Properties

Label 2736.2.dc.e.449.5
Level $2736$
Weight $2$
Character 2736.449
Analytic conductor $21.847$
Analytic rank $0$
Dimension $20$
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] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + 30728 x^{12} - 110512 x^{11} - 211368 x^{10} + 231024 x^{9} + \cdots + 414864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.5
Root \(1.85589 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2736.449
Dual form 2736.2.dc.e.1889.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0242659 + 0.0140099i) q^{5} +1.95697 q^{7} +O(q^{10})\) \(q+(-0.0242659 + 0.0140099i) q^{5} +1.95697 q^{7} -2.26548i q^{11} +(-1.04824 - 0.605203i) q^{13} +(5.68055 - 3.27967i) q^{17} +(-3.90798 - 1.93073i) q^{19} +(2.86319 + 1.65306i) q^{23} +(-2.49961 + 4.32945i) q^{25} +(0.972667 - 1.68471i) q^{29} -7.53692i q^{31} +(-0.0474875 + 0.0274169i) q^{35} +1.60246i q^{37} +(5.51426 + 9.55099i) q^{41} +(-6.35764 - 11.0117i) q^{43} +(-5.51811 - 3.18588i) q^{47} -3.17028 q^{49} +(5.92934 - 10.2699i) q^{53} +(0.0317391 + 0.0549738i) q^{55} +(-2.67047 - 4.62540i) q^{59} +(-0.233163 + 0.403850i) q^{61} +0.0339154 q^{65} +(4.18806 + 2.41798i) q^{67} +(1.11337 + 1.92842i) q^{71} +(1.85674 + 3.21597i) q^{73} -4.43346i q^{77} +(4.79782 - 2.77002i) q^{79} +2.64123i q^{83} +(-0.0918956 + 0.159168i) q^{85} +(1.78874 - 3.09818i) q^{89} +(-2.05138 - 1.18436i) q^{91} +(0.121880 - 0.00789951i) q^{95} +(9.80237 - 5.65940i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 4 q^{19} + 14 q^{25} - 12 q^{35} - 8 q^{41} + 2 q^{43} - 36 q^{47} + 32 q^{49} + 8 q^{53} - 12 q^{55} + 8 q^{59} - 2 q^{61} - 8 q^{65} - 30 q^{67} - 4 q^{71} - 22 q^{73} - 54 q^{79} - 4 q^{85} + 32 q^{89} - 18 q^{91} - 32 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.0242659 + 0.0140099i −0.0108520 + 0.00626542i −0.505416 0.862876i \(-0.668661\pi\)
0.494564 + 0.869141i \(0.335328\pi\)
\(6\) 0 0
\(7\) 1.95697 0.739664 0.369832 0.929099i \(-0.379415\pi\)
0.369832 + 0.929099i \(0.379415\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.26548i 0.683067i −0.939870 0.341534i \(-0.889054\pi\)
0.939870 0.341534i \(-0.110946\pi\)
\(12\) 0 0
\(13\) −1.04824 0.605203i −0.290730 0.167853i 0.347541 0.937665i \(-0.387017\pi\)
−0.638271 + 0.769812i \(0.720350\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.68055 3.27967i 1.37774 0.795436i 0.385849 0.922562i \(-0.373909\pi\)
0.991887 + 0.127126i \(0.0405754\pi\)
\(18\) 0 0
\(19\) −3.90798 1.93073i −0.896551 0.442940i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.86319 + 1.65306i 0.597015 + 0.344687i 0.767867 0.640610i \(-0.221318\pi\)
−0.170851 + 0.985297i \(0.554652\pi\)
\(24\) 0 0
\(25\) −2.49961 + 4.32945i −0.499921 + 0.865889i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.972667 1.68471i 0.180620 0.312843i −0.761472 0.648198i \(-0.775523\pi\)
0.942092 + 0.335355i \(0.108856\pi\)
\(30\) 0 0
\(31\) 7.53692i 1.35367i −0.736134 0.676836i \(-0.763351\pi\)
0.736134 0.676836i \(-0.236649\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.0474875 + 0.0274169i −0.00802685 + 0.00463431i
\(36\) 0 0
\(37\) 1.60246i 0.263444i 0.991287 + 0.131722i \(0.0420505\pi\)
−0.991287 + 0.131722i \(0.957949\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.51426 + 9.55099i 0.861183 + 1.49161i 0.870788 + 0.491659i \(0.163609\pi\)
−0.00960424 + 0.999954i \(0.503057\pi\)
\(42\) 0 0
\(43\) −6.35764 11.0117i −0.969531 1.67928i −0.696916 0.717153i \(-0.745445\pi\)
−0.272615 0.962123i \(-0.587888\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.51811 3.18588i −0.804899 0.464708i 0.0402825 0.999188i \(-0.487174\pi\)
−0.845181 + 0.534480i \(0.820508\pi\)
\(48\) 0 0
\(49\) −3.17028 −0.452898
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.92934 10.2699i 0.814457 1.41068i −0.0952593 0.995452i \(-0.530368\pi\)
0.909717 0.415229i \(-0.136299\pi\)
\(54\) 0 0
\(55\) 0.0317391 + 0.0549738i 0.00427970 + 0.00741266i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.67047 4.62540i −0.347666 0.602175i 0.638168 0.769897i \(-0.279692\pi\)
−0.985834 + 0.167721i \(0.946359\pi\)
\(60\) 0 0
\(61\) −0.233163 + 0.403850i −0.0298534 + 0.0517077i −0.880566 0.473923i \(-0.842837\pi\)
0.850713 + 0.525631i \(0.176171\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.0339154 0.00420669
\(66\) 0 0
\(67\) 4.18806 + 2.41798i 0.511652 + 0.295403i 0.733513 0.679676i \(-0.237880\pi\)
−0.221860 + 0.975078i \(0.571213\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.11337 + 1.92842i 0.132133 + 0.228861i 0.924499 0.381185i \(-0.124484\pi\)
−0.792366 + 0.610047i \(0.791151\pi\)
\(72\) 0 0
\(73\) 1.85674 + 3.21597i 0.217315 + 0.376401i 0.953986 0.299850i \(-0.0969367\pi\)
−0.736671 + 0.676251i \(0.763603\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.43346i 0.505240i
\(78\) 0 0
\(79\) 4.79782 2.77002i 0.539796 0.311652i −0.205200 0.978720i \(-0.565784\pi\)
0.744996 + 0.667068i \(0.232451\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.64123i 0.289913i 0.989438 + 0.144956i \(0.0463041\pi\)
−0.989438 + 0.144956i \(0.953696\pi\)
\(84\) 0 0
\(85\) −0.0918956 + 0.159168i −0.00996748 + 0.0172642i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.78874 3.09818i 0.189606 0.328406i −0.755513 0.655133i \(-0.772612\pi\)
0.945119 + 0.326727i \(0.105946\pi\)
\(90\) 0 0
\(91\) −2.05138 1.18436i −0.215043 0.124155i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.121880 0.00789951i 0.0125046 0.000810473i
\(96\) 0 0
\(97\) 9.80237 5.65940i 0.995279 0.574625i 0.0884312 0.996082i \(-0.471815\pi\)
0.906848 + 0.421457i \(0.138481\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.1242 + 5.84519i 1.00739 + 0.581618i 0.910427 0.413670i \(-0.135753\pi\)
0.0969652 + 0.995288i \(0.469086\pi\)
\(102\) 0 0
\(103\) 10.3345i 1.01829i −0.860682 0.509144i \(-0.829962\pi\)
0.860682 0.509144i \(-0.170038\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.30575 −0.222905 −0.111453 0.993770i \(-0.535550\pi\)
−0.111453 + 0.993770i \(0.535550\pi\)
\(108\) 0 0
\(109\) −11.9111 + 6.87687i −1.14088 + 0.658684i −0.946647 0.322272i \(-0.895553\pi\)
−0.194228 + 0.980956i \(0.562220\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.919294 −0.0864799 −0.0432400 0.999065i \(-0.513768\pi\)
−0.0432400 + 0.999065i \(0.513768\pi\)
\(114\) 0 0
\(115\) −0.0926369 −0.00863844
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.1166 6.41819i 1.01906 0.588355i
\(120\) 0 0
\(121\) 5.86761 0.533419
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.280176i 0.0250597i
\(126\) 0 0
\(127\) −12.9970 7.50381i −1.15330 0.665856i −0.203607 0.979053i \(-0.565267\pi\)
−0.949688 + 0.313197i \(0.898600\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.4272 10.6389i 1.60999 0.929529i 0.620620 0.784111i \(-0.286881\pi\)
0.989370 0.145417i \(-0.0464525\pi\)
\(132\) 0 0
\(133\) −7.64778 3.77838i −0.663146 0.327627i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.71514 0.990237i −0.146534 0.0846017i 0.424940 0.905221i \(-0.360295\pi\)
−0.571475 + 0.820620i \(0.693628\pi\)
\(138\) 0 0
\(139\) −1.31672 + 2.28063i −0.111683 + 0.193441i −0.916449 0.400152i \(-0.868957\pi\)
0.804766 + 0.593592i \(0.202291\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.37107 + 2.37477i −0.114655 + 0.198588i
\(144\) 0 0
\(145\) 0.0545079i 0.00452664i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.5891 9.57774i 1.35903 0.784639i 0.369540 0.929215i \(-0.379515\pi\)
0.989494 + 0.144576i \(0.0461818\pi\)
\(150\) 0 0
\(151\) 11.5023i 0.936041i 0.883718 + 0.468021i \(0.155033\pi\)
−0.883718 + 0.468021i \(0.844967\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.105592 + 0.182890i 0.00848132 + 0.0146901i
\(156\) 0 0
\(157\) 0.536899 + 0.929936i 0.0428492 + 0.0742169i 0.886655 0.462432i \(-0.153023\pi\)
−0.843805 + 0.536649i \(0.819690\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.60316 + 3.23498i 0.441591 + 0.254952i
\(162\) 0 0
\(163\) 6.23141 0.488082 0.244041 0.969765i \(-0.421527\pi\)
0.244041 + 0.969765i \(0.421527\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.41891 + 2.45762i −0.109798 + 0.190176i −0.915688 0.401889i \(-0.868354\pi\)
0.805890 + 0.592065i \(0.201687\pi\)
\(168\) 0 0
\(169\) −5.76746 9.98953i −0.443651 0.768425i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.22395 10.7802i −0.473198 0.819603i 0.526331 0.850280i \(-0.323567\pi\)
−0.999529 + 0.0306763i \(0.990234\pi\)
\(174\) 0 0
\(175\) −4.89165 + 8.47258i −0.369774 + 0.640467i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.78997 −0.208532 −0.104266 0.994549i \(-0.533249\pi\)
−0.104266 + 0.994549i \(0.533249\pi\)
\(180\) 0 0
\(181\) −18.8842 10.9028i −1.40365 0.810401i −0.408889 0.912584i \(-0.634084\pi\)
−0.994766 + 0.102183i \(0.967417\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.0224504 0.0388852i −0.00165059 0.00285890i
\(186\) 0 0
\(187\) −7.43001 12.8691i −0.543336 0.941085i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.08745i 0.0786848i 0.999226 + 0.0393424i \(0.0125263\pi\)
−0.999226 + 0.0393424i \(0.987474\pi\)
\(192\) 0 0
\(193\) 5.15756 2.97772i 0.371249 0.214341i −0.302755 0.953069i \(-0.597906\pi\)
0.674004 + 0.738728i \(0.264573\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.52729i 0.180062i −0.995939 0.0900312i \(-0.971303\pi\)
0.995939 0.0900312i \(-0.0286967\pi\)
\(198\) 0 0
\(199\) 8.26276 14.3115i 0.585731 1.01452i −0.409053 0.912511i \(-0.634141\pi\)
0.994784 0.102006i \(-0.0325259\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.90348 3.29692i 0.133598 0.231398i
\(204\) 0 0
\(205\) −0.267617 0.154509i −0.0186912 0.0107914i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.37403 + 8.85343i −0.302558 + 0.612405i
\(210\) 0 0
\(211\) 6.08255 3.51176i 0.418740 0.241760i −0.275798 0.961216i \(-0.588942\pi\)
0.694538 + 0.719456i \(0.255609\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.308547 + 0.178140i 0.0210427 + 0.0121490i
\(216\) 0 0
\(217\) 14.7495i 1.00126i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.93946 −0.534066
\(222\) 0 0
\(223\) −18.7211 + 10.8086i −1.25366 + 0.723799i −0.971834 0.235668i \(-0.924272\pi\)
−0.281822 + 0.959467i \(0.590939\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.5102 0.697587 0.348794 0.937200i \(-0.386591\pi\)
0.348794 + 0.937200i \(0.386591\pi\)
\(228\) 0 0
\(229\) 22.4754 1.48522 0.742609 0.669726i \(-0.233588\pi\)
0.742609 + 0.669726i \(0.233588\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.2218 11.0977i 1.25926 0.727036i 0.286332 0.958130i \(-0.407564\pi\)
0.972931 + 0.231094i \(0.0742306\pi\)
\(234\) 0 0
\(235\) 0.178536 0.0116464
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.6897i 1.27362i −0.771019 0.636812i \(-0.780253\pi\)
0.771019 0.636812i \(-0.219747\pi\)
\(240\) 0 0
\(241\) 1.25313 + 0.723492i 0.0807209 + 0.0466042i 0.539817 0.841782i \(-0.318493\pi\)
−0.459096 + 0.888387i \(0.651827\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.0769297 0.0444154i 0.00491486 0.00283759i
\(246\) 0 0
\(247\) 2.92802 + 4.38900i 0.186306 + 0.279265i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.4324 + 6.02313i 0.658486 + 0.380177i 0.791700 0.610910i \(-0.209196\pi\)
−0.133214 + 0.991087i \(0.542530\pi\)
\(252\) 0 0
\(253\) 3.74497 6.48648i 0.235444 0.407802i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.6723 + 20.2169i −0.728095 + 1.26110i 0.229592 + 0.973287i \(0.426261\pi\)
−0.957687 + 0.287811i \(0.907072\pi\)
\(258\) 0 0
\(259\) 3.13597i 0.194860i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.1039 11.6070i 1.23966 0.715717i 0.270634 0.962682i \(-0.412767\pi\)
0.969024 + 0.246965i \(0.0794333\pi\)
\(264\) 0 0
\(265\) 0.332278i 0.0204117i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −13.5865 23.5324i −0.828381 1.43480i −0.899307 0.437317i \(-0.855929\pi\)
0.0709262 0.997482i \(-0.477405\pi\)
\(270\) 0 0
\(271\) 13.5687 + 23.5017i 0.824241 + 1.42763i 0.902498 + 0.430693i \(0.141731\pi\)
−0.0782578 + 0.996933i \(0.524936\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.80826 + 5.66280i 0.591461 + 0.341480i
\(276\) 0 0
\(277\) −20.2717 −1.21801 −0.609003 0.793168i \(-0.708430\pi\)
−0.609003 + 0.793168i \(0.708430\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.8887 + 18.8598i −0.649564 + 1.12508i 0.333662 + 0.942693i \(0.391715\pi\)
−0.983227 + 0.182386i \(0.941618\pi\)
\(282\) 0 0
\(283\) 8.51299 + 14.7449i 0.506045 + 0.876495i 0.999976 + 0.00699380i \(0.00222622\pi\)
−0.493931 + 0.869501i \(0.664440\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.7912 + 18.6910i 0.636986 + 1.10329i
\(288\) 0 0
\(289\) 13.0124 22.5382i 0.765436 1.32577i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −11.4499 −0.668911 −0.334456 0.942411i \(-0.608552\pi\)
−0.334456 + 0.942411i \(0.608552\pi\)
\(294\) 0 0
\(295\) 0.129603 + 0.0748262i 0.00754577 + 0.00435655i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.00088 3.46562i −0.115714 0.200422i
\(300\) 0 0
\(301\) −12.4417 21.5496i −0.717127 1.24210i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.0130664i 0.000748178i
\(306\) 0 0
\(307\) −2.02178 + 1.16727i −0.115389 + 0.0666198i −0.556584 0.830792i \(-0.687888\pi\)
0.441195 + 0.897411i \(0.354555\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.3009i 0.584112i 0.956401 + 0.292056i \(0.0943393\pi\)
−0.956401 + 0.292056i \(0.905661\pi\)
\(312\) 0 0
\(313\) −7.62720 + 13.2107i −0.431115 + 0.746713i −0.996970 0.0777919i \(-0.975213\pi\)
0.565855 + 0.824505i \(0.308546\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.14205 10.6384i 0.344972 0.597509i −0.640377 0.768061i \(-0.721222\pi\)
0.985349 + 0.170552i \(0.0545550\pi\)
\(318\) 0 0
\(319\) −3.81667 2.20355i −0.213692 0.123375i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −28.5316 + 1.84925i −1.58754 + 0.102895i
\(324\) 0 0
\(325\) 5.24039 3.02554i 0.290685 0.167827i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.7987 6.23466i −0.595354 0.343728i
\(330\) 0 0
\(331\) 27.8419i 1.53033i 0.643833 + 0.765166i \(0.277343\pi\)
−0.643833 + 0.765166i \(0.722657\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.135502 −0.00740329
\(336\) 0 0
\(337\) −2.04577 + 1.18113i −0.111440 + 0.0643401i −0.554684 0.832061i \(-0.687161\pi\)
0.443244 + 0.896401i \(0.353828\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −17.0747 −0.924648
\(342\) 0 0
\(343\) −19.9029 −1.07466
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.6674 + 9.62291i −0.894751 + 0.516585i −0.875494 0.483230i \(-0.839464\pi\)
−0.0192576 + 0.999815i \(0.506130\pi\)
\(348\) 0 0
\(349\) −4.64786 −0.248794 −0.124397 0.992232i \(-0.539700\pi\)
−0.124397 + 0.992232i \(0.539700\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 26.0686i 1.38749i 0.720220 + 0.693746i \(0.244041\pi\)
−0.720220 + 0.693746i \(0.755959\pi\)
\(354\) 0 0
\(355\) −0.0540339 0.0311965i −0.00286782 0.00165574i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.70064 + 0.981864i −0.0897563 + 0.0518208i −0.544206 0.838951i \(-0.683169\pi\)
0.454450 + 0.890772i \(0.349836\pi\)
\(360\) 0 0
\(361\) 11.5446 + 15.0905i 0.607608 + 0.794237i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.0901109 0.0520256i −0.00471662 0.00272314i
\(366\) 0 0
\(367\) 6.75091 11.6929i 0.352395 0.610365i −0.634274 0.773108i \(-0.718701\pi\)
0.986668 + 0.162743i \(0.0520342\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.6035 20.0979i 0.602425 1.04343i
\(372\) 0 0
\(373\) 7.71743i 0.399593i −0.979837 0.199797i \(-0.935972\pi\)
0.979837 0.199797i \(-0.0640281\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.03918 + 1.17732i −0.105023 + 0.0606352i
\(378\) 0 0
\(379\) 24.4327i 1.25503i 0.778606 + 0.627513i \(0.215927\pi\)
−0.778606 + 0.627513i \(0.784073\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.25602 7.37165i −0.217473 0.376674i 0.736562 0.676370i \(-0.236448\pi\)
−0.954035 + 0.299696i \(0.903115\pi\)
\(384\) 0 0
\(385\) 0.0621124 + 0.107582i 0.00316554 + 0.00548288i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.86282 1.65285i −0.145151 0.0838030i 0.425666 0.904881i \(-0.360040\pi\)
−0.570817 + 0.821078i \(0.693373\pi\)
\(390\) 0 0
\(391\) 21.6859 1.09671
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.0776155 + 0.134434i −0.00390526 + 0.00676411i
\(396\) 0 0
\(397\) 0.136363 + 0.236188i 0.00684388 + 0.0118540i 0.869427 0.494061i \(-0.164488\pi\)
−0.862583 + 0.505915i \(0.831155\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.81087 + 4.86857i 0.140368 + 0.243125i 0.927635 0.373487i \(-0.121838\pi\)
−0.787267 + 0.616612i \(0.788505\pi\)
\(402\) 0 0
\(403\) −4.56137 + 7.90053i −0.227218 + 0.393553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.63035 0.179950
\(408\) 0 0
\(409\) 24.1399 + 13.9372i 1.19364 + 0.689148i 0.959130 0.282966i \(-0.0913182\pi\)
0.234510 + 0.972114i \(0.424652\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.22603 9.05175i −0.257156 0.445407i
\(414\) 0 0
\(415\) −0.0370034 0.0640917i −0.00181642 0.00314614i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.9837i 1.02512i 0.858651 + 0.512560i \(0.171303\pi\)
−0.858651 + 0.512560i \(0.828697\pi\)
\(420\) 0 0
\(421\) −17.5540 + 10.1348i −0.855530 + 0.493941i −0.862513 0.506035i \(-0.831111\pi\)
0.00698280 + 0.999976i \(0.497777\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 32.7915i 1.59062i
\(426\) 0 0
\(427\) −0.456292 + 0.790321i −0.0220815 + 0.0382463i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.56201 + 16.5619i −0.460586 + 0.797758i −0.998990 0.0449285i \(-0.985694\pi\)
0.538404 + 0.842687i \(0.319027\pi\)
\(432\) 0 0
\(433\) 24.4861 + 14.1371i 1.17673 + 0.679383i 0.955255 0.295783i \(-0.0955806\pi\)
0.221472 + 0.975167i \(0.428914\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.99764 11.9882i −0.382579 0.573472i
\(438\) 0 0
\(439\) −22.0618 + 12.7374i −1.05295 + 0.607923i −0.923475 0.383659i \(-0.874664\pi\)
−0.129479 + 0.991582i \(0.541330\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.423087 0.244269i −0.0201015 0.0116056i 0.489916 0.871770i \(-0.337028\pi\)
−0.510017 + 0.860164i \(0.670361\pi\)
\(444\) 0 0
\(445\) 0.100240i 0.00475184i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.5253 0.874263 0.437131 0.899398i \(-0.355994\pi\)
0.437131 + 0.899398i \(0.355994\pi\)
\(450\) 0 0
\(451\) 21.6375 12.4924i 1.01887 0.588246i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.0663713 0.00311153
\(456\) 0 0
\(457\) 0.784145 0.0366808 0.0183404 0.999832i \(-0.494162\pi\)
0.0183404 + 0.999832i \(0.494162\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −7.52710 + 4.34578i −0.350572 + 0.202403i −0.664937 0.746899i \(-0.731542\pi\)
0.314365 + 0.949302i \(0.398208\pi\)
\(462\) 0 0
\(463\) 25.9548 1.20622 0.603112 0.797656i \(-0.293927\pi\)
0.603112 + 0.797656i \(0.293927\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.2996i 1.35583i 0.735143 + 0.677913i \(0.237115\pi\)
−0.735143 + 0.677913i \(0.762885\pi\)
\(468\) 0 0
\(469\) 8.19588 + 4.73190i 0.378451 + 0.218499i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −24.9469 + 14.4031i −1.14706 + 0.662254i
\(474\) 0 0
\(475\) 18.1274 12.0933i 0.831742 0.554879i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −11.5829 6.68741i −0.529237 0.305555i 0.211468 0.977385i \(-0.432175\pi\)
−0.740706 + 0.671829i \(0.765509\pi\)
\(480\) 0 0
\(481\) 0.969817 1.67977i 0.0442198 0.0765910i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.158575 + 0.274661i −0.00720053 + 0.0124717i
\(486\) 0 0
\(487\) 7.50566i 0.340114i 0.985434 + 0.170057i \(0.0543952\pi\)
−0.985434 + 0.170057i \(0.945605\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.62320 0.937156i 0.0732540 0.0422932i −0.462926 0.886397i \(-0.653200\pi\)
0.536180 + 0.844104i \(0.319867\pi\)
\(492\) 0 0
\(493\) 12.7601i 0.574686i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.17883 + 3.77385i 0.0977340 + 0.169280i
\(498\) 0 0
\(499\) −12.6861 21.9729i −0.567907 0.983643i −0.996773 0.0802750i \(-0.974420\pi\)
0.428866 0.903368i \(-0.358913\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15.1031 8.71977i −0.673413 0.388795i 0.123955 0.992288i \(-0.460442\pi\)
−0.797369 + 0.603492i \(0.793775\pi\)
\(504\) 0 0
\(505\) −0.327562 −0.0145763
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.8213 30.8674i 0.789916 1.36817i −0.136102 0.990695i \(-0.543458\pi\)
0.926018 0.377479i \(-0.123209\pi\)
\(510\) 0 0
\(511\) 3.63358 + 6.29355i 0.160740 + 0.278410i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.144785 + 0.250775i 0.00638000 + 0.0110505i
\(516\) 0 0
\(517\) −7.21754 + 12.5011i −0.317427 + 0.549800i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −21.1812 −0.927965 −0.463982 0.885844i \(-0.653580\pi\)
−0.463982 + 0.885844i \(0.653580\pi\)
\(522\) 0 0
\(523\) 19.7491 + 11.4022i 0.863568 + 0.498581i 0.865206 0.501417i \(-0.167188\pi\)
−0.00163745 + 0.999999i \(0.500521\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24.7186 42.8138i −1.07676 1.86500i
\(528\) 0 0
\(529\) −6.03478 10.4525i −0.262382 0.454458i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13.3490i 0.578210i
\(534\) 0 0
\(535\) 0.0559510 0.0323033i 0.00241897 0.00139659i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 7.18220i 0.309359i
\(540\) 0 0
\(541\) −20.5455 + 35.5858i −0.883318 + 1.52995i −0.0356895 + 0.999363i \(0.511363\pi\)
−0.847629 + 0.530589i \(0.821971\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.192689 0.333746i 0.00825387 0.0142961i
\(546\) 0 0
\(547\) −37.0347 21.3820i −1.58349 0.914229i −0.994345 0.106203i \(-0.966131\pi\)
−0.589146 0.808026i \(-0.700536\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.05388 + 4.70584i −0.300505 + 0.200476i
\(552\) 0 0
\(553\) 9.38916 5.42084i 0.399268 0.230517i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.40605 + 3.12119i 0.229062 + 0.132249i 0.610139 0.792294i \(-0.291114\pi\)
−0.381077 + 0.924543i \(0.624447\pi\)
\(558\) 0 0
\(559\) 15.3907i 0.650955i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.28035 −0.264685 −0.132343 0.991204i \(-0.542250\pi\)
−0.132343 + 0.991204i \(0.542250\pi\)
\(564\) 0 0
\(565\) 0.0223075 0.0128792i 0.000938483 0.000541833i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −40.8651 −1.71315 −0.856576 0.516021i \(-0.827413\pi\)
−0.856576 + 0.516021i \(0.827413\pi\)
\(570\) 0 0
\(571\) −10.7423 −0.449551 −0.224775 0.974411i \(-0.572165\pi\)
−0.224775 + 0.974411i \(0.572165\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −14.3137 + 8.26401i −0.596922 + 0.344633i
\(576\) 0 0
\(577\) 24.6355 1.02559 0.512795 0.858511i \(-0.328610\pi\)
0.512795 + 0.858511i \(0.328610\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.16880i 0.214438i
\(582\) 0 0
\(583\) −23.2663 13.4328i −0.963590 0.556329i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.64931 3.26163i 0.233172 0.134622i −0.378863 0.925453i \(-0.623685\pi\)
0.612034 + 0.790831i \(0.290351\pi\)
\(588\) 0 0
\(589\) −14.5518 + 29.4541i −0.599595 + 1.21364i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 14.6643 + 8.46645i 0.602192 + 0.347676i 0.769903 0.638161i \(-0.220304\pi\)
−0.167712 + 0.985836i \(0.553638\pi\)
\(594\) 0 0
\(595\) −0.179837 + 0.311486i −0.00737258 + 0.0127697i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.07972 + 13.9945i −0.330128 + 0.571799i −0.982537 0.186069i \(-0.940425\pi\)
0.652408 + 0.757868i \(0.273759\pi\)
\(600\) 0 0
\(601\) 16.6915i 0.680861i −0.940270 0.340430i \(-0.889427\pi\)
0.940270 0.340430i \(-0.110573\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.142383 + 0.0822047i −0.00578868 + 0.00334210i
\(606\) 0 0
\(607\) 23.6154i 0.958521i 0.877673 + 0.479261i \(0.159095\pi\)
−0.877673 + 0.479261i \(0.840905\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.85621 + 6.67915i 0.156006 + 0.270210i
\(612\) 0 0
\(613\) 10.5746 + 18.3158i 0.427106 + 0.739769i 0.996615 0.0822162i \(-0.0261998\pi\)
−0.569509 + 0.821985i \(0.692866\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0334 + 19.6492i 1.37013 + 0.791046i 0.990944 0.134273i \(-0.0428700\pi\)
0.379188 + 0.925320i \(0.376203\pi\)
\(618\) 0 0
\(619\) −5.65391 −0.227250 −0.113625 0.993524i \(-0.536246\pi\)
−0.113625 + 0.993524i \(0.536246\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.50049 6.06303i 0.140244 0.242910i
\(624\) 0 0
\(625\) −12.4941 21.6404i −0.499764 0.865617i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.25555 + 9.10288i 0.209552 + 0.362955i
\(630\) 0 0
\(631\) 20.9285 36.2492i 0.833151 1.44306i −0.0623768 0.998053i \(-0.519868\pi\)
0.895527 0.445006i \(-0.146799\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.420511 0.0166875
\(636\) 0 0
\(637\) 3.32323 + 1.91867i 0.131671 + 0.0760203i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.91177 + 10.2395i 0.233501 + 0.404435i 0.958836 0.283961i \(-0.0916486\pi\)
−0.725335 + 0.688396i \(0.758315\pi\)
\(642\) 0 0
\(643\) 3.55537 + 6.15808i 0.140210 + 0.242851i 0.927576 0.373635i \(-0.121889\pi\)
−0.787366 + 0.616486i \(0.788556\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 35.9205i 1.41218i 0.708121 + 0.706091i \(0.249543\pi\)
−0.708121 + 0.706091i \(0.750457\pi\)
\(648\) 0 0
\(649\) −10.4787 + 6.04990i −0.411326 + 0.237479i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 35.4760i 1.38828i 0.719839 + 0.694141i \(0.244216\pi\)
−0.719839 + 0.694141i \(0.755784\pi\)
\(654\) 0 0
\(655\) −0.298101 + 0.516326i −0.0116478 + 0.0201745i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10.3388 + 17.9073i −0.402741 + 0.697568i −0.994056 0.108873i \(-0.965276\pi\)
0.591314 + 0.806441i \(0.298609\pi\)
\(660\) 0 0
\(661\) −4.99396 2.88326i −0.194242 0.112146i 0.399725 0.916635i \(-0.369106\pi\)
−0.593967 + 0.804489i \(0.702439\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.238515 0.0154591i 0.00924920 0.000599477i
\(666\) 0 0
\(667\) 5.56985 3.21576i 0.215666 0.124515i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.914913 + 0.528225i 0.0353198 + 0.0203919i
\(672\) 0 0
\(673\) 39.9658i 1.54057i 0.637702 + 0.770284i \(0.279885\pi\)
−0.637702 + 0.770284i \(0.720115\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.9424 −0.881747 −0.440874 0.897569i \(-0.645331\pi\)
−0.440874 + 0.897569i \(0.645331\pi\)
\(678\) 0 0
\(679\) 19.1829 11.0753i 0.736172 0.425029i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −45.4321 −1.73841 −0.869206 0.494449i \(-0.835370\pi\)
−0.869206 + 0.494449i \(0.835370\pi\)
\(684\) 0 0
\(685\) 0.0554925 0.00212026
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.4308 + 7.17691i −0.473575 + 0.273419i
\(690\) 0 0
\(691\) 35.0596 1.33373 0.666865 0.745178i \(-0.267636\pi\)
0.666865 + 0.745178i \(0.267636\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.0737887i 0.00279897i
\(696\) 0 0
\(697\) 62.6481 + 36.1699i 2.37296 + 1.37003i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.0720 15.0527i 0.984725 0.568531i 0.0810317 0.996712i \(-0.474178\pi\)
0.903693 + 0.428180i \(0.140845\pi\)
\(702\) 0 0
\(703\) 3.09393 6.26239i 0.116690 0.236191i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.8127 + 11.4388i 0.745132 + 0.430202i
\(708\) 0 0
\(709\) −16.9216 + 29.3091i −0.635505 + 1.10073i 0.350903 + 0.936412i \(0.385875\pi\)
−0.986408 + 0.164315i \(0.947459\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.4590 21.5796i 0.466593 0.808163i
\(714\) 0 0
\(715\) 0.0768345i 0.00287345i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.3342 + 7.69852i −0.497283 + 0.287106i −0.727591 0.686011i \(-0.759360\pi\)
0.230308 + 0.973118i \(0.426027\pi\)
\(720\) 0 0
\(721\) 20.2242i 0.753190i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.86257 + 8.42222i 0.180591 + 0.312793i
\(726\) 0 0
\(727\) −13.9650 24.1881i −0.517934 0.897088i −0.999783 0.0208336i \(-0.993368\pi\)
0.481849 0.876254i \(-0.339965\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −72.2297 41.7018i −2.67151 1.54240i
\(732\) 0 0
\(733\) −0.0809344 −0.00298938 −0.00149469 0.999999i \(-0.500476\pi\)
−0.00149469 + 0.999999i \(0.500476\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.47787 9.48794i 0.201780 0.349493i
\(738\) 0 0
\(739\) −15.5420 26.9195i −0.571720 0.990249i −0.996389 0.0849002i \(-0.972943\pi\)
0.424669 0.905349i \(-0.360390\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.5752 + 32.1732i 0.681458 + 1.18032i 0.974536 + 0.224232i \(0.0719872\pi\)
−0.293078 + 0.956089i \(0.594679\pi\)
\(744\) 0 0
\(745\) −0.268366 + 0.464824i −0.00983219 + 0.0170298i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.51227 −0.164875
\(750\) 0 0
\(751\) 17.7892 + 10.2706i 0.649136 + 0.374779i 0.788125 0.615515i \(-0.211052\pi\)
−0.138989 + 0.990294i \(0.544385\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.161146 0.279113i −0.00586469 0.0101579i
\(756\) 0 0
\(757\) 17.1466 + 29.6987i 0.623202 + 1.07942i 0.988886 + 0.148679i \(0.0475021\pi\)
−0.365683 + 0.930739i \(0.619165\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.14685i 0.0415734i 0.999784 + 0.0207867i \(0.00661709\pi\)
−0.999784 + 0.0207867i \(0.993383\pi\)
\(762\) 0 0
\(763\) −23.3096 + 13.4578i −0.843864 + 0.487205i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.46472i 0.233428i
\(768\) 0 0
\(769\) −13.3092 + 23.0522i −0.479942 + 0.831285i −0.999735 0.0230077i \(-0.992676\pi\)
0.519793 + 0.854292i \(0.326009\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.8406 39.5611i 0.821519 1.42291i −0.0830322 0.996547i \(-0.526460\pi\)
0.904551 0.426365i \(-0.140206\pi\)
\(774\) 0 0
\(775\) 32.6307 + 18.8393i 1.17213 + 0.676729i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.10923 47.9716i −0.111400 1.71876i
\(780\) 0 0
\(781\) 4.36879 2.52232i 0.156327 0.0902557i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.0260566 0.0150438i −0.000930001 0.000536936i
\(786\) 0 0
\(787\) 32.1982i 1.14774i 0.818946 + 0.573871i \(0.194559\pi\)
−0.818946 + 0.573871i \(0.805441\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.79903 −0.0639661
\(792\) 0 0
\(793\) 0.488823 0.282222i 0.0173586 0.0100220i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −23.6289 −0.836980 −0.418490 0.908221i \(-0.637441\pi\)
−0.418490 + 0.908221i \(0.637441\pi\)
\(798\) 0 0
\(799\) −41.7945 −1.47858
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.28571 4.20641i 0.257107 0.148441i
\(804\) 0 0
\(805\) −0.181287 −0.00638954
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.8880i 0.875015i 0.899215 + 0.437508i \(0.144139\pi\)
−0.899215 + 0.437508i \(0.855861\pi\)
\(810\) 0 0
\(811\) −5.88792 3.39939i −0.206753 0.119369i 0.393049 0.919518i \(-0.371420\pi\)
−0.599801 + 0.800149i \(0.704754\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.151211 + 0.0873015i −0.00529668 + 0.00305804i
\(816\) 0 0
\(817\) 3.58477 + 55.3085i 0.125415 + 1.93500i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32.9682 19.0342i −1.15060 0.664299i −0.201566 0.979475i \(-0.564603\pi\)
−0.949033 + 0.315176i \(0.897936\pi\)
\(822\) 0 0
\(823\) 24.1459 41.8219i 0.841673 1.45782i −0.0468069 0.998904i \(-0.514905\pi\)
0.888480 0.458916i \(-0.151762\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9.50746 + 16.4674i −0.330607 + 0.572628i −0.982631 0.185570i \(-0.940587\pi\)
0.652024 + 0.758198i \(0.273920\pi\)
\(828\) 0 0
\(829\) 38.6461i 1.34223i 0.741351 + 0.671117i \(0.234185\pi\)
−0.741351 + 0.671117i \(0.765815\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18.0089 + 10.3975i −0.623973 + 0.360251i
\(834\) 0 0
\(835\) 0.0795149i 0.00275173i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −13.9272 24.1226i −0.480819 0.832803i 0.518939 0.854811i \(-0.326327\pi\)
−0.999758 + 0.0220085i \(0.992994\pi\)
\(840\) 0 0
\(841\) 12.6078 + 21.8374i 0.434753 + 0.753014i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.279905 + 0.161603i 0.00962902 + 0.00555932i
\(846\) 0 0
\(847\) 11.4827 0.394551
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.64897 + 4.58815i −0.0908056 + 0.157280i
\(852\) 0 0
\(853\) −22.6282 39.1933i −0.774777 1.34195i −0.934920 0.354859i \(-0.884529\pi\)
0.160143 0.987094i \(-0.448804\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.9079 + 32.7495i 0.645883 + 1.11870i 0.984097 + 0.177632i \(0.0568438\pi\)
−0.338214 + 0.941069i \(0.609823\pi\)
\(858\) 0 0
\(859\) −9.59497 + 16.6190i −0.327376 + 0.567032i −0.981990 0.188931i \(-0.939498\pi\)
0.654614 + 0.755963i \(0.272831\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 37.6978 1.28325 0.641624 0.767019i \(-0.278261\pi\)
0.641624 + 0.767019i \(0.278261\pi\)
\(864\) 0 0
\(865\) 0.302059 + 0.174394i 0.0102703 + 0.00592957i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.27542 10.8693i −0.212879 0.368717i
\(870\) 0 0
\(871\) −2.92673 5.06925i −0.0991686 0.171765i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.548295i 0.0185358i
\(876\) 0 0
\(877\) 32.4147 18.7146i 1.09457 0.631948i 0.159778 0.987153i \(-0.448922\pi\)
0.934789 + 0.355205i \(0.115589\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.698543i 0.0235345i 0.999931 + 0.0117673i \(0.00374572\pi\)
−0.999931 + 0.0117673i \(0.996254\pi\)
\(882\) 0 0
\(883\) 29.0805 50.3689i 0.978636 1.69505i 0.311264 0.950323i \(-0.399248\pi\)
0.667372 0.744724i \(-0.267419\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.5940 + 21.8134i −0.422865 + 0.732423i −0.996218 0.0868850i \(-0.972309\pi\)
0.573354 + 0.819308i \(0.305642\pi\)
\(888\) 0 0
\(889\) −25.4347 14.6847i −0.853051 0.492509i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.4136 + 23.1043i 0.515795 + 0.773157i
\(894\) 0 0
\(895\) 0.0677010 0.0390872i 0.00226300 0.00130654i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.6975 7.33092i −0.423486 0.244500i
\(900\) 0 0
\(901\) 77.7850i 2.59139i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.610990 0.0203100
\(906\) 0 0
\(907\) −6.37701 + 3.68177i −0.211745 + 0.122251i −0.602122 0.798404i \(-0.705678\pi\)
0.390377 + 0.920655i \(0.372345\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −32.0294 −1.06118 −0.530590 0.847629i \(-0.678030\pi\)
−0.530590 + 0.847629i \(0.678030\pi\)
\(912\) 0 0
\(913\) 5.98364 0.198030
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.0614 20.8200i 1.19085 0.687539i
\(918\) 0 0
\(919\) −6.20294 −0.204616 −0.102308 0.994753i \(-0.532623\pi\)
−0.102308 + 0.994753i \(0.532623\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.69527i 0.0887158i
\(924\) 0 0
\(925\) −6.93779 4.00553i −0.228113 0.131701i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.902179 0.520873i 0.0295995 0.0170893i −0.485127 0.874444i \(-0.661227\pi\)
0.514727 + 0.857354i \(0.327893\pi\)
\(930\) 0 0
\(931\) 12.3894 + 6.12096i 0.406046 + 0.200606i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.360591 + 0.208187i 0.0117926 + 0.00680846i
\(936\) 0 0
\(937\) 14.1422 24.4950i 0.462006 0.800217i −0.537055 0.843547i \(-0.680463\pi\)
0.999061 + 0.0433298i \(0.0137966\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −19.5894 + 33.9298i −0.638595 + 1.10608i 0.347147 + 0.937811i \(0.387151\pi\)
−0.985741 + 0.168268i \(0.946183\pi\)
\(942\) 0 0
\(943\) 36.4617i 1.18735i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.8474 8.57215i 0.482475 0.278557i −0.238972 0.971026i \(-0.576810\pi\)
0.721448 + 0.692469i \(0.243477\pi\)
\(948\) 0 0
\(949\) 4.49482i 0.145908i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −8.08753 14.0080i −0.261981 0.453764i 0.704787 0.709419i \(-0.251042\pi\)
−0.966768 + 0.255654i \(0.917709\pi\)
\(954\) 0 0
\(955\) −0.0152350 0.0263878i −0.000492994 0.000853890i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.35647 1.93786i −0.108386 0.0625768i
\(960\) 0 0
\(961\) −25.8052 −0.832426
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.0834352 + 0.144514i −0.00268587 + 0.00465207i
\(966\) 0 0
\(967\) 0.0997161 + 0.172713i 0.00320665 + 0.00555409i 0.867624 0.497220i \(-0.165646\pi\)
−0.864418 + 0.502774i \(0.832313\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 11.4475 + 19.8276i 0.367368 + 0.636299i 0.989153 0.146888i \(-0.0469258\pi\)
−0.621785 + 0.783188i \(0.713592\pi\)
\(972\) 0 0
\(973\) −2.57678 + 4.46312i −0.0826079 + 0.143081i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26.3775 0.843891 0.421945 0.906621i \(-0.361347\pi\)
0.421945 + 0.906621i \(0.361347\pi\)
\(978\) 0 0
\(979\) −7.01886 4.05234i −0.224324 0.129513i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 28.0334 + 48.5553i 0.894128 + 1.54867i 0.834880 + 0.550431i \(0.185537\pi\)
0.0592474 + 0.998243i \(0.481130\pi\)
\(984\) 0 0
\(985\) 0.0354072 + 0.0613270i 0.00112817 + 0.00195404i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 42.0382i 1.33674i
\(990\) 0 0
\(991\) −26.3299 + 15.2016i −0.836398 + 0.482895i −0.856038 0.516913i \(-0.827081\pi\)
0.0196403 + 0.999807i \(0.493748\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0.463042i 0.0146794i
\(996\) 0 0
\(997\) −12.0115 + 20.8046i −0.380409 + 0.658888i −0.991121 0.132965i \(-0.957550\pi\)
0.610712 + 0.791853i \(0.290884\pi\)
\(998\) 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 2736.2.dc.e.449.5 20
3.2 odd 2 2736.2.dc.f.449.6 20
4.3 odd 2 1368.2.cu.a.449.5 20
12.11 even 2 1368.2.cu.b.449.6 yes 20
19.8 odd 6 2736.2.dc.f.1889.6 20
57.8 even 6 inner 2736.2.dc.e.1889.5 20
76.27 even 6 1368.2.cu.b.521.6 yes 20
228.179 odd 6 1368.2.cu.a.521.5 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.cu.a.449.5 20 4.3 odd 2
1368.2.cu.a.521.5 yes 20 228.179 odd 6
1368.2.cu.b.449.6 yes 20 12.11 even 2
1368.2.cu.b.521.6 yes 20 76.27 even 6
2736.2.dc.e.449.5 20 1.1 even 1 trivial
2736.2.dc.e.1889.5 20 57.8 even 6 inner
2736.2.dc.f.449.6 20 3.2 odd 2
2736.2.dc.f.1889.6 20 19.8 odd 6