Properties

Label 216.3.h.e.53.4
Level $216$
Weight $3$
Character 216.53
Analytic conductor $5.886$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(53,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.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: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.629407744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.4
Root \(0.767178 + 1.18804i\) of defining polynomial
Character \(\chi\) \(=\) 216.53
Dual form 216.3.h.e.53.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.420861 + 1.95522i) q^{2} +(-3.64575 - 1.64575i) q^{4} -2.22699 q^{5} -8.64575 q^{7} +(4.75216 - 6.43560i) q^{8} +O(q^{10})\) \(q+(-0.420861 + 1.95522i) q^{2} +(-3.64575 - 1.64575i) q^{4} -2.22699 q^{5} -8.64575 q^{7} +(4.75216 - 6.43560i) q^{8} +(0.937254 - 4.35425i) q^{10} +19.8504 q^{11} -14.5203i q^{13} +(3.63866 - 16.9043i) q^{14} +(10.5830 + 12.0000i) q^{16} -5.29570i q^{17} -24.2915i q^{19} +(8.11905 + 3.66507i) q^{20} +(-8.35425 + 38.8118i) q^{22} -29.6528i q^{23} -20.0405 q^{25} +(28.3903 + 6.11102i) q^{26} +(31.5203 + 14.2288i) q^{28} -6.13742 q^{29} +10.1255 q^{31} +(-27.9166 + 15.6417i) q^{32} +(10.3542 + 2.22876i) q^{34} +19.2540 q^{35} -16.6458i q^{37} +(47.4952 + 10.2234i) q^{38} +(-10.5830 + 14.3320i) q^{40} +7.82087i q^{41} +1.54249i q^{43} +(-72.3695 - 32.6687i) q^{44} +(57.9778 + 12.4797i) q^{46} +63.7069i q^{47} +25.7490 q^{49} +(8.43428 - 39.1836i) q^{50} +(-23.8967 + 52.9373i) q^{52} -76.5253 q^{53} -44.2065 q^{55} +(-41.0860 + 55.6406i) q^{56} +(2.58301 - 12.0000i) q^{58} +41.0332 q^{59} -52.0627i q^{61} +(-4.26143 + 19.7975i) q^{62} +(-18.8340 - 61.1660i) q^{64} +32.3365i q^{65} -110.247i q^{67} +(-8.71541 + 19.3068i) q^{68} +(-8.10326 + 37.6458i) q^{70} -100.690i q^{71} -93.9150 q^{73} +(32.5461 + 7.00555i) q^{74} +(-39.9778 + 88.5608i) q^{76} -171.621 q^{77} -10.7712 q^{79} +(-23.5682 - 26.7239i) q^{80} +(-15.2915 - 3.29150i) q^{82} +118.996 q^{83} +11.7935i q^{85} +(-3.01590 - 0.649173i) q^{86} +(94.3320 - 127.749i) q^{88} +146.071i q^{89} +125.539i q^{91} +(-48.8012 + 108.107i) q^{92} +(-124.561 - 26.8118i) q^{94} +54.0969i q^{95} -3.70850 q^{97} +(-10.8368 + 50.3449i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 48 q^{7} - 56 q^{10} - 88 q^{22} + 136 q^{25} + 104 q^{28} + 208 q^{31} + 104 q^{34} + 104 q^{46} - 48 q^{49} - 424 q^{52} + 112 q^{55} - 64 q^{58} - 320 q^{64} + 168 q^{70} - 328 q^{73} + 40 q^{76} - 192 q^{79} - 80 q^{82} + 416 q^{88} - 552 q^{94} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.420861 + 1.95522i −0.210431 + 0.977609i
\(3\) 0 0
\(4\) −3.64575 1.64575i −0.911438 0.411438i
\(5\) −2.22699 −0.445398 −0.222699 0.974887i \(-0.571487\pi\)
−0.222699 + 0.974887i \(0.571487\pi\)
\(6\) 0 0
\(7\) −8.64575 −1.23511 −0.617554 0.786529i \(-0.711876\pi\)
−0.617554 + 0.786529i \(0.711876\pi\)
\(8\) 4.75216 6.43560i 0.594020 0.804450i
\(9\) 0 0
\(10\) 0.937254 4.35425i 0.0937254 0.435425i
\(11\) 19.8504 1.80458 0.902289 0.431132i \(-0.141886\pi\)
0.902289 + 0.431132i \(0.141886\pi\)
\(12\) 0 0
\(13\) 14.5203i 1.11694i −0.829524 0.558472i \(-0.811388\pi\)
0.829524 0.558472i \(-0.188612\pi\)
\(14\) 3.63866 16.9043i 0.259905 1.20745i
\(15\) 0 0
\(16\) 10.5830 + 12.0000i 0.661438 + 0.750000i
\(17\) 5.29570i 0.311512i −0.987796 0.155756i \(-0.950219\pi\)
0.987796 0.155756i \(-0.0497813\pi\)
\(18\) 0 0
\(19\) 24.2915i 1.27850i −0.768999 0.639250i \(-0.779245\pi\)
0.768999 0.639250i \(-0.220755\pi\)
\(20\) 8.11905 + 3.66507i 0.405952 + 0.183254i
\(21\) 0 0
\(22\) −8.35425 + 38.8118i −0.379739 + 1.76417i
\(23\) 29.6528i 1.28925i −0.764497 0.644627i \(-0.777013\pi\)
0.764497 0.644627i \(-0.222987\pi\)
\(24\) 0 0
\(25\) −20.0405 −0.801621
\(26\) 28.3903 + 6.11102i 1.09193 + 0.235039i
\(27\) 0 0
\(28\) 31.5203 + 14.2288i 1.12572 + 0.508170i
\(29\) −6.13742 −0.211635 −0.105818 0.994386i \(-0.533746\pi\)
−0.105818 + 0.994386i \(0.533746\pi\)
\(30\) 0 0
\(31\) 10.1255 0.326629 0.163314 0.986574i \(-0.447782\pi\)
0.163314 + 0.986574i \(0.447782\pi\)
\(32\) −27.9166 + 15.6417i −0.872393 + 0.488804i
\(33\) 0 0
\(34\) 10.3542 + 2.22876i 0.304537 + 0.0655517i
\(35\) 19.2540 0.550114
\(36\) 0 0
\(37\) 16.6458i 0.449885i −0.974372 0.224943i \(-0.927781\pi\)
0.974372 0.224943i \(-0.0722195\pi\)
\(38\) 47.4952 + 10.2234i 1.24987 + 0.269036i
\(39\) 0 0
\(40\) −10.5830 + 14.3320i −0.264575 + 0.358301i
\(41\) 7.82087i 0.190753i 0.995441 + 0.0953765i \(0.0304055\pi\)
−0.995441 + 0.0953765i \(0.969595\pi\)
\(42\) 0 0
\(43\) 1.54249i 0.0358718i 0.999839 + 0.0179359i \(0.00570948\pi\)
−0.999839 + 0.0179359i \(0.994291\pi\)
\(44\) −72.3695 32.6687i −1.64476 0.742472i
\(45\) 0 0
\(46\) 57.9778 + 12.4797i 1.26039 + 0.271299i
\(47\) 63.7069i 1.35547i 0.735308 + 0.677733i \(0.237037\pi\)
−0.735308 + 0.677733i \(0.762963\pi\)
\(48\) 0 0
\(49\) 25.7490 0.525490
\(50\) 8.43428 39.1836i 0.168686 0.783671i
\(51\) 0 0
\(52\) −23.8967 + 52.9373i −0.459553 + 1.01802i
\(53\) −76.5253 −1.44387 −0.721936 0.691959i \(-0.756748\pi\)
−0.721936 + 0.691959i \(0.756748\pi\)
\(54\) 0 0
\(55\) −44.2065 −0.803755
\(56\) −41.0860 + 55.6406i −0.733678 + 0.993583i
\(57\) 0 0
\(58\) 2.58301 12.0000i 0.0445346 0.206897i
\(59\) 41.0332 0.695477 0.347739 0.937592i \(-0.386950\pi\)
0.347739 + 0.937592i \(0.386950\pi\)
\(60\) 0 0
\(61\) 52.0627i 0.853488i −0.904373 0.426744i \(-0.859661\pi\)
0.904373 0.426744i \(-0.140339\pi\)
\(62\) −4.26143 + 19.7975i −0.0687327 + 0.319315i
\(63\) 0 0
\(64\) −18.8340 61.1660i −0.294281 0.955719i
\(65\) 32.3365i 0.497484i
\(66\) 0 0
\(67\) 110.247i 1.64548i −0.568419 0.822739i \(-0.692445\pi\)
0.568419 0.822739i \(-0.307555\pi\)
\(68\) −8.71541 + 19.3068i −0.128168 + 0.283924i
\(69\) 0 0
\(70\) −8.10326 + 37.6458i −0.115761 + 0.537796i
\(71\) 100.690i 1.41817i −0.705125 0.709083i \(-0.749109\pi\)
0.705125 0.709083i \(-0.250891\pi\)
\(72\) 0 0
\(73\) −93.9150 −1.28651 −0.643254 0.765653i \(-0.722416\pi\)
−0.643254 + 0.765653i \(0.722416\pi\)
\(74\) 32.5461 + 7.00555i 0.439812 + 0.0946697i
\(75\) 0 0
\(76\) −39.9778 + 88.5608i −0.526023 + 1.16527i
\(77\) −171.621 −2.22885
\(78\) 0 0
\(79\) −10.7712 −0.136345 −0.0681724 0.997674i \(-0.521717\pi\)
−0.0681724 + 0.997674i \(0.521717\pi\)
\(80\) −23.5682 26.7239i −0.294603 0.334048i
\(81\) 0 0
\(82\) −15.2915 3.29150i −0.186482 0.0401403i
\(83\) 118.996 1.43369 0.716846 0.697231i \(-0.245585\pi\)
0.716846 + 0.697231i \(0.245585\pi\)
\(84\) 0 0
\(85\) 11.7935i 0.138747i
\(86\) −3.01590 0.649173i −0.0350686 0.00754853i
\(87\) 0 0
\(88\) 94.3320 127.749i 1.07195 1.45169i
\(89\) 146.071i 1.64125i 0.571466 + 0.820626i \(0.306375\pi\)
−0.571466 + 0.820626i \(0.693625\pi\)
\(90\) 0 0
\(91\) 125.539i 1.37954i
\(92\) −48.8012 + 108.107i −0.530448 + 1.17508i
\(93\) 0 0
\(94\) −124.561 26.8118i −1.32511 0.285232i
\(95\) 54.0969i 0.569441i
\(96\) 0 0
\(97\) −3.70850 −0.0382319 −0.0191160 0.999817i \(-0.506085\pi\)
−0.0191160 + 0.999817i \(0.506085\pi\)
\(98\) −10.8368 + 50.3449i −0.110579 + 0.513724i
\(99\) 0 0
\(100\) 73.0627 + 32.9817i 0.730627 + 0.329817i
\(101\) −109.598 −1.08513 −0.542563 0.840015i \(-0.682546\pi\)
−0.542563 + 0.840015i \(0.682546\pi\)
\(102\) 0 0
\(103\) 159.391 1.54748 0.773742 0.633501i \(-0.218383\pi\)
0.773742 + 0.633501i \(0.218383\pi\)
\(104\) −93.4466 69.0026i −0.898525 0.663486i
\(105\) 0 0
\(106\) 32.2065 149.624i 0.303835 1.41154i
\(107\) 49.8355 0.465752 0.232876 0.972506i \(-0.425186\pi\)
0.232876 + 0.972506i \(0.425186\pi\)
\(108\) 0 0
\(109\) 115.956i 1.06381i 0.846803 + 0.531906i \(0.178524\pi\)
−0.846803 + 0.531906i \(0.821476\pi\)
\(110\) 18.6048 86.4334i 0.169135 0.785758i
\(111\) 0 0
\(112\) −91.4980 103.749i −0.816947 0.926330i
\(113\) 170.025i 1.50464i −0.658796 0.752322i \(-0.728934\pi\)
0.658796 0.752322i \(-0.271066\pi\)
\(114\) 0 0
\(115\) 66.0366i 0.574231i
\(116\) 22.3755 + 10.1007i 0.192892 + 0.0870748i
\(117\) 0 0
\(118\) −17.2693 + 80.2288i −0.146350 + 0.679905i
\(119\) 45.7853i 0.384751i
\(120\) 0 0
\(121\) 273.037 2.25650
\(122\) 101.794 + 21.9112i 0.834377 + 0.179600i
\(123\) 0 0
\(124\) −36.9150 16.6640i −0.297702 0.134387i
\(125\) 100.305 0.802438
\(126\) 0 0
\(127\) −106.583 −0.839236 −0.419618 0.907701i \(-0.637836\pi\)
−0.419618 + 0.907701i \(0.637836\pi\)
\(128\) 127.519 11.0821i 0.996245 0.0865792i
\(129\) 0 0
\(130\) −63.2248 13.6092i −0.486345 0.104686i
\(131\) 65.5488 0.500372 0.250186 0.968198i \(-0.419508\pi\)
0.250186 + 0.968198i \(0.419508\pi\)
\(132\) 0 0
\(133\) 210.018i 1.57908i
\(134\) 215.557 + 46.3987i 1.60863 + 0.346259i
\(135\) 0 0
\(136\) −34.0810 25.1660i −0.250596 0.185044i
\(137\) 141.829i 1.03525i −0.855609 0.517623i \(-0.826817\pi\)
0.855609 0.517623i \(-0.173183\pi\)
\(138\) 0 0
\(139\) 171.207i 1.23170i 0.787863 + 0.615851i \(0.211188\pi\)
−0.787863 + 0.615851i \(0.788812\pi\)
\(140\) −70.1953 31.6873i −0.501395 0.226338i
\(141\) 0 0
\(142\) 196.871 + 42.3765i 1.38641 + 0.298426i
\(143\) 288.232i 2.01561i
\(144\) 0 0
\(145\) 13.6680 0.0942619
\(146\) 39.5252 183.624i 0.270721 1.25770i
\(147\) 0 0
\(148\) −27.3948 + 60.6863i −0.185100 + 0.410042i
\(149\) 216.916 1.45581 0.727905 0.685678i \(-0.240494\pi\)
0.727905 + 0.685678i \(0.240494\pi\)
\(150\) 0 0
\(151\) −60.7268 −0.402164 −0.201082 0.979574i \(-0.564446\pi\)
−0.201082 + 0.979574i \(0.564446\pi\)
\(152\) −156.330 115.437i −1.02849 0.759454i
\(153\) 0 0
\(154\) 72.2288 335.557i 0.469018 2.17894i
\(155\) −22.5494 −0.145480
\(156\) 0 0
\(157\) 192.207i 1.22425i −0.790763 0.612123i \(-0.790316\pi\)
0.790763 0.612123i \(-0.209684\pi\)
\(158\) 4.53320 21.0601i 0.0286911 0.133292i
\(159\) 0 0
\(160\) 62.1699 34.8340i 0.388562 0.217712i
\(161\) 256.371i 1.59237i
\(162\) 0 0
\(163\) 6.29150i 0.0385982i 0.999814 + 0.0192991i \(0.00614347\pi\)
−0.999814 + 0.0192991i \(0.993857\pi\)
\(164\) 12.8712 28.5129i 0.0784830 0.173859i
\(165\) 0 0
\(166\) −50.0810 + 232.664i −0.301693 + 1.40159i
\(167\) 214.092i 1.28199i 0.767545 + 0.640995i \(0.221478\pi\)
−0.767545 + 0.640995i \(0.778522\pi\)
\(168\) 0 0
\(169\) −41.8379 −0.247562
\(170\) −23.0588 4.96342i −0.135640 0.0291966i
\(171\) 0 0
\(172\) 2.53855 5.62352i 0.0147590 0.0326949i
\(173\) 44.0491 0.254619 0.127309 0.991863i \(-0.459366\pi\)
0.127309 + 0.991863i \(0.459366\pi\)
\(174\) 0 0
\(175\) 173.265 0.990088
\(176\) 210.076 + 238.204i 1.19362 + 1.35343i
\(177\) 0 0
\(178\) −285.601 61.4758i −1.60450 0.345370i
\(179\) −132.570 −0.740613 −0.370306 0.928910i \(-0.620747\pi\)
−0.370306 + 0.928910i \(0.620747\pi\)
\(180\) 0 0
\(181\) 141.516i 0.781858i 0.920421 + 0.390929i \(0.127846\pi\)
−0.920421 + 0.390929i \(0.872154\pi\)
\(182\) −245.455 52.8343i −1.34865 0.290299i
\(183\) 0 0
\(184\) −190.834 140.915i −1.03714 0.765843i
\(185\) 37.0699i 0.200378i
\(186\) 0 0
\(187\) 105.122i 0.562147i
\(188\) 104.846 232.259i 0.557690 1.23542i
\(189\) 0 0
\(190\) −105.771 22.7673i −0.556691 0.119828i
\(191\) 1.45679i 0.00762718i −0.999993 0.00381359i \(-0.998786\pi\)
0.999993 0.00381359i \(-0.00121391\pi\)
\(192\) 0 0
\(193\) −243.539 −1.26186 −0.630929 0.775841i \(-0.717326\pi\)
−0.630929 + 0.775841i \(0.717326\pi\)
\(194\) 1.56076 7.25092i 0.00804517 0.0373759i
\(195\) 0 0
\(196\) −93.8745 42.3765i −0.478952 0.216207i
\(197\) −4.99752 −0.0253681 −0.0126841 0.999920i \(-0.504038\pi\)
−0.0126841 + 0.999920i \(0.504038\pi\)
\(198\) 0 0
\(199\) 102.476 0.514954 0.257477 0.966284i \(-0.417109\pi\)
0.257477 + 0.966284i \(0.417109\pi\)
\(200\) −95.2357 + 128.973i −0.476179 + 0.644864i
\(201\) 0 0
\(202\) 46.1255 214.288i 0.228344 1.06083i
\(203\) 53.0626 0.261392
\(204\) 0 0
\(205\) 17.4170i 0.0849610i
\(206\) −67.0815 + 311.644i −0.325638 + 1.51283i
\(207\) 0 0
\(208\) 174.243 153.668i 0.837707 0.738788i
\(209\) 482.195i 2.30715i
\(210\) 0 0
\(211\) 9.54642i 0.0452437i 0.999744 + 0.0226219i \(0.00720138\pi\)
−0.999744 + 0.0226219i \(0.992799\pi\)
\(212\) 278.992 + 125.942i 1.31600 + 0.594064i
\(213\) 0 0
\(214\) −20.9738 + 97.4392i −0.0980086 + 0.455323i
\(215\) 3.43510i 0.0159772i
\(216\) 0 0
\(217\) −87.5425 −0.403422
\(218\) −226.718 48.8012i −1.03999 0.223859i
\(219\) 0 0
\(220\) 161.166 + 72.7530i 0.732573 + 0.330695i
\(221\) −76.8950 −0.347941
\(222\) 0 0
\(223\) 51.1294 0.229280 0.114640 0.993407i \(-0.463429\pi\)
0.114640 + 0.993407i \(0.463429\pi\)
\(224\) 241.360 135.235i 1.07750 0.603726i
\(225\) 0 0
\(226\) 332.435 + 71.5568i 1.47095 + 0.316623i
\(227\) −235.539 −1.03762 −0.518809 0.854890i \(-0.673624\pi\)
−0.518809 + 0.854890i \(0.673624\pi\)
\(228\) 0 0
\(229\) 206.332i 0.901013i 0.892773 + 0.450507i \(0.148757\pi\)
−0.892773 + 0.450507i \(0.851243\pi\)
\(230\) −129.116 27.7923i −0.561373 0.120836i
\(231\) 0 0
\(232\) −29.1660 + 39.4980i −0.125716 + 0.170250i
\(233\) 210.356i 0.902815i 0.892318 + 0.451407i \(0.149078\pi\)
−0.892318 + 0.451407i \(0.850922\pi\)
\(234\) 0 0
\(235\) 141.875i 0.603721i
\(236\) −149.597 67.5304i −0.633884 0.286146i
\(237\) 0 0
\(238\) −89.5203 19.2693i −0.376136 0.0809633i
\(239\) 23.1457i 0.0968440i −0.998827 0.0484220i \(-0.984581\pi\)
0.998827 0.0484220i \(-0.0154192\pi\)
\(240\) 0 0
\(241\) 81.6275 0.338703 0.169352 0.985556i \(-0.445833\pi\)
0.169352 + 0.985556i \(0.445833\pi\)
\(242\) −114.911 + 533.846i −0.474837 + 2.20597i
\(243\) 0 0
\(244\) −85.6823 + 189.808i −0.351157 + 0.777901i
\(245\) −57.3428 −0.234052
\(246\) 0 0
\(247\) −352.719 −1.42801
\(248\) 48.1179 65.1637i 0.194024 0.262757i
\(249\) 0 0
\(250\) −42.2144 + 196.118i −0.168858 + 0.784470i
\(251\) −232.769 −0.927366 −0.463683 0.886001i \(-0.653472\pi\)
−0.463683 + 0.886001i \(0.653472\pi\)
\(252\) 0 0
\(253\) 588.620i 2.32656i
\(254\) 44.8567 208.393i 0.176601 0.820445i
\(255\) 0 0
\(256\) −32.0000 + 253.992i −0.125000 + 0.992157i
\(257\) 298.491i 1.16145i 0.814102 + 0.580723i \(0.197230\pi\)
−0.814102 + 0.580723i \(0.802770\pi\)
\(258\) 0 0
\(259\) 143.915i 0.555656i
\(260\) 53.2178 117.891i 0.204684 0.453426i
\(261\) 0 0
\(262\) −27.5869 + 128.162i −0.105294 + 0.489168i
\(263\) 400.163i 1.52153i 0.649027 + 0.760766i \(0.275176\pi\)
−0.649027 + 0.760766i \(0.724824\pi\)
\(264\) 0 0
\(265\) 170.421 0.643098
\(266\) −410.631 88.3886i −1.54373 0.332288i
\(267\) 0 0
\(268\) −181.439 + 401.933i −0.677012 + 1.49975i
\(269\) 446.650 1.66041 0.830205 0.557459i \(-0.188224\pi\)
0.830205 + 0.557459i \(0.188224\pi\)
\(270\) 0 0
\(271\) −30.3503 −0.111994 −0.0559969 0.998431i \(-0.517834\pi\)
−0.0559969 + 0.998431i \(0.517834\pi\)
\(272\) 63.5484 56.0444i 0.233634 0.206046i
\(273\) 0 0
\(274\) 277.306 + 59.6902i 1.01207 + 0.217847i
\(275\) −397.811 −1.44659
\(276\) 0 0
\(277\) 149.284i 0.538930i −0.963010 0.269465i \(-0.913153\pi\)
0.963010 0.269465i \(-0.0868469\pi\)
\(278\) −334.746 72.0542i −1.20412 0.259188i
\(279\) 0 0
\(280\) 91.4980 123.911i 0.326779 0.442540i
\(281\) 81.9607i 0.291675i −0.989309 0.145838i \(-0.953412\pi\)
0.989309 0.145838i \(-0.0465877\pi\)
\(282\) 0 0
\(283\) 462.207i 1.63324i −0.577177 0.816619i \(-0.695846\pi\)
0.577177 0.816619i \(-0.304154\pi\)
\(284\) −165.710 + 367.090i −0.583488 + 1.29257i
\(285\) 0 0
\(286\) 563.557 + 121.306i 1.97048 + 0.424146i
\(287\) 67.6173i 0.235600i
\(288\) 0 0
\(289\) 260.956 0.902960
\(290\) −5.75233 + 26.7239i −0.0198356 + 0.0921513i
\(291\) 0 0
\(292\) 342.391 + 154.561i 1.17257 + 0.529318i
\(293\) 335.860 1.14628 0.573139 0.819458i \(-0.305725\pi\)
0.573139 + 0.819458i \(0.305725\pi\)
\(294\) 0 0
\(295\) −91.3804 −0.309764
\(296\) −107.125 79.1032i −0.361910 0.267241i
\(297\) 0 0
\(298\) −91.2915 + 424.118i −0.306347 + 1.42321i
\(299\) −430.567 −1.44002
\(300\) 0 0
\(301\) 13.3360i 0.0443055i
\(302\) 25.5576 118.734i 0.0846277 0.393159i
\(303\) 0 0
\(304\) 291.498 257.077i 0.958875 0.845648i
\(305\) 115.943i 0.380142i
\(306\) 0 0
\(307\) 40.6275i 0.132337i 0.997808 + 0.0661685i \(0.0210775\pi\)
−0.997808 + 0.0661685i \(0.978923\pi\)
\(308\) 625.688 + 282.446i 2.03146 + 0.917032i
\(309\) 0 0
\(310\) 9.49016 44.0889i 0.0306134 0.142222i
\(311\) 30.9512i 0.0995215i 0.998761 + 0.0497608i \(0.0158459\pi\)
−0.998761 + 0.0497608i \(0.984154\pi\)
\(312\) 0 0
\(313\) −285.612 −0.912497 −0.456249 0.889852i \(-0.650807\pi\)
−0.456249 + 0.889852i \(0.650807\pi\)
\(314\) 375.806 + 80.8923i 1.19683 + 0.257619i
\(315\) 0 0
\(316\) 39.2693 + 17.7268i 0.124270 + 0.0560974i
\(317\) 173.426 0.547084 0.273542 0.961860i \(-0.411805\pi\)
0.273542 + 0.961860i \(0.411805\pi\)
\(318\) 0 0
\(319\) −121.830 −0.381912
\(320\) 41.9431 + 136.216i 0.131072 + 0.425675i
\(321\) 0 0
\(322\) −501.261 107.897i −1.55671 0.335083i
\(323\) −128.641 −0.398268
\(324\) 0 0
\(325\) 290.994i 0.895365i
\(326\) −12.3013 2.64785i −0.0377339 0.00812224i
\(327\) 0 0
\(328\) 50.3320 + 37.1660i 0.153451 + 0.113311i
\(329\) 550.794i 1.67414i
\(330\) 0 0
\(331\) 228.705i 0.690950i −0.938428 0.345475i \(-0.887718\pi\)
0.938428 0.345475i \(-0.112282\pi\)
\(332\) −433.832 195.839i −1.30672 0.589875i
\(333\) 0 0
\(334\) −418.597 90.1033i −1.25329 0.269770i
\(335\) 245.519i 0.732893i
\(336\) 0 0
\(337\) −403.207 −1.19646 −0.598229 0.801325i \(-0.704129\pi\)
−0.598229 + 0.801325i \(0.704129\pi\)
\(338\) 17.6080 81.8022i 0.0520946 0.242018i
\(339\) 0 0
\(340\) 19.4091 42.9961i 0.0570856 0.126459i
\(341\) 200.995 0.589427
\(342\) 0 0
\(343\) 201.022 0.586071
\(344\) 9.92683 + 7.33014i 0.0288571 + 0.0213086i
\(345\) 0 0
\(346\) −18.5385 + 86.1255i −0.0535796 + 0.248918i
\(347\) −166.941 −0.481097 −0.240548 0.970637i \(-0.577327\pi\)
−0.240548 + 0.970637i \(0.577327\pi\)
\(348\) 0 0
\(349\) 552.395i 1.58279i −0.611303 0.791397i \(-0.709354\pi\)
0.611303 0.791397i \(-0.290646\pi\)
\(350\) −72.9207 + 338.771i −0.208345 + 0.967918i
\(351\) 0 0
\(352\) −554.154 + 310.494i −1.57430 + 0.882085i
\(353\) 54.0815i 0.153205i −0.997062 0.0766027i \(-0.975593\pi\)
0.997062 0.0766027i \(-0.0244073\pi\)
\(354\) 0 0
\(355\) 224.235i 0.631649i
\(356\) 240.397 532.540i 0.675273 1.49590i
\(357\) 0 0
\(358\) 55.7935 259.203i 0.155848 0.724030i
\(359\) 623.374i 1.73642i −0.496198 0.868209i \(-0.665271\pi\)
0.496198 0.868209i \(-0.334729\pi\)
\(360\) 0 0
\(361\) −229.077 −0.634563
\(362\) −276.695 59.5588i −0.764351 0.164527i
\(363\) 0 0
\(364\) 206.605 457.682i 0.567597 1.25737i
\(365\) 209.148 0.573008
\(366\) 0 0
\(367\) 61.2653 0.166936 0.0834678 0.996510i \(-0.473400\pi\)
0.0834678 + 0.996510i \(0.473400\pi\)
\(368\) 355.834 313.816i 0.966941 0.852762i
\(369\) 0 0
\(370\) −72.4797 15.6013i −0.195891 0.0421657i
\(371\) 661.618 1.78334
\(372\) 0 0
\(373\) 72.7712i 0.195097i 0.995231 + 0.0975486i \(0.0311001\pi\)
−0.995231 + 0.0975486i \(0.968900\pi\)
\(374\) 205.536 + 44.2416i 0.549560 + 0.118293i
\(375\) 0 0
\(376\) 409.992 + 302.745i 1.09040 + 0.805173i
\(377\) 89.1170i 0.236385i
\(378\) 0 0
\(379\) 341.405i 0.900805i −0.892826 0.450403i \(-0.851280\pi\)
0.892826 0.450403i \(-0.148720\pi\)
\(380\) 89.0301 197.224i 0.234290 0.519010i
\(381\) 0 0
\(382\) 2.84834 + 0.613107i 0.00745640 + 0.00160499i
\(383\) 318.059i 0.830441i 0.909721 + 0.415221i \(0.136296\pi\)
−0.909721 + 0.415221i \(0.863704\pi\)
\(384\) 0 0
\(385\) 382.199 0.992724
\(386\) 102.496 476.171i 0.265534 1.23360i
\(387\) 0 0
\(388\) 13.5203 + 6.10326i 0.0348460 + 0.0157301i
\(389\) 308.645 0.793432 0.396716 0.917941i \(-0.370150\pi\)
0.396716 + 0.917941i \(0.370150\pi\)
\(390\) 0 0
\(391\) −157.033 −0.401618
\(392\) 122.363 165.710i 0.312152 0.422731i
\(393\) 0 0
\(394\) 2.10326 9.77124i 0.00533823 0.0248001i
\(395\) 23.9874 0.0607277
\(396\) 0 0
\(397\) 172.044i 0.433361i −0.976243 0.216681i \(-0.930477\pi\)
0.976243 0.216681i \(-0.0695230\pi\)
\(398\) −43.1281 + 200.362i −0.108362 + 0.503423i
\(399\) 0 0
\(400\) −212.089 240.486i −0.530222 0.601216i
\(401\) 329.110i 0.820724i −0.911923 0.410362i \(-0.865402\pi\)
0.911923 0.410362i \(-0.134598\pi\)
\(402\) 0 0
\(403\) 147.025i 0.364826i
\(404\) 399.566 + 180.371i 0.989026 + 0.446462i
\(405\) 0 0
\(406\) −22.3320 + 103.749i −0.0550050 + 0.255539i
\(407\) 330.424i 0.811853i
\(408\) 0 0
\(409\) −353.458 −0.864199 −0.432100 0.901826i \(-0.642227\pi\)
−0.432100 + 0.901826i \(0.642227\pi\)
\(410\) 34.0540 + 7.33014i 0.0830586 + 0.0178784i
\(411\) 0 0
\(412\) −581.099 262.318i −1.41044 0.636693i
\(413\) −354.763 −0.858989
\(414\) 0 0
\(415\) −265.004 −0.638564
\(416\) 227.122 + 405.356i 0.545967 + 0.974414i
\(417\) 0 0
\(418\) 942.796 + 202.937i 2.25549 + 0.485496i
\(419\) 327.287 0.781115 0.390557 0.920579i \(-0.372282\pi\)
0.390557 + 0.920579i \(0.372282\pi\)
\(420\) 0 0
\(421\) 501.759i 1.19183i 0.803048 + 0.595914i \(0.203210\pi\)
−0.803048 + 0.595914i \(0.796790\pi\)
\(422\) −18.6653 4.01772i −0.0442307 0.00952067i
\(423\) 0 0
\(424\) −363.660 + 492.486i −0.857689 + 1.16152i
\(425\) 106.129i 0.249714i
\(426\) 0 0
\(427\) 450.122i 1.05415i
\(428\) −181.688 82.0168i −0.424504 0.191628i
\(429\) 0 0
\(430\) 6.71637 + 1.44570i 0.0156195 + 0.00336210i
\(431\) 29.8267i 0.0692035i −0.999401 0.0346017i \(-0.988984\pi\)
0.999401 0.0346017i \(-0.0110163\pi\)
\(432\) 0 0
\(433\) 519.859 1.20060 0.600299 0.799776i \(-0.295048\pi\)
0.600299 + 0.799776i \(0.295048\pi\)
\(434\) 36.8433 171.165i 0.0848923 0.394388i
\(435\) 0 0
\(436\) 190.834 422.745i 0.437693 0.969599i
\(437\) −720.312 −1.64831
\(438\) 0 0
\(439\) 386.450 0.880295 0.440148 0.897925i \(-0.354926\pi\)
0.440148 + 0.897925i \(0.354926\pi\)
\(440\) −210.076 + 284.496i −0.477446 + 0.646581i
\(441\) 0 0
\(442\) 32.3621 150.346i 0.0732175 0.340150i
\(443\) 378.527 0.854462 0.427231 0.904143i \(-0.359489\pi\)
0.427231 + 0.904143i \(0.359489\pi\)
\(444\) 0 0
\(445\) 325.299i 0.731010i
\(446\) −21.5184 + 99.9692i −0.0482475 + 0.224146i
\(447\) 0 0
\(448\) 162.834 + 528.826i 0.363469 + 1.18042i
\(449\) 289.127i 0.643935i 0.946751 + 0.321968i \(0.104344\pi\)
−0.946751 + 0.321968i \(0.895656\pi\)
\(450\) 0 0
\(451\) 155.247i 0.344228i
\(452\) −279.818 + 619.868i −0.619067 + 1.37139i
\(453\) 0 0
\(454\) 99.1294 460.531i 0.218347 1.01438i
\(455\) 279.573i 0.614446i
\(456\) 0 0
\(457\) 485.365 1.06207 0.531034 0.847351i \(-0.321804\pi\)
0.531034 + 0.847351i \(0.321804\pi\)
\(458\) −403.424 86.8372i −0.880838 0.189601i
\(459\) 0 0
\(460\) 108.680 240.753i 0.236260 0.523376i
\(461\) −215.037 −0.466457 −0.233228 0.972422i \(-0.574929\pi\)
−0.233228 + 0.972422i \(0.574929\pi\)
\(462\) 0 0
\(463\) −374.755 −0.809407 −0.404704 0.914448i \(-0.632625\pi\)
−0.404704 + 0.914448i \(0.632625\pi\)
\(464\) −64.9524 73.6491i −0.139984 0.158726i
\(465\) 0 0
\(466\) −411.292 88.5307i −0.882600 0.189980i
\(467\) −268.050 −0.573982 −0.286991 0.957933i \(-0.592655\pi\)
−0.286991 + 0.957933i \(0.592655\pi\)
\(468\) 0 0
\(469\) 953.169i 2.03234i
\(470\) 277.396 + 59.7095i 0.590203 + 0.127042i
\(471\) 0 0
\(472\) 194.996 264.073i 0.413127 0.559477i
\(473\) 30.6189i 0.0647334i
\(474\) 0 0
\(475\) 486.814i 1.02487i
\(476\) 75.3512 166.922i 0.158301 0.350676i
\(477\) 0 0
\(478\) 45.2549 + 9.74114i 0.0946756 + 0.0203790i
\(479\) 512.109i 1.06912i 0.845130 + 0.534560i \(0.179523\pi\)
−0.845130 + 0.534560i \(0.820477\pi\)
\(480\) 0 0
\(481\) −241.701 −0.502496
\(482\) −34.3538 + 159.599i −0.0712735 + 0.331119i
\(483\) 0 0
\(484\) −995.423 449.350i −2.05666 0.928410i
\(485\) 8.25878 0.0170284
\(486\) 0 0
\(487\) 885.804 1.81890 0.909450 0.415814i \(-0.136503\pi\)
0.909450 + 0.415814i \(0.136503\pi\)
\(488\) −335.055 247.410i −0.686589 0.506989i
\(489\) 0 0
\(490\) 24.1334 112.118i 0.0492518 0.228811i
\(491\) 434.847 0.885636 0.442818 0.896612i \(-0.353979\pi\)
0.442818 + 0.896612i \(0.353979\pi\)
\(492\) 0 0
\(493\) 32.5020i 0.0659269i
\(494\) 148.446 689.642i 0.300498 1.39604i
\(495\) 0 0
\(496\) 107.158 + 121.506i 0.216045 + 0.244972i
\(497\) 870.539i 1.75159i
\(498\) 0 0
\(499\) 712.980i 1.42882i 0.699728 + 0.714409i \(0.253304\pi\)
−0.699728 + 0.714409i \(0.746696\pi\)
\(500\) −365.686 165.077i −0.731372 0.330153i
\(501\) 0 0
\(502\) 97.9634 455.114i 0.195146 0.906601i
\(503\) 68.4403i 0.136064i −0.997683 0.0680321i \(-0.978328\pi\)
0.997683 0.0680321i \(-0.0216720\pi\)
\(504\) 0 0
\(505\) 244.073 0.483313
\(506\) 1150.88 + 247.727i 2.27447 + 0.489580i
\(507\) 0 0
\(508\) 388.575 + 175.409i 0.764912 + 0.345294i
\(509\) −7.45117 −0.0146388 −0.00731942 0.999973i \(-0.502330\pi\)
−0.00731942 + 0.999973i \(0.502330\pi\)
\(510\) 0 0
\(511\) 811.966 1.58897
\(512\) −483.142 169.462i −0.943637 0.330981i
\(513\) 0 0
\(514\) −583.616 125.624i −1.13544 0.244404i
\(515\) −354.962 −0.689246
\(516\) 0 0
\(517\) 1264.60i 2.44604i
\(518\) −281.385 60.5683i −0.543215 0.116927i
\(519\) 0 0
\(520\) 208.105 + 153.668i 0.400201 + 0.295515i
\(521\) 426.007i 0.817673i −0.912608 0.408836i \(-0.865935\pi\)
0.912608 0.408836i \(-0.134065\pi\)
\(522\) 0 0
\(523\) 270.875i 0.517924i 0.965887 + 0.258962i \(0.0833805\pi\)
−0.965887 + 0.258962i \(0.916619\pi\)
\(524\) −238.974 107.877i −0.456058 0.205872i
\(525\) 0 0
\(526\) −782.405 168.413i −1.48746 0.320177i
\(527\) 53.6216i 0.101749i
\(528\) 0 0
\(529\) −350.292 −0.662177
\(530\) −71.7236 + 333.210i −0.135328 + 0.628698i
\(531\) 0 0
\(532\) 345.638 765.674i 0.649695 1.43924i
\(533\) 113.561 0.213060
\(534\) 0 0
\(535\) −110.983 −0.207445
\(536\) −709.506 523.911i −1.32371 0.977447i
\(537\) 0 0
\(538\) −187.978 + 873.298i −0.349401 + 1.62323i
\(539\) 511.127 0.948288
\(540\) 0 0
\(541\) 964.379i 1.78259i −0.453428 0.891293i \(-0.649799\pi\)
0.453428 0.891293i \(-0.350201\pi\)
\(542\) 12.7733 59.3415i 0.0235669 0.109486i
\(543\) 0 0
\(544\) 82.8340 + 147.838i 0.152268 + 0.271761i
\(545\) 258.232i 0.473820i
\(546\) 0 0
\(547\) 485.162i 0.886951i −0.896287 0.443475i \(-0.853745\pi\)
0.896287 0.443475i \(-0.146255\pi\)
\(548\) −233.415 + 517.072i −0.425939 + 0.943562i
\(549\) 0 0
\(550\) 167.423 777.808i 0.304406 1.41420i
\(551\) 149.087i 0.270576i
\(552\) 0 0
\(553\) 93.1255 0.168401
\(554\) 291.882 + 62.8277i 0.526863 + 0.113407i
\(555\) 0 0
\(556\) 281.763 624.176i 0.506769 1.12262i
\(557\) −930.917 −1.67130 −0.835652 0.549259i \(-0.814910\pi\)
−0.835652 + 0.549259i \(0.814910\pi\)
\(558\) 0 0
\(559\) 22.3973 0.0400667
\(560\) 203.765 + 231.048i 0.363866 + 0.412586i
\(561\) 0 0
\(562\) 160.251 + 34.4941i 0.285144 + 0.0613774i
\(563\) −511.376 −0.908305 −0.454153 0.890924i \(-0.650058\pi\)
−0.454153 + 0.890924i \(0.650058\pi\)
\(564\) 0 0
\(565\) 378.643i 0.670165i
\(566\) 903.714 + 194.525i 1.59667 + 0.343684i
\(567\) 0 0
\(568\) −648.000 478.494i −1.14085 0.842419i
\(569\) 88.0640i 0.154770i −0.997001 0.0773849i \(-0.975343\pi\)
0.997001 0.0773849i \(-0.0246570\pi\)
\(570\) 0 0
\(571\) 451.288i 0.790346i −0.918607 0.395173i \(-0.870685\pi\)
0.918607 0.395173i \(-0.129315\pi\)
\(572\) −474.359 + 1050.82i −0.829298 + 1.83710i
\(573\) 0 0
\(574\) 132.207 + 28.4575i 0.230325 + 0.0495775i
\(575\) 594.258i 1.03349i
\(576\) 0 0
\(577\) −177.405 −0.307461 −0.153731 0.988113i \(-0.549129\pi\)
−0.153731 + 0.988113i \(0.549129\pi\)
\(578\) −109.826 + 510.225i −0.190011 + 0.882742i
\(579\) 0 0
\(580\) −49.8301 22.4941i −0.0859139 0.0387829i
\(581\) −1028.81 −1.77076
\(582\) 0 0
\(583\) −1519.05 −2.60558
\(584\) −446.299 + 604.400i −0.764211 + 1.03493i
\(585\) 0 0
\(586\) −141.350 + 656.678i −0.241212 + 1.12061i
\(587\) −74.8077 −0.127441 −0.0637204 0.997968i \(-0.520297\pi\)
−0.0637204 + 0.997968i \(0.520297\pi\)
\(588\) 0 0
\(589\) 245.963i 0.417595i
\(590\) 38.4585 178.669i 0.0651839 0.302828i
\(591\) 0 0
\(592\) 199.749 176.162i 0.337414 0.297571i
\(593\) 149.087i 0.251412i −0.992068 0.125706i \(-0.959880\pi\)
0.992068 0.125706i \(-0.0401196\pi\)
\(594\) 0 0
\(595\) 101.963i 0.171367i
\(596\) −790.821 356.989i −1.32688 0.598976i
\(597\) 0 0
\(598\) 181.209 841.852i 0.303025 1.40778i
\(599\) 234.452i 0.391406i −0.980663 0.195703i \(-0.937301\pi\)
0.980663 0.195703i \(-0.0626989\pi\)
\(600\) 0 0
\(601\) 20.8627 0.0347133 0.0173567 0.999849i \(-0.494475\pi\)
0.0173567 + 0.999849i \(0.494475\pi\)
\(602\) 26.0747 + 5.61259i 0.0433135 + 0.00932324i
\(603\) 0 0
\(604\) 221.395 + 99.9412i 0.366548 + 0.165466i
\(605\) −608.050 −1.00504
\(606\) 0 0
\(607\) 119.103 0.196216 0.0981081 0.995176i \(-0.468721\pi\)
0.0981081 + 0.995176i \(0.468721\pi\)
\(608\) 379.961 + 678.136i 0.624936 + 1.11536i
\(609\) 0 0
\(610\) −226.694 48.7960i −0.371630 0.0799935i
\(611\) 925.040 1.51398
\(612\) 0 0
\(613\) 610.586i 0.996061i −0.867159 0.498031i \(-0.834057\pi\)
0.867159 0.498031i \(-0.165943\pi\)
\(614\) −79.4355 17.0985i −0.129374 0.0278478i
\(615\) 0 0
\(616\) −815.571 + 1104.49i −1.32398 + 1.79300i
\(617\) 1199.40i 1.94393i 0.235127 + 0.971965i \(0.424449\pi\)
−0.235127 + 0.971965i \(0.575551\pi\)
\(618\) 0 0
\(619\) 982.482i 1.58721i 0.608434 + 0.793604i \(0.291798\pi\)
−0.608434 + 0.793604i \(0.708202\pi\)
\(620\) 82.2094 + 37.1106i 0.132596 + 0.0598559i
\(621\) 0 0
\(622\) −60.5163 13.0262i −0.0972931 0.0209424i
\(623\) 1262.90i 2.02712i
\(624\) 0 0
\(625\) 277.635 0.444217
\(626\) 120.203 558.433i 0.192017 0.892066i
\(627\) 0 0
\(628\) −316.324 + 700.737i −0.503701 + 1.11582i
\(629\) −88.1509 −0.140145
\(630\) 0 0
\(631\) −25.8928 −0.0410345 −0.0205173 0.999789i \(-0.506531\pi\)
−0.0205173 + 0.999789i \(0.506531\pi\)
\(632\) −51.1867 + 69.3195i −0.0809915 + 0.109683i
\(633\) 0 0
\(634\) −72.9882 + 339.085i −0.115123 + 0.534834i
\(635\) 237.359 0.373794
\(636\) 0 0
\(637\) 373.882i 0.586943i
\(638\) 51.2736 238.204i 0.0803661 0.373361i
\(639\) 0 0
\(640\) −283.984 + 24.6798i −0.443725 + 0.0385622i
\(641\) 74.7736i 0.116651i 0.998298 + 0.0583257i \(0.0185762\pi\)
−0.998298 + 0.0583257i \(0.981424\pi\)
\(642\) 0 0
\(643\) 804.936i 1.25184i −0.779886 0.625922i \(-0.784723\pi\)
0.779886 0.625922i \(-0.215277\pi\)
\(644\) 421.923 934.666i 0.655160 1.45134i
\(645\) 0 0
\(646\) 54.1398 251.520i 0.0838078 0.389350i
\(647\) 1114.00i 1.72179i 0.508782 + 0.860895i \(0.330096\pi\)
−0.508782 + 0.860895i \(0.669904\pi\)
\(648\) 0 0
\(649\) 814.523 1.25504
\(650\) −568.956 122.468i −0.875316 0.188412i
\(651\) 0 0
\(652\) 10.3542 22.9373i 0.0158807 0.0351798i
\(653\) 789.281 1.20870 0.604350 0.796719i \(-0.293433\pi\)
0.604350 + 0.796719i \(0.293433\pi\)
\(654\) 0 0
\(655\) −145.976 −0.222865
\(656\) −93.8504 + 82.7683i −0.143065 + 0.126171i
\(657\) 0 0
\(658\) 1076.92 + 231.808i 1.63666 + 0.352292i
\(659\) 1233.28 1.87144 0.935722 0.352739i \(-0.114750\pi\)
0.935722 + 0.352739i \(0.114750\pi\)
\(660\) 0 0
\(661\) 4.49943i 0.00680700i 0.999994 + 0.00340350i \(0.00108337\pi\)
−0.999994 + 0.00340350i \(0.998917\pi\)
\(662\) 447.167 + 96.2529i 0.675479 + 0.145397i
\(663\) 0 0
\(664\) 565.490 765.814i 0.851642 1.15333i
\(665\) 467.709i 0.703321i
\(666\) 0 0
\(667\) 181.992i 0.272852i
\(668\) 352.343 780.528i 0.527460 1.16845i
\(669\) 0 0
\(670\) −480.043 103.329i −0.716482 0.154223i
\(671\) 1033.46i 1.54018i
\(672\) 0 0
\(673\) −163.620 −0.243120 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(674\) 169.694 788.356i 0.251772 1.16967i
\(675\) 0 0
\(676\) 152.531 + 68.8548i 0.225637 + 0.101856i
\(677\) −686.078 −1.01341 −0.506704 0.862120i \(-0.669136\pi\)
−0.506704 + 0.862120i \(0.669136\pi\)
\(678\) 0 0
\(679\) 32.0627 0.0472205
\(680\) 75.8981 + 56.0444i 0.111615 + 0.0824183i
\(681\) 0 0
\(682\) −84.5909 + 392.988i −0.124034 + 0.576229i
\(683\) 267.879 0.392210 0.196105 0.980583i \(-0.437171\pi\)
0.196105 + 0.980583i \(0.437171\pi\)
\(684\) 0 0
\(685\) 315.851i 0.461096i
\(686\) −84.6025 + 393.042i −0.123327 + 0.572948i
\(687\) 0 0
\(688\) −18.5098 + 16.3241i −0.0269038 + 0.0237270i
\(689\) 1111.17i 1.61272i
\(690\) 0 0
\(691\) 576.037i 0.833627i −0.908992 0.416814i \(-0.863147\pi\)
0.908992 0.416814i \(-0.136853\pi\)
\(692\) −160.592 72.4938i −0.232069 0.104760i
\(693\) 0 0
\(694\) 70.2589 326.405i 0.101238 0.470324i
\(695\) 381.275i 0.548597i
\(696\) 0 0
\(697\) 41.4170 0.0594218
\(698\) 1080.05 + 232.482i 1.54735 + 0.333068i
\(699\) 0 0
\(700\) −631.682 285.152i −0.902403 0.407360i
\(701\) −356.794 −0.508978 −0.254489 0.967076i \(-0.581907\pi\)
−0.254489 + 0.967076i \(0.581907\pi\)
\(702\) 0 0
\(703\) −404.350 −0.575178
\(704\) −373.861 1214.17i −0.531053 1.72467i
\(705\) 0 0
\(706\) 105.741 + 22.7608i 0.149775 + 0.0322391i
\(707\) 947.555 1.34025
\(708\) 0 0
\(709\) 1146.88i 1.61760i 0.588081 + 0.808802i \(0.299884\pi\)
−0.588081 + 0.808802i \(0.700116\pi\)
\(710\) −438.429 94.3720i −0.617505 0.132918i
\(711\) 0 0
\(712\) 940.057 + 694.154i 1.32031 + 0.974936i
\(713\) 300.250i 0.421108i
\(714\) 0 0
\(715\) 641.890i 0.897749i
\(716\) 483.316 + 218.177i 0.675023 + 0.304716i
\(717\) 0 0
\(718\) 1218.83 + 262.354i 1.69754 + 0.365396i
\(719\) 79.3640i 0.110381i −0.998476 0.0551905i \(-0.982423\pi\)
0.998476 0.0551905i \(-0.0175766\pi\)
\(720\) 0 0
\(721\) −1378.05 −1.91131
\(722\) 96.4097 447.896i 0.133531 0.620354i
\(723\) 0 0
\(724\) 232.901 515.933i 0.321686 0.712615i
\(725\) 122.997 0.169651
\(726\) 0 0
\(727\) 871.940 1.19937 0.599683 0.800237i \(-0.295293\pi\)
0.599683 + 0.800237i \(0.295293\pi\)
\(728\) 807.916 + 596.579i 1.10978 + 0.819477i
\(729\) 0 0
\(730\) −88.0222 + 408.929i −0.120578 + 0.560177i
\(731\) 8.16855 0.0111745
\(732\) 0 0
\(733\) 1266.45i 1.72776i −0.503696 0.863881i \(-0.668027\pi\)
0.503696 0.863881i \(-0.331973\pi\)
\(734\) −25.7842 + 119.787i −0.0351284 + 0.163198i
\(735\) 0 0
\(736\) 463.822 + 827.806i 0.630193 + 1.12474i
\(737\) 2188.44i 2.96939i
\(738\) 0 0
\(739\) 560.199i 0.758050i −0.925387 0.379025i \(-0.876260\pi\)
0.925387 0.379025i \(-0.123740\pi\)
\(740\) 61.0079 135.148i 0.0824430 0.182632i
\(741\) 0 0
\(742\) −278.450 + 1293.61i −0.375269 + 1.74341i
\(743\) 470.187i 0.632823i 0.948622 + 0.316411i \(0.102478\pi\)
−0.948622 + 0.316411i \(0.897522\pi\)
\(744\) 0 0
\(745\) −483.069 −0.648415
\(746\) −142.284 30.6266i −0.190729 0.0410544i
\(747\) 0 0
\(748\) −173.004 + 383.247i −0.231289 + 0.512362i
\(749\) −430.865 −0.575254
\(750\) 0 0
\(751\) 1127.93 1.50190 0.750952 0.660357i \(-0.229595\pi\)
0.750952 + 0.660357i \(0.229595\pi\)
\(752\) −764.482 + 674.210i −1.01660 + 0.896556i
\(753\) 0 0
\(754\) −174.243 37.5059i −0.231092 0.0497426i
\(755\) 135.238 0.179123
\(756\) 0 0
\(757\) 450.055i 0.594524i 0.954796 + 0.297262i \(0.0960735\pi\)
−0.954796 + 0.297262i \(0.903926\pi\)
\(758\) 667.521 + 143.684i 0.880635 + 0.189557i
\(759\) 0 0
\(760\) 348.146 + 257.077i 0.458087 + 0.338259i
\(761\) 223.932i 0.294261i −0.989117 0.147130i \(-0.952996\pi\)
0.989117 0.147130i \(-0.0470037\pi\)
\(762\) 0 0
\(763\) 1002.52i 1.31392i
\(764\) −2.39752 + 5.31110i −0.00313811 + 0.00695170i
\(765\) 0 0
\(766\) −621.875 133.859i −0.811847 0.174750i
\(767\) 595.812i 0.776809i
\(768\) 0 0
\(769\) 562.393 0.731331 0.365665 0.930746i \(-0.380841\pi\)
0.365665 + 0.930746i \(0.380841\pi\)
\(770\) −160.853 + 747.282i −0.208900 + 0.970495i
\(771\) 0 0
\(772\) 887.881 + 400.804i 1.15010 + 0.519176i
\(773\) 857.103 1.10880 0.554400 0.832250i \(-0.312948\pi\)
0.554400 + 0.832250i \(0.312948\pi\)
\(774\) 0 0
\(775\) −202.920 −0.261832
\(776\) −17.6234 + 23.8664i −0.0227105 + 0.0307557i
\(777\) 0 0
\(778\) −129.897 + 603.468i −0.166962 + 0.775666i
\(779\) 189.981 0.243878
\(780\) 0 0
\(781\) 1998.73i 2.55919i
\(782\) 66.0890 307.033i 0.0845128 0.392625i
\(783\) 0 0
\(784\) 272.502 + 308.988i 0.347579 + 0.394118i
\(785\) 428.042i 0.545276i
\(786\) 0 0
\(787\) 386.624i 0.491262i −0.969363 0.245631i \(-0.921005\pi\)
0.969363 0.245631i \(-0.0789952\pi\)
\(788\) 18.2197 + 8.22468i 0.0231215 + 0.0104374i
\(789\) 0 0
\(790\) −10.0954 + 46.9007i −0.0127790 + 0.0593679i
\(791\) 1469.99i 1.85840i
\(792\) 0 0
\(793\) −755.965 −0.953297
\(794\) 336.384 + 72.4069i 0.423658 + 0.0911925i
\(795\) 0 0
\(796\) −373.601 168.650i −0.469348 0.211871i
\(797\) 1049.51 1.31682 0.658410 0.752659i \(-0.271229\pi\)
0.658410 + 0.752659i \(0.271229\pi\)
\(798\) 0 0
\(799\) 337.373 0.422243
\(800\) 559.463 313.469i 0.699329 0.391836i
\(801\) 0 0
\(802\) 643.482 + 138.510i 0.802347 + 0.172706i
\(803\) −1864.25 −2.32160
\(804\) 0 0
\(805\) 570.936i 0.709237i
\(806\) 287.465 + 61.8771i 0.356657 + 0.0767705i
\(807\) 0 0
\(808\) −520.826 + 705.328i −0.644587 + 0.872931i
\(809\) 467.780i 0.578220i 0.957296 + 0.289110i \(0.0933593\pi\)
−0.957296 + 0.289110i \(0.906641\pi\)
\(810\) 0 0
\(811\) 1.80922i 0.00223085i 0.999999 + 0.00111543i \(0.000355051\pi\)
−0.999999 + 0.00111543i \(0.999645\pi\)
\(812\) −193.453 87.3279i −0.238243 0.107547i
\(813\) 0 0
\(814\) 646.051 + 139.063i 0.793674 + 0.170839i
\(815\) 14.0111i 0.0171915i
\(816\) 0 0
\(817\) 37.4693 0.0458621
\(818\) 148.757 691.086i 0.181854 0.844849i
\(819\) 0 0
\(820\) −28.6640 + 63.4980i −0.0349561 + 0.0774366i
\(821\) 737.637 0.898462 0.449231 0.893416i \(-0.351698\pi\)
0.449231 + 0.893416i \(0.351698\pi\)
\(822\) 0 0
\(823\) −290.172 −0.352579 −0.176289 0.984338i \(-0.556409\pi\)
−0.176289 + 0.984338i \(0.556409\pi\)
\(824\) 757.450 1025.78i 0.919236 1.24487i
\(825\) 0 0
\(826\) 149.306 693.638i 0.180758 0.839755i
\(827\) −429.517 −0.519368 −0.259684 0.965694i \(-0.583618\pi\)
−0.259684 + 0.965694i \(0.583618\pi\)
\(828\) 0 0
\(829\) 718.136i 0.866268i 0.901330 + 0.433134i \(0.142592\pi\)
−0.901330 + 0.433134i \(0.857408\pi\)
\(830\) 111.530 518.140i 0.134373 0.624265i
\(831\) 0 0
\(832\) −888.146 + 273.474i −1.06748 + 0.328695i
\(833\) 136.359i 0.163696i
\(834\) 0 0
\(835\) 476.782i 0.570996i
\(836\) −793.573 + 1757.96i −0.949250 + 2.10283i
\(837\) 0 0
\(838\) −137.743 + 639.918i −0.164371 + 0.763625i
\(839\) 1171.31i 1.39608i −0.716059 0.698040i \(-0.754056\pi\)
0.716059 0.698040i \(-0.245944\pi\)
\(840\) 0 0
\(841\) −803.332 −0.955210
\(842\) −981.049 211.171i −1.16514 0.250797i
\(843\) 0 0
\(844\) 15.7110 34.8039i 0.0186150 0.0412368i
\(845\) 93.1726 0.110263
\(846\) 0 0
\(847\) −2360.61 −2.78702
\(848\) −809.867 918.303i −0.955032 1.08290i
\(849\) 0 0
\(850\) −207.505 44.6654i −0.244123 0.0525476i
\(851\) −493.594 −0.580016
\(852\) 0 0
\(853\) 94.3059i 0.110558i 0.998471 + 0.0552789i \(0.0176048\pi\)
−0.998471 + 0.0552789i \(0.982395\pi\)
\(854\) −880.086 189.439i −1.03055 0.221825i
\(855\) 0 0
\(856\) 236.826 320.721i 0.276666 0.374675i
\(857\) 651.269i 0.759940i 0.924999 + 0.379970i \(0.124066\pi\)
−0.924999 + 0.379970i \(0.875934\pi\)
\(858\) 0 0
\(859\) 470.587i 0.547831i −0.961754 0.273916i \(-0.911681\pi\)
0.961754 0.273916i \(-0.0883189\pi\)
\(860\) −5.65332 + 12.5235i −0.00657363 + 0.0145622i
\(861\) 0 0
\(862\) 58.3177 + 12.5529i 0.0676539 + 0.0145625i
\(863\) 269.299i 0.312049i 0.987753 + 0.156025i \(0.0498679\pi\)
−0.987753 + 0.156025i \(0.950132\pi\)
\(864\) 0 0
\(865\) −98.0968 −0.113407
\(866\) −218.789 + 1016.44i −0.252643 + 1.17371i
\(867\) 0 0
\(868\) 319.158 + 144.073i 0.367694 + 0.165983i
\(869\) −213.813 −0.246045
\(870\) 0 0
\(871\) −1600.82 −1.83791
\(872\) 746.244 + 551.039i 0.855784 + 0.631926i
\(873\) 0 0
\(874\) 303.152 1408.37i 0.346855 1.61140i
\(875\) −867.210 −0.991097
\(876\) 0 0
\(877\) 1011.57i 1.15344i −0.816942 0.576719i \(-0.804333\pi\)
0.816942 0.576719i \(-0.195667\pi\)
\(878\) −162.642 + 755.593i −0.185241 + 0.860584i
\(879\) 0 0
\(880\) −467.838 530.478i −0.531634 0.602816i
\(881\) 803.454i 0.911979i 0.889985 + 0.455990i \(0.150715\pi\)
−0.889985 + 0.455990i \(0.849285\pi\)
\(882\) 0 0
\(883\) 458.660i 0.519434i 0.965685 + 0.259717i \(0.0836293\pi\)
−0.965685 + 0.259717i \(0.916371\pi\)
\(884\) 280.340 + 126.550i 0.317127 + 0.143156i
\(885\) 0 0
\(886\) −159.307 + 740.102i −0.179805 + 0.835329i
\(887\) 56.5044i 0.0637028i 0.999493 + 0.0318514i \(0.0101403\pi\)
−0.999493 + 0.0318514i \(0.989860\pi\)
\(888\) 0 0
\(889\) 921.490 1.03655
\(890\) 636.031 + 136.906i 0.714642 + 0.153827i
\(891\) 0 0
\(892\) −186.405 84.1463i −0.208974 0.0943344i
\(893\) 1547.54 1.73296
\(894\) 0 0
\(895\) 295.231 0.329867
\(896\) −1102.50 + 95.8134i −1.23047 + 0.106935i
\(897\) 0 0
\(898\) −565.306 121.682i −0.629517 0.135504i
\(899\) −62.1444 −0.0691262
\(900\) 0 0
\(901\) 405.255i 0.449783i
\(902\) −303.542 65.3375i −0.336521 0.0724362i
\(903\) 0 0
\(904\) −1094.21 807.984i −1.21041 0.893788i
\(905\) 315.155i 0.348238i
\(906\) 0 0
\(907\) 462.461i 0.509880i 0.966957 + 0.254940i \(0.0820558\pi\)
−0.966957 + 0.254940i \(0.917944\pi\)
\(908\) 858.718 + 387.639i 0.945725 + 0.426915i
\(909\) 0 0
\(910\) 546.626 + 117.661i 0.600688 + 0.129298i
\(911\) 199.274i 0.218742i −0.994001 0.109371i \(-0.965116\pi\)
0.994001 0.109371i \(-0.0348836\pi\)
\(912\) 0 0
\(913\) 2362.12 2.58721
\(914\) −204.271 + 948.994i −0.223492 + 1.03829i
\(915\) 0 0
\(916\) 339.571 752.235i 0.370711 0.821218i
\(917\) −566.718 −0.618013
\(918\) 0 0
\(919\) 1663.67 1.81031 0.905154 0.425084i \(-0.139755\pi\)
0.905154 + 0.425084i \(0.139755\pi\)
\(920\) 424.985 + 313.816i 0.461941 + 0.341105i
\(921\) 0 0
\(922\) 90.5006 420.443i 0.0981568 0.456012i
\(923\) −1462.04 −1.58401
\(924\) 0 0
\(925\) 333.589i 0.360637i
\(926\) 157.720 732.729i 0.170324 0.791283i
\(927\) 0 0
\(928\) 171.336 96.0000i 0.184629 0.103448i
\(929\) 826.068i 0.889201i −0.895729 0.444601i \(-0.853346\pi\)
0.895729 0.444601i \(-0.146654\pi\)
\(930\) 0 0
\(931\) 625.482i 0.671839i
\(932\) 346.193 766.905i 0.371452 0.822860i
\(933\) 0 0
\(934\) 112.812 524.095i 0.120783 0.561130i
\(935\) 234.105i 0.250379i
\(936\) 0 0
\(937\) −536.859 −0.572955 −0.286477 0.958087i \(-0.592484\pi\)
−0.286477 + 0.958087i \(0.592484\pi\)
\(938\) −1863.65 401.152i −1.98684 0.427667i
\(939\) 0 0
\(940\) −233.490 + 517.239i −0.248394 + 0.550254i
\(941\) −1261.70 −1.34081 −0.670406 0.741995i \(-0.733880\pi\)
−0.670406 + 0.741995i \(0.733880\pi\)
\(942\) 0 0
\(943\) 231.911 0.245929
\(944\) 434.254 + 492.398i 0.460015 + 0.521608i
\(945\) 0 0
\(946\) −59.8666 12.8863i −0.0632840 0.0136219i
\(947\) −173.218 −0.182912 −0.0914560 0.995809i \(-0.529152\pi\)
−0.0914560 + 0.995809i \(0.529152\pi\)
\(948\) 0 0
\(949\) 1363.67i 1.43696i
\(950\) −951.828 204.881i −1.00192 0.215665i
\(951\) 0 0
\(952\) 294.656 + 217.579i 0.309513 + 0.228549i
\(953\) 1457.32i 1.52919i −0.644510 0.764596i \(-0.722939\pi\)
0.644510 0.764596i \(-0.277061\pi\)
\(954\) 0 0
\(955\) 3.24426i 0.00339713i
\(956\) −38.0921 + 84.3835i −0.0398453 + 0.0882673i
\(957\) 0 0
\(958\) −1001.28 215.527i −1.04518 0.224976i
\(959\) 1226.22i 1.27864i
\(960\) 0 0
\(961\) −858.474 −0.893314
\(962\) 101.722 472.577i 0.105741 0.491245i
\(963\) 0 0
\(964\) −297.593 134.339i −0.308707 0.139355i
\(965\) 542.358 0.562029
\(966\) 0 0
\(967\) 1148.30 1.18748 0.593742 0.804655i \(-0.297650\pi\)
0.593742 + 0.804655i \(0.297650\pi\)
\(968\) 1297.51 1757.16i 1.34041 1.81524i
\(969\) 0 0
\(970\) −3.47580 + 16.1477i −0.00358330 + 0.0166471i
\(971\) −15.1851 −0.0156386 −0.00781932 0.999969i \(-0.502489\pi\)
−0.00781932 + 0.999969i \(0.502489\pi\)
\(972\) 0 0
\(973\) 1480.21i 1.52128i
\(974\) −372.801 + 1731.94i −0.382752 + 1.77817i
\(975\) 0 0
\(976\) 624.753 550.980i 0.640116 0.564529i
\(977\) 149.435i 0.152953i −0.997071 0.0764764i \(-0.975633\pi\)
0.997071 0.0764764i \(-0.0243670\pi\)
\(978\) 0 0
\(979\) 2899.57i 2.96177i
\(980\) 209.058 + 94.3720i 0.213324 + 0.0962979i
\(981\) 0 0
\(982\) −183.010 + 850.221i −0.186365 + 0.865805i
\(983\) 683.140i 0.694954i 0.937688 + 0.347477i \(0.112962\pi\)
−0.937688 + 0.347477i \(0.887038\pi\)
\(984\) 0 0
\(985\) 11.1294 0.0112989
\(986\) −63.5484 13.6788i −0.0644507 0.0138730i
\(987\) 0 0
\(988\) 1285.93 + 580.488i 1.30154 + 0.587538i
\(989\) 45.7391 0.0462479
\(990\) 0 0
\(991\) −1531.80 −1.54571 −0.772854 0.634584i \(-0.781171\pi\)
−0.772854 + 0.634584i \(0.781171\pi\)
\(992\) −282.669 + 158.380i −0.284949 + 0.159658i
\(993\) 0 0
\(994\) −1702.09 366.376i −1.71237 0.368588i
\(995\) −228.213 −0.229359
\(996\) 0 0
\(997\) 194.125i 0.194710i 0.995250 + 0.0973548i \(0.0310382\pi\)
−0.995250 + 0.0973548i \(0.968962\pi\)
\(998\) −1394.03 300.066i −1.39683 0.300667i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 216.3.h.e.53.4 yes 8
3.2 odd 2 inner 216.3.h.e.53.5 yes 8
4.3 odd 2 864.3.h.f.593.3 8
8.3 odd 2 864.3.h.f.593.6 8
8.5 even 2 inner 216.3.h.e.53.6 yes 8
12.11 even 2 864.3.h.f.593.5 8
24.5 odd 2 inner 216.3.h.e.53.3 8
24.11 even 2 864.3.h.f.593.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.h.e.53.3 8 24.5 odd 2 inner
216.3.h.e.53.4 yes 8 1.1 even 1 trivial
216.3.h.e.53.5 yes 8 3.2 odd 2 inner
216.3.h.e.53.6 yes 8 8.5 even 2 inner
864.3.h.f.593.3 8 4.3 odd 2
864.3.h.f.593.4 8 24.11 even 2
864.3.h.f.593.5 8 12.11 even 2
864.3.h.f.593.6 8 8.3 odd 2