Properties

Label 420.2.bv.c.317.7
Level $420$
Weight $2$
Character 420.317
Analytic conductor $3.354$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [420,2,Mod(53,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bv (of order \(12\), degree \(4\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 317.7
Character \(\chi\) \(=\) 420.317
Dual form 420.2.bv.c.53.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.0801247 - 1.73020i) q^{3} +(-0.746108 - 2.10792i) q^{5} +(-2.40557 + 1.10146i) q^{7} +(-2.98716 - 0.277263i) q^{9} +O(q^{10})\) \(q+(0.0801247 - 1.73020i) q^{3} +(-0.746108 - 2.10792i) q^{5} +(-2.40557 + 1.10146i) q^{7} +(-2.98716 - 0.277263i) q^{9} +(2.63993 - 1.52416i) q^{11} +(0.512041 - 0.512041i) q^{13} +(-3.70690 + 1.12202i) q^{15} +(-3.88873 - 1.04198i) q^{17} +(-5.78739 - 3.34135i) q^{19} +(1.71300 + 4.25037i) q^{21} +(-2.29376 + 0.614611i) q^{23} +(-3.88664 + 3.14547i) q^{25} +(-0.719065 + 5.14616i) q^{27} +8.51556 q^{29} +(0.921008 + 1.59523i) q^{31} +(-2.42558 - 4.68971i) q^{33} +(4.11661 + 4.24895i) q^{35} +(2.78938 - 0.747413i) q^{37} +(-0.844905 - 0.926959i) q^{39} -8.91355i q^{41} +(-7.54399 + 7.54399i) q^{43} +(1.64430 + 6.50356i) q^{45} +(-1.95084 - 7.28063i) q^{47} +(4.57357 - 5.29929i) q^{49} +(-2.11442 + 6.64477i) q^{51} +(2.81412 - 10.5024i) q^{53} +(-5.18248 - 4.42756i) q^{55} +(-6.24491 + 9.74561i) q^{57} +(1.06765 + 1.84923i) q^{59} +(7.12318 - 12.3377i) q^{61} +(7.49123 - 2.62326i) q^{63} +(-1.46138 - 0.697303i) q^{65} +(1.03593 - 3.86614i) q^{67} +(0.879611 + 4.01790i) q^{69} -3.41895i q^{71} +(11.0196 + 2.95269i) q^{73} +(5.13087 + 6.97669i) q^{75} +(-4.67173 + 6.57426i) q^{77} +(12.1569 + 7.01879i) q^{79} +(8.84625 + 1.65646i) q^{81} +(-0.995323 - 0.995323i) q^{83} +(0.705000 + 8.97455i) q^{85} +(0.682306 - 14.7336i) q^{87} +(-0.706248 + 1.22326i) q^{89} +(-0.667760 + 1.79575i) q^{91} +(2.83386 - 1.46571i) q^{93} +(-2.72528 + 14.6924i) q^{95} +(-0.556234 - 0.556234i) q^{97} +(-8.30847 + 3.82096i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 2 q^{3} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75} + 20 q^{81} - 56 q^{85} - 24 q^{87} - 24 q^{91} + 6 q^{93} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.0801247 1.73020i 0.0462600 0.998929i
\(4\) 0 0
\(5\) −0.746108 2.10792i −0.333670 0.942690i
\(6\) 0 0
\(7\) −2.40557 + 1.10146i −0.909221 + 0.416313i
\(8\) 0 0
\(9\) −2.98716 0.277263i −0.995720 0.0924210i
\(10\) 0 0
\(11\) 2.63993 1.52416i 0.795968 0.459552i −0.0460916 0.998937i \(-0.514677\pi\)
0.842059 + 0.539385i \(0.181343\pi\)
\(12\) 0 0
\(13\) 0.512041 0.512041i 0.142015 0.142015i −0.632525 0.774540i \(-0.717982\pi\)
0.774540 + 0.632525i \(0.217982\pi\)
\(14\) 0 0
\(15\) −3.70690 + 1.12202i −0.957116 + 0.289704i
\(16\) 0 0
\(17\) −3.88873 1.04198i −0.943155 0.252718i −0.245700 0.969346i \(-0.579018\pi\)
−0.697455 + 0.716628i \(0.745684\pi\)
\(18\) 0 0
\(19\) −5.78739 3.34135i −1.32772 0.766559i −0.342773 0.939418i \(-0.611366\pi\)
−0.984947 + 0.172859i \(0.944699\pi\)
\(20\) 0 0
\(21\) 1.71300 + 4.25037i 0.373806 + 0.927507i
\(22\) 0 0
\(23\) −2.29376 + 0.614611i −0.478282 + 0.128155i −0.489902 0.871777i \(-0.662968\pi\)
0.0116206 + 0.999932i \(0.496301\pi\)
\(24\) 0 0
\(25\) −3.88664 + 3.14547i −0.777329 + 0.629094i
\(26\) 0 0
\(27\) −0.719065 + 5.14616i −0.138384 + 0.990379i
\(28\) 0 0
\(29\) 8.51556 1.58130 0.790650 0.612269i \(-0.209743\pi\)
0.790650 + 0.612269i \(0.209743\pi\)
\(30\) 0 0
\(31\) 0.921008 + 1.59523i 0.165418 + 0.286512i 0.936804 0.349856i \(-0.113769\pi\)
−0.771386 + 0.636368i \(0.780436\pi\)
\(32\) 0 0
\(33\) −2.42558 4.68971i −0.422239 0.816374i
\(34\) 0 0
\(35\) 4.11661 + 4.24895i 0.695834 + 0.718203i
\(36\) 0 0
\(37\) 2.78938 0.747413i 0.458572 0.122874i −0.0221357 0.999755i \(-0.507047\pi\)
0.480707 + 0.876881i \(0.340380\pi\)
\(38\) 0 0
\(39\) −0.844905 0.926959i −0.135293 0.148432i
\(40\) 0 0
\(41\) 8.91355i 1.39206i −0.718012 0.696031i \(-0.754948\pi\)
0.718012 0.696031i \(-0.245052\pi\)
\(42\) 0 0
\(43\) −7.54399 + 7.54399i −1.15045 + 1.15045i −0.163984 + 0.986463i \(0.552435\pi\)
−0.986463 + 0.163984i \(0.947565\pi\)
\(44\) 0 0
\(45\) 1.64430 + 6.50356i 0.245117 + 0.969493i
\(46\) 0 0
\(47\) −1.95084 7.28063i −0.284559 1.06199i −0.949161 0.314791i \(-0.898066\pi\)
0.664602 0.747198i \(-0.268601\pi\)
\(48\) 0 0
\(49\) 4.57357 5.29929i 0.653367 0.757041i
\(50\) 0 0
\(51\) −2.11442 + 6.64477i −0.296077 + 0.930455i
\(52\) 0 0
\(53\) 2.81412 10.5024i 0.386549 1.44262i −0.449161 0.893451i \(-0.648277\pi\)
0.835710 0.549171i \(-0.185056\pi\)
\(54\) 0 0
\(55\) −5.18248 4.42756i −0.698806 0.597012i
\(56\) 0 0
\(57\) −6.24491 + 9.74561i −0.827159 + 1.29084i
\(58\) 0 0
\(59\) 1.06765 + 1.84923i 0.138996 + 0.240749i 0.927117 0.374772i \(-0.122279\pi\)
−0.788121 + 0.615521i \(0.788946\pi\)
\(60\) 0 0
\(61\) 7.12318 12.3377i 0.912029 1.57968i 0.100836 0.994903i \(-0.467848\pi\)
0.811193 0.584778i \(-0.198818\pi\)
\(62\) 0 0
\(63\) 7.49123 2.62326i 0.943806 0.330500i
\(64\) 0 0
\(65\) −1.46138 0.697303i −0.181262 0.0864898i
\(66\) 0 0
\(67\) 1.03593 3.86614i 0.126559 0.472325i −0.873331 0.487126i \(-0.838045\pi\)
0.999890 + 0.0148018i \(0.00471174\pi\)
\(68\) 0 0
\(69\) 0.879611 + 4.01790i 0.105893 + 0.483698i
\(70\) 0 0
\(71\) 3.41895i 0.405755i −0.979204 0.202878i \(-0.934971\pi\)
0.979204 0.202878i \(-0.0650293\pi\)
\(72\) 0 0
\(73\) 11.0196 + 2.95269i 1.28975 + 0.345586i 0.837564 0.546340i \(-0.183979\pi\)
0.452182 + 0.891926i \(0.350646\pi\)
\(74\) 0 0
\(75\) 5.13087 + 6.97669i 0.592462 + 0.805599i
\(76\) 0 0
\(77\) −4.67173 + 6.57426i −0.532393 + 0.749206i
\(78\) 0 0
\(79\) 12.1569 + 7.01879i 1.36776 + 0.789675i 0.990641 0.136490i \(-0.0435823\pi\)
0.377117 + 0.926166i \(0.376916\pi\)
\(80\) 0 0
\(81\) 8.84625 + 1.65646i 0.982917 + 0.184051i
\(82\) 0 0
\(83\) −0.995323 0.995323i −0.109251 0.109251i 0.650368 0.759619i \(-0.274615\pi\)
−0.759619 + 0.650368i \(0.774615\pi\)
\(84\) 0 0
\(85\) 0.705000 + 8.97455i 0.0764680 + 0.973427i
\(86\) 0 0
\(87\) 0.682306 14.7336i 0.0731509 1.57961i
\(88\) 0 0
\(89\) −0.706248 + 1.22326i −0.0748622 + 0.129665i −0.901026 0.433764i \(-0.857185\pi\)
0.826164 + 0.563429i \(0.190518\pi\)
\(90\) 0 0
\(91\) −0.667760 + 1.79575i −0.0700003 + 0.188245i
\(92\) 0 0
\(93\) 2.83386 1.46571i 0.293858 0.151987i
\(94\) 0 0
\(95\) −2.72528 + 14.6924i −0.279608 + 1.50741i
\(96\) 0 0
\(97\) −0.556234 0.556234i −0.0564770 0.0564770i 0.678304 0.734781i \(-0.262715\pi\)
−0.734781 + 0.678304i \(0.762715\pi\)
\(98\) 0 0
\(99\) −8.30847 + 3.82096i −0.835033 + 0.384021i
\(100\) 0 0
\(101\) 2.90475 1.67706i 0.289034 0.166874i −0.348472 0.937319i \(-0.613299\pi\)
0.637506 + 0.770445i \(0.279966\pi\)
\(102\) 0 0
\(103\) 0.448028 + 1.67206i 0.0441456 + 0.164753i 0.984480 0.175499i \(-0.0561540\pi\)
−0.940334 + 0.340253i \(0.889487\pi\)
\(104\) 0 0
\(105\) 7.68135 6.78209i 0.749623 0.661865i
\(106\) 0 0
\(107\) −2.06095 7.69158i −0.199240 0.743573i −0.991128 0.132908i \(-0.957569\pi\)
0.791889 0.610666i \(-0.209098\pi\)
\(108\) 0 0
\(109\) 2.12083 1.22446i 0.203138 0.117282i −0.394980 0.918690i \(-0.629249\pi\)
0.598119 + 0.801408i \(0.295915\pi\)
\(110\) 0 0
\(111\) −1.06967 4.88607i −0.101529 0.463765i
\(112\) 0 0
\(113\) −5.01803 5.01803i −0.472056 0.472056i 0.430523 0.902580i \(-0.358329\pi\)
−0.902580 + 0.430523i \(0.858329\pi\)
\(114\) 0 0
\(115\) 3.00694 + 4.37649i 0.280399 + 0.408110i
\(116\) 0 0
\(117\) −1.67152 + 1.38758i −0.154532 + 0.128282i
\(118\) 0 0
\(119\) 10.5023 1.77671i 0.962746 0.162871i
\(120\) 0 0
\(121\) −0.853861 + 1.47893i −0.0776237 + 0.134448i
\(122\) 0 0
\(123\) −15.4222 0.714195i −1.39057 0.0643968i
\(124\) 0 0
\(125\) 9.53026 + 5.84587i 0.852412 + 0.522870i
\(126\) 0 0
\(127\) −5.74490 5.74490i −0.509777 0.509777i 0.404681 0.914458i \(-0.367383\pi\)
−0.914458 + 0.404681i \(0.867383\pi\)
\(128\) 0 0
\(129\) 12.4481 + 13.6570i 1.09600 + 1.20244i
\(130\) 0 0
\(131\) −1.12384 0.648847i −0.0981900 0.0566900i 0.450101 0.892978i \(-0.351388\pi\)
−0.548291 + 0.836288i \(0.684721\pi\)
\(132\) 0 0
\(133\) 17.6024 + 1.66329i 1.52632 + 0.144226i
\(134\) 0 0
\(135\) 11.3842 2.32386i 0.979795 0.200006i
\(136\) 0 0
\(137\) −0.507990 0.136116i −0.0434006 0.0116291i 0.237053 0.971497i \(-0.423818\pi\)
−0.280454 + 0.959867i \(0.590485\pi\)
\(138\) 0 0
\(139\) 17.6744i 1.49912i 0.661936 + 0.749561i \(0.269735\pi\)
−0.661936 + 0.749561i \(0.730265\pi\)
\(140\) 0 0
\(141\) −12.7532 + 2.79198i −1.07402 + 0.235127i
\(142\) 0 0
\(143\) 0.571317 2.13218i 0.0477759 0.178302i
\(144\) 0 0
\(145\) −6.35353 17.9501i −0.527632 1.49068i
\(146\) 0 0
\(147\) −8.80235 8.33778i −0.726006 0.687689i
\(148\) 0 0
\(149\) −10.3981 + 18.0100i −0.851842 + 1.47543i 0.0277017 + 0.999616i \(0.491181\pi\)
−0.879544 + 0.475818i \(0.842152\pi\)
\(150\) 0 0
\(151\) −5.44655 9.43371i −0.443234 0.767704i 0.554693 0.832055i \(-0.312836\pi\)
−0.997927 + 0.0643508i \(0.979502\pi\)
\(152\) 0 0
\(153\) 11.3273 + 4.19077i 0.915762 + 0.338803i
\(154\) 0 0
\(155\) 2.67545 3.13163i 0.214897 0.251538i
\(156\) 0 0
\(157\) −2.58508 + 9.64763i −0.206312 + 0.769965i 0.782734 + 0.622356i \(0.213824\pi\)
−0.989046 + 0.147609i \(0.952842\pi\)
\(158\) 0 0
\(159\) −17.9458 5.71049i −1.42320 0.452871i
\(160\) 0 0
\(161\) 4.84084 4.00498i 0.381511 0.315636i
\(162\) 0 0
\(163\) −0.883109 3.29581i −0.0691705 0.258148i 0.922678 0.385572i \(-0.125996\pi\)
−0.991848 + 0.127424i \(0.959329\pi\)
\(164\) 0 0
\(165\) −8.07579 + 8.61195i −0.628700 + 0.670440i
\(166\) 0 0
\(167\) −10.6613 + 10.6613i −0.824994 + 0.824994i −0.986819 0.161825i \(-0.948262\pi\)
0.161825 + 0.986819i \(0.448262\pi\)
\(168\) 0 0
\(169\) 12.4756i 0.959664i
\(170\) 0 0
\(171\) 16.3614 + 11.5858i 1.25119 + 0.885987i
\(172\) 0 0
\(173\) 16.7718 4.49399i 1.27514 0.341672i 0.443140 0.896453i \(-0.353865\pi\)
0.831996 + 0.554781i \(0.187198\pi\)
\(174\) 0 0
\(175\) 5.88500 11.8476i 0.444864 0.895598i
\(176\) 0 0
\(177\) 3.28507 1.69908i 0.246921 0.127711i
\(178\) 0 0
\(179\) −7.14090 12.3684i −0.533736 0.924458i −0.999223 0.0394031i \(-0.987454\pi\)
0.465488 0.885054i \(-0.345879\pi\)
\(180\) 0 0
\(181\) −5.90010 −0.438551 −0.219275 0.975663i \(-0.570369\pi\)
−0.219275 + 0.975663i \(0.570369\pi\)
\(182\) 0 0
\(183\) −20.7759 13.3130i −1.53580 0.984129i
\(184\) 0 0
\(185\) −3.65667 5.32214i −0.268844 0.391292i
\(186\) 0 0
\(187\) −11.8541 + 3.17630i −0.866858 + 0.232274i
\(188\) 0 0
\(189\) −3.93852 13.1715i −0.286485 0.958085i
\(190\) 0 0
\(191\) 9.55692 + 5.51769i 0.691514 + 0.399246i 0.804179 0.594387i \(-0.202605\pi\)
−0.112665 + 0.993633i \(0.535939\pi\)
\(192\) 0 0
\(193\) −13.6035 3.64505i −0.979203 0.262377i −0.266494 0.963837i \(-0.585865\pi\)
−0.712709 + 0.701460i \(0.752532\pi\)
\(194\) 0 0
\(195\) −1.32356 + 2.47260i −0.0947824 + 0.177067i
\(196\) 0 0
\(197\) 1.63139 1.63139i 0.116232 0.116232i −0.646599 0.762830i \(-0.723809\pi\)
0.762830 + 0.646599i \(0.223809\pi\)
\(198\) 0 0
\(199\) 9.65354 5.57347i 0.684321 0.395093i −0.117160 0.993113i \(-0.537379\pi\)
0.801481 + 0.598020i \(0.204046\pi\)
\(200\) 0 0
\(201\) −6.60618 2.10214i −0.465964 0.148273i
\(202\) 0 0
\(203\) −20.4848 + 9.37955i −1.43775 + 0.658315i
\(204\) 0 0
\(205\) −18.7890 + 6.65047i −1.31228 + 0.464489i
\(206\) 0 0
\(207\) 7.02223 1.19997i 0.488079 0.0834035i
\(208\) 0 0
\(209\) −20.3711 −1.40910
\(210\) 0 0
\(211\) 2.72171 0.187370 0.0936852 0.995602i \(-0.470135\pi\)
0.0936852 + 0.995602i \(0.470135\pi\)
\(212\) 0 0
\(213\) −5.91546 0.273943i −0.405321 0.0187702i
\(214\) 0 0
\(215\) 21.5307 + 10.2735i 1.46838 + 0.700646i
\(216\) 0 0
\(217\) −3.97264 2.82300i −0.269680 0.191638i
\(218\) 0 0
\(219\) 5.99168 18.8295i 0.404880 1.27238i
\(220\) 0 0
\(221\) −2.52473 + 1.45765i −0.169831 + 0.0980522i
\(222\) 0 0
\(223\) 13.0957 13.0957i 0.876950 0.876950i −0.116268 0.993218i \(-0.537093\pi\)
0.993218 + 0.116268i \(0.0370931\pi\)
\(224\) 0 0
\(225\) 12.4822 8.31841i 0.832143 0.554560i
\(226\) 0 0
\(227\) −13.2362 3.54662i −0.878515 0.235397i −0.208748 0.977969i \(-0.566939\pi\)
−0.669766 + 0.742572i \(0.733606\pi\)
\(228\) 0 0
\(229\) −10.3326 5.96554i −0.682799 0.394214i 0.118110 0.993001i \(-0.462316\pi\)
−0.800909 + 0.598787i \(0.795650\pi\)
\(230\) 0 0
\(231\) 11.0004 + 8.60978i 0.723775 + 0.566482i
\(232\) 0 0
\(233\) 9.76217 2.61577i 0.639541 0.171365i 0.0755455 0.997142i \(-0.475930\pi\)
0.563996 + 0.825778i \(0.309264\pi\)
\(234\) 0 0
\(235\) −13.8914 + 9.54435i −0.906177 + 0.622605i
\(236\) 0 0
\(237\) 13.1180 20.4714i 0.852103 1.32976i
\(238\) 0 0
\(239\) 21.4204 1.38557 0.692784 0.721145i \(-0.256384\pi\)
0.692784 + 0.721145i \(0.256384\pi\)
\(240\) 0 0
\(241\) −5.02811 8.70894i −0.323889 0.560992i 0.657398 0.753544i \(-0.271657\pi\)
−0.981287 + 0.192552i \(0.938324\pi\)
\(242\) 0 0
\(243\) 3.57480 15.1730i 0.229324 0.973350i
\(244\) 0 0
\(245\) −14.5828 5.68688i −0.931664 0.363321i
\(246\) 0 0
\(247\) −4.67429 + 1.25247i −0.297418 + 0.0796930i
\(248\) 0 0
\(249\) −1.80185 + 1.64235i −0.114188 + 0.104080i
\(250\) 0 0
\(251\) 5.13237i 0.323952i 0.986795 + 0.161976i \(0.0517867\pi\)
−0.986795 + 0.161976i \(0.948213\pi\)
\(252\) 0 0
\(253\) −5.11859 + 5.11859i −0.321803 + 0.321803i
\(254\) 0 0
\(255\) 15.5842 0.500705i 0.975922 0.0313554i
\(256\) 0 0
\(257\) −2.90270 10.8330i −0.181065 0.675745i −0.995439 0.0954037i \(-0.969586\pi\)
0.814373 0.580341i \(-0.197081\pi\)
\(258\) 0 0
\(259\) −5.88682 + 4.87035i −0.365789 + 0.302629i
\(260\) 0 0
\(261\) −25.4373 2.36105i −1.57453 0.146145i
\(262\) 0 0
\(263\) −3.68895 + 13.7673i −0.227470 + 0.848931i 0.753930 + 0.656955i \(0.228156\pi\)
−0.981400 + 0.191975i \(0.938511\pi\)
\(264\) 0 0
\(265\) −24.2379 + 1.90402i −1.48893 + 0.116963i
\(266\) 0 0
\(267\) 2.05989 + 1.31996i 0.126063 + 0.0807803i
\(268\) 0 0
\(269\) −7.17746 12.4317i −0.437618 0.757976i 0.559888 0.828568i \(-0.310844\pi\)
−0.997505 + 0.0705927i \(0.977511\pi\)
\(270\) 0 0
\(271\) 0.840888 1.45646i 0.0510803 0.0884737i −0.839355 0.543584i \(-0.817067\pi\)
0.890435 + 0.455110i \(0.150400\pi\)
\(272\) 0 0
\(273\) 3.05349 + 1.29924i 0.184806 + 0.0786335i
\(274\) 0 0
\(275\) −5.46624 + 14.2277i −0.329627 + 0.857962i
\(276\) 0 0
\(277\) −2.03258 + 7.58569i −0.122126 + 0.455780i −0.999721 0.0236225i \(-0.992480\pi\)
0.877595 + 0.479403i \(0.159147\pi\)
\(278\) 0 0
\(279\) −2.30890 5.02058i −0.138230 0.300574i
\(280\) 0 0
\(281\) 8.01304i 0.478018i 0.971017 + 0.239009i \(0.0768225\pi\)
−0.971017 + 0.239009i \(0.923177\pi\)
\(282\) 0 0
\(283\) 1.35822 + 0.363933i 0.0807376 + 0.0216336i 0.298962 0.954265i \(-0.403360\pi\)
−0.218224 + 0.975899i \(0.570026\pi\)
\(284\) 0 0
\(285\) 25.2023 + 5.89249i 1.49286 + 0.349041i
\(286\) 0 0
\(287\) 9.81791 + 21.4422i 0.579533 + 1.26569i
\(288\) 0 0
\(289\) −0.685955 0.396036i −0.0403503 0.0232963i
\(290\) 0 0
\(291\) −1.00696 + 0.917826i −0.0590292 + 0.0538039i
\(292\) 0 0
\(293\) 3.47074 + 3.47074i 0.202763 + 0.202763i 0.801183 0.598420i \(-0.204205\pi\)
−0.598420 + 0.801183i \(0.704205\pi\)
\(294\) 0 0
\(295\) 3.10144 3.63025i 0.180573 0.211361i
\(296\) 0 0
\(297\) 5.94530 + 14.6814i 0.344981 + 0.851904i
\(298\) 0 0
\(299\) −0.859793 + 1.48920i −0.0497231 + 0.0861229i
\(300\) 0 0
\(301\) 9.83822 26.4570i 0.567066 1.52496i
\(302\) 0 0
\(303\) −2.66890 5.16017i −0.153324 0.296444i
\(304\) 0 0
\(305\) −31.3215 5.80981i −1.79347 0.332669i
\(306\) 0 0
\(307\) 19.7303 + 19.7303i 1.12607 + 1.12607i 0.990811 + 0.135254i \(0.0431852\pi\)
0.135254 + 0.990811i \(0.456815\pi\)
\(308\) 0 0
\(309\) 2.92890 0.641204i 0.166619 0.0364768i
\(310\) 0 0
\(311\) −23.7702 + 13.7238i −1.34789 + 0.778203i −0.987950 0.154774i \(-0.950535\pi\)
−0.359937 + 0.932977i \(0.617202\pi\)
\(312\) 0 0
\(313\) 6.10435 + 22.7817i 0.345038 + 1.28770i 0.892568 + 0.450913i \(0.148902\pi\)
−0.547530 + 0.836786i \(0.684432\pi\)
\(314\) 0 0
\(315\) −11.1189 13.8337i −0.626478 0.779439i
\(316\) 0 0
\(317\) 8.31005 + 31.0135i 0.466739 + 1.74189i 0.651059 + 0.759027i \(0.274325\pi\)
−0.184320 + 0.982866i \(0.559008\pi\)
\(318\) 0 0
\(319\) 22.4804 12.9791i 1.25866 0.726690i
\(320\) 0 0
\(321\) −13.4731 + 2.94957i −0.751994 + 0.164629i
\(322\) 0 0
\(323\) 19.0240 + 19.0240i 1.05852 + 1.05852i
\(324\) 0 0
\(325\) −0.379511 + 3.60073i −0.0210515 + 0.199733i
\(326\) 0 0
\(327\) −1.94863 3.76756i −0.107759 0.208346i
\(328\) 0 0
\(329\) 12.7122 + 15.3653i 0.700847 + 0.847117i
\(330\) 0 0
\(331\) 13.9114 24.0952i 0.764637 1.32439i −0.175801 0.984426i \(-0.556252\pi\)
0.940438 0.339965i \(-0.110415\pi\)
\(332\) 0 0
\(333\) −8.53956 + 1.45925i −0.467965 + 0.0799664i
\(334\) 0 0
\(335\) −8.92243 + 0.700906i −0.487485 + 0.0382946i
\(336\) 0 0
\(337\) −5.16934 5.16934i −0.281592 0.281592i 0.552152 0.833744i \(-0.313807\pi\)
−0.833744 + 0.552152i \(0.813807\pi\)
\(338\) 0 0
\(339\) −9.08424 + 8.28011i −0.493388 + 0.449714i
\(340\) 0 0
\(341\) 4.86279 + 2.80753i 0.263335 + 0.152036i
\(342\) 0 0
\(343\) −5.16511 + 17.7854i −0.278890 + 0.960323i
\(344\) 0 0
\(345\) 7.81312 4.85194i 0.420644 0.261220i
\(346\) 0 0
\(347\) 24.8906 + 6.66941i 1.33620 + 0.358033i 0.855022 0.518592i \(-0.173543\pi\)
0.481175 + 0.876625i \(0.340210\pi\)
\(348\) 0 0
\(349\) 18.4866i 0.989563i −0.869017 0.494782i \(-0.835248\pi\)
0.869017 0.494782i \(-0.164752\pi\)
\(350\) 0 0
\(351\) 2.26685 + 3.00323i 0.120996 + 0.160301i
\(352\) 0 0
\(353\) 6.80767 25.4066i 0.362336 1.35226i −0.508661 0.860967i \(-0.669859\pi\)
0.870997 0.491289i \(-0.163474\pi\)
\(354\) 0 0
\(355\) −7.20688 + 2.55091i −0.382501 + 0.135388i
\(356\) 0 0
\(357\) −2.23257 18.3134i −0.118160 0.969250i
\(358\) 0 0
\(359\) −4.69610 + 8.13389i −0.247851 + 0.429290i −0.962929 0.269754i \(-0.913058\pi\)
0.715078 + 0.699044i \(0.246391\pi\)
\(360\) 0 0
\(361\) 12.8293 + 22.2210i 0.675226 + 1.16953i
\(362\) 0 0
\(363\) 2.49043 + 1.59585i 0.130713 + 0.0837602i
\(364\) 0 0
\(365\) −1.99778 25.4314i −0.104568 1.33114i
\(366\) 0 0
\(367\) 3.54474 13.2291i 0.185034 0.690555i −0.809590 0.586996i \(-0.800310\pi\)
0.994623 0.103559i \(-0.0330231\pi\)
\(368\) 0 0
\(369\) −2.47140 + 26.6262i −0.128656 + 1.38610i
\(370\) 0 0
\(371\) 4.79845 + 28.3641i 0.249123 + 1.47259i
\(372\) 0 0
\(373\) 7.60558 + 28.3844i 0.393802 + 1.46969i 0.823811 + 0.566865i \(0.191844\pi\)
−0.430008 + 0.902825i \(0.641489\pi\)
\(374\) 0 0
\(375\) 10.8781 16.0208i 0.561743 0.827312i
\(376\) 0 0
\(377\) 4.36032 4.36032i 0.224568 0.224568i
\(378\) 0 0
\(379\) 0.342505i 0.0175933i −0.999961 0.00879664i \(-0.997200\pi\)
0.999961 0.00879664i \(-0.00280009\pi\)
\(380\) 0 0
\(381\) −10.4001 + 9.47949i −0.532814 + 0.485649i
\(382\) 0 0
\(383\) −25.0334 + 6.70768i −1.27915 + 0.342746i −0.833528 0.552477i \(-0.813683\pi\)
−0.445618 + 0.895223i \(0.647016\pi\)
\(384\) 0 0
\(385\) 17.3436 + 4.94253i 0.883913 + 0.251895i
\(386\) 0 0
\(387\) 24.6268 20.4434i 1.25185 1.03920i
\(388\) 0 0
\(389\) −14.6458 25.3673i −0.742572 1.28617i −0.951321 0.308203i \(-0.900272\pi\)
0.208749 0.977969i \(-0.433061\pi\)
\(390\) 0 0
\(391\) 9.56022 0.483481
\(392\) 0 0
\(393\) −1.21268 + 1.89247i −0.0611716 + 0.0954624i
\(394\) 0 0
\(395\) 5.72467 30.8625i 0.288040 1.55286i
\(396\) 0 0
\(397\) 27.5081 7.37078i 1.38059 0.369929i 0.509257 0.860615i \(-0.329920\pi\)
0.871337 + 0.490686i \(0.163254\pi\)
\(398\) 0 0
\(399\) 4.28820 30.3223i 0.214679 1.51801i
\(400\) 0 0
\(401\) −22.3395 12.8977i −1.11558 0.644081i −0.175311 0.984513i \(-0.556093\pi\)
−0.940269 + 0.340432i \(0.889427\pi\)
\(402\) 0 0
\(403\) 1.28842 + 0.345231i 0.0641807 + 0.0171972i
\(404\) 0 0
\(405\) −3.10858 19.8831i −0.154467 0.987998i
\(406\) 0 0
\(407\) 6.22459 6.22459i 0.308541 0.308541i
\(408\) 0 0
\(409\) 25.1282 14.5078i 1.24251 0.717364i 0.272905 0.962041i \(-0.412015\pi\)
0.969605 + 0.244677i \(0.0786820\pi\)
\(410\) 0 0
\(411\) −0.276209 + 0.868017i −0.0136244 + 0.0428161i
\(412\) 0 0
\(413\) −4.60517 3.27248i −0.226605 0.161028i
\(414\) 0 0
\(415\) −1.35544 + 2.84068i −0.0665360 + 0.139443i
\(416\) 0 0
\(417\) 30.5801 + 1.41615i 1.49752 + 0.0693494i
\(418\) 0 0
\(419\) −20.0060 −0.977359 −0.488679 0.872463i \(-0.662521\pi\)
−0.488679 + 0.872463i \(0.662521\pi\)
\(420\) 0 0
\(421\) −0.405902 −0.0197825 −0.00989124 0.999951i \(-0.503149\pi\)
−0.00989124 + 0.999951i \(0.503149\pi\)
\(422\) 0 0
\(423\) 3.80882 + 22.2893i 0.185191 + 1.08374i
\(424\) 0 0
\(425\) 18.3916 8.18207i 0.892125 0.396889i
\(426\) 0 0
\(427\) −3.54584 + 37.5251i −0.171595 + 1.81597i
\(428\) 0 0
\(429\) −3.64332 1.15933i −0.175901 0.0559730i
\(430\) 0 0
\(431\) 34.8654 20.1295i 1.67940 0.969605i 0.717364 0.696698i \(-0.245348\pi\)
0.962041 0.272907i \(-0.0879850\pi\)
\(432\) 0 0
\(433\) −9.70713 + 9.70713i −0.466495 + 0.466495i −0.900777 0.434282i \(-0.857002\pi\)
0.434282 + 0.900777i \(0.357002\pi\)
\(434\) 0 0
\(435\) −31.5663 + 9.55461i −1.51349 + 0.458108i
\(436\) 0 0
\(437\) 15.3285 + 4.10727i 0.733263 + 0.196477i
\(438\) 0 0
\(439\) 10.6016 + 6.12083i 0.505987 + 0.292131i 0.731182 0.682182i \(-0.238969\pi\)
−0.225196 + 0.974314i \(0.572302\pi\)
\(440\) 0 0
\(441\) −15.1313 + 14.5617i −0.720537 + 0.693416i
\(442\) 0 0
\(443\) −10.5633 + 2.83043i −0.501879 + 0.134478i −0.500872 0.865521i \(-0.666987\pi\)
−0.00100695 + 0.999999i \(0.500321\pi\)
\(444\) 0 0
\(445\) 3.10547 + 0.576031i 0.147213 + 0.0273065i
\(446\) 0 0
\(447\) 30.3276 + 19.4337i 1.43445 + 0.919184i
\(448\) 0 0
\(449\) 22.4970 1.06170 0.530849 0.847467i \(-0.321873\pi\)
0.530849 + 0.847467i \(0.321873\pi\)
\(450\) 0 0
\(451\) −13.5857 23.5311i −0.639725 1.10804i
\(452\) 0 0
\(453\) −16.7586 + 8.66774i −0.787386 + 0.407246i
\(454\) 0 0
\(455\) 4.28351 + 0.0677631i 0.200814 + 0.00317678i
\(456\) 0 0
\(457\) −17.9896 + 4.82030i −0.841517 + 0.225484i −0.653732 0.756726i \(-0.726798\pi\)
−0.187785 + 0.982210i \(0.560131\pi\)
\(458\) 0 0
\(459\) 8.15845 19.2628i 0.380804 0.899108i
\(460\) 0 0
\(461\) 5.92233i 0.275830i 0.990444 + 0.137915i \(0.0440402\pi\)
−0.990444 + 0.137915i \(0.955960\pi\)
\(462\) 0 0
\(463\) 26.3657 26.3657i 1.22532 1.22532i 0.259605 0.965715i \(-0.416408\pi\)
0.965715 0.259605i \(-0.0835922\pi\)
\(464\) 0 0
\(465\) −5.20396 4.87997i −0.241328 0.226303i
\(466\) 0 0
\(467\) −10.4566 39.0245i −0.483873 1.80584i −0.585083 0.810973i \(-0.698938\pi\)
0.101210 0.994865i \(-0.467729\pi\)
\(468\) 0 0
\(469\) 1.76640 + 10.4413i 0.0815646 + 0.482136i
\(470\) 0 0
\(471\) 16.4852 + 5.24570i 0.759597 + 0.241709i
\(472\) 0 0
\(473\) −8.41731 + 31.4138i −0.387028 + 1.44441i
\(474\) 0 0
\(475\) 33.0037 5.21743i 1.51431 0.239392i
\(476\) 0 0
\(477\) −11.3182 + 30.5922i −0.518224 + 1.40072i
\(478\) 0 0
\(479\) −0.363532 0.629656i −0.0166102 0.0287697i 0.857601 0.514316i \(-0.171954\pi\)
−0.874211 + 0.485546i \(0.838621\pi\)
\(480\) 0 0
\(481\) 1.04557 1.81098i 0.0476740 0.0825738i
\(482\) 0 0
\(483\) −6.54152 8.69650i −0.297650 0.395704i
\(484\) 0 0
\(485\) −0.757485 + 1.58751i −0.0343956 + 0.0720850i
\(486\) 0 0
\(487\) 4.72938 17.6503i 0.214309 0.799812i −0.772100 0.635501i \(-0.780793\pi\)
0.986409 0.164310i \(-0.0525398\pi\)
\(488\) 0 0
\(489\) −5.77316 + 1.26388i −0.261071 + 0.0571545i
\(490\) 0 0
\(491\) 37.6269i 1.69808i 0.528330 + 0.849039i \(0.322818\pi\)
−0.528330 + 0.849039i \(0.677182\pi\)
\(492\) 0 0
\(493\) −33.1147 8.87305i −1.49141 0.399622i
\(494\) 0 0
\(495\) 14.2533 + 14.6627i 0.640638 + 0.659041i
\(496\) 0 0
\(497\) 3.76584 + 8.22455i 0.168921 + 0.368921i
\(498\) 0 0
\(499\) 3.10886 + 1.79490i 0.139172 + 0.0803509i 0.567969 0.823050i \(-0.307729\pi\)
−0.428797 + 0.903401i \(0.641063\pi\)
\(500\) 0 0
\(501\) 17.5919 + 19.3003i 0.785947 + 0.862275i
\(502\) 0 0
\(503\) 11.0730 + 11.0730i 0.493722 + 0.493722i 0.909477 0.415754i \(-0.136482\pi\)
−0.415754 + 0.909477i \(0.636482\pi\)
\(504\) 0 0
\(505\) −5.70237 4.87172i −0.253752 0.216789i
\(506\) 0 0
\(507\) 21.5853 + 0.999606i 0.958636 + 0.0443941i
\(508\) 0 0
\(509\) −17.1754 + 29.7487i −0.761287 + 1.31859i 0.180900 + 0.983502i \(0.442099\pi\)
−0.942187 + 0.335087i \(0.891234\pi\)
\(510\) 0 0
\(511\) −29.7607 + 5.03472i −1.31654 + 0.222723i
\(512\) 0 0
\(513\) 21.3566 27.3802i 0.942919 1.20887i
\(514\) 0 0
\(515\) 3.19030 2.19195i 0.140581 0.0965888i
\(516\) 0 0
\(517\) −16.2469 16.2469i −0.714539 0.714539i
\(518\) 0 0
\(519\) −6.43165 29.3786i −0.282318 1.28958i
\(520\) 0 0
\(521\) −23.1158 + 13.3459i −1.01272 + 0.584696i −0.911988 0.410217i \(-0.865453\pi\)
−0.100736 + 0.994913i \(0.532120\pi\)
\(522\) 0 0
\(523\) 6.89247 + 25.7231i 0.301387 + 1.12479i 0.936011 + 0.351970i \(0.114488\pi\)
−0.634625 + 0.772821i \(0.718845\pi\)
\(524\) 0 0
\(525\) −20.0272 11.1315i −0.874060 0.485818i
\(526\) 0 0
\(527\) −1.91935 7.16310i −0.0836080 0.312029i
\(528\) 0 0
\(529\) −15.0350 + 8.68046i −0.653696 + 0.377411i
\(530\) 0 0
\(531\) −2.67653 5.81996i −0.116151 0.252565i
\(532\) 0 0
\(533\) −4.56410 4.56410i −0.197693 0.197693i
\(534\) 0 0
\(535\) −14.6755 + 10.0831i −0.634479 + 0.435929i
\(536\) 0 0
\(537\) −21.9719 + 11.3641i −0.948158 + 0.490399i
\(538\) 0 0
\(539\) 3.99692 20.9606i 0.172160 0.902836i
\(540\) 0 0
\(541\) 4.10742 7.11425i 0.176592 0.305866i −0.764119 0.645075i \(-0.776826\pi\)
0.940711 + 0.339209i \(0.110159\pi\)
\(542\) 0 0
\(543\) −0.472743 + 10.2083i −0.0202874 + 0.438081i
\(544\) 0 0
\(545\) −4.16343 3.55695i −0.178342 0.152363i
\(546\) 0 0
\(547\) −22.4125 22.4125i −0.958288 0.958288i 0.0408761 0.999164i \(-0.486985\pi\)
−0.999164 + 0.0408761i \(0.986985\pi\)
\(548\) 0 0
\(549\) −24.6989 + 34.8797i −1.05412 + 1.48863i
\(550\) 0 0
\(551\) −49.2829 28.4535i −2.09952 1.21216i
\(552\) 0 0
\(553\) −36.9752 3.49388i −1.57235 0.148575i
\(554\) 0 0
\(555\) −9.50134 + 5.90032i −0.403310 + 0.250455i
\(556\) 0 0
\(557\) 35.9630 + 9.63625i 1.52380 + 0.408301i 0.920991 0.389585i \(-0.127382\pi\)
0.602809 + 0.797886i \(0.294048\pi\)
\(558\) 0 0
\(559\) 7.72566i 0.326761i
\(560\) 0 0
\(561\) 4.54581 + 20.7644i 0.191924 + 0.876675i
\(562\) 0 0
\(563\) −5.68038 + 21.1995i −0.239399 + 0.893450i 0.736717 + 0.676201i \(0.236375\pi\)
−0.976116 + 0.217249i \(0.930292\pi\)
\(564\) 0 0
\(565\) −6.83360 + 14.3216i −0.287492 + 0.602514i
\(566\) 0 0
\(567\) −23.1048 + 5.75906i −0.970312 + 0.241858i
\(568\) 0 0
\(569\) 13.0396 22.5852i 0.546648 0.946821i −0.451854 0.892092i \(-0.649237\pi\)
0.998501 0.0547292i \(-0.0174296\pi\)
\(570\) 0 0
\(571\) 0.341798 + 0.592011i 0.0143038 + 0.0247749i 0.873089 0.487561i \(-0.162113\pi\)
−0.858785 + 0.512336i \(0.828780\pi\)
\(572\) 0 0
\(573\) 10.3124 16.0932i 0.430808 0.672305i
\(574\) 0 0
\(575\) 6.98178 9.60373i 0.291161 0.400503i
\(576\) 0 0
\(577\) −7.48627 + 27.9391i −0.311657 + 1.16312i 0.615404 + 0.788212i \(0.288993\pi\)
−0.927061 + 0.374910i \(0.877674\pi\)
\(578\) 0 0
\(579\) −7.39663 + 23.2447i −0.307394 + 0.966017i
\(580\) 0 0
\(581\) 3.49063 + 1.29801i 0.144816 + 0.0538507i
\(582\) 0 0
\(583\) −8.57836 32.0149i −0.355279 1.32592i
\(584\) 0 0
\(585\) 4.17204 + 2.48814i 0.172493 + 0.102872i
\(586\) 0 0
\(587\) 32.5547 32.5547i 1.34367 1.34367i 0.451304 0.892370i \(-0.350959\pi\)
0.892370 0.451304i \(-0.149041\pi\)
\(588\) 0 0
\(589\) 12.3097i 0.507211i
\(590\) 0 0
\(591\) −2.69191 2.95334i −0.110730 0.121484i
\(592\) 0 0
\(593\) 35.3500 9.47201i 1.45165 0.388969i 0.555053 0.831815i \(-0.312698\pi\)
0.896598 + 0.442846i \(0.146031\pi\)
\(594\) 0 0
\(595\) −11.5810 20.8124i −0.474776 0.853226i
\(596\) 0 0
\(597\) −8.86971 17.1491i −0.363013 0.701865i
\(598\) 0 0
\(599\) −16.8528 29.1898i −0.688585 1.19266i −0.972296 0.233754i \(-0.924899\pi\)
0.283711 0.958910i \(-0.408434\pi\)
\(600\) 0 0
\(601\) 7.76968 0.316932 0.158466 0.987364i \(-0.449345\pi\)
0.158466 + 0.987364i \(0.449345\pi\)
\(602\) 0 0
\(603\) −4.16643 + 11.2616i −0.169670 + 0.458606i
\(604\) 0 0
\(605\) 3.75454 + 0.696427i 0.152644 + 0.0283138i
\(606\) 0 0
\(607\) 0.00880159 0.00235838i 0.000357246 9.57237e-5i −0.258640 0.965974i \(-0.583274\pi\)
0.258998 + 0.965878i \(0.416608\pi\)
\(608\) 0 0
\(609\) 14.5871 + 36.1943i 0.591100 + 1.46667i
\(610\) 0 0
\(611\) −4.72689 2.72907i −0.191229 0.110406i
\(612\) 0 0
\(613\) 36.9113 + 9.89035i 1.49083 + 0.399467i 0.910018 0.414568i \(-0.136067\pi\)
0.580815 + 0.814036i \(0.302734\pi\)
\(614\) 0 0
\(615\) 10.0012 + 33.0416i 0.403286 + 1.33237i
\(616\) 0 0
\(617\) 13.3739 13.3739i 0.538413 0.538413i −0.384650 0.923063i \(-0.625678\pi\)
0.923063 + 0.384650i \(0.125678\pi\)
\(618\) 0 0
\(619\) −35.2461 + 20.3493i −1.41666 + 0.817908i −0.996004 0.0893126i \(-0.971533\pi\)
−0.420655 + 0.907221i \(0.638200\pi\)
\(620\) 0 0
\(621\) −1.51352 12.2460i −0.0607356 0.491415i
\(622\) 0 0
\(623\) 0.351563 3.72054i 0.0140851 0.149060i
\(624\) 0 0
\(625\) 5.21201 24.4507i 0.208480 0.978027i
\(626\) 0 0
\(627\) −1.63222 + 35.2459i −0.0651848 + 1.40759i
\(628\) 0 0
\(629\) −11.6259 −0.463557
\(630\) 0 0
\(631\) 12.9314 0.514789 0.257395 0.966306i \(-0.417136\pi\)
0.257395 + 0.966306i \(0.417136\pi\)
\(632\) 0 0
\(633\) 0.218076 4.70910i 0.00866776 0.187170i
\(634\) 0 0
\(635\) −7.82346 + 16.3961i −0.310465 + 0.650659i
\(636\) 0 0
\(637\) −0.371596 5.05531i −0.0147232 0.200299i
\(638\) 0 0
\(639\) −0.947949 + 10.2130i −0.0375003 + 0.404018i
\(640\) 0 0
\(641\) 1.71838 0.992107i 0.0678719 0.0391859i −0.465680 0.884953i \(-0.654190\pi\)
0.533552 + 0.845767i \(0.320857\pi\)
\(642\) 0 0
\(643\) −3.02169 + 3.02169i −0.119164 + 0.119164i −0.764174 0.645010i \(-0.776853\pi\)
0.645010 + 0.764174i \(0.276853\pi\)
\(644\) 0 0
\(645\) 19.5003 36.4293i 0.767823 1.43440i
\(646\) 0 0
\(647\) −20.3963 5.46518i −0.801862 0.214858i −0.165461 0.986216i \(-0.552911\pi\)
−0.636401 + 0.771358i \(0.719578\pi\)
\(648\) 0 0
\(649\) 5.63704 + 3.25455i 0.221273 + 0.127752i
\(650\) 0 0
\(651\) −5.20265 + 6.64725i −0.203908 + 0.260526i
\(652\) 0 0
\(653\) −1.47134 + 0.394244i −0.0575779 + 0.0154279i −0.287493 0.957783i \(-0.592822\pi\)
0.229915 + 0.973211i \(0.426155\pi\)
\(654\) 0 0
\(655\) −0.529213 + 2.85306i −0.0206781 + 0.111478i
\(656\) 0 0
\(657\) −32.0986 11.8755i −1.25229 0.463307i
\(658\) 0 0
\(659\) 28.8543 1.12400 0.562002 0.827136i \(-0.310031\pi\)
0.562002 + 0.827136i \(0.310031\pi\)
\(660\) 0 0
\(661\) −5.34839 9.26369i −0.208028 0.360316i 0.743065 0.669219i \(-0.233371\pi\)
−0.951093 + 0.308903i \(0.900038\pi\)
\(662\) 0 0
\(663\) 2.31973 + 4.48506i 0.0900908 + 0.174185i
\(664\) 0 0
\(665\) −9.62720 38.3454i −0.373327 1.48697i
\(666\) 0 0
\(667\) −19.5326 + 5.23376i −0.756307 + 0.202652i
\(668\) 0 0
\(669\) −21.6088 23.7073i −0.835444 0.916579i
\(670\) 0 0
\(671\) 43.4275i 1.67650i
\(672\) 0 0
\(673\) 25.9024 25.9024i 0.998464 0.998464i −0.00153521 0.999999i \(-0.500489\pi\)
0.999999 + 0.00153521i \(0.000488674\pi\)
\(674\) 0 0
\(675\) −13.3923 22.2631i −0.515472 0.856907i
\(676\) 0 0
\(677\) 3.01599 + 11.2558i 0.115914 + 0.432597i 0.999354 0.0359478i \(-0.0114450\pi\)
−0.883440 + 0.468545i \(0.844778\pi\)
\(678\) 0 0
\(679\) 1.95073 + 0.725393i 0.0748622 + 0.0278380i
\(680\) 0 0
\(681\) −7.19689 + 22.6170i −0.275785 + 0.866685i
\(682\) 0 0
\(683\) 4.94076 18.4392i 0.189053 0.705556i −0.804673 0.593718i \(-0.797660\pi\)
0.993726 0.111838i \(-0.0356737\pi\)
\(684\) 0 0
\(685\) 0.0920952 + 1.17236i 0.00351878 + 0.0447936i
\(686\) 0 0
\(687\) −11.1495 + 17.3995i −0.425378 + 0.663831i
\(688\) 0 0
\(689\) −3.93674 6.81863i −0.149978 0.259769i
\(690\) 0 0
\(691\) −21.0814 + 36.5141i −0.801976 + 1.38906i 0.116338 + 0.993210i \(0.462885\pi\)
−0.918314 + 0.395854i \(0.870449\pi\)
\(692\) 0 0
\(693\) 15.7780 18.3431i 0.599357 0.696795i
\(694\) 0 0
\(695\) 37.2562 13.1870i 1.41321 0.500211i
\(696\) 0 0
\(697\) −9.28775 + 34.6624i −0.351799 + 1.31293i
\(698\) 0 0
\(699\) −3.74360 17.1001i −0.141596 0.646784i
\(700\) 0 0
\(701\) 0.0274102i 0.00103527i −1.00000 0.000517635i \(-0.999835\pi\)
1.00000 0.000517635i \(-0.000164768\pi\)
\(702\) 0 0
\(703\) −18.6406 4.99474i −0.703045 0.188380i
\(704\) 0 0
\(705\) 15.4005 + 24.7997i 0.580018 + 0.934009i
\(706\) 0 0
\(707\) −5.14039 + 7.23376i −0.193324 + 0.272054i
\(708\) 0 0
\(709\) −24.5893 14.1966i −0.923471 0.533166i −0.0387299 0.999250i \(-0.512331\pi\)
−0.884741 + 0.466084i \(0.845665\pi\)
\(710\) 0 0
\(711\) −34.3686 24.3369i −1.28892 0.912705i
\(712\) 0 0
\(713\) −3.09302 3.09302i −0.115834 0.115834i
\(714\) 0 0
\(715\) −4.92074 + 0.386550i −0.184025 + 0.0144562i
\(716\) 0 0
\(717\) 1.71630 37.0614i 0.0640964 1.38408i
\(718\) 0 0
\(719\) 6.78490 11.7518i 0.253034 0.438268i −0.711326 0.702863i \(-0.751905\pi\)
0.964360 + 0.264595i \(0.0852382\pi\)
\(720\) 0 0
\(721\) −2.91948 3.52879i −0.108727 0.131419i
\(722\) 0 0
\(723\) −15.4711 + 8.00181i −0.575375 + 0.297591i
\(724\) 0 0
\(725\) −33.0970 + 26.7855i −1.22919 + 0.994787i
\(726\) 0 0
\(727\) −11.5270 11.5270i −0.427515 0.427515i 0.460266 0.887781i \(-0.347754\pi\)
−0.887781 + 0.460266i \(0.847754\pi\)
\(728\) 0 0
\(729\) −25.9659 7.40084i −0.961700 0.274105i
\(730\) 0 0
\(731\) 37.1972 21.4758i 1.37579 0.794312i
\(732\) 0 0
\(733\) 3.45107 + 12.8796i 0.127468 + 0.475718i 0.999916 0.0129901i \(-0.00413499\pi\)
−0.872447 + 0.488708i \(0.837468\pi\)
\(734\) 0 0
\(735\) −11.0079 + 24.7755i −0.406031 + 0.913859i
\(736\) 0 0
\(737\) −3.15785 11.7853i −0.116321 0.434116i
\(738\) 0 0
\(739\) 25.3089 14.6121i 0.931003 0.537515i 0.0438740 0.999037i \(-0.486030\pi\)
0.887129 + 0.461522i \(0.152697\pi\)
\(740\) 0 0
\(741\) 1.79250 + 8.18780i 0.0658491 + 0.300786i
\(742\) 0 0
\(743\) −12.8645 12.8645i −0.471952 0.471952i 0.430594 0.902546i \(-0.358304\pi\)
−0.902546 + 0.430594i \(0.858304\pi\)
\(744\) 0 0
\(745\) 45.7216 + 8.48088i 1.67511 + 0.310715i
\(746\) 0 0
\(747\) 2.69722 + 3.24916i 0.0986863 + 0.118880i
\(748\) 0 0
\(749\) 13.4297 + 16.2326i 0.490712 + 0.593127i
\(750\) 0 0
\(751\) −10.6527 + 18.4510i −0.388721 + 0.673285i −0.992278 0.124035i \(-0.960416\pi\)
0.603556 + 0.797320i \(0.293750\pi\)
\(752\) 0 0
\(753\) 8.88000 + 0.411229i 0.323605 + 0.0149860i
\(754\) 0 0
\(755\) −15.8218 + 18.5195i −0.575813 + 0.673992i
\(756\) 0 0
\(757\) 18.4939 + 18.4939i 0.672174 + 0.672174i 0.958217 0.286043i \(-0.0923401\pi\)
−0.286043 + 0.958217i \(0.592340\pi\)
\(758\) 0 0
\(759\) 8.44604 + 9.26629i 0.306572 + 0.336345i
\(760\) 0 0
\(761\) −29.7628 17.1835i −1.07890 0.622903i −0.148300 0.988942i \(-0.547380\pi\)
−0.930599 + 0.366039i \(0.880714\pi\)
\(762\) 0 0
\(763\) −3.75311 + 5.28153i −0.135872 + 0.191204i
\(764\) 0 0
\(765\) 0.382363 27.0039i 0.0138244 0.976328i
\(766\) 0 0
\(767\) 1.49356 + 0.400199i 0.0539294 + 0.0144503i
\(768\) 0 0
\(769\) 8.44100i 0.304390i −0.988350 0.152195i \(-0.951366\pi\)
0.988350 0.152195i \(-0.0486342\pi\)
\(770\) 0 0
\(771\) −18.9758 + 4.15425i −0.683398 + 0.149612i
\(772\) 0 0
\(773\) 6.28472 23.4549i 0.226046 0.843614i −0.755937 0.654644i \(-0.772818\pi\)
0.981983 0.188970i \(-0.0605149\pi\)
\(774\) 0 0
\(775\) −8.59739 3.30310i −0.308827 0.118651i
\(776\) 0 0
\(777\) 7.95499 + 10.5756i 0.285384 + 0.379397i
\(778\) 0 0
\(779\) −29.7833 + 51.5862i −1.06710 + 1.84827i
\(780\) 0 0
\(781\) −5.21104 9.02578i −0.186466 0.322968i
\(782\) 0 0
\(783\) −6.12324 + 43.8224i −0.218827 + 1.56609i
\(784\) 0 0
\(785\) 22.2652 1.74905i 0.794678 0.0624263i
\(786\) 0 0
\(787\) 0.522746 1.95091i 0.0186339 0.0695426i −0.955983 0.293423i \(-0.905206\pi\)
0.974617 + 0.223880i \(0.0718724\pi\)
\(788\) 0 0
\(789\) 23.5246 + 7.48571i 0.837499 + 0.266498i
\(790\) 0 0
\(791\) 17.5984 + 6.54408i 0.625727 + 0.232681i
\(792\) 0 0
\(793\) −2.67005 9.96477i −0.0948163 0.353859i
\(794\) 0 0
\(795\) 1.35227 + 42.0890i 0.0479602 + 1.49274i
\(796\) 0 0
\(797\) 10.5999 10.5999i 0.375468 0.375468i −0.493996 0.869464i \(-0.664464\pi\)
0.869464 + 0.493996i \(0.164464\pi\)
\(798\) 0 0
\(799\) 30.3451i 1.07353i
\(800\) 0 0
\(801\) 2.44884 3.45825i 0.0865255 0.122191i
\(802\) 0 0
\(803\) 33.5913 9.00076i 1.18541 0.317630i
\(804\) 0 0
\(805\) −12.0540 7.21595i −0.424846 0.254329i
\(806\) 0 0
\(807\) −22.0844 + 11.4223i −0.777408 + 0.402085i
\(808\) 0 0
\(809\) 2.50317 + 4.33563i 0.0880069 + 0.152432i 0.906669 0.421844i \(-0.138617\pi\)
−0.818662 + 0.574276i \(0.805284\pi\)
\(810\) 0 0
\(811\) 6.45712 0.226740 0.113370 0.993553i \(-0.463835\pi\)
0.113370 + 0.993553i \(0.463835\pi\)
\(812\) 0 0
\(813\) −2.45259 1.57160i −0.0860160 0.0551184i
\(814\) 0 0
\(815\) −6.28840 + 4.32055i −0.220273 + 0.151342i
\(816\) 0 0
\(817\) 68.8672 18.4529i 2.40936 0.645585i
\(818\) 0 0
\(819\) 2.49260 5.17903i 0.0870985 0.180970i
\(820\) 0 0
\(821\) 7.96631 + 4.59935i 0.278026 + 0.160518i 0.632530 0.774536i \(-0.282017\pi\)
−0.354503 + 0.935055i \(0.615350\pi\)
\(822\) 0 0
\(823\) 12.6038 + 3.37717i 0.439340 + 0.117721i 0.471707 0.881755i \(-0.343638\pi\)
−0.0323671 + 0.999476i \(0.510305\pi\)
\(824\) 0 0
\(825\) 24.1787 + 10.5977i 0.841795 + 0.368963i
\(826\) 0 0
\(827\) −8.73603 + 8.73603i −0.303782 + 0.303782i −0.842491 0.538710i \(-0.818912\pi\)
0.538710 + 0.842491i \(0.318912\pi\)
\(828\) 0 0
\(829\) −23.6216 + 13.6379i −0.820411 + 0.473665i −0.850558 0.525881i \(-0.823736\pi\)
0.0301471 + 0.999545i \(0.490402\pi\)
\(830\) 0 0
\(831\) 12.9619 + 4.12457i 0.449643 + 0.143080i
\(832\) 0 0
\(833\) −23.3071 + 15.8419i −0.807544 + 0.548889i
\(834\) 0 0
\(835\) 30.4276 + 14.5186i 1.05299 + 0.502438i
\(836\) 0 0
\(837\) −8.87158 + 3.59258i −0.306647 + 0.124178i
\(838\) 0 0
\(839\) 22.7675 0.786022 0.393011 0.919534i \(-0.371433\pi\)
0.393011 + 0.919534i \(0.371433\pi\)
\(840\) 0 0
\(841\) 43.5147 1.50051
\(842\) 0 0
\(843\) 13.8641 + 0.642042i 0.477506 + 0.0221131i
\(844\) 0 0
\(845\) 26.2976 9.30817i 0.904665 0.320211i
\(846\) 0 0
\(847\) 0.425043 4.49817i 0.0146046 0.154559i
\(848\) 0 0
\(849\) 0.738502 2.32082i 0.0253453 0.0796504i
\(850\) 0 0
\(851\) −5.93881 + 3.42877i −0.203580 + 0.117537i
\(852\) 0 0
\(853\) 7.75252 7.75252i 0.265441 0.265441i −0.561819 0.827260i \(-0.689898\pi\)
0.827260 + 0.561819i \(0.189898\pi\)
\(854\) 0 0
\(855\) 12.2145 43.1328i 0.417727 1.47511i
\(856\) 0 0
\(857\) −54.0758 14.4896i −1.84719 0.494954i −0.847819 0.530285i \(-0.822085\pi\)
−0.999376 + 0.0353310i \(0.988751\pi\)
\(858\) 0 0
\(859\) 14.4633 + 8.35040i 0.493482 + 0.284912i 0.726018 0.687676i \(-0.241369\pi\)
−0.232536 + 0.972588i \(0.574702\pi\)
\(860\) 0 0
\(861\) 37.8859 15.2689i 1.29115 0.520362i
\(862\) 0 0
\(863\) −35.0073 + 9.38019i −1.19166 + 0.319305i −0.799543 0.600609i \(-0.794925\pi\)
−0.392121 + 0.919914i \(0.628258\pi\)
\(864\) 0 0
\(865\) −21.9865 32.0006i −0.747565 1.08805i
\(866\) 0 0
\(867\) −0.740183 + 1.15511i −0.0251379 + 0.0392294i
\(868\) 0 0
\(869\) 42.7911 1.45159
\(870\) 0 0
\(871\) −1.44919 2.51006i −0.0491038 0.0850502i
\(872\) 0 0
\(873\) 1.50734 + 1.81578i 0.0510156 + 0.0614550i
\(874\) 0 0
\(875\) −29.3647 3.56547i −0.992709 0.120535i
\(876\) 0 0
\(877\) 36.2427 9.71119i 1.22383 0.327924i 0.411654 0.911340i \(-0.364951\pi\)
0.812173 + 0.583416i \(0.198284\pi\)
\(878\) 0 0
\(879\) 6.28315 5.72697i 0.211925 0.193166i
\(880\) 0 0
\(881\) 38.3229i 1.29113i 0.763704 + 0.645566i \(0.223379\pi\)
−0.763704 + 0.645566i \(0.776621\pi\)
\(882\) 0 0
\(883\) 14.7904 14.7904i 0.497735 0.497735i −0.412997 0.910732i \(-0.635518\pi\)
0.910732 + 0.412997i \(0.135518\pi\)
\(884\) 0 0
\(885\) −6.03254 5.65697i −0.202782 0.190157i
\(886\) 0 0
\(887\) 0.221528 + 0.826754i 0.00743818 + 0.0277597i 0.969545 0.244912i \(-0.0787592\pi\)
−0.962107 + 0.272672i \(0.912093\pi\)
\(888\) 0 0
\(889\) 20.1475 + 7.49200i 0.675727 + 0.251274i
\(890\) 0 0
\(891\) 25.8782 9.11019i 0.866951 0.305203i
\(892\) 0 0
\(893\) −13.0369 + 48.6543i −0.436263 + 1.62815i
\(894\) 0 0
\(895\) −20.7437 + 24.2806i −0.693385 + 0.811611i
\(896\) 0 0
\(897\) 2.50773 + 1.60693i 0.0837305 + 0.0536539i
\(898\) 0 0
\(899\) 7.84290 + 13.5843i 0.261575 + 0.453062i
\(900\) 0 0
\(901\) −21.8867 + 37.9089i −0.729152 + 1.26293i
\(902\) 0 0
\(903\) −44.9876 19.1419i −1.49709 0.637003i
\(904\) 0 0
\(905\) 4.40211 + 12.4369i 0.146331 + 0.413417i
\(906\) 0 0
\(907\) 12.2353 45.6627i 0.406266 1.51621i −0.395443 0.918491i \(-0.629409\pi\)
0.801709 0.597715i \(-0.203925\pi\)
\(908\) 0 0
\(909\) −9.14195 + 4.20427i −0.303219 + 0.139447i
\(910\) 0 0
\(911\) 32.5777i 1.07935i −0.841875 0.539673i \(-0.818548\pi\)
0.841875 0.539673i \(-0.181452\pi\)
\(912\) 0 0
\(913\) −4.14461 1.11055i −0.137167 0.0367537i
\(914\) 0 0
\(915\) −12.5617 + 53.7269i −0.415279 + 1.77616i
\(916\) 0 0
\(917\) 3.41815 + 0.322989i 0.112877 + 0.0106660i
\(918\) 0 0
\(919\) 17.0821 + 9.86237i 0.563487 + 0.325329i 0.754544 0.656250i \(-0.227858\pi\)
−0.191057 + 0.981579i \(0.561191\pi\)
\(920\) 0 0
\(921\) 35.7181 32.5563i 1.17695 1.07277i
\(922\) 0 0
\(923\) −1.75064 1.75064i −0.0576232 0.0576232i
\(924\) 0 0
\(925\) −8.49037 + 11.6789i −0.279162 + 0.383998i
\(926\) 0 0
\(927\) −0.874731 5.11895i −0.0287299 0.168128i
\(928\) 0 0
\(929\) −22.4160 + 38.8256i −0.735444 + 1.27383i 0.219085 + 0.975706i \(0.429693\pi\)
−0.954528 + 0.298120i \(0.903641\pi\)
\(930\) 0 0
\(931\) −44.1759 + 15.3871i −1.44781 + 0.504293i
\(932\) 0 0
\(933\) 21.8402 + 42.2268i 0.715016 + 1.38244i
\(934\) 0 0
\(935\) 15.5398 + 22.6176i 0.508206 + 0.739675i
\(936\) 0 0
\(937\) −20.1180 20.1180i −0.657228 0.657228i 0.297495 0.954723i \(-0.403849\pi\)
−0.954723 + 0.297495i \(0.903849\pi\)
\(938\) 0 0
\(939\) 39.9060 8.73634i 1.30228 0.285100i
\(940\) 0 0
\(941\) 30.7949 17.7794i 1.00389 0.579593i 0.0944896 0.995526i \(-0.469878\pi\)
0.909395 + 0.415933i \(0.136545\pi\)
\(942\) 0 0
\(943\) 5.47836 + 20.4455i 0.178400 + 0.665798i
\(944\) 0 0
\(945\) −24.8259 + 18.1294i −0.807585 + 0.589751i
\(946\) 0 0
\(947\) −2.02075 7.54155i −0.0656657 0.245068i 0.925289 0.379262i \(-0.123822\pi\)
−0.990955 + 0.134194i \(0.957155\pi\)
\(948\) 0 0
\(949\) 7.15438 4.13058i 0.232241 0.134084i
\(950\) 0 0
\(951\) 54.3253 11.8931i 1.76162 0.385659i
\(952\) 0 0
\(953\) −9.66000 9.66000i −0.312918 0.312918i 0.533121 0.846039i \(-0.321019\pi\)
−0.846039 + 0.533121i \(0.821019\pi\)
\(954\) 0 0
\(955\) 4.50034 24.2620i 0.145628 0.785100i
\(956\) 0 0
\(957\) −20.6551 39.9355i −0.667686 1.29093i
\(958\) 0 0
\(959\) 1.37193 0.232095i 0.0443021 0.00749474i
\(960\) 0 0
\(961\) 13.8035 23.9083i 0.445274 0.771237i
\(962\) 0 0
\(963\) 4.02381 + 23.5474i 0.129665 + 0.758804i
\(964\) 0 0
\(965\) 2.46623 + 31.3947i 0.0793906 + 1.01063i
\(966\) 0 0
\(967\) −11.2478 11.2478i −0.361704 0.361704i 0.502736 0.864440i \(-0.332327\pi\)
−0.864440 + 0.502736i \(0.832327\pi\)
\(968\) 0 0
\(969\) 34.4395 31.3909i 1.10636 1.00842i
\(970\) 0 0
\(971\) 23.4447 + 13.5358i 0.752375 + 0.434384i 0.826551 0.562861i \(-0.190300\pi\)
−0.0741766 + 0.997245i \(0.523633\pi\)
\(972\) 0 0
\(973\) −19.4676 42.5170i −0.624103 1.36303i
\(974\) 0 0
\(975\) 6.19957 + 0.945136i 0.198545 + 0.0302686i
\(976\) 0 0
\(977\) −30.8489 8.26594i −0.986944 0.264451i −0.270978 0.962586i \(-0.587347\pi\)
−0.715966 + 0.698135i \(0.754014\pi\)
\(978\) 0 0
\(979\) 4.30575i 0.137612i
\(980\) 0 0
\(981\) −6.67475 + 3.06963i −0.213108 + 0.0980058i
\(982\) 0 0
\(983\) 7.40257 27.6268i 0.236105 0.881157i −0.741542 0.670906i \(-0.765905\pi\)
0.977648 0.210251i \(-0.0674280\pi\)
\(984\) 0 0
\(985\) −4.65603 2.22164i −0.148353 0.0707874i
\(986\) 0 0
\(987\) 27.6036 20.7635i 0.878632 0.660909i
\(988\) 0 0
\(989\) 12.6675 21.9407i 0.402802 0.697674i
\(990\) 0 0
\(991\) 9.37664 + 16.2408i 0.297859 + 0.515907i 0.975646 0.219352i \(-0.0703942\pi\)
−0.677787 + 0.735258i \(0.737061\pi\)
\(992\) 0 0
\(993\) −40.5747 26.0000i −1.28760 0.825085i
\(994\) 0 0
\(995\) −18.9510 16.1905i −0.600787 0.513272i
\(996\) 0 0
\(997\) −7.26414 + 27.1101i −0.230058 + 0.858587i 0.750257 + 0.661146i \(0.229930\pi\)
−0.980315 + 0.197441i \(0.936737\pi\)
\(998\) 0 0
\(999\) 1.84056 + 14.8920i 0.0582327 + 0.471164i
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 420.2.bv.c.317.7 yes 48
3.2 odd 2 inner 420.2.bv.c.317.5 yes 48
5.3 odd 4 inner 420.2.bv.c.233.12 yes 48
7.4 even 3 inner 420.2.bv.c.137.3 yes 48
15.8 even 4 inner 420.2.bv.c.233.3 yes 48
21.11 odd 6 inner 420.2.bv.c.137.12 yes 48
35.18 odd 12 inner 420.2.bv.c.53.5 48
105.53 even 12 inner 420.2.bv.c.53.7 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bv.c.53.5 48 35.18 odd 12 inner
420.2.bv.c.53.7 yes 48 105.53 even 12 inner
420.2.bv.c.137.3 yes 48 7.4 even 3 inner
420.2.bv.c.137.12 yes 48 21.11 odd 6 inner
420.2.bv.c.233.3 yes 48 15.8 even 4 inner
420.2.bv.c.233.12 yes 48 5.3 odd 4 inner
420.2.bv.c.317.5 yes 48 3.2 odd 2 inner
420.2.bv.c.317.7 yes 48 1.1 even 1 trivial