Properties

Label 2352.2.bl.q.607.4
Level $2352$
Weight $2$
Character 2352.607
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
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] = [2352,2,Mod(31,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("2352.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 607.4
Root \(-1.60021 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 2352.607
Dual form 2352.2.bl.q.31.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(2.78415 + 1.60743i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(2.78415 + 1.60743i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-1.18394 + 0.683549i) q^{11} +2.93015i q^{13} -3.21486i q^{15} +(-5.98456 + 3.45519i) q^{17} +(3.67725 - 6.36918i) q^{19} +(3.14171 + 1.81387i) q^{23} +(2.66765 + 4.62051i) q^{25} +1.00000 q^{27} -1.11185 q^{29} +(4.35159 + 7.53718i) q^{31} +(1.18394 + 0.683549i) q^{33} +(-3.81896 + 6.61463i) q^{37} +(2.53759 - 1.46508i) q^{39} +0.833147i q^{41} +4.82362i q^{43} +(-2.78415 + 1.60743i) q^{45} +(1.47683 - 2.55795i) q^{47} +(5.98456 + 3.45519i) q^{51} +(2.28897 + 3.96462i) q^{53} -4.39502 q^{55} -7.35449 q^{57} +(-7.04923 - 12.2096i) q^{59} +(9.57981 + 5.53091i) q^{61} +(-4.71001 + 8.15797i) q^{65} +(-10.5261 + 6.07723i) q^{67} -3.62774i q^{69} +12.6249i q^{71} +(5.62060 - 3.24505i) q^{73} +(2.66765 - 4.62051i) q^{75} +(-6.74952 - 3.89684i) q^{79} +(-0.500000 - 0.866025i) q^{81} -8.87502 q^{83} -22.2159 q^{85} +(0.555927 + 0.962893i) q^{87} +(10.9246 + 6.30731i) q^{89} +(4.35159 - 7.53718i) q^{93} +(20.4760 - 11.8218i) q^{95} -13.8811i q^{97} -1.36710i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 4 q^{9} + 24 q^{23} + 12 q^{25} + 8 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{47} - 8 q^{53} - 64 q^{55} - 24 q^{59} + 48 q^{61} + 8 q^{65} - 48 q^{67} + 48 q^{73} + 12 q^{75} - 24 q^{79} - 4 q^{81} - 64 q^{85} - 8 q^{87} + 48 q^{89} + 16 q^{93} + 72 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −0.500000 0.866025i −0.288675 0.500000i
\(4\) 0 0
\(5\) 2.78415 + 1.60743i 1.24511 + 0.718864i 0.970130 0.242586i \(-0.0779958\pi\)
0.274979 + 0.961450i \(0.411329\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −1.18394 + 0.683549i −0.356972 + 0.206098i −0.667752 0.744384i \(-0.732743\pi\)
0.310780 + 0.950482i \(0.399410\pi\)
\(12\) 0 0
\(13\) 2.93015i 0.812678i 0.913722 + 0.406339i \(0.133195\pi\)
−0.913722 + 0.406339i \(0.866805\pi\)
\(14\) 0 0
\(15\) 3.21486i 0.830072i
\(16\) 0 0
\(17\) −5.98456 + 3.45519i −1.45147 + 0.838006i −0.998565 0.0535532i \(-0.982945\pi\)
−0.452904 + 0.891559i \(0.649612\pi\)
\(18\) 0 0
\(19\) 3.67725 6.36918i 0.843618 1.46119i −0.0431974 0.999067i \(-0.513754\pi\)
0.886816 0.462123i \(-0.152912\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.14171 + 1.81387i 0.655092 + 0.378218i 0.790405 0.612585i \(-0.209870\pi\)
−0.135312 + 0.990803i \(0.543204\pi\)
\(24\) 0 0
\(25\) 2.66765 + 4.62051i 0.533530 + 0.924102i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.11185 −0.206466 −0.103233 0.994657i \(-0.532919\pi\)
−0.103233 + 0.994657i \(0.532919\pi\)
\(30\) 0 0
\(31\) 4.35159 + 7.53718i 0.781569 + 1.35372i 0.931027 + 0.364949i \(0.118914\pi\)
−0.149458 + 0.988768i \(0.547753\pi\)
\(32\) 0 0
\(33\) 1.18394 + 0.683549i 0.206098 + 0.118991i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.81896 + 6.61463i −0.627833 + 1.08744i 0.360152 + 0.932893i \(0.382725\pi\)
−0.987986 + 0.154546i \(0.950609\pi\)
\(38\) 0 0
\(39\) 2.53759 1.46508i 0.406339 0.234600i
\(40\) 0 0
\(41\) 0.833147i 0.130116i 0.997881 + 0.0650579i \(0.0207232\pi\)
−0.997881 + 0.0650579i \(0.979277\pi\)
\(42\) 0 0
\(43\) 4.82362i 0.735595i 0.929906 + 0.367797i \(0.119888\pi\)
−0.929906 + 0.367797i \(0.880112\pi\)
\(44\) 0 0
\(45\) −2.78415 + 1.60743i −0.415036 + 0.239621i
\(46\) 0 0
\(47\) 1.47683 2.55795i 0.215418 0.373116i −0.737983 0.674819i \(-0.764222\pi\)
0.953402 + 0.301703i \(0.0975551\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.98456 + 3.45519i 0.838006 + 0.483823i
\(52\) 0 0
\(53\) 2.28897 + 3.96462i 0.314414 + 0.544582i 0.979313 0.202352i \(-0.0648585\pi\)
−0.664898 + 0.746934i \(0.731525\pi\)
\(54\) 0 0
\(55\) −4.39502 −0.592625
\(56\) 0 0
\(57\) −7.35449 −0.974127
\(58\) 0 0
\(59\) −7.04923 12.2096i −0.917732 1.58956i −0.802852 0.596179i \(-0.796685\pi\)
−0.114880 0.993379i \(-0.536648\pi\)
\(60\) 0 0
\(61\) 9.57981 + 5.53091i 1.22657 + 0.708160i 0.966311 0.257379i \(-0.0828588\pi\)
0.260259 + 0.965539i \(0.416192\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.71001 + 8.15797i −0.584205 + 1.01187i
\(66\) 0 0
\(67\) −10.5261 + 6.07723i −1.28596 + 0.742452i −0.977932 0.208924i \(-0.933004\pi\)
−0.308032 + 0.951376i \(0.599670\pi\)
\(68\) 0 0
\(69\) 3.62774i 0.436728i
\(70\) 0 0
\(71\) 12.6249i 1.49830i 0.662402 + 0.749148i \(0.269537\pi\)
−0.662402 + 0.749148i \(0.730463\pi\)
\(72\) 0 0
\(73\) 5.62060 3.24505i 0.657841 0.379805i −0.133613 0.991034i \(-0.542658\pi\)
0.791454 + 0.611229i \(0.209324\pi\)
\(74\) 0 0
\(75\) 2.66765 4.62051i 0.308034 0.533530i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.74952 3.89684i −0.759380 0.438428i 0.0696931 0.997568i \(-0.477798\pi\)
−0.829073 + 0.559140i \(0.811131\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −8.87502 −0.974159 −0.487080 0.873358i \(-0.661938\pi\)
−0.487080 + 0.873358i \(0.661938\pi\)
\(84\) 0 0
\(85\) −22.2159 −2.40965
\(86\) 0 0
\(87\) 0.555927 + 0.962893i 0.0596016 + 0.103233i
\(88\) 0 0
\(89\) 10.9246 + 6.30731i 1.15800 + 0.668574i 0.950825 0.309729i \(-0.100238\pi\)
0.207179 + 0.978303i \(0.433572\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.35159 7.53718i 0.451239 0.781569i
\(94\) 0 0
\(95\) 20.4760 11.8218i 2.10079 1.21289i
\(96\) 0 0
\(97\) 13.8811i 1.40942i −0.709497 0.704708i \(-0.751078\pi\)
0.709497 0.704708i \(-0.248922\pi\)
\(98\) 0 0
\(99\) 1.36710i 0.137398i
\(100\) 0 0
\(101\) 1.43666 0.829455i 0.142953 0.0825338i −0.426818 0.904338i \(-0.640365\pi\)
0.569771 + 0.821804i \(0.307032\pi\)
\(102\) 0 0
\(103\) 0.653953 1.13268i 0.0644359 0.111606i −0.832008 0.554764i \(-0.812809\pi\)
0.896444 + 0.443158i \(0.146142\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.19385 + 3.57602i 0.598782 + 0.345707i 0.768562 0.639775i \(-0.220973\pi\)
−0.169780 + 0.985482i \(0.554306\pi\)
\(108\) 0 0
\(109\) 7.52215 + 13.0287i 0.720491 + 1.24793i 0.960803 + 0.277231i \(0.0894168\pi\)
−0.240312 + 0.970696i \(0.577250\pi\)
\(110\) 0 0
\(111\) 7.63792 0.724959
\(112\) 0 0
\(113\) −15.4530 −1.45369 −0.726846 0.686800i \(-0.759015\pi\)
−0.726846 + 0.686800i \(0.759015\pi\)
\(114\) 0 0
\(115\) 5.83133 + 10.1002i 0.543774 + 0.941845i
\(116\) 0 0
\(117\) −2.53759 1.46508i −0.234600 0.135446i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.56552 + 7.90772i −0.415047 + 0.718883i
\(122\) 0 0
\(123\) 0.721527 0.416574i 0.0650579 0.0375612i
\(124\) 0 0
\(125\) 1.07795i 0.0964149i
\(126\) 0 0
\(127\) 2.16478i 0.192094i 0.995377 + 0.0960468i \(0.0306198\pi\)
−0.995377 + 0.0960468i \(0.969380\pi\)
\(128\) 0 0
\(129\) 4.17738 2.41181i 0.367797 0.212348i
\(130\) 0 0
\(131\) −2.48938 + 4.31174i −0.217499 + 0.376718i −0.954043 0.299671i \(-0.903123\pi\)
0.736544 + 0.676390i \(0.236456\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.78415 + 1.60743i 0.239621 + 0.138345i
\(136\) 0 0
\(137\) 1.84490 + 3.19546i 0.157620 + 0.273006i 0.934010 0.357247i \(-0.116284\pi\)
−0.776390 + 0.630253i \(0.782951\pi\)
\(138\) 0 0
\(139\) 12.0577 1.02272 0.511360 0.859367i \(-0.329142\pi\)
0.511360 + 0.859367i \(0.329142\pi\)
\(140\) 0 0
\(141\) −2.95367 −0.248744
\(142\) 0 0
\(143\) −2.00290 3.46913i −0.167491 0.290103i
\(144\) 0 0
\(145\) −3.09556 1.78722i −0.257073 0.148421i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.65131 + 2.86015i −0.135280 + 0.234313i −0.925705 0.378247i \(-0.876527\pi\)
0.790424 + 0.612560i \(0.209860\pi\)
\(150\) 0 0
\(151\) 16.4122 9.47557i 1.33560 0.771111i 0.349452 0.936954i \(-0.386368\pi\)
0.986152 + 0.165843i \(0.0530346\pi\)
\(152\) 0 0
\(153\) 6.91037i 0.558671i
\(154\) 0 0
\(155\) 27.9795i 2.24737i
\(156\) 0 0
\(157\) −0.767676 + 0.443218i −0.0612672 + 0.0353726i −0.530321 0.847797i \(-0.677928\pi\)
0.469053 + 0.883170i \(0.344595\pi\)
\(158\) 0 0
\(159\) 2.28897 3.96462i 0.181527 0.314414i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.7609 + 9.67689i 1.31281 + 0.757953i 0.982561 0.185941i \(-0.0595332\pi\)
0.330251 + 0.943893i \(0.392867\pi\)
\(164\) 0 0
\(165\) 2.19751 + 3.80620i 0.171076 + 0.296312i
\(166\) 0 0
\(167\) −25.4855 −1.97213 −0.986065 0.166360i \(-0.946799\pi\)
−0.986065 + 0.166360i \(0.946799\pi\)
\(168\) 0 0
\(169\) 4.41421 0.339555
\(170\) 0 0
\(171\) 3.67725 + 6.36918i 0.281206 + 0.487063i
\(172\) 0 0
\(173\) 18.7862 + 10.8462i 1.42829 + 0.824623i 0.996986 0.0775839i \(-0.0247206\pi\)
0.431303 + 0.902207i \(0.358054\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.04923 + 12.2096i −0.529853 + 0.917732i
\(178\) 0 0
\(179\) −10.6369 + 6.14122i −0.795039 + 0.459016i −0.841734 0.539893i \(-0.818465\pi\)
0.0466943 + 0.998909i \(0.485131\pi\)
\(180\) 0 0
\(181\) 2.74444i 0.203993i −0.994785 0.101996i \(-0.967477\pi\)
0.994785 0.101996i \(-0.0325230\pi\)
\(182\) 0 0
\(183\) 11.0618i 0.817713i
\(184\) 0 0
\(185\) −21.2651 + 12.2774i −1.56344 + 0.902653i
\(186\) 0 0
\(187\) 4.72358 8.18148i 0.345422 0.598289i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.8196 11.4428i −1.43409 0.827974i −0.436664 0.899625i \(-0.643840\pi\)
−0.997430 + 0.0716507i \(0.977173\pi\)
\(192\) 0 0
\(193\) −7.30816 12.6581i −0.526053 0.911151i −0.999539 0.0303495i \(-0.990338\pi\)
0.473486 0.880801i \(-0.342995\pi\)
\(194\) 0 0
\(195\) 9.42002 0.674581
\(196\) 0 0
\(197\) 24.5430 1.74861 0.874307 0.485374i \(-0.161316\pi\)
0.874307 + 0.485374i \(0.161316\pi\)
\(198\) 0 0
\(199\) 3.02329 + 5.23650i 0.214316 + 0.371206i 0.953061 0.302779i \(-0.0979146\pi\)
−0.738745 + 0.673985i \(0.764581\pi\)
\(200\) 0 0
\(201\) 10.5261 + 6.07723i 0.742452 + 0.428655i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.33922 + 2.31960i −0.0935355 + 0.162008i
\(206\) 0 0
\(207\) −3.14171 + 1.81387i −0.218364 + 0.126073i
\(208\) 0 0
\(209\) 10.0543i 0.695471i
\(210\) 0 0
\(211\) 9.33513i 0.642657i −0.946968 0.321328i \(-0.895871\pi\)
0.946968 0.321328i \(-0.104129\pi\)
\(212\) 0 0
\(213\) 10.9335 6.31243i 0.749148 0.432521i
\(214\) 0 0
\(215\) −7.75362 + 13.4297i −0.528792 + 0.915895i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5.62060 3.24505i −0.379805 0.219280i
\(220\) 0 0
\(221\) −10.1242 17.5357i −0.681029 1.17958i
\(222\) 0 0
\(223\) 5.48794 0.367500 0.183750 0.982973i \(-0.441176\pi\)
0.183750 + 0.982973i \(0.441176\pi\)
\(224\) 0 0
\(225\) −5.33530 −0.355687
\(226\) 0 0
\(227\) 0.519063 + 0.899043i 0.0344514 + 0.0596716i 0.882737 0.469867i \(-0.155698\pi\)
−0.848286 + 0.529539i \(0.822365\pi\)
\(228\) 0 0
\(229\) 13.5133 + 7.80193i 0.892987 + 0.515566i 0.874918 0.484270i \(-0.160915\pi\)
0.0180687 + 0.999837i \(0.494248\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.60780 + 7.98095i −0.301867 + 0.522849i −0.976559 0.215250i \(-0.930943\pi\)
0.674692 + 0.738100i \(0.264277\pi\)
\(234\) 0 0
\(235\) 8.22345 4.74781i 0.536439 0.309713i
\(236\) 0 0
\(237\) 7.79367i 0.506253i
\(238\) 0 0
\(239\) 1.54809i 0.100137i −0.998746 0.0500687i \(-0.984056\pi\)
0.998746 0.0500687i \(-0.0159440\pi\)
\(240\) 0 0
\(241\) −3.19154 + 1.84264i −0.205585 + 0.118695i −0.599258 0.800556i \(-0.704538\pi\)
0.393673 + 0.919251i \(0.371204\pi\)
\(242\) 0 0
\(243\) −0.500000 + 0.866025i −0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 18.6627 + 10.7749i 1.18748 + 0.685590i
\(248\) 0 0
\(249\) 4.43751 + 7.68599i 0.281216 + 0.487080i
\(250\) 0 0
\(251\) 26.1940 1.65335 0.826676 0.562679i \(-0.190229\pi\)
0.826676 + 0.562679i \(0.190229\pi\)
\(252\) 0 0
\(253\) −4.95947 −0.311799
\(254\) 0 0
\(255\) 11.1079 + 19.2395i 0.695606 + 1.20482i
\(256\) 0 0
\(257\) 0.721527 + 0.416574i 0.0450076 + 0.0259851i 0.522335 0.852740i \(-0.325061\pi\)
−0.477327 + 0.878726i \(0.658394\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.555927 0.962893i 0.0344110 0.0596016i
\(262\) 0 0
\(263\) −4.36661 + 2.52106i −0.269256 + 0.155455i −0.628550 0.777769i \(-0.716351\pi\)
0.359293 + 0.933225i \(0.383018\pi\)
\(264\) 0 0
\(265\) 14.7174i 0.904085i
\(266\) 0 0
\(267\) 12.6146i 0.772002i
\(268\) 0 0
\(269\) −4.20537 + 2.42797i −0.256406 + 0.148036i −0.622694 0.782466i \(-0.713962\pi\)
0.366288 + 0.930501i \(0.380628\pi\)
\(270\) 0 0
\(271\) 11.1244 19.2680i 0.675759 1.17045i −0.300487 0.953786i \(-0.597149\pi\)
0.976246 0.216664i \(-0.0695175\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.31669 3.64694i −0.380910 0.219919i
\(276\) 0 0
\(277\) −0.335303 0.580762i −0.0201464 0.0348946i 0.855776 0.517346i \(-0.173080\pi\)
−0.875923 + 0.482451i \(0.839747\pi\)
\(278\) 0 0
\(279\) −8.70319 −0.521046
\(280\) 0 0
\(281\) 25.5971 1.52700 0.763499 0.645810i \(-0.223480\pi\)
0.763499 + 0.645810i \(0.223480\pi\)
\(282\) 0 0
\(283\) −5.55574 9.62283i −0.330255 0.572018i 0.652307 0.757955i \(-0.273801\pi\)
−0.982562 + 0.185937i \(0.940468\pi\)
\(284\) 0 0
\(285\) −20.4760 11.8218i −1.21289 0.700264i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 15.3766 26.6331i 0.904508 1.56665i
\(290\) 0 0
\(291\) −12.0214 + 6.94057i −0.704708 + 0.406864i
\(292\) 0 0
\(293\) 19.2818i 1.12645i 0.826303 + 0.563226i \(0.190440\pi\)
−0.826303 + 0.563226i \(0.809560\pi\)
\(294\) 0 0
\(295\) 45.3245i 2.63890i
\(296\) 0 0
\(297\) −1.18394 + 0.683549i −0.0686992 + 0.0396635i
\(298\) 0 0
\(299\) −5.31491 + 9.20569i −0.307369 + 0.532379i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.43666 0.829455i −0.0825338 0.0476509i
\(304\) 0 0
\(305\) 17.7811 + 30.7977i 1.01814 + 1.76347i
\(306\) 0 0
\(307\) 26.0058 1.48423 0.742115 0.670273i \(-0.233823\pi\)
0.742115 + 0.670273i \(0.233823\pi\)
\(308\) 0 0
\(309\) −1.30791 −0.0744042
\(310\) 0 0
\(311\) −9.53862 16.5214i −0.540885 0.936841i −0.998853 0.0478723i \(-0.984756\pi\)
0.457968 0.888969i \(-0.348577\pi\)
\(312\) 0 0
\(313\) −18.4968 10.6791i −1.04550 0.603620i −0.124114 0.992268i \(-0.539609\pi\)
−0.921386 + 0.388648i \(0.872942\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.87066 + 6.70417i −0.217398 + 0.376544i −0.954012 0.299770i \(-0.903090\pi\)
0.736614 + 0.676313i \(0.236424\pi\)
\(318\) 0 0
\(319\) 1.31637 0.760006i 0.0737025 0.0425522i
\(320\) 0 0
\(321\) 7.15204i 0.399188i
\(322\) 0 0
\(323\) 50.8223i 2.82783i
\(324\) 0 0
\(325\) −13.5388 + 7.81662i −0.750997 + 0.433588i
\(326\) 0 0
\(327\) 7.52215 13.0287i 0.415976 0.720491i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.585423 0.337994i −0.0321778 0.0185778i 0.483825 0.875165i \(-0.339247\pi\)
−0.516003 + 0.856587i \(0.672581\pi\)
\(332\) 0 0
\(333\) −3.81896 6.61463i −0.209278 0.362480i
\(334\) 0 0
\(335\) −39.0748 −2.13489
\(336\) 0 0
\(337\) −0.176755 −0.00962847 −0.00481423 0.999988i \(-0.501532\pi\)
−0.00481423 + 0.999988i \(0.501532\pi\)
\(338\) 0 0
\(339\) 7.72648 + 13.3827i 0.419645 + 0.726846i
\(340\) 0 0
\(341\) −10.3041 5.94905i −0.557996 0.322159i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5.83133 10.1002i 0.313948 0.543774i
\(346\) 0 0
\(347\) 5.39220 3.11319i 0.289468 0.167125i −0.348234 0.937408i \(-0.613218\pi\)
0.637702 + 0.770283i \(0.279885\pi\)
\(348\) 0 0
\(349\) 17.4414i 0.933616i −0.884359 0.466808i \(-0.845404\pi\)
0.884359 0.466808i \(-0.154596\pi\)
\(350\) 0 0
\(351\) 2.93015i 0.156400i
\(352\) 0 0
\(353\) 7.53631 4.35109i 0.401117 0.231585i −0.285849 0.958275i \(-0.592275\pi\)
0.686966 + 0.726690i \(0.258942\pi\)
\(354\) 0 0
\(355\) −20.2936 + 35.1495i −1.07707 + 1.86554i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.661871 + 0.382131i 0.0349322 + 0.0201681i 0.517364 0.855765i \(-0.326913\pi\)
−0.482432 + 0.875933i \(0.660247\pi\)
\(360\) 0 0
\(361\) −17.5443 30.3876i −0.923384 1.59935i
\(362\) 0 0
\(363\) 9.13104 0.479256
\(364\) 0 0
\(365\) 20.8648 1.09211
\(366\) 0 0
\(367\) −13.2581 22.9637i −0.692067 1.19869i −0.971160 0.238431i \(-0.923367\pi\)
0.279093 0.960264i \(-0.409966\pi\)
\(368\) 0 0
\(369\) −0.721527 0.416574i −0.0375612 0.0216860i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0655 17.4340i 0.521173 0.902698i −0.478524 0.878075i \(-0.658828\pi\)
0.999697 0.0246235i \(-0.00783870\pi\)
\(374\) 0 0
\(375\) 0.933533 0.538976i 0.0482074 0.0278326i
\(376\) 0 0
\(377\) 3.25790i 0.167790i
\(378\) 0 0
\(379\) 10.2608i 0.527061i −0.964651 0.263531i \(-0.915113\pi\)
0.964651 0.263531i \(-0.0848870\pi\)
\(380\) 0 0
\(381\) 1.87476 1.08239i 0.0960468 0.0554526i
\(382\) 0 0
\(383\) 1.17157 2.02922i 0.0598646 0.103688i −0.834540 0.550947i \(-0.814266\pi\)
0.894404 + 0.447259i \(0.147600\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.17738 2.41181i −0.212348 0.122599i
\(388\) 0 0
\(389\) −12.0220 20.8227i −0.609540 1.05576i −0.991316 0.131500i \(-0.958021\pi\)
0.381776 0.924255i \(-0.375313\pi\)
\(390\) 0 0
\(391\) −25.0690 −1.26780
\(392\) 0 0
\(393\) 4.97877 0.251146
\(394\) 0 0
\(395\) −12.5278 21.6987i −0.630340 1.09178i
\(396\) 0 0
\(397\) −15.8938 9.17628i −0.797686 0.460544i 0.0449754 0.998988i \(-0.485679\pi\)
−0.842661 + 0.538444i \(0.819012\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.48486 2.57185i 0.0741503 0.128432i −0.826566 0.562840i \(-0.809709\pi\)
0.900716 + 0.434407i \(0.143042\pi\)
\(402\) 0 0
\(403\) −22.0851 + 12.7508i −1.10014 + 0.635164i
\(404\) 0 0
\(405\) 3.21486i 0.159748i
\(406\) 0 0
\(407\) 10.4418i 0.517580i
\(408\) 0 0
\(409\) 20.7415 11.9751i 1.02560 0.592131i 0.109880 0.993945i \(-0.464953\pi\)
0.915721 + 0.401814i \(0.131620\pi\)
\(410\) 0 0
\(411\) 1.84490 3.19546i 0.0910021 0.157620i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −24.7094 14.2660i −1.21293 0.700288i
\(416\) 0 0
\(417\) −6.02884 10.4423i −0.295234 0.511360i
\(418\) 0 0
\(419\) −36.3065 −1.77369 −0.886844 0.462069i \(-0.847107\pi\)
−0.886844 + 0.462069i \(0.847107\pi\)
\(420\) 0 0
\(421\) −22.6274 −1.10279 −0.551396 0.834243i \(-0.685905\pi\)
−0.551396 + 0.834243i \(0.685905\pi\)
\(422\) 0 0
\(423\) 1.47683 + 2.55795i 0.0718061 + 0.124372i
\(424\) 0 0
\(425\) −31.9294 18.4345i −1.54881 0.894203i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.00290 + 3.46913i −0.0967010 + 0.167491i
\(430\) 0 0
\(431\) −26.5038 + 15.3020i −1.27664 + 0.737071i −0.976230 0.216738i \(-0.930458\pi\)
−0.300414 + 0.953809i \(0.597125\pi\)
\(432\) 0 0
\(433\) 14.4650i 0.695146i −0.937653 0.347573i \(-0.887006\pi\)
0.937653 0.347573i \(-0.112994\pi\)
\(434\) 0 0
\(435\) 3.57445i 0.171382i
\(436\) 0 0
\(437\) 23.1057 13.3401i 1.10530 0.638143i
\(438\) 0 0
\(439\) 4.55491 7.88933i 0.217394 0.376537i −0.736617 0.676311i \(-0.763578\pi\)
0.954010 + 0.299773i \(0.0969111\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.45362 1.41660i −0.116575 0.0673045i 0.440579 0.897714i \(-0.354773\pi\)
−0.557154 + 0.830409i \(0.688107\pi\)
\(444\) 0 0
\(445\) 20.2771 + 35.1210i 0.961227 + 1.66489i
\(446\) 0 0
\(447\) 3.30262 0.156208
\(448\) 0 0
\(449\) 10.3630 0.489058 0.244529 0.969642i \(-0.421367\pi\)
0.244529 + 0.969642i \(0.421367\pi\)
\(450\) 0 0
\(451\) −0.569497 0.986397i −0.0268166 0.0464476i
\(452\) 0 0
\(453\) −16.4122 9.47557i −0.771111 0.445201i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.32976 5.76731i 0.155759 0.269783i −0.777576 0.628789i \(-0.783551\pi\)
0.933335 + 0.359006i \(0.116884\pi\)
\(458\) 0 0
\(459\) −5.98456 + 3.45519i −0.279335 + 0.161274i
\(460\) 0 0
\(461\) 1.71311i 0.0797873i −0.999204 0.0398936i \(-0.987298\pi\)
0.999204 0.0398936i \(-0.0127019\pi\)
\(462\) 0 0
\(463\) 24.1733i 1.12343i −0.827331 0.561715i \(-0.810142\pi\)
0.827331 0.561715i \(-0.189858\pi\)
\(464\) 0 0
\(465\) 24.2310 13.9897i 1.12368 0.648759i
\(466\) 0 0
\(467\) −18.6327 + 32.2728i −0.862220 + 1.49341i 0.00756221 + 0.999971i \(0.497593\pi\)
−0.869782 + 0.493437i \(0.835740\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.767676 + 0.443218i 0.0353726 + 0.0204224i
\(472\) 0 0
\(473\) −3.29718 5.71088i −0.151604 0.262587i
\(474\) 0 0
\(475\) 39.2385 1.80038
\(476\) 0 0
\(477\) −4.57794 −0.209610
\(478\) 0 0
\(479\) −10.9298 18.9310i −0.499395 0.864978i 0.500605 0.865676i \(-0.333111\pi\)
−1.00000 0.000698448i \(0.999778\pi\)
\(480\) 0 0
\(481\) −19.3819 11.1901i −0.883737 0.510226i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 22.3129 38.6472i 1.01318 1.75488i
\(486\) 0 0
\(487\) −2.92083 + 1.68634i −0.132356 + 0.0764155i −0.564716 0.825285i \(-0.691014\pi\)
0.432360 + 0.901701i \(0.357681\pi\)
\(488\) 0 0
\(489\) 19.3538i 0.875208i
\(490\) 0 0
\(491\) 9.68824i 0.437224i −0.975812 0.218612i \(-0.929847\pi\)
0.975812 0.218612i \(-0.0701529\pi\)
\(492\) 0 0
\(493\) 6.65395 3.84166i 0.299679 0.173020i
\(494\) 0 0
\(495\) 2.19751 3.80620i 0.0987708 0.171076i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 11.8992 + 6.87003i 0.532683 + 0.307545i 0.742108 0.670280i \(-0.233826\pi\)
−0.209425 + 0.977825i \(0.567159\pi\)
\(500\) 0 0
\(501\) 12.7428 + 22.0711i 0.569305 + 0.986065i
\(502\) 0 0
\(503\) −1.92715 −0.0859273 −0.0429637 0.999077i \(-0.513680\pi\)
−0.0429637 + 0.999077i \(0.513680\pi\)
\(504\) 0 0
\(505\) 5.33316 0.237322
\(506\) 0 0
\(507\) −2.20711 3.82282i −0.0980211 0.169777i
\(508\) 0 0
\(509\) −12.2647 7.08104i −0.543624 0.313862i 0.202922 0.979195i \(-0.434956\pi\)
−0.746546 + 0.665333i \(0.768289\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.67725 6.36918i 0.162354 0.281206i
\(514\) 0 0
\(515\) 3.64140 2.10236i 0.160459 0.0926413i
\(516\) 0 0
\(517\) 4.03795i 0.177589i
\(518\) 0 0
\(519\) 21.6924i 0.952193i
\(520\) 0 0
\(521\) −18.7084 + 10.8013i −0.819630 + 0.473213i −0.850289 0.526316i \(-0.823573\pi\)
0.0306591 + 0.999530i \(0.490239\pi\)
\(522\) 0 0
\(523\) −6.98251 + 12.0941i −0.305324 + 0.528836i −0.977333 0.211706i \(-0.932098\pi\)
0.672010 + 0.740542i \(0.265431\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −52.0847 30.0711i −2.26885 1.30992i
\(528\) 0 0
\(529\) −4.91976 8.52127i −0.213903 0.370490i
\(530\) 0 0
\(531\) 14.0985 0.611821
\(532\) 0 0
\(533\) −2.44125 −0.105742
\(534\) 0 0
\(535\) 11.4964 + 19.9123i 0.497032 + 0.860885i
\(536\) 0 0
\(537\) 10.6369 + 6.14122i 0.459016 + 0.265013i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.94002 + 3.36022i −0.0834081 + 0.144467i −0.904712 0.426024i \(-0.859914\pi\)
0.821304 + 0.570491i \(0.193247\pi\)
\(542\) 0 0
\(543\) −2.37676 + 1.37222i −0.101996 + 0.0588876i
\(544\) 0 0
\(545\) 48.3652i 2.07174i
\(546\) 0 0
\(547\) 35.1640i 1.50351i 0.659445 + 0.751753i \(0.270791\pi\)
−0.659445 + 0.751753i \(0.729209\pi\)
\(548\) 0 0
\(549\) −9.57981 + 5.53091i −0.408856 + 0.236053i
\(550\) 0 0
\(551\) −4.08856 + 7.08159i −0.174178 + 0.301686i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 21.2651 + 12.2774i 0.902653 + 0.521147i
\(556\) 0 0
\(557\) −21.2173 36.7495i −0.899006 1.55712i −0.828767 0.559594i \(-0.810957\pi\)
−0.0702393 0.997530i \(-0.522376\pi\)
\(558\) 0 0
\(559\) −14.1339 −0.597802
\(560\) 0 0
\(561\) −9.44716 −0.398859
\(562\) 0 0
\(563\) 8.27438 + 14.3316i 0.348724 + 0.604007i 0.986023 0.166609i \(-0.0532819\pi\)
−0.637299 + 0.770616i \(0.719949\pi\)
\(564\) 0 0
\(565\) −43.0233 24.8395i −1.81000 1.04501i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.1283 21.0069i 0.508446 0.880654i −0.491506 0.870874i \(-0.663554\pi\)
0.999952 0.00978001i \(-0.00311312\pi\)
\(570\) 0 0
\(571\) −4.00591 + 2.31281i −0.167642 + 0.0967882i −0.581474 0.813565i \(-0.697524\pi\)
0.413831 + 0.910354i \(0.364190\pi\)
\(572\) 0 0
\(573\) 22.8857i 0.956062i
\(574\) 0 0
\(575\) 19.3551i 0.807163i
\(576\) 0 0
\(577\) 23.2014 13.3954i 0.965889 0.557656i 0.0679083 0.997692i \(-0.478367\pi\)
0.897980 + 0.440035i \(0.145034\pi\)
\(578\) 0 0
\(579\) −7.30816 + 12.6581i −0.303717 + 0.526053i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −5.42002 3.12925i −0.224474 0.129600i
\(584\) 0 0
\(585\) −4.71001 8.15797i −0.194735 0.337291i
\(586\) 0 0
\(587\) 27.8591 1.14987 0.574933 0.818200i \(-0.305028\pi\)
0.574933 + 0.818200i \(0.305028\pi\)
\(588\) 0 0
\(589\) 64.0075 2.63738
\(590\) 0 0
\(591\) −12.2715 21.2548i −0.504781 0.874307i
\(592\) 0 0
\(593\) 1.35884 + 0.784528i 0.0558010 + 0.0322167i 0.527641 0.849468i \(-0.323077\pi\)
−0.471840 + 0.881684i \(0.656410\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.02329 5.23650i 0.123735 0.214316i
\(598\) 0 0
\(599\) 34.7412 20.0578i 1.41949 0.819540i 0.423232 0.906021i \(-0.360895\pi\)
0.996254 + 0.0864809i \(0.0275621\pi\)
\(600\) 0 0
\(601\) 24.7827i 1.01091i 0.862854 + 0.505453i \(0.168675\pi\)
−0.862854 + 0.505453i \(0.831325\pi\)
\(602\) 0 0
\(603\) 12.1545i 0.494968i
\(604\) 0 0
\(605\) −25.4222 + 14.6775i −1.03356 + 0.596725i
\(606\) 0 0
\(607\) 2.33120 4.03776i 0.0946205 0.163887i −0.814830 0.579700i \(-0.803170\pi\)
0.909450 + 0.415813i \(0.136503\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.49519 + 4.32735i 0.303223 + 0.175066i
\(612\) 0 0
\(613\) −14.2909 24.7525i −0.577202 0.999743i −0.995799 0.0915710i \(-0.970811\pi\)
0.418596 0.908172i \(-0.362522\pi\)
\(614\) 0 0
\(615\) 2.67845 0.108005
\(616\) 0 0
\(617\) 10.1524 0.408720 0.204360 0.978896i \(-0.434489\pi\)
0.204360 + 0.978896i \(0.434489\pi\)
\(618\) 0 0
\(619\) 4.32011 + 7.48265i 0.173640 + 0.300753i 0.939690 0.342028i \(-0.111114\pi\)
−0.766050 + 0.642781i \(0.777780\pi\)
\(620\) 0 0
\(621\) 3.14171 + 1.81387i 0.126073 + 0.0727881i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.6055 20.1014i 0.464221 0.804055i
\(626\) 0 0
\(627\) 8.70729 5.02716i 0.347736 0.200765i
\(628\) 0 0
\(629\) 52.7809i 2.10451i
\(630\) 0 0
\(631\) 5.46211i 0.217443i −0.994072 0.108722i \(-0.965324\pi\)
0.994072 0.108722i \(-0.0346757\pi\)
\(632\) 0 0
\(633\) −8.08446 + 4.66756i −0.321328 + 0.185519i
\(634\) 0 0
\(635\) −3.47974 + 6.02708i −0.138089 + 0.239177i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −10.9335 6.31243i −0.432521 0.249716i
\(640\) 0 0
\(641\) −0.714111 1.23688i −0.0282057 0.0488537i 0.851578 0.524228i \(-0.175646\pi\)
−0.879784 + 0.475374i \(0.842313\pi\)
\(642\) 0 0
\(643\) −24.6036 −0.970270 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(644\) 0 0
\(645\) 15.5072 0.610597
\(646\) 0 0
\(647\) −10.1392 17.5617i −0.398614 0.690420i 0.594941 0.803770i \(-0.297175\pi\)
−0.993555 + 0.113349i \(0.963842\pi\)
\(648\) 0 0
\(649\) 16.6918 + 9.63699i 0.655209 + 0.378285i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21.4420 + 37.1387i −0.839092 + 1.45335i 0.0515635 + 0.998670i \(0.483580\pi\)
−0.890655 + 0.454680i \(0.849754\pi\)
\(654\) 0 0
\(655\) −13.8616 + 8.00301i −0.541619 + 0.312704i
\(656\) 0 0
\(657\) 6.49011i 0.253203i
\(658\) 0 0
\(659\) 31.4717i 1.22596i −0.790098 0.612981i \(-0.789970\pi\)
0.790098 0.612981i \(-0.210030\pi\)
\(660\) 0 0
\(661\) 10.6655 6.15771i 0.414839 0.239507i −0.278028 0.960573i \(-0.589681\pi\)
0.692867 + 0.721066i \(0.256347\pi\)
\(662\) 0 0
\(663\) −10.1242 + 17.5357i −0.393192 + 0.681029i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.49312 2.01676i −0.135254 0.0780891i
\(668\) 0 0
\(669\) −2.74397 4.75270i −0.106088 0.183750i
\(670\) 0 0
\(671\) −15.1226 −0.583801
\(672\) 0 0
\(673\) 37.1864 1.43343 0.716716 0.697365i \(-0.245645\pi\)
0.716716 + 0.697365i \(0.245645\pi\)
\(674\) 0 0
\(675\) 2.66765 + 4.62051i 0.102678 + 0.177843i
\(676\) 0 0
\(677\) 30.8345 + 17.8023i 1.18507 + 0.684198i 0.957181 0.289490i \(-0.0934857\pi\)
0.227885 + 0.973688i \(0.426819\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0.519063 0.899043i 0.0198905 0.0344514i
\(682\) 0 0
\(683\) −27.8909 + 16.1028i −1.06722 + 0.616157i −0.927419 0.374023i \(-0.877978\pi\)
−0.139796 + 0.990180i \(0.544645\pi\)
\(684\) 0 0
\(685\) 11.8622i 0.453230i
\(686\) 0 0
\(687\) 15.6039i 0.595325i
\(688\) 0 0
\(689\) −11.6169 + 6.70703i −0.442570 + 0.255518i
\(690\) 0 0
\(691\) 19.3084 33.4432i 0.734527 1.27224i −0.220404 0.975409i \(-0.570737\pi\)
0.954931 0.296829i \(-0.0959292\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.5704 + 19.3819i 1.27340 + 0.735196i
\(696\) 0 0
\(697\) −2.87868 4.98602i −0.109038 0.188859i
\(698\) 0 0
\(699\) 9.21561 0.348566
\(700\) 0 0
\(701\) 30.8725 1.16604 0.583018 0.812459i \(-0.301872\pi\)
0.583018 + 0.812459i \(0.301872\pi\)
\(702\) 0 0
\(703\) 28.0865 + 48.6473i 1.05930 + 1.83477i
\(704\) 0 0
\(705\) −8.22345 4.74781i −0.309713 0.178813i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19.9226 + 34.5070i −0.748209 + 1.29594i 0.200471 + 0.979700i \(0.435753\pi\)
−0.948680 + 0.316237i \(0.897581\pi\)
\(710\) 0 0
\(711\) 6.74952 3.89684i 0.253127 0.146143i
\(712\) 0 0
\(713\) 31.5729i 1.18241i
\(714\) 0 0
\(715\) 12.8781i 0.481613i
\(716\) 0 0
\(717\) −1.34068 + 0.774044i −0.0500687 + 0.0289072i
\(718\) 0 0
\(719\) −1.04633 + 1.81230i −0.0390216 + 0.0675873i −0.884877 0.465826i \(-0.845757\pi\)
0.845855 + 0.533413i \(0.179091\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3.19154 + 1.84264i 0.118695 + 0.0685284i
\(724\) 0 0
\(725\) −2.96604 5.13733i −0.110156 0.190796i
\(726\) 0 0
\(727\) 21.5164 0.798001 0.399000 0.916951i \(-0.369357\pi\)
0.399000 + 0.916951i \(0.369357\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.6665 28.8672i −0.616433 1.06769i
\(732\) 0 0
\(733\) 6.69313 + 3.86428i 0.247216 + 0.142730i 0.618489 0.785794i \(-0.287745\pi\)
−0.371273 + 0.928524i \(0.621078\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.30816 14.3902i 0.306035 0.530068i
\(738\) 0 0
\(739\) −13.9485 + 8.05316i −0.513103 + 0.296240i −0.734108 0.679032i \(-0.762400\pi\)
0.221005 + 0.975273i \(0.429066\pi\)
\(740\) 0 0
\(741\) 21.5498i 0.791651i
\(742\) 0 0
\(743\) 29.3296i 1.07600i −0.842945 0.537999i \(-0.819180\pi\)
0.842945 0.537999i \(-0.180820\pi\)
\(744\) 0 0
\(745\) −9.19497 + 5.30872i −0.336878 + 0.194496i
\(746\) 0 0
\(747\) 4.43751 7.68599i 0.162360 0.281216i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −11.7772 6.79955i −0.429755 0.248119i 0.269487 0.963004i \(-0.413146\pi\)
−0.699242 + 0.714885i \(0.746479\pi\)
\(752\) 0 0
\(753\) −13.0970 22.6847i −0.477281 0.826676i
\(754\) 0 0
\(755\) 60.9252 2.21730
\(756\) 0 0
\(757\) −2.59688 −0.0943851 −0.0471926 0.998886i \(-0.515027\pi\)
−0.0471926 + 0.998886i \(0.515027\pi\)
\(758\) 0 0
\(759\) 2.47974 + 4.29503i 0.0900087 + 0.155900i
\(760\) 0 0
\(761\) −14.9753 8.64598i −0.542853 0.313416i 0.203381 0.979100i \(-0.434807\pi\)
−0.746235 + 0.665683i \(0.768140\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 11.1079 19.2395i 0.401608 0.695606i
\(766\) 0 0
\(767\) 35.7761 20.6553i 1.29180 0.745820i
\(768\) 0 0
\(769\) 41.4025i 1.49301i −0.665378 0.746507i \(-0.731730\pi\)
0.665378 0.746507i \(-0.268270\pi\)
\(770\) 0 0
\(771\) 0.833147i 0.0300051i
\(772\) 0 0
\(773\) 25.5191 14.7335i 0.917858 0.529925i 0.0349068 0.999391i \(-0.488887\pi\)
0.882951 + 0.469465i \(0.155553\pi\)
\(774\) 0 0
\(775\) −23.2171 + 40.2131i −0.833982 + 1.44450i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.30646 + 3.06369i 0.190124 + 0.109768i
\(780\) 0 0
\(781\) −8.62971 14.9471i −0.308795 0.534849i
\(782\) 0 0
\(783\) −1.11185 −0.0397344
\(784\) 0 0
\(785\) −2.84976 −0.101712
\(786\) 0 0
\(787\) 27.0921 + 46.9248i 0.965728 + 1.67269i 0.707647 + 0.706566i \(0.249757\pi\)
0.258081 + 0.966123i \(0.416910\pi\)
\(788\) 0 0
\(789\) 4.36661 + 2.52106i 0.155455 + 0.0897521i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −16.2064 + 28.0703i −0.575506 + 0.996806i
\(794\) 0 0
\(795\) 12.7457 7.35872i 0.452042 0.260987i
\(796\) 0 0
\(797\) 0.512549i 0.0181554i 0.999959 + 0.00907771i \(0.00288956\pi\)
−0.999959 + 0.00907771i \(0.997110\pi\)
\(798\) 0 0
\(799\) 20.4110i 0.722088i
\(800\) 0 0
\(801\) −10.9246 + 6.30731i −0.386001 + 0.222858i
\(802\) 0 0
\(803\) −4.43631 + 7.68391i −0.156554 + 0.271159i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4.20537 + 2.42797i 0.148036 + 0.0854686i
\(808\) 0 0
\(809\) 26.5704 + 46.0212i 0.934164 + 1.61802i 0.776118 + 0.630588i \(0.217186\pi\)
0.158046 + 0.987432i \(0.449481\pi\)
\(810\) 0 0
\(811\) 10.8127 0.379687 0.189843 0.981814i \(-0.439202\pi\)
0.189843 + 0.981814i \(0.439202\pi\)
\(812\) 0 0
\(813\) −22.2488 −0.780300
\(814\) 0 0
\(815\) 31.1098 + 53.8838i 1.08973 + 1.88747i
\(816\) 0 0
\(817\) 30.7225 + 17.7376i 1.07484 + 0.620561i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.35039 2.33895i 0.0471290 0.0816298i −0.841499 0.540259i \(-0.818326\pi\)
0.888628 + 0.458630i \(0.151659\pi\)
\(822\) 0 0
\(823\) 10.9234 6.30662i 0.380765 0.219835i −0.297386 0.954757i \(-0.596115\pi\)
0.678151 + 0.734922i \(0.262781\pi\)
\(824\) 0 0
\(825\) 7.29388i 0.253940i
\(826\) 0 0
\(827\) 2.95803i 0.102861i 0.998677 + 0.0514304i \(0.0163780\pi\)
−0.998677 + 0.0514304i \(0.983622\pi\)
\(828\) 0 0
\(829\) −13.6919 + 7.90504i −0.475540 + 0.274553i −0.718556 0.695469i \(-0.755197\pi\)
0.243016 + 0.970022i \(0.421863\pi\)
\(830\) 0 0
\(831\) −0.335303 + 0.580762i −0.0116315 + 0.0201464i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −70.9555 40.9662i −2.45552 1.41769i
\(836\) 0 0
\(837\) 4.35159 + 7.53718i 0.150413 + 0.260523i
\(838\) 0 0
\(839\) 5.13130 0.177152 0.0885761 0.996069i \(-0.471768\pi\)
0.0885761 + 0.996069i \(0.471768\pi\)
\(840\) 0 0
\(841\) −27.7638 −0.957372
\(842\) 0 0
\(843\) −12.7986 22.1678i −0.440806 0.763499i
\(844\) 0 0
\(845\) 12.2898 + 7.09553i 0.422783 + 0.244094i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −5.55574 + 9.62283i −0.190673 + 0.330255i
\(850\) 0 0
\(851\) −23.9962 + 13.8542i −0.822578 + 0.474915i
\(852\) 0 0
\(853\) 45.1627i 1.54634i 0.634198 + 0.773170i \(0.281330\pi\)
−0.634198 + 0.773170i \(0.718670\pi\)
\(854\) 0 0
\(855\) 23.6436i 0.808596i
\(856\) 0 0
\(857\) −1.27933 + 0.738624i −0.0437012 + 0.0252309i −0.521691 0.853134i \(-0.674699\pi\)
0.477990 + 0.878365i \(0.341365\pi\)
\(858\) 0 0
\(859\) −1.54091 + 2.66894i −0.0525753 + 0.0910631i −0.891115 0.453777i \(-0.850076\pi\)
0.838540 + 0.544840i \(0.183410\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.1836 + 23.2000i 1.36787 + 0.789738i 0.990655 0.136390i \(-0.0435500\pi\)
0.377210 + 0.926128i \(0.376883\pi\)
\(864\) 0 0
\(865\) 34.8691 + 60.3950i 1.18558 + 2.05349i
\(866\) 0 0
\(867\) −30.7533 −1.04444
\(868\) 0 0
\(869\) 10.6547 0.361436
\(870\) 0 0
\(871\) −17.8072 30.8430i −0.603374 1.04507i
\(872\) 0 0
\(873\) 12.0214 + 6.94057i 0.406864 + 0.234903i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 25.2387 43.7147i 0.852251 1.47614i −0.0269211 0.999638i \(-0.508570\pi\)
0.879172 0.476504i \(-0.158096\pi\)
\(878\) 0 0
\(879\) 16.6985 9.64088i 0.563226 0.325179i
\(880\) 0 0
\(881\) 3.66920i 0.123619i −0.998088 0.0618093i \(-0.980313\pi\)
0.998088 0.0618093i \(-0.0196870\pi\)
\(882\) 0 0
\(883\) 18.0393i 0.607071i 0.952820 + 0.303535i \(0.0981671\pi\)
−0.952820 + 0.303535i \(0.901833\pi\)
\(884\) 0 0
\(885\) −39.2522 + 22.6623i −1.31945 + 0.761784i
\(886\) 0 0
\(887\) 6.44980 11.1714i 0.216563 0.375098i −0.737192 0.675683i \(-0.763849\pi\)
0.953755 + 0.300585i \(0.0971820\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.18394 + 0.683549i 0.0396635 + 0.0228997i
\(892\) 0 0
\(893\) −10.8614 18.8124i −0.363462 0.629534i
\(894\) 0 0
\(895\) −39.4863 −1.31988
\(896\) 0 0
\(897\) 10.6298 0.354919
\(898\) 0 0
\(899\) −4.83833 8.38024i −0.161367 0.279497i
\(900\) 0 0
\(901\) −27.3970 15.8177i −0.912726 0.526962i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.41149 7.64093i 0.146643 0.253993i
\(906\) 0 0
\(907\) 0.662657 0.382585i 0.0220032 0.0127035i −0.488958 0.872307i \(-0.662623\pi\)
0.510961 + 0.859604i \(0.329290\pi\)
\(908\) 0 0
\(909\) 1.65891i 0.0550226i
\(910\) 0 0
\(911\) 18.7415i 0.620934i 0.950584 + 0.310467i \(0.100485\pi\)
−0.950584 + 0.310467i \(0.899515\pi\)
\(912\) 0 0
\(913\) 10.5075 6.06651i 0.347747 0.200772i
\(914\) 0 0
\(915\) 17.7811 30.7977i 0.587824 1.01814i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −18.2263 10.5230i −0.601231 0.347121i 0.168295 0.985737i \(-0.446174\pi\)
−0.769526 + 0.638616i \(0.779507\pi\)
\(920\) 0 0
\(921\) −13.0029 22.5217i −0.428460 0.742115i
\(922\) 0 0
\(923\) −36.9928 −1.21763
\(924\) 0 0
\(925\) −40.7506 −1.33987
\(926\) 0 0
\(927\) 0.653953 + 1.13268i 0.0214786 + 0.0372021i
\(928\) 0 0
\(929\) −28.3438 16.3643i −0.929929 0.536895i −0.0431400 0.999069i \(-0.513736\pi\)
−0.886789 + 0.462174i \(0.847069\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −9.53862 + 16.5214i −0.312280 + 0.540885i
\(934\) 0 0
\(935\) 26.3023 15.1856i 0.860176 0.496623i
\(936\) 0 0
\(937\) 2.53824i 0.0829207i 0.999140 + 0.0414603i \(0.0132010\pi\)
−0.999140 + 0.0414603i \(0.986799\pi\)
\(938\) 0 0
\(939\) 21.3583i 0.697001i
\(940\) 0 0
\(941\) 39.9880 23.0871i 1.30357 0.752617i 0.322556 0.946550i \(-0.395458\pi\)
0.981015 + 0.193933i \(0.0621246\pi\)
\(942\) 0 0
\(943\) −1.51122 + 2.61751i −0.0492121 + 0.0852378i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.31063 + 0.756690i 0.0425896 + 0.0245891i 0.521144 0.853469i \(-0.325506\pi\)
−0.478554 + 0.878058i \(0.658839\pi\)
\(948\) 0 0
\(949\) 9.50850 + 16.4692i 0.308659 + 0.534613i
\(950\) 0 0
\(951\) 7.74131 0.251029
\(952\) 0 0
\(953\) −43.1132 −1.39657 −0.698287 0.715818i \(-0.746054\pi\)
−0.698287 + 0.715818i \(0.746054\pi\)
\(954\) 0 0
\(955\) −36.7871 63.7171i −1.19040 2.06184i
\(956\) 0 0
\(957\) −1.31637 0.760006i −0.0425522 0.0245675i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −22.3727 + 38.7507i −0.721701 + 1.25002i
\(962\) 0 0
\(963\) −6.19385 + 3.57602i −0.199594 + 0.115236i
\(964\) 0 0
\(965\) 46.9894i 1.51264i
\(966\) 0 0
\(967\) 22.5218i 0.724253i 0.932129 + 0.362127i \(0.117949\pi\)
−0.932129 + 0.362127i \(0.882051\pi\)
\(968\) 0 0
\(969\) 44.0134 25.4112i 1.41391 0.816324i
\(970\) 0 0
\(971\) 12.2447 21.2084i 0.392951 0.680612i −0.599886 0.800085i \(-0.704787\pi\)
0.992837 + 0.119474i \(0.0381208\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 13.5388 + 7.81662i 0.433588 + 0.250332i
\(976\) 0 0
\(977\) −8.31909 14.4091i −0.266151 0.460987i 0.701713 0.712459i \(-0.252419\pi\)
−0.967865 + 0.251472i \(0.919085\pi\)
\(978\) 0 0
\(979\) −17.2454 −0.551166
\(980\) 0 0
\(981\) −15.0443 −0.480327
\(982\) 0 0
\(983\) −2.67809 4.63858i −0.0854177 0.147948i 0.820151 0.572146i \(-0.193889\pi\)
−0.905569 + 0.424199i \(0.860556\pi\)
\(984\) 0 0
\(985\) 68.3312 + 39.4510i 2.17721 + 1.25701i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.74941 + 15.1544i −0.278215 + 0.481883i
\(990\) 0 0
\(991\) 4.04958 2.33803i 0.128639 0.0742698i −0.434300 0.900769i \(-0.643004\pi\)
0.562939 + 0.826499i \(0.309671\pi\)
\(992\) 0 0
\(993\) 0.675988i 0.0214518i
\(994\) 0 0
\(995\) 19.4389i 0.616255i
\(996\) 0 0
\(997\) 0.785425 0.453465i 0.0248746 0.0143614i −0.487511 0.873117i \(-0.662095\pi\)
0.512386 + 0.858755i \(0.328762\pi\)
\(998\) 0 0
\(999\) −3.81896 + 6.61463i −0.120827 + 0.209278i
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 2352.2.bl.q.607.4 8
4.3 odd 2 2352.2.bl.r.607.4 8
7.2 even 3 2352.2.b.l.1567.2 yes 8
7.3 odd 6 2352.2.bl.r.31.4 8
7.4 even 3 2352.2.bl.o.31.1 8
7.5 odd 6 2352.2.b.k.1567.7 yes 8
7.6 odd 2 2352.2.bl.t.607.1 8
21.2 odd 6 7056.2.b.w.1567.7 8
21.5 even 6 7056.2.b.x.1567.2 8
28.3 even 6 inner 2352.2.bl.q.31.4 8
28.11 odd 6 2352.2.bl.t.31.1 8
28.19 even 6 2352.2.b.l.1567.7 yes 8
28.23 odd 6 2352.2.b.k.1567.2 8
28.27 even 2 2352.2.bl.o.607.1 8
84.23 even 6 7056.2.b.x.1567.7 8
84.47 odd 6 7056.2.b.w.1567.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.2 8 28.23 odd 6
2352.2.b.k.1567.7 yes 8 7.5 odd 6
2352.2.b.l.1567.2 yes 8 7.2 even 3
2352.2.b.l.1567.7 yes 8 28.19 even 6
2352.2.bl.o.31.1 8 7.4 even 3
2352.2.bl.o.607.1 8 28.27 even 2
2352.2.bl.q.31.4 8 28.3 even 6 inner
2352.2.bl.q.607.4 8 1.1 even 1 trivial
2352.2.bl.r.31.4 8 7.3 odd 6
2352.2.bl.r.607.4 8 4.3 odd 2
2352.2.bl.t.31.1 8 28.11 odd 6
2352.2.bl.t.607.1 8 7.6 odd 2
7056.2.b.w.1567.2 8 84.47 odd 6
7056.2.b.w.1567.7 8 21.2 odd 6
7056.2.b.x.1567.2 8 21.5 even 6
7056.2.b.x.1567.7 8 84.23 even 6