Properties

Label 756.2.w.a.521.4
Level $756$
Weight $2$
Character 756.521
Analytic conductor $6.037$
Analytic rank $0$
Dimension $16$
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] = [756,2,Mod(341,756)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.341");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.w (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: \(\Z[a_1, \ldots, a_{13}]\)
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 521.4
Root \(1.68124 + 0.416458i\) of defining polynomial
Character \(\chi\) \(=\) 756.521
Dual form 756.2.w.a.341.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.349828 + 0.605920i) q^{5} +(2.48683 - 0.903137i) q^{7} +O(q^{10})\) \(q+(-0.349828 + 0.605920i) q^{5} +(2.48683 - 0.903137i) q^{7} +(-0.229685 + 0.132608i) q^{11} +(1.13823 - 0.657156i) q^{13} +(1.86392 - 3.22840i) q^{17} +(-0.382449 + 0.220807i) q^{19} +(4.29949 + 2.48231i) q^{23} +(2.25524 + 3.90619i) q^{25} +(0.273287 + 0.157782i) q^{29} -5.60632i q^{31} +(-0.322736 + 1.82276i) q^{35} +(-0.351124 - 0.608164i) q^{37} +(5.39354 + 9.34189i) q^{41} +(3.73131 - 6.46283i) q^{43} +7.00570 q^{47} +(5.36869 - 4.49190i) q^{49} +(-8.51919 - 4.91856i) q^{53} -0.185561i q^{55} +13.4636 q^{59} +5.65207i q^{61} +0.919566i q^{65} -5.94120 q^{67} +13.4323i q^{71} +(-6.66182 - 3.84620i) q^{73} +(-0.451424 + 0.537212i) q^{77} +1.39672 q^{79} +(3.72399 - 6.45014i) q^{83} +(1.30410 + 2.25877i) q^{85} +(-5.59261 - 9.68668i) q^{89} +(2.23708 - 2.66221i) q^{91} -0.308978i q^{95} +(-9.18225 - 5.30138i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{7} + 6 q^{11} - 3 q^{13} - 9 q^{17} - 21 q^{23} - 8 q^{25} - 6 q^{29} + 15 q^{35} + q^{37} + 6 q^{41} - 2 q^{43} + 36 q^{47} - 5 q^{49} + 30 q^{59} + 14 q^{67} - 3 q^{77} + 2 q^{79} + 6 q^{85} - 21 q^{89} + 9 q^{91} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −0.349828 + 0.605920i −0.156448 + 0.270975i −0.933585 0.358355i \(-0.883338\pi\)
0.777137 + 0.629331i \(0.216671\pi\)
\(6\) 0 0
\(7\) 2.48683 0.903137i 0.939935 0.341354i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.229685 + 0.132608i −0.0692525 + 0.0399829i −0.534226 0.845341i \(-0.679397\pi\)
0.464974 + 0.885324i \(0.346064\pi\)
\(12\) 0 0
\(13\) 1.13823 0.657156i 0.315688 0.182262i −0.333781 0.942651i \(-0.608325\pi\)
0.649469 + 0.760388i \(0.274991\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.86392 3.22840i 0.452067 0.783003i −0.546447 0.837493i \(-0.684020\pi\)
0.998514 + 0.0544906i \(0.0173535\pi\)
\(18\) 0 0
\(19\) −0.382449 + 0.220807i −0.0877398 + 0.0506566i −0.543228 0.839585i \(-0.682798\pi\)
0.455488 + 0.890242i \(0.349465\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.29949 + 2.48231i 0.896507 + 0.517598i 0.876065 0.482193i \(-0.160160\pi\)
0.0204414 + 0.999791i \(0.493493\pi\)
\(24\) 0 0
\(25\) 2.25524 + 3.90619i 0.451048 + 0.781238i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.273287 + 0.157782i 0.0507480 + 0.0292994i 0.525159 0.851004i \(-0.324006\pi\)
−0.474411 + 0.880303i \(0.657339\pi\)
\(30\) 0 0
\(31\) 5.60632i 1.00692i −0.864017 0.503462i \(-0.832059\pi\)
0.864017 0.503462i \(-0.167941\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.322736 + 1.82276i −0.0545523 + 0.308103i
\(36\) 0 0
\(37\) −0.351124 0.608164i −0.0577244 0.0999816i 0.835719 0.549157i \(-0.185051\pi\)
−0.893444 + 0.449175i \(0.851718\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.39354 + 9.34189i 0.842330 + 1.45896i 0.887920 + 0.459998i \(0.152150\pi\)
−0.0455900 + 0.998960i \(0.514517\pi\)
\(42\) 0 0
\(43\) 3.73131 6.46283i 0.569020 0.985572i −0.427643 0.903948i \(-0.640656\pi\)
0.996663 0.0816240i \(-0.0260106\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.00570 1.02189 0.510943 0.859614i \(-0.329296\pi\)
0.510943 + 0.859614i \(0.329296\pi\)
\(48\) 0 0
\(49\) 5.36869 4.49190i 0.766955 0.641700i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.51919 4.91856i −1.17020 0.675616i −0.216474 0.976288i \(-0.569456\pi\)
−0.953727 + 0.300672i \(0.902789\pi\)
\(54\) 0 0
\(55\) 0.185561i 0.0250210i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.4636 1.75282 0.876408 0.481570i \(-0.159933\pi\)
0.876408 + 0.481570i \(0.159933\pi\)
\(60\) 0 0
\(61\) 5.65207i 0.723674i 0.932241 + 0.361837i \(0.117850\pi\)
−0.932241 + 0.361837i \(0.882150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.919566i 0.114058i
\(66\) 0 0
\(67\) −5.94120 −0.725833 −0.362916 0.931822i \(-0.618219\pi\)
−0.362916 + 0.931822i \(0.618219\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4323i 1.59412i 0.603900 + 0.797060i \(0.293613\pi\)
−0.603900 + 0.797060i \(0.706387\pi\)
\(72\) 0 0
\(73\) −6.66182 3.84620i −0.779707 0.450164i 0.0566194 0.998396i \(-0.481968\pi\)
−0.836326 + 0.548232i \(0.815301\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.451424 + 0.537212i −0.0514445 + 0.0612210i
\(78\) 0 0
\(79\) 1.39672 0.157143 0.0785716 0.996908i \(-0.474964\pi\)
0.0785716 + 0.996908i \(0.474964\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.72399 6.45014i 0.408761 0.707995i −0.585990 0.810318i \(-0.699294\pi\)
0.994751 + 0.102323i \(0.0326276\pi\)
\(84\) 0 0
\(85\) 1.30410 + 2.25877i 0.141450 + 0.244998i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.59261 9.68668i −0.592815 1.02679i −0.993851 0.110724i \(-0.964683\pi\)
0.401036 0.916062i \(-0.368650\pi\)
\(90\) 0 0
\(91\) 2.23708 2.66221i 0.234510 0.279076i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.308978i 0.0317004i
\(96\) 0 0
\(97\) −9.18225 5.30138i −0.932316 0.538273i −0.0447729 0.998997i \(-0.514256\pi\)
−0.887543 + 0.460724i \(0.847590\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.75357 15.1616i −0.871013 1.50864i −0.860950 0.508690i \(-0.830130\pi\)
−0.0100634 0.999949i \(-0.503203\pi\)
\(102\) 0 0
\(103\) 7.39775 + 4.27110i 0.728922 + 0.420844i 0.818028 0.575179i \(-0.195067\pi\)
−0.0891054 + 0.996022i \(0.528401\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.09489 + 5.25093i −0.879236 + 0.507627i −0.870406 0.492334i \(-0.836144\pi\)
−0.00882940 + 0.999961i \(0.502811\pi\)
\(108\) 0 0
\(109\) −7.12110 + 12.3341i −0.682078 + 1.18139i 0.292268 + 0.956337i \(0.405590\pi\)
−0.974346 + 0.225057i \(0.927743\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −13.3783 + 7.72396i −1.25852 + 0.726609i −0.972788 0.231699i \(-0.925572\pi\)
−0.285737 + 0.958308i \(0.592238\pi\)
\(114\) 0 0
\(115\) −3.00817 + 1.73677i −0.280513 + 0.161954i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.71957 9.71188i 0.157633 0.890286i
\(120\) 0 0
\(121\) −5.46483 + 9.46536i −0.496803 + 0.860488i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.65406 −0.595157
\(126\) 0 0
\(127\) 21.8304 1.93713 0.968566 0.248758i \(-0.0800225\pi\)
0.968566 + 0.248758i \(0.0800225\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.60461 + 4.51132i −0.227566 + 0.394156i −0.957086 0.289803i \(-0.906410\pi\)
0.729520 + 0.683959i \(0.239743\pi\)
\(132\) 0 0
\(133\) −0.751668 + 0.894514i −0.0651779 + 0.0775642i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.33589 1.34863i 0.199568 0.115221i −0.396886 0.917868i \(-0.629909\pi\)
0.596454 + 0.802647i \(0.296576\pi\)
\(138\) 0 0
\(139\) −10.1448 + 5.85710i −0.860470 + 0.496793i −0.864170 0.503200i \(-0.832156\pi\)
0.00369951 + 0.999993i \(0.498822\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.174289 + 0.301877i −0.0145748 + 0.0252442i
\(144\) 0 0
\(145\) −0.191206 + 0.110393i −0.0158788 + 0.00916765i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.3055 9.41399i −1.33580 0.771224i −0.349618 0.936892i \(-0.613689\pi\)
−0.986182 + 0.165668i \(0.947022\pi\)
\(150\) 0 0
\(151\) −5.00143 8.66273i −0.407010 0.704963i 0.587543 0.809193i \(-0.300095\pi\)
−0.994553 + 0.104230i \(0.966762\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.39698 + 1.96125i 0.272852 + 0.157531i
\(156\) 0 0
\(157\) 0.252063i 0.0201168i −0.999949 0.0100584i \(-0.996798\pi\)
0.999949 0.0100584i \(-0.00320175\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.9340 + 2.29007i 1.01934 + 0.180483i
\(162\) 0 0
\(163\) 4.29780 + 7.44400i 0.336629 + 0.583059i 0.983796 0.179289i \(-0.0573797\pi\)
−0.647167 + 0.762348i \(0.724046\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.24437 3.88736i −0.173674 0.300813i 0.766027 0.642808i \(-0.222231\pi\)
−0.939702 + 0.341995i \(0.888897\pi\)
\(168\) 0 0
\(169\) −5.63629 + 9.76234i −0.433561 + 0.750949i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.12145 0.541434 0.270717 0.962659i \(-0.412739\pi\)
0.270717 + 0.962659i \(0.412739\pi\)
\(174\) 0 0
\(175\) 9.13624 + 7.67726i 0.690634 + 0.580346i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −22.1270 12.7750i −1.65385 0.954848i −0.975470 0.220134i \(-0.929351\pi\)
−0.678376 0.734715i \(-0.737316\pi\)
\(180\) 0 0
\(181\) 0.943175i 0.0701057i −0.999385 0.0350528i \(-0.988840\pi\)
0.999385 0.0350528i \(-0.0111599\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.491332 0.0361234
\(186\) 0 0
\(187\) 0.988686i 0.0722999i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.97235i 0.215072i 0.994201 + 0.107536i \(0.0342960\pi\)
−0.994201 + 0.107536i \(0.965704\pi\)
\(192\) 0 0
\(193\) −18.5144 −1.33270 −0.666348 0.745641i \(-0.732144\pi\)
−0.666348 + 0.745641i \(0.732144\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.1774i 1.01010i 0.863091 + 0.505048i \(0.168525\pi\)
−0.863091 + 0.505048i \(0.831475\pi\)
\(198\) 0 0
\(199\) −20.5293 11.8526i −1.45529 0.840209i −0.456512 0.889717i \(-0.650901\pi\)
−0.998774 + 0.0495081i \(0.984235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.822117 + 0.145563i 0.0577013 + 0.0102165i
\(204\) 0 0
\(205\) −7.54725 −0.527123
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.0585617 0.101432i 0.00405080 0.00701619i
\(210\) 0 0
\(211\) 3.04004 + 5.26550i 0.209285 + 0.362492i 0.951489 0.307681i \(-0.0995531\pi\)
−0.742205 + 0.670173i \(0.766220\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.61063 + 4.52175i 0.178044 + 0.308381i
\(216\) 0 0
\(217\) −5.06327 13.9420i −0.343717 0.946444i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.89954i 0.329579i
\(222\) 0 0
\(223\) 0.796137 + 0.459650i 0.0533133 + 0.0307804i 0.526420 0.850225i \(-0.323534\pi\)
−0.473106 + 0.881005i \(0.656867\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.00297 8.66540i −0.332059 0.575143i 0.650857 0.759201i \(-0.274410\pi\)
−0.982915 + 0.184058i \(0.941077\pi\)
\(228\) 0 0
\(229\) −2.38179 1.37513i −0.157393 0.0908710i 0.419235 0.907878i \(-0.362298\pi\)
−0.576628 + 0.817007i \(0.695632\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.55513 3.20725i 0.363928 0.210114i −0.306874 0.951750i \(-0.599283\pi\)
0.670803 + 0.741636i \(0.265950\pi\)
\(234\) 0 0
\(235\) −2.45079 + 4.24489i −0.159872 + 0.276906i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.4288 6.59844i 0.739270 0.426818i −0.0825337 0.996588i \(-0.526301\pi\)
0.821804 + 0.569770i \(0.192968\pi\)
\(240\) 0 0
\(241\) −2.20722 + 1.27434i −0.142180 + 0.0820874i −0.569402 0.822059i \(-0.692825\pi\)
0.427223 + 0.904146i \(0.359492\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.843615 + 4.82439i 0.0538966 + 0.308219i
\(246\) 0 0
\(247\) −0.290209 + 0.502657i −0.0184656 + 0.0319833i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.7893 −1.18597 −0.592986 0.805213i \(-0.702051\pi\)
−0.592986 + 0.805213i \(0.702051\pi\)
\(252\) 0 0
\(253\) −1.31670 −0.0827804
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.19727 + 12.4660i −0.448953 + 0.777610i −0.998318 0.0579725i \(-0.981536\pi\)
0.549365 + 0.835583i \(0.314870\pi\)
\(258\) 0 0
\(259\) −1.42244 1.19529i −0.0883863 0.0742718i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.79810 + 3.92488i −0.419189 + 0.242019i −0.694730 0.719271i \(-0.744476\pi\)
0.275542 + 0.961289i \(0.411143\pi\)
\(264\) 0 0
\(265\) 5.96050 3.44130i 0.366151 0.211397i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.72267 13.3760i 0.470859 0.815552i −0.528585 0.848880i \(-0.677277\pi\)
0.999444 + 0.0333281i \(0.0106106\pi\)
\(270\) 0 0
\(271\) 10.9476 6.32057i 0.665016 0.383947i −0.129169 0.991623i \(-0.541231\pi\)
0.794186 + 0.607675i \(0.207898\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.03599 0.598128i −0.0624724 0.0360685i
\(276\) 0 0
\(277\) 5.94531 + 10.2976i 0.357219 + 0.618722i 0.987495 0.157649i \(-0.0503915\pi\)
−0.630276 + 0.776371i \(0.717058\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.75411 + 1.59009i 0.164297 + 0.0948568i 0.579894 0.814692i \(-0.303094\pi\)
−0.415597 + 0.909549i \(0.636427\pi\)
\(282\) 0 0
\(283\) 18.4978i 1.09958i −0.835303 0.549789i \(-0.814708\pi\)
0.835303 0.549789i \(-0.185292\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 21.8499 + 18.3606i 1.28976 + 1.08379i
\(288\) 0 0
\(289\) 1.55161 + 2.68746i 0.0912711 + 0.158086i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.42975 + 2.47639i 0.0835266 + 0.144672i 0.904762 0.425917i \(-0.140048\pi\)
−0.821236 + 0.570589i \(0.806715\pi\)
\(294\) 0 0
\(295\) −4.70995 + 8.15788i −0.274224 + 0.474970i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.52507 0.377355
\(300\) 0 0
\(301\) 3.44234 19.4419i 0.198413 1.12061i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.42470 1.97725i −0.196098 0.113217i
\(306\) 0 0
\(307\) 21.6746i 1.23704i 0.785771 + 0.618518i \(0.212266\pi\)
−0.785771 + 0.618518i \(0.787734\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 23.6925 1.34348 0.671738 0.740789i \(-0.265548\pi\)
0.671738 + 0.740789i \(0.265548\pi\)
\(312\) 0 0
\(313\) 27.2836i 1.54216i −0.636737 0.771081i \(-0.719716\pi\)
0.636737 0.771081i \(-0.280284\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.5544i 1.37911i 0.724232 + 0.689556i \(0.242194\pi\)
−0.724232 + 0.689556i \(0.757806\pi\)
\(318\) 0 0
\(319\) −0.0836929 −0.00468590
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.64626i 0.0916006i
\(324\) 0 0
\(325\) 5.13396 + 2.96409i 0.284781 + 0.164418i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 17.4220 6.32711i 0.960507 0.348825i
\(330\) 0 0
\(331\) 16.3116 0.896566 0.448283 0.893892i \(-0.352036\pi\)
0.448283 + 0.893892i \(0.352036\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.07840 3.59989i 0.113555 0.196683i
\(336\) 0 0
\(337\) 13.6580 + 23.6563i 0.743998 + 1.28864i 0.950661 + 0.310230i \(0.100406\pi\)
−0.206663 + 0.978412i \(0.566261\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.743445 + 1.28768i 0.0402598 + 0.0697320i
\(342\) 0 0
\(343\) 9.29424 16.0193i 0.501842 0.864960i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.21213i 0.333485i −0.986001 0.166742i \(-0.946675\pi\)
0.986001 0.166742i \(-0.0533248\pi\)
\(348\) 0 0
\(349\) 24.6529 + 14.2334i 1.31964 + 0.761896i 0.983671 0.179977i \(-0.0576023\pi\)
0.335971 + 0.941872i \(0.390936\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.49346 2.58674i −0.0794887 0.137678i 0.823541 0.567257i \(-0.191995\pi\)
−0.903029 + 0.429579i \(0.858662\pi\)
\(354\) 0 0
\(355\) −8.13889 4.69899i −0.431968 0.249397i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.5977 15.3562i 1.40377 0.810468i 0.408994 0.912537i \(-0.365880\pi\)
0.994777 + 0.102070i \(0.0325465\pi\)
\(360\) 0 0
\(361\) −9.40249 + 16.2856i −0.494868 + 0.857136i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.66098 2.69102i 0.243967 0.140854i
\(366\) 0 0
\(367\) 16.4877 9.51918i 0.860651 0.496897i −0.00357920 0.999994i \(-0.501139\pi\)
0.864230 + 0.503096i \(0.167806\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −25.6280 4.53764i −1.33054 0.235583i
\(372\) 0 0
\(373\) −2.05869 + 3.56576i −0.106595 + 0.184628i −0.914389 0.404837i \(-0.867328\pi\)
0.807794 + 0.589465i \(0.200661\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.414750 0.0213607
\(378\) 0 0
\(379\) −11.2436 −0.577546 −0.288773 0.957398i \(-0.593247\pi\)
−0.288773 + 0.957398i \(0.593247\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −15.8046 + 27.3745i −0.807580 + 1.39877i 0.106956 + 0.994264i \(0.465890\pi\)
−0.914536 + 0.404505i \(0.867444\pi\)
\(384\) 0 0
\(385\) −0.167586 0.461458i −0.00854100 0.0235181i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.4018 + 10.6243i −0.933007 + 0.538672i −0.887761 0.460304i \(-0.847740\pi\)
−0.0452458 + 0.998976i \(0.514407\pi\)
\(390\) 0 0
\(391\) 16.0278 9.25367i 0.810562 0.467978i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.488611 + 0.846300i −0.0245847 + 0.0425820i
\(396\) 0 0
\(397\) 20.6927 11.9469i 1.03854 0.599599i 0.119118 0.992880i \(-0.461993\pi\)
0.919419 + 0.393281i \(0.128660\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0121 12.7087i −1.09923 0.634642i −0.163213 0.986591i \(-0.552186\pi\)
−0.936019 + 0.351948i \(0.885519\pi\)
\(402\) 0 0
\(403\) −3.68423 6.38127i −0.183524 0.317874i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.161295 + 0.0931240i 0.00799512 + 0.00461598i
\(408\) 0 0
\(409\) 22.3817i 1.10670i −0.832948 0.553351i \(-0.813349\pi\)
0.832948 0.553351i \(-0.186651\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 33.4818 12.1595i 1.64753 0.598330i
\(414\) 0 0
\(415\) 2.60551 + 4.51288i 0.127900 + 0.221528i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7.04181 12.1968i −0.344015 0.595851i 0.641159 0.767408i \(-0.278454\pi\)
−0.985174 + 0.171556i \(0.945120\pi\)
\(420\) 0 0
\(421\) 8.07639 13.9887i 0.393619 0.681768i −0.599305 0.800521i \(-0.704556\pi\)
0.992924 + 0.118753i \(0.0378896\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16.8143 0.815616
\(426\) 0 0
\(427\) 5.10459 + 14.0558i 0.247029 + 0.680206i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.16179 + 4.13486i 0.344971 + 0.199169i 0.662468 0.749090i \(-0.269509\pi\)
−0.317497 + 0.948259i \(0.602842\pi\)
\(432\) 0 0
\(433\) 4.35102i 0.209097i −0.994520 0.104548i \(-0.966660\pi\)
0.994520 0.104548i \(-0.0333397\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.19245 −0.104879
\(438\) 0 0
\(439\) 20.8077i 0.993098i 0.868009 + 0.496549i \(0.165400\pi\)
−0.868009 + 0.496549i \(0.834600\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 30.9376i 1.46989i 0.678127 + 0.734945i \(0.262792\pi\)
−0.678127 + 0.734945i \(0.737208\pi\)
\(444\) 0 0
\(445\) 7.82580 0.370978
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.9215i 0.987346i −0.869648 0.493673i \(-0.835654\pi\)
0.869648 0.493673i \(-0.164346\pi\)
\(450\) 0 0
\(451\) −2.47763 1.43046i −0.116667 0.0673577i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.830494 + 2.28681i 0.0389342 + 0.107207i
\(456\) 0 0
\(457\) 2.30115 0.107643 0.0538217 0.998551i \(-0.482860\pi\)
0.0538217 + 0.998551i \(0.482860\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.92497 + 15.4585i −0.415677 + 0.719974i −0.995499 0.0947688i \(-0.969789\pi\)
0.579822 + 0.814743i \(0.303122\pi\)
\(462\) 0 0
\(463\) −6.24034 10.8086i −0.290013 0.502318i 0.683799 0.729670i \(-0.260326\pi\)
−0.973813 + 0.227353i \(0.926993\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.42799 + 4.20541i 0.112354 + 0.194603i 0.916719 0.399533i \(-0.130828\pi\)
−0.804365 + 0.594136i \(0.797494\pi\)
\(468\) 0 0
\(469\) −14.7748 + 5.36571i −0.682236 + 0.247766i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.97921i 0.0910044i
\(474\) 0 0
\(475\) −1.72503 0.995945i −0.0791497 0.0456971i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.40542 7.63041i −0.201289 0.348642i 0.747655 0.664087i \(-0.231180\pi\)
−0.948944 + 0.315445i \(0.897846\pi\)
\(480\) 0 0
\(481\) −0.799318 0.461486i −0.0364458 0.0210420i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.42441 3.70914i 0.291718 0.168423i
\(486\) 0 0
\(487\) 4.66185 8.07456i 0.211249 0.365893i −0.740857 0.671663i \(-0.765580\pi\)
0.952106 + 0.305770i \(0.0989137\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.9192 15.5418i 1.21485 0.701391i 0.251034 0.967978i \(-0.419229\pi\)
0.963811 + 0.266587i \(0.0858960\pi\)
\(492\) 0 0
\(493\) 1.01877 0.588186i 0.0458830 0.0264906i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.1312 + 33.4039i 0.544159 + 1.49837i
\(498\) 0 0
\(499\) 11.1694 19.3459i 0.500010 0.866043i −0.499990 0.866031i \(-0.666663\pi\)
1.00000 1.16519e-5i \(-3.70891e-6\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.2396 0.545738 0.272869 0.962051i \(-0.412027\pi\)
0.272869 + 0.962051i \(0.412027\pi\)
\(504\) 0 0
\(505\) 12.2490 0.545072
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.05496 12.2195i 0.312706 0.541622i −0.666242 0.745736i \(-0.732098\pi\)
0.978947 + 0.204114i \(0.0654314\pi\)
\(510\) 0 0
\(511\) −20.0405 3.54834i −0.886539 0.156969i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.17588 + 2.98830i −0.228077 + 0.131680i
\(516\) 0 0
\(517\) −1.60910 + 0.929015i −0.0707682 + 0.0408580i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.81632 + 4.87800i −0.123385 + 0.213709i −0.921101 0.389325i \(-0.872708\pi\)
0.797715 + 0.603034i \(0.206042\pi\)
\(522\) 0 0
\(523\) −33.2293 + 19.1849i −1.45302 + 0.838899i −0.998651 0.0519176i \(-0.983467\pi\)
−0.454364 + 0.890816i \(0.650133\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.0995 10.4497i −0.788425 0.455197i
\(528\) 0 0
\(529\) 0.823769 + 1.42681i 0.0358161 + 0.0620352i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.2782 + 7.08880i 0.531826 + 0.307050i
\(534\) 0 0
\(535\) 7.34769i 0.317668i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.637441 + 1.74365i −0.0274565 + 0.0751045i
\(540\) 0 0
\(541\) −3.21673 5.57154i −0.138298 0.239539i 0.788555 0.614965i \(-0.210830\pi\)
−0.926852 + 0.375426i \(0.877496\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.98232 8.62963i −0.213419 0.369653i
\(546\) 0 0
\(547\) −6.52889 + 11.3084i −0.279155 + 0.483511i −0.971175 0.238368i \(-0.923388\pi\)
0.692020 + 0.721878i \(0.256721\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.139357 −0.00593683
\(552\) 0 0
\(553\) 3.47341 1.26143i 0.147704 0.0536414i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 25.5409 + 14.7460i 1.08220 + 0.624809i 0.931489 0.363769i \(-0.118510\pi\)
0.150712 + 0.988578i \(0.451843\pi\)
\(558\) 0 0
\(559\) 9.80822i 0.414844i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.5187 0.443310 0.221655 0.975125i \(-0.428854\pi\)
0.221655 + 0.975125i \(0.428854\pi\)
\(564\) 0 0
\(565\) 10.8082i 0.454706i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.3334i 1.10395i 0.833859 + 0.551977i \(0.186126\pi\)
−0.833859 + 0.551977i \(0.813874\pi\)
\(570\) 0 0
\(571\) −44.0590 −1.84381 −0.921906 0.387413i \(-0.873369\pi\)
−0.921906 + 0.387413i \(0.873369\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 22.3929i 0.933847i
\(576\) 0 0
\(577\) 12.1535 + 7.01684i 0.505957 + 0.292115i 0.731170 0.682195i \(-0.238974\pi\)
−0.225213 + 0.974310i \(0.572308\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.43559 19.4037i 0.142532 0.805001i
\(582\) 0 0
\(583\) 2.60897 0.108052
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.52469 + 2.64085i −0.0629308 + 0.108999i −0.895774 0.444509i \(-0.853378\pi\)
0.832843 + 0.553509i \(0.186711\pi\)
\(588\) 0 0
\(589\) 1.23791 + 2.14413i 0.0510073 + 0.0883473i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −13.3041 23.0434i −0.546334 0.946278i −0.998522 0.0543552i \(-0.982690\pi\)
0.452188 0.891923i \(-0.350644\pi\)
\(594\) 0 0
\(595\) 5.28306 + 4.43941i 0.216585 + 0.181998i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.46099i 0.182271i −0.995838 0.0911356i \(-0.970950\pi\)
0.995838 0.0911356i \(-0.0290497\pi\)
\(600\) 0 0
\(601\) 5.25019 + 3.03120i 0.214160 + 0.123645i 0.603243 0.797557i \(-0.293875\pi\)
−0.389083 + 0.921203i \(0.627208\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.82350 6.62250i −0.155447 0.269243i
\(606\) 0 0
\(607\) 39.2581 + 22.6657i 1.59344 + 0.919971i 0.992711 + 0.120516i \(0.0384548\pi\)
0.600725 + 0.799455i \(0.294879\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.97409 4.60384i 0.322597 0.186251i
\(612\) 0 0
\(613\) 16.6294 28.8029i 0.671654 1.16334i −0.305781 0.952102i \(-0.598917\pi\)
0.977435 0.211237i \(-0.0677492\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.3001 + 18.0711i −1.26010 + 0.727516i −0.973093 0.230414i \(-0.925992\pi\)
−0.287002 + 0.957930i \(0.592659\pi\)
\(618\) 0 0
\(619\) −22.9031 + 13.2231i −0.920554 + 0.531482i −0.883812 0.467843i \(-0.845031\pi\)
−0.0367423 + 0.999325i \(0.511698\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −22.6563 19.0383i −0.907705 0.762752i
\(624\) 0 0
\(625\) −8.94843 + 15.4991i −0.357937 + 0.619965i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.61787 −0.104381
\(630\) 0 0
\(631\) −32.0484 −1.27583 −0.637914 0.770107i \(-0.720203\pi\)
−0.637914 + 0.770107i \(0.720203\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7.63687 + 13.2274i −0.303060 + 0.524915i
\(636\) 0 0
\(637\) 3.15891 8.64088i 0.125161 0.342364i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 21.1444 12.2077i 0.835153 0.482176i −0.0204610 0.999791i \(-0.506513\pi\)
0.855614 + 0.517615i \(0.173180\pi\)
\(642\) 0 0
\(643\) −31.9014 + 18.4183i −1.25807 + 0.726346i −0.972699 0.232071i \(-0.925450\pi\)
−0.285370 + 0.958418i \(0.592116\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13.2847 + 23.0098i −0.522276 + 0.904608i 0.477389 + 0.878692i \(0.341583\pi\)
−0.999664 + 0.0259155i \(0.991750\pi\)
\(648\) 0 0
\(649\) −3.09239 + 1.78539i −0.121387 + 0.0700827i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −26.2767 15.1709i −1.02829 0.593683i −0.111794 0.993731i \(-0.535660\pi\)
−0.916494 + 0.400049i \(0.868993\pi\)
\(654\) 0 0
\(655\) −1.82233 3.15637i −0.0712044 0.123330i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.9873 23.6640i −1.59664 0.921820i −0.992129 0.125223i \(-0.960035\pi\)
−0.604511 0.796597i \(-0.706631\pi\)
\(660\) 0 0
\(661\) 35.1245i 1.36618i 0.730332 + 0.683092i \(0.239365\pi\)
−0.730332 + 0.683092i \(0.760635\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.279049 0.768376i −0.0108211 0.0297963i
\(666\) 0 0
\(667\) 0.783329 + 1.35677i 0.0303306 + 0.0525342i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.749513 1.29819i −0.0289346 0.0501162i
\(672\) 0 0
\(673\) −2.54758 + 4.41254i −0.0982020 + 0.170091i −0.910940 0.412538i \(-0.864642\pi\)
0.812738 + 0.582629i \(0.197976\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.8414 0.647269 0.323635 0.946182i \(-0.395095\pi\)
0.323635 + 0.946182i \(0.395095\pi\)
\(678\) 0 0
\(679\) −27.6226 4.89081i −1.06006 0.187692i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −15.7555 9.09645i −0.602868 0.348066i 0.167301 0.985906i \(-0.446495\pi\)
−0.770169 + 0.637840i \(0.779828\pi\)
\(684\) 0 0
\(685\) 1.88715i 0.0721042i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.9290 −0.492557
\(690\) 0 0
\(691\) 3.52652i 0.134155i 0.997748 + 0.0670775i \(0.0213675\pi\)
−0.997748 + 0.0670775i \(0.978633\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.19591i 0.310888i
\(696\) 0 0
\(697\) 40.2125 1.52316
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.3502i 0.504229i 0.967697 + 0.252114i \(0.0811259\pi\)
−0.967697 + 0.252114i \(0.918874\pi\)
\(702\) 0 0
\(703\) 0.268574 + 0.155061i 0.0101295 + 0.00584824i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −35.4617 29.7988i −1.33368 1.12070i
\(708\) 0 0
\(709\) −42.2894 −1.58821 −0.794107 0.607779i \(-0.792061\pi\)
−0.794107 + 0.607779i \(0.792061\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.9166 24.1043i 0.521182 0.902715i
\(714\) 0 0
\(715\) −0.121942 0.211210i −0.00456038 0.00789881i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15.2035 26.3332i −0.566994 0.982062i −0.996861 0.0791697i \(-0.974773\pi\)
0.429868 0.902892i \(-0.358560\pi\)
\(720\) 0 0
\(721\) 22.2544 + 3.94032i 0.828796 + 0.146745i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.42335i 0.0528617i
\(726\) 0 0
\(727\) −11.3671 6.56280i −0.421583 0.243401i 0.274171 0.961681i \(-0.411596\pi\)
−0.695754 + 0.718280i \(0.744930\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.9097 24.0924i −0.514470 0.891089i
\(732\) 0 0
\(733\) −32.7001 18.8794i −1.20781 0.697327i −0.245527 0.969390i \(-0.578961\pi\)
−0.962280 + 0.272063i \(0.912294\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.36460 0.787853i 0.0502657 0.0290209i
\(738\) 0 0
\(739\) −13.1215 + 22.7271i −0.482683 + 0.836031i −0.999802 0.0198820i \(-0.993671\pi\)
0.517119 + 0.855913i \(0.327004\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.78379 5.07132i 0.322246 0.186049i −0.330147 0.943929i \(-0.607098\pi\)
0.652393 + 0.757881i \(0.273765\pi\)
\(744\) 0 0
\(745\) 11.4082 6.58655i 0.417966 0.241313i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −17.8752 + 21.2721i −0.653144 + 0.777267i
\(750\) 0 0
\(751\) −3.95369 + 6.84798i −0.144272 + 0.249886i −0.929101 0.369826i \(-0.879417\pi\)
0.784829 + 0.619712i \(0.212751\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.99855 0.254703
\(756\) 0 0
\(757\) 29.8903 1.08638 0.543191 0.839609i \(-0.317216\pi\)
0.543191 + 0.839609i \(0.317216\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.05687 5.29465i 0.110811 0.191931i −0.805286 0.592886i \(-0.797988\pi\)
0.916098 + 0.400955i \(0.131322\pi\)
\(762\) 0 0
\(763\) −6.56961 + 37.1042i −0.237836 + 1.34326i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.3247 8.84771i 0.553342 0.319472i
\(768\) 0 0
\(769\) −9.79863 + 5.65724i −0.353348 + 0.204005i −0.666159 0.745810i \(-0.732063\pi\)
0.312811 + 0.949815i \(0.398729\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.2106 + 33.2737i −0.690956 + 1.19677i 0.280569 + 0.959834i \(0.409477\pi\)
−0.971525 + 0.236937i \(0.923856\pi\)
\(774\) 0 0
\(775\) 21.8994 12.6436i 0.786648 0.454172i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.12551 2.38186i −0.147812 0.0853391i
\(780\) 0 0
\(781\) −1.78124 3.08519i −0.0637376 0.110397i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.152730 + 0.0881788i 0.00545117 + 0.00314723i
\(786\) 0 0
\(787\) 47.6600i 1.69889i 0.527674 + 0.849447i \(0.323064\pi\)
−0.527674 + 0.849447i \(0.676936\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −26.2938 + 31.2906i −0.934900 + 1.11257i
\(792\) 0 0
\(793\) 3.71430 + 6.43335i 0.131898 + 0.228455i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.1359 + 26.2161i 0.536139 + 0.928621i 0.999107 + 0.0422457i \(0.0134512\pi\)
−0.462968 + 0.886375i \(0.653215\pi\)
\(798\) 0 0
\(799\) 13.0581 22.6172i 0.461961 0.800140i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.04016 0.0719955
\(804\) 0 0
\(805\) −5.91227 + 7.03583i −0.208380 + 0.247981i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.219373 0.126655i −0.00771273 0.00445295i 0.496139 0.868243i \(-0.334751\pi\)
−0.503851 + 0.863790i \(0.668084\pi\)
\(810\) 0 0
\(811\) 22.0629i 0.774735i −0.921925 0.387367i \(-0.873385\pi\)
0.921925 0.387367i \(-0.126615\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.01395 −0.210660
\(816\) 0 0
\(817\) 3.29560i 0.115298i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 27.9976i 0.977124i −0.872529 0.488562i \(-0.837522\pi\)
0.872529 0.488562i \(-0.162478\pi\)
\(822\) 0 0
\(823\) 48.9542 1.70643 0.853217 0.521556i \(-0.174648\pi\)
0.853217 + 0.521556i \(0.174648\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.641658i 0.0223126i −0.999938 0.0111563i \(-0.996449\pi\)
0.999938 0.0111563i \(-0.00355124\pi\)
\(828\) 0 0
\(829\) 9.57180 + 5.52628i 0.332442 + 0.191936i 0.656925 0.753956i \(-0.271857\pi\)
−0.324483 + 0.945892i \(0.605190\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.49487 25.7048i −0.155738 0.890620i
\(834\) 0 0
\(835\) 3.14057 0.108684
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.62330 8.00780i 0.159614 0.276460i −0.775115 0.631820i \(-0.782308\pi\)
0.934730 + 0.355360i \(0.115642\pi\)
\(840\) 0 0
\(841\) −14.4502 25.0285i −0.498283 0.863052i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.94346 6.83028i −0.135659 0.234969i
\(846\) 0 0
\(847\) −5.04161 + 28.4743i −0.173232 + 0.978388i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.48640i 0.119512i
\(852\) 0 0
\(853\) −34.3256 19.8179i −1.17529 0.678551i −0.220366 0.975417i \(-0.570725\pi\)
−0.954919 + 0.296866i \(0.904059\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.9260 + 20.6565i 0.407385 + 0.705612i 0.994596 0.103822i \(-0.0331074\pi\)
−0.587211 + 0.809434i \(0.699774\pi\)
\(858\) 0 0
\(859\) −9.62480 5.55688i −0.328394 0.189598i 0.326734 0.945116i \(-0.394052\pi\)
−0.655128 + 0.755518i \(0.727385\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.7780 22.3885i 1.32002 0.762113i 0.336287 0.941759i \(-0.390829\pi\)
0.983731 + 0.179646i \(0.0574954\pi\)
\(864\) 0 0
\(865\) −2.49128 + 4.31503i −0.0847061 + 0.146715i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.320805 + 0.185217i −0.0108826 + 0.00628305i
\(870\) 0 0
\(871\) −6.76244 + 3.90429i −0.229136 + 0.132292i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −16.5476 + 6.00953i −0.559409 + 0.203159i
\(876\) 0 0
\(877\) −1.84096 + 3.18863i −0.0621647 + 0.107672i −0.895433 0.445197i \(-0.853134\pi\)
0.833268 + 0.552869i \(0.186467\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −17.3992 −0.586194 −0.293097 0.956083i \(-0.594686\pi\)
−0.293097 + 0.956083i \(0.594686\pi\)
\(882\) 0 0
\(883\) −2.02834 −0.0682592 −0.0341296 0.999417i \(-0.510866\pi\)
−0.0341296 + 0.999417i \(0.510866\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23.1890 + 40.1645i −0.778610 + 1.34859i 0.154132 + 0.988050i \(0.450742\pi\)
−0.932743 + 0.360542i \(0.882592\pi\)
\(888\) 0 0
\(889\) 54.2885 19.7158i 1.82078 0.661247i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.67932 + 1.54691i −0.0896601 + 0.0517653i
\(894\) 0 0
\(895\) 15.4812 8.93810i 0.517481 0.298768i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.884576 1.53213i 0.0295023 0.0510994i
\(900\) 0 0
\(901\) −31.7582 + 18.3356i −1.05802 + 0.610847i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.571488 + 0.329949i 0.0189969 + 0.0109679i
\(906\) 0 0
\(907\) −8.01957 13.8903i −0.266285 0.461220i 0.701614 0.712557i \(-0.252463\pi\)
−0.967900 + 0.251337i \(0.919130\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −17.7833 10.2672i −0.589187 0.340167i 0.175589 0.984464i \(-0.443817\pi\)
−0.764776 + 0.644296i \(0.777150\pi\)
\(912\) 0 0
\(913\) 1.97533i 0.0653739i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.40290 + 13.5712i −0.0793507 + 0.448161i
\(918\) 0 0
\(919\) −17.7069 30.6693i −0.584097 1.01169i −0.994987 0.100001i \(-0.968115\pi\)
0.410890 0.911685i \(-0.365218\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.82712 + 15.2890i 0.290548 + 0.503244i
\(924\) 0 0
\(925\) 1.58374 2.74311i 0.0520730 0.0901930i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39.7251 1.30334 0.651670 0.758503i \(-0.274069\pi\)
0.651670 + 0.758503i \(0.274069\pi\)
\(930\) 0 0
\(931\) −1.06141 + 2.90337i −0.0347861 + 0.0951540i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.599064 0.345870i −0.0195915 0.0113112i
\(936\) 0 0
\(937\) 23.2142i 0.758376i −0.925320 0.379188i \(-0.876203\pi\)
0.925320 0.379188i \(-0.123797\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 35.9232 1.17106 0.585531 0.810650i \(-0.300886\pi\)
0.585531 + 0.810650i \(0.300886\pi\)
\(942\) 0 0
\(943\) 53.5539i 1.74395i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.8726i 1.00322i 0.865093 + 0.501612i \(0.167260\pi\)
−0.865093 + 0.501612i \(0.832740\pi\)
\(948\) 0 0
\(949\) −10.1102 −0.328192
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 30.4640i 0.986826i −0.869795 0.493413i \(-0.835749\pi\)
0.869795 0.493413i \(-0.164251\pi\)
\(954\) 0 0
\(955\) −1.80100 1.03981i −0.0582791 0.0336475i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.59098 5.46344i 0.148250 0.176424i
\(960\) 0 0
\(961\) −0.430813 −0.0138972
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.47686 11.2182i 0.208497 0.361128i
\(966\) 0 0
\(967\) −6.75865 11.7063i −0.217343 0.376450i 0.736652 0.676272i \(-0.236406\pi\)
−0.953995 + 0.299823i \(0.903072\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −16.6428 28.8261i −0.534092 0.925074i −0.999207 0.0398238i \(-0.987320\pi\)
0.465115 0.885250i \(-0.346013\pi\)
\(972\) 0 0
\(973\) −19.9387 + 23.7278i −0.639204 + 0.760677i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 44.9325i 1.43752i 0.695260 + 0.718758i \(0.255289\pi\)
−0.695260 + 0.718758i \(0.744711\pi\)
\(978\) 0 0
\(979\) 2.56907 + 1.48325i 0.0821079 + 0.0474050i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.4474 + 23.2916i 0.428907 + 0.742888i 0.996776 0.0802305i \(-0.0255656\pi\)
−0.567870 + 0.823118i \(0.692232\pi\)
\(984\) 0 0
\(985\) −8.59035 4.95964i −0.273711 0.158027i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0855 18.5246i 1.02026 0.589048i
\(990\) 0 0
\(991\) 17.7201 30.6920i 0.562896 0.974965i −0.434346 0.900746i \(-0.643020\pi\)
0.997242 0.0742186i \(-0.0236463\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.3635 8.29275i 0.455352 0.262898i
\(996\) 0 0
\(997\) −28.1418 + 16.2477i −0.891259 + 0.514568i −0.874354 0.485289i \(-0.838714\pi\)
−0.0169046 + 0.999857i \(0.505381\pi\)
\(998\) 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.w.a.521.4 16
3.2 odd 2 252.2.w.a.101.6 yes 16
4.3 odd 2 3024.2.ca.d.2033.4 16
7.2 even 3 5292.2.bm.a.4625.5 16
7.3 odd 6 5292.2.x.b.4409.5 16
7.4 even 3 5292.2.x.a.4409.4 16
7.5 odd 6 756.2.bm.a.89.4 16
7.6 odd 2 5292.2.w.b.521.5 16
9.2 odd 6 2268.2.t.b.1781.5 16
9.4 even 3 252.2.bm.a.185.8 yes 16
9.5 odd 6 756.2.bm.a.17.4 16
9.7 even 3 2268.2.t.a.1781.4 16
12.11 even 2 1008.2.ca.d.353.3 16
21.2 odd 6 1764.2.bm.a.1685.1 16
21.5 even 6 252.2.bm.a.173.8 yes 16
21.11 odd 6 1764.2.x.a.1469.6 16
21.17 even 6 1764.2.x.b.1469.3 16
21.20 even 2 1764.2.w.b.1109.3 16
28.19 even 6 3024.2.df.d.1601.4 16
36.23 even 6 3024.2.df.d.17.4 16
36.31 odd 6 1008.2.df.d.689.1 16
63.4 even 3 1764.2.x.b.293.3 16
63.5 even 6 inner 756.2.w.a.341.4 16
63.13 odd 6 1764.2.bm.a.1697.1 16
63.23 odd 6 5292.2.w.b.1097.5 16
63.31 odd 6 1764.2.x.a.293.6 16
63.32 odd 6 5292.2.x.b.881.5 16
63.40 odd 6 252.2.w.a.5.6 16
63.41 even 6 5292.2.bm.a.2285.5 16
63.47 even 6 2268.2.t.a.2105.4 16
63.58 even 3 1764.2.w.b.509.3 16
63.59 even 6 5292.2.x.a.881.4 16
63.61 odd 6 2268.2.t.b.2105.5 16
84.47 odd 6 1008.2.df.d.929.1 16
252.103 even 6 1008.2.ca.d.257.3 16
252.131 odd 6 3024.2.ca.d.2609.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.6 16 63.40 odd 6
252.2.w.a.101.6 yes 16 3.2 odd 2
252.2.bm.a.173.8 yes 16 21.5 even 6
252.2.bm.a.185.8 yes 16 9.4 even 3
756.2.w.a.341.4 16 63.5 even 6 inner
756.2.w.a.521.4 16 1.1 even 1 trivial
756.2.bm.a.17.4 16 9.5 odd 6
756.2.bm.a.89.4 16 7.5 odd 6
1008.2.ca.d.257.3 16 252.103 even 6
1008.2.ca.d.353.3 16 12.11 even 2
1008.2.df.d.689.1 16 36.31 odd 6
1008.2.df.d.929.1 16 84.47 odd 6
1764.2.w.b.509.3 16 63.58 even 3
1764.2.w.b.1109.3 16 21.20 even 2
1764.2.x.a.293.6 16 63.31 odd 6
1764.2.x.a.1469.6 16 21.11 odd 6
1764.2.x.b.293.3 16 63.4 even 3
1764.2.x.b.1469.3 16 21.17 even 6
1764.2.bm.a.1685.1 16 21.2 odd 6
1764.2.bm.a.1697.1 16 63.13 odd 6
2268.2.t.a.1781.4 16 9.7 even 3
2268.2.t.a.2105.4 16 63.47 even 6
2268.2.t.b.1781.5 16 9.2 odd 6
2268.2.t.b.2105.5 16 63.61 odd 6
3024.2.ca.d.2033.4 16 4.3 odd 2
3024.2.ca.d.2609.4 16 252.131 odd 6
3024.2.df.d.17.4 16 36.23 even 6
3024.2.df.d.1601.4 16 28.19 even 6
5292.2.w.b.521.5 16 7.6 odd 2
5292.2.w.b.1097.5 16 63.23 odd 6
5292.2.x.a.881.4 16 63.59 even 6
5292.2.x.a.4409.4 16 7.4 even 3
5292.2.x.b.881.5 16 63.32 odd 6
5292.2.x.b.4409.5 16 7.3 odd 6
5292.2.bm.a.2285.5 16 63.41 even 6
5292.2.bm.a.4625.5 16 7.2 even 3