Properties

Label 2736.2.dc.c.449.5
Level $2736$
Weight $2$
Character 2736.449
Analytic conductor $21.847$
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] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \( x^{16} + 16x^{14} + 174x^{12} + 1012x^{10} + 4243x^{8} + 9708x^{6} + 15858x^{4} + 12150x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.5
Root \(0.484374 + 0.838961i\) of defining polynomial
Character \(\chi\) \(=\) 2736.449
Dual form 2736.2.dc.c.1889.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.557544 - 0.321898i) q^{5} -2.15106 q^{7} +O(q^{10})\) \(q+(0.557544 - 0.321898i) q^{5} -2.15106 q^{7} +2.27164i q^{11} +(-0.313530 - 0.181017i) q^{13} +(1.40976 - 0.813922i) q^{17} +(2.64170 + 3.46719i) q^{19} +(-5.64153 - 3.25714i) q^{23} +(-2.29276 + 3.97118i) q^{25} +(-0.968749 + 1.67792i) q^{29} -6.67274i q^{31} +(-1.19931 + 0.692422i) q^{35} -3.30902i q^{37} +(3.49359 + 6.05108i) q^{41} +(-2.71723 - 4.70639i) q^{43} +(-7.69557 - 4.44304i) q^{47} -2.37294 q^{49} +(-1.96730 + 3.40746i) q^{53} +(0.731237 + 1.26654i) q^{55} +(2.93605 + 5.08538i) q^{59} +(-3.91982 + 6.78933i) q^{61} -0.233076 q^{65} +(-10.4184 - 6.01507i) q^{67} +(7.60883 + 13.1789i) q^{71} +(-7.63029 - 13.2161i) q^{73} -4.88644i q^{77} +(-2.77876 + 1.60432i) q^{79} +12.6883i q^{83} +(0.524000 - 0.907595i) q^{85} +(-8.16638 + 14.1446i) q^{89} +(0.674422 + 0.389378i) q^{91} +(2.58895 + 1.08275i) q^{95} +(-7.37105 + 4.25568i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} - 24 q^{13} + 12 q^{19} + 20 q^{25} + 4 q^{55} - 44 q^{61} + 24 q^{67} - 20 q^{73} + 48 q^{79} - 56 q^{85} + 24 q^{91} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.557544 0.321898i 0.249341 0.143957i −0.370121 0.928983i \(-0.620684\pi\)
0.619463 + 0.785026i \(0.287351\pi\)
\(6\) 0 0
\(7\) −2.15106 −0.813024 −0.406512 0.913645i \(-0.633255\pi\)
−0.406512 + 0.913645i \(0.633255\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.27164i 0.684926i 0.939531 + 0.342463i \(0.111261\pi\)
−0.939531 + 0.342463i \(0.888739\pi\)
\(12\) 0 0
\(13\) −0.313530 0.181017i −0.0869575 0.0502050i 0.455891 0.890036i \(-0.349321\pi\)
−0.542848 + 0.839831i \(0.682654\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.40976 0.813922i 0.341916 0.197405i −0.319203 0.947686i \(-0.603415\pi\)
0.661119 + 0.750281i \(0.270082\pi\)
\(18\) 0 0
\(19\) 2.64170 + 3.46719i 0.606048 + 0.795428i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.64153 3.25714i −1.17634 0.679161i −0.221176 0.975234i \(-0.570989\pi\)
−0.955165 + 0.296073i \(0.904323\pi\)
\(24\) 0 0
\(25\) −2.29276 + 3.97118i −0.458553 + 0.794236i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.968749 + 1.67792i −0.179892 + 0.311582i −0.941843 0.336052i \(-0.890908\pi\)
0.761951 + 0.647634i \(0.224242\pi\)
\(30\) 0 0
\(31\) 6.67274i 1.19846i −0.800577 0.599229i \(-0.795474\pi\)
0.800577 0.599229i \(-0.204526\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.19931 + 0.692422i −0.202721 + 0.117041i
\(36\) 0 0
\(37\) 3.30902i 0.544000i −0.962297 0.272000i \(-0.912315\pi\)
0.962297 0.272000i \(-0.0876851\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.49359 + 6.05108i 0.545607 + 0.945020i 0.998568 + 0.0534894i \(0.0170343\pi\)
−0.452961 + 0.891530i \(0.649632\pi\)
\(42\) 0 0
\(43\) −2.71723 4.70639i −0.414374 0.717717i 0.580988 0.813912i \(-0.302666\pi\)
−0.995363 + 0.0961948i \(0.969333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.69557 4.44304i −1.12251 0.648084i −0.180473 0.983580i \(-0.557763\pi\)
−0.942042 + 0.335496i \(0.891096\pi\)
\(48\) 0 0
\(49\) −2.37294 −0.338991
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.96730 + 3.40746i −0.270229 + 0.468051i −0.968920 0.247373i \(-0.920433\pi\)
0.698691 + 0.715423i \(0.253766\pi\)
\(54\) 0 0
\(55\) 0.731237 + 1.26654i 0.0986000 + 0.170780i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.93605 + 5.08538i 0.382241 + 0.662061i 0.991382 0.131001i \(-0.0418191\pi\)
−0.609141 + 0.793062i \(0.708486\pi\)
\(60\) 0 0
\(61\) −3.91982 + 6.78933i −0.501882 + 0.869285i 0.498116 + 0.867111i \(0.334025\pi\)
−0.999998 + 0.00217438i \(0.999308\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.233076 −0.0289095
\(66\) 0 0
\(67\) −10.4184 6.01507i −1.27281 0.734858i −0.297295 0.954786i \(-0.596085\pi\)
−0.975516 + 0.219928i \(0.929418\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.60883 + 13.1789i 0.903002 + 1.56405i 0.823577 + 0.567205i \(0.191975\pi\)
0.0794258 + 0.996841i \(0.474691\pi\)
\(72\) 0 0
\(73\) −7.63029 13.2161i −0.893058 1.54682i −0.836189 0.548441i \(-0.815222\pi\)
−0.0568688 0.998382i \(-0.518112\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.88644i 0.556861i
\(78\) 0 0
\(79\) −2.77876 + 1.60432i −0.312635 + 0.180500i −0.648105 0.761551i \(-0.724438\pi\)
0.335470 + 0.942051i \(0.391105\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.6883i 1.39272i 0.717691 + 0.696362i \(0.245199\pi\)
−0.717691 + 0.696362i \(0.754801\pi\)
\(84\) 0 0
\(85\) 0.524000 0.907595i 0.0568358 0.0984425i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.16638 + 14.1446i −0.865634 + 1.49932i 0.000781849 1.00000i \(0.499751\pi\)
−0.866416 + 0.499323i \(0.833582\pi\)
\(90\) 0 0
\(91\) 0.674422 + 0.389378i 0.0706986 + 0.0408179i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.58895 + 1.08275i 0.265620 + 0.111088i
\(96\) 0 0
\(97\) −7.37105 + 4.25568i −0.748416 + 0.432098i −0.825121 0.564955i \(-0.808893\pi\)
0.0767050 + 0.997054i \(0.475560\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −13.0399 7.52860i −1.29752 0.749124i −0.317545 0.948243i \(-0.602859\pi\)
−0.979975 + 0.199119i \(0.936192\pi\)
\(102\) 0 0
\(103\) 2.56268i 0.252509i 0.991998 + 0.126254i \(0.0402955\pi\)
−0.991998 + 0.126254i \(0.959704\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.3982 −1.19857 −0.599287 0.800534i \(-0.704549\pi\)
−0.599287 + 0.800534i \(0.704549\pi\)
\(108\) 0 0
\(109\) 11.5522 6.66965i 1.10650 0.638836i 0.168577 0.985688i \(-0.446083\pi\)
0.937920 + 0.346852i \(0.112749\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15.1581 −1.42595 −0.712975 0.701189i \(-0.752653\pi\)
−0.712975 + 0.701189i \(0.752653\pi\)
\(114\) 0 0
\(115\) −4.19387 −0.391081
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.03247 + 1.75080i −0.277986 + 0.160495i
\(120\) 0 0
\(121\) 5.83965 0.530877
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.17113i 0.551962i
\(126\) 0 0
\(127\) 1.36570 + 0.788486i 0.121186 + 0.0699669i 0.559368 0.828920i \(-0.311044\pi\)
−0.438182 + 0.898886i \(0.644377\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.88308 1.08719i 0.164525 0.0949886i −0.415477 0.909604i \(-0.636385\pi\)
0.580002 + 0.814615i \(0.303052\pi\)
\(132\) 0 0
\(133\) −5.68246 7.45813i −0.492732 0.646702i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.8919 + 10.9072i 1.61404 + 0.931869i 0.988420 + 0.151744i \(0.0484890\pi\)
0.625624 + 0.780125i \(0.284844\pi\)
\(138\) 0 0
\(139\) 0.200474 0.347232i 0.0170040 0.0294518i −0.857398 0.514654i \(-0.827921\pi\)
0.874402 + 0.485202i \(0.161254\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.411205 0.712227i 0.0343867 0.0595595i
\(144\) 0 0
\(145\) 1.24735i 0.103587i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.5975 + 7.85050i −1.11395 + 0.643138i −0.939849 0.341590i \(-0.889035\pi\)
−0.174099 + 0.984728i \(0.555701\pi\)
\(150\) 0 0
\(151\) 2.82433i 0.229840i 0.993375 + 0.114920i \(0.0366612\pi\)
−0.993375 + 0.114920i \(0.963339\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.14794 3.72034i −0.172527 0.298825i
\(156\) 0 0
\(157\) 2.57082 + 4.45279i 0.205174 + 0.355371i 0.950188 0.311677i \(-0.100891\pi\)
−0.745014 + 0.667048i \(0.767557\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.1353 + 7.00631i 0.956394 + 0.552174i
\(162\) 0 0
\(163\) 6.09504 0.477401 0.238700 0.971093i \(-0.423279\pi\)
0.238700 + 0.971093i \(0.423279\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.411205 + 0.712227i −0.0318200 + 0.0551138i −0.881497 0.472190i \(-0.843464\pi\)
0.849677 + 0.527304i \(0.176797\pi\)
\(168\) 0 0
\(169\) −6.43447 11.1448i −0.494959 0.857294i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.822409 + 1.42445i 0.0625266 + 0.108299i 0.895594 0.444872i \(-0.146751\pi\)
−0.833068 + 0.553171i \(0.813417\pi\)
\(174\) 0 0
\(175\) 4.93187 8.54225i 0.372814 0.645734i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.8099 1.03220 0.516100 0.856528i \(-0.327383\pi\)
0.516100 + 0.856528i \(0.327383\pi\)
\(180\) 0 0
\(181\) −15.0783 8.70549i −1.12076 0.647074i −0.179169 0.983818i \(-0.557341\pi\)
−0.941596 + 0.336745i \(0.890674\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.06517 1.84493i −0.0783128 0.135642i
\(186\) 0 0
\(187\) 1.84894 + 3.20246i 0.135208 + 0.234187i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.85988i 0.279291i 0.990202 + 0.139645i \(0.0445963\pi\)
−0.990202 + 0.139645i \(0.955404\pi\)
\(192\) 0 0
\(193\) −0.766710 + 0.442660i −0.0551890 + 0.0318634i −0.527341 0.849654i \(-0.676811\pi\)
0.472152 + 0.881517i \(0.343477\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.1414i 1.15003i −0.818144 0.575014i \(-0.804997\pi\)
0.818144 0.575014i \(-0.195003\pi\)
\(198\) 0 0
\(199\) 0.978594 1.69498i 0.0693707 0.120154i −0.829254 0.558872i \(-0.811234\pi\)
0.898624 + 0.438719i \(0.144568\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.08384 3.60931i 0.146257 0.253324i
\(204\) 0 0
\(205\) 3.89566 + 2.24916i 0.272085 + 0.157088i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.87621 + 6.00100i −0.544809 + 0.415098i
\(210\) 0 0
\(211\) −13.9778 + 8.07010i −0.962273 + 0.555568i −0.896872 0.442291i \(-0.854166\pi\)
−0.0654010 + 0.997859i \(0.520833\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.02995 1.74934i −0.206641 0.119304i
\(216\) 0 0
\(217\) 14.3535i 0.974376i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.589334 −0.0396429
\(222\) 0 0
\(223\) −15.1589 + 8.75202i −1.01512 + 0.586079i −0.912686 0.408662i \(-0.865996\pi\)
−0.102432 + 0.994740i \(0.532662\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.35137 −0.355183 −0.177591 0.984104i \(-0.556831\pi\)
−0.177591 + 0.984104i \(0.556831\pi\)
\(228\) 0 0
\(229\) 16.2233 1.07206 0.536032 0.844197i \(-0.319922\pi\)
0.536032 + 0.844197i \(0.319922\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.728503 + 0.420601i −0.0477258 + 0.0275545i −0.523673 0.851919i \(-0.675439\pi\)
0.475947 + 0.879474i \(0.342105\pi\)
\(234\) 0 0
\(235\) −5.72083 −0.373185
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.9723i 1.42127i 0.703561 + 0.710634i \(0.251592\pi\)
−0.703561 + 0.710634i \(0.748408\pi\)
\(240\) 0 0
\(241\) −17.0648 9.85234i −1.09924 0.634645i −0.163217 0.986590i \(-0.552187\pi\)
−0.936021 + 0.351945i \(0.885520\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.32302 + 0.763845i −0.0845246 + 0.0488003i
\(246\) 0 0
\(247\) −0.200634 1.56526i −0.0127660 0.0995951i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.17073 2.98532i −0.326373 0.188432i 0.327856 0.944728i \(-0.393674\pi\)
−0.654230 + 0.756296i \(0.727007\pi\)
\(252\) 0 0
\(253\) 7.39906 12.8155i 0.465175 0.805706i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.41242 + 7.64254i −0.275239 + 0.476728i −0.970195 0.242324i \(-0.922090\pi\)
0.694956 + 0.719052i \(0.255424\pi\)
\(258\) 0 0
\(259\) 7.11791i 0.442285i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.1378 7.00776i 0.748448 0.432117i −0.0766847 0.997055i \(-0.524433\pi\)
0.825133 + 0.564939i \(0.191100\pi\)
\(264\) 0 0
\(265\) 2.53308i 0.155606i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.3673 19.6887i −0.693076 1.20044i −0.970825 0.239789i \(-0.922922\pi\)
0.277749 0.960654i \(-0.410412\pi\)
\(270\) 0 0
\(271\) −10.5147 18.2120i −0.638723 1.10630i −0.985713 0.168432i \(-0.946130\pi\)
0.346990 0.937869i \(-0.387204\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.02110 5.20834i −0.543993 0.314074i
\(276\) 0 0
\(277\) 10.4400 0.627277 0.313639 0.949542i \(-0.398452\pi\)
0.313639 + 0.949542i \(0.398452\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.90311 17.1527i 0.590770 1.02324i −0.403359 0.915042i \(-0.632157\pi\)
0.994129 0.108202i \(-0.0345094\pi\)
\(282\) 0 0
\(283\) −0.123052 0.213132i −0.00731466 0.0126694i 0.862345 0.506321i \(-0.168995\pi\)
−0.869660 + 0.493652i \(0.835662\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.51493 13.0162i −0.443592 0.768324i
\(288\) 0 0
\(289\) −7.17506 + 12.4276i −0.422062 + 0.731033i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.24330 0.0726342 0.0363171 0.999340i \(-0.488437\pi\)
0.0363171 + 0.999340i \(0.488437\pi\)
\(294\) 0 0
\(295\) 3.27395 + 1.89022i 0.190617 + 0.110053i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.17919 + 2.04242i 0.0681945 + 0.118116i
\(300\) 0 0
\(301\) 5.84493 + 10.1237i 0.336896 + 0.583521i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.04714i 0.288998i
\(306\) 0 0
\(307\) 26.2943 15.1810i 1.50069 0.866425i 0.500693 0.865625i \(-0.333078\pi\)
1.00000 0.000800631i \(-0.000254849\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.5851i 0.713634i 0.934174 + 0.356817i \(0.116138\pi\)
−0.934174 + 0.356817i \(0.883862\pi\)
\(312\) 0 0
\(313\) −10.9773 + 19.0133i −0.620476 + 1.07470i 0.368921 + 0.929461i \(0.379727\pi\)
−0.989397 + 0.145235i \(0.953606\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.84229 10.1192i 0.328136 0.568348i −0.654006 0.756489i \(-0.726913\pi\)
0.982142 + 0.188141i \(0.0602463\pi\)
\(318\) 0 0
\(319\) −3.81164 2.20065i −0.213411 0.123213i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.54618 + 2.73775i 0.364239 + 0.152332i
\(324\) 0 0
\(325\) 1.43770 0.830056i 0.0797492 0.0460432i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16.5536 + 9.55724i 0.912631 + 0.526908i
\(330\) 0 0
\(331\) 2.64080i 0.145151i 0.997363 + 0.0725757i \(0.0231219\pi\)
−0.997363 + 0.0725757i \(0.976878\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.74496 −0.423153
\(336\) 0 0
\(337\) 9.25123 5.34120i 0.503947 0.290954i −0.226395 0.974036i \(-0.572694\pi\)
0.730342 + 0.683082i \(0.239361\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15.1581 0.820855
\(342\) 0 0
\(343\) 20.1618 1.08863
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.6462 + 11.9201i −1.10835 + 0.639905i −0.938401 0.345548i \(-0.887693\pi\)
−0.169947 + 0.985453i \(0.554360\pi\)
\(348\) 0 0
\(349\) −27.0252 −1.44662 −0.723312 0.690521i \(-0.757381\pi\)
−0.723312 + 0.690521i \(0.757381\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.2312i 1.13002i 0.825082 + 0.565012i \(0.191129\pi\)
−0.825082 + 0.565012i \(0.808871\pi\)
\(354\) 0 0
\(355\) 8.48452 + 4.89854i 0.450312 + 0.259987i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.6102 8.43520i 0.771097 0.445193i −0.0621688 0.998066i \(-0.519802\pi\)
0.833266 + 0.552873i \(0.186468\pi\)
\(360\) 0 0
\(361\) −5.04281 + 18.3186i −0.265411 + 0.964135i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.50845 4.91236i −0.445353 0.257124i
\(366\) 0 0
\(367\) −0.413060 + 0.715441i −0.0215616 + 0.0373457i −0.876605 0.481211i \(-0.840197\pi\)
0.855043 + 0.518557i \(0.173530\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.23178 7.32966i 0.219703 0.380537i
\(372\) 0 0
\(373\) 13.0097i 0.673616i −0.941573 0.336808i \(-0.890653\pi\)
0.941573 0.336808i \(-0.109347\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.607463 0.350719i 0.0312860 0.0180630i
\(378\) 0 0
\(379\) 21.3820i 1.09832i −0.835717 0.549161i \(-0.814947\pi\)
0.835717 0.549161i \(-0.185053\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.1554 21.0538i −0.621112 1.07580i −0.989279 0.146038i \(-0.953348\pi\)
0.368167 0.929760i \(-0.379985\pi\)
\(384\) 0 0
\(385\) −1.57294 2.72440i −0.0801642 0.138848i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.74124 + 1.00530i 0.0882843 + 0.0509710i 0.543492 0.839414i \(-0.317102\pi\)
−0.455208 + 0.890385i \(0.650435\pi\)
\(390\) 0 0
\(391\) −10.6042 −0.536279
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.03285 + 1.78895i −0.0519685 + 0.0900121i
\(396\) 0 0
\(397\) 7.37506 + 12.7740i 0.370143 + 0.641107i 0.989587 0.143934i \(-0.0459752\pi\)
−0.619444 + 0.785041i \(0.712642\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.2748 + 31.6528i 0.912598 + 1.58067i 0.810381 + 0.585903i \(0.199260\pi\)
0.102217 + 0.994762i \(0.467407\pi\)
\(402\) 0 0
\(403\) −1.20788 + 2.09210i −0.0601686 + 0.104215i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.51691 0.372600
\(408\) 0 0
\(409\) 7.08559 + 4.09087i 0.350360 + 0.202280i 0.664844 0.746982i \(-0.268498\pi\)
−0.314484 + 0.949263i \(0.601831\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.31562 10.9390i −0.310771 0.538271i
\(414\) 0 0
\(415\) 4.08434 + 7.07429i 0.200493 + 0.347263i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 30.7373i 1.50161i −0.660522 0.750807i \(-0.729665\pi\)
0.660522 0.750807i \(-0.270335\pi\)
\(420\) 0 0
\(421\) 31.2702 18.0538i 1.52401 0.879890i 0.524419 0.851460i \(-0.324283\pi\)
0.999596 0.0284299i \(-0.00905073\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.46453i 0.362083i
\(426\) 0 0
\(427\) 8.43177 14.6043i 0.408042 0.706750i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −15.5648 + 26.9590i −0.749728 + 1.29857i 0.198224 + 0.980157i \(0.436483\pi\)
−0.947953 + 0.318411i \(0.896851\pi\)
\(432\) 0 0
\(433\) 0.962315 + 0.555593i 0.0462459 + 0.0267001i 0.522945 0.852367i \(-0.324833\pi\)
−0.476699 + 0.879067i \(0.658167\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.61013 28.1647i −0.172696 1.34730i
\(438\) 0 0
\(439\) −6.24345 + 3.60465i −0.297983 + 0.172041i −0.641537 0.767092i \(-0.721703\pi\)
0.343553 + 0.939133i \(0.388369\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.55411 0.897264i −0.0738378 0.0426303i 0.462627 0.886553i \(-0.346907\pi\)
−0.536464 + 0.843923i \(0.680240\pi\)
\(444\) 0 0
\(445\) 10.5150i 0.498457i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.922248 0.0435236 0.0217618 0.999763i \(-0.493072\pi\)
0.0217618 + 0.999763i \(0.493072\pi\)
\(450\) 0 0
\(451\) −13.7459 + 7.93619i −0.647268 + 0.373700i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.501360 0.0235041
\(456\) 0 0
\(457\) −14.3504 −0.671285 −0.335643 0.941989i \(-0.608953\pi\)
−0.335643 + 0.941989i \(0.608953\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.90876 1.67937i 0.135474 0.0782162i −0.430731 0.902480i \(-0.641744\pi\)
0.566205 + 0.824264i \(0.308411\pi\)
\(462\) 0 0
\(463\) 14.1766 0.658841 0.329420 0.944183i \(-0.393147\pi\)
0.329420 + 0.944183i \(0.393147\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.04097i 0.418366i −0.977876 0.209183i \(-0.932920\pi\)
0.977876 0.209183i \(-0.0670805\pi\)
\(468\) 0 0
\(469\) 22.4106 + 12.9388i 1.03483 + 0.597457i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.6912 6.17258i 0.491583 0.283815i
\(474\) 0 0
\(475\) −19.8256 + 2.54124i −0.909663 + 0.116600i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.8635 + 10.3135i 0.816206 + 0.471237i 0.849107 0.528222i \(-0.177141\pi\)
−0.0329001 + 0.999459i \(0.510474\pi\)
\(480\) 0 0
\(481\) −0.598988 + 1.03748i −0.0273115 + 0.0473049i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.73979 + 4.74545i −0.124407 + 0.215480i
\(486\) 0 0
\(487\) 23.1950i 1.05107i 0.850773 + 0.525533i \(0.176134\pi\)
−0.850773 + 0.525533i \(0.823866\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.6050 8.43221i 0.659115 0.380540i −0.132824 0.991140i \(-0.542405\pi\)
0.791940 + 0.610599i \(0.209071\pi\)
\(492\) 0 0
\(493\) 3.15395i 0.142047i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.3671 28.3486i −0.734163 1.27161i
\(498\) 0 0
\(499\) −0.912997 1.58136i −0.0408714 0.0707913i 0.844866 0.534978i \(-0.179680\pi\)
−0.885737 + 0.464187i \(0.846347\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.54206 + 2.62236i 0.202521 + 0.116925i 0.597831 0.801622i \(-0.296030\pi\)
−0.395310 + 0.918548i \(0.629363\pi\)
\(504\) 0 0
\(505\) −9.69377 −0.431367
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.9161 36.2278i 0.927091 1.60577i 0.138929 0.990302i \(-0.455634\pi\)
0.788163 0.615467i \(-0.211033\pi\)
\(510\) 0 0
\(511\) 16.4132 + 28.4285i 0.726078 + 1.25760i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.824923 + 1.42881i 0.0363505 + 0.0629608i
\(516\) 0 0
\(517\) 10.0930 17.4816i 0.443889 0.768839i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.461124 −0.0202022 −0.0101011 0.999949i \(-0.503215\pi\)
−0.0101011 + 0.999949i \(0.503215\pi\)
\(522\) 0 0
\(523\) 33.5957 + 19.3965i 1.46904 + 0.848150i 0.999398 0.0347068i \(-0.0110497\pi\)
0.469642 + 0.882857i \(0.344383\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.43109 9.40692i −0.236582 0.409772i
\(528\) 0 0
\(529\) 9.71793 + 16.8320i 0.422519 + 0.731824i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.52959i 0.109569i
\(534\) 0 0
\(535\) −6.91252 + 3.99094i −0.298854 + 0.172544i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.39047i 0.232184i
\(540\) 0 0
\(541\) 15.4679 26.7912i 0.665016 1.15184i −0.314264 0.949335i \(-0.601758\pi\)
0.979281 0.202507i \(-0.0649088\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.29390 7.43724i 0.183930 0.318577i
\(546\) 0 0
\(547\) −14.9526 8.63291i −0.639329 0.369117i 0.145027 0.989428i \(-0.453673\pi\)
−0.784356 + 0.620311i \(0.787006\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.37682 + 1.07374i −0.356865 + 0.0457427i
\(552\) 0 0
\(553\) 5.97728 3.45098i 0.254180 0.146751i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.3157 10.5746i −0.776060 0.448058i 0.0589723 0.998260i \(-0.481218\pi\)
−0.835032 + 0.550201i \(0.814551\pi\)
\(558\) 0 0
\(559\) 1.96746i 0.0832146i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 43.9292 1.85140 0.925698 0.378264i \(-0.123479\pi\)
0.925698 + 0.378264i \(0.123479\pi\)
\(564\) 0 0
\(565\) −8.45129 + 4.87935i −0.355548 + 0.205276i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.6631 1.03393 0.516965 0.856007i \(-0.327062\pi\)
0.516965 + 0.856007i \(0.327062\pi\)
\(570\) 0 0
\(571\) 33.9160 1.41934 0.709670 0.704535i \(-0.248844\pi\)
0.709670 + 0.704535i \(0.248844\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 25.8694 14.9357i 1.07883 0.622862i
\(576\) 0 0
\(577\) 7.08575 0.294984 0.147492 0.989063i \(-0.452880\pi\)
0.147492 + 0.989063i \(0.452880\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 27.2933i 1.13232i
\(582\) 0 0
\(583\) −7.74053 4.46900i −0.320580 0.185087i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.8990 6.29254i 0.449850 0.259721i −0.257917 0.966167i \(-0.583036\pi\)
0.707767 + 0.706446i \(0.249703\pi\)
\(588\) 0 0
\(589\) 23.1356 17.6274i 0.953288 0.726324i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 26.8796 + 15.5190i 1.10381 + 0.637287i 0.937220 0.348738i \(-0.113390\pi\)
0.166594 + 0.986026i \(0.446723\pi\)
\(594\) 0 0
\(595\) −1.12716 + 1.95229i −0.0462089 + 0.0800362i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.5973 18.3551i 0.432994 0.749968i −0.564135 0.825683i \(-0.690790\pi\)
0.997130 + 0.0757140i \(0.0241236\pi\)
\(600\) 0 0
\(601\) 6.03577i 0.246204i 0.992394 + 0.123102i \(0.0392843\pi\)
−0.992394 + 0.123102i \(0.960716\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.25586 1.87977i 0.132370 0.0764236i
\(606\) 0 0
\(607\) 46.3226i 1.88018i −0.340930 0.940089i \(-0.610742\pi\)
0.340930 0.940089i \(-0.389258\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.60853 + 2.78605i 0.0650741 + 0.112712i
\(612\) 0 0
\(613\) −4.55012 7.88103i −0.183777 0.318312i 0.759386 0.650640i \(-0.225499\pi\)
−0.943164 + 0.332328i \(0.892166\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.99382 3.46053i −0.241302 0.139316i 0.374473 0.927238i \(-0.377824\pi\)
−0.615775 + 0.787922i \(0.711157\pi\)
\(618\) 0 0
\(619\) 0.717872 0.0288537 0.0144269 0.999896i \(-0.495408\pi\)
0.0144269 + 0.999896i \(0.495408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.5664 30.4258i 0.703782 1.21899i
\(624\) 0 0
\(625\) −9.47734 16.4152i −0.379094 0.656609i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.69329 4.66491i −0.107388 0.186002i
\(630\) 0 0
\(631\) −14.4205 + 24.9770i −0.574070 + 0.994318i 0.422073 + 0.906562i \(0.361303\pi\)
−0.996142 + 0.0877555i \(0.972031\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.01525 0.0402889
\(636\) 0 0
\(637\) 0.743988 + 0.429541i 0.0294779 + 0.0170191i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.88506 3.26503i −0.0744556 0.128961i 0.826394 0.563093i \(-0.190389\pi\)
−0.900849 + 0.434132i \(0.857055\pi\)
\(642\) 0 0
\(643\) 21.8957 + 37.9244i 0.863482 + 1.49559i 0.868547 + 0.495607i \(0.165054\pi\)
−0.00506540 + 0.999987i \(0.501612\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 40.9746i 1.61088i −0.592680 0.805438i \(-0.701930\pi\)
0.592680 0.805438i \(-0.298070\pi\)
\(648\) 0 0
\(649\) −11.5522 + 6.66965i −0.453462 + 0.261807i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.5028i 1.35020i 0.737727 + 0.675099i \(0.235899\pi\)
−0.737727 + 0.675099i \(0.764101\pi\)
\(654\) 0 0
\(655\) 0.699932 1.21232i 0.0273486 0.0473692i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −3.53041 + 6.11485i −0.137525 + 0.238201i −0.926559 0.376149i \(-0.877248\pi\)
0.789034 + 0.614349i \(0.210581\pi\)
\(660\) 0 0
\(661\) 32.3836 + 18.6967i 1.25958 + 0.727217i 0.972992 0.230837i \(-0.0741465\pi\)
0.286585 + 0.958055i \(0.407480\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.56898 2.32906i −0.215956 0.0903172i
\(666\) 0 0
\(667\) 10.9305 6.31070i 0.423229 0.244351i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.4229 8.90443i −0.595396 0.343752i
\(672\) 0 0
\(673\) 13.5910i 0.523895i 0.965082 + 0.261948i \(0.0843648\pi\)
−0.965082 + 0.261948i \(0.915635\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −39.5478 −1.51995 −0.759973 0.649954i \(-0.774788\pi\)
−0.759973 + 0.649954i \(0.774788\pi\)
\(678\) 0 0
\(679\) 15.8556 9.15422i 0.608481 0.351307i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −43.0705 −1.64805 −0.824024 0.566555i \(-0.808276\pi\)
−0.824024 + 0.566555i \(0.808276\pi\)
\(684\) 0 0
\(685\) 14.0441 0.536597
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.23361 0.712227i 0.0469970 0.0271337i
\(690\) 0 0
\(691\) 7.41460 0.282065 0.141032 0.990005i \(-0.454958\pi\)
0.141032 + 0.990005i \(0.454958\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.258129i 0.00979141i
\(696\) 0 0
\(697\) 9.85022 + 5.68703i 0.373104 + 0.215411i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.85633 1.64910i 0.107882 0.0622857i −0.445088 0.895487i \(-0.646828\pi\)
0.552970 + 0.833201i \(0.313494\pi\)
\(702\) 0 0
\(703\) 11.4730 8.74146i 0.432713 0.329690i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.0497 + 16.1945i 1.05492 + 0.609056i
\(708\) 0 0
\(709\) −25.1244 + 43.5167i −0.943566 + 1.63430i −0.184970 + 0.982744i \(0.559219\pi\)
−0.758597 + 0.651561i \(0.774115\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −21.7340 + 37.6445i −0.813946 + 1.40980i
\(714\) 0 0
\(715\) 0.529464i 0.0198008i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.54366 + 3.77799i −0.244037 + 0.140895i −0.617031 0.786939i \(-0.711665\pi\)
0.372994 + 0.927834i \(0.378331\pi\)
\(720\) 0 0
\(721\) 5.51249i 0.205296i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.44222 7.69416i −0.164980 0.285754i
\(726\) 0 0
\(727\) 15.3363 + 26.5632i 0.568791 + 0.985175i 0.996686 + 0.0813464i \(0.0259220\pi\)
−0.427895 + 0.903829i \(0.640745\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.66127 4.42323i −0.283362 0.163599i
\(732\) 0 0
\(733\) −8.24989 −0.304717 −0.152358 0.988325i \(-0.548687\pi\)
−0.152358 + 0.988325i \(0.548687\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.6641 23.6669i 0.503323 0.871781i
\(738\) 0 0
\(739\) −0.605814 1.04930i −0.0222852 0.0385991i 0.854668 0.519175i \(-0.173761\pi\)
−0.876953 + 0.480576i \(0.840428\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.1052 + 29.6272i 0.627531 + 1.08691i 0.988046 + 0.154161i \(0.0492676\pi\)
−0.360515 + 0.932753i \(0.617399\pi\)
\(744\) 0 0
\(745\) −5.05412 + 8.75400i −0.185169 + 0.320722i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 26.6692 0.974471
\(750\) 0 0
\(751\) −4.95075 2.85831i −0.180655 0.104301i 0.406945 0.913453i \(-0.366594\pi\)
−0.587601 + 0.809151i \(0.699927\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.909146 + 1.57469i 0.0330872 + 0.0573087i
\(756\) 0 0
\(757\) −13.2419 22.9357i −0.481286 0.833612i 0.518483 0.855088i \(-0.326497\pi\)
−0.999769 + 0.0214755i \(0.993164\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0147i 0.544282i −0.962257 0.272141i \(-0.912268\pi\)
0.962257 0.272141i \(-0.0877317\pi\)
\(762\) 0 0
\(763\) −24.8494 + 14.3468i −0.899609 + 0.519390i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.12589i 0.0767616i
\(768\) 0 0
\(769\) −6.42918 + 11.1357i −0.231842 + 0.401562i −0.958350 0.285596i \(-0.907809\pi\)
0.726508 + 0.687158i \(0.241142\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 21.7780 37.7207i 0.783301 1.35672i −0.146707 0.989180i \(-0.546868\pi\)
0.930009 0.367538i \(-0.119799\pi\)
\(774\) 0 0
\(775\) 26.4986 + 15.2990i 0.951860 + 0.549556i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11.7512 + 28.0981i −0.421031 + 1.00672i
\(780\) 0 0
\(781\) −29.9377 + 17.2845i −1.07126 + 0.618489i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.86669 + 1.65508i 0.102317 + 0.0590725i
\(786\) 0 0
\(787\) 22.5389i 0.803426i 0.915766 + 0.401713i \(0.131585\pi\)
−0.915766 + 0.401713i \(0.868415\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 32.6059 1.15933
\(792\) 0 0
\(793\) 2.45796 1.41911i 0.0872848 0.0503939i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19.1329 −0.677722 −0.338861 0.940836i \(-0.610042\pi\)
−0.338861 + 0.940836i \(0.610042\pi\)
\(798\) 0 0
\(799\) −14.4652 −0.511740
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 30.0221 17.3333i 1.05946 0.611679i
\(804\) 0 0
\(805\) 9.02127 0.317958
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 13.1712i 0.463076i 0.972826 + 0.231538i \(0.0743757\pi\)
−0.972826 + 0.231538i \(0.925624\pi\)
\(810\) 0 0
\(811\) −48.4914 27.9965i −1.70276 0.983092i −0.942940 0.332964i \(-0.891951\pi\)
−0.759825 0.650128i \(-0.774715\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.39826 1.96198i 0.119036 0.0687253i
\(816\) 0 0
\(817\) 9.13981 21.8540i 0.319761 0.764576i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −17.2744 9.97336i −0.602880 0.348073i 0.167294 0.985907i \(-0.446497\pi\)
−0.770174 + 0.637834i \(0.779830\pi\)
\(822\) 0 0
\(823\) 17.2253 29.8352i 0.600437 1.03999i −0.392317 0.919830i \(-0.628326\pi\)
0.992755 0.120158i \(-0.0383402\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.21205 3.83138i 0.0769204 0.133230i −0.824999 0.565134i \(-0.808825\pi\)
0.901920 + 0.431904i \(0.142158\pi\)
\(828\) 0 0
\(829\) 46.9000i 1.62890i −0.580230 0.814452i \(-0.697038\pi\)
0.580230 0.814452i \(-0.302962\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.34526 + 1.93139i −0.115907 + 0.0669187i
\(834\) 0 0
\(835\) 0.529464i 0.0183229i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.97765 + 17.2818i 0.344467 + 0.596634i 0.985257 0.171082i \(-0.0547264\pi\)
−0.640790 + 0.767716i \(0.721393\pi\)
\(840\) 0 0
\(841\) 12.6231 + 21.8638i 0.435278 + 0.753923i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7.17500 4.14249i −0.246827 0.142506i
\(846\) 0 0
\(847\) −12.5614 −0.431616
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −10.7780 + 18.6680i −0.369464 + 0.639930i
\(852\) 0 0
\(853\) 11.4564 + 19.8431i 0.392260 + 0.679415i 0.992747 0.120220i \(-0.0383600\pi\)
−0.600487 + 0.799634i \(0.705027\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.32820 + 7.49666i 0.147848 + 0.256081i 0.930432 0.366465i \(-0.119432\pi\)
−0.782584 + 0.622546i \(0.786099\pi\)
\(858\) 0 0
\(859\) 7.64029 13.2334i 0.260683 0.451517i −0.705740 0.708471i \(-0.749385\pi\)
0.966424 + 0.256954i \(0.0827188\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36.4317 −1.24015 −0.620074 0.784543i \(-0.712897\pi\)
−0.620074 + 0.784543i \(0.712897\pi\)
\(864\) 0 0
\(865\) 0.917059 + 0.529464i 0.0311809 + 0.0180023i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3.64443 6.31234i −0.123629 0.214132i
\(870\) 0 0
\(871\) 2.17766 + 3.77181i 0.0737870 + 0.127803i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 13.2745i 0.448759i
\(876\) 0 0
\(877\) 36.5420 21.0976i 1.23394 0.712414i 0.266089 0.963949i \(-0.414269\pi\)
0.967848 + 0.251535i \(0.0809353\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22.3580i 0.753258i 0.926364 + 0.376629i \(0.122917\pi\)
−0.926364 + 0.376629i \(0.877083\pi\)
\(882\) 0 0
\(883\) −20.7843 + 35.9994i −0.699446 + 1.21148i 0.269212 + 0.963081i \(0.413237\pi\)
−0.968659 + 0.248396i \(0.920097\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.77689 16.9341i 0.328276 0.568591i −0.653894 0.756586i \(-0.726866\pi\)
0.982170 + 0.187996i \(0.0601991\pi\)
\(888\) 0 0
\(889\) −2.93770 1.69608i −0.0985273 0.0568848i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.92455 38.4192i −0.164794 1.28565i
\(894\) 0 0
\(895\) 7.69963 4.44538i 0.257370 0.148593i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 11.1963 + 6.46420i 0.373419 + 0.215593i
\(900\) 0 0
\(901\) 6.40492i 0.213379i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.2091 −0.372604
\(906\) 0 0
\(907\) 13.7459 7.93619i 0.456424 0.263517i −0.254115 0.967174i \(-0.581784\pi\)
0.710540 + 0.703657i \(0.248451\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −52.4664 −1.73829 −0.869145 0.494558i \(-0.835330\pi\)
−0.869145 + 0.494558i \(0.835330\pi\)
\(912\) 0 0
\(913\) −28.8233 −0.953912
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.05061 + 2.33862i −0.133763 + 0.0772281i
\(918\) 0 0
\(919\) −42.6867 −1.40810 −0.704052 0.710148i \(-0.748628\pi\)
−0.704052 + 0.710148i \(0.748628\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.50930i 0.181341i
\(924\) 0 0
\(925\) 13.1407 + 7.58681i 0.432065 + 0.249453i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 22.6161 13.0574i 0.742009 0.428399i −0.0807905 0.996731i \(-0.525744\pi\)
0.822799 + 0.568332i \(0.192411\pi\)
\(930\) 0 0
\(931\) −6.26860 8.22743i −0.205445 0.269643i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.06173 + 1.19034i 0.0674258 + 0.0389283i
\(936\) 0 0
\(937\) 19.3858 33.5771i 0.633305 1.09692i −0.353567 0.935409i \(-0.615031\pi\)
0.986872 0.161507i \(-0.0516354\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.50473 + 4.33831i −0.0816517 + 0.141425i −0.903960 0.427618i \(-0.859353\pi\)
0.822308 + 0.569043i \(0.192686\pi\)
\(942\) 0 0
\(943\) 45.5165i 1.48222i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.5887 6.11340i 0.344087 0.198659i −0.317991 0.948094i \(-0.603008\pi\)
0.662078 + 0.749435i \(0.269675\pi\)
\(948\) 0 0
\(949\) 5.52484i 0.179344i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 10.4783 + 18.1489i 0.339424 + 0.587900i 0.984325 0.176367i \(-0.0564345\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(954\) 0 0
\(955\) 1.24249 + 2.15205i 0.0402059 + 0.0696387i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −40.6376 23.4621i −1.31226 0.757632i
\(960\) 0 0
\(961\) −13.5254 −0.436303
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.284983 + 0.493605i −0.00917393 + 0.0158897i
\(966\) 0 0
\(967\) 13.4393 + 23.2776i 0.432180 + 0.748558i 0.997061 0.0766149i \(-0.0244112\pi\)
−0.564881 + 0.825172i \(0.691078\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.60883 + 13.1789i 0.244179 + 0.422931i 0.961900 0.273400i \(-0.0881482\pi\)
−0.717721 + 0.696330i \(0.754815\pi\)
\(972\) 0 0
\(973\) −0.431232 + 0.746917i −0.0138247 + 0.0239450i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.1283 0.963890 0.481945 0.876201i \(-0.339930\pi\)
0.481945 + 0.876201i \(0.339930\pi\)
\(978\) 0 0
\(979\) −32.1314 18.5511i −1.02692 0.592895i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.52751 + 9.57393i 0.176300 + 0.305361i 0.940610 0.339488i \(-0.110254\pi\)
−0.764310 + 0.644849i \(0.776920\pi\)
\(984\) 0 0
\(985\) −5.19589 8.99955i −0.165555 0.286749i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 35.4016i 1.12571i
\(990\) 0 0
\(991\) 39.9706 23.0770i 1.26971 0.733066i 0.294775 0.955567i \(-0.404755\pi\)
0.974932 + 0.222501i \(0.0714220\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.26003i 0.0399457i
\(996\) 0 0
\(997\) −5.17506 + 8.96347i −0.163896 + 0.283876i −0.936263 0.351301i \(-0.885739\pi\)
0.772367 + 0.635177i \(0.219073\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2736.2.dc.c.449.5 16
3.2 odd 2 inner 2736.2.dc.c.449.4 16
4.3 odd 2 171.2.m.a.107.4 yes 16
12.11 even 2 171.2.m.a.107.5 yes 16
19.8 odd 6 inner 2736.2.dc.c.1889.4 16
57.8 even 6 inner 2736.2.dc.c.1889.5 16
76.27 even 6 171.2.m.a.8.5 yes 16
228.179 odd 6 171.2.m.a.8.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.m.a.8.4 16 228.179 odd 6
171.2.m.a.8.5 yes 16 76.27 even 6
171.2.m.a.107.4 yes 16 4.3 odd 2
171.2.m.a.107.5 yes 16 12.11 even 2
2736.2.dc.c.449.4 16 3.2 odd 2 inner
2736.2.dc.c.449.5 16 1.1 even 1 trivial
2736.2.dc.c.1889.4 16 19.8 odd 6 inner
2736.2.dc.c.1889.5 16 57.8 even 6 inner