Properties

Label 864.2.y.b.193.2
Level $864$
Weight $2$
Character 864.193
Analytic conductor $6.899$
Analytic rank $0$
Dimension $54$
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] = [864,2,Mod(97,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.y (of order \(9\), degree \(6\), 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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 193.2
Character \(\chi\) \(=\) 864.193
Dual form 864.2.y.b.385.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.67597 + 0.437160i) q^{3} +(-0.158606 + 0.899497i) q^{5} +(-1.25116 + 1.04985i) q^{7} +(2.61778 - 1.46534i) q^{9} +O(q^{10})\) \(q+(-1.67597 + 0.437160i) q^{3} +(-0.158606 + 0.899497i) q^{5} +(-1.25116 + 1.04985i) q^{7} +(2.61778 - 1.46534i) q^{9} +(-0.887648 - 5.03410i) q^{11} +(1.45896 - 0.531018i) q^{13} +(-0.127405 - 1.57687i) q^{15} +(-0.661128 + 1.14511i) q^{17} +(-1.16059 - 2.01020i) q^{19} +(1.63796 - 2.30648i) q^{21} +(2.86651 + 2.40528i) q^{23} +(3.91452 + 1.42477i) q^{25} +(-3.74675 + 3.60026i) q^{27} +(0.0747537 + 0.0272081i) q^{29} +(4.09516 + 3.43625i) q^{31} +(3.68838 + 8.04898i) q^{33} +(-0.745894 - 1.29193i) q^{35} +(2.08453 - 3.61052i) q^{37} +(-2.21304 + 1.52777i) q^{39} +(7.69988 - 2.80253i) q^{41} +(1.29485 + 7.34345i) q^{43} +(0.902872 + 2.58710i) q^{45} +(4.78889 - 4.01836i) q^{47} +(-0.752316 + 4.26660i) q^{49} +(0.607438 - 2.20819i) q^{51} +10.3219 q^{53} +4.66894 q^{55} +(2.82390 + 2.86169i) q^{57} +(1.03999 - 5.89807i) q^{59} +(-1.68954 + 1.41769i) q^{61} +(-1.73688 + 4.58165i) q^{63} +(0.246250 + 1.39655i) q^{65} +(-0.0698559 + 0.0254255i) q^{67} +(-5.85568 - 2.77807i) q^{69} +(3.73567 - 6.47037i) q^{71} +(-3.83421 - 6.64105i) q^{73} +(-7.18350 - 0.676606i) q^{75} +(6.39564 + 5.36658i) q^{77} +(14.7441 + 5.36643i) q^{79} +(4.70557 - 7.67187i) q^{81} +(-1.07899 - 0.392719i) q^{83} +(-0.925161 - 0.776302i) q^{85} +(-0.137180 - 0.0129208i) q^{87} +(-9.09078 - 15.7457i) q^{89} +(-1.26790 + 2.19608i) q^{91} +(-8.36557 - 3.96882i) q^{93} +(1.99225 - 0.725119i) q^{95} +(1.75201 + 9.93616i) q^{97} +(-9.70033 - 11.8775i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 54 q+O(q^{10}) \) Copy content Toggle raw display \( 54 q - 9 q^{11} + 12 q^{17} + 18 q^{19} + 12 q^{21} - 21 q^{27} + 6 q^{29} + 36 q^{31} - 9 q^{33} + 24 q^{39} + 3 q^{41} - 21 q^{43} + 42 q^{45} - 18 q^{49} + 24 q^{51} + 36 q^{53} - 72 q^{55} + 39 q^{57} + 18 q^{59} - 18 q^{61} - 30 q^{63} + 48 q^{65} - 27 q^{67} + 24 q^{69} - 84 q^{75} + 36 q^{77} + 72 q^{79} + 36 q^{81} + 6 q^{87} + 33 q^{89} + 36 q^{91} + 72 q^{93} + 36 q^{95} + 9 q^{97} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{9}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.67597 + 0.437160i −0.967624 + 0.252394i
\(4\) 0 0
\(5\) −0.158606 + 0.899497i −0.0709306 + 0.402267i 0.928584 + 0.371122i \(0.121027\pi\)
−0.999515 + 0.0311456i \(0.990084\pi\)
\(6\) 0 0
\(7\) −1.25116 + 1.04985i −0.472894 + 0.396805i −0.847849 0.530238i \(-0.822103\pi\)
0.374955 + 0.927043i \(0.377658\pi\)
\(8\) 0 0
\(9\) 2.61778 1.46534i 0.872594 0.488446i
\(10\) 0 0
\(11\) −0.887648 5.03410i −0.267636 1.51784i −0.761423 0.648256i \(-0.775499\pi\)
0.493787 0.869583i \(-0.335612\pi\)
\(12\) 0 0
\(13\) 1.45896 0.531018i 0.404643 0.147278i −0.131677 0.991293i \(-0.542036\pi\)
0.536320 + 0.844015i \(0.319814\pi\)
\(14\) 0 0
\(15\) −0.127405 1.57687i −0.0328958 0.407146i
\(16\) 0 0
\(17\) −0.661128 + 1.14511i −0.160347 + 0.277729i −0.934993 0.354666i \(-0.884595\pi\)
0.774646 + 0.632395i \(0.217928\pi\)
\(18\) 0 0
\(19\) −1.16059 2.01020i −0.266258 0.461173i 0.701634 0.712537i \(-0.252454\pi\)
−0.967892 + 0.251365i \(0.919121\pi\)
\(20\) 0 0
\(21\) 1.63796 2.30648i 0.357433 0.503315i
\(22\) 0 0
\(23\) 2.86651 + 2.40528i 0.597708 + 0.501536i 0.890708 0.454576i \(-0.150209\pi\)
−0.293000 + 0.956112i \(0.594654\pi\)
\(24\) 0 0
\(25\) 3.91452 + 1.42477i 0.782905 + 0.284954i
\(26\) 0 0
\(27\) −3.74675 + 3.60026i −0.721062 + 0.692870i
\(28\) 0 0
\(29\) 0.0747537 + 0.0272081i 0.0138814 + 0.00505242i 0.348952 0.937141i \(-0.386538\pi\)
−0.335070 + 0.942193i \(0.608760\pi\)
\(30\) 0 0
\(31\) 4.09516 + 3.43625i 0.735512 + 0.617168i 0.931628 0.363413i \(-0.118388\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(32\) 0 0
\(33\) 3.68838 + 8.04898i 0.642065 + 1.40115i
\(34\) 0 0
\(35\) −0.745894 1.29193i −0.126079 0.218376i
\(36\) 0 0
\(37\) 2.08453 3.61052i 0.342695 0.593566i −0.642237 0.766506i \(-0.721993\pi\)
0.984932 + 0.172940i \(0.0553268\pi\)
\(38\) 0 0
\(39\) −2.21304 + 1.52777i −0.354370 + 0.244639i
\(40\) 0 0
\(41\) 7.69988 2.80253i 1.20252 0.437681i 0.338417 0.940996i \(-0.390109\pi\)
0.864103 + 0.503315i \(0.167887\pi\)
\(42\) 0 0
\(43\) 1.29485 + 7.34345i 0.197463 + 1.11987i 0.908868 + 0.417084i \(0.136948\pi\)
−0.711405 + 0.702782i \(0.751941\pi\)
\(44\) 0 0
\(45\) 0.902872 + 2.58710i 0.134592 + 0.385662i
\(46\) 0 0
\(47\) 4.78889 4.01836i 0.698531 0.586137i −0.222824 0.974859i \(-0.571528\pi\)
0.921355 + 0.388721i \(0.127083\pi\)
\(48\) 0 0
\(49\) −0.752316 + 4.26660i −0.107474 + 0.609514i
\(50\) 0 0
\(51\) 0.607438 2.20819i 0.0850584 0.309208i
\(52\) 0 0
\(53\) 10.3219 1.41782 0.708911 0.705298i \(-0.249187\pi\)
0.708911 + 0.705298i \(0.249187\pi\)
\(54\) 0 0
\(55\) 4.66894 0.629560
\(56\) 0 0
\(57\) 2.82390 + 2.86169i 0.374035 + 0.379040i
\(58\) 0 0
\(59\) 1.03999 5.89807i 0.135395 0.767864i −0.839189 0.543840i \(-0.816970\pi\)
0.974584 0.224023i \(-0.0719192\pi\)
\(60\) 0 0
\(61\) −1.68954 + 1.41769i −0.216324 + 0.181517i −0.744510 0.667612i \(-0.767317\pi\)
0.528186 + 0.849129i \(0.322872\pi\)
\(62\) 0 0
\(63\) −1.73688 + 4.58165i −0.218827 + 0.577233i
\(64\) 0 0
\(65\) 0.246250 + 1.39655i 0.0305435 + 0.173221i
\(66\) 0 0
\(67\) −0.0698559 + 0.0254255i −0.00853425 + 0.00310621i −0.346284 0.938130i \(-0.612557\pi\)
0.337749 + 0.941236i \(0.390334\pi\)
\(68\) 0 0
\(69\) −5.85568 2.77807i −0.704942 0.334441i
\(70\) 0 0
\(71\) 3.73567 6.47037i 0.443343 0.767892i −0.554592 0.832122i \(-0.687126\pi\)
0.997935 + 0.0642300i \(0.0204591\pi\)
\(72\) 0 0
\(73\) −3.83421 6.64105i −0.448760 0.777276i 0.549545 0.835464i \(-0.314801\pi\)
−0.998306 + 0.0581881i \(0.981468\pi\)
\(74\) 0 0
\(75\) −7.18350 0.676606i −0.829479 0.0781277i
\(76\) 0 0
\(77\) 6.39564 + 5.36658i 0.728850 + 0.611578i
\(78\) 0 0
\(79\) 14.7441 + 5.36643i 1.65884 + 0.603770i 0.990180 0.139795i \(-0.0446443\pi\)
0.668664 + 0.743565i \(0.266867\pi\)
\(80\) 0 0
\(81\) 4.70557 7.67187i 0.522841 0.852430i
\(82\) 0 0
\(83\) −1.07899 0.392719i −0.118434 0.0431066i 0.282123 0.959378i \(-0.408961\pi\)
−0.400558 + 0.916272i \(0.631184\pi\)
\(84\) 0 0
\(85\) −0.925161 0.776302i −0.100348 0.0842018i
\(86\) 0 0
\(87\) −0.137180 0.0129208i −0.0147072 0.00138526i
\(88\) 0 0
\(89\) −9.09078 15.7457i −0.963621 1.66904i −0.713276 0.700883i \(-0.752789\pi\)
−0.250344 0.968157i \(-0.580544\pi\)
\(90\) 0 0
\(91\) −1.26790 + 2.19608i −0.132913 + 0.230211i
\(92\) 0 0
\(93\) −8.36557 3.96882i −0.867470 0.411548i
\(94\) 0 0
\(95\) 1.99225 0.725119i 0.204400 0.0743957i
\(96\) 0 0
\(97\) 1.75201 + 9.93616i 0.177890 + 1.00886i 0.934755 + 0.355293i \(0.115619\pi\)
−0.756865 + 0.653571i \(0.773270\pi\)
\(98\) 0 0
\(99\) −9.70033 11.8775i −0.974920 1.19373i
\(100\) 0 0
\(101\) −9.92348 + 8.32679i −0.987424 + 0.828547i −0.985193 0.171451i \(-0.945155\pi\)
−0.00223089 + 0.999998i \(0.500710\pi\)
\(102\) 0 0
\(103\) −1.35040 + 7.65849i −0.133059 + 0.754613i 0.843133 + 0.537705i \(0.180708\pi\)
−0.976192 + 0.216909i \(0.930403\pi\)
\(104\) 0 0
\(105\) 1.81488 + 1.83916i 0.177114 + 0.179484i
\(106\) 0 0
\(107\) 8.66656 0.837828 0.418914 0.908026i \(-0.362411\pi\)
0.418914 + 0.908026i \(0.362411\pi\)
\(108\) 0 0
\(109\) 17.3261 1.65954 0.829768 0.558109i \(-0.188473\pi\)
0.829768 + 0.558109i \(0.188473\pi\)
\(110\) 0 0
\(111\) −1.91525 + 6.96241i −0.181788 + 0.660843i
\(112\) 0 0
\(113\) 0.381238 2.16211i 0.0358639 0.203394i −0.961611 0.274417i \(-0.911515\pi\)
0.997475 + 0.0710227i \(0.0226263\pi\)
\(114\) 0 0
\(115\) −2.61819 + 2.19692i −0.244147 + 0.204864i
\(116\) 0 0
\(117\) 3.04112 3.52796i 0.281151 0.326160i
\(118\) 0 0
\(119\) −0.375012 2.12680i −0.0343773 0.194963i
\(120\) 0 0
\(121\) −14.2176 + 5.17480i −1.29251 + 0.470436i
\(122\) 0 0
\(123\) −11.6797 + 8.06304i −1.05312 + 0.727020i
\(124\) 0 0
\(125\) −4.18587 + 7.25015i −0.374396 + 0.648473i
\(126\) 0 0
\(127\) 10.0273 + 17.3678i 0.889777 + 1.54114i 0.840139 + 0.542372i \(0.182473\pi\)
0.0496382 + 0.998767i \(0.484193\pi\)
\(128\) 0 0
\(129\) −5.38040 11.7414i −0.473718 1.03377i
\(130\) 0 0
\(131\) −9.91431 8.31909i −0.866217 0.726843i 0.0970808 0.995277i \(-0.469049\pi\)
−0.963298 + 0.268434i \(0.913494\pi\)
\(132\) 0 0
\(133\) 3.56250 + 1.29664i 0.308908 + 0.112433i
\(134\) 0 0
\(135\) −2.64417 3.94121i −0.227574 0.339205i
\(136\) 0 0
\(137\) −12.8377 4.67256i −1.09680 0.399204i −0.270666 0.962673i \(-0.587244\pi\)
−0.826137 + 0.563470i \(0.809466\pi\)
\(138\) 0 0
\(139\) 4.92501 + 4.13257i 0.417734 + 0.350520i 0.827300 0.561760i \(-0.189876\pi\)
−0.409566 + 0.912280i \(0.634320\pi\)
\(140\) 0 0
\(141\) −6.26939 + 8.82817i −0.527978 + 0.743466i
\(142\) 0 0
\(143\) −3.96824 6.87320i −0.331841 0.574765i
\(144\) 0 0
\(145\) −0.0363300 + 0.0629254i −0.00301704 + 0.00522567i
\(146\) 0 0
\(147\) −0.604322 7.47959i −0.0498436 0.616906i
\(148\) 0 0
\(149\) 2.52311 0.918335i 0.206701 0.0752329i −0.236595 0.971608i \(-0.576031\pi\)
0.443296 + 0.896375i \(0.353809\pi\)
\(150\) 0 0
\(151\) −3.61660 20.5107i −0.294314 1.66914i −0.669975 0.742383i \(-0.733695\pi\)
0.375661 0.926757i \(-0.377416\pi\)
\(152\) 0 0
\(153\) −0.0527201 + 3.96641i −0.00426217 + 0.320666i
\(154\) 0 0
\(155\) −3.74041 + 3.13858i −0.300437 + 0.252096i
\(156\) 0 0
\(157\) 1.34143 7.60761i 0.107058 0.607154i −0.883321 0.468769i \(-0.844698\pi\)
0.990379 0.138385i \(-0.0441911\pi\)
\(158\) 0 0
\(159\) −17.2992 + 4.51232i −1.37192 + 0.357850i
\(160\) 0 0
\(161\) −6.11164 −0.481665
\(162\) 0 0
\(163\) 4.18421 0.327733 0.163866 0.986483i \(-0.447603\pi\)
0.163866 + 0.986483i \(0.447603\pi\)
\(164\) 0 0
\(165\) −7.82503 + 2.04108i −0.609178 + 0.158898i
\(166\) 0 0
\(167\) 1.87330 10.6240i 0.144960 0.822111i −0.822439 0.568854i \(-0.807387\pi\)
0.967399 0.253257i \(-0.0815018\pi\)
\(168\) 0 0
\(169\) −8.11199 + 6.80677i −0.624000 + 0.523598i
\(170\) 0 0
\(171\) −5.98381 3.56162i −0.457593 0.272364i
\(172\) 0 0
\(173\) 2.50365 + 14.1989i 0.190349 + 1.07952i 0.918888 + 0.394519i \(0.129089\pi\)
−0.728538 + 0.685005i \(0.759800\pi\)
\(174\) 0 0
\(175\) −6.39349 + 2.32704i −0.483303 + 0.175908i
\(176\) 0 0
\(177\) 0.835405 + 10.3397i 0.0627929 + 0.777177i
\(178\) 0 0
\(179\) 5.33967 9.24858i 0.399106 0.691271i −0.594510 0.804088i \(-0.702654\pi\)
0.993616 + 0.112817i \(0.0359873\pi\)
\(180\) 0 0
\(181\) −11.4877 19.8973i −0.853876 1.47896i −0.877684 0.479239i \(-0.840913\pi\)
0.0238086 0.999717i \(-0.492421\pi\)
\(182\) 0 0
\(183\) 2.21187 3.11462i 0.163506 0.230239i
\(184\) 0 0
\(185\) 2.91703 + 2.44768i 0.214464 + 0.179957i
\(186\) 0 0
\(187\) 6.35143 + 2.31173i 0.464463 + 0.169051i
\(188\) 0 0
\(189\) 0.908060 8.43802i 0.0660517 0.613776i
\(190\) 0 0
\(191\) 4.56758 + 1.66246i 0.330498 + 0.120292i 0.501939 0.864903i \(-0.332620\pi\)
−0.171441 + 0.985194i \(0.554842\pi\)
\(192\) 0 0
\(193\) −15.0728 12.6476i −1.08496 0.910392i −0.0886394 0.996064i \(-0.528252\pi\)
−0.996323 + 0.0856719i \(0.972696\pi\)
\(194\) 0 0
\(195\) −1.02322 2.23294i −0.0732747 0.159904i
\(196\) 0 0
\(197\) −8.86378 15.3525i −0.631518 1.09382i −0.987242 0.159230i \(-0.949099\pi\)
0.355724 0.934591i \(-0.384234\pi\)
\(198\) 0 0
\(199\) −2.78507 + 4.82388i −0.197428 + 0.341956i −0.947694 0.319181i \(-0.896592\pi\)
0.750266 + 0.661137i \(0.229926\pi\)
\(200\) 0 0
\(201\) 0.105962 0.0731506i 0.00747396 0.00515965i
\(202\) 0 0
\(203\) −0.122093 + 0.0444383i −0.00856927 + 0.00311896i
\(204\) 0 0
\(205\) 1.29962 + 7.37051i 0.0907694 + 0.514779i
\(206\) 0 0
\(207\) 11.0284 + 2.09611i 0.766530 + 0.145690i
\(208\) 0 0
\(209\) −9.08938 + 7.62689i −0.628725 + 0.527563i
\(210\) 0 0
\(211\) 0.818207 4.64028i 0.0563277 0.319450i −0.943605 0.331074i \(-0.892589\pi\)
0.999932 + 0.0116239i \(0.00370009\pi\)
\(212\) 0 0
\(213\) −3.43230 + 12.4773i −0.235178 + 0.854928i
\(214\) 0 0
\(215\) −6.81078 −0.464491
\(216\) 0 0
\(217\) −8.73124 −0.592715
\(218\) 0 0
\(219\) 9.32924 + 9.45407i 0.630412 + 0.638847i
\(220\) 0 0
\(221\) −0.356486 + 2.02173i −0.0239799 + 0.135997i
\(222\) 0 0
\(223\) −9.44885 + 7.92852i −0.632742 + 0.530933i −0.901780 0.432196i \(-0.857739\pi\)
0.269038 + 0.963130i \(0.413294\pi\)
\(224\) 0 0
\(225\) 12.3351 2.00636i 0.822343 0.133757i
\(226\) 0 0
\(227\) 0.621004 + 3.52189i 0.0412175 + 0.233756i 0.998456 0.0555429i \(-0.0176890\pi\)
−0.957239 + 0.289299i \(0.906578\pi\)
\(228\) 0 0
\(229\) −19.6799 + 7.16289i −1.30048 + 0.473337i −0.897154 0.441718i \(-0.854369\pi\)
−0.403329 + 0.915055i \(0.632147\pi\)
\(230\) 0 0
\(231\) −13.0650 6.19833i −0.859612 0.407820i
\(232\) 0 0
\(233\) 12.4263 21.5230i 0.814074 1.41002i −0.0959174 0.995389i \(-0.530578\pi\)
0.909991 0.414628i \(-0.136088\pi\)
\(234\) 0 0
\(235\) 2.85495 + 4.94492i 0.186237 + 0.322571i
\(236\) 0 0
\(237\) −27.0568 2.54845i −1.75753 0.165540i
\(238\) 0 0
\(239\) 11.3987 + 9.56464i 0.737320 + 0.618685i 0.932117 0.362158i \(-0.117960\pi\)
−0.194796 + 0.980844i \(0.562405\pi\)
\(240\) 0 0
\(241\) 3.15858 + 1.14963i 0.203462 + 0.0740542i 0.441741 0.897143i \(-0.354361\pi\)
−0.238279 + 0.971197i \(0.576583\pi\)
\(242\) 0 0
\(243\) −4.53258 + 14.9149i −0.290765 + 0.956794i
\(244\) 0 0
\(245\) −3.71847 1.35341i −0.237564 0.0864663i
\(246\) 0 0
\(247\) −2.76071 2.31651i −0.175660 0.147396i
\(248\) 0 0
\(249\) 1.98004 + 0.186498i 0.125480 + 0.0118188i
\(250\) 0 0
\(251\) 4.93370 + 8.54542i 0.311412 + 0.539382i 0.978668 0.205446i \(-0.0658646\pi\)
−0.667256 + 0.744828i \(0.732531\pi\)
\(252\) 0 0
\(253\) 9.56400 16.5653i 0.601283 1.04145i
\(254\) 0 0
\(255\) 1.88991 + 0.896620i 0.118351 + 0.0561485i
\(256\) 0 0
\(257\) 3.22778 1.17482i 0.201344 0.0732831i −0.239380 0.970926i \(-0.576944\pi\)
0.440724 + 0.897643i \(0.354722\pi\)
\(258\) 0 0
\(259\) 1.18241 + 6.70579i 0.0734714 + 0.416677i
\(260\) 0 0
\(261\) 0.235558 0.0383145i 0.0145807 0.00237161i
\(262\) 0 0
\(263\) 12.4722 10.4654i 0.769066 0.645323i −0.171403 0.985201i \(-0.554830\pi\)
0.940470 + 0.339878i \(0.110386\pi\)
\(264\) 0 0
\(265\) −1.63711 + 9.28452i −0.100567 + 0.570343i
\(266\) 0 0
\(267\) 22.1193 + 22.4153i 1.35368 + 1.37179i
\(268\) 0 0
\(269\) −16.8944 −1.03007 −0.515035 0.857169i \(-0.672221\pi\)
−0.515035 + 0.857169i \(0.672221\pi\)
\(270\) 0 0
\(271\) −2.50352 −0.152078 −0.0760389 0.997105i \(-0.524227\pi\)
−0.0760389 + 0.997105i \(0.524227\pi\)
\(272\) 0 0
\(273\) 1.16494 4.23484i 0.0705054 0.256304i
\(274\) 0 0
\(275\) 3.69772 20.9708i 0.222981 1.26459i
\(276\) 0 0
\(277\) 16.6397 13.9624i 0.999785 0.838919i 0.0128303 0.999918i \(-0.495916\pi\)
0.986955 + 0.160998i \(0.0514714\pi\)
\(278\) 0 0
\(279\) 15.7555 + 2.99455i 0.943257 + 0.179279i
\(280\) 0 0
\(281\) −1.76595 10.0152i −0.105348 0.597456i −0.991081 0.133261i \(-0.957455\pi\)
0.885733 0.464194i \(-0.153656\pi\)
\(282\) 0 0
\(283\) 21.3886 7.78482i 1.27142 0.462760i 0.383836 0.923401i \(-0.374603\pi\)
0.887586 + 0.460642i \(0.152381\pi\)
\(284\) 0 0
\(285\) −3.02197 + 2.08621i −0.179006 + 0.123577i
\(286\) 0 0
\(287\) −6.69156 + 11.5901i −0.394990 + 0.684143i
\(288\) 0 0
\(289\) 7.62582 + 13.2083i 0.448578 + 0.776959i
\(290\) 0 0
\(291\) −7.28002 15.8868i −0.426762 0.931303i
\(292\) 0 0
\(293\) 25.9472 + 21.7723i 1.51585 + 1.27195i 0.851265 + 0.524737i \(0.175836\pi\)
0.664585 + 0.747213i \(0.268608\pi\)
\(294\) 0 0
\(295\) 5.14035 + 1.87093i 0.299283 + 0.108930i
\(296\) 0 0
\(297\) 21.4499 + 15.6658i 1.24465 + 0.909020i
\(298\) 0 0
\(299\) 5.45936 + 1.98705i 0.315723 + 0.114914i
\(300\) 0 0
\(301\) −9.32958 7.82844i −0.537748 0.451224i
\(302\) 0 0
\(303\) 12.9914 18.2936i 0.746335 1.05094i
\(304\) 0 0
\(305\) −1.00724 1.74459i −0.0576744 0.0998950i
\(306\) 0 0
\(307\) 3.49176 6.04791i 0.199285 0.345173i −0.749011 0.662557i \(-0.769471\pi\)
0.948297 + 0.317384i \(0.102805\pi\)
\(308\) 0 0
\(309\) −1.08475 13.4258i −0.0617093 0.763766i
\(310\) 0 0
\(311\) −8.44915 + 3.07524i −0.479107 + 0.174381i −0.570273 0.821455i \(-0.693163\pi\)
0.0911662 + 0.995836i \(0.470941\pi\)
\(312\) 0 0
\(313\) 4.11285 + 23.3251i 0.232472 + 1.31841i 0.847873 + 0.530199i \(0.177883\pi\)
−0.615401 + 0.788214i \(0.711006\pi\)
\(314\) 0 0
\(315\) −3.84570 2.28900i −0.216681 0.128970i
\(316\) 0 0
\(317\) −13.0106 + 10.9172i −0.730749 + 0.613171i −0.930336 0.366709i \(-0.880484\pi\)
0.199587 + 0.979880i \(0.436040\pi\)
\(318\) 0 0
\(319\) 0.0706135 0.400469i 0.00395360 0.0224220i
\(320\) 0 0
\(321\) −14.5249 + 3.78867i −0.810703 + 0.211463i
\(322\) 0 0
\(323\) 3.06920 0.170775
\(324\) 0 0
\(325\) 6.46771 0.358764
\(326\) 0 0
\(327\) −29.0380 + 7.57426i −1.60581 + 0.418858i
\(328\) 0 0
\(329\) −1.77301 + 10.0552i −0.0977490 + 0.554362i
\(330\) 0 0
\(331\) −15.7632 + 13.2269i −0.866423 + 0.727015i −0.963342 0.268277i \(-0.913546\pi\)
0.0969185 + 0.995292i \(0.469101\pi\)
\(332\) 0 0
\(333\) 0.166226 12.5061i 0.00910915 0.685330i
\(334\) 0 0
\(335\) −0.0117906 0.0668678i −0.000644189 0.00365338i
\(336\) 0 0
\(337\) −9.92735 + 3.61326i −0.540777 + 0.196827i −0.597944 0.801538i \(-0.704016\pi\)
0.0571669 + 0.998365i \(0.481793\pi\)
\(338\) 0 0
\(339\) 0.306242 + 3.79030i 0.0166328 + 0.205861i
\(340\) 0 0
\(341\) 13.6634 23.6656i 0.739912 1.28157i
\(342\) 0 0
\(343\) −9.25447 16.0292i −0.499694 0.865496i
\(344\) 0 0
\(345\) 3.42761 4.82655i 0.184536 0.259853i
\(346\) 0 0
\(347\) 13.9395 + 11.6966i 0.748312 + 0.627908i 0.935056 0.354500i \(-0.115349\pi\)
−0.186744 + 0.982409i \(0.559793\pi\)
\(348\) 0 0
\(349\) 32.8524 + 11.9573i 1.75855 + 0.640060i 0.999934 0.0114776i \(-0.00365353\pi\)
0.758616 + 0.651538i \(0.225876\pi\)
\(350\) 0 0
\(351\) −3.55456 + 7.24222i −0.189728 + 0.386561i
\(352\) 0 0
\(353\) 3.45455 + 1.25735i 0.183867 + 0.0669222i 0.432313 0.901724i \(-0.357698\pi\)
−0.248446 + 0.968646i \(0.579920\pi\)
\(354\) 0 0
\(355\) 5.22758 + 4.38646i 0.277451 + 0.232809i
\(356\) 0 0
\(357\) 1.55826 + 3.40052i 0.0824719 + 0.179974i
\(358\) 0 0
\(359\) −10.2575 17.7665i −0.541370 0.937680i −0.998826 0.0484479i \(-0.984573\pi\)
0.457456 0.889232i \(-0.348761\pi\)
\(360\) 0 0
\(361\) 6.80605 11.7884i 0.358213 0.620444i
\(362\) 0 0
\(363\) 21.5662 14.8882i 1.13193 0.781429i
\(364\) 0 0
\(365\) 6.58173 2.39555i 0.344503 0.125389i
\(366\) 0 0
\(367\) −5.80904 32.9447i −0.303229 1.71970i −0.631725 0.775193i \(-0.717653\pi\)
0.328496 0.944505i \(-0.393458\pi\)
\(368\) 0 0
\(369\) 16.0500 18.6193i 0.835528 0.969284i
\(370\) 0 0
\(371\) −12.9144 + 10.8364i −0.670480 + 0.562600i
\(372\) 0 0
\(373\) −2.34279 + 13.2866i −0.121305 + 0.687954i 0.862129 + 0.506688i \(0.169130\pi\)
−0.983434 + 0.181266i \(0.941981\pi\)
\(374\) 0 0
\(375\) 3.84595 13.9810i 0.198604 0.721974i
\(376\) 0 0
\(377\) 0.123511 0.00636112
\(378\) 0 0
\(379\) −30.7527 −1.57966 −0.789830 0.613325i \(-0.789831\pi\)
−0.789830 + 0.613325i \(0.789831\pi\)
\(380\) 0 0
\(381\) −24.3979 24.7244i −1.24994 1.26667i
\(382\) 0 0
\(383\) −5.78899 + 32.8310i −0.295804 + 1.67759i 0.368115 + 0.929780i \(0.380003\pi\)
−0.663918 + 0.747805i \(0.731108\pi\)
\(384\) 0 0
\(385\) −5.84160 + 4.90168i −0.297716 + 0.249813i
\(386\) 0 0
\(387\) 14.1503 + 17.3262i 0.719299 + 0.880739i
\(388\) 0 0
\(389\) 4.57033 + 25.9196i 0.231725 + 1.31418i 0.849402 + 0.527746i \(0.176963\pi\)
−0.617677 + 0.786432i \(0.711926\pi\)
\(390\) 0 0
\(391\) −4.64943 + 1.69225i −0.235132 + 0.0855810i
\(392\) 0 0
\(393\) 20.2529 + 9.60845i 1.02162 + 0.484682i
\(394\) 0 0
\(395\) −7.16558 + 12.4112i −0.360540 + 0.624473i
\(396\) 0 0
\(397\) −5.56953 9.64672i −0.279527 0.484155i 0.691740 0.722146i \(-0.256844\pi\)
−0.971267 + 0.237992i \(0.923511\pi\)
\(398\) 0 0
\(399\) −6.53750 0.615760i −0.327284 0.0308266i
\(400\) 0 0
\(401\) −5.96288 5.00345i −0.297772 0.249860i 0.481644 0.876367i \(-0.340040\pi\)
−0.779416 + 0.626506i \(0.784484\pi\)
\(402\) 0 0
\(403\) 7.79938 + 2.83874i 0.388515 + 0.141408i
\(404\) 0 0
\(405\) 6.15449 + 5.44945i 0.305819 + 0.270785i
\(406\) 0 0
\(407\) −20.0261 7.28889i −0.992655 0.361297i
\(408\) 0 0
\(409\) 1.53837 + 1.29084i 0.0760674 + 0.0638281i 0.680029 0.733186i \(-0.261967\pi\)
−0.603961 + 0.797014i \(0.706412\pi\)
\(410\) 0 0
\(411\) 23.5584 + 2.21894i 1.16205 + 0.109452i
\(412\) 0 0
\(413\) 4.89089 + 8.47127i 0.240665 + 0.416844i
\(414\) 0 0
\(415\) 0.524383 0.908259i 0.0257410 0.0445847i
\(416\) 0 0
\(417\) −10.0608 4.77307i −0.492679 0.233738i
\(418\) 0 0
\(419\) 1.54093 0.560851i 0.0752792 0.0273994i −0.304106 0.952638i \(-0.598358\pi\)
0.379385 + 0.925239i \(0.376136\pi\)
\(420\) 0 0
\(421\) 2.80867 + 15.9288i 0.136886 + 0.776321i 0.973528 + 0.228570i \(0.0734049\pi\)
−0.836641 + 0.547751i \(0.815484\pi\)
\(422\) 0 0
\(423\) 6.64802 17.5365i 0.323238 0.852655i
\(424\) 0 0
\(425\) −4.21951 + 3.54059i −0.204676 + 0.171744i
\(426\) 0 0
\(427\) 0.625524 3.54752i 0.0302712 0.171677i
\(428\) 0 0
\(429\) 9.65536 + 9.78455i 0.466165 + 0.472402i
\(430\) 0 0
\(431\) −34.7870 −1.67563 −0.837816 0.545952i \(-0.816168\pi\)
−0.837816 + 0.545952i \(0.816168\pi\)
\(432\) 0 0
\(433\) 28.2579 1.35799 0.678994 0.734144i \(-0.262416\pi\)
0.678994 + 0.734144i \(0.262416\pi\)
\(434\) 0 0
\(435\) 0.0333797 0.121343i 0.00160043 0.00581797i
\(436\) 0 0
\(437\) 1.50827 8.55382i 0.0721503 0.409185i
\(438\) 0 0
\(439\) −11.0220 + 9.24858i −0.526053 + 0.441411i −0.866736 0.498768i \(-0.833786\pi\)
0.340683 + 0.940178i \(0.389342\pi\)
\(440\) 0 0
\(441\) 4.28260 + 12.2714i 0.203934 + 0.584353i
\(442\) 0 0
\(443\) 1.80438 + 10.2331i 0.0857285 + 0.486191i 0.997197 + 0.0748208i \(0.0238385\pi\)
−0.911468 + 0.411370i \(0.865050\pi\)
\(444\) 0 0
\(445\) 15.6050 5.67977i 0.739750 0.269247i
\(446\) 0 0
\(447\) −3.82720 + 2.64211i −0.181020 + 0.124967i
\(448\) 0 0
\(449\) 11.5067 19.9301i 0.543033 0.940560i −0.455695 0.890136i \(-0.650609\pi\)
0.998728 0.0504241i \(-0.0160573\pi\)
\(450\) 0 0
\(451\) −20.9430 36.2743i −0.986167 1.70809i
\(452\) 0 0
\(453\) 15.0278 + 32.7945i 0.706068 + 1.54082i
\(454\) 0 0
\(455\) −1.77427 1.48879i −0.0831789 0.0697954i
\(456\) 0 0
\(457\) −1.42490 0.518622i −0.0666541 0.0242601i 0.308478 0.951231i \(-0.400180\pi\)
−0.375132 + 0.926971i \(0.622403\pi\)
\(458\) 0 0
\(459\) −1.64560 6.67066i −0.0768100 0.311360i
\(460\) 0 0
\(461\) 7.20103 + 2.62096i 0.335385 + 0.122070i 0.504223 0.863574i \(-0.331779\pi\)
−0.168837 + 0.985644i \(0.554001\pi\)
\(462\) 0 0
\(463\) −25.2827 21.2147i −1.17499 0.985931i −0.999999 0.00136057i \(-0.999567\pi\)
−0.174988 0.984571i \(-0.555989\pi\)
\(464\) 0 0
\(465\) 4.89677 6.89533i 0.227082 0.319763i
\(466\) 0 0
\(467\) 15.9624 + 27.6477i 0.738653 + 1.27938i 0.953102 + 0.302649i \(0.0978710\pi\)
−0.214449 + 0.976735i \(0.568796\pi\)
\(468\) 0 0
\(469\) 0.0607081 0.105149i 0.00280324 0.00485535i
\(470\) 0 0
\(471\) 1.07754 + 13.3366i 0.0496507 + 0.614517i
\(472\) 0 0
\(473\) 35.8183 13.0368i 1.64693 0.599433i
\(474\) 0 0
\(475\) −1.67909 9.52257i −0.0770418 0.436926i
\(476\) 0 0
\(477\) 27.0205 15.1251i 1.23718 0.692530i
\(478\) 0 0
\(479\) 8.68278 7.28571i 0.396726 0.332893i −0.422500 0.906363i \(-0.638848\pi\)
0.819227 + 0.573470i \(0.194403\pi\)
\(480\) 0 0
\(481\) 1.12400 6.37453i 0.0512500 0.290653i
\(482\) 0 0
\(483\) 10.2430 2.67177i 0.466071 0.121570i
\(484\) 0 0
\(485\) −9.21542 −0.418451
\(486\) 0 0
\(487\) 31.9729 1.44883 0.724415 0.689365i \(-0.242110\pi\)
0.724415 + 0.689365i \(0.242110\pi\)
\(488\) 0 0
\(489\) −7.01263 + 1.82917i −0.317122 + 0.0827179i
\(490\) 0 0
\(491\) −2.40328 + 13.6297i −0.108458 + 0.615098i 0.881324 + 0.472512i \(0.156653\pi\)
−0.989782 + 0.142586i \(0.954458\pi\)
\(492\) 0 0
\(493\) −0.0805779 + 0.0676129i −0.00362905 + 0.00304513i
\(494\) 0 0
\(495\) 12.2223 6.84158i 0.549351 0.307506i
\(496\) 0 0
\(497\) 2.11899 + 12.0174i 0.0950495 + 0.539053i
\(498\) 0 0
\(499\) 7.12831 2.59449i 0.319107 0.116145i −0.177500 0.984121i \(-0.556801\pi\)
0.496607 + 0.867975i \(0.334579\pi\)
\(500\) 0 0
\(501\) 1.50479 + 18.6245i 0.0672290 + 0.832082i
\(502\) 0 0
\(503\) −8.50373 + 14.7289i −0.379162 + 0.656728i −0.990941 0.134301i \(-0.957121\pi\)
0.611778 + 0.791029i \(0.290454\pi\)
\(504\) 0 0
\(505\) −5.91600 10.2468i −0.263259 0.455977i
\(506\) 0 0
\(507\) 10.6198 14.9542i 0.471644 0.664140i
\(508\) 0 0
\(509\) 18.9563 + 15.9063i 0.840225 + 0.705033i 0.957614 0.288053i \(-0.0930080\pi\)
−0.117389 + 0.993086i \(0.537452\pi\)
\(510\) 0 0
\(511\) 11.7693 + 4.28368i 0.520644 + 0.189499i
\(512\) 0 0
\(513\) 11.5857 + 3.35330i 0.511521 + 0.148052i
\(514\) 0 0
\(515\) −6.67461 2.42936i −0.294118 0.107050i
\(516\) 0 0
\(517\) −24.4797 20.5409i −1.07661 0.903387i
\(518\) 0 0
\(519\) −10.4033 22.7025i −0.456652 0.996531i
\(520\) 0 0
\(521\) −4.92213 8.52538i −0.215643 0.373504i 0.737829 0.674988i \(-0.235851\pi\)
−0.953471 + 0.301484i \(0.902518\pi\)
\(522\) 0 0
\(523\) 5.49604 9.51943i 0.240325 0.416256i −0.720482 0.693474i \(-0.756079\pi\)
0.960807 + 0.277218i \(0.0894126\pi\)
\(524\) 0 0
\(525\) 9.69804 6.69504i 0.423257 0.292196i
\(526\) 0 0
\(527\) −6.64229 + 2.41760i −0.289343 + 0.105312i
\(528\) 0 0
\(529\) −1.56244 8.86106i −0.0679323 0.385263i
\(530\) 0 0
\(531\) −5.92021 16.9638i −0.256915 0.736167i
\(532\) 0 0
\(533\) 9.74562 8.17755i 0.422130 0.354209i
\(534\) 0 0
\(535\) −1.37456 + 7.79554i −0.0594276 + 0.337031i
\(536\) 0 0
\(537\) −4.90604 + 17.8347i −0.211711 + 0.769623i
\(538\) 0 0
\(539\) 22.1463 0.953907
\(540\) 0 0
\(541\) 6.17773 0.265601 0.132801 0.991143i \(-0.457603\pi\)
0.132801 + 0.991143i \(0.457603\pi\)
\(542\) 0 0
\(543\) 27.9514 + 28.3254i 1.19951 + 1.21556i
\(544\) 0 0
\(545\) −2.74801 + 15.5847i −0.117712 + 0.667577i
\(546\) 0 0
\(547\) −26.0849 + 21.8878i −1.11531 + 0.935857i −0.998358 0.0572808i \(-0.981757\pi\)
−0.116952 + 0.993138i \(0.537313\pi\)
\(548\) 0 0
\(549\) −2.34545 + 6.18696i −0.100101 + 0.264053i
\(550\) 0 0
\(551\) −0.0320647 0.181848i −0.00136600 0.00774698i
\(552\) 0 0
\(553\) −24.0812 + 8.76484i −1.02404 + 0.372719i
\(554\) 0 0
\(555\) −5.95890 2.82704i −0.252941 0.120001i
\(556\) 0 0
\(557\) 6.11418 10.5901i 0.259066 0.448716i −0.706926 0.707288i \(-0.749919\pi\)
0.965992 + 0.258572i \(0.0832519\pi\)
\(558\) 0 0
\(559\) 5.78864 + 10.0262i 0.244833 + 0.424064i
\(560\) 0 0
\(561\) −11.6554 1.09781i −0.492093 0.0463497i
\(562\) 0 0
\(563\) −18.5963 15.6042i −0.783742 0.657637i 0.160446 0.987045i \(-0.448707\pi\)
−0.944188 + 0.329407i \(0.893151\pi\)
\(564\) 0 0
\(565\) 1.88434 + 0.685845i 0.0792750 + 0.0288537i
\(566\) 0 0
\(567\) 2.16688 + 14.5389i 0.0910004 + 0.610576i
\(568\) 0 0
\(569\) −11.5281 4.19590i −0.483284 0.175901i 0.0888763 0.996043i \(-0.471672\pi\)
−0.572161 + 0.820142i \(0.693895\pi\)
\(570\) 0 0
\(571\) −8.91500 7.48058i −0.373081 0.313052i 0.436898 0.899511i \(-0.356077\pi\)
−0.809979 + 0.586459i \(0.800522\pi\)
\(572\) 0 0
\(573\) −8.38191 0.789483i −0.350159 0.0329811i
\(574\) 0 0
\(575\) 7.79403 + 13.4997i 0.325033 + 0.562974i
\(576\) 0 0
\(577\) 18.3725 31.8220i 0.764856 1.32477i −0.175467 0.984485i \(-0.556144\pi\)
0.940323 0.340283i \(-0.110523\pi\)
\(578\) 0 0
\(579\) 30.7906 + 14.6078i 1.27961 + 0.607079i
\(580\) 0 0
\(581\) 1.76228 0.641419i 0.0731118 0.0266105i
\(582\) 0 0
\(583\) −9.16222 51.9615i −0.379460 2.15203i
\(584\) 0 0
\(585\) 2.69105 + 3.29503i 0.111261 + 0.136233i
\(586\) 0 0
\(587\) −31.8718 + 26.7436i −1.31549 + 1.10383i −0.328249 + 0.944591i \(0.606458\pi\)
−0.987241 + 0.159236i \(0.949097\pi\)
\(588\) 0 0
\(589\) 2.15475 12.2202i 0.0887849 0.503524i
\(590\) 0 0
\(591\) 21.5670 + 21.8555i 0.887146 + 0.899016i
\(592\) 0 0
\(593\) −18.6260 −0.764877 −0.382438 0.923981i \(-0.624916\pi\)
−0.382438 + 0.923981i \(0.624916\pi\)
\(594\) 0 0
\(595\) 1.97253 0.0808656
\(596\) 0 0
\(597\) 2.55890 9.30223i 0.104729 0.380715i
\(598\) 0 0
\(599\) −6.05175 + 34.3212i −0.247268 + 1.40232i 0.567899 + 0.823098i \(0.307756\pi\)
−0.815167 + 0.579227i \(0.803355\pi\)
\(600\) 0 0
\(601\) 4.05833 3.40534i 0.165543 0.138907i −0.556253 0.831013i \(-0.687761\pi\)
0.721796 + 0.692106i \(0.243317\pi\)
\(602\) 0 0
\(603\) −0.145611 + 0.168921i −0.00592972 + 0.00687899i
\(604\) 0 0
\(605\) −2.39972 13.6095i −0.0975624 0.553304i
\(606\) 0 0
\(607\) −42.8968 + 15.6132i −1.74113 + 0.633719i −0.999319 0.0368918i \(-0.988254\pi\)
−0.741809 + 0.670611i \(0.766032\pi\)
\(608\) 0 0
\(609\) 0.185199 0.127852i 0.00750463 0.00518082i
\(610\) 0 0
\(611\) 4.85298 8.40561i 0.196330 0.340054i
\(612\) 0 0
\(613\) 14.3555 + 24.8644i 0.579812 + 1.00426i 0.995500 + 0.0947573i \(0.0302075\pi\)
−0.415688 + 0.909507i \(0.636459\pi\)
\(614\) 0 0
\(615\) −5.40023 11.7847i −0.217758 0.475203i
\(616\) 0 0
\(617\) −8.30054 6.96498i −0.334167 0.280399i 0.460228 0.887801i \(-0.347768\pi\)
−0.794395 + 0.607401i \(0.792212\pi\)
\(618\) 0 0
\(619\) 45.1107 + 16.4189i 1.81315 + 0.659933i 0.996575 + 0.0826968i \(0.0263533\pi\)
0.816577 + 0.577236i \(0.195869\pi\)
\(620\) 0 0
\(621\) −19.3997 + 1.30816i −0.778484 + 0.0524948i
\(622\) 0 0
\(623\) 27.9046 + 10.1564i 1.11797 + 0.406910i
\(624\) 0 0
\(625\) 10.0982 + 8.47337i 0.403927 + 0.338935i
\(626\) 0 0
\(627\) 11.8994 16.7560i 0.475216 0.669170i
\(628\) 0 0
\(629\) 2.75629 + 4.77403i 0.109900 + 0.190353i
\(630\) 0 0
\(631\) −1.97244 + 3.41636i −0.0785214 + 0.136003i −0.902612 0.430455i \(-0.858353\pi\)
0.824091 + 0.566458i \(0.191687\pi\)
\(632\) 0 0
\(633\) 0.657251 + 8.13468i 0.0261234 + 0.323325i
\(634\) 0 0
\(635\) −17.2126 + 6.26488i −0.683062 + 0.248614i
\(636\) 0 0
\(637\) 1.16804 + 6.62428i 0.0462794 + 0.262464i
\(638\) 0 0
\(639\) 0.297893 22.4121i 0.0117844 0.886607i
\(640\) 0 0
\(641\) −30.9739 + 25.9902i −1.22339 + 1.02655i −0.224754 + 0.974416i \(0.572158\pi\)
−0.998640 + 0.0521339i \(0.983398\pi\)
\(642\) 0 0
\(643\) 3.48821 19.7826i 0.137561 0.780150i −0.835480 0.549520i \(-0.814810\pi\)
0.973042 0.230629i \(-0.0740784\pi\)
\(644\) 0 0
\(645\) 11.4147 2.97740i 0.449453 0.117235i
\(646\) 0 0
\(647\) 24.0065 0.943794 0.471897 0.881654i \(-0.343569\pi\)
0.471897 + 0.881654i \(0.343569\pi\)
\(648\) 0 0
\(649\) −30.6147 −1.20173
\(650\) 0 0
\(651\) 14.6333 3.81695i 0.573526 0.149598i
\(652\) 0 0
\(653\) −1.65099 + 9.36326i −0.0646084 + 0.366412i 0.935312 + 0.353823i \(0.115119\pi\)
−0.999921 + 0.0125893i \(0.995993\pi\)
\(654\) 0 0
\(655\) 9.05546 7.59844i 0.353826 0.296895i
\(656\) 0 0
\(657\) −19.7685 11.7664i −0.771243 0.459051i
\(658\) 0 0
\(659\) 4.02748 + 22.8410i 0.156888 + 0.889758i 0.957039 + 0.289959i \(0.0936416\pi\)
−0.800151 + 0.599799i \(0.795247\pi\)
\(660\) 0 0
\(661\) −18.4498 + 6.71518i −0.717614 + 0.261190i −0.674912 0.737898i \(-0.735819\pi\)
−0.0427014 + 0.999088i \(0.513596\pi\)
\(662\) 0 0
\(663\) −0.286359 3.54422i −0.0111213 0.137646i
\(664\) 0 0
\(665\) −1.73136 + 2.99880i −0.0671392 + 0.116289i
\(666\) 0 0
\(667\) 0.148839 + 0.257796i 0.00576306 + 0.00998190i
\(668\) 0 0
\(669\) 12.3700 17.4187i 0.478252 0.673444i
\(670\) 0 0
\(671\) 8.63653 + 7.24691i 0.333410 + 0.279764i
\(672\) 0 0
\(673\) 37.7532 + 13.7410i 1.45528 + 0.529678i 0.944060 0.329773i \(-0.106972\pi\)
0.511218 + 0.859451i \(0.329194\pi\)
\(674\) 0 0
\(675\) −19.7963 + 8.75504i −0.761959 + 0.336982i
\(676\) 0 0
\(677\) −22.7982 8.29787i −0.876206 0.318913i −0.135529 0.990773i \(-0.543273\pi\)
−0.740678 + 0.671860i \(0.765495\pi\)
\(678\) 0 0
\(679\) −12.6235 10.5924i −0.484446 0.406498i
\(680\) 0 0
\(681\) −2.58042 5.63112i −0.0988818 0.215785i
\(682\) 0 0
\(683\) −5.13229 8.88938i −0.196382 0.340143i 0.750971 0.660335i \(-0.229586\pi\)
−0.947353 + 0.320192i \(0.896252\pi\)
\(684\) 0 0
\(685\) 6.23909 10.8064i 0.238383 0.412892i
\(686\) 0 0
\(687\) 29.8517 20.6081i 1.13891 0.786247i
\(688\) 0 0
\(689\) 15.0592 5.48112i 0.573711 0.208814i
\(690\) 0 0
\(691\) −8.11802 46.0396i −0.308824 1.75143i −0.604936 0.796274i \(-0.706801\pi\)
0.296112 0.955153i \(-0.404310\pi\)
\(692\) 0 0
\(693\) 24.6062 + 4.67676i 0.934713 + 0.177655i
\(694\) 0 0
\(695\) −4.49837 + 3.77458i −0.170633 + 0.143178i
\(696\) 0 0
\(697\) −1.88141 + 10.6700i −0.0712635 + 0.404156i
\(698\) 0 0
\(699\) −11.4172 + 41.5042i −0.431837 + 1.56983i
\(700\) 0 0
\(701\) −10.7090 −0.404475 −0.202237 0.979337i \(-0.564821\pi\)
−0.202237 + 0.979337i \(0.564821\pi\)
\(702\) 0 0
\(703\) −9.67718 −0.364982
\(704\) 0 0
\(705\) −6.94655 7.03950i −0.261622 0.265123i
\(706\) 0 0
\(707\) 3.67400 20.8363i 0.138175 0.783630i
\(708\) 0 0
\(709\) −39.4897 + 33.1358i −1.48307 + 1.24444i −0.580230 + 0.814453i \(0.697037\pi\)
−0.902835 + 0.429987i \(0.858518\pi\)
\(710\) 0 0
\(711\) 46.4606 7.55700i 1.74241 0.283410i
\(712\) 0 0
\(713\) 3.47365 + 19.7000i 0.130089 + 0.737772i
\(714\) 0 0
\(715\) 6.81180 2.47929i 0.254747 0.0927203i
\(716\) 0 0
\(717\) −23.2852 11.0470i −0.869602 0.412560i
\(718\) 0 0
\(719\) 7.78782 13.4889i 0.290437 0.503051i −0.683476 0.729973i \(-0.739533\pi\)
0.973913 + 0.226922i \(0.0728661\pi\)
\(720\) 0 0
\(721\) −6.35069 10.9997i −0.236512 0.409651i
\(722\) 0 0
\(723\) −5.79628 0.545945i −0.215566 0.0203039i
\(724\) 0 0
\(725\) 0.253860 + 0.213014i 0.00942812 + 0.00791113i
\(726\) 0 0
\(727\) −9.61987 3.50135i −0.356781 0.129858i 0.157410 0.987533i \(-0.449686\pi\)
−0.514191 + 0.857676i \(0.671908\pi\)
\(728\) 0 0
\(729\) 1.07627 26.9785i 0.0398620 0.999205i
\(730\) 0 0
\(731\) −9.26510 3.37222i −0.342682 0.124726i
\(732\) 0 0
\(733\) 28.4760 + 23.8942i 1.05179 + 0.882553i 0.993280 0.115734i \(-0.0369221\pi\)
0.0585052 + 0.998287i \(0.481367\pi\)
\(734\) 0 0
\(735\) 6.82371 + 0.642718i 0.251697 + 0.0237070i
\(736\) 0 0
\(737\) 0.190002 + 0.329093i 0.00699881 + 0.0121223i
\(738\) 0 0
\(739\) −4.39183 + 7.60687i −0.161556 + 0.279823i −0.935427 0.353520i \(-0.884985\pi\)
0.773871 + 0.633343i \(0.218318\pi\)
\(740\) 0 0
\(741\) 5.63957 + 2.67554i 0.207175 + 0.0982885i
\(742\) 0 0
\(743\) −11.7938 + 4.29260i −0.432673 + 0.157480i −0.549170 0.835711i \(-0.685056\pi\)
0.116496 + 0.993191i \(0.462834\pi\)
\(744\) 0 0
\(745\) 0.425861 + 2.41518i 0.0156023 + 0.0884853i
\(746\) 0 0
\(747\) −3.40002 + 0.553028i −0.124400 + 0.0202342i
\(748\) 0 0
\(749\) −10.8433 + 9.09858i −0.396204 + 0.332455i
\(750\) 0 0
\(751\) 0.723810 4.10493i 0.0264122 0.149791i −0.968750 0.248041i \(-0.920213\pi\)
0.995162 + 0.0982497i \(0.0313244\pi\)
\(752\) 0 0
\(753\) −12.0045 12.1651i −0.437467 0.443321i
\(754\) 0 0
\(755\) 19.0230 0.692316
\(756\) 0 0
\(757\) −22.7574 −0.827131 −0.413566 0.910474i \(-0.635717\pi\)
−0.413566 + 0.910474i \(0.635717\pi\)
\(758\) 0 0
\(759\) −8.78732 + 31.9441i −0.318959 + 1.15950i
\(760\) 0 0
\(761\) −0.391625 + 2.22102i −0.0141964 + 0.0805118i −0.991083 0.133247i \(-0.957460\pi\)
0.976886 + 0.213759i \(0.0685707\pi\)
\(762\) 0 0
\(763\) −21.6777 + 18.1897i −0.784785 + 0.658513i
\(764\) 0 0
\(765\) −3.55942 0.676517i −0.128691 0.0244595i
\(766\) 0 0
\(767\) −1.61468 9.15731i −0.0583027 0.330651i
\(768\) 0 0
\(769\) 18.2524 6.64334i 0.658200 0.239565i 0.00874088 0.999962i \(-0.497218\pi\)
0.649459 + 0.760397i \(0.274995\pi\)
\(770\) 0 0
\(771\) −4.89610 + 3.38002i −0.176329 + 0.121729i
\(772\) 0 0
\(773\) 0.866208 1.50032i 0.0311553 0.0539626i −0.850027 0.526739i \(-0.823415\pi\)
0.881183 + 0.472776i \(0.156748\pi\)
\(774\) 0 0
\(775\) 11.1347 + 19.2859i 0.399972 + 0.692771i
\(776\) 0 0
\(777\) −4.91319 10.7218i −0.176260 0.384643i
\(778\) 0 0
\(779\) −14.5701 12.2257i −0.522027 0.438033i
\(780\) 0 0
\(781\) −35.8885 13.0623i −1.28419 0.467407i
\(782\) 0 0
\(783\) −0.378040 + 0.167191i −0.0135100 + 0.00597491i
\(784\) 0 0
\(785\) 6.63027 + 2.41322i 0.236644 + 0.0861315i
\(786\) 0 0
\(787\) 7.34653 + 6.16447i 0.261875 + 0.219740i 0.764266 0.644901i \(-0.223102\pi\)
−0.502390 + 0.864641i \(0.667546\pi\)
\(788\) 0 0
\(789\) −16.3280 + 22.9920i −0.581291 + 0.818539i
\(790\) 0 0
\(791\) 1.79290 + 3.10539i 0.0637481 + 0.110415i
\(792\) 0 0
\(793\) −1.71215 + 2.96553i −0.0608003 + 0.105309i
\(794\) 0 0
\(795\) −1.31506 16.2763i −0.0466405 0.577261i
\(796\) 0 0
\(797\) −49.4162 + 17.9860i −1.75041 + 0.637097i −0.999722 0.0235589i \(-0.992500\pi\)
−0.750688 + 0.660656i \(0.770278\pi\)
\(798\) 0 0
\(799\) 1.43538 + 8.14043i 0.0507800 + 0.287988i
\(800\) 0 0
\(801\) −46.8704 27.8977i −1.65609 0.985718i
\(802\) 0 0
\(803\) −30.0283 + 25.1967i −1.05968 + 0.889173i
\(804\) 0 0
\(805\) 0.969341 5.49740i 0.0341648 0.193758i
\(806\) 0 0
\(807\) 28.3146 7.38556i 0.996721 0.259984i
\(808\) 0 0
\(809\) 10.1781 0.357842 0.178921 0.983863i \(-0.442739\pi\)
0.178921 + 0.983863i \(0.442739\pi\)
\(810\) 0 0
\(811\) −49.5069 −1.73842 −0.869211 0.494441i \(-0.835373\pi\)
−0.869211 + 0.494441i \(0.835373\pi\)
\(812\) 0 0
\(813\) 4.19583 1.09444i 0.147154 0.0383836i
\(814\) 0 0
\(815\) −0.663639 + 3.76368i −0.0232463 + 0.131836i
\(816\) 0 0
\(817\) 13.2591 11.1257i 0.463875 0.389238i
\(818\) 0 0
\(819\) −0.101106 + 7.60676i −0.00353293 + 0.265802i
\(820\) 0 0
\(821\) −8.44421 47.8895i −0.294705 1.67136i −0.668399 0.743803i \(-0.733020\pi\)
0.373694 0.927552i \(-0.378091\pi\)
\(822\) 0 0
\(823\) −20.9283 + 7.61729i −0.729516 + 0.265522i −0.679960 0.733249i \(-0.738003\pi\)
−0.0495559 + 0.998771i \(0.515781\pi\)
\(824\) 0 0
\(825\) 2.97031 + 36.7630i 0.103413 + 1.27992i
\(826\) 0 0
\(827\) −6.93673 + 12.0148i −0.241214 + 0.417795i −0.961060 0.276339i \(-0.910879\pi\)
0.719847 + 0.694133i \(0.244212\pi\)
\(828\) 0 0
\(829\) 0.222661 + 0.385659i 0.00773332 + 0.0133945i 0.869866 0.493288i \(-0.164205\pi\)
−0.862133 + 0.506682i \(0.830872\pi\)
\(830\) 0 0
\(831\) −21.7840 + 30.6749i −0.755678 + 1.06410i
\(832\) 0 0
\(833\) −4.38833 3.68225i −0.152047 0.127582i
\(834\) 0 0
\(835\) 9.25915 + 3.37005i 0.320426 + 0.116626i
\(836\) 0 0
\(837\) −27.7149 + 1.86888i −0.957968 + 0.0645978i
\(838\) 0 0
\(839\) 14.3991 + 5.24083i 0.497111 + 0.180934i 0.578394 0.815757i \(-0.303680\pi\)
−0.0812832 + 0.996691i \(0.525902\pi\)
\(840\) 0 0
\(841\) −22.2104 18.6368i −0.765877 0.642647i
\(842\) 0 0
\(843\) 7.33792 + 16.0132i 0.252731 + 0.551523i
\(844\) 0 0
\(845\) −4.83606 8.37630i −0.166366 0.288154i
\(846\) 0 0
\(847\) 12.3558 21.4009i 0.424551 0.735343i
\(848\) 0 0
\(849\) −32.4436 + 22.3974i −1.11346 + 0.768677i
\(850\) 0 0
\(851\) 14.6597 5.33568i 0.502526 0.182905i
\(852\) 0 0
\(853\) −3.21377 18.2262i −0.110037 0.624052i −0.989088 0.147324i \(-0.952934\pi\)
0.879051 0.476728i \(-0.158177\pi\)
\(854\) 0 0
\(855\) 4.15273 4.81752i 0.142020 0.164756i
\(856\) 0 0
\(857\) 28.1960 23.6592i 0.963156 0.808184i −0.0183076 0.999832i \(-0.505828\pi\)
0.981464 + 0.191648i \(0.0613834\pi\)
\(858\) 0 0
\(859\) 3.38976 19.2243i 0.115657 0.655924i −0.870766 0.491698i \(-0.836376\pi\)
0.986423 0.164226i \(-0.0525126\pi\)
\(860\) 0 0
\(861\) 6.14815 22.3500i 0.209528 0.761687i
\(862\) 0 0
\(863\) −44.0808 −1.50053 −0.750263 0.661139i \(-0.770073\pi\)
−0.750263 + 0.661139i \(0.770073\pi\)
\(864\) 0 0
\(865\) −13.1690 −0.447759
\(866\) 0 0
\(867\) −18.5548 18.8031i −0.630155 0.638586i
\(868\) 0 0
\(869\) 13.9275 78.9870i 0.472459 2.67945i
\(870\) 0 0
\(871\) −0.0884155 + 0.0741894i −0.00299585 + 0.00251381i
\(872\) 0 0
\(873\) 19.1462 + 23.4434i 0.648001 + 0.793439i
\(874\) 0 0
\(875\) −2.37435 13.4656i −0.0802678 0.455221i
\(876\) 0 0
\(877\) −23.5133 + 8.55813i −0.793987 + 0.288987i −0.706992 0.707221i \(-0.749948\pi\)
−0.0869944 + 0.996209i \(0.527726\pi\)
\(878\) 0 0
\(879\) −53.0048 25.1467i −1.78781 0.848177i
\(880\) 0 0
\(881\) −22.3706 + 38.7470i −0.753684 + 1.30542i 0.192343 + 0.981328i \(0.438392\pi\)
−0.946026 + 0.324090i \(0.894942\pi\)
\(882\) 0 0
\(883\) −19.1927 33.2427i −0.645885 1.11871i −0.984096 0.177636i \(-0.943155\pi\)
0.338211 0.941070i \(-0.390178\pi\)
\(884\) 0 0
\(885\) −9.43300 0.888484i −0.317087 0.0298661i
\(886\) 0 0
\(887\) 6.35934 + 5.33612i 0.213526 + 0.179169i 0.743277 0.668984i \(-0.233270\pi\)
−0.529751 + 0.848153i \(0.677715\pi\)
\(888\) 0 0
\(889\) −30.7792 11.2027i −1.03230 0.375727i
\(890\) 0 0
\(891\) −42.7979 16.8784i −1.43378 0.565448i
\(892\) 0 0
\(893\) −13.6357 4.96298i −0.456300 0.166080i
\(894\) 0 0
\(895\) 7.47217 + 6.26989i 0.249767 + 0.209579i
\(896\) 0 0
\(897\) −10.0184 0.943624i −0.334505 0.0315067i
\(898\) 0 0
\(899\) 0.212635 + 0.368294i 0.00709176 + 0.0122833i
\(900\) 0 0
\(901\) −6.82409 + 11.8197i −0.227344 + 0.393771i
\(902\) 0 0
\(903\) 19.0584 + 9.04176i 0.634224 + 0.300891i
\(904\) 0 0
\(905\) 19.7196 7.17734i 0.655501 0.238583i
\(906\) 0 0
\(907\) −5.55783 31.5200i −0.184545 1.04661i −0.926539 0.376199i \(-0.877231\pi\)
0.741994 0.670407i \(-0.233880\pi\)
\(908\) 0 0
\(909\) −13.7760 + 36.3390i −0.456920 + 1.20529i
\(910\) 0 0
\(911\) 38.8843 32.6278i 1.28829 1.08101i 0.296250 0.955111i \(-0.404264\pi\)
0.992043 0.125896i \(-0.0401805\pi\)
\(912\) 0 0
\(913\) −1.01923 + 5.78033i −0.0337315 + 0.191301i
\(914\) 0 0
\(915\) 2.45077 + 2.48357i 0.0810201 + 0.0821041i
\(916\) 0 0
\(917\) 21.1382 0.698044
\(918\) 0 0
\(919\) 22.2511 0.733995 0.366998 0.930222i \(-0.380386\pi\)
0.366998 + 0.930222i \(0.380386\pi\)
\(920\) 0 0
\(921\) −3.20820 + 11.6626i −0.105714 + 0.384296i
\(922\) 0 0
\(923\) 2.01431 11.4237i 0.0663018 0.376016i
\(924\) 0 0
\(925\) 13.3041 11.1635i 0.437437 0.367053i
\(926\) 0 0
\(927\) 7.68723 + 22.0270i 0.252482 + 0.723463i
\(928\) 0 0
\(929\) 3.10163 + 17.5902i 0.101761 + 0.577116i 0.992465 + 0.122532i \(0.0391013\pi\)
−0.890703 + 0.454585i \(0.849788\pi\)
\(930\) 0 0
\(931\) 9.44986 3.43947i 0.309707 0.112724i
\(932\) 0 0
\(933\) 12.8162 8.84765i 0.419583 0.289659i
\(934\) 0 0
\(935\) −3.08677 + 5.34644i −0.100948 + 0.174847i
\(936\) 0 0
\(937\) 6.68749 + 11.5831i 0.218471 + 0.378402i 0.954341 0.298721i \(-0.0965598\pi\)
−0.735870 + 0.677123i \(0.763227\pi\)
\(938\) 0 0
\(939\) −17.0898 37.2943i −0.557706 1.21705i
\(940\) 0 0
\(941\) 6.69590 + 5.61853i 0.218280 + 0.183159i 0.745371 0.666650i \(-0.232273\pi\)
−0.527090 + 0.849809i \(0.676717\pi\)
\(942\) 0 0
\(943\) 28.8126 + 10.4869i 0.938268 + 0.341502i
\(944\) 0 0
\(945\) 7.44595 + 2.15511i 0.242217 + 0.0701059i
\(946\) 0 0
\(947\) −19.9691 7.26815i −0.648908 0.236183i −0.00346769 0.999994i \(-0.501104\pi\)
−0.645440 + 0.763811i \(0.723326\pi\)
\(948\) 0 0
\(949\) −9.12047 7.65299i −0.296063 0.248426i
\(950\) 0 0
\(951\) 17.0329 23.9847i 0.552329 0.777756i
\(952\) 0 0
\(953\) 3.80567 + 6.59162i 0.123278 + 0.213523i 0.921058 0.389424i \(-0.127326\pi\)
−0.797781 + 0.602948i \(0.793993\pi\)
\(954\) 0 0
\(955\) −2.21982 + 3.84485i −0.0718318 + 0.124416i
\(956\) 0 0
\(957\) 0.0567226 + 0.702045i 0.00183358 + 0.0226939i
\(958\) 0 0
\(959\) 20.9676 7.63157i 0.677078 0.246436i
\(960\) 0 0
\(961\) −0.420554 2.38508i −0.0135663 0.0769381i
\(962\) 0 0
\(963\) 22.6872 12.6994i 0.731084 0.409234i
\(964\) 0 0
\(965\) 13.7671 11.5519i 0.443178 0.371870i
\(966\) 0 0
\(967\) −2.95220 + 16.7428i −0.0949364 + 0.538411i 0.899830 + 0.436240i \(0.143690\pi\)
−0.994767 + 0.102171i \(0.967421\pi\)
\(968\) 0 0
\(969\) −5.14390 + 1.34173i −0.165246 + 0.0431026i
\(970\) 0 0
\(971\) −33.0729 −1.06136 −0.530679 0.847573i \(-0.678063\pi\)
−0.530679 + 0.847573i \(0.678063\pi\)
\(972\) 0 0
\(973\) −10.5006 −0.336632
\(974\) 0 0
\(975\) −10.8397 + 2.82742i −0.347149 + 0.0905500i
\(976\) 0 0
\(977\) −8.76021 + 49.6816i −0.280264 + 1.58946i 0.441464 + 0.897279i \(0.354459\pi\)
−0.721728 + 0.692177i \(0.756652\pi\)
\(978\) 0 0
\(979\) −71.1960 + 59.7405i −2.27543 + 1.90932i
\(980\) 0 0
\(981\) 45.3559 25.3885i 1.44810 0.810594i
\(982\) 0 0
\(983\) 2.02177 + 11.4660i 0.0644843 + 0.365709i 0.999925 + 0.0122258i \(0.00389168\pi\)
−0.935441 + 0.353483i \(0.884997\pi\)
\(984\) 0 0
\(985\) 15.2154 5.53794i 0.484802 0.176454i
\(986\) 0 0
\(987\) −1.42422 17.6274i −0.0453336 0.561086i
\(988\) 0 0
\(989\) −13.9514 + 24.1645i −0.443629 + 0.768387i
\(990\) 0 0
\(991\) 16.9339 + 29.3304i 0.537923 + 0.931710i 0.999016 + 0.0443581i \(0.0141242\pi\)
−0.461093 + 0.887352i \(0.652542\pi\)
\(992\) 0 0
\(993\) 20.6364 29.0590i 0.654878 0.922158i
\(994\) 0 0
\(995\) −3.89734 3.27026i −0.123554 0.103674i
\(996\) 0 0
\(997\) −20.9654 7.63077i −0.663980 0.241669i −0.0120262 0.999928i \(-0.503828\pi\)
−0.651954 + 0.758259i \(0.726050\pi\)
\(998\) 0 0
\(999\) 5.18858 + 21.0326i 0.164159 + 0.665441i
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 864.2.y.b.193.2 54
4.3 odd 2 864.2.y.c.193.8 yes 54
27.7 even 9 inner 864.2.y.b.385.2 yes 54
108.7 odd 18 864.2.y.c.385.8 yes 54
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.y.b.193.2 54 1.1 even 1 trivial
864.2.y.b.385.2 yes 54 27.7 even 9 inner
864.2.y.c.193.8 yes 54 4.3 odd 2
864.2.y.c.385.8 yes 54 108.7 odd 18