Properties

Label 912.2.q.d.49.1
Level $912$
Weight $2$
Character 912.49
Analytic conductor $7.282$
Analytic rank $0$
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] = [912,2,Mod(49,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.q (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: \(7.28235666434\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 912.49
Dual form 912.2.q.d.577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(2.00000 - 3.46410i) q^{5} +3.00000 q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{3} +(2.00000 - 3.46410i) q^{5} +3.00000 q^{7} +(-0.500000 - 0.866025i) q^{9} -2.00000 q^{11} +(3.50000 + 6.06218i) q^{13} +(2.00000 + 3.46410i) q^{15} +(4.00000 - 1.73205i) q^{19} +(-1.50000 + 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +1.00000 q^{27} +(-2.00000 - 3.46410i) q^{29} -1.00000 q^{31} +(1.00000 - 1.73205i) q^{33} +(6.00000 - 10.3923i) q^{35} +7.00000 q^{37} -7.00000 q^{39} +(-2.00000 + 3.46410i) q^{41} +(3.50000 - 6.06218i) q^{43} -4.00000 q^{45} +(1.00000 + 1.73205i) q^{47} +2.00000 q^{49} +(2.00000 + 3.46410i) q^{53} +(-4.00000 + 6.92820i) q^{55} +(-0.500000 + 4.33013i) q^{57} +(-3.00000 + 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-1.50000 - 2.59808i) q^{63} +28.0000 q^{65} +(1.50000 + 2.59808i) q^{67} +4.00000 q^{69} +(1.00000 - 1.73205i) q^{71} +(1.50000 - 2.59808i) q^{73} +11.0000 q^{75} -6.00000 q^{77} +(2.50000 - 4.33013i) q^{79} +(-0.500000 + 0.866025i) q^{81} +12.0000 q^{83} +4.00000 q^{87} +(-9.00000 - 15.5885i) q^{89} +(10.5000 + 18.1865i) q^{91} +(0.500000 - 0.866025i) q^{93} +(2.00000 - 17.3205i) q^{95} +(-5.00000 + 8.66025i) q^{97} +(1.00000 + 1.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 4 q^{5} + 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 4 q^{5} + 6 q^{7} - q^{9} - 4 q^{11} + 7 q^{13} + 4 q^{15} + 8 q^{19} - 3 q^{21} - 4 q^{23} - 11 q^{25} + 2 q^{27} - 4 q^{29} - 2 q^{31} + 2 q^{33} + 12 q^{35} + 14 q^{37} - 14 q^{39} - 4 q^{41} + 7 q^{43} - 8 q^{45} + 2 q^{47} + 4 q^{49} + 4 q^{53} - 8 q^{55} - q^{57} - 6 q^{59} + q^{61} - 3 q^{63} + 56 q^{65} + 3 q^{67} + 8 q^{69} + 2 q^{71} + 3 q^{73} + 22 q^{75} - 12 q^{77} + 5 q^{79} - q^{81} + 24 q^{83} + 8 q^{87} - 18 q^{89} + 21 q^{91} + q^{93} + 4 q^{95} - 10 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 2.00000 3.46410i 0.894427 1.54919i 0.0599153 0.998203i \(-0.480917\pi\)
0.834512 0.550990i \(-0.185750\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 3.50000 + 6.06218i 0.970725 + 1.68135i 0.693375 + 0.720577i \(0.256123\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.00000 + 3.46410i 0.516398 + 0.894427i
\(16\) 0 0
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 4.00000 1.73205i 0.917663 0.397360i
\(20\) 0 0
\(21\) −1.50000 + 2.59808i −0.327327 + 0.566947i
\(22\) 0 0
\(23\) −2.00000 3.46410i −0.417029 0.722315i 0.578610 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 0 0
\(25\) −5.50000 9.52628i −1.10000 1.90526i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 3.46410i −0.371391 0.643268i 0.618389 0.785872i \(-0.287786\pi\)
−0.989780 + 0.142605i \(0.954452\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 1.00000 1.73205i 0.174078 0.301511i
\(34\) 0 0
\(35\) 6.00000 10.3923i 1.01419 1.75662i
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) −7.00000 −1.12090
\(40\) 0 0
\(41\) −2.00000 + 3.46410i −0.312348 + 0.541002i −0.978870 0.204483i \(-0.934449\pi\)
0.666523 + 0.745485i \(0.267782\pi\)
\(42\) 0 0
\(43\) 3.50000 6.06218i 0.533745 0.924473i −0.465478 0.885059i \(-0.654118\pi\)
0.999223 0.0394140i \(-0.0125491\pi\)
\(44\) 0 0
\(45\) −4.00000 −0.596285
\(46\) 0 0
\(47\) 1.00000 + 1.73205i 0.145865 + 0.252646i 0.929695 0.368329i \(-0.120070\pi\)
−0.783830 + 0.620975i \(0.786737\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 + 3.46410i 0.274721 + 0.475831i 0.970065 0.242846i \(-0.0780811\pi\)
−0.695344 + 0.718677i \(0.744748\pi\)
\(54\) 0 0
\(55\) −4.00000 + 6.92820i −0.539360 + 0.934199i
\(56\) 0 0
\(57\) −0.500000 + 4.33013i −0.0662266 + 0.573539i
\(58\) 0 0
\(59\) −3.00000 + 5.19615i −0.390567 + 0.676481i −0.992524 0.122047i \(-0.961054\pi\)
0.601958 + 0.798528i \(0.294388\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.0640184 + 0.110883i 0.896258 0.443533i \(-0.146275\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) −1.50000 2.59808i −0.188982 0.327327i
\(64\) 0 0
\(65\) 28.0000 3.47297
\(66\) 0 0
\(67\) 1.50000 + 2.59808i 0.183254 + 0.317406i 0.942987 0.332830i \(-0.108004\pi\)
−0.759733 + 0.650236i \(0.774670\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 1.00000 1.73205i 0.118678 0.205557i −0.800566 0.599245i \(-0.795468\pi\)
0.919244 + 0.393688i \(0.128801\pi\)
\(72\) 0 0
\(73\) 1.50000 2.59808i 0.175562 0.304082i −0.764794 0.644275i \(-0.777159\pi\)
0.940356 + 0.340193i \(0.110493\pi\)
\(74\) 0 0
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 2.50000 4.33013i 0.281272 0.487177i −0.690426 0.723403i \(-0.742577\pi\)
0.971698 + 0.236225i \(0.0759104\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −9.00000 15.5885i −0.953998 1.65237i −0.736644 0.676280i \(-0.763591\pi\)
−0.217354 0.976093i \(-0.569742\pi\)
\(90\) 0 0
\(91\) 10.5000 + 18.1865i 1.10070 + 1.90647i
\(92\) 0 0
\(93\) 0.500000 0.866025i 0.0518476 0.0898027i
\(94\) 0 0
\(95\) 2.00000 17.3205i 0.205196 1.77705i
\(96\) 0 0
\(97\) −5.00000 + 8.66025i −0.507673 + 0.879316i 0.492287 + 0.870433i \(0.336161\pi\)
−0.999961 + 0.00888289i \(0.997172\pi\)
\(98\) 0 0
\(99\) 1.00000 + 1.73205i 0.100504 + 0.174078i
\(100\) 0 0
\(101\) −1.00000 1.73205i −0.0995037 0.172345i 0.811976 0.583691i \(-0.198392\pi\)
−0.911479 + 0.411346i \(0.865059\pi\)
\(102\) 0 0
\(103\) −9.00000 −0.886796 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 0 0
\(105\) 6.00000 + 10.3923i 0.585540 + 1.01419i
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) −3.00000 + 5.19615i −0.287348 + 0.497701i −0.973176 0.230063i \(-0.926107\pi\)
0.685828 + 0.727764i \(0.259440\pi\)
\(110\) 0 0
\(111\) −3.50000 + 6.06218i −0.332205 + 0.575396i
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) 0 0
\(117\) 3.50000 6.06218i 0.323575 0.560449i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −2.00000 3.46410i −0.180334 0.312348i
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(128\) 0 0
\(129\) 3.50000 + 6.06218i 0.308158 + 0.533745i
\(130\) 0 0
\(131\) −9.00000 + 15.5885i −0.786334 + 1.36197i 0.141865 + 0.989886i \(0.454690\pi\)
−0.928199 + 0.372084i \(0.878643\pi\)
\(132\) 0 0
\(133\) 12.0000 5.19615i 1.04053 0.450564i
\(134\) 0 0
\(135\) 2.00000 3.46410i 0.172133 0.298142i
\(136\) 0 0
\(137\) 1.00000 + 1.73205i 0.0854358 + 0.147979i 0.905577 0.424182i \(-0.139438\pi\)
−0.820141 + 0.572161i \(0.806105\pi\)
\(138\) 0 0
\(139\) 10.5000 + 18.1865i 0.890598 + 1.54256i 0.839159 + 0.543885i \(0.183047\pi\)
0.0514389 + 0.998676i \(0.483619\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) −7.00000 12.1244i −0.585369 1.01389i
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) 0 0
\(147\) −1.00000 + 1.73205i −0.0824786 + 0.142857i
\(148\) 0 0
\(149\) 9.00000 15.5885i 0.737309 1.27706i −0.216394 0.976306i \(-0.569430\pi\)
0.953703 0.300750i \(-0.0972370\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 + 3.46410i −0.160644 + 0.278243i
\(156\) 0 0
\(157\) −3.50000 + 6.06218i −0.279330 + 0.483814i −0.971219 0.238190i \(-0.923446\pi\)
0.691888 + 0.722005i \(0.256779\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) −6.00000 10.3923i −0.472866 0.819028i
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 0 0
\(165\) −4.00000 6.92820i −0.311400 0.539360i
\(166\) 0 0
\(167\) 3.00000 + 5.19615i 0.232147 + 0.402090i 0.958440 0.285295i \(-0.0920916\pi\)
−0.726293 + 0.687386i \(0.758758\pi\)
\(168\) 0 0
\(169\) −18.0000 + 31.1769i −1.38462 + 2.39822i
\(170\) 0 0
\(171\) −3.50000 2.59808i −0.267652 0.198680i
\(172\) 0 0
\(173\) −6.00000 + 10.3923i −0.456172 + 0.790112i −0.998755 0.0498898i \(-0.984113\pi\)
0.542583 + 0.840002i \(0.317446\pi\)
\(174\) 0 0
\(175\) −16.5000 28.5788i −1.24728 2.16036i
\(176\) 0 0
\(177\) −3.00000 5.19615i −0.225494 0.390567i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 1.00000 + 1.73205i 0.0743294 + 0.128742i 0.900794 0.434246i \(-0.142985\pi\)
−0.826465 + 0.562988i \(0.809652\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 14.0000 24.2487i 1.02930 1.78280i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) 10.5000 18.1865i 0.755807 1.30910i −0.189166 0.981945i \(-0.560578\pi\)
0.944972 0.327150i \(-0.106088\pi\)
\(194\) 0 0
\(195\) −14.0000 + 24.2487i −1.00256 + 1.73649i
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −3.50000 6.06218i −0.248108 0.429736i 0.714893 0.699234i \(-0.246476\pi\)
−0.963001 + 0.269498i \(0.913142\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 0 0
\(203\) −6.00000 10.3923i −0.421117 0.729397i
\(204\) 0 0
\(205\) 8.00000 + 13.8564i 0.558744 + 0.967773i
\(206\) 0 0
\(207\) −2.00000 + 3.46410i −0.139010 + 0.240772i
\(208\) 0 0
\(209\) −8.00000 + 3.46410i −0.553372 + 0.239617i
\(210\) 0 0
\(211\) 4.50000 7.79423i 0.309793 0.536577i −0.668524 0.743690i \(-0.733074\pi\)
0.978317 + 0.207114i \(0.0664070\pi\)
\(212\) 0 0
\(213\) 1.00000 + 1.73205i 0.0685189 + 0.118678i
\(214\) 0 0
\(215\) −14.0000 24.2487i −0.954792 1.65375i
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 1.50000 + 2.59808i 0.101361 + 0.175562i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.50000 4.33013i 0.167412 0.289967i −0.770097 0.637927i \(-0.779792\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) −5.50000 + 9.52628i −0.366667 + 0.635085i
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 3.00000 5.19615i 0.197386 0.341882i
\(232\) 0 0
\(233\) −3.00000 + 5.19615i −0.196537 + 0.340411i −0.947403 0.320043i \(-0.896303\pi\)
0.750867 + 0.660454i \(0.229636\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 2.50000 + 4.33013i 0.162392 + 0.281272i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −0.500000 0.866025i −0.0322078 0.0557856i 0.849472 0.527633i \(-0.176921\pi\)
−0.881680 + 0.471848i \(0.843587\pi\)
\(242\) 0 0
\(243\) −0.500000 0.866025i −0.0320750 0.0555556i
\(244\) 0 0
\(245\) 4.00000 6.92820i 0.255551 0.442627i
\(246\) 0 0
\(247\) 24.5000 + 18.1865i 1.55890 + 1.15718i
\(248\) 0 0
\(249\) −6.00000 + 10.3923i −0.380235 + 0.658586i
\(250\) 0 0
\(251\) 7.00000 + 12.1244i 0.441836 + 0.765283i 0.997826 0.0659066i \(-0.0209939\pi\)
−0.555990 + 0.831189i \(0.687661\pi\)
\(252\) 0 0
\(253\) 4.00000 + 6.92820i 0.251478 + 0.435572i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.0000 + 24.2487i 0.873296 + 1.51259i 0.858567 + 0.512702i \(0.171355\pi\)
0.0147291 + 0.999892i \(0.495311\pi\)
\(258\) 0 0
\(259\) 21.0000 1.30488
\(260\) 0 0
\(261\) −2.00000 + 3.46410i −0.123797 + 0.214423i
\(262\) 0 0
\(263\) −9.00000 + 15.5885i −0.554964 + 0.961225i 0.442943 + 0.896550i \(0.353935\pi\)
−0.997906 + 0.0646755i \(0.979399\pi\)
\(264\) 0 0
\(265\) 16.0000 0.982872
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 0 0
\(269\) −9.00000 + 15.5885i −0.548740 + 0.950445i 0.449622 + 0.893219i \(0.351559\pi\)
−0.998361 + 0.0572259i \(0.981774\pi\)
\(270\) 0 0
\(271\) −4.00000 + 6.92820i −0.242983 + 0.420858i −0.961563 0.274586i \(-0.911459\pi\)
0.718580 + 0.695444i \(0.244792\pi\)
\(272\) 0 0
\(273\) −21.0000 −1.27098
\(274\) 0 0
\(275\) 11.0000 + 19.0526i 0.663325 + 1.14891i
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 0.500000 + 0.866025i 0.0299342 + 0.0518476i
\(280\) 0 0
\(281\) −2.00000 3.46410i −0.119310 0.206651i 0.800184 0.599754i \(-0.204735\pi\)
−0.919494 + 0.393103i \(0.871402\pi\)
\(282\) 0 0
\(283\) −6.00000 + 10.3923i −0.356663 + 0.617758i −0.987401 0.158237i \(-0.949419\pi\)
0.630738 + 0.775996i \(0.282752\pi\)
\(284\) 0 0
\(285\) 14.0000 + 10.3923i 0.829288 + 0.615587i
\(286\) 0 0
\(287\) −6.00000 + 10.3923i −0.354169 + 0.613438i
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) −5.00000 8.66025i −0.293105 0.507673i
\(292\) 0 0
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 12.0000 + 20.7846i 0.698667 + 1.21013i
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) 14.0000 24.2487i 0.809641 1.40234i
\(300\) 0 0
\(301\) 10.5000 18.1865i 0.605210 1.04825i
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 6.00000 10.3923i 0.342438 0.593120i −0.642447 0.766330i \(-0.722081\pi\)
0.984885 + 0.173210i \(0.0554140\pi\)
\(308\) 0 0
\(309\) 4.50000 7.79423i 0.255996 0.443398i
\(310\) 0 0
\(311\) −34.0000 −1.92796 −0.963982 0.265969i \(-0.914308\pi\)
−0.963982 + 0.265969i \(0.914308\pi\)
\(312\) 0 0
\(313\) −7.00000 12.1244i −0.395663 0.685309i 0.597522 0.801852i \(-0.296152\pi\)
−0.993186 + 0.116543i \(0.962819\pi\)
\(314\) 0 0
\(315\) −12.0000 −0.676123
\(316\) 0 0
\(317\) 12.0000 + 20.7846i 0.673987 + 1.16738i 0.976764 + 0.214318i \(0.0687530\pi\)
−0.302777 + 0.953062i \(0.597914\pi\)
\(318\) 0 0
\(319\) 4.00000 + 6.92820i 0.223957 + 0.387905i
\(320\) 0 0
\(321\) 1.00000 1.73205i 0.0558146 0.0966736i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 38.5000 66.6840i 2.13560 3.69896i
\(326\) 0 0
\(327\) −3.00000 5.19615i −0.165900 0.287348i
\(328\) 0 0
\(329\) 3.00000 + 5.19615i 0.165395 + 0.286473i
\(330\) 0 0
\(331\) −23.0000 −1.26419 −0.632097 0.774889i \(-0.717806\pi\)
−0.632097 + 0.774889i \(0.717806\pi\)
\(332\) 0 0
\(333\) −3.50000 6.06218i −0.191799 0.332205i
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −6.50000 + 11.2583i −0.354078 + 0.613280i −0.986960 0.160968i \(-0.948538\pi\)
0.632882 + 0.774248i \(0.281872\pi\)
\(338\) 0 0
\(339\) −1.00000 + 1.73205i −0.0543125 + 0.0940721i
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 8.00000 13.8564i 0.430706 0.746004i
\(346\) 0 0
\(347\) 14.0000 24.2487i 0.751559 1.30174i −0.195507 0.980702i \(-0.562635\pi\)
0.947067 0.321037i \(-0.104031\pi\)
\(348\) 0 0
\(349\) 31.0000 1.65939 0.829696 0.558216i \(-0.188514\pi\)
0.829696 + 0.558216i \(0.188514\pi\)
\(350\) 0 0
\(351\) 3.50000 + 6.06218i 0.186816 + 0.323575i
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) −4.00000 6.92820i −0.212298 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.00000 + 10.3923i −0.316668 + 0.548485i −0.979791 0.200026i \(-0.935897\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(360\) 0 0
\(361\) 13.0000 13.8564i 0.684211 0.729285i
\(362\) 0 0
\(363\) 3.50000 6.06218i 0.183702 0.318182i
\(364\) 0 0
\(365\) −6.00000 10.3923i −0.314054 0.543958i
\(366\) 0 0
\(367\) 9.50000 + 16.4545i 0.495896 + 0.858917i 0.999989 0.00473247i \(-0.00150640\pi\)
−0.504093 + 0.863649i \(0.668173\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) 6.00000 + 10.3923i 0.311504 + 0.539542i
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 12.0000 20.7846i 0.619677 1.07331i
\(376\) 0 0
\(377\) 14.0000 24.2487i 0.721037 1.24887i
\(378\) 0 0
\(379\) 21.0000 1.07870 0.539349 0.842082i \(-0.318670\pi\)
0.539349 + 0.842082i \(0.318670\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.00000 + 12.1244i −0.357683 + 0.619526i −0.987573 0.157159i \(-0.949767\pi\)
0.629890 + 0.776684i \(0.283100\pi\)
\(384\) 0 0
\(385\) −12.0000 + 20.7846i −0.611577 + 1.05928i
\(386\) 0 0
\(387\) −7.00000 −0.355830
\(388\) 0 0
\(389\) 12.0000 + 20.7846i 0.608424 + 1.05382i 0.991500 + 0.130105i \(0.0415314\pi\)
−0.383076 + 0.923717i \(0.625135\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −9.00000 15.5885i −0.453990 0.786334i
\(394\) 0 0
\(395\) −10.0000 17.3205i −0.503155 0.871489i
\(396\) 0 0
\(397\) 6.50000 11.2583i 0.326226 0.565039i −0.655534 0.755166i \(-0.727556\pi\)
0.981760 + 0.190126i \(0.0608897\pi\)
\(398\) 0 0
\(399\) −1.50000 + 12.9904i −0.0750939 + 0.650332i
\(400\) 0 0
\(401\) −5.00000 + 8.66025i −0.249688 + 0.432472i −0.963439 0.267927i \(-0.913661\pi\)
0.713751 + 0.700399i \(0.246995\pi\)
\(402\) 0 0
\(403\) −3.50000 6.06218i −0.174347 0.301979i
\(404\) 0 0
\(405\) 2.00000 + 3.46410i 0.0993808 + 0.172133i
\(406\) 0 0
\(407\) −14.0000 −0.693954
\(408\) 0 0
\(409\) −5.00000 8.66025i −0.247234 0.428222i 0.715523 0.698589i \(-0.246188\pi\)
−0.962757 + 0.270367i \(0.912855\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) −9.00000 + 15.5885i −0.442861 + 0.767058i
\(414\) 0 0
\(415\) 24.0000 41.5692i 1.17811 2.04055i
\(416\) 0 0
\(417\) −21.0000 −1.02837
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −13.0000 + 22.5167i −0.633581 + 1.09739i 0.353233 + 0.935536i \(0.385082\pi\)
−0.986814 + 0.161859i \(0.948251\pi\)
\(422\) 0 0
\(423\) 1.00000 1.73205i 0.0486217 0.0842152i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.50000 + 2.59808i 0.0725901 + 0.125730i
\(428\) 0 0
\(429\) 14.0000 0.675926
\(430\) 0 0
\(431\) 3.00000 + 5.19615i 0.144505 + 0.250290i 0.929188 0.369607i \(-0.120508\pi\)
−0.784683 + 0.619897i \(0.787174\pi\)
\(432\) 0 0
\(433\) −11.5000 19.9186i −0.552655 0.957226i −0.998082 0.0619079i \(-0.980282\pi\)
0.445427 0.895318i \(-0.353052\pi\)
\(434\) 0 0
\(435\) 8.00000 13.8564i 0.383571 0.664364i
\(436\) 0 0
\(437\) −14.0000 10.3923i −0.669711 0.497131i
\(438\) 0 0
\(439\) 11.5000 19.9186i 0.548865 0.950662i −0.449488 0.893287i \(-0.648393\pi\)
0.998353 0.0573756i \(-0.0182733\pi\)
\(440\) 0 0
\(441\) −1.00000 1.73205i −0.0476190 0.0824786i
\(442\) 0 0
\(443\) 2.00000 + 3.46410i 0.0950229 + 0.164584i 0.909618 0.415445i \(-0.136374\pi\)
−0.814595 + 0.580030i \(0.803041\pi\)
\(444\) 0 0
\(445\) −72.0000 −3.41313
\(446\) 0 0
\(447\) 9.00000 + 15.5885i 0.425685 + 0.737309i
\(448\) 0 0
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 0 0
\(451\) 4.00000 6.92820i 0.188353 0.326236i
\(452\) 0 0
\(453\) 10.0000 17.3205i 0.469841 0.813788i
\(454\) 0 0
\(455\) 84.0000 3.93798
\(456\) 0 0
\(457\) 29.0000 1.35656 0.678281 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17.0000 29.4449i 0.791769 1.37138i −0.133102 0.991102i \(-0.542494\pi\)
0.924871 0.380282i \(-0.124173\pi\)
\(462\) 0 0
\(463\) −19.0000 −0.883005 −0.441502 0.897260i \(-0.645554\pi\)
−0.441502 + 0.897260i \(0.645554\pi\)
\(464\) 0 0
\(465\) −2.00000 3.46410i −0.0927478 0.160644i
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 4.50000 + 7.79423i 0.207791 + 0.359904i
\(470\) 0 0
\(471\) −3.50000 6.06218i −0.161271 0.279330i
\(472\) 0 0
\(473\) −7.00000 + 12.1244i −0.321860 + 0.557478i
\(474\) 0 0
\(475\) −38.5000 28.5788i −1.76650 1.31129i
\(476\) 0 0
\(477\) 2.00000 3.46410i 0.0915737 0.158610i
\(478\) 0 0
\(479\) 17.0000 + 29.4449i 0.776750 + 1.34537i 0.933806 + 0.357780i \(0.116466\pi\)
−0.157056 + 0.987590i \(0.550200\pi\)
\(480\) 0 0
\(481\) 24.5000 + 42.4352i 1.11710 + 1.93488i
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) 20.0000 + 34.6410i 0.908153 + 1.57297i
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 5.50000 9.52628i 0.248719 0.430793i
\(490\) 0 0
\(491\) 13.0000 22.5167i 0.586682 1.01616i −0.407982 0.912990i \(-0.633767\pi\)
0.994663 0.103173i \(-0.0328994\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 3.00000 5.19615i 0.134568 0.233079i
\(498\) 0 0
\(499\) −5.50000 + 9.52628i −0.246214 + 0.426455i −0.962472 0.271380i \(-0.912520\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) −6.00000 10.3923i −0.267527 0.463370i 0.700696 0.713460i \(-0.252873\pi\)
−0.968223 + 0.250090i \(0.919540\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) −18.0000 31.1769i −0.799408 1.38462i
\(508\) 0 0
\(509\) 3.00000 + 5.19615i 0.132973 + 0.230315i 0.924821 0.380402i \(-0.124214\pi\)
−0.791849 + 0.610718i \(0.790881\pi\)
\(510\) 0 0
\(511\) 4.50000 7.79423i 0.199068 0.344796i
\(512\) 0 0
\(513\) 4.00000 1.73205i 0.176604 0.0764719i
\(514\) 0 0
\(515\) −18.0000 + 31.1769i −0.793175 + 1.37382i
\(516\) 0 0
\(517\) −2.00000 3.46410i −0.0879599 0.152351i
\(518\) 0 0
\(519\) −6.00000 10.3923i −0.263371 0.456172i
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) 14.5000 + 25.1147i 0.634041 + 1.09819i 0.986718 + 0.162446i \(0.0519382\pi\)
−0.352677 + 0.935745i \(0.614728\pi\)
\(524\) 0 0
\(525\) 33.0000 1.44024
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −28.0000 −1.21281
\(534\) 0 0
\(535\) −4.00000 + 6.92820i −0.172935 + 0.299532i
\(536\) 0 0
\(537\) −6.00000 + 10.3923i −0.258919 + 0.448461i
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 9.50000 + 16.4545i 0.408437 + 0.707433i 0.994715 0.102677i \(-0.0327407\pi\)
−0.586278 + 0.810110i \(0.699407\pi\)
\(542\) 0 0
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 12.0000 + 20.7846i 0.514024 + 0.890315i
\(546\) 0 0
\(547\) −14.5000 25.1147i −0.619975 1.07383i −0.989490 0.144604i \(-0.953809\pi\)
0.369514 0.929225i \(-0.379524\pi\)
\(548\) 0 0
\(549\) 0.500000 0.866025i 0.0213395 0.0369611i
\(550\) 0 0
\(551\) −14.0000 10.3923i −0.596420 0.442727i
\(552\) 0 0
\(553\) 7.50000 12.9904i 0.318932 0.552407i
\(554\) 0 0
\(555\) 14.0000 + 24.2487i 0.594267 + 1.02930i
\(556\) 0 0
\(557\) −11.0000 19.0526i −0.466085 0.807283i 0.533165 0.846011i \(-0.321003\pi\)
−0.999250 + 0.0387286i \(0.987669\pi\)
\(558\) 0 0
\(559\) 49.0000 2.07248
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 0 0
\(565\) 4.00000 6.92820i 0.168281 0.291472i
\(566\) 0 0
\(567\) −1.50000 + 2.59808i −0.0629941 + 0.109109i
\(568\) 0 0
\(569\) 16.0000 0.670755 0.335377 0.942084i \(-0.391136\pi\)
0.335377 + 0.942084i \(0.391136\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 0 0
\(573\) −10.0000 + 17.3205i −0.417756 + 0.723575i
\(574\) 0 0
\(575\) −22.0000 + 38.1051i −0.917463 + 1.58909i
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 10.5000 + 18.1865i 0.436365 + 0.755807i
\(580\) 0 0
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) −4.00000 6.92820i −0.165663 0.286937i
\(584\) 0 0
\(585\) −14.0000 24.2487i −0.578829 1.00256i
\(586\) 0 0
\(587\) −9.00000 + 15.5885i −0.371470 + 0.643404i −0.989792 0.142520i \(-0.954479\pi\)
0.618322 + 0.785925i \(0.287813\pi\)
\(588\) 0 0
\(589\) −4.00000 + 1.73205i −0.164817 + 0.0713679i
\(590\) 0 0
\(591\) 5.00000 8.66025i 0.205673 0.356235i
\(592\) 0 0
\(593\) −17.0000 29.4449i −0.698106 1.20916i −0.969122 0.246581i \(-0.920693\pi\)
0.271016 0.962575i \(-0.412640\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.00000 0.286491
\(598\) 0 0
\(599\) 15.0000 + 25.9808i 0.612883 + 1.06155i 0.990752 + 0.135686i \(0.0433238\pi\)
−0.377869 + 0.925859i \(0.623343\pi\)
\(600\) 0 0
\(601\) 19.0000 0.775026 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) 0 0
\(603\) 1.50000 2.59808i 0.0610847 0.105802i
\(604\) 0 0
\(605\) −14.0000 + 24.2487i −0.569181 + 0.985850i
\(606\) 0 0
\(607\) −41.0000 −1.66414 −0.832069 0.554672i \(-0.812844\pi\)
−0.832069 + 0.554672i \(0.812844\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −7.00000 + 12.1244i −0.283190 + 0.490499i
\(612\) 0 0
\(613\) 19.0000 32.9090i 0.767403 1.32918i −0.171564 0.985173i \(-0.554882\pi\)
0.938967 0.344008i \(-0.111785\pi\)
\(614\) 0 0
\(615\) −16.0000 −0.645182
\(616\) 0 0
\(617\) −9.00000 15.5885i −0.362326 0.627568i 0.626017 0.779809i \(-0.284684\pi\)
−0.988343 + 0.152242i \(0.951351\pi\)
\(618\) 0 0
\(619\) 23.0000 0.924448 0.462224 0.886763i \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(620\) 0 0
\(621\) −2.00000 3.46410i −0.0802572 0.139010i
\(622\) 0 0
\(623\) −27.0000 46.7654i −1.08173 1.87362i
\(624\) 0 0
\(625\) −20.5000 + 35.5070i −0.820000 + 1.42028i
\(626\) 0 0
\(627\) 1.00000 8.66025i 0.0399362 0.345857i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −9.50000 16.4545i −0.378189 0.655043i 0.612610 0.790386i \(-0.290120\pi\)
−0.990799 + 0.135343i \(0.956786\pi\)
\(632\) 0 0
\(633\) 4.50000 + 7.79423i 0.178859 + 0.309793i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 7.00000 + 12.1244i 0.277350 + 0.480384i
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −9.00000 + 15.5885i −0.355479 + 0.615707i −0.987200 0.159489i \(-0.949015\pi\)
0.631721 + 0.775196i \(0.282349\pi\)
\(642\) 0 0
\(643\) 6.50000 11.2583i 0.256335 0.443985i −0.708922 0.705287i \(-0.750818\pi\)
0.965257 + 0.261301i \(0.0841516\pi\)
\(644\) 0 0
\(645\) 28.0000 1.10250
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 6.00000 10.3923i 0.235521 0.407934i
\(650\) 0 0
\(651\) 1.50000 2.59808i 0.0587896 0.101827i
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 36.0000 + 62.3538i 1.40664 + 2.43637i
\(656\) 0 0
\(657\) −3.00000 −0.117041
\(658\) 0 0
\(659\) −5.00000 8.66025i −0.194772 0.337356i 0.752054 0.659102i \(-0.229063\pi\)
−0.946826 + 0.321746i \(0.895730\pi\)
\(660\) 0 0
\(661\) −5.00000 8.66025i −0.194477 0.336845i 0.752252 0.658876i \(-0.228968\pi\)
−0.946729 + 0.322031i \(0.895634\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 51.9615i 0.232670 2.01498i
\(666\) 0 0
\(667\) −8.00000 + 13.8564i −0.309761 + 0.536522i
\(668\) 0 0
\(669\) 2.50000 + 4.33013i 0.0966556 + 0.167412i
\(670\) 0 0
\(671\) −1.00000 1.73205i −0.0386046 0.0668651i
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) −5.50000 9.52628i −0.211695 0.366667i
\(676\) 0 0
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) −15.0000 + 25.9808i −0.575647 + 0.997050i
\(680\) 0 0
\(681\) 2.00000 3.46410i 0.0766402 0.132745i
\(682\) 0 0
\(683\) −40.0000 −1.53056 −0.765279 0.643699i \(-0.777399\pi\)
−0.765279 + 0.643699i \(0.777399\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) 3.50000 6.06218i 0.133533 0.231287i
\(688\) 0 0
\(689\) −14.0000 + 24.2487i −0.533358 + 0.923802i
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 0 0
\(693\) 3.00000 + 5.19615i 0.113961 + 0.197386i
\(694\) 0 0
\(695\) 84.0000 3.18630
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −3.00000 5.19615i −0.113470 0.196537i
\(700\) 0 0
\(701\) −4.00000 + 6.92820i −0.151078 + 0.261675i −0.931624 0.363424i \(-0.881608\pi\)
0.780546 + 0.625098i \(0.214941\pi\)
\(702\) 0 0
\(703\) 28.0000 12.1244i 1.05604 0.457279i
\(704\) 0 0
\(705\) −4.00000 + 6.92820i −0.150649 + 0.260931i
\(706\) 0 0
\(707\) −3.00000 5.19615i −0.112827 0.195421i
\(708\) 0 0
\(709\) 1.50000 + 2.59808i 0.0563337 + 0.0975728i 0.892817 0.450420i \(-0.148726\pi\)
−0.836483 + 0.547992i \(0.815392\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 0 0
\(713\) 2.00000 + 3.46410i 0.0749006 + 0.129732i
\(714\) 0 0
\(715\) −56.0000 −2.09428
\(716\) 0 0
\(717\) 12.0000 20.7846i 0.448148 0.776215i
\(718\) 0 0
\(719\) 8.00000 13.8564i 0.298350 0.516757i −0.677409 0.735607i \(-0.736897\pi\)
0.975759 + 0.218850i \(0.0702305\pi\)
\(720\) 0 0
\(721\) −27.0000 −1.00553
\(722\) 0 0
\(723\) 1.00000 0.0371904
\(724\) 0 0
\(725\) −22.0000 + 38.1051i −0.817059 + 1.41519i
\(726\) 0 0
\(727\) 18.5000 32.0429i 0.686127 1.18841i −0.286954 0.957944i \(-0.592643\pi\)
0.973081 0.230463i \(-0.0740239\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 0 0
\(735\) 4.00000 + 6.92820i 0.147542 + 0.255551i
\(736\) 0 0
\(737\) −3.00000 5.19615i −0.110506 0.191403i
\(738\) 0 0
\(739\) 17.5000 30.3109i 0.643748 1.11500i −0.340841 0.940121i \(-0.610712\pi\)
0.984589 0.174883i \(-0.0559548\pi\)
\(740\) 0 0
\(741\) −28.0000 + 12.1244i −1.02861 + 0.445399i
\(742\) 0 0
\(743\) 13.0000 22.5167i 0.476924 0.826056i −0.522727 0.852500i \(-0.675085\pi\)
0.999650 + 0.0264443i \(0.00841845\pi\)
\(744\) 0 0
\(745\) −36.0000 62.3538i −1.31894 2.28447i
\(746\) 0 0
\(747\) −6.00000 10.3923i −0.219529 0.380235i
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −15.5000 26.8468i −0.565603 0.979653i −0.996993 0.0774878i \(-0.975310\pi\)
0.431390 0.902165i \(-0.358023\pi\)
\(752\) 0 0
\(753\) −14.0000 −0.510188
\(754\) 0 0
\(755\) −40.0000 + 69.2820i −1.45575 + 2.52143i
\(756\) 0 0
\(757\) 11.5000 19.9186i 0.417975 0.723953i −0.577761 0.816206i \(-0.696073\pi\)
0.995736 + 0.0922527i \(0.0294068\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) −9.00000 + 15.5885i −0.325822 + 0.564340i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −42.0000 −1.51653
\(768\) 0 0
\(769\) 2.50000 + 4.33013i 0.0901523 + 0.156148i 0.907575 0.419890i \(-0.137931\pi\)
−0.817423 + 0.576038i \(0.804598\pi\)
\(770\) 0 0
\(771\) −28.0000 −1.00840
\(772\) 0 0
\(773\) −3.00000 5.19615i −0.107903 0.186893i 0.807018 0.590527i \(-0.201080\pi\)
−0.914920 + 0.403634i \(0.867747\pi\)
\(774\) 0 0
\(775\) 5.50000 + 9.52628i 0.197566 + 0.342194i
\(776\) 0 0
\(777\) −10.5000 + 18.1865i −0.376685 + 0.652438i
\(778\) 0 0
\(779\) −2.00000 + 17.3205i −0.0716574 + 0.620572i
\(780\) 0 0
\(781\) −2.00000 + 3.46410i −0.0715656 + 0.123955i
\(782\) 0 0
\(783\) −2.00000 3.46410i −0.0714742 0.123797i
\(784\) 0 0
\(785\) 14.0000 + 24.2487i 0.499681 + 0.865474i
\(786\) 0 0
\(787\) −49.0000 −1.74666 −0.873331 0.487128i \(-0.838045\pi\)
−0.873331 + 0.487128i \(0.838045\pi\)
\(788\) 0 0
\(789\) −9.00000 15.5885i −0.320408 0.554964i
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −3.50000 + 6.06218i −0.124289 + 0.215274i
\(794\) 0 0
\(795\) −8.00000 + 13.8564i −0.283731 + 0.491436i
\(796\) 0 0
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −9.00000 + 15.5885i −0.317999 + 0.550791i
\(802\) 0 0
\(803\) −3.00000 + 5.19615i −0.105868 + 0.183368i
\(804\) 0 0
\(805\) −48.0000 −1.69178
\(806\) 0 0
\(807\) −9.00000 15.5885i −0.316815 0.548740i
\(808\) 0 0
\(809\) 36.0000 1.26569 0.632846 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) 0 0
\(811\) 18.0000 + 31.1769i 0.632065 + 1.09477i 0.987129 + 0.159927i \(0.0511260\pi\)
−0.355063 + 0.934842i \(0.615541\pi\)
\(812\) 0 0
\(813\) −4.00000 6.92820i −0.140286 0.242983i
\(814\) 0 0
\(815\) −22.0000 + 38.1051i −0.770626 + 1.33476i
\(816\) 0 0
\(817\) 3.50000 30.3109i 0.122449 1.06044i
\(818\) 0 0
\(819\) 10.5000 18.1865i 0.366900 0.635489i
\(820\) 0 0
\(821\) −6.00000 10.3923i −0.209401 0.362694i 0.742125 0.670262i \(-0.233818\pi\)
−0.951526 + 0.307568i \(0.900485\pi\)
\(822\) 0 0
\(823\) 22.0000 + 38.1051i 0.766872 + 1.32826i 0.939251 + 0.343230i \(0.111521\pi\)
−0.172379 + 0.985031i \(0.555146\pi\)
\(824\) 0 0
\(825\) −22.0000 −0.765942
\(826\) 0 0
\(827\) 4.00000 + 6.92820i 0.139094 + 0.240917i 0.927154 0.374681i \(-0.122248\pi\)
−0.788060 + 0.615598i \(0.788914\pi\)
\(828\) 0 0
\(829\) 45.0000 1.56291 0.781457 0.623959i \(-0.214477\pi\)
0.781457 + 0.623959i \(0.214477\pi\)
\(830\) 0 0
\(831\) 13.0000 22.5167i 0.450965 0.781094i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −20.0000 + 34.6410i −0.690477 + 1.19594i 0.281205 + 0.959648i \(0.409266\pi\)
−0.971682 + 0.236293i \(0.924067\pi\)
\(840\) 0 0
\(841\) 6.50000 11.2583i 0.224138 0.388218i
\(842\) 0 0
\(843\) 4.00000 0.137767
\(844\) 0 0
\(845\) 72.0000 + 124.708i 2.47688 + 4.29007i
\(846\) 0 0
\(847\) −21.0000 −0.721569
\(848\) 0 0
\(849\) −6.00000 10.3923i −0.205919 0.356663i
\(850\) 0 0
\(851\) −14.0000 24.2487i −0.479914 0.831235i
\(852\) 0 0
\(853\) 4.50000 7.79423i 0.154077 0.266869i −0.778646 0.627464i \(-0.784093\pi\)
0.932723 + 0.360595i \(0.117426\pi\)
\(854\) 0 0
\(855\) −16.0000 + 6.92820i −0.547188 + 0.236940i
\(856\) 0 0
\(857\) 3.00000 5.19615i 0.102478 0.177497i −0.810227 0.586116i \(-0.800656\pi\)
0.912705 + 0.408619i \(0.133990\pi\)
\(858\) 0 0
\(859\) 3.50000 + 6.06218i 0.119418 + 0.206839i 0.919537 0.393003i \(-0.128564\pi\)
−0.800119 + 0.599841i \(0.795230\pi\)
\(860\) 0 0
\(861\) −6.00000 10.3923i −0.204479 0.354169i
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) 24.0000 + 41.5692i 0.816024 + 1.41340i
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) −5.00000 + 8.66025i −0.169613 + 0.293779i
\(870\) 0 0
\(871\) −10.5000 + 18.1865i −0.355779 + 0.616227i
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) −72.0000 −2.43404
\(876\) 0 0
\(877\) −9.50000 + 16.4545i −0.320792 + 0.555628i −0.980652 0.195761i \(-0.937282\pi\)
0.659860 + 0.751389i \(0.270616\pi\)
\(878\) 0 0
\(879\) 9.00000 15.5885i 0.303562 0.525786i
\(880\) 0 0
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0 0
\(883\) −20.5000 35.5070i −0.689880 1.19491i −0.971876 0.235492i \(-0.924330\pi\)
0.281996 0.959415i \(-0.409003\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 0 0
\(887\) 5.00000 + 8.66025i 0.167884 + 0.290783i 0.937676 0.347512i \(-0.112973\pi\)
−0.769792 + 0.638295i \(0.779640\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 1.73205i 0.0335013 0.0580259i
\(892\) 0 0
\(893\) 7.00000 + 5.19615i 0.234246 + 0.173883i
\(894\) 0 0
\(895\) 24.0000 41.5692i 0.802232 1.38951i
\(896\) 0 0
\(897\) 14.0000 + 24.2487i 0.467446 + 0.809641i
\(898\) 0 0
\(899\) 2.00000 + 3.46410i 0.0667037 + 0.115534i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 10.5000 + 18.1865i 0.349418 + 0.605210i
\(904\) 0 0
\(905\) 8.00000 0.265929
\(906\) 0 0
\(907\) 10.0000 17.3205i 0.332045 0.575118i −0.650868 0.759191i \(-0.725595\pi\)
0.982913 + 0.184073i \(0.0589282\pi\)
\(908\) 0 0
\(909\) −1.00000 + 1.73205i −0.0331679 + 0.0574485i
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −24.0000 −0.794284
\(914\) 0 0
\(915\) −2.00000 + 3.46410i −0.0661180 + 0.114520i
\(916\) 0 0
\(917\) −27.0000 + 46.7654i −0.891619 + 1.54433i
\(918\) 0 0
\(919\) −11.0000 −0.362857 −0.181428 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 0 0
\(921\) 6.00000 + 10.3923i 0.197707 + 0.342438i
\(922\) 0 0
\(923\) 14.0000 0.460816
\(924\) 0 0
\(925\) −38.5000 66.6840i −1.26587 2.19255i
\(926\) 0 0
\(927\) 4.50000 + 7.79423i 0.147799 + 0.255996i
\(928\) 0 0
\(929\) −4.00000 + 6.92820i −0.131236 + 0.227307i −0.924153 0.382022i \(-0.875228\pi\)
0.792917 + 0.609329i \(0.208561\pi\)
\(930\) 0 0
\(931\) 8.00000 3.46410i 0.262189 0.113531i
\(932\) 0 0
\(933\) 17.0000 29.4449i 0.556555 0.963982i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 12.5000 + 21.6506i 0.408357 + 0.707295i 0.994706 0.102763i \(-0.0327685\pi\)
−0.586349 + 0.810059i \(0.699435\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) −8.00000 13.8564i −0.260793 0.451706i 0.705660 0.708550i \(-0.250651\pi\)
−0.966453 + 0.256844i \(0.917317\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 6.00000 10.3923i 0.195180 0.338062i
\(946\) 0 0
\(947\) 27.0000 46.7654i 0.877382 1.51967i 0.0231788 0.999731i \(-0.492621\pi\)
0.854203 0.519939i \(-0.174045\pi\)
\(948\) 0 0
\(949\) 21.0000 0.681689
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 5.00000 8.66025i 0.161966 0.280533i −0.773608 0.633665i \(-0.781550\pi\)
0.935574 + 0.353132i \(0.114883\pi\)
\(954\) 0 0
\(955\) 40.0000 69.2820i 1.29437 2.24191i
\(956\) 0 0
\(957\) −8.00000 −0.258603
\(958\) 0 0
\(959\) 3.00000 + 5.19615i 0.0968751 + 0.167793i
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 1.00000 + 1.73205i 0.0322245 + 0.0558146i
\(964\) 0 0
\(965\) −42.0000 72.7461i −1.35203 2.34178i
\(966\) 0 0
\(967\) −4.50000 + 7.79423i −0.144710 + 0.250645i −0.929265 0.369414i \(-0.879558\pi\)
0.784555 + 0.620060i \(0.212892\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.00000 12.1244i 0.224641 0.389089i −0.731571 0.681765i \(-0.761212\pi\)
0.956212 + 0.292676i \(0.0945458\pi\)
\(972\) 0 0
\(973\) 31.5000 + 54.5596i 1.00984 + 1.74910i
\(974\) 0 0
\(975\) 38.5000 + 66.6840i 1.23299 + 2.13560i
\(976\) 0 0
\(977\) 14.0000 0.447900 0.223950 0.974601i \(-0.428105\pi\)
0.223950 + 0.974601i \(0.428105\pi\)
\(978\) 0 0
\(979\) 18.0000 + 31.1769i 0.575282 + 0.996419i
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 0 0
\(983\) 3.00000 5.19615i 0.0956851 0.165732i −0.814209 0.580572i \(-0.802829\pi\)
0.909894 + 0.414840i \(0.136162\pi\)
\(984\) 0 0
\(985\) −20.0000 + 34.6410i −0.637253 + 1.10375i
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) −28.0000 −0.890348
\(990\) 0 0
\(991\) −14.5000 + 25.1147i −0.460608 + 0.797796i −0.998991 0.0449040i \(-0.985702\pi\)
0.538384 + 0.842700i \(0.319035\pi\)
\(992\) 0 0
\(993\) 11.5000 19.9186i 0.364941 0.632097i
\(994\) 0 0
\(995\) −28.0000 −0.887660
\(996\) 0 0
\(997\) 26.5000 + 45.8993i 0.839263 + 1.45365i 0.890511 + 0.454961i \(0.150347\pi\)
−0.0512480 + 0.998686i \(0.516320\pi\)
\(998\) 0 0
\(999\) 7.00000 0.221470
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 912.2.q.d.49.1 2
3.2 odd 2 2736.2.s.c.1873.1 2
4.3 odd 2 114.2.e.a.49.1 yes 2
12.11 even 2 342.2.g.d.163.1 2
19.7 even 3 inner 912.2.q.d.577.1 2
57.26 odd 6 2736.2.s.c.577.1 2
76.7 odd 6 114.2.e.a.7.1 2
76.11 odd 6 2166.2.a.f.1.1 1
76.27 even 6 2166.2.a.c.1.1 1
228.11 even 6 6498.2.a.l.1.1 1
228.83 even 6 342.2.g.d.235.1 2
228.179 odd 6 6498.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.e.a.7.1 2 76.7 odd 6
114.2.e.a.49.1 yes 2 4.3 odd 2
342.2.g.d.163.1 2 12.11 even 2
342.2.g.d.235.1 2 228.83 even 6
912.2.q.d.49.1 2 1.1 even 1 trivial
912.2.q.d.577.1 2 19.7 even 3 inner
2166.2.a.c.1.1 1 76.27 even 6
2166.2.a.f.1.1 1 76.11 odd 6
2736.2.s.c.577.1 2 57.26 odd 6
2736.2.s.c.1873.1 2 3.2 odd 2
6498.2.a.l.1.1 1 228.11 even 6
6498.2.a.x.1.1 1 228.179 odd 6