Properties

Label 1456.2.cc.c.673.1
Level $1456$
Weight $2$
Character 1456.673
Analytic conductor $11.626$
Analytic rank $0$
Dimension $12$
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] = [1456,2,Mod(225,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.cc (of order \(6\), degree \(2\), not 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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.58891012706304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 673.1
Root \(0.759479 - 1.19298i\) of defining polynomial
Character \(\chi\) \(=\) 1456.673
Dual form 1456.2.cc.c.225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.41289 - 2.44719i) q^{3} -0.518957i q^{5} +(0.866025 + 0.500000i) q^{7} +(-2.49250 + 4.31714i) q^{9} +O(q^{10})\) \(q+(-1.41289 - 2.44719i) q^{3} -0.518957i q^{5} +(0.866025 + 0.500000i) q^{7} +(-2.49250 + 4.31714i) q^{9} +(-1.40656 + 0.812080i) q^{11} +(1.42641 + 3.31140i) q^{13} +(-1.26999 + 0.733228i) q^{15} +(0.974127 - 1.68724i) q^{17} +(-2.15740 - 1.24558i) q^{19} -2.82577i q^{21} +(4.57029 + 7.91598i) q^{23} +4.73068 q^{25} +5.60916 q^{27} +(2.61498 + 4.52928i) q^{29} +5.79391i q^{31} +(3.97463 + 2.29475i) q^{33} +(0.259479 - 0.449430i) q^{35} +(-8.85879 + 5.11463i) q^{37} +(6.08826 - 8.16934i) q^{39} +(3.64513 - 2.10452i) q^{41} +(0.498655 - 0.863697i) q^{43} +(2.24041 + 1.29350i) q^{45} +4.51725i q^{47} +(0.500000 + 0.866025i) q^{49} -5.50532 q^{51} -8.89651 q^{53} +(0.421434 + 0.729946i) q^{55} +7.03944i q^{57} +(5.37392 + 3.10263i) q^{59} +(6.73536 - 11.6660i) q^{61} +(-4.31714 + 2.49250i) q^{63} +(1.71847 - 0.740247i) q^{65} +(7.25094 - 4.18633i) q^{67} +(12.9146 - 22.3688i) q^{69} +(4.50168 + 2.59905i) q^{71} -11.8395i q^{73} +(-6.68392 - 11.5769i) q^{75} -1.62416 q^{77} -0.982310 q^{79} +(-0.447609 - 0.775281i) q^{81} -8.91851i q^{83} +(-0.875603 - 0.505530i) q^{85} +(7.38934 - 12.7987i) q^{87} +(-10.4087 + 6.00949i) q^{89} +(-0.420388 + 3.58096i) q^{91} +(14.1788 - 8.18614i) q^{93} +(-0.646401 + 1.11960i) q^{95} +(3.82981 + 2.21114i) q^{97} -8.09643i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{9} - 6 q^{11} + 4 q^{13} - 6 q^{15} - 4 q^{17} + 12 q^{23} - 20 q^{25} - 12 q^{27} + 8 q^{29} - 30 q^{33} - 6 q^{35} - 42 q^{37} + 4 q^{39} + 30 q^{41} - 2 q^{43} + 6 q^{49} - 52 q^{51} - 44 q^{53} + 6 q^{55} - 18 q^{59} + 14 q^{61} - 12 q^{63} + 60 q^{65} + 24 q^{67} + 4 q^{69} + 24 q^{71} - 46 q^{75} + 8 q^{77} + 56 q^{79} + 2 q^{81} - 48 q^{85} + 2 q^{87} - 12 q^{89} - 14 q^{91} - 18 q^{93} + 22 q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.41289 2.44719i −0.815731 1.41289i −0.908802 0.417228i \(-0.863002\pi\)
0.0930713 0.995659i \(-0.470332\pi\)
\(4\) 0 0
\(5\) 0.518957i 0.232085i −0.993244 0.116042i \(-0.962979\pi\)
0.993244 0.116042i \(-0.0370208\pi\)
\(6\) 0 0
\(7\) 0.866025 + 0.500000i 0.327327 + 0.188982i
\(8\) 0 0
\(9\) −2.49250 + 4.31714i −0.830833 + 1.43905i
\(10\) 0 0
\(11\) −1.40656 + 0.812080i −0.424095 + 0.244851i −0.696828 0.717239i \(-0.745406\pi\)
0.272733 + 0.962090i \(0.412072\pi\)
\(12\) 0 0
\(13\) 1.42641 + 3.31140i 0.395616 + 0.918416i
\(14\) 0 0
\(15\) −1.26999 + 0.733228i −0.327909 + 0.189319i
\(16\) 0 0
\(17\) 0.974127 1.68724i 0.236260 0.409215i −0.723378 0.690452i \(-0.757411\pi\)
0.959638 + 0.281237i \(0.0907448\pi\)
\(18\) 0 0
\(19\) −2.15740 1.24558i −0.494942 0.285755i 0.231680 0.972792i \(-0.425578\pi\)
−0.726622 + 0.687037i \(0.758911\pi\)
\(20\) 0 0
\(21\) 2.82577i 0.616634i
\(22\) 0 0
\(23\) 4.57029 + 7.91598i 0.952971 + 1.65059i 0.738943 + 0.673767i \(0.235325\pi\)
0.214028 + 0.976828i \(0.431342\pi\)
\(24\) 0 0
\(25\) 4.73068 0.946137
\(26\) 0 0
\(27\) 5.60916 1.07948
\(28\) 0 0
\(29\) 2.61498 + 4.52928i 0.485589 + 0.841065i 0.999863 0.0165608i \(-0.00527172\pi\)
−0.514274 + 0.857626i \(0.671938\pi\)
\(30\) 0 0
\(31\) 5.79391i 1.04062i 0.853978 + 0.520308i \(0.174183\pi\)
−0.853978 + 0.520308i \(0.825817\pi\)
\(32\) 0 0
\(33\) 3.97463 + 2.29475i 0.691894 + 0.399465i
\(34\) 0 0
\(35\) 0.259479 0.449430i 0.0438599 0.0759675i
\(36\) 0 0
\(37\) −8.85879 + 5.11463i −1.45638 + 0.840840i −0.998831 0.0483462i \(-0.984605\pi\)
−0.457546 + 0.889186i \(0.651272\pi\)
\(38\) 0 0
\(39\) 6.08826 8.16934i 0.974902 1.30814i
\(40\) 0 0
\(41\) 3.64513 2.10452i 0.569273 0.328670i −0.187586 0.982248i \(-0.560066\pi\)
0.756859 + 0.653578i \(0.226733\pi\)
\(42\) 0 0
\(43\) 0.498655 0.863697i 0.0760442 0.131712i −0.825496 0.564408i \(-0.809104\pi\)
0.901540 + 0.432696i \(0.142438\pi\)
\(44\) 0 0
\(45\) 2.24041 + 1.29350i 0.333980 + 0.192824i
\(46\) 0 0
\(47\) 4.51725i 0.658909i 0.944171 + 0.329455i \(0.106865\pi\)
−0.944171 + 0.329455i \(0.893135\pi\)
\(48\) 0 0
\(49\) 0.500000 + 0.866025i 0.0714286 + 0.123718i
\(50\) 0 0
\(51\) −5.50532 −0.770899
\(52\) 0 0
\(53\) −8.89651 −1.22203 −0.611015 0.791619i \(-0.709238\pi\)
−0.611015 + 0.791619i \(0.709238\pi\)
\(54\) 0 0
\(55\) 0.421434 + 0.729946i 0.0568262 + 0.0984259i
\(56\) 0 0
\(57\) 7.03944i 0.932397i
\(58\) 0 0
\(59\) 5.37392 + 3.10263i 0.699624 + 0.403928i 0.807207 0.590268i \(-0.200978\pi\)
−0.107583 + 0.994196i \(0.534311\pi\)
\(60\) 0 0
\(61\) 6.73536 11.6660i 0.862375 1.49368i −0.00725571 0.999974i \(-0.502310\pi\)
0.869630 0.493703i \(-0.164357\pi\)
\(62\) 0 0
\(63\) −4.31714 + 2.49250i −0.543908 + 0.314025i
\(64\) 0 0
\(65\) 1.71847 0.740247i 0.213150 0.0918164i
\(66\) 0 0
\(67\) 7.25094 4.18633i 0.885843 0.511442i 0.0132624 0.999912i \(-0.495778\pi\)
0.872580 + 0.488470i \(0.162445\pi\)
\(68\) 0 0
\(69\) 12.9146 22.3688i 1.55474 2.69288i
\(70\) 0 0
\(71\) 4.50168 + 2.59905i 0.534251 + 0.308450i 0.742746 0.669573i \(-0.233523\pi\)
−0.208495 + 0.978023i \(0.566856\pi\)
\(72\) 0 0
\(73\) 11.8395i 1.38571i −0.721076 0.692856i \(-0.756352\pi\)
0.721076 0.692856i \(-0.243648\pi\)
\(74\) 0 0
\(75\) −6.68392 11.5769i −0.771793 1.33678i
\(76\) 0 0
\(77\) −1.62416 −0.185090
\(78\) 0 0
\(79\) −0.982310 −0.110518 −0.0552592 0.998472i \(-0.517599\pi\)
−0.0552592 + 0.998472i \(0.517599\pi\)
\(80\) 0 0
\(81\) −0.447609 0.775281i −0.0497343 0.0861423i
\(82\) 0 0
\(83\) 8.91851i 0.978934i −0.872022 0.489467i \(-0.837191\pi\)
0.872022 0.489467i \(-0.162809\pi\)
\(84\) 0 0
\(85\) −0.875603 0.505530i −0.0949725 0.0548324i
\(86\) 0 0
\(87\) 7.38934 12.7987i 0.792220 1.37217i
\(88\) 0 0
\(89\) −10.4087 + 6.00949i −1.10332 + 0.637005i −0.937092 0.349082i \(-0.886494\pi\)
−0.166233 + 0.986087i \(0.553160\pi\)
\(90\) 0 0
\(91\) −0.420388 + 3.58096i −0.0440686 + 0.375387i
\(92\) 0 0
\(93\) 14.1788 8.18614i 1.47027 0.848863i
\(94\) 0 0
\(95\) −0.646401 + 1.11960i −0.0663194 + 0.114869i
\(96\) 0 0
\(97\) 3.82981 + 2.21114i 0.388858 + 0.224507i 0.681665 0.731664i \(-0.261256\pi\)
−0.292807 + 0.956172i \(0.594589\pi\)
\(98\) 0 0
\(99\) 8.09643i 0.813722i
\(100\) 0 0
\(101\) 9.15132 + 15.8506i 0.910591 + 1.57719i 0.813231 + 0.581940i \(0.197706\pi\)
0.0973594 + 0.995249i \(0.468960\pi\)
\(102\) 0 0
\(103\) 5.02046 0.494680 0.247340 0.968929i \(-0.420444\pi\)
0.247340 + 0.968929i \(0.420444\pi\)
\(104\) 0 0
\(105\) −1.46646 −0.143111
\(106\) 0 0
\(107\) 3.07228 + 5.32134i 0.297008 + 0.514434i 0.975450 0.220221i \(-0.0706777\pi\)
−0.678442 + 0.734654i \(0.737344\pi\)
\(108\) 0 0
\(109\) 11.8962i 1.13945i −0.821834 0.569727i \(-0.807049\pi\)
0.821834 0.569727i \(-0.192951\pi\)
\(110\) 0 0
\(111\) 25.0330 + 14.4528i 2.37602 + 1.37180i
\(112\) 0 0
\(113\) −1.77806 + 3.07969i −0.167266 + 0.289713i −0.937458 0.348099i \(-0.886827\pi\)
0.770192 + 0.637812i \(0.220161\pi\)
\(114\) 0 0
\(115\) 4.10805 2.37178i 0.383078 0.221170i
\(116\) 0 0
\(117\) −17.8511 2.09563i −1.65033 0.193741i
\(118\) 0 0
\(119\) 1.68724 0.974127i 0.154669 0.0892980i
\(120\) 0 0
\(121\) −4.18105 + 7.24180i −0.380096 + 0.658345i
\(122\) 0 0
\(123\) −10.3003 5.94689i −0.928748 0.536213i
\(124\) 0 0
\(125\) 5.04981i 0.451668i
\(126\) 0 0
\(127\) 0.711749 + 1.23279i 0.0631575 + 0.109392i 0.895875 0.444306i \(-0.146550\pi\)
−0.832718 + 0.553698i \(0.813216\pi\)
\(128\) 0 0
\(129\) −2.81818 −0.248127
\(130\) 0 0
\(131\) −8.67374 −0.757828 −0.378914 0.925432i \(-0.623702\pi\)
−0.378914 + 0.925432i \(0.623702\pi\)
\(132\) 0 0
\(133\) −1.24558 2.15740i −0.108005 0.187071i
\(134\) 0 0
\(135\) 2.91091i 0.250531i
\(136\) 0 0
\(137\) 7.37667 + 4.25892i 0.630231 + 0.363864i 0.780842 0.624729i \(-0.214791\pi\)
−0.150611 + 0.988593i \(0.548124\pi\)
\(138\) 0 0
\(139\) −2.51922 + 4.36342i −0.213677 + 0.370100i −0.952863 0.303402i \(-0.901877\pi\)
0.739185 + 0.673502i \(0.235211\pi\)
\(140\) 0 0
\(141\) 11.0546 6.38237i 0.930964 0.537493i
\(142\) 0 0
\(143\) −4.69546 3.49933i −0.392654 0.292628i
\(144\) 0 0
\(145\) 2.35050 1.35706i 0.195198 0.112698i
\(146\) 0 0
\(147\) 1.41289 2.44719i 0.116533 0.201841i
\(148\) 0 0
\(149\) 2.91409 + 1.68245i 0.238732 + 0.137832i 0.614594 0.788844i \(-0.289320\pi\)
−0.375862 + 0.926676i \(0.622653\pi\)
\(150\) 0 0
\(151\) 12.6566i 1.02998i −0.857196 0.514991i \(-0.827795\pi\)
0.857196 0.514991i \(-0.172205\pi\)
\(152\) 0 0
\(153\) 4.85602 + 8.41087i 0.392586 + 0.679979i
\(154\) 0 0
\(155\) 3.00679 0.241511
\(156\) 0 0
\(157\) 10.3691 0.827547 0.413773 0.910380i \(-0.364211\pi\)
0.413773 + 0.910380i \(0.364211\pi\)
\(158\) 0 0
\(159\) 12.5698 + 21.7715i 0.996847 + 1.72659i
\(160\) 0 0
\(161\) 9.14058i 0.720379i
\(162\) 0 0
\(163\) −13.6428 7.87669i −1.06859 0.616950i −0.140794 0.990039i \(-0.544965\pi\)
−0.927796 + 0.373089i \(0.878299\pi\)
\(164\) 0 0
\(165\) 1.19088 2.06266i 0.0927098 0.160578i
\(166\) 0 0
\(167\) 14.2016 8.19930i 1.09895 0.634481i 0.163007 0.986625i \(-0.447881\pi\)
0.935946 + 0.352144i \(0.114547\pi\)
\(168\) 0 0
\(169\) −8.93069 + 9.44684i −0.686976 + 0.726680i
\(170\) 0 0
\(171\) 10.7547 6.20920i 0.822429 0.474830i
\(172\) 0 0
\(173\) −0.150677 + 0.260981i −0.0114558 + 0.0198420i −0.871696 0.490046i \(-0.836980\pi\)
0.860241 + 0.509888i \(0.170313\pi\)
\(174\) 0 0
\(175\) 4.09689 + 2.36534i 0.309696 + 0.178803i
\(176\) 0 0
\(177\) 17.5347i 1.31799i
\(178\) 0 0
\(179\) 4.90791 + 8.50075i 0.366834 + 0.635376i 0.989069 0.147455i \(-0.0471083\pi\)
−0.622234 + 0.782831i \(0.713775\pi\)
\(180\) 0 0
\(181\) 12.4320 0.924062 0.462031 0.886864i \(-0.347121\pi\)
0.462031 + 0.886864i \(0.347121\pi\)
\(182\) 0 0
\(183\) −38.0652 −2.81386
\(184\) 0 0
\(185\) 2.65427 + 4.59733i 0.195146 + 0.338003i
\(186\) 0 0
\(187\) 3.16427i 0.231395i
\(188\) 0 0
\(189\) 4.85767 + 2.80458i 0.353344 + 0.204003i
\(190\) 0 0
\(191\) −6.12346 + 10.6061i −0.443078 + 0.767434i −0.997916 0.0645248i \(-0.979447\pi\)
0.554838 + 0.831958i \(0.312780\pi\)
\(192\) 0 0
\(193\) −10.0752 + 5.81692i −0.725229 + 0.418711i −0.816674 0.577099i \(-0.804185\pi\)
0.0914452 + 0.995810i \(0.470851\pi\)
\(194\) 0 0
\(195\) −4.23953 3.15955i −0.303599 0.226260i
\(196\) 0 0
\(197\) −1.55984 + 0.900572i −0.111134 + 0.0641631i −0.554537 0.832159i \(-0.687104\pi\)
0.443403 + 0.896322i \(0.353771\pi\)
\(198\) 0 0
\(199\) −3.29657 + 5.70982i −0.233687 + 0.404759i −0.958890 0.283777i \(-0.908413\pi\)
0.725203 + 0.688535i \(0.241746\pi\)
\(200\) 0 0
\(201\) −20.4895 11.8296i −1.44522 0.834397i
\(202\) 0 0
\(203\) 5.22996i 0.367071i
\(204\) 0 0
\(205\) −1.09215 1.89166i −0.0762793 0.132120i
\(206\) 0 0
\(207\) −45.5658 −3.16704
\(208\) 0 0
\(209\) 4.04603 0.279870
\(210\) 0 0
\(211\) 5.35996 + 9.28373i 0.368995 + 0.639118i 0.989409 0.145157i \(-0.0463686\pi\)
−0.620414 + 0.784275i \(0.713035\pi\)
\(212\) 0 0
\(213\) 14.6886i 1.00645i
\(214\) 0 0
\(215\) −0.448221 0.258781i −0.0305684 0.0176487i
\(216\) 0 0
\(217\) −2.89695 + 5.01767i −0.196658 + 0.340622i
\(218\) 0 0
\(219\) −28.9736 + 16.7279i −1.95785 + 1.13037i
\(220\) 0 0
\(221\) 6.97662 + 0.819021i 0.469298 + 0.0550933i
\(222\) 0 0
\(223\) 11.1612 6.44392i 0.747409 0.431517i −0.0773480 0.997004i \(-0.524645\pi\)
0.824757 + 0.565487i \(0.191312\pi\)
\(224\) 0 0
\(225\) −11.7912 + 20.4230i −0.786082 + 1.36153i
\(226\) 0 0
\(227\) 0.605486 + 0.349577i 0.0401875 + 0.0232023i 0.519959 0.854191i \(-0.325947\pi\)
−0.479772 + 0.877393i \(0.659280\pi\)
\(228\) 0 0
\(229\) 18.2868i 1.20843i 0.796822 + 0.604214i \(0.206513\pi\)
−0.796822 + 0.604214i \(0.793487\pi\)
\(230\) 0 0
\(231\) 2.29475 + 3.97463i 0.150984 + 0.261511i
\(232\) 0 0
\(233\) −26.7796 −1.75439 −0.877194 0.480137i \(-0.840587\pi\)
−0.877194 + 0.480137i \(0.840587\pi\)
\(234\) 0 0
\(235\) 2.34426 0.152923
\(236\) 0 0
\(237\) 1.38789 + 2.40390i 0.0901533 + 0.156150i
\(238\) 0 0
\(239\) 16.6177i 1.07491i −0.843293 0.537454i \(-0.819386\pi\)
0.843293 0.537454i \(-0.180614\pi\)
\(240\) 0 0
\(241\) 15.0800 + 8.70643i 0.971387 + 0.560830i 0.899659 0.436594i \(-0.143815\pi\)
0.0717279 + 0.997424i \(0.477149\pi\)
\(242\) 0 0
\(243\) 7.14890 12.3823i 0.458602 0.794322i
\(244\) 0 0
\(245\) 0.449430 0.259479i 0.0287130 0.0165775i
\(246\) 0 0
\(247\) 1.04725 8.92073i 0.0666350 0.567612i
\(248\) 0 0
\(249\) −21.8253 + 12.6008i −1.38312 + 0.798546i
\(250\) 0 0
\(251\) 3.22491 5.58571i 0.203554 0.352567i −0.746117 0.665815i \(-0.768084\pi\)
0.949671 + 0.313249i \(0.101417\pi\)
\(252\) 0 0
\(253\) −12.8568 7.42288i −0.808300 0.466672i
\(254\) 0 0
\(255\) 2.85703i 0.178914i
\(256\) 0 0
\(257\) −1.83578 3.17966i −0.114513 0.198342i 0.803072 0.595882i \(-0.203197\pi\)
−0.917585 + 0.397540i \(0.869864\pi\)
\(258\) 0 0
\(259\) −10.2293 −0.635615
\(260\) 0 0
\(261\) −26.0713 −1.61377
\(262\) 0 0
\(263\) 9.15964 + 15.8650i 0.564807 + 0.978275i 0.997068 + 0.0765263i \(0.0243829\pi\)
−0.432260 + 0.901749i \(0.642284\pi\)
\(264\) 0 0
\(265\) 4.61690i 0.283614i
\(266\) 0 0
\(267\) 29.4128 + 16.9815i 1.80003 + 1.03925i
\(268\) 0 0
\(269\) −13.7715 + 23.8529i −0.839661 + 1.45434i 0.0505171 + 0.998723i \(0.483913\pi\)
−0.890178 + 0.455613i \(0.849420\pi\)
\(270\) 0 0
\(271\) −5.64582 + 3.25961i −0.342959 + 0.198007i −0.661580 0.749875i \(-0.730114\pi\)
0.318621 + 0.947882i \(0.396780\pi\)
\(272\) 0 0
\(273\) 9.35726 4.03072i 0.566327 0.243950i
\(274\) 0 0
\(275\) −6.65401 + 3.84169i −0.401252 + 0.231663i
\(276\) 0 0
\(277\) 2.72093 4.71279i 0.163485 0.283164i −0.772631 0.634855i \(-0.781060\pi\)
0.936116 + 0.351691i \(0.114393\pi\)
\(278\) 0 0
\(279\) −25.0131 14.4413i −1.49749 0.864579i
\(280\) 0 0
\(281\) 3.54237i 0.211320i 0.994402 + 0.105660i \(0.0336955\pi\)
−0.994402 + 0.105660i \(0.966304\pi\)
\(282\) 0 0
\(283\) 7.06956 + 12.2448i 0.420242 + 0.727880i 0.995963 0.0897658i \(-0.0286119\pi\)
−0.575721 + 0.817646i \(0.695279\pi\)
\(284\) 0 0
\(285\) 3.65317 0.216395
\(286\) 0 0
\(287\) 4.20903 0.248451
\(288\) 0 0
\(289\) 6.60215 + 11.4353i 0.388362 + 0.672663i
\(290\) 0 0
\(291\) 12.4964i 0.732551i
\(292\) 0 0
\(293\) −7.23071 4.17465i −0.422423 0.243886i 0.273691 0.961818i \(-0.411756\pi\)
−0.696113 + 0.717932i \(0.745089\pi\)
\(294\) 0 0
\(295\) 1.61013 2.78883i 0.0937455 0.162372i
\(296\) 0 0
\(297\) −7.88964 + 4.55508i −0.457803 + 0.264313i
\(298\) 0 0
\(299\) −19.6938 + 26.4255i −1.13892 + 1.52823i
\(300\) 0 0
\(301\) 0.863697 0.498655i 0.0497826 0.0287420i
\(302\) 0 0
\(303\) 25.8596 44.7901i 1.48559 2.57312i
\(304\) 0 0
\(305\) −6.05415 3.49536i −0.346659 0.200144i
\(306\) 0 0
\(307\) 8.33362i 0.475625i 0.971311 + 0.237813i \(0.0764304\pi\)
−0.971311 + 0.237813i \(0.923570\pi\)
\(308\) 0 0
\(309\) −7.09334 12.2860i −0.403526 0.698927i
\(310\) 0 0
\(311\) −14.6227 −0.829176 −0.414588 0.910009i \(-0.636074\pi\)
−0.414588 + 0.910009i \(0.636074\pi\)
\(312\) 0 0
\(313\) 17.1328 0.968404 0.484202 0.874956i \(-0.339110\pi\)
0.484202 + 0.874956i \(0.339110\pi\)
\(314\) 0 0
\(315\) 1.29350 + 2.24041i 0.0728805 + 0.126233i
\(316\) 0 0
\(317\) 14.0000i 0.786320i 0.919470 + 0.393160i \(0.128618\pi\)
−0.919470 + 0.393160i \(0.871382\pi\)
\(318\) 0 0
\(319\) −7.35627 4.24714i −0.411872 0.237794i
\(320\) 0 0
\(321\) 8.68157 15.0369i 0.484558 0.839279i
\(322\) 0 0
\(323\) −4.20317 + 2.42670i −0.233871 + 0.135025i
\(324\) 0 0
\(325\) 6.74791 + 15.6652i 0.374307 + 0.868947i
\(326\) 0 0
\(327\) −29.1124 + 16.8081i −1.60992 + 0.929487i
\(328\) 0 0
\(329\) −2.25863 + 3.91206i −0.124522 + 0.215679i
\(330\) 0 0
\(331\) −5.99286 3.45998i −0.329397 0.190178i 0.326176 0.945309i \(-0.394240\pi\)
−0.655574 + 0.755131i \(0.727573\pi\)
\(332\) 0 0
\(333\) 50.9928i 2.79439i
\(334\) 0 0
\(335\) −2.17253 3.76292i −0.118698 0.205591i
\(336\) 0 0
\(337\) 11.1559 0.607703 0.303852 0.952719i \(-0.401727\pi\)
0.303852 + 0.952719i \(0.401727\pi\)
\(338\) 0 0
\(339\) 10.0488 0.545776
\(340\) 0 0
\(341\) −4.70512 8.14950i −0.254796 0.441320i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −11.6084 6.70213i −0.624977 0.360830i
\(346\) 0 0
\(347\) −2.46255 + 4.26527i −0.132197 + 0.228971i −0.924523 0.381126i \(-0.875536\pi\)
0.792326 + 0.610097i \(0.208870\pi\)
\(348\) 0 0
\(349\) 1.31926 0.761675i 0.0706183 0.0407715i −0.464275 0.885691i \(-0.653685\pi\)
0.534893 + 0.844920i \(0.320352\pi\)
\(350\) 0 0
\(351\) 8.00098 + 18.5741i 0.427061 + 0.991415i
\(352\) 0 0
\(353\) 15.5261 8.96401i 0.826372 0.477106i −0.0262367 0.999656i \(-0.508352\pi\)
0.852609 + 0.522550i \(0.175019\pi\)
\(354\) 0 0
\(355\) 1.34879 2.33618i 0.0715865 0.123991i
\(356\) 0 0
\(357\) −4.76775 2.75266i −0.252336 0.145686i
\(358\) 0 0
\(359\) 20.0014i 1.05563i 0.849359 + 0.527816i \(0.176989\pi\)
−0.849359 + 0.527816i \(0.823011\pi\)
\(360\) 0 0
\(361\) −6.39707 11.0801i −0.336688 0.583161i
\(362\) 0 0
\(363\) 23.6294 1.24022
\(364\) 0 0
\(365\) −6.14421 −0.321602
\(366\) 0 0
\(367\) 13.7078 + 23.7427i 0.715544 + 1.23936i 0.962749 + 0.270395i \(0.0871543\pi\)
−0.247206 + 0.968963i \(0.579512\pi\)
\(368\) 0 0
\(369\) 20.9820i 1.09228i
\(370\) 0 0
\(371\) −7.70460 4.44825i −0.400003 0.230942i
\(372\) 0 0
\(373\) 7.94643 13.7636i 0.411451 0.712653i −0.583598 0.812043i \(-0.698356\pi\)
0.995049 + 0.0993893i \(0.0316889\pi\)
\(374\) 0 0
\(375\) −12.3578 + 7.13481i −0.638156 + 0.368440i
\(376\) 0 0
\(377\) −11.2682 + 15.1199i −0.580341 + 0.778712i
\(378\) 0 0
\(379\) −7.60284 + 4.38950i −0.390532 + 0.225474i −0.682390 0.730988i \(-0.739060\pi\)
0.291859 + 0.956461i \(0.405726\pi\)
\(380\) 0 0
\(381\) 2.01124 3.48357i 0.103039 0.178469i
\(382\) 0 0
\(383\) −6.89562 3.98119i −0.352349 0.203429i 0.313370 0.949631i \(-0.398542\pi\)
−0.665720 + 0.746202i \(0.731875\pi\)
\(384\) 0 0
\(385\) 0.842869i 0.0429566i
\(386\) 0 0
\(387\) 2.48580 + 4.30553i 0.126360 + 0.218862i
\(388\) 0 0
\(389\) −32.0434 −1.62467 −0.812333 0.583194i \(-0.801803\pi\)
−0.812333 + 0.583194i \(0.801803\pi\)
\(390\) 0 0
\(391\) 17.8082 0.900598
\(392\) 0 0
\(393\) 12.2550 + 21.2263i 0.618184 + 1.07073i
\(394\) 0 0
\(395\) 0.509777i 0.0256496i
\(396\) 0 0
\(397\) 5.57251 + 3.21729i 0.279676 + 0.161471i 0.633277 0.773925i \(-0.281710\pi\)
−0.353601 + 0.935396i \(0.615043\pi\)
\(398\) 0 0
\(399\) −3.51972 + 6.09633i −0.176206 + 0.305198i
\(400\) 0 0
\(401\) −0.462092 + 0.266789i −0.0230758 + 0.0133228i −0.511494 0.859287i \(-0.670908\pi\)
0.488418 + 0.872610i \(0.337574\pi\)
\(402\) 0 0
\(403\) −19.1859 + 8.26451i −0.955719 + 0.411685i
\(404\) 0 0
\(405\) −0.402337 + 0.232290i −0.0199923 + 0.0115426i
\(406\) 0 0
\(407\) 8.30697 14.3881i 0.411761 0.713191i
\(408\) 0 0
\(409\) −34.4269 19.8764i −1.70230 0.982824i −0.943424 0.331590i \(-0.892415\pi\)
−0.758877 0.651234i \(-0.774252\pi\)
\(410\) 0 0
\(411\) 24.0695i 1.18726i
\(412\) 0 0
\(413\) 3.10263 + 5.37392i 0.152671 + 0.264433i
\(414\) 0 0
\(415\) −4.62832 −0.227195
\(416\) 0 0
\(417\) 14.2375 0.697213
\(418\) 0 0
\(419\) 11.9088 + 20.6266i 0.581783 + 1.00768i 0.995268 + 0.0971665i \(0.0309779\pi\)
−0.413485 + 0.910511i \(0.635689\pi\)
\(420\) 0 0
\(421\) 23.2419i 1.13274i −0.824151 0.566370i \(-0.808347\pi\)
0.824151 0.566370i \(-0.191653\pi\)
\(422\) 0 0
\(423\) −19.5016 11.2593i −0.948200 0.547444i
\(424\) 0 0
\(425\) 4.60828 7.98178i 0.223535 0.387173i
\(426\) 0 0
\(427\) 11.6660 6.73536i 0.564557 0.325947i
\(428\) 0 0
\(429\) −1.92937 + 16.4348i −0.0931510 + 0.793482i
\(430\) 0 0
\(431\) 2.34424 1.35345i 0.112918 0.0651932i −0.442477 0.896780i \(-0.645900\pi\)
0.555395 + 0.831586i \(0.312567\pi\)
\(432\) 0 0
\(433\) −2.90945 + 5.03932i −0.139819 + 0.242174i −0.927428 0.374002i \(-0.877985\pi\)
0.787609 + 0.616176i \(0.211319\pi\)
\(434\) 0 0
\(435\) −6.64198 3.83475i −0.318459 0.183862i
\(436\) 0 0
\(437\) 22.7706i 1.08927i
\(438\) 0 0
\(439\) −19.0851 33.0563i −0.910882 1.57769i −0.812822 0.582513i \(-0.802070\pi\)
−0.0980599 0.995181i \(-0.531264\pi\)
\(440\) 0 0
\(441\) −4.98500 −0.237381
\(442\) 0 0
\(443\) 31.6740 1.50488 0.752440 0.658661i \(-0.228877\pi\)
0.752440 + 0.658661i \(0.228877\pi\)
\(444\) 0 0
\(445\) 3.11867 + 5.40169i 0.147839 + 0.256065i
\(446\) 0 0
\(447\) 9.50845i 0.449734i
\(448\) 0 0
\(449\) 27.1975 + 15.7025i 1.28353 + 0.741045i 0.977491 0.210975i \(-0.0676638\pi\)
0.306036 + 0.952020i \(0.400997\pi\)
\(450\) 0 0
\(451\) −3.41807 + 5.92027i −0.160951 + 0.278775i
\(452\) 0 0
\(453\) −30.9732 + 17.8824i −1.45525 + 0.840187i
\(454\) 0 0
\(455\) 1.85836 + 0.218163i 0.0871215 + 0.0102276i
\(456\) 0 0
\(457\) 27.5640 15.9141i 1.28939 0.744429i 0.310844 0.950461i \(-0.399388\pi\)
0.978545 + 0.206032i \(0.0660551\pi\)
\(458\) 0 0
\(459\) 5.46403 9.46398i 0.255039 0.441741i
\(460\) 0 0
\(461\) 1.01005 + 0.583153i 0.0470427 + 0.0271601i 0.523337 0.852126i \(-0.324687\pi\)
−0.476294 + 0.879286i \(0.658020\pi\)
\(462\) 0 0
\(463\) 20.3441i 0.945469i −0.881205 0.472734i \(-0.843267\pi\)
0.881205 0.472734i \(-0.156733\pi\)
\(464\) 0 0
\(465\) −4.24825 7.35819i −0.197008 0.341228i
\(466\) 0 0
\(467\) −1.56939 −0.0726229 −0.0363114 0.999341i \(-0.511561\pi\)
−0.0363114 + 0.999341i \(0.511561\pi\)
\(468\) 0 0
\(469\) 8.37266 0.386614
\(470\) 0 0
\(471\) −14.6504 25.3753i −0.675055 1.16923i
\(472\) 0 0
\(473\) 1.61979i 0.0744781i
\(474\) 0 0
\(475\) −10.2060 5.89243i −0.468283 0.270363i
\(476\) 0 0
\(477\) 22.1745 38.4074i 1.01530 1.75856i
\(478\) 0 0
\(479\) −6.68501 + 3.85959i −0.305446 + 0.176349i −0.644887 0.764278i \(-0.723095\pi\)
0.339441 + 0.940627i \(0.389762\pi\)
\(480\) 0 0
\(481\) −29.5729 22.0394i −1.34841 1.00491i
\(482\) 0 0
\(483\) 22.3688 12.9146i 1.01781 0.587635i
\(484\) 0 0
\(485\) 1.14749 1.98751i 0.0521047 0.0902481i
\(486\) 0 0
\(487\) 0.0659739 + 0.0380900i 0.00298956 + 0.00172602i 0.501494 0.865161i \(-0.332784\pi\)
−0.498504 + 0.866887i \(0.666117\pi\)
\(488\) 0 0
\(489\) 44.5155i 2.01306i
\(490\) 0 0
\(491\) −0.893574 1.54772i −0.0403264 0.0698474i 0.845158 0.534517i \(-0.179506\pi\)
−0.885484 + 0.464670i \(0.846173\pi\)
\(492\) 0 0
\(493\) 10.1893 0.458902
\(494\) 0 0
\(495\) −4.20170 −0.188852
\(496\) 0 0
\(497\) 2.59905 + 4.50168i 0.116583 + 0.201928i
\(498\) 0 0
\(499\) 8.33493i 0.373123i 0.982443 + 0.186561i \(0.0597343\pi\)
−0.982443 + 0.186561i \(0.940266\pi\)
\(500\) 0 0
\(501\) −40.1305 23.1694i −1.79290 1.03513i
\(502\) 0 0
\(503\) 0.720238 1.24749i 0.0321138 0.0556228i −0.849522 0.527554i \(-0.823109\pi\)
0.881636 + 0.471931i \(0.156443\pi\)
\(504\) 0 0
\(505\) 8.22576 4.74914i 0.366041 0.211334i
\(506\) 0 0
\(507\) 35.7363 + 8.50779i 1.58710 + 0.377844i
\(508\) 0 0
\(509\) 12.8394 7.41282i 0.569096 0.328568i −0.187692 0.982228i \(-0.560101\pi\)
0.756788 + 0.653660i \(0.226767\pi\)
\(510\) 0 0
\(511\) 5.91976 10.2533i 0.261875 0.453581i
\(512\) 0 0
\(513\) −12.1012 6.98664i −0.534282 0.308468i
\(514\) 0 0
\(515\) 2.60540i 0.114808i
\(516\) 0 0
\(517\) −3.66837 6.35380i −0.161335 0.279440i
\(518\) 0 0
\(519\) 0.851561 0.0373794
\(520\) 0 0
\(521\) −0.334388 −0.0146498 −0.00732489 0.999973i \(-0.502332\pi\)
−0.00732489 + 0.999973i \(0.502332\pi\)
\(522\) 0 0
\(523\) −16.2533 28.1515i −0.710705 1.23098i −0.964593 0.263744i \(-0.915043\pi\)
0.253887 0.967234i \(-0.418291\pi\)
\(524\) 0 0
\(525\) 13.3678i 0.583420i
\(526\) 0 0
\(527\) 9.77570 + 5.64400i 0.425836 + 0.245857i
\(528\) 0 0
\(529\) −30.2751 + 52.4380i −1.31631 + 2.27991i
\(530\) 0 0
\(531\) −26.7890 + 15.4666i −1.16254 + 0.671194i
\(532\) 0 0
\(533\) 12.1683 + 9.06855i 0.527070 + 0.392803i
\(534\) 0 0
\(535\) 2.76155 1.59438i 0.119392 0.0689311i
\(536\) 0 0
\(537\) 13.8686 24.0212i 0.598476 1.03659i
\(538\) 0 0
\(539\) −1.40656 0.812080i −0.0605850 0.0349787i
\(540\) 0 0
\(541\) 10.6015i 0.455796i 0.973685 + 0.227898i \(0.0731852\pi\)
−0.973685 + 0.227898i \(0.926815\pi\)
\(542\) 0 0
\(543\) −17.5650 30.4235i −0.753786 1.30560i
\(544\) 0 0
\(545\) −6.17364 −0.264450
\(546\) 0 0
\(547\) −10.2327 −0.437519 −0.218760 0.975779i \(-0.570201\pi\)
−0.218760 + 0.975779i \(0.570201\pi\)
\(548\) 0 0
\(549\) 33.5758 + 58.1549i 1.43298 + 2.48199i
\(550\) 0 0
\(551\) 13.0286i 0.555038i
\(552\) 0 0
\(553\) −0.850705 0.491155i −0.0361757 0.0208860i
\(554\) 0 0
\(555\) 7.50037 12.9910i 0.318373 0.551438i
\(556\) 0 0
\(557\) 27.7067 15.9965i 1.17397 0.677793i 0.219359 0.975644i \(-0.429603\pi\)
0.954612 + 0.297851i \(0.0962700\pi\)
\(558\) 0 0
\(559\) 3.57133 + 0.419257i 0.151051 + 0.0177327i
\(560\) 0 0
\(561\) 7.74359 4.47076i 0.326934 0.188756i
\(562\) 0 0
\(563\) −5.39566 + 9.34556i −0.227400 + 0.393868i −0.957037 0.289967i \(-0.906356\pi\)
0.729637 + 0.683835i \(0.239689\pi\)
\(564\) 0 0
\(565\) 1.59823 + 0.922737i 0.0672380 + 0.0388199i
\(566\) 0 0
\(567\) 0.895217i 0.0375956i
\(568\) 0 0
\(569\) −12.3007 21.3054i −0.515672 0.893170i −0.999835 0.0181917i \(-0.994209\pi\)
0.484163 0.874978i \(-0.339124\pi\)
\(570\) 0 0
\(571\) 16.5724 0.693534 0.346767 0.937951i \(-0.387279\pi\)
0.346767 + 0.937951i \(0.387279\pi\)
\(572\) 0 0
\(573\) 34.6070 1.44573
\(574\) 0 0
\(575\) 21.6206 + 37.4480i 0.901641 + 1.56169i
\(576\) 0 0
\(577\) 14.6611i 0.610348i −0.952297 0.305174i \(-0.901285\pi\)
0.952297 0.305174i \(-0.0987147\pi\)
\(578\) 0 0
\(579\) 28.4702 + 16.4373i 1.18318 + 0.683111i
\(580\) 0 0
\(581\) 4.45926 7.72366i 0.185001 0.320431i
\(582\) 0 0
\(583\) 12.5135 7.22467i 0.518256 0.299215i
\(584\) 0 0
\(585\) −1.08754 + 9.26394i −0.0449644 + 0.383017i
\(586\) 0 0
\(587\) −30.5998 + 17.6668i −1.26299 + 0.729186i −0.973652 0.228041i \(-0.926768\pi\)
−0.289336 + 0.957227i \(0.593435\pi\)
\(588\) 0 0
\(589\) 7.21676 12.4998i 0.297362 0.515045i
\(590\) 0 0
\(591\) 4.40775 + 2.54481i 0.181310 + 0.104680i
\(592\) 0 0
\(593\) 16.4294i 0.674675i 0.941384 + 0.337338i \(0.109526\pi\)
−0.941384 + 0.337338i \(0.890474\pi\)
\(594\) 0 0
\(595\) −0.505530 0.875603i −0.0207247 0.0358962i
\(596\) 0 0
\(597\) 18.6307 0.762504
\(598\) 0 0
\(599\) −12.0819 −0.493653 −0.246826 0.969060i \(-0.579388\pi\)
−0.246826 + 0.969060i \(0.579388\pi\)
\(600\) 0 0
\(601\) −3.90743 6.76787i −0.159387 0.276067i 0.775261 0.631642i \(-0.217619\pi\)
−0.934648 + 0.355574i \(0.884285\pi\)
\(602\) 0 0
\(603\) 41.7377i 1.69969i
\(604\) 0 0
\(605\) 3.75818 + 2.16979i 0.152792 + 0.0882144i
\(606\) 0 0
\(607\) 17.7825 30.8001i 0.721768 1.25014i −0.238523 0.971137i \(-0.576663\pi\)
0.960291 0.279002i \(-0.0900035\pi\)
\(608\) 0 0
\(609\) 12.7987 7.38934i 0.518630 0.299431i
\(610\) 0 0
\(611\) −14.9584 + 6.44347i −0.605153 + 0.260675i
\(612\) 0 0
\(613\) −10.3376 + 5.96839i −0.417530 + 0.241061i −0.694020 0.719956i \(-0.744162\pi\)
0.276490 + 0.961017i \(0.410829\pi\)
\(614\) 0 0
\(615\) −3.08618 + 5.34542i −0.124447 + 0.215548i
\(616\) 0 0
\(617\) −20.4124 11.7851i −0.821772 0.474450i 0.0292550 0.999572i \(-0.490687\pi\)
−0.851027 + 0.525122i \(0.824020\pi\)
\(618\) 0 0
\(619\) 28.5571i 1.14781i −0.818923 0.573904i \(-0.805428\pi\)
0.818923 0.573904i \(-0.194572\pi\)
\(620\) 0 0
\(621\) 25.6355 + 44.4020i 1.02872 + 1.78179i
\(622\) 0 0
\(623\) −12.0190 −0.481530
\(624\) 0 0
\(625\) 21.0328 0.841311
\(626\) 0 0
\(627\) −5.71659 9.90142i −0.228298 0.395425i
\(628\) 0 0
\(629\) 19.9292i 0.794628i
\(630\) 0 0
\(631\) 38.9646 + 22.4962i 1.55116 + 0.895561i 0.998048 + 0.0624526i \(0.0198922\pi\)
0.553109 + 0.833109i \(0.313441\pi\)
\(632\) 0 0
\(633\) 15.1460 26.2337i 0.602001 1.04270i
\(634\) 0 0
\(635\) 0.639763 0.369367i 0.0253882 0.0146579i
\(636\) 0 0
\(637\) −2.15455 + 2.89101i −0.0853662 + 0.114546i
\(638\) 0 0
\(639\) −22.4409 + 12.9562i −0.887747 + 0.512541i
\(640\) 0 0
\(641\) −1.26650 + 2.19364i −0.0500238 + 0.0866437i −0.889953 0.456052i \(-0.849263\pi\)
0.839929 + 0.542696i \(0.182596\pi\)
\(642\) 0 0
\(643\) −15.9150 9.18853i −0.627627 0.362360i 0.152206 0.988349i \(-0.451362\pi\)
−0.779832 + 0.625988i \(0.784696\pi\)
\(644\) 0 0
\(645\) 1.46251i 0.0575863i
\(646\) 0 0
\(647\) −10.4643 18.1248i −0.411396 0.712558i 0.583647 0.812008i \(-0.301625\pi\)
−0.995043 + 0.0994494i \(0.968292\pi\)
\(648\) 0 0
\(649\) −10.0783 −0.395609
\(650\) 0 0
\(651\) 16.3723 0.641680
\(652\) 0 0
\(653\) 24.0580 + 41.6696i 0.941461 + 1.63066i 0.762686 + 0.646769i \(0.223880\pi\)
0.178775 + 0.983890i \(0.442786\pi\)
\(654\) 0 0
\(655\) 4.50130i 0.175880i
\(656\) 0 0
\(657\) 51.1129 + 29.5100i 1.99410 + 1.15130i
\(658\) 0 0
\(659\) −1.10819 + 1.91944i −0.0431690 + 0.0747708i −0.886803 0.462148i \(-0.847079\pi\)
0.843634 + 0.536919i \(0.180412\pi\)
\(660\) 0 0
\(661\) −0.552034 + 0.318717i −0.0214716 + 0.0123966i −0.510697 0.859761i \(-0.670613\pi\)
0.489226 + 0.872157i \(0.337279\pi\)
\(662\) 0 0
\(663\) −7.85287 18.2303i −0.304980 0.708006i
\(664\) 0 0
\(665\) −1.11960 + 0.646401i −0.0434162 + 0.0250664i
\(666\) 0 0
\(667\) −23.9024 + 41.4002i −0.925506 + 1.60302i
\(668\) 0 0
\(669\) −31.5390 18.2091i −1.21937 0.704003i
\(670\) 0 0
\(671\) 21.8786i 0.844614i
\(672\) 0 0
\(673\) 7.70343 + 13.3427i 0.296945 + 0.514324i 0.975436 0.220285i \(-0.0706988\pi\)
−0.678490 + 0.734609i \(0.737365\pi\)
\(674\) 0 0
\(675\) 26.5352 1.02134
\(676\) 0 0
\(677\) −11.6812 −0.448945 −0.224473 0.974480i \(-0.572066\pi\)
−0.224473 + 0.974480i \(0.572066\pi\)
\(678\) 0 0
\(679\) 2.21114 + 3.82981i 0.0848559 + 0.146975i
\(680\) 0 0
\(681\) 1.97565i 0.0757072i
\(682\) 0 0
\(683\) −19.8419 11.4557i −0.759227 0.438340i 0.0697909 0.997562i \(-0.477767\pi\)
−0.829018 + 0.559221i \(0.811100\pi\)
\(684\) 0 0
\(685\) 2.21020 3.82817i 0.0844473 0.146267i
\(686\) 0 0
\(687\) 44.7514 25.8372i 1.70737 0.985752i
\(688\) 0 0
\(689\) −12.6901 29.4599i −0.483454 1.12233i
\(690\) 0 0
\(691\) 40.9046 23.6163i 1.55608 0.898405i 0.558458 0.829533i \(-0.311393\pi\)
0.997625 0.0688729i \(-0.0219403\pi\)
\(692\) 0 0
\(693\) 4.04822 7.01172i 0.153779 0.266353i
\(694\) 0 0
\(695\) 2.26443 + 1.30737i 0.0858946 + 0.0495913i
\(696\) 0 0
\(697\) 8.20026i 0.310607i
\(698\) 0 0
\(699\) 37.8365 + 65.5347i 1.43111 + 2.47875i
\(700\) 0 0
\(701\) 12.2098 0.461158 0.230579 0.973054i \(-0.425938\pi\)
0.230579 + 0.973054i \(0.425938\pi\)
\(702\) 0 0
\(703\) 25.4827 0.961097
\(704\) 0 0
\(705\) −3.31218 5.73686i −0.124744 0.216063i
\(706\) 0 0
\(707\) 18.3026i 0.688342i
\(708\) 0 0
\(709\) −15.4910 8.94374i −0.581777 0.335889i 0.180062 0.983655i \(-0.442370\pi\)
−0.761839 + 0.647766i \(0.775703\pi\)
\(710\) 0 0
\(711\) 2.44841 4.24076i 0.0918224 0.159041i
\(712\) 0 0
\(713\) −45.8644 + 26.4798i −1.71764 + 0.991678i
\(714\) 0 0
\(715\) −1.81600 + 2.43674i −0.0679146 + 0.0911290i
\(716\) 0 0
\(717\) −40.6667 + 23.4789i −1.51872 + 0.876836i
\(718\) 0 0
\(719\) −4.56317 + 7.90364i −0.170178 + 0.294756i −0.938482 0.345329i \(-0.887767\pi\)
0.768304 + 0.640085i \(0.221101\pi\)
\(720\) 0 0
\(721\) 4.34784 + 2.51023i 0.161922 + 0.0934858i
\(722\) 0 0
\(723\) 49.2048i 1.82995i
\(724\) 0 0
\(725\) 12.3706 + 21.4266i 0.459434 + 0.795763i
\(726\) 0 0
\(727\) −33.6859 −1.24934 −0.624670 0.780889i \(-0.714767\pi\)
−0.624670 + 0.780889i \(0.714767\pi\)
\(728\) 0 0
\(729\) −43.0880 −1.59585
\(730\) 0 0
\(731\) −0.971507 1.68270i −0.0359325 0.0622369i
\(732\) 0 0
\(733\) 46.4344i 1.71509i −0.514406 0.857547i \(-0.671987\pi\)
0.514406 0.857547i \(-0.328013\pi\)
\(734\) 0 0
\(735\) −1.26999 0.733228i −0.0468442 0.0270455i
\(736\) 0 0
\(737\) −6.79927 + 11.7767i −0.250454 + 0.433799i
\(738\) 0 0
\(739\) 1.60237 0.925127i 0.0589440 0.0340314i −0.470238 0.882539i \(-0.655832\pi\)
0.529182 + 0.848508i \(0.322499\pi\)
\(740\) 0 0
\(741\) −23.3104 + 10.0412i −0.856328 + 0.368871i
\(742\) 0 0
\(743\) −28.7095 + 16.5755i −1.05325 + 0.608094i −0.923558 0.383460i \(-0.874732\pi\)
−0.129693 + 0.991554i \(0.541399\pi\)
\(744\) 0 0
\(745\) 0.873120 1.51229i 0.0319886 0.0554059i
\(746\) 0 0
\(747\) 38.5024 + 22.2294i 1.40873 + 0.813331i
\(748\) 0 0
\(749\) 6.14456i 0.224517i
\(750\) 0 0
\(751\) 10.3871 + 17.9910i 0.379032 + 0.656503i 0.990922 0.134441i \(-0.0429237\pi\)
−0.611890 + 0.790943i \(0.709590\pi\)
\(752\) 0 0
\(753\) −18.2257 −0.664182
\(754\) 0 0
\(755\) −6.56824 −0.239043
\(756\) 0 0
\(757\) −21.8075 37.7717i −0.792607 1.37283i −0.924348 0.381551i \(-0.875390\pi\)
0.131741 0.991284i \(-0.457943\pi\)
\(758\) 0 0
\(759\) 41.9508i 1.52272i
\(760\) 0 0
\(761\) 10.7302 + 6.19511i 0.388971 + 0.224573i 0.681714 0.731619i \(-0.261235\pi\)
−0.292743 + 0.956191i \(0.594568\pi\)
\(762\) 0 0
\(763\) 5.94812 10.3025i 0.215337 0.372974i
\(764\) 0 0
\(765\) 4.36488 2.52007i 0.157813 0.0911132i
\(766\) 0 0
\(767\) −2.60862 + 22.2208i −0.0941917 + 0.802347i
\(768\) 0 0
\(769\) −4.80955 + 2.77680i −0.173437 + 0.100134i −0.584205 0.811606i \(-0.698594\pi\)
0.410769 + 0.911740i \(0.365260\pi\)
\(770\) 0 0
\(771\) −5.18750 + 8.98501i −0.186823 + 0.323587i
\(772\) 0 0
\(773\) 37.9355 + 21.9021i 1.36445 + 0.787764i 0.990212 0.139570i \(-0.0445721\pi\)
0.374235 + 0.927334i \(0.377905\pi\)
\(774\) 0 0
\(775\) 27.4092i 0.984566i
\(776\) 0 0
\(777\) 14.4528 + 25.0330i 0.518491 + 0.898052i
\(778\) 0 0
\(779\) −10.4853 −0.375677
\(780\) 0 0
\(781\) −8.44253 −0.302098
\(782\) 0 0
\(783\) 14.6678 + 25.4054i 0.524185 + 0.907916i
\(784\) 0 0
\(785\) 5.38113i 0.192061i
\(786\) 0 0
\(787\) −21.4782 12.4005i −0.765616 0.442029i 0.0656923 0.997840i \(-0.479074\pi\)
−0.831309 + 0.555811i \(0.812408\pi\)
\(788\) 0 0
\(789\) 25.8831 44.8308i 0.921462 1.59602i
\(790\) 0 0
\(791\) −3.07969 + 1.77806i −0.109501 + 0.0632206i
\(792\) 0 0
\(793\) 48.2381 + 5.66293i 1.71299 + 0.201096i
\(794\) 0 0
\(795\) 11.2985 6.52316i 0.400715 0.231353i
\(796\) 0 0
\(797\) −8.23575 + 14.2647i −0.291725 + 0.505283i −0.974218 0.225610i \(-0.927563\pi\)
0.682492 + 0.730893i \(0.260896\pi\)
\(798\) 0 0
\(799\) 7.62168 + 4.40038i 0.269636 + 0.155674i
\(800\) 0 0
\(801\) 59.9146i 2.11698i
\(802\) 0 0
\(803\) 9.61464 + 16.6530i 0.339293 + 0.587673i
\(804\) 0 0
\(805\) 4.74357 0.167189
\(806\) 0 0
\(807\) 77.8301 2.73975
\(808\) 0 0
\(809\) −0.690968 1.19679i −0.0242932 0.0420770i 0.853623 0.520891i \(-0.174400\pi\)
−0.877916 + 0.478814i \(0.841067\pi\)
\(810\) 0 0
\(811\) 6.83571i 0.240034i −0.992772 0.120017i \(-0.961705\pi\)
0.992772 0.120017i \(-0.0382950\pi\)
\(812\) 0 0
\(813\) 15.9538 + 9.21093i 0.559524 + 0.323041i
\(814\) 0 0
\(815\) −4.08767 + 7.08004i −0.143185 + 0.248003i
\(816\) 0 0
\(817\) −2.15160 + 1.24223i −0.0752750 + 0.0434601i
\(818\) 0 0
\(819\) −14.4117 10.7404i −0.503585 0.375300i
\(820\) 0 0
\(821\) −9.13009 + 5.27126i −0.318642 + 0.183968i −0.650787 0.759260i \(-0.725561\pi\)
0.332145 + 0.943228i \(0.392228\pi\)
\(822\) 0 0
\(823\) −7.41652 + 12.8458i −0.258524 + 0.447776i −0.965847 0.259114i \(-0.916569\pi\)
0.707323 + 0.706891i \(0.249903\pi\)
\(824\) 0 0
\(825\) 18.8027 + 10.8558i 0.654627 + 0.377949i
\(826\) 0 0
\(827\) 55.6758i 1.93604i −0.250879 0.968018i \(-0.580720\pi\)
0.250879 0.968018i \(-0.419280\pi\)
\(828\) 0 0
\(829\) −0.0232424 0.0402570i −0.000807242 0.00139818i 0.865622 0.500699i \(-0.166924\pi\)
−0.866429 + 0.499301i \(0.833590\pi\)
\(830\) 0 0
\(831\) −15.3775 −0.533438
\(832\) 0 0
\(833\) 1.94825 0.0675030
\(834\) 0 0
\(835\) −4.25508 7.37002i −0.147253 0.255050i
\(836\) 0 0
\(837\) 32.4990i 1.12333i
\(838\) 0 0
\(839\) −22.1248 12.7738i −0.763833 0.440999i 0.0668370 0.997764i \(-0.478709\pi\)
−0.830670 + 0.556765i \(0.812043\pi\)
\(840\) 0 0
\(841\) 0.823775 1.42682i 0.0284060 0.0492007i
\(842\) 0 0
\(843\) 8.66886 5.00497i 0.298572 0.172380i
\(844\) 0 0
\(845\) 4.90250 + 4.63464i 0.168651 + 0.159437i
\(846\) 0 0
\(847\) −7.24180 + 4.18105i −0.248831 + 0.143663i
\(848\) 0 0
\(849\) 19.9770 34.6012i 0.685609 1.18751i
\(850\) 0 0
\(851\) −80.9745 46.7507i −2.77577 1.60259i
\(852\) 0 0
\(853\) 22.6671i 0.776105i −0.921637 0.388053i \(-0.873148\pi\)
0.921637 0.388053i \(-0.126852\pi\)
\(854\) 0 0
\(855\) −3.22231 5.58120i −0.110201 0.190873i
\(856\) 0 0
\(857\) 37.0535 1.26572 0.632862 0.774264i \(-0.281880\pi\)
0.632862 + 0.774264i \(0.281880\pi\)
\(858\) 0 0
\(859\) 4.24339 0.144782 0.0723912 0.997376i \(-0.476937\pi\)
0.0723912 + 0.997376i \(0.476937\pi\)
\(860\) 0 0
\(861\) −5.94689 10.3003i −0.202669 0.351034i
\(862\) 0 0
\(863\) 7.50051i 0.255320i −0.991818 0.127660i \(-0.959253\pi\)
0.991818 0.127660i \(-0.0407467\pi\)
\(864\) 0 0
\(865\) 0.135438 + 0.0781951i 0.00460503 + 0.00265871i
\(866\) 0 0
\(867\) 18.6562 32.3135i 0.633598 1.09742i
\(868\) 0 0
\(869\) 1.38168 0.797714i 0.0468703 0.0270606i
\(870\) 0 0
\(871\) 24.2054 + 18.0393i 0.820170 + 0.611238i
\(872\) 0 0
\(873\) −19.0916 + 11.0225i −0.646153 + 0.373057i
\(874\) 0 0
\(875\) 2.52490 4.37326i 0.0853573 0.147843i
\(876\) 0 0
\(877\) −24.4996 14.1448i −0.827291 0.477637i 0.0256330 0.999671i \(-0.491840\pi\)
−0.852924 + 0.522035i \(0.825173\pi\)
\(878\) 0 0
\(879\) 23.5933i 0.795781i
\(880\) 0 0
\(881\) −19.7860 34.2704i −0.666609 1.15460i −0.978846 0.204596i \(-0.934412\pi\)
0.312238 0.950004i \(-0.398921\pi\)
\(882\) 0 0
\(883\) 28.3609 0.954419 0.477209 0.878790i \(-0.341648\pi\)
0.477209 + 0.878790i \(0.341648\pi\)
\(884\) 0 0
\(885\) −9.09974 −0.305884
\(886\) 0 0
\(887\) −21.8593 37.8614i −0.733963 1.27126i −0.955177 0.296036i \(-0.904335\pi\)
0.221214 0.975225i \(-0.428998\pi\)
\(888\) 0 0
\(889\) 1.42350i 0.0477426i
\(890\) 0 0
\(891\) 1.25918 + 0.726988i 0.0421841 + 0.0243550i
\(892\) 0 0
\(893\) 5.62659 9.74554i 0.188287 0.326122i
\(894\) 0 0
\(895\) 4.41152 2.54699i 0.147461 0.0851366i
\(896\) 0 0
\(897\) 92.4934 + 10.8583i 3.08826 + 0.362547i
\(898\) 0 0
\(899\) −26.2422 + 15.1509i −0.875227 + 0.505312i
\(900\) 0 0
\(901\) −8.66632 + 15.0105i −0.288717 + 0.500073i
\(902\) 0 0
\(903\) −2.44061 1.40909i −0.0812185 0.0468915i
\(904\) 0 0
\(905\) 6.45167i 0.214461i
\(906\) 0 0
\(907\) 18.3493 + 31.7818i 0.609277 + 1.05530i 0.991360 + 0.131171i \(0.0418736\pi\)
−0.382083 + 0.924128i \(0.624793\pi\)
\(908\) 0 0
\(909\) −91.2387 −3.02620
\(910\) 0 0
\(911\) 35.5211 1.17686 0.588432 0.808546i \(-0.299745\pi\)
0.588432 + 0.808546i \(0.299745\pi\)
\(912\) 0 0
\(913\) 7.24254 + 12.5444i 0.239693 + 0.415161i
\(914\) 0 0
\(915\) 19.7542i 0.653054i
\(916\) 0 0
\(917\) −7.51168 4.33687i −0.248057 0.143216i
\(918\) 0 0
\(919\) 10.9667 18.9949i 0.361758 0.626583i −0.626492 0.779428i \(-0.715510\pi\)
0.988250 + 0.152844i \(0.0488434\pi\)
\(920\) 0 0
\(921\) 20.3940 11.7745i 0.672005 0.387982i
\(922\) 0 0
\(923\) −2.18521 + 18.6142i −0.0719272 + 0.612693i
\(924\) 0 0
\(925\) −41.9082 + 24.1957i −1.37793 + 0.795549i
\(926\) 0 0
\(927\) −12.5135 + 21.6740i −0.410997 + 0.711867i
\(928\) 0 0
\(929\) −13.4383 7.75858i −0.440895 0.254551i 0.263082 0.964773i \(-0.415261\pi\)
−0.703977 + 0.710223i \(0.748594\pi\)
\(930\) 0 0
\(931\) 2.49115i 0.0816443i
\(932\) 0 0
\(933\) 20.6602 + 35.7845i 0.676385 + 1.17153i
\(934\) 0 0
\(935\) 1.64212 0.0537031
\(936\) 0 0
\(937\) 40.8110 1.33324 0.666618 0.745399i \(-0.267741\pi\)
0.666618 + 0.745399i \(0.267741\pi\)
\(938\) 0 0
\(939\) −24.2067 41.9273i −0.789957 1.36825i
\(940\) 0 0
\(941\) 51.5936i 1.68190i 0.541109 + 0.840952i \(0.318004\pi\)
−0.541109 + 0.840952i \(0.681996\pi\)
\(942\) 0 0
\(943\) 33.3186 + 19.2365i 1.08500 + 0.626427i
\(944\) 0 0
\(945\) 1.45546 2.52092i 0.0473460 0.0820057i
\(946\) 0 0
\(947\) 4.31462 2.49105i 0.140206 0.0809482i −0.428256 0.903657i \(-0.640872\pi\)
0.568462 + 0.822709i \(0.307539\pi\)
\(948\) 0 0
\(949\) 39.2054 16.8881i 1.27266 0.548210i
\(950\) 0 0
\(951\) 34.2608 19.7805i 1.11098 0.641426i
\(952\) 0 0
\(953\) 15.1163 26.1822i 0.489664 0.848123i −0.510265 0.860017i \(-0.670453\pi\)
0.999929 + 0.0118941i \(0.00378608\pi\)
\(954\) 0 0
\(955\) 5.50413 + 3.17781i 0.178110 + 0.102832i
\(956\) 0 0
\(957\) 24.0029i 0.775904i
\(958\) 0 0
\(959\) 4.25892 + 7.37667i 0.137528 + 0.238205i
\(960\) 0 0
\(961\) −2.56939 −0.0828835
\(962\) 0 0
\(963\) −30.6306 −0.987058
\(964\) 0 0
\(965\) 3.01873 + 5.22860i 0.0971764 + 0.168315i
\(966\) 0 0
\(967\) 29.9990i 0.964703i −0.875978 0.482352i \(-0.839783\pi\)
0.875978 0.482352i \(-0.160217\pi\)
\(968\) 0 0
\(969\) 11.8772 + 6.85731i 0.381551 + 0.220288i
\(970\) 0 0
\(971\) 22.0620 38.2125i 0.708003 1.22630i −0.257594 0.966253i \(-0.582930\pi\)
0.965597 0.260044i \(-0.0837371\pi\)
\(972\) 0 0
\(973\) −4.36342 + 2.51922i −0.139885 + 0.0807625i
\(974\) 0 0
\(975\) 28.8016 38.6465i 0.922391 1.23768i
\(976\) 0 0
\(977\) −12.9925 + 7.50121i −0.415666 + 0.239985i −0.693221 0.720725i \(-0.743809\pi\)
0.277555 + 0.960710i \(0.410476\pi\)
\(978\) 0 0
\(979\) 9.76037 16.9055i 0.311943 0.540301i
\(980\) 0 0
\(981\) 51.3577 + 29.6514i 1.63973 + 0.946696i
\(982\) 0 0
\(983\) 6.01856i 0.191962i 0.995383 + 0.0959812i \(0.0305989\pi\)
−0.995383 + 0.0959812i \(0.969401\pi\)
\(984\) 0 0
\(985\) 0.467358 + 0.809488i 0.0148913 + 0.0257924i
\(986\) 0 0
\(987\) 12.7647 0.406306
\(988\) 0 0
\(989\) 9.11600 0.289872
\(990\) 0 0
\(991\) 16.9200 + 29.3063i 0.537482 + 0.930946i 0.999039 + 0.0438356i \(0.0139578\pi\)
−0.461557 + 0.887111i \(0.652709\pi\)
\(992\) 0 0
\(993\) 19.5542i 0.620535i
\(994\) 0 0
\(995\) 2.96315 + 1.71078i 0.0939382 + 0.0542353i
\(996\) 0 0
\(997\) −13.1608 + 22.7951i −0.416805 + 0.721928i −0.995616 0.0935340i \(-0.970184\pi\)
0.578811 + 0.815462i \(0.303517\pi\)
\(998\) 0 0
\(999\) −49.6904 + 28.6888i −1.57213 + 0.907672i
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 1456.2.cc.c.673.1 12
4.3 odd 2 91.2.q.a.36.2 12
12.11 even 2 819.2.ct.a.127.5 12
13.4 even 6 inner 1456.2.cc.c.225.1 12
28.3 even 6 637.2.k.g.569.2 12
28.11 odd 6 637.2.k.h.569.2 12
28.19 even 6 637.2.u.i.361.5 12
28.23 odd 6 637.2.u.h.361.5 12
28.27 even 2 637.2.q.h.491.2 12
52.3 odd 6 1183.2.c.i.337.3 12
52.11 even 12 1183.2.a.m.1.6 6
52.15 even 12 1183.2.a.p.1.1 6
52.23 odd 6 1183.2.c.i.337.10 12
52.43 odd 6 91.2.q.a.43.2 yes 12
156.95 even 6 819.2.ct.a.316.5 12
364.95 odd 6 637.2.u.h.30.5 12
364.167 odd 12 8281.2.a.by.1.6 6
364.199 even 6 637.2.u.i.30.5 12
364.223 odd 12 8281.2.a.ch.1.1 6
364.251 even 6 637.2.q.h.589.2 12
364.303 odd 6 637.2.k.h.459.5 12
364.355 even 6 637.2.k.g.459.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.q.a.36.2 12 4.3 odd 2
91.2.q.a.43.2 yes 12 52.43 odd 6
637.2.k.g.459.5 12 364.355 even 6
637.2.k.g.569.2 12 28.3 even 6
637.2.k.h.459.5 12 364.303 odd 6
637.2.k.h.569.2 12 28.11 odd 6
637.2.q.h.491.2 12 28.27 even 2
637.2.q.h.589.2 12 364.251 even 6
637.2.u.h.30.5 12 364.95 odd 6
637.2.u.h.361.5 12 28.23 odd 6
637.2.u.i.30.5 12 364.199 even 6
637.2.u.i.361.5 12 28.19 even 6
819.2.ct.a.127.5 12 12.11 even 2
819.2.ct.a.316.5 12 156.95 even 6
1183.2.a.m.1.6 6 52.11 even 12
1183.2.a.p.1.1 6 52.15 even 12
1183.2.c.i.337.3 12 52.3 odd 6
1183.2.c.i.337.10 12 52.23 odd 6
1456.2.cc.c.225.1 12 13.4 even 6 inner
1456.2.cc.c.673.1 12 1.1 even 1 trivial
8281.2.a.by.1.6 6 364.167 odd 12
8281.2.a.ch.1.1 6 364.223 odd 12