) −15.7858 −0.561280
\(792\) 0 0
\(793\) 2.64911 0.0940727
\(794\) 0 0
\(795\) 10.1290 22.6491i 0.359238 0.803281i
\(796\) 0 0
\(797\) 35.8143i 1.26861i 0.773083 + 0.634304i \(0.218713\pi\)
−0.773083 + 0.634304i \(0.781287\pi\)
\(798\) 0 0
\(799\) 14.7018i 0.520112i
\(800\) 0 0
\(801\) 22.6491 20.2580i 0.800267 0.715781i
\(802\) 0 0
\(803\) 37.2285 1.31377
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 38.7389 + 17.3246i 1.36367 + 0.609853i
\(808\) 0 0
\(809\) 0.266737i 0.00937798i 0.999989 + 0.00468899i \(0.00149256\pi\)
−0.999989 + 0.00468899i \(0.998507\pi\)
\(810\) 0 0
\(811\) 14.5132i 0.509626i −0.966990 0.254813i \(-0.917986\pi\)
0.966990 0.254813i \(-0.0820140\pi\)
\(812\) 0 0
\(813\) −22.6491 10.1290i −0.794339 0.355239i
\(814\) 0 0
\(815\) −11.7727 −0.412380
\(816\) 0 0
\(817\) −30.9737 −1.08363
\(818\) 0 0
\(819\) −1.87320 + 1.67544i −0.0654550 + 0.0585448i
\(820\) 0 0
\(821\) 22.3607i 0.780393i −0.920732 0.390197i \(-0.872407\pi\)
0.920732 0.390197i \(-0.127593\pi\)
\(822\) 0 0
\(823\) 53.9473i 1.88049i −0.340504 0.940243i \(-0.610598\pi\)
0.340504 0.940243i \(-0.389402\pi\)
\(824\) 0 0
\(825\) 9.48683 21.2132i 0.330289 0.738549i
\(826\) 0 0
\(827\) −21.9017 −0.761596 −0.380798 0.924658i \(-0.624351\pi\)
−0.380798 + 0.924658i \(0.624351\pi\)
\(828\) 0 0
\(829\) 28.8377 1.00158 0.500788 0.865570i \(-0.333044\pi\)
0.500788 + 0.865570i \(0.333044\pi\)
\(830\) 0 0
\(831\) −1.41421 + 3.16228i −0.0490585 + 0.109698i
\(832\) 0 0
\(833\) 1.64371i 0.0569511i
\(834\) 0 0
\(835\) 24.6491i 0.853018i
\(836\) 0 0
\(837\) −10.0000 31.3050i −0.345651 1.08206i
\(838\) 0 0
\(839\) 13.2242 0.456549 0.228274 0.973597i \(-0.426692\pi\)
0.228274 + 0.973597i \(0.426692\pi\)
\(840\) 0 0
\(841\) −24.2982 −0.837870
\(842\) 0 0
\(843\) 19.3400 + 8.64911i 0.666105 + 0.297891i
\(844\) 0 0
\(845\) 17.3923i 0.598314i
\(846\) 0 0
\(847\) 9.00000i 0.309244i
\(848\) 0 0
\(849\) −40.2982 18.0219i −1.38303 0.618510i
\(850\) 0 0
\(851\) 47.0908 1.61425
\(852\) 0 0
\(853\) 18.1359 0.620963 0.310481 0.950579i \(-0.399510\pi\)
0.310481 + 0.950579i \(0.399510\pi\)
\(854\) 0 0
\(855\) 20.2580 + 22.6491i 0.692808 + 0.774583i
\(856\) 0 0
\(857\) 7.75955i 0.265061i 0.991179 + 0.132531i \(0.0423103\pi\)
−0.991179 + 0.132531i \(0.957690\pi\)
\(858\) 0 0
\(859\) 8.46050i 0.288668i −0.989529 0.144334i \(-0.953896\pi\)
0.989529 0.144334i \(-0.0461040\pi\)
\(860\) 0 0
\(861\) 0.837722 1.87320i 0.0285495 0.0638386i
\(862\) 0 0
\(863\) −33.6744 −1.14629 −0.573145 0.819454i \(-0.694277\pi\)
−0.573145 + 0.819454i \(0.694277\pi\)
\(864\) 0 0
\(865\) −30.6491 −1.04210
\(866\) 0 0
\(867\) −10.1104 + 22.6075i −0.343366 + 0.767790i
\(868\) 0 0
\(869\) 17.8885i 0.606827i
\(870\) 0 0
\(871\) 1.67544i 0.0567703i
\(872\) 0 0
\(873\) −29.2982 32.7564i −0.991595 1.10864i
\(874\) 0 0
\(875\) −11.3137 −0.382473
\(876\) 0 0
\(877\) 3.02633 0.102192 0.0510960 0.998694i \(-0.483729\pi\)
0.0510960 + 0.998694i \(0.483729\pi\)
\(878\) 0 0
\(879\) 21.5761 + 9.64911i 0.727743 + 0.325456i
\(880\) 0 0
\(881\) 15.3269i 0.516375i 0.966095 + 0.258187i \(0.0831252\pi\)
−0.966095 + 0.258187i \(0.916875\pi\)
\(882\) 0 0
\(883\) 18.0000i 0.605748i 0.953031 + 0.302874i \(0.0979462\pi\)
−0.953031 + 0.302874i \(0.902054\pi\)
\(884\) 0 0
\(885\) −3.16228 1.41421i −0.106299 0.0475383i
\(886\) 0 0
\(887\) −34.4001 −1.15504 −0.577521 0.816376i \(-0.695980\pi\)
−0.577521 + 0.816376i \(0.695980\pi\)
\(888\) 0 0
\(889\) 6.00000 0.201234
\(890\) 0 0
\(891\) 4.47214 40.0000i 0.149822 1.34005i
\(892\) 0 0
\(893\) 64.0614i 2.14373i
\(894\) 0 0
\(895\) 35.6228i 1.19074i
\(896\) 0 0
\(897\) −3.35089 + 7.49282i −0.111883 + 0.250178i
\(898\) 0 0
\(899\) −46.1728 −1.53995
\(900\) 0 0
\(901\) 16.6491 0.554662
\(902\) 0 0
\(903\) 3.05792 6.83772i 0.101761 0.227545i
\(904\) 0 0
\(905\) 36.9618i 1.22865i
\(906\) 0 0
\(907\) 47.9473i 1.59207i −0.605254 0.796033i \(-0.706928\pi\)
0.605254 0.796033i \(-0.293072\pi\)
\(908\) 0 0
\(909\) 3.16228 2.82843i 0.104886 0.0938130i
\(910\) 0 0
\(911\) −31.5717 −1.04602 −0.523008 0.852328i \(-0.675190\pi\)
−0.523008 + 0.852328i \(0.675190\pi\)
\(912\) 0 0
\(913\) −46.3246 −1.53312
\(914\) 0 0
\(915\) 7.07107 + 3.16228i 0.233762 + 0.104542i
\(916\) 0 0
\(917\) 13.1869i 0.435470i
\(918\) 0 0
\(919\) 4.64911i 0.153360i 0.997056 + 0.0766800i \(0.0244320\pi\)
−0.997056 + 0.0766800i \(0.975568\pi\)
\(920\) 0 0
\(921\) 33.9737 + 15.1935i 1.11947 + 0.500642i
\(922\) 0 0
\(923\) 0.992465 0.0326674
\(924\) 0 0
\(925\) −24.9737 −0.821129
\(926\) 0 0
\(927\) −17.8885 + 16.0000i −0.587537 + 0.525509i
\(928\) 0 0
\(929\) 37.4208i 1.22774i −0.789408 0.613868i \(-0.789613\pi\)
0.789408 0.613868i \(-0.210387\pi\)
\(930\) 0 0
\(931\) 7.16228i 0.234734i
\(932\) 0 0
\(933\) 0.324555 0.725728i 0.0106255 0.0237593i
\(934\) 0 0
\(935\) −10.3957 −0.339977
\(936\) 0 0
\(937\) −20.3246 −0.663974 −0.331987 0.943284i \(-0.607719\pi\)
−0.331987 + 0.943284i \(0.607719\pi\)
\(938\) 0 0
\(939\) −23.3159 + 52.1359i −0.760886 + 1.70139i
\(940\) 0 0
\(941\) 48.5050i 1.58122i 0.612321 + 0.790609i \(0.290236\pi\)
−0.612321 + 0.790609i \(0.709764\pi\)
\(942\) 0 0
\(943\) 6.70178i 0.218240i
\(944\) 0 0
\(945\) −7.00000 + 2.23607i −0.227710 + 0.0727393i
\(946\) 0 0
\(947\) −4.47214 −0.145325 −0.0726624 0.997357i \(-0.523150\pi\)
−0.0726624 + 0.997357i \(0.523150\pi\)
\(948\) 0 0
\(949\) 6.97367 0.226375
\(950\) 0 0
\(951\) 30.1575 + 13.4868i 0.977923 + 0.437341i
\(952\) 0 0
\(953\) 26.8328i 0.869200i 0.900624 + 0.434600i \(0.143110\pi\)
−0.900624 + 0.434600i \(0.856890\pi\)
\(954\) 0 0
\(955\) 30.3246i 0.981280i
\(956\) 0 0
\(957\) −51.6228 23.0864i −1.66873 0.746278i
\(958\) 0 0
\(959\) −17.8885 −0.577651
\(960\) 0 0
\(961\) −9.00000 −0.290323
\(962\) 0 0
\(963\) 37.2285 + 41.6228i 1.19967 + 1.34128i
\(964\) 0 0
\(965\) 22.6274i 0.728402i
\(966\) 0 0
\(967\) 9.35089i 0.300704i 0.988633 + 0.150352i \(0.0480408\pi\)
−0.988633 + 0.150352i \(0.951959\pi\)
\(968\) 0 0
\(969\) −8.32456 + 18.6143i −0.267423 + 0.597977i
\(970\) 0 0
\(971\) −43.8406 −1.40691 −0.703456 0.710739i \(-0.748361\pi\)
−0.703456 + 0.710739i \(0.748361\pi\)
\(972\) 0 0
\(973\) 0.837722 0.0268561
\(974\) 0 0
\(975\) 1.77708 3.97367i 0.0569120 0.127259i
\(976\) 0 0
\(977\) 4.20540i 0.134543i −0.997735 0.0672713i \(-0.978571\pi\)
0.997735 0.0672713i \(-0.0214293\pi\)
\(978\) 0 0
\(979\) 45.2982i 1.44774i
\(980\) 0 0
\(981\) −16.6491 18.6143i −0.531565 0.594308i
\(982\) 0 0
\(983\) 34.4001 1.09719 0.548597 0.836087i \(-0.315162\pi\)
0.548597 + 0.836087i \(0.315162\pi\)
\(984\) 0 0
\(985\) −1.67544 −0.0533841
\(986\) 0 0
\(987\) 14.1421 + 6.32456i 0.450149 + 0.201313i
\(988\) 0 0
\(989\) 24.4634i 0.777890i
\(990\) 0 0
\(991\) 54.5964i 1.73431i 0.498036 + 0.867157i \(0.334055\pi\)
−0.498036 + 0.867157i \(0.665945\pi\)
\(992\) 0 0
\(993\) 14.1886 + 6.34534i 0.450262 + 0.201363i
\(994\) 0 0
\(995\) −0.917981 −0.0291020
\(996\) 0 0
\(997\) −6.51317 −0.206274 −0.103137 0.994667i \(-0.532888\pi\)
−0.103137 + 0.994667i \(0.532888\pi\)
\(998\) 0 0
\(999\) −41.2044 + 13.1623i −1.30365 + 0.416436i
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 672.2.h.e.575.4 yes 8
3.2 odd 2 inner 672.2.h.e.575.7 yes 8
4.3 odd 2 inner 672.2.h.e.575.6 yes 8
8.3 odd 2 1344.2.h.f.575.3 8
8.5 even 2 1344.2.h.f.575.5 8
12.11 even 2 inner 672.2.h.e.575.1 8
24.5 odd 2 1344.2.h.f.575.2 8
24.11 even 2 1344.2.h.f.575.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.h.e.575.1 8 12.11 even 2 inner
672.2.h.e.575.4 yes 8 1.1 even 1 trivial
672.2.h.e.575.6 yes 8 4.3 odd 2 inner
672.2.h.e.575.7 yes 8 3.2 odd 2 inner
1344.2.h.f.575.2 8 24.5 odd 2
1344.2.h.f.575.3 8 8.3 odd 2
1344.2.h.f.575.5 8 8.5 even 2
1344.2.h.f.575.8 8 24.11 even 2