Properties

Label 1134.2.e.c.865.1
Level $1134$
Weight $2$
Character 1134.865
Analytic conductor $9.055$
Analytic rank $1$
Dimension $2$
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] = [1134,2,Mod(865,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.e (of order \(3\), 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: \(9.05503558921\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1134.865
Dual form 1134.2.e.c.919.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-0.500000 - 2.59808i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-0.500000 - 2.59808i) q^{7} -1.00000 q^{8} +(-3.00000 - 5.19615i) q^{11} +(-2.50000 - 4.33013i) q^{13} +(0.500000 + 2.59808i) q^{14} +1.00000 q^{16} +(-3.00000 + 5.19615i) q^{17} +(2.00000 + 3.46410i) q^{19} +(3.00000 + 5.19615i) q^{22} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +(2.50000 + 4.33013i) q^{26} +(-0.500000 - 2.59808i) q^{28} +(-3.00000 + 5.19615i) q^{29} -1.00000 q^{31} -1.00000 q^{32} +(3.00000 - 5.19615i) q^{34} +(0.500000 + 0.866025i) q^{37} +(-2.00000 - 3.46410i) q^{38} +(3.00000 + 5.19615i) q^{41} +(0.500000 - 0.866025i) q^{43} +(-3.00000 - 5.19615i) q^{44} +(3.00000 - 5.19615i) q^{46} -6.00000 q^{47} +(-6.50000 + 2.59808i) q^{49} +(-2.50000 - 4.33013i) q^{50} +(-2.50000 - 4.33013i) q^{52} +(3.00000 - 5.19615i) q^{53} +(0.500000 + 2.59808i) q^{56} +(3.00000 - 5.19615i) q^{58} -6.00000 q^{59} -1.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -1.00000 q^{67} +(-3.00000 + 5.19615i) q^{68} +12.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-0.500000 - 0.866025i) q^{74} +(2.00000 + 3.46410i) q^{76} +(-12.0000 + 10.3923i) q^{77} -1.00000 q^{79} +(-3.00000 - 5.19615i) q^{82} +(-3.00000 + 5.19615i) q^{83} +(-0.500000 + 0.866025i) q^{86} +(3.00000 + 5.19615i) q^{88} +(-10.0000 + 8.66025i) q^{91} +(-3.00000 + 5.19615i) q^{92} +6.00000 q^{94} +(-8.50000 + 14.7224i) q^{97} +(6.50000 - 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - q^{7} - 2 q^{8} - 6 q^{11} - 5 q^{13} + q^{14} + 2 q^{16} - 6 q^{17} + 4 q^{19} + 6 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - q^{28} - 6 q^{29} - 2 q^{31} - 2 q^{32} + 6 q^{34} + q^{37} - 4 q^{38} + 6 q^{41} + q^{43} - 6 q^{44} + 6 q^{46} - 12 q^{47} - 13 q^{49} - 5 q^{50} - 5 q^{52} + 6 q^{53} + q^{56} + 6 q^{58} - 12 q^{59} - 2 q^{61} + 2 q^{62} + 2 q^{64} - 2 q^{67} - 6 q^{68} + 24 q^{71} - 2 q^{73} - q^{74} + 4 q^{76} - 24 q^{77} - 2 q^{79} - 6 q^{82} - 6 q^{83} - q^{86} + 6 q^{88} - 20 q^{91} - 6 q^{92} + 12 q^{94} - 17 q^{97} + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −0.500000 2.59808i −0.188982 0.981981i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 5.19615i −0.904534 1.56670i −0.821541 0.570149i \(-0.806886\pi\)
−0.0829925 0.996550i \(-0.526448\pi\)
\(12\) 0 0
\(13\) −2.50000 4.33013i −0.693375 1.20096i −0.970725 0.240192i \(-0.922790\pi\)
0.277350 0.960769i \(-0.410544\pi\)
\(14\) 0.500000 + 2.59808i 0.133631 + 0.694365i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 + 5.19615i −0.727607 + 1.26025i 0.230285 + 0.973123i \(0.426034\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 + 5.19615i 0.639602 + 1.10782i
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 2.50000 + 4.33013i 0.490290 + 0.849208i
\(27\) 0 0
\(28\) −0.500000 2.59808i −0.0944911 0.490990i
\(29\) −3.00000 + 5.19615i −0.557086 + 0.964901i 0.440652 + 0.897678i \(0.354747\pi\)
−0.997738 + 0.0672232i \(0.978586\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.00000 5.19615i 0.514496 0.891133i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.00000 3.46410i −0.324443 0.561951i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 + 5.19615i 0.468521 + 0.811503i 0.999353 0.0359748i \(-0.0114536\pi\)
−0.530831 + 0.847477i \(0.678120\pi\)
\(42\) 0 0
\(43\) 0.500000 0.866025i 0.0762493 0.132068i −0.825380 0.564578i \(-0.809039\pi\)
0.901629 + 0.432511i \(0.142372\pi\)
\(44\) −3.00000 5.19615i −0.452267 0.783349i
\(45\) 0 0
\(46\) 3.00000 5.19615i 0.442326 0.766131i
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) −2.50000 4.33013i −0.353553 0.612372i
\(51\) 0 0
\(52\) −2.50000 4.33013i −0.346688 0.600481i
\(53\) 3.00000 5.19615i 0.412082 0.713746i −0.583036 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.500000 + 2.59808i 0.0668153 + 0.347183i
\(57\) 0 0
\(58\) 3.00000 5.19615i 0.393919 0.682288i
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 −0.122169 −0.0610847 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) −3.00000 + 5.19615i −0.363803 + 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) −0.500000 0.866025i −0.0581238 0.100673i
\(75\) 0 0
\(76\) 2.00000 + 3.46410i 0.229416 + 0.397360i
\(77\) −12.0000 + 10.3923i −1.36753 + 1.18431i
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.00000 5.19615i −0.331295 0.573819i
\(83\) −3.00000 + 5.19615i −0.329293 + 0.570352i −0.982372 0.186938i \(-0.940144\pi\)
0.653079 + 0.757290i \(0.273477\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.500000 + 0.866025i −0.0539164 + 0.0933859i
\(87\) 0 0
\(88\) 3.00000 + 5.19615i 0.319801 + 0.553912i
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) −10.0000 + 8.66025i −1.04828 + 0.907841i
\(92\) −3.00000 + 5.19615i −0.312772 + 0.541736i
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) −8.50000 + 14.7224i −0.863044 + 1.49484i 0.00593185 + 0.999982i \(0.498112\pi\)
−0.868976 + 0.494854i \(0.835222\pi\)
\(98\) 6.50000 2.59808i 0.656599 0.262445i
\(99\) 0 0
\(100\) 2.50000 + 4.33013i 0.250000 + 0.433013i
\(101\) −6.00000 10.3923i −0.597022 1.03407i −0.993258 0.115924i \(-0.963017\pi\)
0.396236 0.918149i \(-0.370316\pi\)
\(102\) 0 0
\(103\) 9.50000 16.4545i 0.936063 1.62131i 0.163335 0.986571i \(-0.447775\pi\)
0.772728 0.634738i \(-0.218892\pi\)
\(104\) 2.50000 + 4.33013i 0.245145 + 0.424604i
\(105\) 0 0
\(106\) −3.00000 + 5.19615i −0.291386 + 0.504695i
\(107\) −9.00000 15.5885i −0.870063 1.50699i −0.861931 0.507026i \(-0.830745\pi\)
−0.00813215 0.999967i \(-0.502589\pi\)
\(108\) 0 0
\(109\) −2.50000 + 4.33013i −0.239457 + 0.414751i −0.960558 0.278078i \(-0.910303\pi\)
0.721102 + 0.692829i \(0.243636\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.500000 2.59808i −0.0472456 0.245495i
\(113\) −6.00000 10.3923i −0.564433 0.977626i −0.997102 0.0760733i \(-0.975762\pi\)
0.432670 0.901553i \(-0.357572\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.00000 + 5.19615i −0.278543 + 0.482451i
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 15.0000 + 5.19615i 1.37505 + 0.476331i
\(120\) 0 0
\(121\) −12.5000 + 21.6506i −1.13636 + 1.96824i
\(122\) 1.00000 0.0905357
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) 8.00000 6.92820i 0.693688 0.600751i
\(134\) 1.00000 0.0863868
\(135\) 0 0
\(136\) 3.00000 5.19615i 0.257248 0.445566i
\(137\) 3.00000 + 5.19615i 0.256307 + 0.443937i 0.965250 0.261329i \(-0.0841608\pi\)
−0.708942 + 0.705266i \(0.750827\pi\)
\(138\) 0 0
\(139\) 3.50000 + 6.06218i 0.296866 + 0.514187i 0.975417 0.220366i \(-0.0707252\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −15.0000 + 25.9808i −1.25436 + 2.17262i
\(144\) 0 0
\(145\) 0 0
\(146\) 1.00000 1.73205i 0.0827606 0.143346i
\(147\) 0 0
\(148\) 0.500000 + 0.866025i 0.0410997 + 0.0711868i
\(149\) 6.00000 10.3923i 0.491539 0.851371i −0.508413 0.861113i \(-0.669768\pi\)
0.999953 + 0.00974235i \(0.00310113\pi\)
\(150\) 0 0
\(151\) −8.50000 14.7224i −0.691720 1.19809i −0.971274 0.237964i \(-0.923520\pi\)
0.279554 0.960130i \(-0.409814\pi\)
\(152\) −2.00000 3.46410i −0.162221 0.280976i
\(153\) 0 0
\(154\) 12.0000 10.3923i 0.966988 0.837436i
\(155\) 0 0
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 1.00000 0.0795557
\(159\) 0 0
\(160\) 0 0
\(161\) 15.0000 + 5.19615i 1.18217 + 0.409514i
\(162\) 0 0
\(163\) −5.50000 9.52628i −0.430793 0.746156i 0.566149 0.824303i \(-0.308433\pi\)
−0.996942 + 0.0781474i \(0.975100\pi\)
\(164\) 3.00000 + 5.19615i 0.234261 + 0.405751i
\(165\) 0 0
\(166\) 3.00000 5.19615i 0.232845 0.403300i
\(167\) 9.00000 + 15.5885i 0.696441 + 1.20627i 0.969693 + 0.244328i \(0.0785675\pi\)
−0.273252 + 0.961943i \(0.588099\pi\)
\(168\) 0 0
\(169\) −6.00000 + 10.3923i −0.461538 + 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.500000 0.866025i 0.0381246 0.0660338i
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 10.0000 8.66025i 0.755929 0.654654i
\(176\) −3.00000 5.19615i −0.226134 0.391675i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 10.0000 8.66025i 0.741249 0.641941i
\(183\) 0 0
\(184\) 3.00000 5.19615i 0.221163 0.383065i
\(185\) 0 0
\(186\) 0 0
\(187\) 36.0000 2.63258
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 8.50000 14.7224i 0.610264 1.05701i
\(195\) 0 0
\(196\) −6.50000 + 2.59808i −0.464286 + 0.185577i
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −5.50000 + 9.52628i −0.389885 + 0.675300i −0.992434 0.122782i \(-0.960818\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) −2.50000 4.33013i −0.176777 0.306186i
\(201\) 0 0
\(202\) 6.00000 + 10.3923i 0.422159 + 0.731200i
\(203\) 15.0000 + 5.19615i 1.05279 + 0.364698i
\(204\) 0 0
\(205\) 0 0
\(206\) −9.50000 + 16.4545i −0.661896 + 1.14644i
\(207\) 0 0
\(208\) −2.50000 4.33013i −0.173344 0.300240i
\(209\) 12.0000 20.7846i 0.830057 1.43770i
\(210\) 0 0
\(211\) 6.50000 + 11.2583i 0.447478 + 0.775055i 0.998221 0.0596196i \(-0.0189888\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 3.00000 5.19615i 0.206041 0.356873i
\(213\) 0 0
\(214\) 9.00000 + 15.5885i 0.615227 + 1.06561i
\(215\) 0 0
\(216\) 0 0
\(217\) 0.500000 + 2.59808i 0.0339422 + 0.176369i
\(218\) 2.50000 4.33013i 0.169321 0.293273i
\(219\) 0 0
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) −4.00000 + 6.92820i −0.267860 + 0.463947i −0.968309 0.249756i \(-0.919650\pi\)
0.700449 + 0.713702i \(0.252983\pi\)
\(224\) 0.500000 + 2.59808i 0.0334077 + 0.173591i
\(225\) 0 0
\(226\) 6.00000 + 10.3923i 0.399114 + 0.691286i
\(227\) 3.00000 + 5.19615i 0.199117 + 0.344881i 0.948242 0.317547i \(-0.102859\pi\)
−0.749125 + 0.662428i \(0.769526\pi\)
\(228\) 0 0
\(229\) −2.50000 + 4.33013i −0.165205 + 0.286143i −0.936728 0.350058i \(-0.886162\pi\)
0.771523 + 0.636201i \(0.219495\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.00000 5.19615i 0.196960 0.341144i
\(233\) −12.0000 20.7846i −0.786146 1.36165i −0.928312 0.371802i \(-0.878740\pi\)
0.142166 0.989843i \(-0.454593\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) −15.0000 5.19615i −0.972306 0.336817i
\(239\) −12.0000 20.7846i −0.776215 1.34444i −0.934109 0.356988i \(-0.883804\pi\)
0.157893 0.987456i \(-0.449530\pi\)
\(240\) 0 0
\(241\) −8.50000 14.7224i −0.547533 0.948355i −0.998443 0.0557856i \(-0.982234\pi\)
0.450910 0.892570i \(-0.351100\pi\)
\(242\) 12.5000 21.6506i 0.803530 1.39176i
\(243\) 0 0
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) 0 0
\(247\) 10.0000 17.3205i 0.636285 1.10208i
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) 13.0000 0.815693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.00000 5.19615i 0.187135 0.324127i −0.757159 0.653231i \(-0.773413\pi\)
0.944294 + 0.329104i \(0.106747\pi\)
\(258\) 0 0
\(259\) 2.00000 1.73205i 0.124274 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) −6.00000 + 10.3923i −0.370681 + 0.642039i
\(263\) −3.00000 5.19615i −0.184988 0.320408i 0.758585 0.651575i \(-0.225891\pi\)
−0.943572 + 0.331166i \(0.892558\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −8.00000 + 6.92820i −0.490511 + 0.424795i
\(267\) 0 0
\(268\) −1.00000 −0.0610847
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 3.50000 + 6.06218i 0.212610 + 0.368251i 0.952531 0.304443i \(-0.0984703\pi\)
−0.739921 + 0.672694i \(0.765137\pi\)
\(272\) −3.00000 + 5.19615i −0.181902 + 0.315063i
\(273\) 0 0
\(274\) −3.00000 5.19615i −0.181237 0.313911i
\(275\) 15.0000 25.9808i 0.904534 1.56670i
\(276\) 0 0
\(277\) −8.50000 14.7224i −0.510716 0.884585i −0.999923 0.0124177i \(-0.996047\pi\)
0.489207 0.872167i \(-0.337286\pi\)
\(278\) −3.50000 6.06218i −0.209916 0.363585i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 + 10.3923i −0.357930 + 0.619953i −0.987615 0.156898i \(-0.949851\pi\)
0.629685 + 0.776851i \(0.283184\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 15.0000 25.9808i 0.886969 1.53627i
\(287\) 12.0000 10.3923i 0.708338 0.613438i
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) −1.00000 + 1.73205i −0.0585206 + 0.101361i
\(293\) −9.00000 15.5885i −0.525786 0.910687i −0.999549 0.0300351i \(-0.990438\pi\)
0.473763 0.880652i \(-0.342895\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.500000 0.866025i −0.0290619 0.0503367i
\(297\) 0 0
\(298\) −6.00000 + 10.3923i −0.347571 + 0.602010i
\(299\) 30.0000 1.73494
\(300\) 0 0
\(301\) −2.50000 0.866025i −0.144098 0.0499169i
\(302\) 8.50000 + 14.7224i 0.489120 + 0.847181i
\(303\) 0 0
\(304\) 2.00000 + 3.46410i 0.114708 + 0.198680i
\(305\) 0 0
\(306\) 0 0
\(307\) 17.0000 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) −12.0000 + 10.3923i −0.683763 + 0.592157i
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 36.0000 2.01561
\(320\) 0 0
\(321\) 0 0
\(322\) −15.0000 5.19615i −0.835917 0.289570i
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 12.5000 21.6506i 0.693375 1.20096i
\(326\) 5.50000 + 9.52628i 0.304617 + 0.527612i
\(327\) 0 0
\(328\) −3.00000 5.19615i −0.165647 0.286910i
\(329\) 3.00000 + 15.5885i 0.165395 + 0.859419i
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −3.00000 + 5.19615i −0.164646 + 0.285176i
\(333\) 0 0
\(334\) −9.00000 15.5885i −0.492458 0.852962i
\(335\) 0 0
\(336\) 0 0
\(337\) −1.00000 1.73205i −0.0544735 0.0943508i 0.837503 0.546433i \(-0.184015\pi\)
−0.891976 + 0.452082i \(0.850681\pi\)
\(338\) 6.00000 10.3923i 0.326357 0.565267i
\(339\) 0 0
\(340\) 0 0
\(341\) 3.00000 + 5.19615i 0.162459 + 0.281387i
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) −0.500000 + 0.866025i −0.0269582 + 0.0466930i
\(345\) 0 0
\(346\) 0 0
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 0.500000 0.866025i 0.0267644 0.0463573i −0.852333 0.523000i \(-0.824813\pi\)
0.879097 + 0.476642i \(0.158146\pi\)
\(350\) −10.0000 + 8.66025i −0.534522 + 0.462910i
\(351\) 0 0
\(352\) 3.00000 + 5.19615i 0.159901 + 0.276956i
\(353\) 18.0000 + 31.1769i 0.958043 + 1.65938i 0.727245 + 0.686378i \(0.240800\pi\)
0.230799 + 0.973002i \(0.425866\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.00000 5.19615i −0.158334 0.274242i 0.775934 0.630814i \(-0.217279\pi\)
−0.934268 + 0.356572i \(0.883946\pi\)
\(360\) 0 0
\(361\) 1.50000 2.59808i 0.0789474 0.136741i
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) −10.0000 + 8.66025i −0.524142 + 0.453921i
\(365\) 0 0
\(366\) 0 0
\(367\) 14.0000 + 24.2487i 0.730794 + 1.26577i 0.956544 + 0.291587i \(0.0941834\pi\)
−0.225750 + 0.974185i \(0.572483\pi\)
\(368\) −3.00000 + 5.19615i −0.156386 + 0.270868i
\(369\) 0 0
\(370\) 0 0
\(371\) −15.0000 5.19615i −0.778761 0.269771i
\(372\) 0 0
\(373\) −13.0000 + 22.5167i −0.673114 + 1.16587i 0.303902 + 0.952703i \(0.401711\pi\)
−0.977016 + 0.213165i \(0.931623\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 30.0000 1.54508
\(378\) 0 0
\(379\) −1.00000 −0.0513665 −0.0256833 0.999670i \(-0.508176\pi\)
−0.0256833 + 0.999670i \(0.508176\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.00000 + 5.19615i −0.153293 + 0.265511i −0.932436 0.361335i \(-0.882321\pi\)
0.779143 + 0.626846i \(0.215654\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.00000 0.0508987
\(387\) 0 0
\(388\) −8.50000 + 14.7224i −0.431522 + 0.747418i
\(389\) 6.00000 + 10.3923i 0.304212 + 0.526911i 0.977086 0.212847i \(-0.0682735\pi\)
−0.672874 + 0.739758i \(0.734940\pi\)
\(390\) 0 0
\(391\) −18.0000 31.1769i −0.910299 1.57668i
\(392\) 6.50000 2.59808i 0.328300 0.131223i
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) 6.50000 + 11.2583i 0.326226 + 0.565039i 0.981760 0.190126i \(-0.0608897\pi\)
−0.655534 + 0.755166i \(0.727556\pi\)
\(398\) 5.50000 9.52628i 0.275690 0.477509i
\(399\) 0 0
\(400\) 2.50000 + 4.33013i 0.125000 + 0.216506i
\(401\) 3.00000 5.19615i 0.149813 0.259483i −0.781345 0.624099i \(-0.785466\pi\)
0.931158 + 0.364615i \(0.118800\pi\)
\(402\) 0 0
\(403\) 2.50000 + 4.33013i 0.124534 + 0.215699i
\(404\) −6.00000 10.3923i −0.298511 0.517036i
\(405\) 0 0
\(406\) −15.0000 5.19615i −0.744438 0.257881i
\(407\) 3.00000 5.19615i 0.148704 0.257564i
\(408\) 0 0
\(409\) 29.0000 1.43396 0.716979 0.697095i \(-0.245524\pi\)
0.716979 + 0.697095i \(0.245524\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 9.50000 16.4545i 0.468031 0.810654i
\(413\) 3.00000 + 15.5885i 0.147620 + 0.767058i
\(414\) 0 0
\(415\) 0 0
\(416\) 2.50000 + 4.33013i 0.122573 + 0.212302i
\(417\) 0 0
\(418\) −12.0000 + 20.7846i −0.586939 + 1.01661i
\(419\) 12.0000 + 20.7846i 0.586238 + 1.01539i 0.994720 + 0.102628i \(0.0327251\pi\)
−0.408481 + 0.912767i \(0.633942\pi\)
\(420\) 0 0
\(421\) −7.00000 + 12.1244i −0.341159 + 0.590905i −0.984648 0.174550i \(-0.944153\pi\)
0.643489 + 0.765455i \(0.277486\pi\)
\(422\) −6.50000 11.2583i −0.316415 0.548047i
\(423\) 0 0
\(424\) −3.00000 + 5.19615i −0.145693 + 0.252347i
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 0.500000 + 2.59808i 0.0241967 + 0.125730i
\(428\) −9.00000 15.5885i −0.435031 0.753497i
\(429\) 0 0
\(430\) 0 0
\(431\) 9.00000 15.5885i 0.433515 0.750870i −0.563658 0.826008i \(-0.690607\pi\)
0.997173 + 0.0751385i \(0.0239399\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) −0.500000 2.59808i −0.0240008 0.124712i
\(435\) 0 0
\(436\) −2.50000 + 4.33013i −0.119728 + 0.207375i
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −30.0000 −1.42695
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4.00000 6.92820i 0.189405 0.328060i
\(447\) 0 0
\(448\) −0.500000 2.59808i −0.0236228 0.122748i
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 18.0000 31.1769i 0.847587 1.46806i
\(452\) −6.00000 10.3923i −0.282216 0.488813i
\(453\) 0 0
\(454\) −3.00000 5.19615i −0.140797 0.243868i
\(455\) 0 0
\(456\) 0 0
\(457\) −19.0000 −0.888783 −0.444391 0.895833i \(-0.646580\pi\)
−0.444391 + 0.895833i \(0.646580\pi\)
\(458\) 2.50000 4.33013i 0.116817 0.202334i
\(459\) 0 0
\(460\) 0 0
\(461\) −9.00000 + 15.5885i −0.419172 + 0.726027i −0.995856 0.0909401i \(-0.971013\pi\)
0.576685 + 0.816967i \(0.304346\pi\)
\(462\) 0 0
\(463\) −4.00000 6.92820i −0.185896 0.321981i 0.757982 0.652275i \(-0.226185\pi\)
−0.943878 + 0.330294i \(0.892852\pi\)
\(464\) −3.00000 + 5.19615i −0.139272 + 0.241225i
\(465\) 0 0
\(466\) 12.0000 + 20.7846i 0.555889 + 0.962828i
\(467\) −6.00000 10.3923i −0.277647 0.480899i 0.693153 0.720791i \(-0.256221\pi\)
−0.970799 + 0.239892i \(0.922888\pi\)
\(468\) 0 0
\(469\) 0.500000 + 2.59808i 0.0230879 + 0.119968i
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −10.0000 + 17.3205i −0.458831 + 0.794719i
\(476\) 15.0000 + 5.19615i 0.687524 + 0.238165i
\(477\) 0 0
\(478\) 12.0000 + 20.7846i 0.548867 + 0.950666i
\(479\) −6.00000 10.3923i −0.274147 0.474837i 0.695773 0.718262i \(-0.255062\pi\)
−0.969920 + 0.243426i \(0.921729\pi\)
\(480\) 0 0
\(481\) 2.50000 4.33013i 0.113990 0.197437i
\(482\) 8.50000 + 14.7224i 0.387164 + 0.670588i
\(483\) 0 0
\(484\) −12.5000 + 21.6506i −0.568182 + 0.984120i
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000 34.6410i 0.906287 1.56973i 0.0871056 0.996199i \(-0.472238\pi\)
0.819181 0.573535i \(-0.194428\pi\)
\(488\) 1.00000 0.0452679
\(489\) 0 0
\(490\) 0 0
\(491\) 15.0000 + 25.9808i 0.676941 + 1.17250i 0.975898 + 0.218229i \(0.0700279\pi\)
−0.298957 + 0.954267i \(0.596639\pi\)
\(492\) 0 0
\(493\) −18.0000 31.1769i −0.810679 1.40414i
\(494\) −10.0000 + 17.3205i −0.449921 + 0.779287i
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) −6.00000 31.1769i −0.269137 1.39848i
\(498\) 0 0
\(499\) 12.5000 21.6506i 0.559577 0.969216i −0.437955 0.898997i \(-0.644297\pi\)
0.997532 0.0702185i \(-0.0223697\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −18.0000 −0.803379
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −36.0000 −1.60040
\(507\) 0 0
\(508\) −13.0000 −0.576782
\(509\) −15.0000 + 25.9808i −0.664863 + 1.15158i 0.314459 + 0.949271i \(0.398177\pi\)
−0.979322 + 0.202306i \(0.935156\pi\)
\(510\) 0 0
\(511\) 5.00000 + 1.73205i 0.221187 + 0.0766214i
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −3.00000 + 5.19615i −0.132324 + 0.229192i
\(515\) 0 0
\(516\) 0 0
\(517\) 18.0000 + 31.1769i 0.791639 + 1.37116i
\(518\) −2.00000 + 1.73205i −0.0878750 + 0.0761019i
\(519\) 0 0
\(520\) 0 0
\(521\) −21.0000 + 36.3731i −0.920027 + 1.59353i −0.120656 + 0.992694i \(0.538500\pi\)
−0.799370 + 0.600839i \(0.794833\pi\)
\(522\) 0 0
\(523\) −11.5000 19.9186i −0.502860 0.870979i −0.999995 0.00330547i \(-0.998948\pi\)
0.497135 0.867673i \(-0.334385\pi\)
\(524\) 6.00000 10.3923i 0.262111 0.453990i
\(525\) 0 0
\(526\) 3.00000 + 5.19615i 0.130806 + 0.226563i
\(527\) 3.00000 5.19615i 0.130682 0.226348i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 8.00000 6.92820i 0.346844 0.300376i
\(533\) 15.0000 25.9808i 0.649722 1.12535i
\(534\) 0 0
\(535\) 0 0
\(536\) 1.00000 0.0431934
\(537\) 0 0
\(538\) 0 0
\(539\) 33.0000 + 25.9808i 1.42141 + 1.11907i
\(540\) 0 0
\(541\) 5.00000 + 8.66025i 0.214967 + 0.372333i 0.953262 0.302144i \(-0.0977023\pi\)
−0.738296 + 0.674477i \(0.764369\pi\)
\(542\) −3.50000 6.06218i −0.150338 0.260393i
\(543\) 0 0
\(544\) 3.00000 5.19615i 0.128624 0.222783i
\(545\) 0 0
\(546\) 0 0
\(547\) 18.5000 32.0429i 0.791003 1.37006i −0.134344 0.990935i \(-0.542893\pi\)
0.925347 0.379122i \(-0.123774\pi\)
\(548\) 3.00000 + 5.19615i 0.128154 + 0.221969i
\(549\) 0 0
\(550\) −15.0000 + 25.9808i −0.639602 + 1.10782i
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0.500000 + 2.59808i 0.0212622 + 0.110481i
\(554\) 8.50000 + 14.7224i 0.361130 + 0.625496i
\(555\) 0 0
\(556\) 3.50000 + 6.06218i 0.148433 + 0.257094i
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 10.3923i 0.253095 0.438373i
\(563\) −42.0000 −1.77009 −0.885044 0.465506i \(-0.845872\pi\)
−0.885044 + 0.465506i \(0.845872\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 13.0000 0.546431
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −15.0000 + 25.9808i −0.627182 + 1.08631i
\(573\) 0 0
\(574\) −12.0000 + 10.3923i −0.500870 + 0.433766i
\(575\) −30.0000 −1.25109
\(576\) 0 0
\(577\) −11.5000 + 19.9186i −0.478751 + 0.829222i −0.999703 0.0243645i \(-0.992244\pi\)
0.520952 + 0.853586i \(0.325577\pi\)
\(578\) 9.50000 + 16.4545i 0.395148 + 0.684416i
\(579\) 0 0
\(580\) 0 0
\(581\) 15.0000 + 5.19615i 0.622305 + 0.215573i
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 1.00000 1.73205i 0.0413803 0.0716728i
\(585\) 0 0
\(586\) 9.00000 + 15.5885i 0.371787 + 0.643953i
\(587\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(588\) 0 0
\(589\) −2.00000 3.46410i −0.0824086 0.142736i
\(590\) 0 0
\(591\) 0 0
\(592\) 0.500000 + 0.866025i 0.0205499 + 0.0355934i
\(593\) −18.0000 31.1769i −0.739171 1.28028i −0.952869 0.303383i \(-0.901884\pi\)
0.213697 0.976900i \(-0.431449\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 10.3923i 0.245770 0.425685i
\(597\) 0 0
\(598\) −30.0000 −1.22679
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −23.5000 + 40.7032i −0.958585 + 1.66032i −0.232643 + 0.972562i \(0.574737\pi\)
−0.725942 + 0.687756i \(0.758596\pi\)
\(602\) 2.50000 + 0.866025i 0.101892 + 0.0352966i
\(603\) 0 0
\(604\) −8.50000 14.7224i −0.345860 0.599047i
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000 13.8564i 0.324710 0.562414i −0.656744 0.754114i \(-0.728067\pi\)
0.981454 + 0.191700i \(0.0614000\pi\)
\(608\) −2.00000 3.46410i −0.0811107 0.140488i
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 + 25.9808i 0.606835 + 1.05107i
\(612\) 0 0
\(613\) −5.50000 + 9.52628i −0.222143 + 0.384763i −0.955458 0.295126i \(-0.904638\pi\)
0.733316 + 0.679888i \(0.237972\pi\)
\(614\) −17.0000 −0.686064
\(615\) 0 0
\(616\) 12.0000 10.3923i 0.483494 0.418718i
\(617\) 15.0000 + 25.9808i 0.603877 + 1.04595i 0.992228 + 0.124434i \(0.0397116\pi\)
−0.388351 + 0.921512i \(0.626955\pi\)
\(618\) 0 0
\(619\) 12.5000 + 21.6506i 0.502417 + 0.870212i 0.999996 + 0.00279365i \(0.000889247\pi\)
−0.497579 + 0.867419i \(0.665777\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 1.00000 0.0397779
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 0 0
\(637\) 27.5000 + 21.6506i 1.08959 + 0.857829i
\(638\) −36.0000 −1.42525
\(639\) 0 0
\(640\) 0 0
\(641\) −9.00000 15.5885i −0.355479 0.615707i 0.631721 0.775196i \(-0.282349\pi\)
−0.987200 + 0.159489i \(0.949015\pi\)
\(642\) 0 0
\(643\) 12.5000 + 21.6506i 0.492952 + 0.853818i 0.999967 0.00811944i \(-0.00258453\pi\)
−0.507015 + 0.861937i \(0.669251\pi\)
\(644\) 15.0000 + 5.19615i 0.591083 + 0.204757i
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 6.00000 10.3923i 0.235884 0.408564i −0.723645 0.690172i \(-0.757535\pi\)
0.959529 + 0.281609i \(0.0908680\pi\)
\(648\) 0 0
\(649\) 18.0000 + 31.1769i 0.706562 + 1.22380i
\(650\) −12.5000 + 21.6506i −0.490290 + 0.849208i
\(651\) 0 0
\(652\) −5.50000 9.52628i −0.215397 0.373078i
\(653\) 9.00000 15.5885i 0.352197 0.610023i −0.634437 0.772975i \(-0.718768\pi\)
0.986634 + 0.162951i \(0.0521013\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.00000 + 5.19615i 0.117130 + 0.202876i
\(657\) 0 0
\(658\) −3.00000 15.5885i −0.116952 0.607701i
\(659\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(660\) 0 0
\(661\) 50.0000 1.94477 0.972387 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 3.00000 5.19615i 0.116423 0.201650i
\(665\) 0 0
\(666\) 0 0
\(667\) −18.0000 31.1769i −0.696963 1.20717i
\(668\) 9.00000 + 15.5885i 0.348220 + 0.603136i
\(669\) 0 0
\(670\) 0 0
\(671\) 3.00000 + 5.19615i 0.115814 + 0.200595i
\(672\) 0 0
\(673\) 17.0000 29.4449i 0.655302 1.13502i −0.326516 0.945192i \(-0.605875\pi\)
0.981818 0.189824i \(-0.0607919\pi\)
\(674\) 1.00000 + 1.73205i 0.0385186 + 0.0667161i
\(675\) 0 0
\(676\) −6.00000 + 10.3923i −0.230769 + 0.399704i
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 42.5000 + 14.7224i 1.63100 + 0.564995i
\(680\) 0 0
\(681\) 0 0
\(682\) −3.00000 5.19615i −0.114876 0.198971i
\(683\) −6.00000 + 10.3923i −0.229584 + 0.397650i −0.957685 0.287819i \(-0.907070\pi\)
0.728101 + 0.685470i \(0.240403\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −10.0000 15.5885i −0.381802 0.595170i
\(687\) 0 0
\(688\) 0.500000 0.866025i 0.0190623 0.0330169i
\(689\) −30.0000 −1.14291
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −0.500000 + 0.866025i −0.0189253 + 0.0327795i
\(699\) 0 0
\(700\) 10.0000 8.66025i 0.377964 0.327327i
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) −2.00000 + 3.46410i −0.0754314 + 0.130651i
\(704\) −3.00000 5.19615i −0.113067 0.195837i
\(705\) 0 0
\(706\) −18.0000 31.1769i −0.677439 1.17336i
\(707\) −24.0000 + 20.7846i −0.902613 + 0.781686i
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.00000 5.19615i 0.112351 0.194597i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 3.00000 + 5.19615i 0.111959 + 0.193919i
\(719\) −6.00000 10.3923i −0.223762 0.387568i 0.732185 0.681106i \(-0.238501\pi\)
−0.955947 + 0.293538i \(0.905167\pi\)
\(720\) 0 0
\(721\) −47.5000 16.4545i −1.76899 0.612797i
\(722\) −1.50000 + 2.59808i −0.0558242 + 0.0966904i
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) −30.0000 −1.11417
\(726\) 0 0
\(727\) −2.50000 + 4.33013i −0.0927199 + 0.160596i −0.908655 0.417548i \(-0.862889\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 10.0000 8.66025i 0.370625 0.320970i
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 + 5.19615i 0.110959 + 0.192187i
\(732\) 0 0
\(733\) −8.50000 + 14.7224i −0.313955 + 0.543785i −0.979215 0.202826i \(-0.934987\pi\)
0.665260 + 0.746612i \(0.268321\pi\)
\(734\) −14.0000 24.2487i −0.516749 0.895036i
\(735\) 0 0
\(736\) 3.00000 5.19615i 0.110581 0.191533i
\(737\) 3.00000 + 5.19615i 0.110506 + 0.191403i
\(738\) 0 0
\(739\) 24.5000 42.4352i 0.901247 1.56101i 0.0753699 0.997156i \(-0.475986\pi\)
0.825877 0.563850i \(-0.190680\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 15.0000 + 5.19615i 0.550667 + 0.190757i
\(743\) −9.00000 15.5885i −0.330178 0.571885i 0.652369 0.757902i \(-0.273775\pi\)
−0.982547 + 0.186017i \(0.940442\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 13.0000 22.5167i 0.475964 0.824394i
\(747\) 0 0
\(748\) 36.0000 1.31629
\(749\) −36.0000 + 31.1769i −1.31541 + 1.13918i
\(750\) 0 0
\(751\) 8.00000 13.8564i 0.291924 0.505627i −0.682341 0.731034i \(-0.739038\pi\)
0.974265 + 0.225407i \(0.0723712\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −30.0000 −1.09254
\(755\) 0 0
\(756\) 0 0
\(757\) 29.0000 1.05402 0.527011 0.849858i \(-0.323312\pi\)
0.527011 + 0.849858i \(0.323312\pi\)
\(758\) 1.00000 0.0363216
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 12.5000 + 4.33013i 0.452530 + 0.156761i
\(764\) 0 0
\(765\) 0 0
\(766\) 3.00000 5.19615i 0.108394 0.187745i
\(767\) 15.0000 + 25.9808i 0.541619 + 0.938111i
\(768\) 0 0
\(769\) 5.00000 + 8.66025i 0.180305 + 0.312297i 0.941984 0.335657i \(-0.108958\pi\)
−0.761680 + 0.647954i \(0.775625\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.00000 −0.0359908
\(773\) 12.0000 20.7846i 0.431610 0.747570i −0.565402 0.824815i \(-0.691279\pi\)
0.997012 + 0.0772449i \(0.0246123\pi\)
\(774\) 0 0
\(775\) −2.50000 4.33013i −0.0898027 0.155543i
\(776\) 8.50000 14.7224i 0.305132 0.528505i
\(777\) 0 0
\(778\) −6.00000 10.3923i −0.215110 0.372582i
\(779\) −12.0000 + 20.7846i −0.429945 + 0.744686i
\(780\) 0 0
\(781\) −36.0000 62.3538i −1.28818 2.23120i
\(782\) 18.0000 + 31.1769i 0.643679 + 1.11488i
\(783\) 0 0
\(784\) −6.50000 + 2.59808i −0.232143 + 0.0927884i
\(785\) 0 0
\(786\) 0 0
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) −12.0000 −0.427482
\(789\) 0 0
\(790\) 0 0
\(791\) −24.0000 + 20.7846i −0.853342 + 0.739016i
\(792\) 0 0
\(793\) 2.50000 + 4.33013i 0.0887776 + 0.153767i
\(794\) −6.50000 11.2583i −0.230676 0.399543i
\(795\) 0 0
\(796\) −5.50000 + 9.52628i −0.194942 + 0.337650i
\(797\) 15.0000 + 25.9808i 0.531327 + 0.920286i 0.999331 + 0.0365596i \(0.0116399\pi\)
−0.468004 + 0.883726i \(0.655027\pi\)
\(798\) 0 0
\(799\) 18.0000 31.1769i 0.636794 1.10296i
\(800\) −2.50000 4.33013i −0.0883883 0.153093i
\(801\) 0 0
\(802\) −3.00000 + 5.19615i −0.105934 + 0.183483i
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) −2.50000 4.33013i −0.0880587 0.152522i
\(807\) 0 0
\(808\) 6.00000 + 10.3923i 0.211079 + 0.365600i
\(809\) 15.0000 25.9808i 0.527372 0.913435i −0.472119 0.881535i \(-0.656511\pi\)
0.999491 0.0319002i \(-0.0101559\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 15.0000 + 5.19615i 0.526397 + 0.182349i
\(813\) 0 0
\(814\) −3.00000 + 5.19615i −0.105150 + 0.182125i
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) −29.0000 −1.01396
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) −9.50000 + 16.4545i −0.330948 + 0.573219i
\(825\) 0 0
\(826\) −3.00000 15.5885i −0.104383 0.542392i
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −7.00000 + 12.1244i −0.243120 + 0.421096i −0.961601 0.274450i \(-0.911504\pi\)
0.718481 + 0.695546i \(0.244838\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.50000 4.33013i −0.0866719 0.150120i
\(833\) 6.00000 41.5692i 0.207888 1.44029i
\(834\) 0 0
\(835\) 0 0
\(836\) 12.0000 20.7846i 0.415029 0.718851i
\(837\) 0 0
\(838\) −12.0000 20.7846i −0.414533 0.717992i
\(839\) −9.00000 + 15.5885i −0.310715 + 0.538173i −0.978517 0.206165i \(-0.933902\pi\)
0.667803 + 0.744338i \(0.267235\pi\)
\(840\) 0 0
\(841\) −3.50000 6.06218i −0.120690 0.209041i
\(842\) 7.00000 12.1244i 0.241236 0.417833i
\(843\) 0 0
\(844\) 6.50000 + 11.2583i 0.223739 + 0.387528i
\(845\) 0 0
\(846\) 0 0
\(847\) 62.5000 + 21.6506i 2.14753 + 0.743925i
\(848\) 3.00000 5.19615i 0.103020 0.178437i
\(849\) 0 0
\(850\) 30.0000 1.02899
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 23.0000 39.8372i 0.787505 1.36400i −0.139986 0.990153i \(-0.544706\pi\)
0.927491 0.373845i \(-0.121961\pi\)
\(854\) −0.500000 2.59808i −0.0171096 0.0889043i
\(855\) 0 0
\(856\) 9.00000 + 15.5885i 0.307614 + 0.532803i
\(857\) 6.00000 + 10.3923i 0.204956 + 0.354994i 0.950119 0.311888i \(-0.100962\pi\)
−0.745163 + 0.666883i \(0.767628\pi\)
\(858\) 0 0
\(859\) −11.5000 + 19.9186i −0.392375 + 0.679613i −0.992762 0.120096i \(-0.961680\pi\)
0.600387 + 0.799709i \(0.295013\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −9.00000 + 15.5885i −0.306541 + 0.530945i
\(863\) −18.0000 31.1769i −0.612727 1.06127i −0.990779 0.135490i \(-0.956739\pi\)
0.378052 0.925785i \(-0.376594\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 25.0000 0.849535
\(867\) 0 0
\(868\) 0.500000 + 2.59808i 0.0169711 + 0.0881845i
\(869\) 3.00000 + 5.19615i 0.101768 + 0.176267i
\(870\) 0 0
\(871\) 2.50000 + 4.33013i 0.0847093 + 0.146721i
\(872\) 2.50000 4.33013i 0.0846607 0.146637i
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 6.50000 11.2583i 0.219489 0.380167i −0.735163 0.677891i \(-0.762894\pi\)
0.954652 + 0.297724i \(0.0962275\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −40.0000 −1.34611 −0.673054 0.739594i \(-0.735018\pi\)
−0.673054 + 0.739594i \(0.735018\pi\)
\(884\) 30.0000 1.00901
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 9.00000 15.5885i 0.302190 0.523409i −0.674441 0.738328i \(-0.735615\pi\)
0.976632 + 0.214919i \(0.0689488\pi\)
\(888\) 0 0
\(889\) 6.50000 + 33.7750i 0.218003 + 1.13278i
\(890\) 0 0
\(891\) 0 0
\(892\) −4.00000 + 6.92820i −0.133930 + 0.231973i
\(893\) −12.0000 20.7846i −0.401565 0.695530i
\(894\) 0 0
\(895\) 0 0
\(896\) 0.500000 + 2.59808i 0.0167038 + 0.0867956i
\(897\) 0 0
\(898\) 24.0000 0.800890
\(899\) 3.00000 5.19615i 0.100056 0.173301i
\(900\) 0 0
\(901\) 18.0000 + 31.1769i 0.599667 + 1.03865i
\(902\) −18.0000 + 31.1769i −0.599334 + 1.03808i
\(903\) 0 0
\(904\) 6.00000 + 10.3923i 0.199557 + 0.345643i
\(905\) 0 0
\(906\) 0 0
\(907\) 18.5000 + 32.0429i 0.614282 + 1.06397i 0.990510 + 0.137441i \(0.0438878\pi\)
−0.376228 + 0.926527i \(0.622779\pi\)
\(908\) 3.00000 + 5.19615i 0.0995585 + 0.172440i
\(909\) 0 0
\(910\) 0 0
\(911\) 3.00000 5.19615i 0.0993944 0.172156i −0.812040 0.583602i \(-0.801643\pi\)
0.911434 + 0.411446i \(0.134976\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 19.0000 0.628464
\(915\) 0 0
\(916\) −2.50000 + 4.33013i −0.0826023 + 0.143071i
\(917\) −30.0000 10.3923i −0.990687 0.343184i
\(918\) 0 0
\(919\) 27.5000 + 47.6314i 0.907141 + 1.57121i 0.818017 + 0.575194i \(0.195074\pi\)
0.0891245 + 0.996020i \(0.471593\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 9.00000 15.5885i 0.296399 0.513378i
\(923\) −30.0000 51.9615i −0.987462 1.71033i
\(924\) 0 0
\(925\) −2.50000 + 4.33013i −0.0821995 + 0.142374i
\(926\) 4.00000 + 6.92820i 0.131448 + 0.227675i
\(927\) 0 0
\(928\) 3.00000 5.19615i 0.0984798 0.170572i
\(929\) 48.0000 1.57483 0.787414 0.616424i \(-0.211419\pi\)
0.787414 + 0.616424i \(0.211419\pi\)
\(930\) 0 0
\(931\) −22.0000 17.3205i −0.721021 0.567657i
\(932\) −12.0000 20.7846i −0.393073 0.680823i
\(933\) 0 0
\(934\) 6.00000 + 10.3923i 0.196326 + 0.340047i
\(935\) 0 0
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) −0.500000 2.59808i −0.0163256 0.0848302i
\(939\) 0 0
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −36.0000 −1.17232
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 10.0000 17.3205i 0.324443 0.561951i
\(951\) 0 0
\(952\) −15.0000 5.19615i −0.486153 0.168408i
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −12.0000 20.7846i −0.388108 0.672222i
\(957\) 0 0
\(958\) 6.00000 + 10.3923i 0.193851 + 0.335760i
\(959\) 12.0000 10.3923i 0.387500 0.335585i
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) −2.50000 + 4.33013i −0.0806032 + 0.139609i
\(963\) 0 0
\(964\) −8.50000 14.7224i −0.273767 0.474178i
\(965\) 0 0
\(966\) 0 0
\(967\) −17.5000 30.3109i −0.562762 0.974732i −0.997254 0.0740568i \(-0.976405\pi\)
0.434492 0.900676i \(-0.356928\pi\)
\(968\) 12.5000 21.6506i 0.401765 0.695878i
\(969\) 0 0
\(970\) 0 0
\(971\) 18.0000 + 31.1769i 0.577647 + 1.00051i 0.995748 + 0.0921142i \(0.0293625\pi\)
−0.418101 + 0.908401i \(0.637304\pi\)
\(972\) 0 0
\(973\) 14.0000 12.1244i 0.448819 0.388689i
\(974\) −20.0000 + 34.6410i −0.640841 + 1.10997i
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −15.0000 25.9808i −0.478669 0.829079i
\(983\) 12.0000 + 20.7846i 0.382741 + 0.662926i 0.991453 0.130465i \(-0.0416470\pi\)
−0.608712 + 0.793391i \(0.708314\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.0000 + 31.1769i 0.573237 + 0.992875i
\(987\) 0 0
\(988\) 10.0000 17.3205i 0.318142 0.551039i
\(989\) 3.00000 + 5.19615i 0.0953945 + 0.165228i
\(990\) 0 0
\(991\) −2.50000 + 4.33013i −0.0794151 + 0.137551i −0.902998 0.429645i \(-0.858639\pi\)
0.823583 + 0.567196i \(0.191972\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) 6.00000 + 31.1769i 0.190308 + 0.988872i
\(995\) 0 0
\(996\) 0 0
\(997\) −11.5000 19.9186i −0.364209 0.630828i 0.624440 0.781073i \(-0.285327\pi\)
−0.988649 + 0.150245i \(0.951994\pi\)
\(998\) −12.5000 + 21.6506i −0.395681 + 0.685339i
\(999\) 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 1134.2.e.c.865.1 2
3.2 odd 2 1134.2.e.m.865.1 2
7.2 even 3 1134.2.h.n.541.1 2
9.2 odd 6 378.2.g.b.109.1 2
9.4 even 3 1134.2.h.n.109.1 2
9.5 odd 6 1134.2.h.d.109.1 2
9.7 even 3 378.2.g.e.109.1 yes 2
21.2 odd 6 1134.2.h.d.541.1 2
63.2 odd 6 378.2.g.b.163.1 yes 2
63.11 odd 6 2646.2.a.x.1.1 1
63.16 even 3 378.2.g.e.163.1 yes 2
63.23 odd 6 1134.2.e.m.919.1 2
63.25 even 3 2646.2.a.h.1.1 1
63.38 even 6 2646.2.a.w.1.1 1
63.52 odd 6 2646.2.a.g.1.1 1
63.58 even 3 inner 1134.2.e.c.919.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.g.b.109.1 2 9.2 odd 6
378.2.g.b.163.1 yes 2 63.2 odd 6
378.2.g.e.109.1 yes 2 9.7 even 3
378.2.g.e.163.1 yes 2 63.16 even 3
1134.2.e.c.865.1 2 1.1 even 1 trivial
1134.2.e.c.919.1 2 63.58 even 3 inner
1134.2.e.m.865.1 2 3.2 odd 2
1134.2.e.m.919.1 2 63.23 odd 6
1134.2.h.d.109.1 2 9.5 odd 6
1134.2.h.d.541.1 2 21.2 odd 6
1134.2.h.n.109.1 2 9.4 even 3
1134.2.h.n.541.1 2 7.2 even 3
2646.2.a.g.1.1 1 63.52 odd 6
2646.2.a.h.1.1 1 63.25 even 3
2646.2.a.w.1.1 1 63.38 even 6
2646.2.a.x.1.1 1 63.11 odd 6