Properties

Label 756.2.bm.a.17.8
Level $756$
Weight $2$
Character 756.17
Analytic conductor $6.037$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,2,Mod(17,756)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.bm (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: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 17.8
Root \(-0.544978 - 1.64408i\) of defining polynomial
Character \(\chi\) \(=\) 756.17
Dual form 756.2.bm.a.89.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.91482 q^{5} +(-2.51757 - 0.813537i) q^{7} +O(q^{10})\) \(q+3.91482 q^{5} +(-2.51757 - 0.813537i) q^{7} -3.69456i q^{11} +(-0.480242 - 0.277268i) q^{13} +(2.91916 - 5.05613i) q^{17} +(4.62434 - 2.66986i) q^{19} +2.27435i q^{23} +10.3258 q^{25} +(-3.53638 + 2.04173i) q^{29} +(-7.00132 + 4.04222i) q^{31} +(-9.85583 - 3.18485i) q^{35} +(3.89849 + 6.75239i) q^{37} +(3.59234 - 6.22212i) q^{41} +(-0.754009 - 1.30598i) q^{43} +(1.41416 - 2.44940i) q^{47} +(5.67631 + 4.09627i) q^{49} +(-0.0415658 - 0.0239980i) q^{53} -14.4635i q^{55} +(4.45656 + 7.71900i) q^{59} +(6.03343 + 3.48340i) q^{61} +(-1.88006 - 1.08545i) q^{65} +(-0.587402 - 1.01741i) q^{67} -6.71061i q^{71} +(-3.52692 - 2.03627i) q^{73} +(-3.00566 + 9.30131i) q^{77} +(1.97374 - 3.41861i) q^{79} +(3.84674 + 6.66275i) q^{83} +(11.4280 - 19.7938i) q^{85} +(-2.71300 - 4.69905i) q^{89} +(0.983474 + 1.08874i) q^{91} +(18.1035 - 10.4520i) q^{95} +(-13.9874 + 8.07563i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{7} + 3 q^{13} + 9 q^{17} + 16 q^{25} - 6 q^{29} + 6 q^{31} - 15 q^{35} + q^{37} - 6 q^{41} - 2 q^{43} + 18 q^{47} + 13 q^{49} + 15 q^{59} + 3 q^{61} + 39 q^{65} - 7 q^{67} + 45 q^{77} - q^{79} + 6 q^{85} + 21 q^{89} + 9 q^{91} - 6 q^{95} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(5\) 3.91482 1.75076 0.875381 0.483434i \(-0.160611\pi\)
0.875381 + 0.483434i \(0.160611\pi\)
\(6\) 0 0
\(7\) −2.51757 0.813537i −0.951552 0.307488i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.69456i 1.11395i −0.830529 0.556976i \(-0.811962\pi\)
0.830529 0.556976i \(-0.188038\pi\)
\(12\) 0 0
\(13\) −0.480242 0.277268i −0.133195 0.0769002i 0.431922 0.901911i \(-0.357836\pi\)
−0.565117 + 0.825011i \(0.691169\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.91916 5.05613i 0.708000 1.22629i −0.257598 0.966252i \(-0.582931\pi\)
0.965598 0.260040i \(-0.0837356\pi\)
\(18\) 0 0
\(19\) 4.62434 2.66986i 1.06090 0.612509i 0.135216 0.990816i \(-0.456827\pi\)
0.925680 + 0.378307i \(0.123494\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.27435i 0.474236i 0.971481 + 0.237118i \(0.0762027\pi\)
−0.971481 + 0.237118i \(0.923797\pi\)
\(24\) 0 0
\(25\) 10.3258 2.06516
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.53638 + 2.04173i −0.656690 + 0.379140i −0.791014 0.611797i \(-0.790447\pi\)
0.134325 + 0.990937i \(0.457113\pi\)
\(30\) 0 0
\(31\) −7.00132 + 4.04222i −1.25748 + 0.726004i −0.972583 0.232556i \(-0.925291\pi\)
−0.284892 + 0.958560i \(0.591958\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.85583 3.18485i −1.66594 0.538338i
\(36\) 0 0
\(37\) 3.89849 + 6.75239i 0.640909 + 1.11009i 0.985230 + 0.171235i \(0.0547756\pi\)
−0.344322 + 0.938852i \(0.611891\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.59234 6.22212i 0.561030 0.971732i −0.436377 0.899764i \(-0.643739\pi\)
0.997407 0.0719684i \(-0.0229281\pi\)
\(42\) 0 0
\(43\) −0.754009 1.30598i −0.114985 0.199160i 0.802789 0.596264i \(-0.203349\pi\)
−0.917774 + 0.397103i \(0.870015\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.41416 2.44940i 0.206277 0.357282i −0.744262 0.667888i \(-0.767199\pi\)
0.950539 + 0.310606i \(0.100532\pi\)
\(48\) 0 0
\(49\) 5.67631 + 4.09627i 0.810902 + 0.585182i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.0415658 0.0239980i −0.00570950 0.00329638i 0.497143 0.867669i \(-0.334383\pi\)
−0.502852 + 0.864373i \(0.667716\pi\)
\(54\) 0 0
\(55\) 14.4635i 1.95026i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.45656 + 7.71900i 0.580195 + 1.00493i 0.995456 + 0.0952251i \(0.0303571\pi\)
−0.415261 + 0.909703i \(0.636310\pi\)
\(60\) 0 0
\(61\) 6.03343 + 3.48340i 0.772501 + 0.446004i 0.833766 0.552118i \(-0.186180\pi\)
−0.0612648 + 0.998122i \(0.519513\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.88006 1.08545i −0.233193 0.134634i
\(66\) 0 0
\(67\) −0.587402 1.01741i −0.0717626 0.124296i 0.827911 0.560859i \(-0.189529\pi\)
−0.899674 + 0.436563i \(0.856196\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.71061i 0.796403i −0.917298 0.398202i \(-0.869634\pi\)
0.917298 0.398202i \(-0.130366\pi\)
\(72\) 0 0
\(73\) −3.52692 2.03627i −0.412795 0.238327i 0.279195 0.960234i \(-0.409932\pi\)
−0.691990 + 0.721907i \(0.743266\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00566 + 9.30131i −0.342527 + 1.05998i
\(78\) 0 0
\(79\) 1.97374 3.41861i 0.222063 0.384624i −0.733371 0.679828i \(-0.762054\pi\)
0.955434 + 0.295204i \(0.0953877\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.84674 + 6.66275i 0.422235 + 0.731332i 0.996158 0.0875774i \(-0.0279125\pi\)
−0.573923 + 0.818909i \(0.694579\pi\)
\(84\) 0 0
\(85\) 11.4280 19.7938i 1.23954 2.14694i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.71300 4.69905i −0.287577 0.498099i 0.685654 0.727928i \(-0.259517\pi\)
−0.973231 + 0.229829i \(0.926183\pi\)
\(90\) 0 0
\(91\) 0.983474 + 1.08874i 0.103096 + 0.114130i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.1035 10.4520i 1.85738 1.07236i
\(96\) 0 0
\(97\) −13.9874 + 8.07563i −1.42021 + 0.819956i −0.996316 0.0857571i \(-0.972669\pi\)
−0.423890 + 0.905714i \(0.639336\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.62350 −0.161544 −0.0807722 0.996733i \(-0.525739\pi\)
−0.0807722 + 0.996733i \(0.525739\pi\)
\(102\) 0 0
\(103\) 0.395662i 0.0389857i 0.999810 + 0.0194929i \(0.00620517\pi\)
−0.999810 + 0.0194929i \(0.993795\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.90777 + 2.83350i −0.474452 + 0.273925i −0.718101 0.695938i \(-0.754989\pi\)
0.243650 + 0.969863i \(0.421655\pi\)
\(108\) 0 0
\(109\) −6.75667 + 11.7029i −0.647171 + 1.12093i 0.336624 + 0.941639i \(0.390715\pi\)
−0.983795 + 0.179294i \(0.942619\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.13651 0.656162i −0.106913 0.0617265i 0.445590 0.895237i \(-0.352994\pi\)
−0.552503 + 0.833511i \(0.686327\pi\)
\(114\) 0 0
\(115\) 8.90369i 0.830273i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.4625 + 10.3543i −1.05077 + 0.949179i
\(120\) 0 0
\(121\) −2.64977 −0.240888
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 20.8496 1.86485
\(126\) 0 0
\(127\) −17.3935 −1.54342 −0.771710 0.635975i \(-0.780598\pi\)
−0.771710 + 0.635975i \(0.780598\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.9072 −0.952968 −0.476484 0.879183i \(-0.658089\pi\)
−0.476484 + 0.879183i \(0.658089\pi\)
\(132\) 0 0
\(133\) −13.8141 + 2.95950i −1.19784 + 0.256621i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.80514i 0.752274i 0.926564 + 0.376137i \(0.122748\pi\)
−0.926564 + 0.376137i \(0.877252\pi\)
\(138\) 0 0
\(139\) 14.2352 + 8.21869i 1.20741 + 0.697100i 0.962193 0.272367i \(-0.0878066\pi\)
0.245220 + 0.969468i \(0.421140\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.02438 + 1.77428i −0.0856631 + 0.148373i
\(144\) 0 0
\(145\) −13.8443 + 7.99301i −1.14971 + 0.663783i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.5278i 1.19016i −0.803665 0.595082i \(-0.797120\pi\)
0.803665 0.595082i \(-0.202880\pi\)
\(150\) 0 0
\(151\) 5.60613 0.456221 0.228110 0.973635i \(-0.426745\pi\)
0.228110 + 0.973635i \(0.426745\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −27.4089 + 15.8246i −2.20154 + 1.27106i
\(156\) 0 0
\(157\) −15.4411 + 8.91493i −1.23233 + 0.711489i −0.967516 0.252809i \(-0.918645\pi\)
−0.264819 + 0.964298i \(0.585312\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.85027 5.72584i 0.145822 0.451260i
\(162\) 0 0
\(163\) −0.576994 0.999383i −0.0451937 0.0782777i 0.842544 0.538628i \(-0.181057\pi\)
−0.887737 + 0.460350i \(0.847724\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.95550 + 15.5114i −0.692997 + 1.20031i 0.277854 + 0.960623i \(0.410377\pi\)
−0.970851 + 0.239683i \(0.922957\pi\)
\(168\) 0 0
\(169\) −6.34625 10.9920i −0.488173 0.845540i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.74814 + 6.49197i −0.284966 + 0.493576i −0.972601 0.232481i \(-0.925316\pi\)
0.687635 + 0.726057i \(0.258649\pi\)
\(174\) 0 0
\(175\) −25.9960 8.40044i −1.96511 0.635014i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.624382 + 0.360487i 0.0466685 + 0.0269441i 0.523153 0.852239i \(-0.324756\pi\)
−0.476484 + 0.879183i \(0.658089\pi\)
\(180\) 0 0
\(181\) 5.07121i 0.376940i 0.982079 + 0.188470i \(0.0603529\pi\)
−0.982079 + 0.188470i \(0.939647\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.2619 + 26.4344i 1.12208 + 1.94350i
\(186\) 0 0
\(187\) −18.6802 10.7850i −1.36603 0.788678i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.0005 + 6.35111i 0.795965 + 0.459551i 0.842058 0.539387i \(-0.181344\pi\)
−0.0460934 + 0.998937i \(0.514677\pi\)
\(192\) 0 0
\(193\) 11.4076 + 19.7586i 0.821140 + 1.42226i 0.904834 + 0.425765i \(0.139995\pi\)
−0.0836931 + 0.996492i \(0.526672\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.0311360i 0.00221835i −0.999999 0.00110918i \(-0.999647\pi\)
0.999999 0.00110918i \(-0.000353062\pi\)
\(198\) 0 0
\(199\) 19.9144 + 11.4976i 1.41169 + 0.815042i 0.995548 0.0942556i \(-0.0300471\pi\)
0.416146 + 0.909298i \(0.363380\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.5641 2.26322i 0.741455 0.158847i
\(204\) 0 0
\(205\) 14.0634 24.3585i 0.982229 1.70127i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.86397 17.0849i −0.682305 1.18179i
\(210\) 0 0
\(211\) 8.55841 14.8236i 0.589185 1.02050i −0.405154 0.914248i \(-0.632782\pi\)
0.994339 0.106250i \(-0.0338845\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.95181 5.11268i −0.201312 0.348682i
\(216\) 0 0
\(217\) 20.9148 4.48072i 1.41979 0.304171i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.80380 + 1.61878i −0.188604 + 0.108891i
\(222\) 0 0
\(223\) −1.25230 + 0.723016i −0.0838602 + 0.0484167i −0.541344 0.840801i \(-0.682084\pi\)
0.457484 + 0.889218i \(0.348751\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.47193 0.296812 0.148406 0.988926i \(-0.452586\pi\)
0.148406 + 0.988926i \(0.452586\pi\)
\(228\) 0 0
\(229\) 2.58736i 0.170978i −0.996339 0.0854888i \(-0.972755\pi\)
0.996339 0.0854888i \(-0.0272452\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.0756 8.70389i 0.987634 0.570211i 0.0830679 0.996544i \(-0.473528\pi\)
0.904566 + 0.426333i \(0.140195\pi\)
\(234\) 0 0
\(235\) 5.53620 9.58898i 0.361142 0.625516i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.23642 2.44590i −0.274031 0.158212i 0.356687 0.934224i \(-0.383906\pi\)
−0.630718 + 0.776012i \(0.717240\pi\)
\(240\) 0 0
\(241\) 8.13235i 0.523851i 0.965088 + 0.261925i \(0.0843574\pi\)
−0.965088 + 0.261925i \(0.915643\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 22.2218 + 16.0362i 1.41970 + 1.02451i
\(246\) 0 0
\(247\) −2.96107 −0.188408
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −25.9341 −1.63694 −0.818472 0.574546i \(-0.805179\pi\)
−0.818472 + 0.574546i \(0.805179\pi\)
\(252\) 0 0
\(253\) 8.40274 0.528276
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −30.8230 −1.92269 −0.961344 0.275349i \(-0.911207\pi\)
−0.961344 + 0.275349i \(0.911207\pi\)
\(258\) 0 0
\(259\) −4.32141 20.1712i −0.268519 1.25338i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0855i 1.11520i −0.830110 0.557600i \(-0.811722\pi\)
0.830110 0.557600i \(-0.188278\pi\)
\(264\) 0 0
\(265\) −0.162723 0.0939479i −0.00999597 0.00577117i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.8203 + 18.7413i −0.659725 + 1.14268i 0.320961 + 0.947092i \(0.395994\pi\)
−0.980687 + 0.195585i \(0.937339\pi\)
\(270\) 0 0
\(271\) −12.3453 + 7.12756i −0.749923 + 0.432968i −0.825666 0.564159i \(-0.809200\pi\)
0.0757430 + 0.997127i \(0.475867\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 38.1494i 2.30049i
\(276\) 0 0
\(277\) 8.80327 0.528937 0.264469 0.964394i \(-0.414803\pi\)
0.264469 + 0.964394i \(0.414803\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.6889 9.63537i 0.995579 0.574798i 0.0886417 0.996064i \(-0.471747\pi\)
0.906937 + 0.421266i \(0.138414\pi\)
\(282\) 0 0
\(283\) −8.32822 + 4.80830i −0.495061 + 0.285824i −0.726672 0.686985i \(-0.758934\pi\)
0.231611 + 0.972809i \(0.425601\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.1059 + 12.7421i −0.832645 + 0.752144i
\(288\) 0 0
\(289\) −8.54297 14.7969i −0.502528 0.870404i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.22598 2.12346i 0.0716225 0.124054i −0.827990 0.560743i \(-0.810516\pi\)
0.899613 + 0.436689i \(0.143849\pi\)
\(294\) 0 0
\(295\) 17.4467 + 30.2185i 1.01578 + 1.75939i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.630605 1.09224i 0.0364688 0.0631658i
\(300\) 0 0
\(301\) 0.835805 + 3.90131i 0.0481750 + 0.224868i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 23.6198 + 13.6369i 1.35247 + 0.780846i
\(306\) 0 0
\(307\) 10.6839i 0.609760i 0.952391 + 0.304880i \(0.0986163\pi\)
−0.952391 + 0.304880i \(0.901384\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.3833 + 17.9843i 0.588780 + 1.01980i 0.994393 + 0.105752i \(0.0337250\pi\)
−0.405612 + 0.914045i \(0.632942\pi\)
\(312\) 0 0
\(313\) 3.40449 + 1.96558i 0.192433 + 0.111101i 0.593121 0.805113i \(-0.297896\pi\)
−0.400688 + 0.916215i \(0.631229\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.98369 1.14528i −0.111415 0.0643256i 0.443257 0.896395i \(-0.353823\pi\)
−0.554672 + 0.832069i \(0.687156\pi\)
\(318\) 0 0
\(319\) 7.54330 + 13.0654i 0.422344 + 0.731521i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 31.1750i 1.73462i
\(324\) 0 0
\(325\) −4.95889 2.86302i −0.275070 0.158812i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.55294 + 5.01607i −0.306143 + 0.276545i
\(330\) 0 0
\(331\) −3.46788 + 6.00655i −0.190612 + 0.330150i −0.945453 0.325758i \(-0.894381\pi\)
0.754841 + 0.655908i \(0.227714\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.29957 3.98298i −0.125639 0.217613i
\(336\) 0 0
\(337\) −9.59771 + 16.6237i −0.522821 + 0.905552i 0.476827 + 0.878997i \(0.341787\pi\)
−0.999647 + 0.0265545i \(0.991546\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14.9342 + 25.8668i 0.808733 + 1.40077i
\(342\) 0 0
\(343\) −10.9580 14.9305i −0.591679 0.806174i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.35287 4.24518i 0.394723 0.227893i −0.289482 0.957184i \(-0.593483\pi\)
0.684204 + 0.729290i \(0.260150\pi\)
\(348\) 0 0
\(349\) 16.5478 9.55386i 0.885782 0.511407i 0.0132216 0.999913i \(-0.495791\pi\)
0.872560 + 0.488506i \(0.162458\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.6590 −0.726996 −0.363498 0.931595i \(-0.618418\pi\)
−0.363498 + 0.931595i \(0.618418\pi\)
\(354\) 0 0
\(355\) 26.2708i 1.39431i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −14.8909 + 8.59724i −0.785909 + 0.453745i −0.838520 0.544870i \(-0.816579\pi\)
0.0526113 + 0.998615i \(0.483246\pi\)
\(360\) 0 0
\(361\) 4.75635 8.23824i 0.250334 0.433592i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.8073 7.97162i −0.722705 0.417254i
\(366\) 0 0
\(367\) 16.8587i 0.880018i 0.897993 + 0.440009i \(0.145025\pi\)
−0.897993 + 0.440009i \(0.854975\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.0851215 + 0.0942320i 0.00441929 + 0.00489228i
\(372\) 0 0
\(373\) −1.40858 −0.0729333 −0.0364667 0.999335i \(-0.511610\pi\)
−0.0364667 + 0.999335i \(0.511610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.26442 0.116624
\(378\) 0 0
\(379\) −0.598572 −0.0307466 −0.0153733 0.999882i \(-0.504894\pi\)
−0.0153733 + 0.999882i \(0.504894\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.52077 0.435391 0.217696 0.976017i \(-0.430146\pi\)
0.217696 + 0.976017i \(0.430146\pi\)
\(384\) 0 0
\(385\) −11.7666 + 36.4130i −0.599683 + 1.85578i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 34.5976i 1.75417i −0.480336 0.877084i \(-0.659485\pi\)
0.480336 0.877084i \(-0.340515\pi\)
\(390\) 0 0
\(391\) 11.4994 + 6.63920i 0.581551 + 0.335759i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.72683 13.3833i 0.388779 0.673385i
\(396\) 0 0
\(397\) 27.9571 16.1411i 1.40313 0.810097i 0.408416 0.912796i \(-0.366081\pi\)
0.994712 + 0.102699i \(0.0327478\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.1028i 0.654321i −0.944969 0.327161i \(-0.893908\pi\)
0.944969 0.327161i \(-0.106092\pi\)
\(402\) 0 0
\(403\) 4.48310 0.223319
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.9471 14.4032i 1.23658 0.713941i
\(408\) 0 0
\(409\) 32.3493 18.6769i 1.59957 0.923513i 0.608002 0.793936i \(-0.291971\pi\)
0.991569 0.129577i \(-0.0413620\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.94002 23.0587i −0.243082 1.13464i
\(414\) 0 0
\(415\) 15.0593 + 26.0835i 0.739232 + 1.28039i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.1954 24.5871i 0.693490 1.20116i −0.277198 0.960813i \(-0.589406\pi\)
0.970687 0.240346i \(-0.0772610\pi\)
\(420\) 0 0
\(421\) −17.3359 30.0267i −0.844901 1.46341i −0.885707 0.464245i \(-0.846326\pi\)
0.0408054 0.999167i \(-0.487008\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 30.1427 52.2087i 1.46214 2.53249i
\(426\) 0 0
\(427\) −12.3557 13.6781i −0.597934 0.661931i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 13.1844 + 7.61200i 0.635069 + 0.366657i 0.782713 0.622383i \(-0.213835\pi\)
−0.147643 + 0.989041i \(0.547169\pi\)
\(432\) 0 0
\(433\) 3.97041i 0.190806i −0.995439 0.0954028i \(-0.969586\pi\)
0.995439 0.0954028i \(-0.0304139\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.07222 + 10.5174i 0.290474 + 0.503115i
\(438\) 0 0
\(439\) 8.21910 + 4.74530i 0.392276 + 0.226481i 0.683146 0.730282i \(-0.260611\pi\)
−0.290870 + 0.956763i \(0.593945\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.3955 + 16.3942i 1.34911 + 0.778910i 0.988124 0.153660i \(-0.0491060\pi\)
0.360989 + 0.932570i \(0.382439\pi\)
\(444\) 0 0
\(445\) −10.6209 18.3960i −0.503479 0.872052i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.658896i 0.0310952i 0.999879 + 0.0155476i \(0.00494916\pi\)
−0.999879 + 0.0155476i \(0.995051\pi\)
\(450\) 0 0
\(451\) −22.9880 13.2721i −1.08246 0.624960i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.85013 + 4.26220i 0.180497 + 0.199815i
\(456\) 0 0
\(457\) −7.94514 + 13.7614i −0.371658 + 0.643730i −0.989821 0.142320i \(-0.954544\pi\)
0.618163 + 0.786050i \(0.287877\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.81626 17.0023i −0.457189 0.791874i 0.541622 0.840622i \(-0.317810\pi\)
−0.998811 + 0.0487477i \(0.984477\pi\)
\(462\) 0 0
\(463\) 0.600159 1.03951i 0.0278918 0.0483099i −0.851743 0.523960i \(-0.824454\pi\)
0.879634 + 0.475651i \(0.157787\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 19.2809 + 33.3955i 0.892213 + 1.54536i 0.837216 + 0.546872i \(0.184182\pi\)
0.0549972 + 0.998487i \(0.482485\pi\)
\(468\) 0 0
\(469\) 0.651124 + 3.03927i 0.0300661 + 0.140341i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.82503 + 2.78573i −0.221855 + 0.128088i
\(474\) 0 0
\(475\) 47.7501 27.5685i 2.19093 1.26493i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.22578 0.330154 0.165077 0.986281i \(-0.447213\pi\)
0.165077 + 0.986281i \(0.447213\pi\)
\(480\) 0 0
\(481\) 4.32371i 0.197144i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −54.7582 + 31.6147i −2.48644 + 1.43555i
\(486\) 0 0
\(487\) 4.85770 8.41378i 0.220123 0.381265i −0.734722 0.678368i \(-0.762687\pi\)
0.954845 + 0.297104i \(0.0960207\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.2480 9.95814i −0.778392 0.449405i 0.0574682 0.998347i \(-0.481697\pi\)
−0.835860 + 0.548943i \(0.815031\pi\)
\(492\) 0 0
\(493\) 23.8405i 1.07372i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.45933 + 16.8944i −0.244885 + 0.757819i
\(498\) 0 0
\(499\) 34.3840 1.53924 0.769619 0.638503i \(-0.220446\pi\)
0.769619 + 0.638503i \(0.220446\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.22542 0.0546388 0.0273194 0.999627i \(-0.491303\pi\)
0.0273194 + 0.999627i \(0.491303\pi\)
\(504\) 0 0
\(505\) −6.35571 −0.282826
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10.1016 −0.447744 −0.223872 0.974619i \(-0.571870\pi\)
−0.223872 + 0.974619i \(0.571870\pi\)
\(510\) 0 0
\(511\) 7.22268 + 7.99572i 0.319513 + 0.353710i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.54895i 0.0682547i
\(516\) 0 0
\(517\) −9.04947 5.22471i −0.397995 0.229783i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.5390 18.2541i 0.461723 0.799728i −0.537324 0.843376i \(-0.680565\pi\)
0.999047 + 0.0436480i \(0.0138980\pi\)
\(522\) 0 0
\(523\) 17.0733 9.85727i 0.746563 0.431028i −0.0778877 0.996962i \(-0.524818\pi\)
0.824451 + 0.565934i \(0.191484\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 47.1995i 2.05604i
\(528\) 0 0
\(529\) 17.8273 0.775101
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.45039 + 1.99208i −0.149453 + 0.0862866i
\(534\) 0 0
\(535\) −19.2130 + 11.0926i −0.830652 + 0.479577i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 15.1339 20.9715i 0.651864 0.903306i
\(540\) 0 0
\(541\) −4.22475 7.31748i −0.181636 0.314603i 0.760802 0.648984i \(-0.224806\pi\)
−0.942438 + 0.334381i \(0.891473\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −26.4511 + 45.8147i −1.13304 + 1.96249i
\(546\) 0 0
\(547\) −4.02889 6.97824i −0.172263 0.298368i 0.766948 0.641709i \(-0.221774\pi\)
−0.939211 + 0.343342i \(0.888441\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10.9023 + 18.8833i −0.464453 + 0.804456i
\(552\) 0 0
\(553\) −7.75019 + 7.00089i −0.329572 + 0.297708i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.2294 10.5247i −0.772403 0.445947i 0.0613279 0.998118i \(-0.480466\pi\)
−0.833731 + 0.552170i \(0.813800\pi\)
\(558\) 0 0
\(559\) 0.836249i 0.0353696i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −20.6410 35.7513i −0.869916 1.50674i −0.862082 0.506769i \(-0.830840\pi\)
−0.00783378 0.999969i \(-0.502494\pi\)
\(564\) 0 0
\(565\) −4.44922 2.56876i −0.187180 0.108068i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −31.2691 18.0532i −1.31087 0.756829i −0.328627 0.944460i \(-0.606586\pi\)
−0.982240 + 0.187630i \(0.939919\pi\)
\(570\) 0 0
\(571\) 9.62111 + 16.6642i 0.402631 + 0.697377i 0.994043 0.108993i \(-0.0347625\pi\)
−0.591412 + 0.806370i \(0.701429\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.4846i 0.979375i
\(576\) 0 0
\(577\) −25.8102 14.9015i −1.07449 0.620359i −0.145088 0.989419i \(-0.546346\pi\)
−0.929406 + 0.369060i \(0.879680\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −4.26404 19.9034i −0.176902 0.825732i
\(582\) 0 0
\(583\) −0.0886621 + 0.153567i −0.00367201 + 0.00636010i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.72218 + 8.17905i 0.194905 + 0.337586i 0.946869 0.321618i \(-0.104227\pi\)
−0.751964 + 0.659204i \(0.770893\pi\)
\(588\) 0 0
\(589\) −21.5843 + 37.3852i −0.889367 + 1.54043i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.4176 + 21.5079i 0.509929 + 0.883223i 0.999934 + 0.0115033i \(0.00366171\pi\)
−0.490005 + 0.871720i \(0.663005\pi\)
\(594\) 0 0
\(595\) −44.8738 + 40.5353i −1.83965 + 1.66179i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.3052 5.94974i 0.421061 0.243100i −0.274470 0.961596i \(-0.588502\pi\)
0.695531 + 0.718496i \(0.255169\pi\)
\(600\) 0 0
\(601\) −22.1276 + 12.7754i −0.902604 + 0.521118i −0.878044 0.478580i \(-0.841152\pi\)
−0.0245596 + 0.999698i \(0.507818\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.3734 −0.421738
\(606\) 0 0
\(607\) 22.5794i 0.916471i 0.888831 + 0.458235i \(0.151518\pi\)
−0.888831 + 0.458235i \(0.848482\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.35828 + 0.784204i −0.0549502 + 0.0317255i
\(612\) 0 0
\(613\) −11.4294 + 19.7963i −0.461628 + 0.799564i −0.999042 0.0437549i \(-0.986068\pi\)
0.537414 + 0.843319i \(0.319401\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.78792 1.03226i −0.0719791 0.0415572i 0.463578 0.886056i \(-0.346565\pi\)
−0.535558 + 0.844499i \(0.679899\pi\)
\(618\) 0 0
\(619\) 32.5894i 1.30988i −0.755681 0.654940i \(-0.772694\pi\)
0.755681 0.654940i \(-0.227306\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.00731 + 14.0373i 0.120485 + 0.562393i
\(624\) 0 0
\(625\) 29.9935 1.19974
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 45.5213 1.81505
\(630\) 0 0
\(631\) 38.4706 1.53149 0.765744 0.643145i \(-0.222371\pi\)
0.765744 + 0.643145i \(0.222371\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −68.0923 −2.70216
\(636\) 0 0
\(637\) −1.59024 3.54106i −0.0630075 0.140302i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 47.7636i 1.88655i 0.332014 + 0.943274i \(0.392272\pi\)
−0.332014 + 0.943274i \(0.607728\pi\)
\(642\) 0 0
\(643\) −29.2346 16.8786i −1.15290 0.665626i −0.203306 0.979115i \(-0.565169\pi\)
−0.949592 + 0.313489i \(0.898502\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.536008 0.928393i 0.0210727 0.0364989i −0.855297 0.518138i \(-0.826625\pi\)
0.876369 + 0.481640i \(0.159959\pi\)
\(648\) 0 0
\(649\) 28.5183 16.4650i 1.11944 0.646309i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.3135i 1.30366i −0.758367 0.651828i \(-0.774002\pi\)
0.758367 0.651828i \(-0.225998\pi\)
\(654\) 0 0
\(655\) −42.6998 −1.66842
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.41890 4.86065i 0.327954 0.189344i −0.326979 0.945032i \(-0.606031\pi\)
0.654932 + 0.755688i \(0.272697\pi\)
\(660\) 0 0
\(661\) −14.7856 + 8.53647i −0.575093 + 0.332030i −0.759181 0.650880i \(-0.774400\pi\)
0.184088 + 0.982910i \(0.441067\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −54.0799 + 11.5859i −2.09713 + 0.449282i
\(666\) 0 0
\(667\) −4.64362 8.04298i −0.179802 0.311426i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.8696 22.2909i 0.496827 0.860529i
\(672\) 0 0
\(673\) −18.3359 31.7588i −0.706798 1.22421i −0.966039 0.258398i \(-0.916805\pi\)
0.259240 0.965813i \(-0.416528\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20.1769 34.9474i 0.775461 1.34314i −0.159073 0.987267i \(-0.550851\pi\)
0.934535 0.355872i \(-0.115816\pi\)
\(678\) 0 0
\(679\) 41.7841 8.95169i 1.60353 0.343534i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.23662 + 4.75541i 0.315165 + 0.181961i 0.649236 0.760587i \(-0.275089\pi\)
−0.334070 + 0.942548i \(0.608422\pi\)
\(684\) 0 0
\(685\) 34.4705i 1.31705i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.0133077 + 0.0230497i 0.000506985 + 0.000878123i
\(690\) 0 0
\(691\) −6.67519 3.85392i −0.253936 0.146610i 0.367629 0.929972i \(-0.380170\pi\)
−0.621565 + 0.783362i \(0.713503\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 55.7282 + 32.1747i 2.11389 + 1.22046i
\(696\) 0 0
\(697\) −20.9732 36.3267i −0.794418 1.37597i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.6388i 0.590671i 0.955394 + 0.295336i \(0.0954314\pi\)
−0.955394 + 0.295336i \(0.904569\pi\)
\(702\) 0 0
\(703\) 36.0559 + 20.8169i 1.35988 + 0.785124i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.08728 + 1.32078i 0.153718 + 0.0496730i
\(708\) 0 0
\(709\) −6.72025 + 11.6398i −0.252384 + 0.437142i −0.964182 0.265242i \(-0.914548\pi\)
0.711797 + 0.702385i \(0.247881\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9.19343 15.9235i −0.344297 0.596339i
\(714\) 0 0
\(715\) −4.01027 + 6.94599i −0.149976 + 0.259765i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.0309 + 34.6946i 0.747027 + 1.29389i 0.949242 + 0.314548i \(0.101853\pi\)
−0.202214 + 0.979341i \(0.564814\pi\)
\(720\) 0 0
\(721\) 0.321886 0.996107i 0.0119877 0.0370970i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −36.5161 + 21.0826i −1.35617 + 0.782986i
\(726\) 0 0
\(727\) −43.2091 + 24.9468i −1.60254 + 0.925225i −0.611560 + 0.791198i \(0.709458\pi\)
−0.990978 + 0.134027i \(0.957209\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.80428 −0.325638
\(732\) 0 0
\(733\) 11.4480i 0.422843i 0.977395 + 0.211422i \(0.0678093\pi\)
−0.977395 + 0.211422i \(0.932191\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.75888 + 2.17019i −0.138460 + 0.0799400i
\(738\) 0 0
\(739\) 4.46303 7.73020i 0.164175 0.284360i −0.772187 0.635396i \(-0.780837\pi\)
0.936362 + 0.351036i \(0.114170\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.8621 + 26.4785i 1.68252 + 0.971403i 0.959979 + 0.280074i \(0.0903589\pi\)
0.722540 + 0.691329i \(0.242974\pi\)
\(744\) 0 0
\(745\) 56.8737i 2.08369i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 14.6608 3.14088i 0.535694 0.114765i
\(750\) 0 0
\(751\) −26.4652 −0.965729 −0.482865 0.875695i \(-0.660404\pi\)
−0.482865 + 0.875695i \(0.660404\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.9470 0.798733
\(756\) 0 0
\(757\) 8.46749 0.307756 0.153878 0.988090i \(-0.450824\pi\)
0.153878 + 0.988090i \(0.450824\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 53.9937 1.95727 0.978635 0.205605i \(-0.0659162\pi\)
0.978635 + 0.205605i \(0.0659162\pi\)
\(762\) 0 0
\(763\) 26.5311 23.9660i 0.960491 0.867629i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.94264i 0.178469i
\(768\) 0 0
\(769\) −30.1912 17.4309i −1.08872 0.628575i −0.155487 0.987838i \(-0.549695\pi\)
−0.933236 + 0.359263i \(0.883028\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.06375 + 1.84246i −0.0382603 + 0.0662688i −0.884521 0.466499i \(-0.845515\pi\)
0.846261 + 0.532768i \(0.178848\pi\)
\(774\) 0 0
\(775\) −72.2944 + 41.7392i −2.59689 + 1.49932i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 38.3643i 1.37454i
\(780\) 0 0
\(781\) −24.7928 −0.887155
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −60.4492 + 34.9004i −2.15752 + 1.24565i
\(786\) 0 0
\(787\) 24.5457 14.1715i 0.874959 0.505158i 0.00596615 0.999982i \(-0.498101\pi\)
0.868993 + 0.494824i \(0.164768\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.32742 + 2.57652i 0.0827535 + 0.0916106i
\(792\) 0 0
\(793\) −1.93167 3.34575i −0.0685956 0.118811i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18.9123 + 32.7570i −0.669907 + 1.16031i 0.308022 + 0.951379i \(0.400333\pi\)
−0.977930 + 0.208935i \(0.933000\pi\)
\(798\) 0 0
\(799\) −8.25634 14.3004i −0.292088 0.505912i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −7.52311 + 13.0304i −0.265485 + 0.459833i
\(804\) 0 0
\(805\) 7.24348 22.4157i 0.255299 0.790048i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 39.2475 + 22.6595i 1.37987 + 0.796667i 0.992143 0.125109i \(-0.0399280\pi\)
0.387724 + 0.921776i \(0.373261\pi\)
\(810\) 0 0
\(811\) 5.45145i 0.191426i 0.995409 + 0.0957132i \(0.0305132\pi\)
−0.995409 + 0.0957132i \(0.969487\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.25883 3.91241i −0.0791233 0.137046i
\(816\) 0 0
\(817\) −6.97359 4.02620i −0.243975 0.140859i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.7121 + 24.6598i 1.49066 + 0.860634i 0.999943 0.0106847i \(-0.00340111\pi\)
0.490718 + 0.871318i \(0.336734\pi\)
\(822\) 0 0
\(823\) 11.8496 + 20.5241i 0.413050 + 0.715424i 0.995222 0.0976419i \(-0.0311300\pi\)
−0.582171 + 0.813066i \(0.697797\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 19.9706i 0.694445i −0.937783 0.347222i \(-0.887125\pi\)
0.937783 0.347222i \(-0.112875\pi\)
\(828\) 0 0
\(829\) −13.3741 7.72155i −0.464503 0.268181i 0.249433 0.968392i \(-0.419756\pi\)
−0.713936 + 0.700211i \(0.753089\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 37.2814 16.7425i 1.29172 0.580094i
\(834\) 0 0
\(835\) −35.0592 + 60.7242i −1.21327 + 2.10145i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5.53910 9.59401i −0.191231 0.331222i 0.754427 0.656383i \(-0.227915\pi\)
−0.945658 + 0.325162i \(0.894581\pi\)
\(840\) 0 0
\(841\) −6.16267 + 10.6741i −0.212506 + 0.368071i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −24.8444 43.0318i −0.854674 1.48034i
\(846\) 0 0
\(847\) 6.67098 + 2.15569i 0.229218 + 0.0740703i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −15.3573 + 8.86656i −0.526442 + 0.303942i
\(852\) 0 0
\(853\) 42.1706 24.3472i 1.44389 0.833633i 0.445788 0.895139i \(-0.352924\pi\)
0.998107 + 0.0615058i \(0.0195903\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −16.7826 −0.573283 −0.286641 0.958038i \(-0.592539\pi\)
−0.286641 + 0.958038i \(0.592539\pi\)
\(858\) 0 0
\(859\) 25.1358i 0.857622i 0.903394 + 0.428811i \(0.141067\pi\)
−0.903394 + 0.428811i \(0.858933\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 5.87377 3.39122i 0.199945 0.115438i −0.396685 0.917955i \(-0.629839\pi\)
0.596630 + 0.802516i \(0.296506\pi\)
\(864\) 0 0
\(865\) −14.6733 + 25.4149i −0.498907 + 0.864133i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12.6303 7.29209i −0.428452 0.247367i
\(870\) 0 0
\(871\) 0.651470i 0.0220742i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −52.4904 16.9620i −1.77450 0.573419i
\(876\) 0 0
\(877\) −43.7259 −1.47652 −0.738260 0.674517i \(-0.764352\pi\)
−0.738260 + 0.674517i \(0.764352\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −27.5307 −0.927531 −0.463766 0.885958i \(-0.653502\pi\)
−0.463766 + 0.885958i \(0.653502\pi\)
\(882\) 0 0
\(883\) 5.56040 0.187122 0.0935612 0.995614i \(-0.470175\pi\)
0.0935612 + 0.995614i \(0.470175\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.6185 −0.826607 −0.413303 0.910593i \(-0.635625\pi\)
−0.413303 + 0.910593i \(0.635625\pi\)
\(888\) 0 0
\(889\) 43.7892 + 14.1502i 1.46864 + 0.474583i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.1025i 0.505386i
\(894\) 0 0
\(895\) 2.44434 + 1.41124i 0.0817054 + 0.0471726i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.5062 28.5896i 0.550514 0.953518i
\(900\) 0 0
\(901\) −0.242674 + 0.140108i −0.00808465 + 0.00466767i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.8529i 0.659932i
\(906\) 0 0
\(907\) −10.0867 −0.334925 −0.167462 0.985878i \(-0.553557\pi\)
−0.167462 + 0.985878i \(0.553557\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −23.5808 + 13.6144i −0.781267 + 0.451065i −0.836879 0.547388i \(-0.815622\pi\)
0.0556121 + 0.998452i \(0.482289\pi\)
\(912\) 0 0
\(913\) 24.6159 14.2120i 0.814668 0.470349i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 27.4597 + 8.87343i 0.906799 + 0.293026i
\(918\) 0 0
\(919\) −19.8493 34.3800i −0.654769 1.13409i −0.981952 0.189132i \(-0.939433\pi\)
0.327183 0.944961i \(-0.393901\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.86064 + 3.22272i −0.0612436 + 0.106077i
\(924\) 0 0
\(925\) 40.2552 + 69.7240i 1.32358 + 2.29251i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.142283 0.246442i 0.00466816 0.00808550i −0.863682 0.504037i \(-0.831847\pi\)
0.868350 + 0.495952i \(0.165181\pi\)
\(930\) 0 0
\(931\) 37.1857 + 3.78758i 1.21871 + 0.124133i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −73.1295 42.2214i −2.39159 1.38079i
\(936\) 0 0
\(937\) 21.7298i 0.709881i −0.934889 0.354940i \(-0.884501\pi\)
0.934889 0.354940i \(-0.115499\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.64242 + 9.77295i 0.183938 + 0.318589i 0.943218 0.332174i \(-0.107782\pi\)
−0.759280 + 0.650764i \(0.774449\pi\)
\(942\) 0 0
\(943\) 14.1513 + 8.17026i 0.460830 + 0.266060i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.6701 11.3566i −0.639194 0.369039i 0.145110 0.989416i \(-0.453646\pi\)
−0.784304 + 0.620377i \(0.786980\pi\)
\(948\) 0 0
\(949\) 1.12918 + 1.95580i 0.0366548 + 0.0634880i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 16.5638i 0.536554i 0.963342 + 0.268277i \(0.0864543\pi\)
−0.963342 + 0.268277i \(0.913546\pi\)
\(954\) 0 0
\(955\) 43.0648 + 24.8635i 1.39354 + 0.804563i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 7.16331 22.1675i 0.231315 0.715827i
\(960\) 0 0
\(961\) 17.1790 29.7550i 0.554162 0.959837i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 44.6589 + 77.3515i 1.43762 + 2.49003i
\(966\) 0 0
\(967\) 8.38867 14.5296i 0.269762 0.467241i −0.699039 0.715084i \(-0.746388\pi\)
0.968800 + 0.247843i \(0.0797218\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −15.6820 27.1620i −0.503259 0.871670i −0.999993 0.00376705i \(-0.998801\pi\)
0.496734 0.867903i \(-0.334532\pi\)
\(972\) 0 0
\(973\) −29.1519 32.2720i −0.934566 1.03459i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −49.0953 + 28.3452i −1.57070 + 0.906843i −0.574614 + 0.818424i \(0.694848\pi\)
−0.996083 + 0.0884183i \(0.971819\pi\)
\(978\) 0 0
\(979\) −17.3609 + 10.0233i −0.554858 + 0.320347i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −39.8408 −1.27072 −0.635362 0.772214i \(-0.719149\pi\)
−0.635362 + 0.772214i \(0.719149\pi\)
\(984\) 0 0
\(985\) 0.121892i 0.00388380i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.97026 1.71488i 0.0944489 0.0545301i
\(990\) 0 0
\(991\) 31.2975 54.2089i 0.994199 1.72200i 0.403952 0.914780i \(-0.367636\pi\)
0.590247 0.807223i \(-0.299030\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 77.9613 + 45.0110i 2.47154 + 1.42694i
\(996\) 0 0
\(997\) 45.1041i 1.42846i 0.699911 + 0.714230i \(0.253223\pi\)
−0.699911 + 0.714230i \(0.746777\pi\)
\(998\) 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.bm.a.17.8 16
3.2 odd 2 252.2.bm.a.185.6 yes 16
4.3 odd 2 3024.2.df.d.17.8 16
7.2 even 3 5292.2.w.b.1097.1 16
7.3 odd 6 5292.2.x.a.881.8 16
7.4 even 3 5292.2.x.b.881.1 16
7.5 odd 6 756.2.w.a.341.8 16
7.6 odd 2 5292.2.bm.a.2285.1 16
9.2 odd 6 756.2.w.a.521.8 16
9.4 even 3 2268.2.t.b.1781.1 16
9.5 odd 6 2268.2.t.a.1781.8 16
9.7 even 3 252.2.w.a.101.8 yes 16
12.11 even 2 1008.2.df.d.689.3 16
21.2 odd 6 1764.2.w.b.509.1 16
21.5 even 6 252.2.w.a.5.8 16
21.11 odd 6 1764.2.x.b.293.6 16
21.17 even 6 1764.2.x.a.293.3 16
21.20 even 2 1764.2.bm.a.1697.3 16
28.19 even 6 3024.2.ca.d.2609.8 16
36.7 odd 6 1008.2.ca.d.353.1 16
36.11 even 6 3024.2.ca.d.2033.8 16
63.2 odd 6 5292.2.bm.a.4625.1 16
63.5 even 6 2268.2.t.b.2105.1 16
63.11 odd 6 5292.2.x.a.4409.8 16
63.16 even 3 1764.2.bm.a.1685.3 16
63.20 even 6 5292.2.w.b.521.1 16
63.25 even 3 1764.2.x.a.1469.3 16
63.34 odd 6 1764.2.w.b.1109.1 16
63.38 even 6 5292.2.x.b.4409.1 16
63.40 odd 6 2268.2.t.a.2105.8 16
63.47 even 6 inner 756.2.bm.a.89.8 16
63.52 odd 6 1764.2.x.b.1469.6 16
63.61 odd 6 252.2.bm.a.173.6 yes 16
84.47 odd 6 1008.2.ca.d.257.1 16
252.47 odd 6 3024.2.df.d.1601.8 16
252.187 even 6 1008.2.df.d.929.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.8 16 21.5 even 6
252.2.w.a.101.8 yes 16 9.7 even 3
252.2.bm.a.173.6 yes 16 63.61 odd 6
252.2.bm.a.185.6 yes 16 3.2 odd 2
756.2.w.a.341.8 16 7.5 odd 6
756.2.w.a.521.8 16 9.2 odd 6
756.2.bm.a.17.8 16 1.1 even 1 trivial
756.2.bm.a.89.8 16 63.47 even 6 inner
1008.2.ca.d.257.1 16 84.47 odd 6
1008.2.ca.d.353.1 16 36.7 odd 6
1008.2.df.d.689.3 16 12.11 even 2
1008.2.df.d.929.3 16 252.187 even 6
1764.2.w.b.509.1 16 21.2 odd 6
1764.2.w.b.1109.1 16 63.34 odd 6
1764.2.x.a.293.3 16 21.17 even 6
1764.2.x.a.1469.3 16 63.25 even 3
1764.2.x.b.293.6 16 21.11 odd 6
1764.2.x.b.1469.6 16 63.52 odd 6
1764.2.bm.a.1685.3 16 63.16 even 3
1764.2.bm.a.1697.3 16 21.20 even 2
2268.2.t.a.1781.8 16 9.5 odd 6
2268.2.t.a.2105.8 16 63.40 odd 6
2268.2.t.b.1781.1 16 9.4 even 3
2268.2.t.b.2105.1 16 63.5 even 6
3024.2.ca.d.2033.8 16 36.11 even 6
3024.2.ca.d.2609.8 16 28.19 even 6
3024.2.df.d.17.8 16 4.3 odd 2
3024.2.df.d.1601.8 16 252.47 odd 6
5292.2.w.b.521.1 16 63.20 even 6
5292.2.w.b.1097.1 16 7.2 even 3
5292.2.x.a.881.8 16 7.3 odd 6
5292.2.x.a.4409.8 16 63.11 odd 6
5292.2.x.b.881.1 16 7.4 even 3
5292.2.x.b.4409.1 16 63.38 even 6
5292.2.bm.a.2285.1 16 7.6 odd 2
5292.2.bm.a.4625.1 16 63.2 odd 6