Properties

Label 1584.2.cd.c.305.4
Level $1584$
Weight $2$
Character 1584.305
Analytic conductor $12.648$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1584,2,Mod(17,1584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1584, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1584.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1584.cd (of order \(10\), degree \(4\), 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: \(12.6483036802\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 305.4
Root \(0.752864 - 0.902863i\) of defining polynomial
Character \(\chi\) \(=\) 1584.305
Dual form 1584.2.cd.c.161.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.23109 - 3.07083i) q^{5} +(0.349790 + 0.113654i) q^{7} +O(q^{10})\) \(q+(2.23109 - 3.07083i) q^{5} +(0.349790 + 0.113654i) q^{7} +(-2.97756 - 1.46086i) q^{11} +(-0.557375 - 0.767161i) q^{13} +(-2.77873 - 2.01886i) q^{17} +(-4.05368 + 1.31712i) q^{19} -4.96800i q^{23} +(-2.90717 - 8.94734i) q^{25} +(0.767418 - 2.36187i) q^{29} +(2.84281 - 2.06543i) q^{31} +(1.12943 - 0.820576i) q^{35} +(-2.21947 + 6.83082i) q^{37} +(0.840249 + 2.58602i) q^{41} -1.88749i q^{43} +(0.0195991 - 0.00636813i) q^{47} +(-5.55368 - 4.03499i) q^{49} +(3.25941 + 4.48619i) q^{53} +(-11.1293 + 5.88428i) q^{55} +(6.29998 + 2.04699i) q^{59} +(-5.73238 + 7.88994i) q^{61} -3.59938 q^{65} -4.46351 q^{67} +(6.06985 - 8.35443i) q^{71} +(-4.18072 - 1.35840i) q^{73} +(-0.875490 - 0.849407i) q^{77} +(-6.42867 - 8.84831i) q^{79} +(7.42940 + 5.39778i) q^{83} +(-12.3992 + 4.02874i) q^{85} -3.04837i q^{89} +(-0.107774 - 0.331693i) q^{91} +(-4.99948 + 15.3868i) q^{95} +(12.2055 - 8.86782i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{25} - 16 q^{31} - 12 q^{37} - 24 q^{49} - 16 q^{55} - 96 q^{67} - 20 q^{73} - 100 q^{85} + 72 q^{91} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.23109 3.07083i 0.997774 1.37332i 0.0710932 0.997470i \(-0.477351\pi\)
0.926681 0.375849i \(-0.122649\pi\)
\(6\) 0 0
\(7\) 0.349790 + 0.113654i 0.132208 + 0.0429571i 0.374374 0.927278i \(-0.377858\pi\)
−0.242165 + 0.970235i \(0.577858\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.97756 1.46086i −0.897769 0.440467i
\(12\) 0 0
\(13\) −0.557375 0.767161i −0.154588 0.212772i 0.724698 0.689067i \(-0.241979\pi\)
−0.879286 + 0.476295i \(0.841979\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.77873 2.01886i −0.673940 0.489646i 0.197402 0.980323i \(-0.436750\pi\)
−0.871342 + 0.490676i \(0.836750\pi\)
\(18\) 0 0
\(19\) −4.05368 + 1.31712i −0.929979 + 0.302168i −0.734554 0.678550i \(-0.762609\pi\)
−0.195425 + 0.980719i \(0.562609\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.96800i 1.03590i −0.855411 0.517950i \(-0.826695\pi\)
0.855411 0.517950i \(-0.173305\pi\)
\(24\) 0 0
\(25\) −2.90717 8.94734i −0.581433 1.78947i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.767418 2.36187i 0.142506 0.438588i −0.854176 0.519984i \(-0.825938\pi\)
0.996682 + 0.0813958i \(0.0259378\pi\)
\(30\) 0 0
\(31\) 2.84281 2.06543i 0.510585 0.370961i −0.302461 0.953162i \(-0.597808\pi\)
0.813045 + 0.582200i \(0.197808\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.12943 0.820576i 0.190908 0.138703i
\(36\) 0 0
\(37\) −2.21947 + 6.83082i −0.364878 + 1.12298i 0.585179 + 0.810904i \(0.301024\pi\)
−0.950057 + 0.312075i \(0.898976\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.840249 + 2.58602i 0.131225 + 0.403869i 0.994984 0.100037i \(-0.0318960\pi\)
−0.863759 + 0.503905i \(0.831896\pi\)
\(42\) 0 0
\(43\) 1.88749i 0.287839i −0.989589 0.143919i \(-0.954029\pi\)
0.989589 0.143919i \(-0.0459706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.0195991 0.00636813i 0.00285882 0.000928888i −0.307587 0.951520i \(-0.599522\pi\)
0.310446 + 0.950591i \(0.399522\pi\)
\(48\) 0 0
\(49\) −5.55368 4.03499i −0.793383 0.576427i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.25941 + 4.48619i 0.447714 + 0.616226i 0.971905 0.235375i \(-0.0756319\pi\)
−0.524190 + 0.851601i \(0.675632\pi\)
\(54\) 0 0
\(55\) −11.1293 + 5.88428i −1.50067 + 0.793436i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.29998 + 2.04699i 0.820187 + 0.266495i 0.688906 0.724850i \(-0.258091\pi\)
0.131281 + 0.991345i \(0.458091\pi\)
\(60\) 0 0
\(61\) −5.73238 + 7.88994i −0.733956 + 1.01020i 0.264988 + 0.964252i \(0.414632\pi\)
−0.998944 + 0.0459514i \(0.985368\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.59938 −0.446448
\(66\) 0 0
\(67\) −4.46351 −0.545305 −0.272652 0.962113i \(-0.587901\pi\)
−0.272652 + 0.962113i \(0.587901\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.06985 8.35443i 0.720359 0.991489i −0.279153 0.960247i \(-0.590054\pi\)
0.999512 0.0312420i \(-0.00994626\pi\)
\(72\) 0 0
\(73\) −4.18072 1.35840i −0.489317 0.158989i 0.0539588 0.998543i \(-0.482816\pi\)
−0.543275 + 0.839555i \(0.682816\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.875490 0.849407i −0.0997714 0.0967989i
\(78\) 0 0
\(79\) −6.42867 8.84831i −0.723282 0.995513i −0.999408 0.0343911i \(-0.989051\pi\)
0.276126 0.961121i \(-0.410949\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.42940 + 5.39778i 0.815483 + 0.592483i 0.915415 0.402511i \(-0.131863\pi\)
−0.0999323 + 0.994994i \(0.531863\pi\)
\(84\) 0 0
\(85\) −12.3992 + 4.02874i −1.34488 + 0.436978i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.04837i 0.323127i −0.986862 0.161563i \(-0.948346\pi\)
0.986862 0.161563i \(-0.0516536\pi\)
\(90\) 0 0
\(91\) −0.107774 0.331693i −0.0112978 0.0347709i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.99948 + 15.3868i −0.512935 + 1.57865i
\(96\) 0 0
\(97\) 12.2055 8.86782i 1.23928 0.900391i 0.241732 0.970343i \(-0.422285\pi\)
0.997550 + 0.0699519i \(0.0222846\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.1253 10.2626i 1.40552 1.02117i 0.411562 0.911382i \(-0.364983\pi\)
0.993955 0.109787i \(-0.0350168\pi\)
\(102\) 0 0
\(103\) 2.87868 8.85967i 0.283645 0.872969i −0.703157 0.711035i \(-0.748227\pi\)
0.986802 0.161934i \(-0.0517733\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.76675 14.6705i −0.460819 1.41825i −0.864166 0.503207i \(-0.832153\pi\)
0.403347 0.915047i \(-0.367847\pi\)
\(108\) 0 0
\(109\) 10.6286i 1.01803i 0.860757 + 0.509016i \(0.169991\pi\)
−0.860757 + 0.509016i \(0.830009\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.995774 0.323547i 0.0936746 0.0304367i −0.261805 0.965121i \(-0.584318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(114\) 0 0
\(115\) −15.2559 11.0841i −1.42262 1.03359i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.742521 1.02199i −0.0680668 0.0936859i
\(120\) 0 0
\(121\) 6.73176 + 8.69962i 0.611978 + 0.790875i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −15.9120 5.17013i −1.42321 0.462430i
\(126\) 0 0
\(127\) −10.3127 + 14.1942i −0.915101 + 1.25953i 0.0502930 + 0.998735i \(0.483984\pi\)
−0.965394 + 0.260794i \(0.916016\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.8890 −1.21349 −0.606744 0.794897i \(-0.707525\pi\)
−0.606744 + 0.794897i \(0.707525\pi\)
\(132\) 0 0
\(133\) −1.56764 −0.135931
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.609963 0.839541i 0.0521126 0.0717269i −0.782165 0.623071i \(-0.785885\pi\)
0.834278 + 0.551344i \(0.185885\pi\)
\(138\) 0 0
\(139\) 18.8643 + 6.12938i 1.60005 + 0.519887i 0.967119 0.254324i \(-0.0818528\pi\)
0.632928 + 0.774210i \(0.281853\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.538902 + 3.09852i 0.0450652 + 0.259111i
\(144\) 0 0
\(145\) −5.54073 7.62616i −0.460133 0.633318i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.50125 4.72344i −0.532603 0.386959i 0.288727 0.957411i \(-0.406768\pi\)
−0.821331 + 0.570452i \(0.806768\pi\)
\(150\) 0 0
\(151\) 9.58050 3.11289i 0.779650 0.253324i 0.107959 0.994155i \(-0.465568\pi\)
0.671691 + 0.740832i \(0.265568\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 13.3380i 1.07133i
\(156\) 0 0
\(157\) −5.00062 15.3903i −0.399093 1.22828i −0.925728 0.378190i \(-0.876547\pi\)
0.526635 0.850091i \(-0.323453\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.564633 1.73776i 0.0444993 0.136955i
\(162\) 0 0
\(163\) 3.45923 2.51328i 0.270948 0.196855i −0.444012 0.896021i \(-0.646445\pi\)
0.714959 + 0.699166i \(0.246445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.8478 10.7875i 1.14896 0.834765i 0.160614 0.987017i \(-0.448653\pi\)
0.988342 + 0.152252i \(0.0486526\pi\)
\(168\) 0 0
\(169\) 3.73935 11.5085i 0.287642 0.885272i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.91046 5.87980i −0.145250 0.447033i 0.851793 0.523878i \(-0.175515\pi\)
−0.997043 + 0.0768452i \(0.975515\pi\)
\(174\) 0 0
\(175\) 3.46010i 0.261559i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.77414 2.85089i 0.655810 0.213086i 0.0378357 0.999284i \(-0.487954\pi\)
0.617974 + 0.786198i \(0.287954\pi\)
\(180\) 0 0
\(181\) 3.38790 + 2.46145i 0.251821 + 0.182958i 0.706533 0.707680i \(-0.250258\pi\)
−0.454713 + 0.890638i \(0.650258\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.0245 + 22.0558i 1.17814 + 1.62157i
\(186\) 0 0
\(187\) 5.32455 + 10.0706i 0.389370 + 0.736438i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.1951 6.23688i −1.38891 0.451285i −0.483323 0.875442i \(-0.660570\pi\)
−0.905589 + 0.424157i \(0.860570\pi\)
\(192\) 0 0
\(193\) 4.71049 6.48343i 0.339068 0.466687i −0.605101 0.796149i \(-0.706867\pi\)
0.944169 + 0.329462i \(0.106867\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.1710 −1.50837 −0.754187 0.656659i \(-0.771969\pi\)
−0.754187 + 0.656659i \(0.771969\pi\)
\(198\) 0 0
\(199\) 10.5160 0.745457 0.372729 0.927940i \(-0.378422\pi\)
0.372729 + 0.927940i \(0.378422\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.536871 0.738940i 0.0376810 0.0518634i
\(204\) 0 0
\(205\) 9.81591 + 3.18938i 0.685573 + 0.222756i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.9942 + 2.00007i 0.968001 + 0.138347i
\(210\) 0 0
\(211\) 2.17376 + 2.99193i 0.149648 + 0.205973i 0.877259 0.480017i \(-0.159369\pi\)
−0.727611 + 0.685990i \(0.759369\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.79615 4.21115i −0.395294 0.287198i
\(216\) 0 0
\(217\) 1.22913 0.399369i 0.0834390 0.0271110i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.25699i 0.219089i
\(222\) 0 0
\(223\) 5.53051 + 17.0212i 0.370351 + 1.13982i 0.946562 + 0.322521i \(0.104530\pi\)
−0.576212 + 0.817301i \(0.695470\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.37022 + 16.5278i −0.356434 + 1.09699i 0.598740 + 0.800944i \(0.295668\pi\)
−0.955173 + 0.296047i \(0.904332\pi\)
\(228\) 0 0
\(229\) −3.45989 + 2.51375i −0.228636 + 0.166114i −0.696205 0.717843i \(-0.745130\pi\)
0.467570 + 0.883956i \(0.345130\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.9171 13.7441i 1.23930 0.900404i 0.241748 0.970339i \(-0.422279\pi\)
0.997552 + 0.0699355i \(0.0222793\pi\)
\(234\) 0 0
\(235\) 0.0241719 0.0743935i 0.00157680 0.00485289i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.17902 + 19.0171i 0.399687 + 1.23011i 0.925251 + 0.379356i \(0.123855\pi\)
−0.525563 + 0.850754i \(0.676145\pi\)
\(240\) 0 0
\(241\) 2.32570i 0.149811i 0.997191 + 0.0749057i \(0.0238656\pi\)
−0.997191 + 0.0749057i \(0.976134\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −24.7815 + 8.05201i −1.58323 + 0.514424i
\(246\) 0 0
\(247\) 3.26987 + 2.37570i 0.208057 + 0.151162i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.88681 + 2.59698i 0.119095 + 0.163920i 0.864402 0.502801i \(-0.167697\pi\)
−0.745307 + 0.666721i \(0.767697\pi\)
\(252\) 0 0
\(253\) −7.25758 + 14.7925i −0.456280 + 0.929999i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.1597 + 5.57552i 1.07039 + 0.347792i 0.790641 0.612281i \(-0.209748\pi\)
0.279752 + 0.960072i \(0.409748\pi\)
\(258\) 0 0
\(259\) −1.55270 + 2.13710i −0.0964798 + 0.132793i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.05966 0.127004 0.0635019 0.997982i \(-0.479773\pi\)
0.0635019 + 0.997982i \(0.479773\pi\)
\(264\) 0 0
\(265\) 21.0484 1.29299
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.69280 + 2.32994i −0.103212 + 0.142059i −0.857499 0.514486i \(-0.827983\pi\)
0.754287 + 0.656545i \(0.227983\pi\)
\(270\) 0 0
\(271\) −9.67272 3.14286i −0.587576 0.190915i 0.000115761 1.00000i \(-0.499963\pi\)
−0.587692 + 0.809085i \(0.699963\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.41457 + 30.8882i −0.266208 + 1.86263i
\(276\) 0 0
\(277\) 4.29714 + 5.91450i 0.258190 + 0.355368i 0.918358 0.395750i \(-0.129515\pi\)
−0.660169 + 0.751117i \(0.729515\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.86513 + 1.35510i 0.111264 + 0.0808384i 0.642026 0.766683i \(-0.278094\pi\)
−0.530762 + 0.847521i \(0.678094\pi\)
\(282\) 0 0
\(283\) −23.2370 + 7.55016i −1.38130 + 0.448810i −0.903095 0.429441i \(-0.858711\pi\)
−0.478200 + 0.878251i \(0.658711\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00006i 0.0590318i
\(288\) 0 0
\(289\) −1.60777 4.94822i −0.0945749 0.291072i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.29456 + 10.1396i −0.192470 + 0.592362i 0.807527 + 0.589831i \(0.200806\pi\)
−0.999997 + 0.00253112i \(0.999194\pi\)
\(294\) 0 0
\(295\) 20.3418 14.7792i 1.18434 0.860476i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.81126 + 2.76904i −0.220411 + 0.160138i
\(300\) 0 0
\(301\) 0.214520 0.660224i 0.0123647 0.0380547i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.4392 + 35.2063i 0.655009 + 2.01591i
\(306\) 0 0
\(307\) 4.56848i 0.260737i −0.991466 0.130369i \(-0.958384\pi\)
0.991466 0.130369i \(-0.0416160\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.92302 + 2.89926i −0.505978 + 0.164402i −0.550872 0.834590i \(-0.685705\pi\)
0.0448944 + 0.998992i \(0.485705\pi\)
\(312\) 0 0
\(313\) 4.05804 + 2.94834i 0.229374 + 0.166650i 0.696536 0.717522i \(-0.254724\pi\)
−0.467162 + 0.884172i \(0.654724\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.86310 9.44624i −0.385470 0.530554i 0.571553 0.820565i \(-0.306341\pi\)
−0.957023 + 0.290011i \(0.906341\pi\)
\(318\) 0 0
\(319\) −5.73541 + 5.91153i −0.321121 + 0.330982i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.9232 + 4.52391i 0.774706 + 0.251717i
\(324\) 0 0
\(325\) −5.24366 + 7.21728i −0.290866 + 0.400343i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.00757934 0.000417863
\(330\) 0 0
\(331\) 29.0516 1.59682 0.798411 0.602113i \(-0.205674\pi\)
0.798411 + 0.602113i \(0.205674\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.95850 + 13.7067i −0.544091 + 0.748877i
\(336\) 0 0
\(337\) 27.4668 + 8.92451i 1.49621 + 0.486149i 0.938911 0.344161i \(-0.111837\pi\)
0.557302 + 0.830310i \(0.311837\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −11.4820 + 1.99697i −0.621783 + 0.108142i
\(342\) 0 0
\(343\) −2.99731 4.12544i −0.161840 0.222753i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.9141 + 13.0153i 0.961678 + 0.698700i 0.953540 0.301267i \(-0.0974096\pi\)
0.00813801 + 0.999967i \(0.497410\pi\)
\(348\) 0 0
\(349\) −14.8959 + 4.83997i −0.797359 + 0.259078i −0.679235 0.733921i \(-0.737688\pi\)
−0.118125 + 0.992999i \(0.537688\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.1636i 0.594180i −0.954850 0.297090i \(-0.903984\pi\)
0.954850 0.297090i \(-0.0960161\pi\)
\(354\) 0 0
\(355\) −12.1127 37.2790i −0.642874 1.97856i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.11867 15.7536i 0.270153 0.831445i −0.720308 0.693654i \(-0.756000\pi\)
0.990461 0.137791i \(-0.0440002\pi\)
\(360\) 0 0
\(361\) −0.673787 + 0.489535i −0.0354625 + 0.0257650i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.4990 + 9.80759i −0.706569 + 0.513353i
\(366\) 0 0
\(367\) −3.03154 + 9.33012i −0.158245 + 0.487028i −0.998475 0.0552016i \(-0.982420\pi\)
0.840230 + 0.542230i \(0.182420\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.630238 + 1.93967i 0.0327203 + 0.100703i
\(372\) 0 0
\(373\) 12.6350i 0.654216i −0.944987 0.327108i \(-0.893926\pi\)
0.944987 0.327108i \(-0.106074\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.23967 + 0.727714i −0.115349 + 0.0374792i
\(378\) 0 0
\(379\) 28.3981 + 20.6325i 1.45871 + 1.05982i 0.983695 + 0.179843i \(0.0575589\pi\)
0.475019 + 0.879975i \(0.342441\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.2760 + 18.2729i 0.678374 + 0.933701i 0.999913 0.0131951i \(-0.00420024\pi\)
−0.321539 + 0.946896i \(0.604200\pi\)
\(384\) 0 0
\(385\) −4.56169 + 0.793379i −0.232485 + 0.0404344i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.99867 2.27401i −0.354847 0.115297i 0.126169 0.992009i \(-0.459732\pi\)
−0.481015 + 0.876712i \(0.659732\pi\)
\(390\) 0 0
\(391\) −10.0297 + 13.8047i −0.507225 + 0.698135i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −41.5146 −2.08883
\(396\) 0 0
\(397\) 16.7327 0.839790 0.419895 0.907573i \(-0.362067\pi\)
0.419895 + 0.907573i \(0.362067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.98209 4.10449i 0.148918 0.204968i −0.728040 0.685534i \(-0.759569\pi\)
0.876959 + 0.480566i \(0.159569\pi\)
\(402\) 0 0
\(403\) −3.16903 1.02968i −0.157860 0.0512920i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.5875 17.0968i 0.822211 0.847459i
\(408\) 0 0
\(409\) 19.0135 + 26.1698i 0.940156 + 1.29401i 0.955764 + 0.294136i \(0.0950319\pi\)
−0.0156071 + 0.999878i \(0.504968\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.97102 + 1.43203i 0.0969877 + 0.0704657i
\(414\) 0 0
\(415\) 33.1513 10.7715i 1.62734 0.528753i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 27.9075i 1.36337i −0.731645 0.681686i \(-0.761247\pi\)
0.731645 0.681686i \(-0.238753\pi\)
\(420\) 0 0
\(421\) 8.25960 + 25.4204i 0.402549 + 1.23892i 0.922925 + 0.384980i \(0.125792\pi\)
−0.520376 + 0.853937i \(0.674208\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −9.98523 + 30.7314i −0.484355 + 1.49069i
\(426\) 0 0
\(427\) −2.90185 + 2.10832i −0.140430 + 0.102029i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.579760 + 0.421220i −0.0279260 + 0.0202895i −0.601661 0.798752i \(-0.705494\pi\)
0.573735 + 0.819041i \(0.305494\pi\)
\(432\) 0 0
\(433\) 3.53281 10.8729i 0.169776 0.522517i −0.829580 0.558387i \(-0.811420\pi\)
0.999356 + 0.0358703i \(0.0114203\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.54346 + 20.1387i 0.313016 + 0.963365i
\(438\) 0 0
\(439\) 8.57525i 0.409274i −0.978838 0.204637i \(-0.934399\pi\)
0.978838 0.204637i \(-0.0656015\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.2928 4.31910i 0.631561 0.205207i 0.0242947 0.999705i \(-0.492266\pi\)
0.607267 + 0.794498i \(0.292266\pi\)
\(444\) 0 0
\(445\) −9.36104 6.80119i −0.443756 0.322407i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.7008 17.4811i −0.599388 0.824986i 0.396264 0.918136i \(-0.370306\pi\)
−0.995652 + 0.0931501i \(0.970306\pi\)
\(450\) 0 0
\(451\) 1.27593 8.92753i 0.0600811 0.420381i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.25903 0.409083i −0.0590241 0.0191781i
\(456\) 0 0
\(457\) 5.76327 7.93246i 0.269594 0.371065i −0.652658 0.757652i \(-0.726346\pi\)
0.922253 + 0.386587i \(0.126346\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.3585 0.575595 0.287797 0.957691i \(-0.407077\pi\)
0.287797 + 0.957691i \(0.407077\pi\)
\(462\) 0 0
\(463\) 33.8717 1.57415 0.787076 0.616856i \(-0.211594\pi\)
0.787076 + 0.616856i \(0.211594\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.24994 12.7315i 0.428036 0.589142i −0.539465 0.842008i \(-0.681373\pi\)
0.967501 + 0.252867i \(0.0813734\pi\)
\(468\) 0 0
\(469\) −1.56129 0.507295i −0.0720939 0.0234247i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.75736 + 5.62011i −0.126783 + 0.258413i
\(474\) 0 0
\(475\) 23.5695 + 32.4406i 1.08144 + 1.48848i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.0877 + 7.32913i 0.460918 + 0.334877i 0.793891 0.608060i \(-0.208052\pi\)
−0.332973 + 0.942936i \(0.608052\pi\)
\(480\) 0 0
\(481\) 6.47741 2.10464i 0.295344 0.0959632i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 57.2660i 2.60032i
\(486\) 0 0
\(487\) 2.14955 + 6.61563i 0.0974054 + 0.299783i 0.987873 0.155263i \(-0.0496227\pi\)
−0.890468 + 0.455046i \(0.849623\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.0932984 0.287143i 0.00421050 0.0129586i −0.948929 0.315489i \(-0.897831\pi\)
0.953140 + 0.302531i \(0.0978314\pi\)
\(492\) 0 0
\(493\) −6.90074 + 5.01368i −0.310794 + 0.225805i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.07269 2.23244i 0.137829 0.100139i
\(498\) 0 0
\(499\) −2.04468 + 6.29287i −0.0915323 + 0.281707i −0.986334 0.164755i \(-0.947317\pi\)
0.894802 + 0.446463i \(0.147317\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.19188 9.82361i −0.142319 0.438013i 0.854337 0.519719i \(-0.173963\pi\)
−0.996657 + 0.0817056i \(0.973963\pi\)
\(504\) 0 0
\(505\) 66.2732i 2.94912i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.3302 + 6.28078i −0.856798 + 0.278390i −0.704291 0.709912i \(-0.748735\pi\)
−0.152507 + 0.988302i \(0.548735\pi\)
\(510\) 0 0
\(511\) −1.30799 0.950310i −0.0578621 0.0420392i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −20.7840 28.6067i −0.915851 1.26056i
\(516\) 0 0
\(517\) −0.0676605 0.00967009i −0.00297571 0.000425290i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −11.4110 3.70764i −0.499923 0.162435i 0.0481915 0.998838i \(-0.484654\pi\)
−0.548115 + 0.836403i \(0.684654\pi\)
\(522\) 0 0
\(523\) 22.2395 30.6101i 0.972466 1.33848i 0.0316743 0.999498i \(-0.489916\pi\)
0.940791 0.338986i \(-0.110084\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0692 −0.525743
\(528\) 0 0
\(529\) −1.68106 −0.0730898
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.51556 2.08599i 0.0656462 0.0903542i
\(534\) 0 0
\(535\) −55.6858 18.0934i −2.40751 0.782247i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.6419 + 20.1276i 0.458378 + 0.866957i
\(540\) 0 0
\(541\) 21.6491 + 29.7975i 0.930769 + 1.28109i 0.959559 + 0.281508i \(0.0908345\pi\)
−0.0287904 + 0.999585i \(0.509166\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 32.6386 + 23.7133i 1.39808 + 1.01577i
\(546\) 0 0
\(547\) −34.8795 + 11.3330i −1.49134 + 0.484565i −0.937478 0.348044i \(-0.886846\pi\)
−0.553860 + 0.832610i \(0.686846\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.5851i 0.450939i
\(552\) 0 0
\(553\) −1.24304 3.82570i −0.0528596 0.162685i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.30765 7.10223i 0.0977784 0.300931i −0.890189 0.455591i \(-0.849428\pi\)
0.987968 + 0.154660i \(0.0494281\pi\)
\(558\) 0 0
\(559\) −1.44800 + 1.05204i −0.0612441 + 0.0444964i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.16715 2.30107i 0.133479 0.0969784i −0.519042 0.854749i \(-0.673711\pi\)
0.652521 + 0.757770i \(0.273711\pi\)
\(564\) 0 0
\(565\) 1.22811 3.77972i 0.0516668 0.159014i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.6410 35.8274i −0.488017 1.50196i −0.827563 0.561373i \(-0.810273\pi\)
0.339546 0.940590i \(-0.389727\pi\)
\(570\) 0 0
\(571\) 30.3411i 1.26974i 0.772621 + 0.634868i \(0.218946\pi\)
−0.772621 + 0.634868i \(0.781054\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −44.4504 + 14.4428i −1.85371 + 0.602307i
\(576\) 0 0
\(577\) 2.68563 + 1.95122i 0.111804 + 0.0812305i 0.642282 0.766468i \(-0.277988\pi\)
−0.530478 + 0.847699i \(0.677988\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.98526 + 2.73247i 0.0823623 + 0.113362i
\(582\) 0 0
\(583\) −3.15138 18.1195i −0.130517 0.750432i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −38.9053 12.6411i −1.60579 0.521754i −0.637263 0.770647i \(-0.719933\pi\)
−0.968531 + 0.248893i \(0.919933\pi\)
\(588\) 0 0
\(589\) −8.80345 + 12.1169i −0.362740 + 0.499269i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.62924 −0.190100 −0.0950500 0.995472i \(-0.530301\pi\)
−0.0950500 + 0.995472i \(0.530301\pi\)
\(594\) 0 0
\(595\) −4.79500 −0.196576
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.9738 + 24.7388i −0.734390 + 1.01080i 0.264532 + 0.964377i \(0.414782\pi\)
−0.998922 + 0.0464237i \(0.985218\pi\)
\(600\) 0 0
\(601\) −29.7015 9.65061i −1.21155 0.393657i −0.367552 0.930003i \(-0.619804\pi\)
−0.843998 + 0.536346i \(0.819804\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 41.7343 1.26245i 1.69674 0.0513259i
\(606\) 0 0
\(607\) 18.0842 + 24.8908i 0.734016 + 1.01029i 0.998941 + 0.0460167i \(0.0146527\pi\)
−0.264925 + 0.964269i \(0.585347\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.0158094 0.0114862i −0.000639581 0.000464683i
\(612\) 0 0
\(613\) 24.2244 7.87097i 0.978413 0.317906i 0.224205 0.974542i \(-0.428021\pi\)
0.754207 + 0.656636i \(0.228021\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.20930i 0.169460i 0.996404 + 0.0847300i \(0.0270028\pi\)
−0.996404 + 0.0847300i \(0.972997\pi\)
\(618\) 0 0
\(619\) −11.8981 36.6187i −0.478227 1.47183i −0.841556 0.540170i \(-0.818360\pi\)
0.363329 0.931661i \(-0.381640\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0.346459 1.06629i 0.0138806 0.0427200i
\(624\) 0 0
\(625\) −13.3225 + 9.67935i −0.532899 + 0.387174i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 19.9578 14.5002i 0.795769 0.578160i
\(630\) 0 0
\(631\) 3.65655 11.2537i 0.145565 0.448003i −0.851518 0.524325i \(-0.824318\pi\)
0.997083 + 0.0763220i \(0.0243177\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.5794 + 63.3370i 0.816670 + 2.51345i
\(636\) 0 0
\(637\) 6.50957i 0.257918i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −34.1810 + 11.1061i −1.35007 + 0.438664i −0.892716 0.450620i \(-0.851203\pi\)
−0.457354 + 0.889285i \(0.651203\pi\)
\(642\) 0 0
\(643\) 2.63357 + 1.91340i 0.103858 + 0.0754571i 0.638502 0.769620i \(-0.279554\pi\)
−0.534644 + 0.845077i \(0.679554\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.14454 + 9.83361i 0.280881 + 0.386599i 0.926025 0.377461i \(-0.123203\pi\)
−0.645145 + 0.764060i \(0.723203\pi\)
\(648\) 0 0
\(649\) −15.7682 15.2984i −0.618956 0.600516i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.2713 + 7.88623i 0.949811 + 0.308612i 0.742639 0.669692i \(-0.233574\pi\)
0.207172 + 0.978305i \(0.433574\pi\)
\(654\) 0 0
\(655\) −30.9877 + 42.6508i −1.21079 + 1.66651i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27.9441 1.08855 0.544273 0.838908i \(-0.316806\pi\)
0.544273 + 0.838908i \(0.316806\pi\)
\(660\) 0 0
\(661\) −27.3798 −1.06495 −0.532475 0.846446i \(-0.678738\pi\)
−0.532475 + 0.846446i \(0.678738\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.49754 + 4.81395i −0.135629 + 0.186677i
\(666\) 0 0
\(667\) −11.7338 3.81254i −0.454334 0.147622i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 28.5946 15.1186i 1.10388 0.583646i
\(672\) 0 0
\(673\) −7.96377 10.9612i −0.306981 0.422523i 0.627456 0.778652i \(-0.284096\pi\)
−0.934437 + 0.356129i \(0.884096\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.6885 + 21.5700i 1.14102 + 0.829001i 0.987261 0.159107i \(-0.0508616\pi\)
0.153760 + 0.988108i \(0.450862\pi\)
\(678\) 0 0
\(679\) 5.27723 1.71468i 0.202522 0.0658033i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.8347i 1.17986i −0.807455 0.589929i \(-0.799156\pi\)
0.807455 0.589929i \(-0.200844\pi\)
\(684\) 0 0
\(685\) −1.21721 3.74619i −0.0465072 0.143134i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.62492 5.00098i 0.0619045 0.190522i
\(690\) 0 0
\(691\) −38.6062 + 28.0491i −1.46865 + 1.06704i −0.487649 + 0.873040i \(0.662145\pi\)
−0.981002 + 0.193997i \(0.937855\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 60.9102 44.2539i 2.31046 1.67864i
\(696\) 0 0
\(697\) 2.88600 8.88220i 0.109315 0.336437i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.34678 + 13.3780i 0.164176 + 0.505281i 0.998975 0.0452742i \(-0.0144162\pi\)
−0.834799 + 0.550555i \(0.814416\pi\)
\(702\) 0 0
\(703\) 30.6133i 1.15460i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.10727 1.98437i 0.229688 0.0746300i
\(708\) 0 0
\(709\) 7.34115 + 5.33366i 0.275703 + 0.200310i 0.717041 0.697031i \(-0.245496\pi\)
−0.441338 + 0.897341i \(0.645496\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10.2610 14.1231i −0.384279 0.528915i
\(714\) 0 0
\(715\) 10.7174 + 5.25820i 0.400807 + 0.196645i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32.2802 10.4885i −1.20385 0.391155i −0.362674 0.931916i \(-0.618136\pi\)
−0.841176 + 0.540761i \(0.818136\pi\)
\(720\) 0 0
\(721\) 2.01387 2.77186i 0.0750005 0.103229i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −23.3635 −0.867697
\(726\) 0 0
\(727\) −30.8992 −1.14599 −0.572993 0.819560i \(-0.694218\pi\)
−0.572993 + 0.819560i \(0.694218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.81058 + 5.24481i −0.140939 + 0.193986i
\(732\) 0 0
\(733\) −39.5648 12.8554i −1.46136 0.474825i −0.532876 0.846194i \(-0.678889\pi\)
−0.928485 + 0.371369i \(0.878889\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.2904 + 6.52058i 0.489558 + 0.240189i
\(738\) 0 0
\(739\) 8.44687 + 11.6261i 0.310723 + 0.427674i 0.935607 0.353044i \(-0.114854\pi\)
−0.624883 + 0.780718i \(0.714854\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.8649 + 7.89384i 0.398596 + 0.289597i 0.768969 0.639286i \(-0.220770\pi\)
−0.370373 + 0.928883i \(0.620770\pi\)
\(744\) 0 0
\(745\) −29.0098 + 9.42585i −1.06284 + 0.345336i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.67337i 0.207301i
\(750\) 0 0
\(751\) 11.2373 + 34.5847i 0.410054 + 1.26202i 0.916601 + 0.399803i \(0.130922\pi\)
−0.506547 + 0.862212i \(0.669078\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.8158 36.3653i 0.430021 1.32347i
\(756\) 0 0
\(757\) −32.2811 + 23.4536i −1.17328 + 0.852435i −0.991397 0.130887i \(-0.958218\pi\)
−0.181878 + 0.983321i \(0.558218\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.75887 + 2.00443i −0.100009 + 0.0726607i −0.636666 0.771140i \(-0.719687\pi\)
0.536657 + 0.843801i \(0.319687\pi\)
\(762\) 0 0
\(763\) −1.20798 + 3.71777i −0.0437317 + 0.134592i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.94108 5.97403i −0.0700884 0.215710i
\(768\) 0 0
\(769\) 21.4154i 0.772260i 0.922444 + 0.386130i \(0.126188\pi\)
−0.922444 + 0.386130i \(0.873812\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.1302 11.0896i 1.22758 0.398864i 0.377741 0.925911i \(-0.376701\pi\)
0.849836 + 0.527048i \(0.176701\pi\)
\(774\) 0 0
\(775\) −26.7446 19.4311i −0.960694 0.697985i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.81221 9.37620i −0.244073 0.335937i
\(780\) 0 0
\(781\) −30.2780 + 16.0086i −1.08343 + 0.572833i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −58.4180 18.9811i −2.08503 0.677466i
\(786\) 0 0
\(787\) 28.2510 38.8842i 1.00704 1.38607i 0.0861340 0.996284i \(-0.472549\pi\)
0.920905 0.389787i \(-0.127451\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.385085 0.0136920
\(792\) 0 0
\(793\) 9.24793 0.328404
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.55363 + 4.89115i −0.125876 + 0.173253i −0.867304 0.497779i \(-0.834149\pi\)
0.741428 + 0.671033i \(0.234149\pi\)
\(798\) 0 0
\(799\) −0.0673169 0.0218726i −0.00238150 0.000773797i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.4639 + 10.1522i 0.369264 + 0.358263i
\(804\) 0 0
\(805\) −4.07663 5.61099i −0.143682 0.197762i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.01444 + 4.36974i 0.211456 + 0.153632i 0.688472 0.725263i \(-0.258282\pi\)
−0.477016 + 0.878895i \(0.658282\pi\)
\(810\) 0 0
\(811\) −27.7357 + 9.01188i −0.973933 + 0.316450i −0.752402 0.658704i \(-0.771105\pi\)
−0.221530 + 0.975154i \(0.571105\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 16.2301i 0.568514i
\(816\) 0 0
\(817\) 2.48605 + 7.65127i 0.0869758 + 0.267684i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.331389 1.01991i 0.0115656 0.0355952i −0.945107 0.326760i \(-0.894043\pi\)
0.956673 + 0.291165i \(0.0940430\pi\)
\(822\) 0 0
\(823\) −7.23762 + 5.25844i −0.252288 + 0.183298i −0.706740 0.707473i \(-0.749835\pi\)
0.454452 + 0.890771i \(0.349835\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.5608 + 20.7507i −0.993157 + 0.721571i −0.960610 0.277899i \(-0.910362\pi\)
−0.0325471 + 0.999470i \(0.510362\pi\)
\(828\) 0 0
\(829\) 8.36614 25.7483i 0.290568 0.894276i −0.694106 0.719873i \(-0.744200\pi\)
0.984674 0.174404i \(-0.0557998\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.28608 + 22.4243i 0.252448 + 0.776954i
\(834\) 0 0
\(835\) 69.6630i 2.41079i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −35.4106 + 11.5056i −1.22251 + 0.397218i −0.847996 0.530003i \(-0.822191\pi\)
−0.374515 + 0.927221i \(0.622191\pi\)
\(840\) 0 0
\(841\) 18.4720 + 13.4207i 0.636965 + 0.462782i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26.9980 37.1595i −0.928759 1.27833i
\(846\) 0 0
\(847\) 1.36596 + 3.80813i 0.0469349 + 0.130849i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 33.9355 + 11.0263i 1.16329 + 0.377977i
\(852\) 0 0
\(853\) 20.6424 28.4118i 0.706782 0.972802i −0.293078 0.956089i \(-0.594680\pi\)
0.999860 0.0167139i \(-0.00532045\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29.9637 −1.02354 −0.511770 0.859122i \(-0.671010\pi\)
−0.511770 + 0.859122i \(0.671010\pi\)
\(858\) 0 0
\(859\) −17.2684 −0.589189 −0.294595 0.955622i \(-0.595185\pi\)
−0.294595 + 0.955622i \(0.595185\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.4637 28.1659i 0.696593 0.958778i −0.303389 0.952867i \(-0.598118\pi\)
0.999983 0.00591175i \(-0.00188178\pi\)
\(864\) 0 0
\(865\) −22.3183 7.25166i −0.758845 0.246564i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.21560 + 35.7378i 0.210850 + 1.21232i
\(870\) 0 0
\(871\) 2.48785 + 3.42423i 0.0842976 + 0.116026i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.97827 3.61692i −0.168296 0.122274i
\(876\) 0 0
\(877\) −1.88584 + 0.612748i −0.0636804 + 0.0206910i −0.340684 0.940178i \(-0.610659\pi\)
0.277004 + 0.960869i \(0.410659\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.9571i 1.64941i −0.565566 0.824703i \(-0.691342\pi\)
0.565566 0.824703i \(-0.308658\pi\)
\(882\) 0 0
\(883\) −10.7400 33.0542i −0.361429 1.11236i −0.952187 0.305515i \(-0.901171\pi\)
0.590758 0.806848i \(-0.298829\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.47860 13.7837i 0.150377 0.462812i −0.847287 0.531136i \(-0.821765\pi\)
0.997663 + 0.0683244i \(0.0217653\pi\)
\(888\) 0 0
\(889\) −5.22049 + 3.79291i −0.175090 + 0.127210i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.0710609 + 0.0516288i −0.00237796 + 0.00172769i
\(894\) 0 0
\(895\) 10.8213 33.3045i 0.361716 1.11325i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.69664 8.29941i −0.0899380 0.276801i
\(900\) 0 0
\(901\) 19.0462i 0.634521i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.1174 4.91195i 0.502520 0.163279i
\(906\) 0 0
\(907\) 18.4206 + 13.3834i 0.611647 + 0.444387i 0.849994 0.526793i \(-0.176606\pi\)
−0.238347 + 0.971180i \(0.576606\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −13.3395 18.3602i −0.441956 0.608300i 0.528689 0.848815i \(-0.322684\pi\)
−0.970645 + 0.240515i \(0.922684\pi\)
\(912\) 0 0
\(913\) −14.2361 26.9256i −0.471146 0.891106i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.85824 1.57854i −0.160433 0.0521280i
\(918\) 0 0
\(919\) 0.737746 1.01542i 0.0243360 0.0334956i −0.796676 0.604407i \(-0.793410\pi\)
0.821012 + 0.570911i \(0.193410\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −9.79237 −0.322320
\(924\) 0 0
\(925\) 67.5699 2.22169
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.68995 2.32602i 0.0554456 0.0763143i −0.780394 0.625288i \(-0.784981\pi\)
0.835839 + 0.548974i \(0.184981\pi\)
\(930\) 0 0
\(931\) 27.8274 + 9.04168i 0.912007 + 0.296329i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 42.8048 + 6.11769i 1.39987 + 0.200070i
\(936\) 0 0
\(937\) 18.1303 + 24.9542i 0.592291 + 0.815219i 0.994975 0.100120i \(-0.0319227\pi\)
−0.402684 + 0.915339i \(0.631923\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −5.29167 3.84462i −0.172503 0.125331i 0.498184 0.867072i \(-0.334001\pi\)
−0.670687 + 0.741741i \(0.734001\pi\)
\(942\) 0 0
\(943\) 12.8474 4.17436i 0.418368 0.135936i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.2579i 1.08074i −0.841429 0.540368i \(-0.818285\pi\)
0.841429 0.540368i \(-0.181715\pi\)
\(948\) 0 0
\(949\) 1.28812 + 3.96442i 0.0418141 + 0.128691i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 16.9820 52.2651i 0.550100 1.69303i −0.158445 0.987368i \(-0.550648\pi\)
0.708545 0.705666i \(-0.249352\pi\)
\(954\) 0 0
\(955\) −61.9785 + 45.0300i −2.00558 + 1.45714i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.308776 0.224339i 0.00997090 0.00724428i
\(960\) 0 0
\(961\) −5.76392 + 17.7395i −0.185933 + 0.572242i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −9.40000 28.9302i −0.302597 0.931297i
\(966\) 0 0
\(967\) 34.0395i 1.09464i −0.836925 0.547318i \(-0.815649\pi\)
0.836925 0.547318i \(-0.184351\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.2008 + 11.7624i −1.16174 + 0.377472i −0.825554 0.564323i \(-0.809137\pi\)
−0.336186 + 0.941796i \(0.609137\pi\)
\(972\) 0 0
\(973\) 5.90192 + 4.28799i 0.189207 + 0.137467i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.8682 25.9699i −0.603648 0.830850i 0.392388 0.919800i \(-0.371649\pi\)
−0.996036 + 0.0889499i \(0.971649\pi\)
\(978\) 0 0
\(979\) −4.45325 + 9.07671i −0.142327 + 0.290093i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −20.3816 6.62238i −0.650071 0.211221i −0.0346258 0.999400i \(-0.511024\pi\)
−0.615446 + 0.788179i \(0.711024\pi\)
\(984\) 0 0
\(985\) −47.2345 + 65.0128i −1.50502 + 2.07148i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.37704 −0.298172
\(990\) 0 0
\(991\) 29.3068 0.930962 0.465481 0.885058i \(-0.345881\pi\)
0.465481 + 0.885058i \(0.345881\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 23.4621 32.2928i 0.743798 1.02375i
\(996\) 0 0
\(997\) −31.1587 10.1241i −0.986807 0.320633i −0.229225 0.973373i \(-0.573619\pi\)
−0.757582 + 0.652740i \(0.773619\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1584.2.cd.c.305.4 16
3.2 odd 2 inner 1584.2.cd.c.305.1 16
4.3 odd 2 99.2.j.a.8.1 16
11.7 odd 10 inner 1584.2.cd.c.161.1 16
12.11 even 2 99.2.j.a.8.4 yes 16
33.29 even 10 inner 1584.2.cd.c.161.4 16
36.7 odd 6 891.2.u.c.701.4 32
36.11 even 6 891.2.u.c.701.1 32
36.23 even 6 891.2.u.c.107.4 32
36.31 odd 6 891.2.u.c.107.1 32
44.7 even 10 99.2.j.a.62.4 yes 16
44.31 odd 10 1089.2.d.g.1088.16 16
44.35 even 10 1089.2.d.g.1088.2 16
132.35 odd 10 1089.2.d.g.1088.15 16
132.95 odd 10 99.2.j.a.62.1 yes 16
132.119 even 10 1089.2.d.g.1088.1 16
396.7 even 30 891.2.u.c.458.4 32
396.95 odd 30 891.2.u.c.755.4 32
396.139 even 30 891.2.u.c.755.1 32
396.227 odd 30 891.2.u.c.458.1 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.j.a.8.1 16 4.3 odd 2
99.2.j.a.8.4 yes 16 12.11 even 2
99.2.j.a.62.1 yes 16 132.95 odd 10
99.2.j.a.62.4 yes 16 44.7 even 10
891.2.u.c.107.1 32 36.31 odd 6
891.2.u.c.107.4 32 36.23 even 6
891.2.u.c.458.1 32 396.227 odd 30
891.2.u.c.458.4 32 396.7 even 30
891.2.u.c.701.1 32 36.11 even 6
891.2.u.c.701.4 32 36.7 odd 6
891.2.u.c.755.1 32 396.139 even 30
891.2.u.c.755.4 32 396.95 odd 30
1089.2.d.g.1088.1 16 132.119 even 10
1089.2.d.g.1088.2 16 44.35 even 10
1089.2.d.g.1088.15 16 132.35 odd 10
1089.2.d.g.1088.16 16 44.31 odd 10
1584.2.cd.c.161.1 16 11.7 odd 10 inner
1584.2.cd.c.161.4 16 33.29 even 10 inner
1584.2.cd.c.305.1 16 3.2 odd 2 inner
1584.2.cd.c.305.4 16 1.1 even 1 trivial