Properties

Label 756.2.x.a.629.8
Level $756$
Weight $2$
Character 756.629
Analytic conductor $6.037$
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] = [756,2,Mod(125,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.x (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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 629.8
Root \(-1.71965 + 0.206851i\) of defining polynomial
Character \(\chi\) \(=\) 756.629
Dual form 756.2.x.a.125.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.09336 + 3.62580i) q^{5} +(1.36869 + 2.26422i) q^{7} +O(q^{10})\) \(q+(2.09336 + 3.62580i) q^{5} +(1.36869 + 2.26422i) q^{7} +(1.23222 + 0.711425i) q^{11} +(-0.850739 + 0.491174i) q^{13} -0.370947 q^{17} -4.97471i q^{19} +(-4.98775 + 2.87968i) q^{23} +(-6.26427 + 10.8500i) q^{25} +(-7.31732 - 4.22466i) q^{29} +(6.28007 - 3.62580i) q^{31} +(-5.34444 + 9.70241i) q^{35} +3.46445 q^{37} +(1.06981 + 1.85297i) q^{41} +(3.00875 - 5.21130i) q^{43} +(4.13542 - 7.16276i) q^{47} +(-3.25337 + 6.19803i) q^{49} +4.97245i q^{53} +5.95706i q^{55} +(2.27883 + 3.94705i) q^{59} +(6.50416 + 3.75518i) q^{61} +(-3.56180 - 2.05640i) q^{65} +(5.03205 + 8.71577i) q^{67} -10.9555i q^{71} +9.52848i q^{73} +(0.0757141 + 3.76374i) q^{77} +(4.25553 - 7.37079i) q^{79} +(-0.972254 + 1.68399i) q^{83} +(-0.776524 - 1.34498i) q^{85} +7.80735 q^{89} +(-2.27652 - 1.25399i) q^{91} +(18.0373 - 10.4138i) q^{95} +(-3.34099 - 1.92892i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{7} - 6 q^{11} - 6 q^{23} - 8 q^{25} + 12 q^{29} + 4 q^{37} + 4 q^{43} - 5 q^{49} + 24 q^{65} + 14 q^{67} + 21 q^{77} + 20 q^{79} + 6 q^{85} - 18 q^{91} + 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.09336 + 3.62580i 0.936177 + 1.62151i 0.772521 + 0.634989i \(0.218995\pi\)
0.163656 + 0.986518i \(0.447671\pi\)
\(6\) 0 0
\(7\) 1.36869 + 2.26422i 0.517317 + 0.855794i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.23222 + 0.711425i 0.371529 + 0.214503i 0.674126 0.738616i \(-0.264520\pi\)
−0.302597 + 0.953119i \(0.597854\pi\)
\(12\) 0 0
\(13\) −0.850739 + 0.491174i −0.235952 + 0.136227i −0.613315 0.789838i \(-0.710164\pi\)
0.377363 + 0.926066i \(0.376831\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.370947 −0.0899679 −0.0449840 0.998988i \(-0.514324\pi\)
−0.0449840 + 0.998988i \(0.514324\pi\)
\(18\) 0 0
\(19\) 4.97471i 1.14128i −0.821201 0.570638i \(-0.806696\pi\)
0.821201 0.570638i \(-0.193304\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.98775 + 2.87968i −1.04002 + 0.600455i −0.919838 0.392298i \(-0.871680\pi\)
−0.120179 + 0.992752i \(0.538347\pi\)
\(24\) 0 0
\(25\) −6.26427 + 10.8500i −1.25285 + 2.17001i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.31732 4.22466i −1.35879 0.784499i −0.369332 0.929298i \(-0.620413\pi\)
−0.989461 + 0.144798i \(0.953747\pi\)
\(30\) 0 0
\(31\) 6.28007 3.62580i 1.12793 0.651213i 0.184519 0.982829i \(-0.440927\pi\)
0.943414 + 0.331616i \(0.107594\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.34444 + 9.70241i −0.903375 + 1.64001i
\(36\) 0 0
\(37\) 3.46445 0.569552 0.284776 0.958594i \(-0.408081\pi\)
0.284776 + 0.958594i \(0.408081\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.06981 + 1.85297i 0.167077 + 0.289386i 0.937391 0.348279i \(-0.113234\pi\)
−0.770314 + 0.637665i \(0.779901\pi\)
\(42\) 0 0
\(43\) 3.00875 5.21130i 0.458830 0.794716i −0.540070 0.841620i \(-0.681602\pi\)
0.998899 + 0.0469039i \(0.0149354\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.13542 7.16276i 0.603213 1.04480i −0.389118 0.921188i \(-0.627220\pi\)
0.992331 0.123608i \(-0.0394466\pi\)
\(48\) 0 0
\(49\) −3.25337 + 6.19803i −0.464766 + 0.885433i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.97245i 0.683019i 0.939878 + 0.341509i \(0.110938\pi\)
−0.939878 + 0.341509i \(0.889062\pi\)
\(54\) 0 0
\(55\) 5.95706i 0.803250i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.27883 + 3.94705i 0.296678 + 0.513862i 0.975374 0.220558i \(-0.0707878\pi\)
−0.678696 + 0.734420i \(0.737454\pi\)
\(60\) 0 0
\(61\) 6.50416 + 3.75518i 0.832772 + 0.480801i 0.854801 0.518956i \(-0.173679\pi\)
−0.0220288 + 0.999757i \(0.507013\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.56180 2.05640i −0.441786 0.255066i
\(66\) 0 0
\(67\) 5.03205 + 8.71577i 0.614763 + 1.06480i 0.990426 + 0.138044i \(0.0440816\pi\)
−0.375663 + 0.926756i \(0.622585\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.9555i 1.30018i −0.759857 0.650090i \(-0.774731\pi\)
0.759857 0.650090i \(-0.225269\pi\)
\(72\) 0 0
\(73\) 9.52848i 1.11522i 0.830102 + 0.557612i \(0.188282\pi\)
−0.830102 + 0.557612i \(0.811718\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.0757141 + 3.76374i 0.00862842 + 0.428918i
\(78\) 0 0
\(79\) 4.25553 7.37079i 0.478784 0.829278i −0.520920 0.853606i \(-0.674411\pi\)
0.999704 + 0.0243272i \(0.00774434\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.972254 + 1.68399i −0.106719 + 0.184842i −0.914439 0.404724i \(-0.867368\pi\)
0.807720 + 0.589566i \(0.200701\pi\)
\(84\) 0 0
\(85\) −0.776524 1.34498i −0.0842259 0.145884i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.80735 0.827578 0.413789 0.910373i \(-0.364205\pi\)
0.413789 + 0.910373i \(0.364205\pi\)
\(90\) 0 0
\(91\) −2.27652 1.25399i −0.238645 0.131454i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.0373 10.4138i 1.85059 1.06844i
\(96\) 0 0
\(97\) −3.34099 1.92892i −0.339226 0.195852i 0.320704 0.947180i \(-0.396081\pi\)
−0.659930 + 0.751327i \(0.729414\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.50637 7.80526i 0.448401 0.776653i −0.549882 0.835243i \(-0.685327\pi\)
0.998282 + 0.0585901i \(0.0186605\pi\)
\(102\) 0 0
\(103\) 5.96343 3.44299i 0.587594 0.339248i −0.176551 0.984291i \(-0.556494\pi\)
0.764146 + 0.645044i \(0.223161\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3630i 1.09850i 0.835658 + 0.549250i \(0.185087\pi\)
−0.835658 + 0.549250i \(0.814913\pi\)
\(108\) 0 0
\(109\) −15.9930 −1.53185 −0.765926 0.642929i \(-0.777719\pi\)
−0.765926 + 0.642929i \(0.777719\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.17043 + 1.25310i −0.204177 + 0.117881i −0.598602 0.801046i \(-0.704277\pi\)
0.394426 + 0.918928i \(0.370944\pi\)
\(114\) 0 0
\(115\) −20.8823 12.0564i −1.94728 1.12426i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.507712 0.839905i −0.0465419 0.0769940i
\(120\) 0 0
\(121\) −4.48775 7.77301i −0.407977 0.706637i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −31.5199 −2.81922
\(126\) 0 0
\(127\) −1.91140 −0.169609 −0.0848046 0.996398i \(-0.527027\pi\)
−0.0848046 + 0.996398i \(0.527027\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.32591 + 7.49270i 0.377957 + 0.654640i 0.990765 0.135592i \(-0.0432935\pi\)
−0.612808 + 0.790232i \(0.709960\pi\)
\(132\) 0 0
\(133\) 11.2638 6.80885i 0.976698 0.590402i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0973 5.82971i −0.862675 0.498065i 0.00223233 0.999998i \(-0.499289\pi\)
−0.864907 + 0.501932i \(0.832623\pi\)
\(138\) 0 0
\(139\) −4.82663 + 2.78666i −0.409390 + 0.236361i −0.690528 0.723306i \(-0.742622\pi\)
0.281138 + 0.959667i \(0.409288\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.39773 −0.116884
\(144\) 0 0
\(145\) 35.3748i 2.93772i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.4024 7.73789i 1.09797 0.633913i 0.162282 0.986744i \(-0.448114\pi\)
0.935687 + 0.352832i \(0.114781\pi\)
\(150\) 0 0
\(151\) 8.31732 14.4060i 0.676854 1.17235i −0.299069 0.954231i \(-0.596676\pi\)
0.975923 0.218114i \(-0.0699905\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 26.2928 + 15.1802i 2.11189 + 1.21930i
\(156\) 0 0
\(157\) 14.5559 8.40387i 1.16169 0.670702i 0.209981 0.977705i \(-0.432660\pi\)
0.951708 + 0.307004i \(0.0993264\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −13.3469 7.35196i −1.05188 0.579416i
\(162\) 0 0
\(163\) 3.51105 0.275007 0.137503 0.990501i \(-0.456092\pi\)
0.137503 + 0.990501i \(0.456092\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.05253 7.01918i −0.313594 0.543160i 0.665544 0.746359i \(-0.268200\pi\)
−0.979138 + 0.203198i \(0.934866\pi\)
\(168\) 0 0
\(169\) −6.01750 + 10.4226i −0.462884 + 0.801739i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.54844 11.3422i 0.497868 0.862334i −0.502128 0.864793i \(-0.667450\pi\)
0.999997 + 0.00245951i \(0.000782886\pi\)
\(174\) 0 0
\(175\) −33.1407 + 0.666682i −2.50520 + 0.0503964i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.7179i 0.950578i −0.879830 0.475289i \(-0.842344\pi\)
0.879830 0.475289i \(-0.157656\pi\)
\(180\) 0 0
\(181\) 26.5518i 1.97358i 0.161998 + 0.986791i \(0.448206\pi\)
−0.161998 + 0.986791i \(0.551794\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.25232 + 12.5614i 0.533201 + 0.923532i
\(186\) 0 0
\(187\) −0.457090 0.263901i −0.0334257 0.0192983i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.09735 + 0.633555i 0.0794014 + 0.0458424i 0.539175 0.842194i \(-0.318736\pi\)
−0.459774 + 0.888036i \(0.652069\pi\)
\(192\) 0 0
\(193\) −9.31732 16.1381i −0.670676 1.16164i −0.977713 0.209947i \(-0.932671\pi\)
0.307037 0.951697i \(-0.400662\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.94312i 0.423430i 0.977331 + 0.211715i \(0.0679049\pi\)
−0.977331 + 0.211715i \(0.932095\pi\)
\(198\) 0 0
\(199\) 5.62675i 0.398870i 0.979911 + 0.199435i \(0.0639106\pi\)
−0.979911 + 0.199435i \(0.936089\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.449613 22.3503i −0.0315567 1.56868i
\(204\) 0 0
\(205\) −4.47900 + 7.75786i −0.312827 + 0.541832i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.53913 6.12996i 0.244807 0.424018i
\(210\) 0 0
\(211\) 1.05305 + 1.82393i 0.0724948 + 0.125565i 0.899994 0.435902i \(-0.143571\pi\)
−0.827499 + 0.561467i \(0.810237\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 25.1935 1.71818
\(216\) 0 0
\(217\) 16.8051 + 9.25684i 1.14080 + 0.628395i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.315579 0.182200i 0.0212281 0.0122561i
\(222\) 0 0
\(223\) 5.52351 + 3.18900i 0.369882 + 0.213551i 0.673407 0.739272i \(-0.264830\pi\)
−0.303525 + 0.952823i \(0.598164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.72365 9.91365i 0.379892 0.657992i −0.611154 0.791511i \(-0.709295\pi\)
0.991046 + 0.133520i \(0.0426279\pi\)
\(228\) 0 0
\(229\) −4.88696 + 2.82149i −0.322940 + 0.186449i −0.652702 0.757615i \(-0.726365\pi\)
0.329762 + 0.944064i \(0.393031\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.4463i 0.815385i −0.913119 0.407693i \(-0.866333\pi\)
0.913119 0.407693i \(-0.133667\pi\)
\(234\) 0 0
\(235\) 34.6276 2.25886
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.52450 + 2.03487i −0.227981 + 0.131625i −0.609640 0.792678i \(-0.708686\pi\)
0.381659 + 0.924303i \(0.375353\pi\)
\(240\) 0 0
\(241\) 4.26195 + 2.46064i 0.274537 + 0.158504i 0.630947 0.775826i \(-0.282666\pi\)
−0.356411 + 0.934329i \(0.616000\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −29.2833 + 1.17864i −1.87084 + 0.0753007i
\(246\) 0 0
\(247\) 2.44345 + 4.23218i 0.155473 + 0.269287i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.65020 0.293518 0.146759 0.989172i \(-0.453116\pi\)
0.146759 + 0.989172i \(0.453116\pi\)
\(252\) 0 0
\(253\) −8.19470 −0.515196
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.44701 + 9.43450i 0.339775 + 0.588508i 0.984390 0.175999i \(-0.0563156\pi\)
−0.644615 + 0.764507i \(0.722982\pi\)
\(258\) 0 0
\(259\) 4.74176 + 7.84426i 0.294639 + 0.487419i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.6489 9.03488i −0.964951 0.557114i −0.0672574 0.997736i \(-0.521425\pi\)
−0.897693 + 0.440621i \(0.854758\pi\)
\(264\) 0 0
\(265\) −18.0291 + 10.4091i −1.10752 + 0.639427i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.98410 −0.181944 −0.0909718 0.995853i \(-0.528997\pi\)
−0.0909718 + 0.995853i \(0.528997\pi\)
\(270\) 0 0
\(271\) 10.3608i 0.629375i −0.949195 0.314688i \(-0.898100\pi\)
0.949195 0.314688i \(-0.101900\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.4380 + 8.91312i −0.930945 + 0.537481i
\(276\) 0 0
\(277\) 5.94345 10.2944i 0.357107 0.618528i −0.630369 0.776296i \(-0.717096\pi\)
0.987476 + 0.157768i \(0.0504297\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0740 + 6.97095i 0.720277 + 0.415852i 0.814855 0.579665i \(-0.196817\pi\)
−0.0945775 + 0.995518i \(0.530150\pi\)
\(282\) 0 0
\(283\) 2.19593 1.26782i 0.130535 0.0753642i −0.433311 0.901245i \(-0.642655\pi\)
0.563845 + 0.825880i \(0.309321\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.73129 + 4.95844i −0.161223 + 0.292687i
\(288\) 0 0
\(289\) −16.8624 −0.991906
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.95496 6.85020i −0.231052 0.400193i 0.727066 0.686567i \(-0.240883\pi\)
−0.958118 + 0.286374i \(0.907550\pi\)
\(294\) 0 0
\(295\) −9.54080 + 16.5251i −0.555487 + 0.962131i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.82885 4.89971i 0.163596 0.283357i
\(300\) 0 0
\(301\) 15.9176 0.320209i 0.917474 0.0184565i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 31.4437i 1.80046i
\(306\) 0 0
\(307\) 13.9676i 0.797170i −0.917131 0.398585i \(-0.869501\pi\)
0.917131 0.398585i \(-0.130499\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.3163 28.2607i −0.925215 1.60252i −0.791215 0.611539i \(-0.790551\pi\)
−0.134001 0.990981i \(-0.542782\pi\)
\(312\) 0 0
\(313\) 13.6110 + 7.85832i 0.769340 + 0.444178i 0.832639 0.553816i \(-0.186829\pi\)
−0.0632994 + 0.997995i \(0.520162\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.1366 + 14.5126i 1.41181 + 0.815111i 0.995559 0.0941377i \(-0.0300094\pi\)
0.416254 + 0.909248i \(0.363343\pi\)
\(318\) 0 0
\(319\) −6.01105 10.4114i −0.336554 0.582929i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.84535i 0.102678i
\(324\) 0 0
\(325\) 12.3074i 0.682692i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 21.8782 0.440117i 1.20618 0.0242644i
\(330\) 0 0
\(331\) −6.58510 + 11.4057i −0.361950 + 0.626915i −0.988282 0.152641i \(-0.951222\pi\)
0.626332 + 0.779556i \(0.284555\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −21.0677 + 36.4904i −1.15105 + 1.99368i
\(336\) 0 0
\(337\) 8.31732 + 14.4060i 0.453073 + 0.784746i 0.998575 0.0533635i \(-0.0169942\pi\)
−0.545502 + 0.838110i \(0.683661\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.3179 0.558747
\(342\) 0 0
\(343\) −18.4866 + 1.11687i −0.998180 + 0.0603053i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.4012 + 12.3560i −1.14888 + 0.663305i −0.948614 0.316437i \(-0.897513\pi\)
−0.200264 + 0.979742i \(0.564180\pi\)
\(348\) 0 0
\(349\) −26.7994 15.4727i −1.43454 0.828232i −0.437078 0.899424i \(-0.643987\pi\)
−0.997463 + 0.0711915i \(0.977320\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11.6758 + 20.2231i −0.621440 + 1.07637i 0.367778 + 0.929914i \(0.380119\pi\)
−0.989218 + 0.146452i \(0.953215\pi\)
\(354\) 0 0
\(355\) 39.7225 22.9338i 2.10825 1.21720i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.2351i 1.12075i −0.828240 0.560374i \(-0.810657\pi\)
0.828240 0.560374i \(-0.189343\pi\)
\(360\) 0 0
\(361\) −5.74775 −0.302513
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −34.5483 + 19.9465i −1.80834 + 1.04405i
\(366\) 0 0
\(367\) 12.0178 + 6.93846i 0.627322 + 0.362185i 0.779714 0.626135i \(-0.215364\pi\)
−0.152392 + 0.988320i \(0.548698\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.2587 + 6.80576i −0.584523 + 0.353337i
\(372\) 0 0
\(373\) 16.0728 + 27.8390i 0.832221 + 1.44145i 0.896273 + 0.443502i \(0.146264\pi\)
−0.0640529 + 0.997947i \(0.520403\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.30017 0.427481
\(378\) 0 0
\(379\) 1.95340 0.100339 0.0501696 0.998741i \(-0.484024\pi\)
0.0501696 + 0.998741i \(0.484024\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.788167 1.36514i −0.0402734 0.0697557i 0.845186 0.534472i \(-0.179490\pi\)
−0.885460 + 0.464717i \(0.846156\pi\)
\(384\) 0 0
\(385\) −13.4881 + 8.15338i −0.687416 + 0.415535i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.74447 5.62597i −0.494064 0.285248i 0.232195 0.972669i \(-0.425409\pi\)
−0.726259 + 0.687421i \(0.758743\pi\)
\(390\) 0 0
\(391\) 1.85019 1.06821i 0.0935682 0.0540216i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 35.6333 1.79291
\(396\) 0 0
\(397\) 0.632961i 0.0317674i 0.999874 + 0.0158837i \(0.00505615\pi\)
−0.999874 + 0.0158837i \(0.994944\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.38032 + 4.26103i −0.368555 + 0.212786i −0.672827 0.739800i \(-0.734920\pi\)
0.304272 + 0.952585i \(0.401587\pi\)
\(402\) 0 0
\(403\) −3.56180 + 6.16921i −0.177426 + 0.307310i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.26897 + 2.46469i 0.211605 + 0.122170i
\(408\) 0 0
\(409\) 12.1822 7.03338i 0.602370 0.347778i −0.167603 0.985855i \(-0.553603\pi\)
0.769973 + 0.638076i \(0.220269\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.81796 + 10.5621i −0.286283 + 0.519725i
\(414\) 0 0
\(415\) −8.14109 −0.399630
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.9339 + 18.9381i 0.534156 + 0.925185i 0.999204 + 0.0398995i \(0.0127038\pi\)
−0.465048 + 0.885286i \(0.653963\pi\)
\(420\) 0 0
\(421\) 13.3616 23.1430i 0.651206 1.12792i −0.331625 0.943411i \(-0.607597\pi\)
0.982831 0.184510i \(-0.0590698\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.32371 4.02479i 0.112717 0.195231i
\(426\) 0 0
\(427\) 0.399648 + 19.8665i 0.0193403 + 0.961408i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.9400i 0.960476i 0.877138 + 0.480238i \(0.159450\pi\)
−0.877138 + 0.480238i \(0.840550\pi\)
\(432\) 0 0
\(433\) 20.2826i 0.974719i 0.873201 + 0.487359i \(0.162040\pi\)
−0.873201 + 0.487359i \(0.837960\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.3256 + 24.8126i 0.685285 + 1.18695i
\(438\) 0 0
\(439\) −24.5936 14.1991i −1.17379 0.677687i −0.219219 0.975676i \(-0.570351\pi\)
−0.954569 + 0.297989i \(0.903684\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.6570 6.15281i −0.506328 0.292329i 0.224995 0.974360i \(-0.427764\pi\)
−0.731323 + 0.682031i \(0.761097\pi\)
\(444\) 0 0
\(445\) 16.3436 + 28.3079i 0.774759 + 1.34192i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 36.1924i 1.70803i −0.520251 0.854013i \(-0.674162\pi\)
0.520251 0.854013i \(-0.325838\pi\)
\(450\) 0 0
\(451\) 3.04437i 0.143354i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.218855 10.8793i −0.0102601 0.510028i
\(456\) 0 0
\(457\) 4.20892 7.29007i 0.196885 0.341015i −0.750632 0.660721i \(-0.770251\pi\)
0.947517 + 0.319706i \(0.103584\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.07730 15.7224i 0.422772 0.732263i −0.573437 0.819249i \(-0.694390\pi\)
0.996209 + 0.0869865i \(0.0277237\pi\)
\(462\) 0 0
\(463\) −7.64690 13.2448i −0.355381 0.615539i 0.631802 0.775130i \(-0.282316\pi\)
−0.987183 + 0.159591i \(0.948982\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.7864 1.23953 0.619763 0.784789i \(-0.287229\pi\)
0.619763 + 0.784789i \(0.287229\pi\)
\(468\) 0 0
\(469\) −12.8471 + 23.3229i −0.593223 + 1.07695i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.41490 4.28100i 0.340938 0.196840i
\(474\) 0 0
\(475\) 53.9758 + 31.1630i 2.47658 + 1.42985i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.2775 + 24.7294i −0.652357 + 1.12992i 0.330193 + 0.943914i \(0.392886\pi\)
−0.982549 + 0.186002i \(0.940447\pi\)
\(480\) 0 0
\(481\) −2.94734 + 1.70165i −0.134387 + 0.0775884i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.1517i 0.733409i
\(486\) 0 0
\(487\) −13.1527 −0.596006 −0.298003 0.954565i \(-0.596321\pi\)
−0.298003 + 0.954565i \(0.596321\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −28.6854 + 16.5615i −1.29455 + 0.747411i −0.979458 0.201649i \(-0.935370\pi\)
−0.315096 + 0.949060i \(0.602037\pi\)
\(492\) 0 0
\(493\) 2.71434 + 1.56712i 0.122248 + 0.0705798i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.8057 14.9947i 1.11269 0.672605i
\(498\) 0 0
\(499\) 3.27652 + 5.67511i 0.146677 + 0.254053i 0.929997 0.367566i \(-0.119809\pi\)
−0.783320 + 0.621619i \(0.786475\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.4969 −0.557210 −0.278605 0.960406i \(-0.589872\pi\)
−0.278605 + 0.960406i \(0.589872\pi\)
\(504\) 0 0
\(505\) 37.7337 1.67913
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.30237 5.71987i −0.146375 0.253529i 0.783510 0.621379i \(-0.213427\pi\)
−0.929885 + 0.367850i \(0.880094\pi\)
\(510\) 0 0
\(511\) −21.5746 + 13.0416i −0.954402 + 0.576924i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 24.9672 + 14.4148i 1.10018 + 0.635192i
\(516\) 0 0
\(517\) 10.1915 5.88408i 0.448223 0.258782i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.9784 0.700026 0.350013 0.936745i \(-0.386177\pi\)
0.350013 + 0.936745i \(0.386177\pi\)
\(522\) 0 0
\(523\) 11.7615i 0.514293i 0.966372 + 0.257147i \(0.0827824\pi\)
−0.966372 + 0.257147i \(0.917218\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.32957 + 1.34498i −0.101478 + 0.0585882i
\(528\) 0 0
\(529\) 5.08510 8.80765i 0.221091 0.382941i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.82026 1.05093i −0.0788444 0.0455208i
\(534\) 0 0
\(535\) −41.1999 + 23.7867i −1.78122 + 1.02839i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8.41831 + 5.32284i −0.362602 + 0.229271i
\(540\) 0 0
\(541\) −0.411960 −0.0177115 −0.00885576 0.999961i \(-0.502819\pi\)
−0.00885576 + 0.999961i \(0.502819\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −33.4790 57.9874i −1.43408 2.48391i
\(546\) 0 0
\(547\) 11.9166 20.6402i 0.509519 0.882513i −0.490420 0.871486i \(-0.663157\pi\)
0.999939 0.0110266i \(-0.00350995\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −21.0165 + 36.4016i −0.895331 + 1.55076i
\(552\) 0 0
\(553\) 22.5136 0.452899i 0.957374 0.0192592i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 42.9474i 1.81974i 0.414896 + 0.909869i \(0.363818\pi\)
−0.414896 + 0.909869i \(0.636182\pi\)
\(558\) 0 0
\(559\) 5.91128i 0.250020i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.9454 27.6182i −0.672019 1.16397i −0.977331 0.211718i \(-0.932094\pi\)
0.305312 0.952252i \(-0.401239\pi\)
\(564\) 0 0
\(565\) −9.08695 5.24635i −0.382291 0.220716i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.428895 0.247623i −0.0179802 0.0103809i 0.490983 0.871169i \(-0.336638\pi\)
−0.508963 + 0.860788i \(0.669971\pi\)
\(570\) 0 0
\(571\) −1.34182 2.32411i −0.0561535 0.0972608i 0.836582 0.547841i \(-0.184550\pi\)
−0.892736 + 0.450581i \(0.851217\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 72.1564i 3.00913i
\(576\) 0 0
\(577\) 42.0259i 1.74956i −0.484518 0.874781i \(-0.661005\pi\)
0.484518 0.874781i \(-0.338995\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.14364 + 0.103473i −0.213394 + 0.00429279i
\(582\) 0 0
\(583\) −3.53753 + 6.12717i −0.146509 + 0.253762i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.71916 + 2.97768i −0.0709575 + 0.122902i −0.899321 0.437289i \(-0.855939\pi\)
0.828364 + 0.560191i \(0.189272\pi\)
\(588\) 0 0
\(589\) −18.0373 31.2415i −0.743214 1.28728i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −9.10251 −0.373795 −0.186898 0.982379i \(-0.559843\pi\)
−0.186898 + 0.982379i \(0.559843\pi\)
\(594\) 0 0
\(595\) 1.98250 3.59908i 0.0812747 0.147548i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.71186 + 3.87510i −0.274239 + 0.158332i −0.630813 0.775935i \(-0.717278\pi\)
0.356573 + 0.934267i \(0.383945\pi\)
\(600\) 0 0
\(601\) −22.0034 12.7037i −0.897536 0.518193i −0.0211361 0.999777i \(-0.506728\pi\)
−0.876400 + 0.481584i \(0.840062\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 18.7889 32.5433i 0.763878 1.32308i
\(606\) 0 0
\(607\) 11.7094 6.76042i 0.475270 0.274397i −0.243173 0.969983i \(-0.578188\pi\)
0.718443 + 0.695586i \(0.244855\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.12485i 0.328696i
\(612\) 0 0
\(613\) 4.83635 0.195338 0.0976691 0.995219i \(-0.468861\pi\)
0.0976691 + 0.995219i \(0.468861\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.30333 + 3.06188i −0.213504 + 0.123267i −0.602939 0.797787i \(-0.706004\pi\)
0.389435 + 0.921054i \(0.372670\pi\)
\(618\) 0 0
\(619\) 4.94313 + 2.85392i 0.198681 + 0.114709i 0.596040 0.802955i \(-0.296740\pi\)
−0.397359 + 0.917663i \(0.630073\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.6859 + 17.6775i 0.428120 + 0.708236i
\(624\) 0 0
\(625\) −34.6609 60.0344i −1.38644 2.40138i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.28513 −0.0512414
\(630\) 0 0
\(631\) −19.0525 −0.758468 −0.379234 0.925301i \(-0.623812\pi\)
−0.379234 + 0.925301i \(0.623812\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.00124 6.93035i −0.158784 0.275022i
\(636\) 0 0
\(637\) −0.276550 6.87087i −0.0109573 0.272234i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13.2820 7.66837i −0.524607 0.302882i 0.214210 0.976788i \(-0.431282\pi\)
−0.738818 + 0.673905i \(0.764616\pi\)
\(642\) 0 0
\(643\) −11.3209 + 6.53612i −0.446453 + 0.257759i −0.706331 0.707882i \(-0.749651\pi\)
0.259878 + 0.965641i \(0.416318\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −35.7066 −1.40377 −0.701885 0.712290i \(-0.747658\pi\)
−0.701885 + 0.712290i \(0.747658\pi\)
\(648\) 0 0
\(649\) 6.48486i 0.254553i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23.6131 13.6330i 0.924051 0.533501i 0.0391261 0.999234i \(-0.487543\pi\)
0.884925 + 0.465733i \(0.154209\pi\)
\(654\) 0 0
\(655\) −18.1113 + 31.3698i −0.707669 + 1.22572i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 14.7911 + 8.53963i 0.576179 + 0.332657i 0.759613 0.650375i \(-0.225388\pi\)
−0.183435 + 0.983032i \(0.558722\pi\)
\(660\) 0 0
\(661\) −40.6657 + 23.4784i −1.58171 + 0.913203i −0.587105 + 0.809511i \(0.699732\pi\)
−0.994609 + 0.103692i \(0.966934\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 48.2667 + 26.5870i 1.87170 + 1.03100i
\(666\) 0 0
\(667\) 48.6626 1.88422
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.34305 + 9.25444i 0.206266 + 0.357264i
\(672\) 0 0
\(673\) 7.76077 13.4421i 0.299156 0.518153i −0.676787 0.736179i \(-0.736628\pi\)
0.975943 + 0.218026i \(0.0699616\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.07869 15.7248i 0.348922 0.604351i −0.637136 0.770751i \(-0.719881\pi\)
0.986058 + 0.166400i \(0.0532143\pi\)
\(678\) 0 0
\(679\) −0.205287 10.2048i −0.00787820 0.391625i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 20.1917i 0.772615i −0.922370 0.386308i \(-0.873750\pi\)
0.922370 0.386308i \(-0.126250\pi\)
\(684\) 0 0
\(685\) 48.8146i 1.86511i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.44234 4.23026i −0.0930458 0.161160i
\(690\) 0 0
\(691\) −0.0695792 0.0401716i −0.00264692 0.00152820i 0.498676 0.866788i \(-0.333820\pi\)
−0.501323 + 0.865260i \(0.667153\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20.2077 11.6669i −0.766523 0.442552i
\(696\) 0 0
\(697\) −0.396844 0.687355i −0.0150316 0.0260354i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 35.1490i 1.32756i −0.747928 0.663780i \(-0.768951\pi\)
0.747928 0.663780i \(-0.231049\pi\)
\(702\) 0 0
\(703\) 17.2346i 0.650016i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 23.8406 0.479595i 0.896620 0.0180370i
\(708\) 0 0
\(709\) −0.782968 + 1.35614i −0.0294050 + 0.0509309i −0.880353 0.474318i \(-0.842695\pi\)
0.850948 + 0.525249i \(0.176028\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −20.8823 + 36.1691i −0.782047 + 1.35455i
\(714\) 0 0
\(715\) −2.92595 5.06790i −0.109424 0.189529i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −25.8787 −0.965112 −0.482556 0.875865i \(-0.660291\pi\)
−0.482556 + 0.875865i \(0.660291\pi\)
\(720\) 0 0
\(721\) 15.9578 + 8.79012i 0.594299 + 0.327361i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 91.6754 52.9288i 3.40474 1.96573i
\(726\) 0 0
\(727\) 0.990545 + 0.571891i 0.0367373 + 0.0212103i 0.518256 0.855225i \(-0.326581\pi\)
−0.481519 + 0.876436i \(0.659915\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.11609 + 1.93312i −0.0412800 + 0.0714990i
\(732\) 0 0
\(733\) −20.1408 + 11.6283i −0.743916 + 0.429500i −0.823491 0.567329i \(-0.807977\pi\)
0.0795755 + 0.996829i \(0.474644\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14.3197i 0.527473i
\(738\) 0 0
\(739\) −36.1516 −1.32986 −0.664929 0.746907i \(-0.731538\pi\)
−0.664929 + 0.746907i \(0.731538\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.43940 + 3.71779i −0.236239 + 0.136392i −0.613447 0.789736i \(-0.710217\pi\)
0.377208 + 0.926129i \(0.376884\pi\)
\(744\) 0 0
\(745\) 56.1121 + 32.3963i 2.05579 + 1.18691i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −25.7283 + 15.5524i −0.940090 + 0.568273i
\(750\) 0 0
\(751\) −3.67798 6.37044i −0.134211 0.232461i 0.791085 0.611707i \(-0.209517\pi\)
−0.925296 + 0.379246i \(0.876183\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 69.6445 2.53462
\(756\) 0 0
\(757\) 7.65326 0.278163 0.139081 0.990281i \(-0.455585\pi\)
0.139081 + 0.990281i \(0.455585\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21.7203 37.6207i −0.787362 1.36375i −0.927578 0.373629i \(-0.878113\pi\)
0.140217 0.990121i \(-0.455220\pi\)
\(762\) 0 0
\(763\) −21.8895 36.2116i −0.792452 1.31095i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3.87738 2.23860i −0.140004 0.0808313i
\(768\) 0 0
\(769\) −18.8491 + 10.8825i −0.679716 + 0.392434i −0.799748 0.600336i \(-0.795034\pi\)
0.120032 + 0.992770i \(0.461700\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14.5147 0.522056 0.261028 0.965331i \(-0.415938\pi\)
0.261028 + 0.965331i \(0.415938\pi\)
\(774\) 0 0
\(775\) 90.8520i 3.26350i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.21800 5.32202i 0.330269 0.190681i
\(780\) 0 0
\(781\) 7.79402 13.4996i 0.278892 0.483055i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 60.9415 + 35.1846i 2.17509 + 1.25579i
\(786\) 0 0
\(787\) 11.4291 6.59861i 0.407405 0.235215i −0.282269 0.959335i \(-0.591087\pi\)
0.689674 + 0.724120i \(0.257754\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.80793 3.19922i −0.206506 0.113751i
\(792\) 0 0
\(793\) −7.37778 −0.261993
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.8899 + 43.1106i 0.881646 + 1.52706i 0.849511 + 0.527572i \(0.176897\pi\)
0.0321352 + 0.999484i \(0.489769\pi\)
\(798\) 0 0
\(799\) −1.53402 + 2.65701i −0.0542698 + 0.0939981i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.77880 + 11.7412i −0.239219 + 0.414339i
\(804\) 0 0
\(805\) −1.28311 63.7835i −0.0452238 2.24807i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 28.1889i 0.991068i −0.868589 0.495534i \(-0.834972\pi\)
0.868589 0.495534i \(-0.165028\pi\)
\(810\) 0 0
\(811\) 33.5981i 1.17979i −0.807480 0.589894i \(-0.799169\pi\)
0.807480 0.589894i \(-0.200831\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.34988 + 12.7304i 0.257455 + 0.445925i
\(816\) 0 0
\(817\) −25.9247 14.9677i −0.906992 0.523652i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 46.9123 + 27.0848i 1.63725 + 0.945267i 0.981774 + 0.190050i \(0.0608650\pi\)
0.655475 + 0.755217i \(0.272468\pi\)
\(822\) 0 0
\(823\) −14.2695 24.7155i −0.497404 0.861529i 0.502591 0.864524i \(-0.332380\pi\)
−0.999996 + 0.00299479i \(0.999047\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.2062i 1.29379i 0.762581 + 0.646893i \(0.223932\pi\)
−0.762581 + 0.646893i \(0.776068\pi\)
\(828\) 0 0
\(829\) 2.22345i 0.0772238i −0.999254 0.0386119i \(-0.987706\pi\)
0.999254 0.0386119i \(-0.0122936\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.20683 2.29914i 0.0418141 0.0796606i
\(834\) 0 0
\(835\) 16.9668 29.3873i 0.587159 1.01699i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 27.8383 48.2173i 0.961084 1.66465i 0.241297 0.970451i \(-0.422427\pi\)
0.719787 0.694195i \(-0.244240\pi\)
\(840\) 0 0
\(841\) 21.1955 + 36.7116i 0.730878 + 1.26592i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −50.3870 −1.73337
\(846\) 0 0
\(847\) 11.4574 20.8001i 0.393682 0.714700i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −17.2798 + 9.97650i −0.592344 + 0.341990i
\(852\) 0 0
\(853\) −16.2574 9.38622i −0.556643 0.321378i 0.195154 0.980773i \(-0.437479\pi\)
−0.751797 + 0.659395i \(0.770813\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −11.8516 + 20.5276i −0.404844 + 0.701210i −0.994303 0.106588i \(-0.966007\pi\)
0.589460 + 0.807798i \(0.299341\pi\)
\(858\) 0 0
\(859\) −14.5237 + 8.38527i −0.495543 + 0.286102i −0.726871 0.686774i \(-0.759026\pi\)
0.231328 + 0.972876i \(0.425693\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 33.7443i 1.14867i −0.818620 0.574335i \(-0.805261\pi\)
0.818620 0.574335i \(-0.194739\pi\)
\(864\) 0 0
\(865\) 54.8328 1.86437
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 10.4875 6.05497i 0.355765 0.205401i
\(870\) 0 0
\(871\) −8.56192 4.94323i −0.290110 0.167495i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −43.1410 71.3678i −1.45843 2.41267i
\(876\) 0 0
\(877\) 19.7087 + 34.1365i 0.665515 + 1.15271i 0.979145 + 0.203161i \(0.0651215\pi\)
−0.313630 + 0.949545i \(0.601545\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −26.2496 −0.884372 −0.442186 0.896923i \(-0.645797\pi\)
−0.442186 + 0.896923i \(0.645797\pi\)
\(882\) 0 0
\(883\) −43.5087 −1.46418 −0.732091 0.681206i \(-0.761456\pi\)
−0.732091 + 0.681206i \(0.761456\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.83888 4.91708i −0.0953202 0.165099i 0.814422 0.580273i \(-0.197054\pi\)
−0.909742 + 0.415174i \(0.863721\pi\)
\(888\) 0 0
\(889\) −2.61612 4.32782i −0.0877417 0.145151i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −35.6327 20.5725i −1.19240 0.688434i
\(894\) 0 0
\(895\) 46.1124 26.6230i 1.54137 0.889909i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −61.2710 −2.04350
\(900\) 0 0
\(901\) 1.84452i 0.0614498i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −96.2716 + 55.5824i −3.20018 + 1.84762i
\(906\) 0 0
\(907\) −7.32741 + 12.6914i −0.243303 + 0.421412i −0.961653 0.274269i \(-0.911564\pi\)
0.718350 + 0.695681i \(0.244897\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7.19133 4.15192i −0.238260 0.137559i 0.376117 0.926572i \(-0.377259\pi\)
−0.614377 + 0.789013i \(0.710592\pi\)
\(912\) 0 0
\(913\) −2.39607 + 1.38337i −0.0792983 + 0.0457829i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −11.0443 + 20.0500i −0.364714 + 0.662109i
\(918\) 0 0
\(919\) −37.1107 −1.22417 −0.612085 0.790792i \(-0.709669\pi\)
−0.612085 + 0.790792i \(0.709669\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.38106 + 9.32027i 0.177120 + 0.306781i
\(924\) 0 0
\(925\) −21.7022 + 37.5894i −0.713566 + 1.23593i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.94493 + 6.83283i −0.129429 + 0.224178i −0.923456 0.383705i \(-0.874648\pi\)
0.794026 + 0.607883i \(0.207981\pi\)
\(930\) 0 0
\(931\) 30.8334 + 16.1846i 1.01052 + 0.530427i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.20975i 0.0722667i
\(936\) 0 0
\(937\) 13.2688i 0.433472i 0.976230 + 0.216736i \(0.0695411\pi\)
−0.976230 + 0.216736i \(0.930459\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.96461 + 6.86691i 0.129243 + 0.223855i 0.923383 0.383879i \(-0.125412\pi\)
−0.794141 + 0.607734i \(0.792079\pi\)
\(942\) 0 0
\(943\) −10.6719 6.16144i −0.347526 0.200644i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.8152 + 15.4817i 0.871375 + 0.503089i 0.867805 0.496905i \(-0.165530\pi\)
0.00357041 + 0.999994i \(0.498864\pi\)
\(948\) 0 0
\(949\) −4.68014 8.10625i −0.151924 0.263140i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.1087i 1.10489i 0.833549 + 0.552445i \(0.186305\pi\)
−0.833549 + 0.552445i \(0.813695\pi\)
\(954\) 0 0
\(955\) 5.30502i 0.171666i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.620432 30.8417i −0.0200348 0.995929i
\(960\) 0 0
\(961\) 10.7928 18.6937i 0.348156 0.603023i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 39.0089 67.5655i 1.25574 2.17501i
\(966\) 0 0
\(967\) 29.1066 + 50.4142i 0.936007 + 1.62121i 0.772829 + 0.634614i \(0.218841\pi\)
0.163177 + 0.986597i \(0.447826\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −9.83792 −0.315714 −0.157857 0.987462i \(-0.550458\pi\)
−0.157857 + 0.987462i \(0.550458\pi\)
\(972\) 0 0
\(973\) −12.9158 7.11447i −0.414061 0.228080i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.0356 16.7637i 0.928930 0.536318i 0.0424567 0.999098i \(-0.486482\pi\)
0.886473 + 0.462781i \(0.153148\pi\)
\(978\) 0 0
\(979\) 9.62041 + 5.55434i 0.307470 + 0.177518i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.2359 41.9779i 0.773006 1.33889i −0.162902 0.986642i \(-0.552086\pi\)
0.935908 0.352243i \(-0.114581\pi\)
\(984\) 0 0
\(985\) −21.5486 + 12.4411i −0.686594 + 0.396405i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 34.6569i 1.10203i
\(990\) 0 0
\(991\) −53.0011 −1.68364 −0.841818 0.539762i \(-0.818514\pi\)
−0.841818 + 0.539762i \(0.818514\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −20.4015 + 11.7788i −0.646770 + 0.373413i
\(996\) 0 0
\(997\) −7.57384 4.37276i −0.239866 0.138487i 0.375249 0.926924i \(-0.377557\pi\)
−0.615115 + 0.788437i \(0.710890\pi\)
\(998\) 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 756.2.x.a.629.8 16
3.2 odd 2 252.2.x.a.209.5 yes 16
4.3 odd 2 3024.2.cc.c.2897.8 16
7.2 even 3 5292.2.w.a.521.8 16
7.3 odd 6 5292.2.bm.b.4625.8 16
7.4 even 3 5292.2.bm.b.4625.1 16
7.5 odd 6 5292.2.w.a.521.1 16
7.6 odd 2 inner 756.2.x.a.629.1 16
9.2 odd 6 2268.2.f.b.1133.15 16
9.4 even 3 252.2.x.a.41.4 16
9.5 odd 6 inner 756.2.x.a.125.1 16
9.7 even 3 2268.2.f.b.1133.1 16
12.11 even 2 1008.2.cc.c.209.4 16
21.2 odd 6 1764.2.w.a.1109.7 16
21.5 even 6 1764.2.w.a.1109.2 16
21.11 odd 6 1764.2.bm.b.1685.1 16
21.17 even 6 1764.2.bm.b.1685.8 16
21.20 even 2 252.2.x.a.209.4 yes 16
28.27 even 2 3024.2.cc.c.2897.1 16
36.23 even 6 3024.2.cc.c.881.1 16
36.31 odd 6 1008.2.cc.c.545.5 16
63.4 even 3 1764.2.w.a.509.2 16
63.5 even 6 5292.2.bm.b.2285.1 16
63.13 odd 6 252.2.x.a.41.5 yes 16
63.20 even 6 2268.2.f.b.1133.2 16
63.23 odd 6 5292.2.bm.b.2285.8 16
63.31 odd 6 1764.2.w.a.509.7 16
63.32 odd 6 5292.2.w.a.1097.1 16
63.34 odd 6 2268.2.f.b.1133.16 16
63.40 odd 6 1764.2.bm.b.1697.1 16
63.41 even 6 inner 756.2.x.a.125.8 16
63.58 even 3 1764.2.bm.b.1697.8 16
63.59 even 6 5292.2.w.a.1097.8 16
84.83 odd 2 1008.2.cc.c.209.5 16
252.139 even 6 1008.2.cc.c.545.4 16
252.167 odd 6 3024.2.cc.c.881.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.x.a.41.4 16 9.4 even 3
252.2.x.a.41.5 yes 16 63.13 odd 6
252.2.x.a.209.4 yes 16 21.20 even 2
252.2.x.a.209.5 yes 16 3.2 odd 2
756.2.x.a.125.1 16 9.5 odd 6 inner
756.2.x.a.125.8 16 63.41 even 6 inner
756.2.x.a.629.1 16 7.6 odd 2 inner
756.2.x.a.629.8 16 1.1 even 1 trivial
1008.2.cc.c.209.4 16 12.11 even 2
1008.2.cc.c.209.5 16 84.83 odd 2
1008.2.cc.c.545.4 16 252.139 even 6
1008.2.cc.c.545.5 16 36.31 odd 6
1764.2.w.a.509.2 16 63.4 even 3
1764.2.w.a.509.7 16 63.31 odd 6
1764.2.w.a.1109.2 16 21.5 even 6
1764.2.w.a.1109.7 16 21.2 odd 6
1764.2.bm.b.1685.1 16 21.11 odd 6
1764.2.bm.b.1685.8 16 21.17 even 6
1764.2.bm.b.1697.1 16 63.40 odd 6
1764.2.bm.b.1697.8 16 63.58 even 3
2268.2.f.b.1133.1 16 9.7 even 3
2268.2.f.b.1133.2 16 63.20 even 6
2268.2.f.b.1133.15 16 9.2 odd 6
2268.2.f.b.1133.16 16 63.34 odd 6
3024.2.cc.c.881.1 16 36.23 even 6
3024.2.cc.c.881.8 16 252.167 odd 6
3024.2.cc.c.2897.1 16 28.27 even 2
3024.2.cc.c.2897.8 16 4.3 odd 2
5292.2.w.a.521.1 16 7.5 odd 6
5292.2.w.a.521.8 16 7.2 even 3
5292.2.w.a.1097.1 16 63.32 odd 6
5292.2.w.a.1097.8 16 63.59 even 6
5292.2.bm.b.2285.1 16 63.5 even 6
5292.2.bm.b.2285.8 16 63.23 odd 6
5292.2.bm.b.4625.1 16 7.4 even 3
5292.2.bm.b.4625.8 16 7.3 odd 6