Properties

Label 4410.2.d.b.4409.8
Level $4410$
Weight $2$
Character 4410.4409
Analytic conductor $35.214$
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] = [4410,2,Mod(4409,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4410.4409");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.d (of order \(2\), degree \(1\), 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: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} - 6 x^{8} - 420 x^{7} + 4650 x^{6} - 21000 x^{5} + 70625 x^{4} - 168750 x^{3} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4409.8
Root \(2.11423 - 0.728019i\) of defining polynomial
Character \(\chi\) \(=\) 4410.4409
Dual form 4410.2.d.b.4409.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +(-0.426635 + 2.19499i) q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +(-0.426635 + 2.19499i) q^{5} +1.00000 q^{8} +(-0.426635 + 2.19499i) q^{10} +1.28310i q^{11} +6.14864 q^{13} +1.00000 q^{16} -6.52257i q^{17} -6.03756i q^{19} +(-0.426635 + 2.19499i) q^{20} +1.28310i q^{22} +2.87917 q^{23} +(-4.63597 - 1.87292i) q^{25} +6.14864 q^{26} +1.35052i q^{29} -8.65515i q^{31} +1.00000 q^{32} -6.52257i q^{34} -9.53381i q^{37} -6.03756i q^{38} +(-0.426635 + 2.19499i) q^{40} +8.71759 q^{41} +5.35859i q^{43} +1.28310i q^{44} +2.87917 q^{46} -0.806172i q^{47} +(-4.63597 - 1.87292i) q^{50} +6.14864 q^{52} -6.66826 q^{53} +(-2.81639 - 0.547415i) q^{55} +1.35052i q^{58} +1.59622 q^{59} +6.35361i q^{61} -8.65515i q^{62} +1.00000 q^{64} +(-2.62322 + 13.4962i) q^{65} +5.39463i q^{67} -6.52257i q^{68} +15.6787i q^{71} +12.4167 q^{73} -9.53381i q^{74} -6.03756i q^{76} +11.1819 q^{79} +(-0.426635 + 2.19499i) q^{80} +8.71759 q^{82} -3.74493i q^{83} +(14.3170 + 2.78275i) q^{85} +5.35859i q^{86} +1.28310i q^{88} -3.63941 q^{89} +2.87917 q^{92} -0.806172i q^{94} +(13.2524 + 2.57583i) q^{95} -8.76818 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} + 16 q^{16} + 16 q^{23} + 12 q^{25} + 16 q^{32} + 16 q^{46} + 12 q^{50} - 32 q^{53} + 16 q^{64} + 40 q^{65} - 8 q^{79} + 64 q^{85} + 16 q^{92} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(2647\) \(3431\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −0.426635 + 2.19499i −0.190797 + 0.981630i
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −0.426635 + 2.19499i −0.134914 + 0.694117i
\(11\) 1.28310i 0.386869i 0.981113 + 0.193435i \(0.0619627\pi\)
−0.981113 + 0.193435i \(0.938037\pi\)
\(12\) 0 0
\(13\) 6.14864 1.70533 0.852663 0.522462i \(-0.174986\pi\)
0.852663 + 0.522462i \(0.174986\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.52257i 1.58195i −0.611846 0.790977i \(-0.709573\pi\)
0.611846 0.790977i \(-0.290427\pi\)
\(18\) 0 0
\(19\) 6.03756i 1.38511i −0.721365 0.692555i \(-0.756485\pi\)
0.721365 0.692555i \(-0.243515\pi\)
\(20\) −0.426635 + 2.19499i −0.0953985 + 0.490815i
\(21\) 0 0
\(22\) 1.28310i 0.273558i
\(23\) 2.87917 0.600348 0.300174 0.953884i \(-0.402955\pi\)
0.300174 + 0.953884i \(0.402955\pi\)
\(24\) 0 0
\(25\) −4.63597 1.87292i −0.927193 0.374584i
\(26\) 6.14864 1.20585
\(27\) 0 0
\(28\) 0 0
\(29\) 1.35052i 0.250785i 0.992107 + 0.125393i \(0.0400191\pi\)
−0.992107 + 0.125393i \(0.959981\pi\)
\(30\) 0 0
\(31\) 8.65515i 1.55451i −0.629185 0.777255i \(-0.716611\pi\)
0.629185 0.777255i \(-0.283389\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.52257i 1.11861i
\(35\) 0 0
\(36\) 0 0
\(37\) 9.53381i 1.56735i −0.621172 0.783674i \(-0.713343\pi\)
0.621172 0.783674i \(-0.286657\pi\)
\(38\) 6.03756i 0.979421i
\(39\) 0 0
\(40\) −0.426635 + 2.19499i −0.0674569 + 0.347058i
\(41\) 8.71759 1.36146 0.680730 0.732535i \(-0.261663\pi\)
0.680730 + 0.732535i \(0.261663\pi\)
\(42\) 0 0
\(43\) 5.35859i 0.817177i 0.912719 + 0.408588i \(0.133979\pi\)
−0.912719 + 0.408588i \(0.866021\pi\)
\(44\) 1.28310i 0.193435i
\(45\) 0 0
\(46\) 2.87917 0.424510
\(47\) 0.806172i 0.117592i −0.998270 0.0587961i \(-0.981274\pi\)
0.998270 0.0587961i \(-0.0187262\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −4.63597 1.87292i −0.655625 0.264871i
\(51\) 0 0
\(52\) 6.14864 0.852663
\(53\) −6.66826 −0.915956 −0.457978 0.888964i \(-0.651426\pi\)
−0.457978 + 0.888964i \(0.651426\pi\)
\(54\) 0 0
\(55\) −2.81639 0.547415i −0.379762 0.0738134i
\(56\) 0 0
\(57\) 0 0
\(58\) 1.35052i 0.177332i
\(59\) 1.59622 0.207810 0.103905 0.994587i \(-0.466866\pi\)
0.103905 + 0.994587i \(0.466866\pi\)
\(60\) 0 0
\(61\) 6.35361i 0.813497i 0.913540 + 0.406748i \(0.133337\pi\)
−0.913540 + 0.406748i \(0.866663\pi\)
\(62\) 8.65515i 1.09921i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.62322 + 13.4962i −0.325371 + 1.67400i
\(66\) 0 0
\(67\) 5.39463i 0.659059i 0.944145 + 0.329529i \(0.106890\pi\)
−0.944145 + 0.329529i \(0.893110\pi\)
\(68\) 6.52257i 0.790977i
\(69\) 0 0
\(70\) 0 0
\(71\) 15.6787i 1.86072i 0.366650 + 0.930359i \(0.380505\pi\)
−0.366650 + 0.930359i \(0.619495\pi\)
\(72\) 0 0
\(73\) 12.4167 1.45327 0.726635 0.687024i \(-0.241083\pi\)
0.726635 + 0.687024i \(0.241083\pi\)
\(74\) 9.53381i 1.10828i
\(75\) 0 0
\(76\) 6.03756i 0.692555i
\(77\) 0 0
\(78\) 0 0
\(79\) 11.1819 1.25806 0.629028 0.777382i \(-0.283453\pi\)
0.629028 + 0.777382i \(0.283453\pi\)
\(80\) −0.426635 + 2.19499i −0.0476992 + 0.245407i
\(81\) 0 0
\(82\) 8.71759 0.962697
\(83\) 3.74493i 0.411059i −0.978651 0.205530i \(-0.934108\pi\)
0.978651 0.205530i \(-0.0658917\pi\)
\(84\) 0 0
\(85\) 14.3170 + 2.78275i 1.55289 + 0.301832i
\(86\) 5.35859i 0.577831i
\(87\) 0 0
\(88\) 1.28310i 0.136779i
\(89\) −3.63941 −0.385777 −0.192889 0.981221i \(-0.561786\pi\)
−0.192889 + 0.981221i \(0.561786\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.87917 0.300174
\(93\) 0 0
\(94\) 0.806172i 0.0831503i
\(95\) 13.2524 + 2.57583i 1.35967 + 0.264275i
\(96\) 0 0
\(97\) −8.76818 −0.890274 −0.445137 0.895463i \(-0.646845\pi\)
−0.445137 + 0.895463i \(0.646845\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.63597 1.87292i −0.463597 0.187292i
\(101\) −1.70654 −0.169807 −0.0849035 0.996389i \(-0.527058\pi\)
−0.0849035 + 0.996389i \(0.527058\pi\)
\(102\) 0 0
\(103\) −10.3644 −1.02123 −0.510617 0.859808i \(-0.670583\pi\)
−0.510617 + 0.859808i \(0.670583\pi\)
\(104\) 6.14864 0.602923
\(105\) 0 0
\(106\) −6.66826 −0.647679
\(107\) −11.5136 −1.11306 −0.556531 0.830827i \(-0.687868\pi\)
−0.556531 + 0.830827i \(0.687868\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −2.81639 0.547415i −0.268532 0.0521940i
\(111\) 0 0
\(112\) 0 0
\(113\) 13.3875 1.25939 0.629695 0.776843i \(-0.283180\pi\)
0.629695 + 0.776843i \(0.283180\pi\)
\(114\) 0 0
\(115\) −1.22835 + 6.31975i −0.114545 + 0.589319i
\(116\) 1.35052i 0.125393i
\(117\) 0 0
\(118\) 1.59622 0.146944
\(119\) 0 0
\(120\) 0 0
\(121\) 9.35366 0.850332
\(122\) 6.35361i 0.575229i
\(123\) 0 0
\(124\) 8.65515i 0.777255i
\(125\) 6.08890 9.37685i 0.544608 0.838691i
\(126\) 0 0
\(127\) 5.65313i 0.501634i −0.968035 0.250817i \(-0.919301\pi\)
0.968035 0.250817i \(-0.0806992\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.62322 + 13.4962i −0.230072 + 1.18369i
\(131\) 12.4673 1.08928 0.544638 0.838671i \(-0.316667\pi\)
0.544638 + 0.838671i \(0.316667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 5.39463i 0.466025i
\(135\) 0 0
\(136\) 6.52257i 0.559305i
\(137\) −0.0509686 −0.00435454 −0.00217727 0.999998i \(-0.500693\pi\)
−0.00217727 + 0.999998i \(0.500693\pi\)
\(138\) 0 0
\(139\) 1.05069i 0.0891184i −0.999007 0.0445592i \(-0.985812\pi\)
0.999007 0.0445592i \(-0.0141883\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.6787i 1.31573i
\(143\) 7.88931i 0.659738i
\(144\) 0 0
\(145\) −2.96438 0.576179i −0.246178 0.0478491i
\(146\) 12.4167 1.02762
\(147\) 0 0
\(148\) 9.53381i 0.783674i
\(149\) 8.22305i 0.673659i 0.941566 + 0.336829i \(0.109355\pi\)
−0.941566 + 0.336829i \(0.890645\pi\)
\(150\) 0 0
\(151\) −4.26715 −0.347256 −0.173628 0.984811i \(-0.555549\pi\)
−0.173628 + 0.984811i \(0.555549\pi\)
\(152\) 6.03756i 0.489710i
\(153\) 0 0
\(154\) 0 0
\(155\) 18.9980 + 3.69259i 1.52595 + 0.296596i
\(156\) 0 0
\(157\) −4.83203 −0.385638 −0.192819 0.981234i \(-0.561763\pi\)
−0.192819 + 0.981234i \(0.561763\pi\)
\(158\) 11.1819 0.889581
\(159\) 0 0
\(160\) −0.426635 + 2.19499i −0.0337285 + 0.173529i
\(161\) 0 0
\(162\) 0 0
\(163\) 10.7532i 0.842257i −0.907001 0.421128i \(-0.861634\pi\)
0.907001 0.421128i \(-0.138366\pi\)
\(164\) 8.71759 0.680730
\(165\) 0 0
\(166\) 3.74493i 0.290663i
\(167\) 18.9267i 1.46459i 0.680988 + 0.732295i \(0.261551\pi\)
−0.680988 + 0.732295i \(0.738449\pi\)
\(168\) 0 0
\(169\) 24.8057 1.90813
\(170\) 14.3170 + 2.78275i 1.09806 + 0.213427i
\(171\) 0 0
\(172\) 5.35859i 0.408588i
\(173\) 11.8435i 0.900447i 0.892916 + 0.450223i \(0.148656\pi\)
−0.892916 + 0.450223i \(0.851344\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.28310i 0.0967173i
\(177\) 0 0
\(178\) −3.63941 −0.272786
\(179\) 3.78561i 0.282949i −0.989942 0.141475i \(-0.954816\pi\)
0.989942 0.141475i \(-0.0451844\pi\)
\(180\) 0 0
\(181\) 19.0033i 1.41251i −0.707960 0.706253i \(-0.750384\pi\)
0.707960 0.706253i \(-0.249616\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.87917 0.212255
\(185\) 20.9266 + 4.06745i 1.53856 + 0.299045i
\(186\) 0 0
\(187\) 8.36910 0.612009
\(188\) 0.806172i 0.0587961i
\(189\) 0 0
\(190\) 13.2524 + 2.57583i 0.961428 + 0.186871i
\(191\) 2.82843i 0.204658i 0.994751 + 0.102329i \(0.0326294\pi\)
−0.994751 + 0.102329i \(0.967371\pi\)
\(192\) 0 0
\(193\) 3.53892i 0.254737i 0.991855 + 0.127368i \(0.0406530\pi\)
−0.991855 + 0.127368i \(0.959347\pi\)
\(194\) −8.76818 −0.629519
\(195\) 0 0
\(196\) 0 0
\(197\) −2.36728 −0.168662 −0.0843308 0.996438i \(-0.526875\pi\)
−0.0843308 + 0.996438i \(0.526875\pi\)
\(198\) 0 0
\(199\) 3.46410i 0.245564i −0.992434 0.122782i \(-0.960818\pi\)
0.992434 0.122782i \(-0.0391815\pi\)
\(200\) −4.63597 1.87292i −0.327812 0.132435i
\(201\) 0 0
\(202\) −1.70654 −0.120072
\(203\) 0 0
\(204\) 0 0
\(205\) −3.71923 + 19.1350i −0.259762 + 1.33645i
\(206\) −10.3644 −0.722122
\(207\) 0 0
\(208\) 6.14864 0.426331
\(209\) 7.74679 0.535856
\(210\) 0 0
\(211\) −14.3620 −0.988721 −0.494361 0.869257i \(-0.664598\pi\)
−0.494361 + 0.869257i \(0.664598\pi\)
\(212\) −6.66826 −0.457978
\(213\) 0 0
\(214\) −11.5136 −0.787053
\(215\) −11.7620 2.28616i −0.802165 0.155915i
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −2.81639 0.547415i −0.189881 0.0369067i
\(221\) 40.1049i 2.69775i
\(222\) 0 0
\(223\) −13.5975 −0.910557 −0.455279 0.890349i \(-0.650460\pi\)
−0.455279 + 0.890349i \(0.650460\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 13.3875 0.890523
\(227\) 12.3559i 0.820092i −0.912065 0.410046i \(-0.865513\pi\)
0.912065 0.410046i \(-0.134487\pi\)
\(228\) 0 0
\(229\) 2.41341i 0.159483i 0.996816 + 0.0797414i \(0.0254094\pi\)
−0.996816 + 0.0797414i \(0.974591\pi\)
\(230\) −1.22835 + 6.31975i −0.0809952 + 0.416712i
\(231\) 0 0
\(232\) 1.35052i 0.0886660i
\(233\) −12.8756 −0.843509 −0.421754 0.906710i \(-0.638586\pi\)
−0.421754 + 0.906710i \(0.638586\pi\)
\(234\) 0 0
\(235\) 1.76954 + 0.343941i 0.115432 + 0.0224362i
\(236\) 1.59622 0.103905
\(237\) 0 0
\(238\) 0 0
\(239\) 27.2546i 1.76296i 0.472226 + 0.881478i \(0.343451\pi\)
−0.472226 + 0.881478i \(0.656549\pi\)
\(240\) 0 0
\(241\) 2.61759i 0.168614i 0.996440 + 0.0843070i \(0.0268677\pi\)
−0.996440 + 0.0843070i \(0.973132\pi\)
\(242\) 9.35366 0.601276
\(243\) 0 0
\(244\) 6.35361i 0.406748i
\(245\) 0 0
\(246\) 0 0
\(247\) 37.1227i 2.36206i
\(248\) 8.65515i 0.549603i
\(249\) 0 0
\(250\) 6.08890 9.37685i 0.385096 0.593044i
\(251\) 26.7173 1.68638 0.843190 0.537615i \(-0.180675\pi\)
0.843190 + 0.537615i \(0.180675\pi\)
\(252\) 0 0
\(253\) 3.69426i 0.232256i
\(254\) 5.65313i 0.354709i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.5613i 0.658793i −0.944192 0.329397i \(-0.893155\pi\)
0.944192 0.329397i \(-0.106845\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.62322 + 13.4962i −0.162685 + 0.836999i
\(261\) 0 0
\(262\) 12.4673 0.770234
\(263\) −8.33174 −0.513757 −0.256879 0.966444i \(-0.582694\pi\)
−0.256879 + 0.966444i \(0.582694\pi\)
\(264\) 0 0
\(265\) 2.84491 14.6368i 0.174762 0.899130i
\(266\) 0 0
\(267\) 0 0
\(268\) 5.39463i 0.329529i
\(269\) 14.2834 0.870876 0.435438 0.900219i \(-0.356594\pi\)
0.435438 + 0.900219i \(0.356594\pi\)
\(270\) 0 0
\(271\) 7.60446i 0.461938i 0.972961 + 0.230969i \(0.0741896\pi\)
−0.972961 + 0.230969i \(0.925810\pi\)
\(272\) 6.52257i 0.395489i
\(273\) 0 0
\(274\) −0.0509686 −0.00307912
\(275\) 2.40314 5.94841i 0.144915 0.358702i
\(276\) 0 0
\(277\) 12.0400i 0.723416i −0.932291 0.361708i \(-0.882194\pi\)
0.932291 0.361708i \(-0.117806\pi\)
\(278\) 1.05069i 0.0630162i
\(279\) 0 0
\(280\) 0 0
\(281\) 12.7879i 0.762861i −0.924397 0.381431i \(-0.875431\pi\)
0.924397 0.381431i \(-0.124569\pi\)
\(282\) 0 0
\(283\) −15.5521 −0.924479 −0.462239 0.886755i \(-0.652954\pi\)
−0.462239 + 0.886755i \(0.652954\pi\)
\(284\) 15.6787i 0.930359i
\(285\) 0 0
\(286\) 7.88931i 0.466505i
\(287\) 0 0
\(288\) 0 0
\(289\) −25.5439 −1.50258
\(290\) −2.96438 0.576179i −0.174074 0.0338344i
\(291\) 0 0
\(292\) 12.4167 0.726635
\(293\) 11.5536i 0.674967i −0.941331 0.337483i \(-0.890424\pi\)
0.941331 0.337483i \(-0.109576\pi\)
\(294\) 0 0
\(295\) −0.681003 + 3.50369i −0.0396495 + 0.203993i
\(296\) 9.53381i 0.554141i
\(297\) 0 0
\(298\) 8.22305i 0.476349i
\(299\) 17.7030 1.02379
\(300\) 0 0
\(301\) 0 0
\(302\) −4.26715 −0.245547
\(303\) 0 0
\(304\) 6.03756i 0.346278i
\(305\) −13.9461 2.71067i −0.798553 0.155213i
\(306\) 0 0
\(307\) −2.57617 −0.147030 −0.0735150 0.997294i \(-0.523422\pi\)
−0.0735150 + 0.997294i \(0.523422\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 18.9980 + 3.69259i 1.07901 + 0.209725i
\(311\) −24.7140 −1.40140 −0.700702 0.713454i \(-0.747130\pi\)
−0.700702 + 0.713454i \(0.747130\pi\)
\(312\) 0 0
\(313\) −9.00173 −0.508808 −0.254404 0.967098i \(-0.581879\pi\)
−0.254404 + 0.967098i \(0.581879\pi\)
\(314\) −4.83203 −0.272687
\(315\) 0 0
\(316\) 11.1819 0.629028
\(317\) 23.9059 1.34269 0.671344 0.741146i \(-0.265717\pi\)
0.671344 + 0.741146i \(0.265717\pi\)
\(318\) 0 0
\(319\) −1.73285 −0.0970211
\(320\) −0.426635 + 2.19499i −0.0238496 + 0.122704i
\(321\) 0 0
\(322\) 0 0
\(323\) −39.3804 −2.19118
\(324\) 0 0
\(325\) −28.5049 11.5159i −1.58117 0.638787i
\(326\) 10.7532i 0.595565i
\(327\) 0 0
\(328\) 8.71759 0.481349
\(329\) 0 0
\(330\) 0 0
\(331\) 4.36728 0.240047 0.120024 0.992771i \(-0.461703\pi\)
0.120024 + 0.992771i \(0.461703\pi\)
\(332\) 3.74493i 0.205530i
\(333\) 0 0
\(334\) 18.9267i 1.03562i
\(335\) −11.8412 2.30154i −0.646951 0.125746i
\(336\) 0 0
\(337\) 19.5159i 1.06310i 0.847028 + 0.531549i \(0.178390\pi\)
−0.847028 + 0.531549i \(0.821610\pi\)
\(338\) 24.8057 1.34925
\(339\) 0 0
\(340\) 14.3170 + 2.78275i 0.776447 + 0.150916i
\(341\) 11.1054 0.601392
\(342\) 0 0
\(343\) 0 0
\(344\) 5.35859i 0.288916i
\(345\) 0 0
\(346\) 11.8435i 0.636712i
\(347\) −26.9147 −1.44486 −0.722429 0.691446i \(-0.756974\pi\)
−0.722429 + 0.691446i \(0.756974\pi\)
\(348\) 0 0
\(349\) 10.3821i 0.555741i −0.960619 0.277870i \(-0.910371\pi\)
0.960619 0.277870i \(-0.0896286\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.28310i 0.0683894i
\(353\) 0.168951i 0.00899236i −0.999990 0.00449618i \(-0.998569\pi\)
0.999990 0.00449618i \(-0.00143118\pi\)
\(354\) 0 0
\(355\) −34.4146 6.68908i −1.82654 0.355019i
\(356\) −3.63941 −0.192889
\(357\) 0 0
\(358\) 3.78561i 0.200075i
\(359\) 37.6661i 1.98794i 0.109655 + 0.993970i \(0.465025\pi\)
−0.109655 + 0.993970i \(0.534975\pi\)
\(360\) 0 0
\(361\) −17.4521 −0.918531
\(362\) 19.0033i 0.998792i
\(363\) 0 0
\(364\) 0 0
\(365\) −5.29741 + 27.2546i −0.277279 + 1.42657i
\(366\) 0 0
\(367\) 15.1557 0.791124 0.395562 0.918439i \(-0.370550\pi\)
0.395562 + 0.918439i \(0.370550\pi\)
\(368\) 2.87917 0.150087
\(369\) 0 0
\(370\) 20.9266 + 4.06745i 1.08792 + 0.211457i
\(371\) 0 0
\(372\) 0 0
\(373\) 12.0400i 0.623410i 0.950179 + 0.311705i \(0.100900\pi\)
−0.950179 + 0.311705i \(0.899100\pi\)
\(374\) 8.36910 0.432756
\(375\) 0 0
\(376\) 0.806172i 0.0415751i
\(377\) 8.30386i 0.427671i
\(378\) 0 0
\(379\) 18.0918 0.929312 0.464656 0.885491i \(-0.346178\pi\)
0.464656 + 0.885491i \(0.346178\pi\)
\(380\) 13.2524 + 2.57583i 0.679833 + 0.132137i
\(381\) 0 0
\(382\) 2.82843i 0.144715i
\(383\) 29.1538i 1.48969i 0.667237 + 0.744846i \(0.267477\pi\)
−0.667237 + 0.744846i \(0.732523\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.53892i 0.180126i
\(387\) 0 0
\(388\) −8.76818 −0.445137
\(389\) 19.6641i 0.997012i 0.866886 + 0.498506i \(0.166118\pi\)
−0.866886 + 0.498506i \(0.833882\pi\)
\(390\) 0 0
\(391\) 18.7796i 0.949723i
\(392\) 0 0
\(393\) 0 0
\(394\) −2.36728 −0.119262
\(395\) −4.77057 + 24.5441i −0.240033 + 1.23495i
\(396\) 0 0
\(397\) −5.68454 −0.285299 −0.142649 0.989773i \(-0.545562\pi\)
−0.142649 + 0.989773i \(0.545562\pi\)
\(398\) 3.46410i 0.173640i
\(399\) 0 0
\(400\) −4.63597 1.87292i −0.231798 0.0936460i
\(401\) 10.2166i 0.510195i 0.966915 + 0.255097i \(0.0821075\pi\)
−0.966915 + 0.255097i \(0.917892\pi\)
\(402\) 0 0
\(403\) 53.2174i 2.65095i
\(404\) −1.70654 −0.0849035
\(405\) 0 0
\(406\) 0 0
\(407\) 12.2328 0.606359
\(408\) 0 0
\(409\) 13.9683i 0.690687i 0.938476 + 0.345344i \(0.112238\pi\)
−0.938476 + 0.345344i \(0.887762\pi\)
\(410\) −3.71923 + 19.1350i −0.183680 + 0.945012i
\(411\) 0 0
\(412\) −10.3644 −0.510617
\(413\) 0 0
\(414\) 0 0
\(415\) 8.22008 + 1.59772i 0.403508 + 0.0784288i
\(416\) 6.14864 0.301462
\(417\) 0 0
\(418\) 7.74679 0.378908
\(419\) 11.3181 0.552924 0.276462 0.961025i \(-0.410838\pi\)
0.276462 + 0.961025i \(0.410838\pi\)
\(420\) 0 0
\(421\) −13.9099 −0.677928 −0.338964 0.940799i \(-0.610077\pi\)
−0.338964 + 0.940799i \(0.610077\pi\)
\(422\) −14.3620 −0.699132
\(423\) 0 0
\(424\) −6.66826 −0.323839
\(425\) −12.2162 + 30.2384i −0.592575 + 1.46678i
\(426\) 0 0
\(427\) 0 0
\(428\) −11.5136 −0.556531
\(429\) 0 0
\(430\) −11.7620 2.28616i −0.567216 0.110248i
\(431\) 26.0901i 1.25672i 0.777924 + 0.628359i \(0.216273\pi\)
−0.777924 + 0.628359i \(0.783727\pi\)
\(432\) 0 0
\(433\) 8.42614 0.404935 0.202467 0.979289i \(-0.435104\pi\)
0.202467 + 0.979289i \(0.435104\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 17.3831i 0.831548i
\(438\) 0 0
\(439\) 27.6585i 1.32007i 0.751236 + 0.660033i \(0.229458\pi\)
−0.751236 + 0.660033i \(0.770542\pi\)
\(440\) −2.81639 0.547415i −0.134266 0.0260970i
\(441\) 0 0
\(442\) 40.1049i 1.90759i
\(443\) 2.35722 0.111995 0.0559975 0.998431i \(-0.482166\pi\)
0.0559975 + 0.998431i \(0.482166\pi\)
\(444\) 0 0
\(445\) 1.55270 7.98848i 0.0736051 0.378690i
\(446\) −13.5975 −0.643861
\(447\) 0 0
\(448\) 0 0
\(449\) 12.8021i 0.604169i −0.953281 0.302084i \(-0.902318\pi\)
0.953281 0.302084i \(-0.0976824\pi\)
\(450\) 0 0
\(451\) 11.1855i 0.526707i
\(452\) 13.3875 0.629695
\(453\) 0 0
\(454\) 12.3559i 0.579893i
\(455\) 0 0
\(456\) 0 0
\(457\) 34.5242i 1.61498i −0.589884 0.807488i \(-0.700827\pi\)
0.589884 0.807488i \(-0.299173\pi\)
\(458\) 2.41341i 0.112771i
\(459\) 0 0
\(460\) −1.22835 + 6.31975i −0.0572723 + 0.294660i
\(461\) 20.2692 0.944032 0.472016 0.881590i \(-0.343526\pi\)
0.472016 + 0.881590i \(0.343526\pi\)
\(462\) 0 0
\(463\) 13.1246i 0.609952i −0.952360 0.304976i \(-0.901352\pi\)
0.952360 0.304976i \(-0.0986485\pi\)
\(464\) 1.35052i 0.0626963i
\(465\) 0 0
\(466\) −12.8756 −0.596451
\(467\) 1.53167i 0.0708773i −0.999372 0.0354387i \(-0.988717\pi\)
0.999372 0.0354387i \(-0.0112828\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.76954 + 0.343941i 0.0816228 + 0.0158648i
\(471\) 0 0
\(472\) 1.59622 0.0734720
\(473\) −6.87560 −0.316140
\(474\) 0 0
\(475\) −11.3079 + 27.9899i −0.518840 + 1.28426i
\(476\) 0 0
\(477\) 0 0
\(478\) 27.2546i 1.24660i
\(479\) −29.3170 −1.33953 −0.669764 0.742574i \(-0.733605\pi\)
−0.669764 + 0.742574i \(0.733605\pi\)
\(480\) 0 0
\(481\) 58.6199i 2.67284i
\(482\) 2.61759i 0.119228i
\(483\) 0 0
\(484\) 9.35366 0.425166
\(485\) 3.74081 19.2461i 0.169862 0.873919i
\(486\) 0 0
\(487\) 8.19540i 0.371369i −0.982609 0.185685i \(-0.940550\pi\)
0.982609 0.185685i \(-0.0594502\pi\)
\(488\) 6.35361i 0.287615i
\(489\) 0 0
\(490\) 0 0
\(491\) 40.4383i 1.82495i −0.409129 0.912477i \(-0.634167\pi\)
0.409129 0.912477i \(-0.365833\pi\)
\(492\) 0 0
\(493\) 8.80886 0.396731
\(494\) 37.1227i 1.67023i
\(495\) 0 0
\(496\) 8.65515i 0.388628i
\(497\) 0 0
\(498\) 0 0
\(499\) 17.4490 0.781125 0.390562 0.920576i \(-0.372281\pi\)
0.390562 + 0.920576i \(0.372281\pi\)
\(500\) 6.08890 9.37685i 0.272304 0.419345i
\(501\) 0 0
\(502\) 26.7173 1.19245
\(503\) 1.91949i 0.0855860i −0.999084 0.0427930i \(-0.986374\pi\)
0.999084 0.0427930i \(-0.0136256\pi\)
\(504\) 0 0
\(505\) 0.728069 3.74584i 0.0323987 0.166688i
\(506\) 3.69426i 0.164230i
\(507\) 0 0
\(508\) 5.65313i 0.250817i
\(509\) 11.0910 0.491599 0.245800 0.969321i \(-0.420949\pi\)
0.245800 + 0.969321i \(0.420949\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 10.5613i 0.465837i
\(515\) 4.42181 22.7498i 0.194848 1.00247i
\(516\) 0 0
\(517\) 1.03440 0.0454928
\(518\) 0 0
\(519\) 0 0
\(520\) −2.62322 + 13.4962i −0.115036 + 0.591847i
\(521\) 39.7412 1.74109 0.870546 0.492088i \(-0.163766\pi\)
0.870546 + 0.492088i \(0.163766\pi\)
\(522\) 0 0
\(523\) −31.2808 −1.36781 −0.683907 0.729570i \(-0.739720\pi\)
−0.683907 + 0.729570i \(0.739720\pi\)
\(524\) 12.4673 0.544638
\(525\) 0 0
\(526\) −8.33174 −0.363281
\(527\) −56.4538 −2.45917
\(528\) 0 0
\(529\) −14.7104 −0.639582
\(530\) 2.84491 14.6368i 0.123575 0.635781i
\(531\) 0 0
\(532\) 0 0
\(533\) 53.6013 2.32173
\(534\) 0 0
\(535\) 4.91210 25.2722i 0.212369 1.09261i
\(536\) 5.39463i 0.233012i
\(537\) 0 0
\(538\) 14.2834 0.615802
\(539\) 0 0
\(540\) 0 0
\(541\) −24.7974 −1.06612 −0.533061 0.846077i \(-0.678959\pi\)
−0.533061 + 0.846077i \(0.678959\pi\)
\(542\) 7.60446i 0.326640i
\(543\) 0 0
\(544\) 6.52257i 0.279653i
\(545\) −0.853270 + 4.38998i −0.0365501 + 0.188046i
\(546\) 0 0
\(547\) 40.5380i 1.73328i −0.498934 0.866640i \(-0.666275\pi\)
0.498934 0.866640i \(-0.333725\pi\)
\(548\) −0.0509686 −0.00217727
\(549\) 0 0
\(550\) 2.40314 5.94841i 0.102470 0.253641i
\(551\) 8.15384 0.347365
\(552\) 0 0
\(553\) 0 0
\(554\) 12.0400i 0.511532i
\(555\) 0 0
\(556\) 1.05069i 0.0445592i
\(557\) −23.7464 −1.00617 −0.503084 0.864238i \(-0.667801\pi\)
−0.503084 + 0.864238i \(0.667801\pi\)
\(558\) 0 0
\(559\) 32.9480i 1.39355i
\(560\) 0 0
\(561\) 0 0
\(562\) 12.7879i 0.539424i
\(563\) 8.42867i 0.355226i 0.984100 + 0.177613i \(0.0568375\pi\)
−0.984100 + 0.177613i \(0.943162\pi\)
\(564\) 0 0
\(565\) −5.71157 + 29.3854i −0.240288 + 1.23625i
\(566\) −15.5521 −0.653705
\(567\) 0 0
\(568\) 15.6787i 0.657863i
\(569\) 23.4565i 0.983348i −0.870779 0.491674i \(-0.836385\pi\)
0.870779 0.491674i \(-0.163615\pi\)
\(570\) 0 0
\(571\) 15.3589 0.642751 0.321376 0.946952i \(-0.395855\pi\)
0.321376 + 0.946952i \(0.395855\pi\)
\(572\) 7.88931i 0.329869i
\(573\) 0 0
\(574\) 0 0
\(575\) −13.3477 5.39245i −0.556639 0.224881i
\(576\) 0 0
\(577\) 14.2408 0.592853 0.296426 0.955056i \(-0.404205\pi\)
0.296426 + 0.955056i \(0.404205\pi\)
\(578\) −25.5439 −1.06248
\(579\) 0 0
\(580\) −2.96438 0.576179i −0.123089 0.0239245i
\(581\) 0 0
\(582\) 0 0
\(583\) 8.55604i 0.354355i
\(584\) 12.4167 0.513808
\(585\) 0 0
\(586\) 11.5536i 0.477274i
\(587\) 12.5249i 0.516958i 0.966017 + 0.258479i \(0.0832212\pi\)
−0.966017 + 0.258479i \(0.916779\pi\)
\(588\) 0 0
\(589\) −52.2560 −2.15317
\(590\) −0.681003 + 3.50369i −0.0280365 + 0.144245i
\(591\) 0 0
\(592\) 9.53381i 0.391837i
\(593\) 14.7150i 0.604273i −0.953265 0.302136i \(-0.902300\pi\)
0.953265 0.302136i \(-0.0976998\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.22305i 0.336829i
\(597\) 0 0
\(598\) 17.7030 0.723928
\(599\) 27.6324i 1.12903i 0.825423 + 0.564515i \(0.190937\pi\)
−0.825423 + 0.564515i \(0.809063\pi\)
\(600\) 0 0
\(601\) 8.82137i 0.359831i −0.983682 0.179916i \(-0.942418\pi\)
0.983682 0.179916i \(-0.0575825\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4.26715 −0.173628
\(605\) −3.99060 + 20.5312i −0.162241 + 0.834711i
\(606\) 0 0
\(607\) 26.5354 1.07704 0.538519 0.842613i \(-0.318984\pi\)
0.538519 + 0.842613i \(0.318984\pi\)
\(608\) 6.03756i 0.244855i
\(609\) 0 0
\(610\) −13.9461 2.71067i −0.564662 0.109752i
\(611\) 4.95686i 0.200533i
\(612\) 0 0
\(613\) 0.913683i 0.0369033i 0.999830 + 0.0184517i \(0.00587368\pi\)
−0.999830 + 0.0184517i \(0.994126\pi\)
\(614\) −2.57617 −0.103966
\(615\) 0 0
\(616\) 0 0
\(617\) −35.0558 −1.41129 −0.705646 0.708565i \(-0.749343\pi\)
−0.705646 + 0.708565i \(0.749343\pi\)
\(618\) 0 0
\(619\) 19.0967i 0.767561i 0.923424 + 0.383781i \(0.125378\pi\)
−0.923424 + 0.383781i \(0.874622\pi\)
\(620\) 18.9980 + 3.69259i 0.762977 + 0.148298i
\(621\) 0 0
\(622\) −24.7140 −0.990942
\(623\) 0 0
\(624\) 0 0
\(625\) 17.9843 + 17.3656i 0.719374 + 0.694623i
\(626\) −9.00173 −0.359782
\(627\) 0 0
\(628\) −4.83203 −0.192819
\(629\) −62.1849 −2.47947
\(630\) 0 0
\(631\) 21.5350 0.857296 0.428648 0.903472i \(-0.358990\pi\)
0.428648 + 0.903472i \(0.358990\pi\)
\(632\) 11.1819 0.444790
\(633\) 0 0
\(634\) 23.9059 0.949423
\(635\) 12.4086 + 2.41182i 0.492419 + 0.0957102i
\(636\) 0 0
\(637\) 0 0
\(638\) −1.73285 −0.0686043
\(639\) 0 0
\(640\) −0.426635 + 2.19499i −0.0168642 + 0.0867646i
\(641\) 33.1131i 1.30789i 0.756543 + 0.653943i \(0.226887\pi\)
−0.756543 + 0.653943i \(0.773113\pi\)
\(642\) 0 0
\(643\) −3.31191 −0.130609 −0.0653044 0.997865i \(-0.520802\pi\)
−0.0653044 + 0.997865i \(0.520802\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −39.3804 −1.54940
\(647\) 23.3338i 0.917348i 0.888605 + 0.458674i \(0.151675\pi\)
−0.888605 + 0.458674i \(0.848325\pi\)
\(648\) 0 0
\(649\) 2.04811i 0.0803953i
\(650\) −28.5049 11.5159i −1.11805 0.451691i
\(651\) 0 0
\(652\) 10.7532i 0.421128i
\(653\) −17.4604 −0.683277 −0.341639 0.939831i \(-0.610982\pi\)
−0.341639 + 0.939831i \(0.610982\pi\)
\(654\) 0 0
\(655\) −5.31900 + 27.3657i −0.207830 + 1.06926i
\(656\) 8.71759 0.340365
\(657\) 0 0
\(658\) 0 0
\(659\) 1.29864i 0.0505877i 0.999680 + 0.0252939i \(0.00805215\pi\)
−0.999680 + 0.0252939i \(0.991948\pi\)
\(660\) 0 0
\(661\) 5.07917i 0.197557i 0.995109 + 0.0987785i \(0.0314935\pi\)
−0.995109 + 0.0987785i \(0.968506\pi\)
\(662\) 4.36728 0.169739
\(663\) 0 0
\(664\) 3.74493i 0.145331i
\(665\) 0 0
\(666\) 0 0
\(667\) 3.88838i 0.150559i
\(668\) 18.9267i 0.732295i
\(669\) 0 0
\(670\) −11.8412 2.30154i −0.457464 0.0889161i
\(671\) −8.15232 −0.314717
\(672\) 0 0
\(673\) 18.4124i 0.709747i 0.934914 + 0.354873i \(0.115476\pi\)
−0.934914 + 0.354873i \(0.884524\pi\)
\(674\) 19.5159i 0.751723i
\(675\) 0 0
\(676\) 24.8057 0.954067
\(677\) 41.2795i 1.58650i −0.608896 0.793250i \(-0.708388\pi\)
0.608896 0.793250i \(-0.291612\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 14.3170 + 2.78275i 0.549031 + 0.106714i
\(681\) 0 0
\(682\) 11.1054 0.425249
\(683\) 22.9793 0.879278 0.439639 0.898174i \(-0.355106\pi\)
0.439639 + 0.898174i \(0.355106\pi\)
\(684\) 0 0
\(685\) 0.0217450 0.111876i 0.000830833 0.00427454i
\(686\) 0 0
\(687\) 0 0
\(688\) 5.35859i 0.204294i
\(689\) −41.0007 −1.56200
\(690\) 0 0
\(691\) 12.7072i 0.483406i −0.970350 0.241703i \(-0.922294\pi\)
0.970350 0.241703i \(-0.0777059\pi\)
\(692\) 11.8435i 0.450223i
\(693\) 0 0
\(694\) −26.9147 −1.02167
\(695\) 2.30625 + 0.448261i 0.0874812 + 0.0170035i
\(696\) 0 0
\(697\) 56.8611i 2.15377i
\(698\) 10.3821i 0.392968i
\(699\) 0 0
\(700\) 0 0
\(701\) 26.5441i 1.00256i −0.865286 0.501279i \(-0.832863\pi\)
0.865286 0.501279i \(-0.167137\pi\)
\(702\) 0 0
\(703\) −57.5609 −2.17095
\(704\) 1.28310i 0.0483586i
\(705\) 0 0
\(706\) 0.168951i 0.00635856i
\(707\) 0 0
\(708\) 0 0
\(709\) 42.3532 1.59061 0.795303 0.606212i \(-0.207312\pi\)
0.795303 + 0.606212i \(0.207312\pi\)
\(710\) −34.4146 6.68908i −1.29156 0.251037i
\(711\) 0 0
\(712\) −3.63941 −0.136393
\(713\) 24.9196i 0.933248i
\(714\) 0 0
\(715\) −17.3170 3.36586i −0.647618 0.125876i
\(716\) 3.78561i 0.141475i
\(717\) 0 0
\(718\) 37.6661i 1.40569i
\(719\) −23.9014 −0.891373 −0.445686 0.895189i \(-0.647040\pi\)
−0.445686 + 0.895189i \(0.647040\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.4521 −0.649499
\(723\) 0 0
\(724\) 19.0033i 0.706253i
\(725\) 2.52942 6.26097i 0.0939401 0.232526i
\(726\) 0 0
\(727\) −22.4529 −0.832731 −0.416365 0.909197i \(-0.636696\pi\)
−0.416365 + 0.909197i \(0.636696\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −5.29741 + 27.2546i −0.196066 + 1.00874i
\(731\) 34.9517 1.29274
\(732\) 0 0
\(733\) −27.5450 −1.01740 −0.508700 0.860944i \(-0.669874\pi\)
−0.508700 + 0.860944i \(0.669874\pi\)
\(734\) 15.1557 0.559409
\(735\) 0 0
\(736\) 2.87917 0.106128
\(737\) −6.92184 −0.254969
\(738\) 0 0
\(739\) 31.0913 1.14371 0.571856 0.820354i \(-0.306224\pi\)
0.571856 + 0.820354i \(0.306224\pi\)
\(740\) 20.9266 + 4.06745i 0.769278 + 0.149523i
\(741\) 0 0
\(742\) 0 0
\(743\) 1.12610 0.0413127 0.0206564 0.999787i \(-0.493424\pi\)
0.0206564 + 0.999787i \(0.493424\pi\)
\(744\) 0 0
\(745\) −18.0495 3.50824i −0.661283 0.128532i
\(746\) 12.0400i 0.440817i
\(747\) 0 0
\(748\) 8.36910 0.306005
\(749\) 0 0
\(750\) 0 0
\(751\) −9.54216 −0.348198 −0.174099 0.984728i \(-0.555701\pi\)
−0.174099 + 0.984728i \(0.555701\pi\)
\(752\) 0.806172i 0.0293981i
\(753\) 0 0
\(754\) 8.30386i 0.302409i
\(755\) 1.82051 9.36635i 0.0662553 0.340876i
\(756\) 0 0
\(757\) 33.1687i 1.20554i 0.797917 + 0.602768i \(0.205935\pi\)
−0.797917 + 0.602768i \(0.794065\pi\)
\(758\) 18.0918 0.657123
\(759\) 0 0
\(760\) 13.2524 + 2.57583i 0.480714 + 0.0934353i
\(761\) 4.34245 0.157414 0.0787068 0.996898i \(-0.474921\pi\)
0.0787068 + 0.996898i \(0.474921\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) 29.1538i 1.05337i
\(767\) 9.81458 0.354384
\(768\) 0 0
\(769\) 46.1795i 1.66528i 0.553818 + 0.832638i \(0.313170\pi\)
−0.553818 + 0.832638i \(0.686830\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.53892i 0.127368i
\(773\) 17.9745i 0.646497i 0.946314 + 0.323248i \(0.104775\pi\)
−0.946314 + 0.323248i \(0.895225\pi\)
\(774\) 0 0
\(775\) −16.2104 + 40.1250i −0.582295 + 1.44133i
\(776\) −8.76818 −0.314759
\(777\) 0 0
\(778\) 19.6641i 0.704994i
\(779\) 52.6330i 1.88577i
\(780\) 0 0
\(781\) −20.1173 −0.719854
\(782\) 18.7796i 0.671556i
\(783\) 0 0
\(784\) 0 0
\(785\) 2.06151 10.6063i 0.0735785 0.378554i
\(786\) 0 0
\(787\) 13.8417 0.493405 0.246702 0.969091i \(-0.420653\pi\)
0.246702 + 0.969091i \(0.420653\pi\)
\(788\) −2.36728 −0.0843308
\(789\) 0 0
\(790\) −4.77057 + 24.5441i −0.169729 + 0.873239i
\(791\) 0 0
\(792\) 0 0
\(793\) 39.0661i 1.38728i
\(794\) −5.68454 −0.201737
\(795\) 0 0
\(796\) 3.46410i 0.122782i
\(797\) 7.46138i 0.264296i −0.991230 0.132148i \(-0.957813\pi\)
0.991230 0.132148i \(-0.0421874\pi\)
\(798\) 0 0
\(799\) −5.25831 −0.186026
\(800\) −4.63597 1.87292i −0.163906 0.0662177i
\(801\) 0 0
\(802\) 10.2166i 0.360762i
\(803\) 15.9319i 0.562225i
\(804\) 0 0
\(805\) 0 0
\(806\) 53.2174i 1.87450i
\(807\) 0 0
\(808\) −1.70654 −0.0600359
\(809\) 1.48668i 0.0522689i 0.999658 + 0.0261344i \(0.00831980\pi\)
−0.999658 + 0.0261344i \(0.991680\pi\)
\(810\) 0 0
\(811\) 31.5494i 1.10785i 0.832567 + 0.553925i \(0.186871\pi\)
−0.832567 + 0.553925i \(0.813129\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 12.2328 0.428760
\(815\) 23.6032 + 4.58770i 0.826784 + 0.160700i
\(816\) 0 0
\(817\) 32.3528 1.13188
\(818\) 13.9683i 0.488390i
\(819\) 0 0
\(820\) −3.71923 + 19.1350i −0.129881 + 0.668224i
\(821\) 31.8039i 1.10996i −0.831862 0.554982i \(-0.812725\pi\)
0.831862 0.554982i \(-0.187275\pi\)
\(822\) 0 0
\(823\) 36.6371i 1.27709i 0.769585 + 0.638545i \(0.220463\pi\)
−0.769585 + 0.638545i \(0.779537\pi\)
\(824\) −10.3644 −0.361061
\(825\) 0 0
\(826\) 0 0
\(827\) −21.9864 −0.764541 −0.382271 0.924050i \(-0.624858\pi\)
−0.382271 + 0.924050i \(0.624858\pi\)
\(828\) 0 0
\(829\) 24.3062i 0.844191i −0.906551 0.422095i \(-0.861295\pi\)
0.906551 0.422095i \(-0.138705\pi\)
\(830\) 8.22008 + 1.59772i 0.285323 + 0.0554576i
\(831\) 0 0
\(832\) 6.14864 0.213166
\(833\) 0 0
\(834\) 0 0
\(835\) −41.5439 8.07478i −1.43768 0.279439i
\(836\) 7.74679 0.267928
\(837\) 0 0
\(838\) 11.3181 0.390977
\(839\) −27.3914 −0.945655 −0.472827 0.881155i \(-0.656767\pi\)
−0.472827 + 0.881155i \(0.656767\pi\)
\(840\) 0 0
\(841\) 27.1761 0.937107
\(842\) −13.9099 −0.479368
\(843\) 0 0
\(844\) −14.3620 −0.494361
\(845\) −10.5830 + 54.4484i −0.364066 + 1.87308i
\(846\) 0 0
\(847\) 0 0
\(848\) −6.66826 −0.228989
\(849\) 0 0
\(850\) −12.2162 + 30.2384i −0.419013 + 1.03717i
\(851\) 27.4494i 0.940954i
\(852\) 0 0
\(853\) 22.6073 0.774060 0.387030 0.922067i \(-0.373501\pi\)
0.387030 + 0.922067i \(0.373501\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −11.5136 −0.393527
\(857\) 47.8578i 1.63479i −0.576076 0.817396i \(-0.695417\pi\)
0.576076 0.817396i \(-0.304583\pi\)
\(858\) 0 0
\(859\) 48.0090i 1.63804i −0.573762 0.819022i \(-0.694516\pi\)
0.573762 0.819022i \(-0.305484\pi\)
\(860\) −11.7620 2.28616i −0.401082 0.0779574i
\(861\) 0 0
\(862\) 26.0901i 0.888634i
\(863\) 47.5569 1.61886 0.809428 0.587218i \(-0.199777\pi\)
0.809428 + 0.587218i \(0.199777\pi\)
\(864\) 0 0
\(865\) −25.9964 5.05286i −0.883905 0.171802i
\(866\) 8.42614 0.286332
\(867\) 0 0
\(868\) 0 0
\(869\) 14.3474i 0.486703i
\(870\) 0 0
\(871\) 33.1696i 1.12391i
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 17.3831i 0.587993i
\(875\) 0 0
\(876\) 0 0
\(877\) 37.0004i 1.24941i −0.780859 0.624707i \(-0.785218\pi\)
0.780859 0.624707i \(-0.214782\pi\)
\(878\) 27.6585i 0.933428i
\(879\) 0 0
\(880\) −2.81639 0.547415i −0.0949405 0.0184534i
\(881\) 5.97213 0.201206 0.100603 0.994927i \(-0.467923\pi\)
0.100603 + 0.994927i \(0.467923\pi\)
\(882\) 0 0
\(883\) 11.0589i 0.372163i −0.982534 0.186081i \(-0.940421\pi\)
0.982534 0.186081i \(-0.0595788\pi\)
\(884\) 40.1049i 1.34887i
\(885\) 0 0
\(886\) 2.35722 0.0791925
\(887\) 5.95070i 0.199805i 0.994997 + 0.0999026i \(0.0318531\pi\)
−0.994997 + 0.0999026i \(0.968147\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.55270 7.98848i 0.0520467 0.267774i
\(891\) 0 0
\(892\) −13.5975 −0.455279
\(893\) −4.86731 −0.162878
\(894\) 0 0
\(895\) 8.30937 + 1.61507i 0.277752 + 0.0539859i
\(896\) 0 0
\(897\) 0 0
\(898\) 12.8021i 0.427212i
\(899\) 11.6890 0.389849
\(900\) 0 0
\(901\) 43.4942i 1.44900i
\(902\) 11.1855i 0.372438i
\(903\) 0 0
\(904\) 13.3875 0.445261
\(905\) 41.7121 + 8.10748i 1.38656 + 0.269502i
\(906\) 0 0
\(907\) 55.3144i 1.83669i 0.395786 + 0.918343i \(0.370472\pi\)
−0.395786 + 0.918343i \(0.629528\pi\)
\(908\) 12.3559i 0.410046i
\(909\) 0 0
\(910\) 0 0
\(911\) 35.2080i 1.16649i −0.812295 0.583246i \(-0.801782\pi\)
0.812295 0.583246i \(-0.198218\pi\)
\(912\) 0 0
\(913\) 4.80512 0.159026
\(914\) 34.5242i 1.14196i
\(915\) 0 0
\(916\) 2.41341i 0.0797414i
\(917\) 0 0
\(918\) 0 0
\(919\) −50.8875 −1.67862 −0.839311 0.543651i \(-0.817041\pi\)
−0.839311 + 0.543651i \(0.817041\pi\)
\(920\) −1.22835 + 6.31975i −0.0404976 + 0.208356i
\(921\) 0 0
\(922\) 20.2692 0.667531
\(923\) 96.4026i 3.17313i
\(924\) 0 0
\(925\) −17.8560 + 44.1984i −0.587103 + 1.45323i
\(926\) 13.1246i 0.431301i
\(927\) 0 0
\(928\) 1.35052i 0.0443330i
\(929\) −59.6810 −1.95807 −0.979036 0.203689i \(-0.934707\pi\)
−0.979036 + 0.203689i \(0.934707\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −12.8756 −0.421754
\(933\) 0 0
\(934\) 1.53167i 0.0501179i
\(935\) −3.57055 + 18.3701i −0.116770 + 0.600766i
\(936\) 0 0
\(937\) 24.2579 0.792471 0.396235 0.918149i \(-0.370316\pi\)
0.396235 + 0.918149i \(0.370316\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.76954 + 0.343941i 0.0577160 + 0.0112181i
\(941\) −30.1079 −0.981488 −0.490744 0.871304i \(-0.663275\pi\)
−0.490744 + 0.871304i \(0.663275\pi\)
\(942\) 0 0
\(943\) 25.0994 0.817350
\(944\) 1.59622 0.0519525
\(945\) 0 0
\(946\) −6.87560 −0.223545
\(947\) −46.1625 −1.50008 −0.750039 0.661393i \(-0.769965\pi\)
−0.750039 + 0.661393i \(0.769965\pi\)
\(948\) 0 0
\(949\) 76.3460 2.47830
\(950\) −11.3079 + 27.9899i −0.366875 + 0.908112i
\(951\) 0 0
\(952\) 0 0
\(953\) 23.7121 0.768110 0.384055 0.923310i \(-0.374527\pi\)
0.384055 + 0.923310i \(0.374527\pi\)
\(954\) 0 0
\(955\) −6.20837 1.20671i −0.200898 0.0390481i
\(956\) 27.2546i 0.881478i
\(957\) 0 0
\(958\) −29.3170 −0.947190
\(959\) 0 0
\(960\) 0 0
\(961\) −43.9116 −1.41650
\(962\) 58.6199i 1.88998i
\(963\) 0 0
\(964\) 2.61759i 0.0843070i
\(965\) −7.76789 1.50983i −0.250057 0.0486030i
\(966\) 0 0
\(967\) 0.0922780i 0.00296746i 0.999999 + 0.00148373i \(0.000472286\pi\)
−0.999999 + 0.00148373i \(0.999528\pi\)
\(968\) 9.35366 0.300638
\(969\) 0 0
\(970\) 3.74081 19.2461i 0.120110 0.617954i
\(971\) −39.7005 −1.27405 −0.637024 0.770844i \(-0.719835\pi\)
−0.637024 + 0.770844i \(0.719835\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 8.19540i 0.262598i
\(975\) 0 0
\(976\) 6.35361i 0.203374i
\(977\) 36.8070 1.17756 0.588779 0.808294i \(-0.299609\pi\)
0.588779 + 0.808294i \(0.299609\pi\)
\(978\) 0 0
\(979\) 4.66973i 0.149245i
\(980\) 0 0
\(981\) 0 0
\(982\) 40.4383i 1.29044i
\(983\) 62.5731i 1.99577i −0.0650009 0.997885i \(-0.520705\pi\)
0.0650009 0.997885i \(-0.479295\pi\)
\(984\) 0 0
\(985\) 1.00996 5.19615i 0.0321801 0.165563i
\(986\) 8.80886 0.280531
\(987\) 0 0
\(988\) 37.1227i 1.18103i
\(989\) 15.4283i 0.490591i
\(990\) 0 0
\(991\) −34.8039 −1.10558 −0.552791 0.833320i \(-0.686437\pi\)
−0.552791 + 0.833320i \(0.686437\pi\)
\(992\) 8.65515i 0.274801i
\(993\) 0 0
\(994\) 0 0
\(995\) 7.60367 + 1.47791i 0.241053 + 0.0468528i
\(996\) 0 0
\(997\) 48.8631 1.54751 0.773755 0.633485i \(-0.218376\pi\)
0.773755 + 0.633485i \(0.218376\pi\)
\(998\) 17.4490 0.552339
\(999\) 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 4410.2.d.b.4409.8 16
3.2 odd 2 4410.2.d.a.4409.9 16
5.4 even 2 4410.2.d.a.4409.7 16
7.4 even 3 630.2.bo.a.89.3 yes 16
7.5 odd 6 630.2.bo.a.269.1 yes 16
7.6 odd 2 inner 4410.2.d.b.4409.9 16
15.14 odd 2 inner 4410.2.d.b.4409.10 16
21.5 even 6 630.2.bo.b.269.8 yes 16
21.11 odd 6 630.2.bo.b.89.6 yes 16
21.20 even 2 4410.2.d.a.4409.8 16
35.4 even 6 630.2.bo.b.89.8 yes 16
35.12 even 12 3150.2.bf.f.1151.9 32
35.18 odd 12 3150.2.bf.f.1601.10 32
35.19 odd 6 630.2.bo.b.269.6 yes 16
35.32 odd 12 3150.2.bf.f.1601.7 32
35.33 even 12 3150.2.bf.f.1151.8 32
35.34 odd 2 4410.2.d.a.4409.10 16
105.32 even 12 3150.2.bf.f.1601.9 32
105.47 odd 12 3150.2.bf.f.1151.7 32
105.53 even 12 3150.2.bf.f.1601.8 32
105.68 odd 12 3150.2.bf.f.1151.10 32
105.74 odd 6 630.2.bo.a.89.1 16
105.89 even 6 630.2.bo.a.269.3 yes 16
105.104 even 2 inner 4410.2.d.b.4409.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.bo.a.89.1 16 105.74 odd 6
630.2.bo.a.89.3 yes 16 7.4 even 3
630.2.bo.a.269.1 yes 16 7.5 odd 6
630.2.bo.a.269.3 yes 16 105.89 even 6
630.2.bo.b.89.6 yes 16 21.11 odd 6
630.2.bo.b.89.8 yes 16 35.4 even 6
630.2.bo.b.269.6 yes 16 35.19 odd 6
630.2.bo.b.269.8 yes 16 21.5 even 6
3150.2.bf.f.1151.7 32 105.47 odd 12
3150.2.bf.f.1151.8 32 35.33 even 12
3150.2.bf.f.1151.9 32 35.12 even 12
3150.2.bf.f.1151.10 32 105.68 odd 12
3150.2.bf.f.1601.7 32 35.32 odd 12
3150.2.bf.f.1601.8 32 105.53 even 12
3150.2.bf.f.1601.9 32 105.32 even 12
3150.2.bf.f.1601.10 32 35.18 odd 12
4410.2.d.a.4409.7 16 5.4 even 2
4410.2.d.a.4409.8 16 21.20 even 2
4410.2.d.a.4409.9 16 3.2 odd 2
4410.2.d.a.4409.10 16 35.34 odd 2
4410.2.d.b.4409.7 16 105.104 even 2 inner
4410.2.d.b.4409.8 16 1.1 even 1 trivial
4410.2.d.b.4409.9 16 7.6 odd 2 inner
4410.2.d.b.4409.10 16 15.14 odd 2 inner