Properties

Label 2268.2.i.j.865.3
Level $2268$
Weight $2$
Character 2268.865
Analytic conductor $18.110$
Analytic rank $0$
Dimension $6$
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] = [2268,2,Mod(865,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("2268.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.i (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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.3
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 2268.865
Dual form 2268.2.i.j.2053.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.21053 - 2.09671i) q^{5} +(-2.56238 + 0.658939i) q^{7} +O(q^{10})\) \(q+(1.21053 - 2.09671i) q^{5} +(-2.56238 + 0.658939i) q^{7} +(-2.35185 - 4.07352i) q^{11} +(1.71053 + 2.96273i) q^{13} +(-0.851848 + 1.47544i) q^{17} +(-0.641315 - 1.11079i) q^{19} +(0.562382 - 0.974074i) q^{23} +(-0.430782 - 0.746136i) q^{25} +(2.35185 - 4.07352i) q^{29} -3.42107 q^{31} +(-1.72025 + 6.17023i) q^{35} +(-4.27292 - 7.40091i) q^{37} +(-1.85868 - 3.21934i) q^{41} +(-2.77292 + 4.80283i) q^{43} -11.8285 q^{47} +(6.13160 - 3.37690i) q^{49} +(-5.13160 + 8.88819i) q^{53} -11.3880 q^{55} +4.12476 q^{59} -9.24953 q^{61} +8.28263 q^{65} -11.1248 q^{67} +14.9669 q^{71} +(-1.06922 + 1.85194i) q^{73} +(8.71053 + 8.88819i) q^{77} -14.5322 q^{79} +(-4.21053 + 7.29286i) q^{83} +(2.06238 + 3.57215i) q^{85} +(-8.04583 - 13.9358i) q^{89} +(-6.33530 - 6.46451i) q^{91} -3.10533 q^{95} +(-8.12476 + 14.0725i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{5} + 2 q^{7} - 5 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} - 14 q^{23} - 10 q^{25} + 5 q^{29} - 4 q^{31} - 26 q^{35} - 12 q^{41} + 9 q^{43} - 18 q^{47} + 12 q^{49} - 6 q^{53} + 16 q^{55} - 10 q^{59} + 14 q^{61} + 48 q^{65} - 32 q^{67} + 22 q^{71} + q^{73} + 44 q^{77} - 16 q^{79} - 17 q^{83} - 5 q^{85} + 3 q^{89} + 5 q^{91} + 64 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.21053 2.09671i 0.541367 0.937675i −0.457459 0.889231i \(-0.651240\pi\)
0.998826 0.0484443i \(-0.0154263\pi\)
\(6\) 0 0
\(7\) −2.56238 + 0.658939i −0.968489 + 0.249055i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.35185 4.07352i −0.709109 1.22821i −0.965188 0.261557i \(-0.915764\pi\)
0.256079 0.966656i \(-0.417569\pi\)
\(12\) 0 0
\(13\) 1.71053 + 2.96273i 0.474417 + 0.821714i 0.999571 0.0292934i \(-0.00932572\pi\)
−0.525154 + 0.851007i \(0.675992\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.851848 + 1.47544i −0.206604 + 0.357848i −0.950642 0.310288i \(-0.899574\pi\)
0.744039 + 0.668136i \(0.232908\pi\)
\(18\) 0 0
\(19\) −0.641315 1.11079i −0.147128 0.254833i 0.783037 0.621975i \(-0.213670\pi\)
−0.930165 + 0.367142i \(0.880336\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.562382 0.974074i 0.117265 0.203108i −0.801418 0.598105i \(-0.795921\pi\)
0.918683 + 0.394996i \(0.129254\pi\)
\(24\) 0 0
\(25\) −0.430782 0.746136i −0.0861564 0.149227i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.35185 4.07352i 0.436727 0.756434i −0.560708 0.828014i \(-0.689471\pi\)
0.997435 + 0.0715801i \(0.0228041\pi\)
\(30\) 0 0
\(31\) −3.42107 −0.614442 −0.307221 0.951638i \(-0.599399\pi\)
−0.307221 + 0.951638i \(0.599399\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.72025 + 6.17023i −0.290775 + 1.04296i
\(36\) 0 0
\(37\) −4.27292 7.40091i −0.702463 1.21670i −0.967599 0.252491i \(-0.918750\pi\)
0.265136 0.964211i \(-0.414583\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.85868 3.21934i −0.290278 0.502776i 0.683598 0.729859i \(-0.260414\pi\)
−0.973875 + 0.227083i \(0.927081\pi\)
\(42\) 0 0
\(43\) −2.77292 + 4.80283i −0.422866 + 0.732425i −0.996218 0.0868839i \(-0.972309\pi\)
0.573353 + 0.819309i \(0.305642\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.8285 −1.72536 −0.862679 0.505752i \(-0.831215\pi\)
−0.862679 + 0.505752i \(0.831215\pi\)
\(48\) 0 0
\(49\) 6.13160 3.37690i 0.875943 0.482415i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.13160 + 8.88819i −0.704879 + 1.22089i 0.261856 + 0.965107i \(0.415666\pi\)
−0.966735 + 0.255780i \(0.917668\pi\)
\(54\) 0 0
\(55\) −11.3880 −1.53555
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.12476 0.536998 0.268499 0.963280i \(-0.413472\pi\)
0.268499 + 0.963280i \(0.413472\pi\)
\(60\) 0 0
\(61\) −9.24953 −1.18428 −0.592140 0.805835i \(-0.701717\pi\)
−0.592140 + 0.805835i \(0.701717\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.28263 1.02733
\(66\) 0 0
\(67\) −11.1248 −1.35911 −0.679553 0.733626i \(-0.737826\pi\)
−0.679553 + 0.733626i \(0.737826\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.9669 1.77624 0.888122 0.459608i \(-0.152010\pi\)
0.888122 + 0.459608i \(0.152010\pi\)
\(72\) 0 0
\(73\) −1.06922 + 1.85194i −0.125143 + 0.216753i −0.921789 0.387693i \(-0.873272\pi\)
0.796646 + 0.604446i \(0.206605\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.71053 + 8.88819i 0.992657 + 1.01290i
\(78\) 0 0
\(79\) −14.5322 −1.63500 −0.817498 0.575932i \(-0.804639\pi\)
−0.817498 + 0.575932i \(0.804639\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.21053 + 7.29286i −0.462166 + 0.800495i −0.999069 0.0431491i \(-0.986261\pi\)
0.536903 + 0.843644i \(0.319594\pi\)
\(84\) 0 0
\(85\) 2.06238 + 3.57215i 0.223697 + 0.387454i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.04583 13.9358i −0.852856 1.47719i −0.878620 0.477522i \(-0.841535\pi\)
0.0257633 0.999668i \(-0.491798\pi\)
\(90\) 0 0
\(91\) −6.33530 6.46451i −0.664120 0.677665i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.10533 −0.318600
\(96\) 0 0
\(97\) −8.12476 + 14.0725i −0.824945 + 1.42885i 0.0770168 + 0.997030i \(0.475460\pi\)
−0.901962 + 0.431816i \(0.857873\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.49316 + 11.2465i 0.646094 + 1.11907i 0.984048 + 0.177905i \(0.0569319\pi\)
−0.337954 + 0.941163i \(0.609735\pi\)
\(102\) 0 0
\(103\) 3.50000 6.06218i 0.344865 0.597324i −0.640464 0.767988i \(-0.721258\pi\)
0.985329 + 0.170664i \(0.0545913\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.35868 11.0136i −0.614717 1.06472i −0.990434 0.137987i \(-0.955937\pi\)
0.375717 0.926735i \(-0.377397\pi\)
\(108\) 0 0
\(109\) 3.70370 6.41499i 0.354750 0.614445i −0.632325 0.774703i \(-0.717899\pi\)
0.987075 + 0.160258i \(0.0512327\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.34213 + 5.78874i 0.314401 + 0.544559i 0.979310 0.202365i \(-0.0648629\pi\)
−0.664909 + 0.746925i \(0.731530\pi\)
\(114\) 0 0
\(115\) −1.36156 2.35830i −0.126966 0.219912i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.21053 4.34197i 0.110969 0.398028i
\(120\) 0 0
\(121\) −5.56238 + 9.63433i −0.505671 + 0.875848i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.0194 0.896165
\(126\) 0 0
\(127\) −11.5322 −1.02331 −0.511657 0.859190i \(-0.670968\pi\)
−0.511657 + 0.859190i \(0.670968\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.56922 7.91412i 0.399214 0.691460i −0.594415 0.804159i \(-0.702616\pi\)
0.993629 + 0.112699i \(0.0359496\pi\)
\(132\) 0 0
\(133\) 2.37524 + 2.42368i 0.205959 + 0.210160i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.12476 + 1.94815i 0.0960950 + 0.166441i 0.910065 0.414465i \(-0.136031\pi\)
−0.813970 + 0.580907i \(0.802698\pi\)
\(138\) 0 0
\(139\) 5.28947 + 9.16163i 0.448647 + 0.777079i 0.998298 0.0583152i \(-0.0185728\pi\)
−0.549652 + 0.835394i \(0.685239\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.04583 13.9358i 0.672826 1.16537i
\(144\) 0 0
\(145\) −5.69398 9.86227i −0.472859 0.819017i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.1871 + 21.1088i −0.998410 + 1.72930i −0.450385 + 0.892834i \(0.648713\pi\)
−0.548025 + 0.836462i \(0.684620\pi\)
\(150\) 0 0
\(151\) −5.56922 9.64617i −0.453217 0.784994i 0.545367 0.838197i \(-0.316390\pi\)
−0.998584 + 0.0532032i \(0.983057\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.14132 + 7.17297i −0.332638 + 0.576147i
\(156\) 0 0
\(157\) 3.26320 0.260432 0.130216 0.991486i \(-0.458433\pi\)
0.130216 + 0.991486i \(0.458433\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.799182 + 2.86652i −0.0629844 + 0.225914i
\(162\) 0 0
\(163\) 0.0760548 + 0.131731i 0.00595707 + 0.0103179i 0.868989 0.494832i \(-0.164770\pi\)
−0.863032 + 0.505150i \(0.831437\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.1179 20.9889i −0.937713 1.62417i −0.769723 0.638379i \(-0.779605\pi\)
−0.167991 0.985789i \(-0.553728\pi\)
\(168\) 0 0
\(169\) 0.648152 1.12263i 0.0498578 0.0863563i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.95322 0.528644 0.264322 0.964435i \(-0.414852\pi\)
0.264322 + 0.964435i \(0.414852\pi\)
\(174\) 0 0
\(175\) 1.59549 + 1.62803i 0.120607 + 0.123067i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −7.34897 + 12.7288i −0.549288 + 0.951394i 0.449036 + 0.893514i \(0.351768\pi\)
−0.998324 + 0.0578806i \(0.981566\pi\)
\(180\) 0 0
\(181\) 17.1053 1.27143 0.635715 0.771924i \(-0.280705\pi\)
0.635715 + 0.771924i \(0.280705\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −20.6900 −1.52116
\(186\) 0 0
\(187\) 8.01367 0.586018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.9806 1.66282 0.831408 0.555663i \(-0.187535\pi\)
0.831408 + 0.555663i \(0.187535\pi\)
\(192\) 0 0
\(193\) 5.37429 0.386850 0.193425 0.981115i \(-0.438040\pi\)
0.193425 + 0.981115i \(0.438040\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.57893 −0.254988 −0.127494 0.991839i \(-0.540693\pi\)
−0.127494 + 0.991839i \(0.540693\pi\)
\(198\) 0 0
\(199\) 3.50000 6.06218i 0.248108 0.429736i −0.714893 0.699234i \(-0.753524\pi\)
0.963001 + 0.269498i \(0.0868577\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.34213 + 11.9876i −0.234572 + 0.841367i
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.01655 + 5.22482i −0.208659 + 0.361408i
\(210\) 0 0
\(211\) −9.55555 16.5507i −0.657831 1.13940i −0.981176 0.193115i \(-0.938141\pi\)
0.323345 0.946281i \(-0.395193\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.71341 + 11.6280i 0.457851 + 0.793021i
\(216\) 0 0
\(217\) 8.76608 2.25427i 0.595080 0.153030i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.82846 −0.392065
\(222\) 0 0
\(223\) −12.1316 + 21.0125i −0.812392 + 1.40710i 0.0987935 + 0.995108i \(0.468502\pi\)
−0.911186 + 0.411996i \(0.864832\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.92395 3.33237i −0.127697 0.221177i 0.795087 0.606495i \(-0.207425\pi\)
−0.922784 + 0.385318i \(0.874092\pi\)
\(228\) 0 0
\(229\) −1.76608 + 3.05894i −0.116706 + 0.202140i −0.918460 0.395513i \(-0.870567\pi\)
0.801755 + 0.597653i \(0.203900\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.6248 20.1347i −0.761564 1.31907i −0.942044 0.335488i \(-0.891099\pi\)
0.180481 0.983579i \(-0.442235\pi\)
\(234\) 0 0
\(235\) −14.3187 + 24.8008i −0.934052 + 1.61783i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.14132 7.17297i −0.267879 0.463981i 0.700435 0.713717i \(-0.252990\pi\)
−0.968314 + 0.249736i \(0.919656\pi\)
\(240\) 0 0
\(241\) −6.76320 11.7142i −0.435656 0.754578i 0.561693 0.827346i \(-0.310150\pi\)
−0.997349 + 0.0727675i \(0.976817\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.342133 16.9440i 0.0218581 1.08251i
\(246\) 0 0
\(247\) 2.19398 3.80009i 0.139600 0.241794i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.11109 −0.322609 −0.161305 0.986905i \(-0.551570\pi\)
−0.161305 + 0.986905i \(0.551570\pi\)
\(252\) 0 0
\(253\) −5.29055 −0.332614
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.04583 13.9358i 0.501885 0.869290i −0.498113 0.867112i \(-0.665973\pi\)
0.999998 0.00217808i \(-0.000693305\pi\)
\(258\) 0 0
\(259\) 15.8256 + 16.1484i 0.983354 + 1.00341i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.35868 11.0136i −0.392093 0.679126i 0.600632 0.799525i \(-0.294916\pi\)
−0.992725 + 0.120400i \(0.961582\pi\)
\(264\) 0 0
\(265\) 12.4239 + 21.5189i 0.763197 + 1.32190i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.3518 19.6620i 0.692134 1.19881i −0.279003 0.960290i \(-0.590004\pi\)
0.971137 0.238522i \(-0.0766628\pi\)
\(270\) 0 0
\(271\) −7.56238 13.0984i −0.459382 0.795673i 0.539546 0.841956i \(-0.318596\pi\)
−0.998928 + 0.0462830i \(0.985262\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.02627 + 3.50960i −0.122188 + 0.211637i
\(276\) 0 0
\(277\) −11.7729 20.3913i −0.707366 1.22519i −0.965831 0.259173i \(-0.916550\pi\)
0.258465 0.966021i \(-0.416783\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.77975 4.81467i 0.165826 0.287219i −0.771122 0.636687i \(-0.780304\pi\)
0.936948 + 0.349468i \(0.113638\pi\)
\(282\) 0 0
\(283\) −13.9532 −0.829433 −0.414717 0.909951i \(-0.636119\pi\)
−0.414717 + 0.909951i \(0.636119\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.88401 + 7.02441i 0.406350 + 0.414638i
\(288\) 0 0
\(289\) 7.04871 + 12.2087i 0.414630 + 0.718160i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.07605 7.05993i −0.238126 0.412446i 0.722051 0.691840i \(-0.243200\pi\)
−0.960176 + 0.279394i \(0.909866\pi\)
\(294\) 0 0
\(295\) 4.99316 8.64841i 0.290713 0.503530i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.84789 0.222529
\(300\) 0 0
\(301\) 3.94050 14.1339i 0.227126 0.814662i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −11.1969 + 19.3935i −0.641130 + 1.11047i
\(306\) 0 0
\(307\) 2.13844 0.122047 0.0610235 0.998136i \(-0.480564\pi\)
0.0610235 + 0.998136i \(0.480564\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.8421 0.614801 0.307400 0.951580i \(-0.400541\pi\)
0.307400 + 0.951580i \(0.400541\pi\)
\(312\) 0 0
\(313\) 32.3412 1.82803 0.914016 0.405678i \(-0.132965\pi\)
0.914016 + 0.405678i \(0.132965\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.9201 −1.23116 −0.615578 0.788076i \(-0.711078\pi\)
−0.615578 + 0.788076i \(0.711078\pi\)
\(318\) 0 0
\(319\) −22.1248 −1.23875
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.18521 0.121588
\(324\) 0 0
\(325\) 1.47373 2.55258i 0.0817480 0.141592i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 30.3090 7.79423i 1.67099 0.429710i
\(330\) 0 0
\(331\) 12.3937 0.681220 0.340610 0.940205i \(-0.389366\pi\)
0.340610 + 0.940205i \(0.389366\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −13.4669 + 23.3253i −0.735775 + 1.27440i
\(336\) 0 0
\(337\) −3.86552 6.69528i −0.210568 0.364715i 0.741324 0.671147i \(-0.234198\pi\)
−0.951893 + 0.306432i \(0.900865\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.04583 + 13.9358i 0.435706 + 0.754665i
\(342\) 0 0
\(343\) −13.4863 + 12.6933i −0.728193 + 0.685372i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.2222 1.51505 0.757523 0.652808i \(-0.226409\pi\)
0.757523 + 0.652808i \(0.226409\pi\)
\(348\) 0 0
\(349\) 0.617927 1.07028i 0.0330769 0.0572908i −0.849013 0.528372i \(-0.822803\pi\)
0.882090 + 0.471081i \(0.156136\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.46006 16.3853i −0.503508 0.872102i −0.999992 0.00405566i \(-0.998709\pi\)
0.496484 0.868046i \(-0.334624\pi\)
\(354\) 0 0
\(355\) 18.1179 31.3812i 0.961600 1.66554i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.93078 10.2724i −0.313015 0.542157i 0.665999 0.745953i \(-0.268006\pi\)
−0.979013 + 0.203795i \(0.934672\pi\)
\(360\) 0 0
\(361\) 8.67743 15.0297i 0.456707 0.791039i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.58865 + 4.48367i 0.135496 + 0.234686i
\(366\) 0 0
\(367\) 7.11109 + 12.3168i 0.371196 + 0.642930i 0.989750 0.142812i \(-0.0456145\pi\)
−0.618554 + 0.785742i \(0.712281\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.29235 26.1563i 0.378600 1.35797i
\(372\) 0 0
\(373\) −11.8421 + 20.5112i −0.613162 + 1.06203i 0.377542 + 0.925993i \(0.376770\pi\)
−0.990704 + 0.136036i \(0.956564\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.0917 0.828763
\(378\) 0 0
\(379\) 4.42107 0.227095 0.113547 0.993533i \(-0.463779\pi\)
0.113547 + 0.993533i \(0.463779\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.20082 + 3.81193i −0.112457 + 0.194780i −0.916760 0.399438i \(-0.869205\pi\)
0.804304 + 0.594219i \(0.202539\pi\)
\(384\) 0 0
\(385\) 29.1803 7.50397i 1.48717 0.382438i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.65499 + 2.86652i 0.0839112 + 0.145339i 0.904927 0.425567i \(-0.139925\pi\)
−0.821016 + 0.570906i \(0.806592\pi\)
\(390\) 0 0
\(391\) 0.958128 + 1.65953i 0.0484546 + 0.0839258i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −17.5917 + 30.4696i −0.885132 + 1.53309i
\(396\) 0 0
\(397\) 15.5848 + 26.9937i 0.782180 + 1.35478i 0.930669 + 0.365861i \(0.119226\pi\)
−0.148490 + 0.988914i \(0.547441\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.293425 + 0.508226i −0.0146529 + 0.0253796i −0.873259 0.487257i \(-0.837998\pi\)
0.858606 + 0.512636i \(0.171331\pi\)
\(402\) 0 0
\(403\) −5.85185 10.1357i −0.291501 0.504895i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.0985 + 34.8116i −0.996245 + 1.72555i
\(408\) 0 0
\(409\) 29.0312 1.43550 0.717750 0.696300i \(-0.245172\pi\)
0.717750 + 0.696300i \(0.245172\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −10.5692 + 2.71797i −0.520077 + 0.133742i
\(414\) 0 0
\(415\) 10.1940 + 17.6565i 0.500403 + 0.866723i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.5917 + 20.0773i 0.566290 + 0.980842i 0.996928 + 0.0783181i \(0.0249550\pi\)
−0.430639 + 0.902524i \(0.641712\pi\)
\(420\) 0 0
\(421\) 14.5293 25.1654i 0.708114 1.22649i −0.257442 0.966294i \(-0.582880\pi\)
0.965556 0.260195i \(-0.0837869\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.46784 0.0712008
\(426\) 0 0
\(427\) 23.7008 6.09487i 1.14696 0.294951i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.990285 1.71522i 0.0477003 0.0826194i −0.841189 0.540741i \(-0.818144\pi\)
0.888890 + 0.458121i \(0.151477\pi\)
\(432\) 0 0
\(433\) 29.5048 1.41791 0.708955 0.705253i \(-0.249167\pi\)
0.708955 + 0.705253i \(0.249167\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.44266 −0.0690116
\(438\) 0 0
\(439\) 18.9727 0.905515 0.452758 0.891634i \(-0.350440\pi\)
0.452758 + 0.891634i \(0.350440\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.2690 1.01052 0.505259 0.862968i \(-0.331397\pi\)
0.505259 + 0.862968i \(0.331397\pi\)
\(444\) 0 0
\(445\) −38.9590 −1.84683
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −21.2690 −1.00374 −0.501872 0.864942i \(-0.667355\pi\)
−0.501872 + 0.864942i \(0.667355\pi\)
\(450\) 0 0
\(451\) −8.74269 + 15.1428i −0.411677 + 0.713046i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −21.2233 + 5.45774i −0.994962 + 0.255863i
\(456\) 0 0
\(457\) −11.4678 −0.536443 −0.268222 0.963357i \(-0.586436\pi\)
−0.268222 + 0.963357i \(0.586436\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.70082 6.41001i 0.172364 0.298544i −0.766882 0.641788i \(-0.778193\pi\)
0.939246 + 0.343245i \(0.111526\pi\)
\(462\) 0 0
\(463\) −2.84213 4.92272i −0.132085 0.228778i 0.792395 0.610008i \(-0.208834\pi\)
−0.924480 + 0.381230i \(0.875501\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.6150 + 21.8499i 0.583755 + 1.01109i 0.995029 + 0.0995815i \(0.0317504\pi\)
−0.411275 + 0.911511i \(0.634916\pi\)
\(468\) 0 0
\(469\) 28.5059 7.33054i 1.31628 0.338493i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 26.0859 1.19943
\(474\) 0 0
\(475\) −0.552534 + 0.957016i −0.0253520 + 0.0439109i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.11793 + 10.5966i 0.279535 + 0.484169i 0.971269 0.237983i \(-0.0764863\pi\)
−0.691734 + 0.722152i \(0.743153\pi\)
\(480\) 0 0
\(481\) 14.6179 25.3190i 0.666520 1.15445i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 19.6706 + 34.0705i 0.893196 + 1.54706i
\(486\) 0 0
\(487\) 2.99028 5.17933i 0.135503 0.234698i −0.790287 0.612737i \(-0.790068\pi\)
0.925789 + 0.378040i \(0.123402\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.60138 + 7.96982i 0.207657 + 0.359673i 0.950976 0.309264i \(-0.100083\pi\)
−0.743319 + 0.668937i \(0.766749\pi\)
\(492\) 0 0
\(493\) 4.00684 + 6.94004i 0.180459 + 0.312564i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −38.3509 + 9.86227i −1.72027 + 0.442383i
\(498\) 0 0
\(499\) −9.83242 + 17.0302i −0.440159 + 0.762379i −0.997701 0.0677705i \(-0.978411\pi\)
0.557541 + 0.830149i \(0.311745\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.54583 0.425628 0.212814 0.977093i \(-0.431737\pi\)
0.212814 + 0.977093i \(0.431737\pi\)
\(504\) 0 0
\(505\) 31.4408 1.39910
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.77292 + 3.07078i −0.0785831 + 0.136110i −0.902639 0.430399i \(-0.858373\pi\)
0.824056 + 0.566509i \(0.191706\pi\)
\(510\) 0 0
\(511\) 1.51943 5.44993i 0.0672156 0.241091i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.47373 14.6769i −0.373397 0.646743i
\(516\) 0 0
\(517\) 27.8187 + 48.1835i 1.22347 + 2.11911i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.77292 13.4631i 0.340538 0.589828i −0.643995 0.765030i \(-0.722724\pi\)
0.984533 + 0.175201i \(0.0560576\pi\)
\(522\) 0 0
\(523\) 15.6871 + 27.1709i 0.685951 + 1.18810i 0.973137 + 0.230227i \(0.0739469\pi\)
−0.287186 + 0.957875i \(0.592720\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.91423 5.04759i 0.126946 0.219877i
\(528\) 0 0
\(529\) 10.8675 + 18.8230i 0.472498 + 0.818391i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.35868 11.0136i 0.275425 0.477050i
\(534\) 0 0
\(535\) −30.7896 −1.33115
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −28.1765 17.0352i −1.21365 0.733759i
\(540\) 0 0
\(541\) 2.13448 + 3.69703i 0.0917684 + 0.158948i 0.908255 0.418416i \(-0.137415\pi\)
−0.816487 + 0.577364i \(0.804081\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.96690 15.5311i −0.384100 0.665280i
\(546\) 0 0
\(547\) 10.2008 17.6683i 0.436155 0.755443i −0.561234 0.827657i \(-0.689673\pi\)
0.997389 + 0.0722139i \(0.0230064\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.03310 −0.257019
\(552\) 0 0
\(553\) 37.2369 9.57580i 1.58348 0.407204i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.7466 30.7381i 0.751950 1.30241i −0.194927 0.980818i \(-0.562447\pi\)
0.946877 0.321597i \(-0.104220\pi\)
\(558\) 0 0
\(559\) −18.9727 −0.802458
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.29055 0.222970 0.111485 0.993766i \(-0.464439\pi\)
0.111485 + 0.993766i \(0.464439\pi\)
\(564\) 0 0
\(565\) 16.1831 0.680826
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.06045 −0.0444564 −0.0222282 0.999753i \(-0.507076\pi\)
−0.0222282 + 0.999753i \(0.507076\pi\)
\(570\) 0 0
\(571\) −29.0996 −1.21778 −0.608890 0.793255i \(-0.708385\pi\)
−0.608890 + 0.793255i \(0.708385\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.969055 −0.0404124
\(576\) 0 0
\(577\) 0.617927 1.07028i 0.0257246 0.0445564i −0.852876 0.522113i \(-0.825144\pi\)
0.878601 + 0.477556i \(0.158477\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.98345 21.4616i 0.248235 0.890376i
\(582\) 0 0
\(583\) 48.2750 1.99935
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.9863 22.4930i 0.536003 0.928385i −0.463111 0.886300i \(-0.653267\pi\)
0.999114 0.0420843i \(-0.0133998\pi\)
\(588\) 0 0
\(589\) 2.19398 + 3.80009i 0.0904014 + 0.156580i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.15103 10.6539i −0.252593 0.437503i 0.711646 0.702538i \(-0.247950\pi\)
−0.964239 + 0.265035i \(0.914617\pi\)
\(594\) 0 0
\(595\) −7.63844 7.79423i −0.313145 0.319532i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −23.7874 −0.971928 −0.485964 0.873979i \(-0.661531\pi\)
−0.485964 + 0.873979i \(0.661531\pi\)
\(600\) 0 0
\(601\) 3.96006 6.85903i 0.161534 0.279785i −0.773885 0.633326i \(-0.781689\pi\)
0.935419 + 0.353541i \(0.115022\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.4669 + 23.3253i 0.547507 + 0.948310i
\(606\) 0 0
\(607\) 0.0233882 0.0405096i 0.000949298 0.00164423i −0.865550 0.500822i \(-0.833031\pi\)
0.866500 + 0.499178i \(0.166364\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20.2330 35.0445i −0.818539 1.41775i
\(612\) 0 0
\(613\) −9.91027 + 17.1651i −0.400272 + 0.693292i −0.993759 0.111552i \(-0.964418\pi\)
0.593486 + 0.804844i \(0.297751\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17.2164 29.8197i −0.693107 1.20050i −0.970815 0.239831i \(-0.922908\pi\)
0.277708 0.960666i \(-0.410425\pi\)
\(618\) 0 0
\(619\) −23.7564 41.1472i −0.954849 1.65385i −0.734714 0.678377i \(-0.762683\pi\)
−0.220135 0.975469i \(-0.570650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 29.7993 + 30.4071i 1.19388 + 1.21823i
\(624\) 0 0
\(625\) 14.2828 24.7385i 0.571311 0.989539i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.5595 0.580525
\(630\) 0 0
\(631\) 9.26320 0.368762 0.184381 0.982855i \(-0.440972\pi\)
0.184381 + 0.982855i \(0.440972\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13.9601 + 24.1795i −0.553988 + 0.959536i
\(636\) 0 0
\(637\) 20.4932 + 12.3900i 0.811969 + 0.490909i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.55267 + 2.68930i 0.0613266 + 0.106221i 0.895059 0.445948i \(-0.147134\pi\)
−0.833732 + 0.552169i \(0.813800\pi\)
\(642\) 0 0
\(643\) 8.86840 + 15.3605i 0.349736 + 0.605760i 0.986202 0.165544i \(-0.0529381\pi\)
−0.636467 + 0.771304i \(0.719605\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.460060 0.796847i 0.0180868 0.0313273i −0.856840 0.515582i \(-0.827576\pi\)
0.874927 + 0.484255i \(0.160909\pi\)
\(648\) 0 0
\(649\) −9.70082 16.8023i −0.380790 0.659548i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.60138 + 7.96982i −0.180066 + 0.311883i −0.941903 0.335886i \(-0.890964\pi\)
0.761837 + 0.647769i \(0.224298\pi\)
\(654\) 0 0
\(655\) −11.0624 19.1606i −0.432243 0.748667i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.6999 42.7814i 0.962170 1.66653i 0.245138 0.969488i \(-0.421167\pi\)
0.717032 0.697040i \(-0.245500\pi\)
\(660\) 0 0
\(661\) −49.9844 −1.94417 −0.972085 0.234631i \(-0.924612\pi\)
−0.972085 + 0.234631i \(0.924612\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.95705 2.04622i 0.308561 0.0793492i
\(666\) 0 0
\(667\) −2.64527 4.58175i −0.102425 0.177406i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 21.7535 + 37.6781i 0.839784 + 1.45455i
\(672\) 0 0
\(673\) −4.71737 + 8.17072i −0.181841 + 0.314958i −0.942508 0.334185i \(-0.891539\pi\)
0.760666 + 0.649143i \(0.224872\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −23.7874 −0.914226 −0.457113 0.889409i \(-0.651116\pi\)
−0.457113 + 0.889409i \(0.651116\pi\)
\(678\) 0 0
\(679\) 11.5458 41.4128i 0.443088 1.58928i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.3187 29.9969i 0.662683 1.14780i −0.317224 0.948350i \(-0.602751\pi\)
0.979908 0.199451i \(-0.0639158\pi\)
\(684\) 0 0
\(685\) 5.44625 0.208091
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −35.1111 −1.33763
\(690\) 0 0
\(691\) 35.9201 1.36647 0.683233 0.730201i \(-0.260573\pi\)
0.683233 + 0.730201i \(0.260573\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 25.6123 0.971530
\(696\) 0 0
\(697\) 6.33327 0.239890
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.72529 −0.367319 −0.183659 0.982990i \(-0.558794\pi\)
−0.183659 + 0.982990i \(0.558794\pi\)
\(702\) 0 0
\(703\) −5.48057 + 9.49263i −0.206704 + 0.358021i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −24.0487 24.5392i −0.904445 0.922892i
\(708\) 0 0
\(709\) −24.1111 −0.905511 −0.452756 0.891635i \(-0.649559\pi\)
−0.452756 + 0.891635i \(0.649559\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.92395 + 3.33237i −0.0720523 + 0.124798i
\(714\) 0 0
\(715\) −19.4795 33.7395i −0.728492 1.26178i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.6764 21.9561i −0.472748 0.818824i 0.526765 0.850011i \(-0.323405\pi\)
−0.999514 + 0.0311869i \(0.990071\pi\)
\(720\) 0 0
\(721\) −4.97373 + 17.8399i −0.185232 + 0.664393i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.05253 −0.150507
\(726\) 0 0
\(727\) −23.7564 + 41.1472i −0.881075 + 1.52607i −0.0309272 + 0.999522i \(0.509846\pi\)
−0.850148 + 0.526545i \(0.823487\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.72421 8.18257i −0.174731 0.302643i
\(732\) 0 0
\(733\) 15.4182 26.7051i 0.569484 0.986375i −0.427133 0.904189i \(-0.640476\pi\)
0.996617 0.0821861i \(-0.0261902\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.1638 + 45.3170i 0.963754 + 1.66927i
\(738\) 0 0
\(739\) −10.4971 + 18.1815i −0.386143 + 0.668819i −0.991927 0.126810i \(-0.959526\pi\)
0.605784 + 0.795629i \(0.292859\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −22.6871 39.2953i −0.832311 1.44160i −0.896201 0.443647i \(-0.853684\pi\)
0.0638908 0.997957i \(-0.479649\pi\)
\(744\) 0 0
\(745\) 29.5059 + 51.1057i 1.08101 + 1.87237i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 23.5506 + 24.0310i 0.860522 + 0.878073i
\(750\) 0 0
\(751\) −6.42107 + 11.1216i −0.234308 + 0.405833i −0.959071 0.283164i \(-0.908616\pi\)
0.724763 + 0.688998i \(0.241949\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −26.9669 −0.981426
\(756\) 0 0
\(757\) 12.9727 0.471499 0.235750 0.971814i \(-0.424245\pi\)
0.235750 + 0.971814i \(0.424245\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20.7466 + 35.9342i −0.752065 + 1.30262i 0.194755 + 0.980852i \(0.437609\pi\)
−0.946820 + 0.321764i \(0.895724\pi\)
\(762\) 0 0
\(763\) −5.26320 + 18.8782i −0.190541 + 0.683435i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.05555 + 12.2206i 0.254761 + 0.441259i
\(768\) 0 0
\(769\) 12.3450 + 21.3822i 0.445173 + 0.771061i 0.998064 0.0621917i \(-0.0198090\pi\)
−0.552892 + 0.833253i \(0.686476\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.8256 + 34.3389i −0.713077 + 1.23508i 0.250620 + 0.968086i \(0.419366\pi\)
−0.963697 + 0.266999i \(0.913968\pi\)
\(774\) 0 0
\(775\) 1.47373 + 2.55258i 0.0529381 + 0.0916914i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.38401 + 4.12922i −0.0854159 + 0.147945i
\(780\) 0 0
\(781\) −35.1999 60.9680i −1.25955 2.18161i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.95021 6.84197i 0.140989 0.244200i
\(786\) 0 0
\(787\) 18.0255 0.642538 0.321269 0.946988i \(-0.395891\pi\)
0.321269 + 0.946988i \(0.395891\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.3782 12.6307i −0.440120 0.449096i
\(792\) 0 0
\(793\) −15.8216 27.4039i −0.561842 0.973139i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19.9396 34.5363i −0.706295 1.22334i −0.966222 0.257711i \(-0.917032\pi\)
0.259927 0.965628i \(-0.416301\pi\)
\(798\) 0 0
\(799\) 10.0761 17.4522i 0.356465 0.617416i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.0586 0.354959
\(804\) 0 0
\(805\) 5.04282 + 5.14567i 0.177736 + 0.181361i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 22.5848 39.1181i 0.794040 1.37532i −0.129407 0.991592i \(-0.541307\pi\)
0.923447 0.383726i \(-0.125359\pi\)
\(810\) 0 0
\(811\) −1.70945 −0.0600271 −0.0300135 0.999549i \(-0.509555\pi\)
−0.0300135 + 0.999549i \(0.509555\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.368267 0.0128998
\(816\) 0 0
\(817\) 7.11325 0.248861
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −52.9144 −1.84672 −0.923362 0.383930i \(-0.874570\pi\)
−0.923362 + 0.383930i \(0.874570\pi\)
\(822\) 0 0
\(823\) −31.2359 −1.08881 −0.544407 0.838821i \(-0.683245\pi\)
−0.544407 + 0.838821i \(0.683245\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.2243 1.22487 0.612435 0.790521i \(-0.290190\pi\)
0.612435 + 0.790521i \(0.290190\pi\)
\(828\) 0 0
\(829\) −9.05842 + 15.6897i −0.314612 + 0.544924i −0.979355 0.202148i \(-0.935208\pi\)
0.664743 + 0.747072i \(0.268541\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.240758 + 11.9234i −0.00834177 + 0.413123i
\(834\) 0 0
\(835\) −58.6766 −2.03059
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.77004 16.9222i 0.337299 0.584219i −0.646625 0.762808i \(-0.723820\pi\)
0.983924 + 0.178589i \(0.0571533\pi\)
\(840\) 0 0
\(841\) 3.43762 + 5.95413i 0.118539 + 0.205315i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.56922 2.71797i −0.0539827 0.0935009i
\(846\) 0 0
\(847\) 7.90451 28.3521i 0.271602 0.974189i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −9.61204 −0.329496
\(852\) 0 0
\(853\) −8.68715 + 15.0466i −0.297442 + 0.515185i −0.975550 0.219777i \(-0.929467\pi\)
0.678108 + 0.734962i \(0.262800\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.7729 + 18.6592i 0.367996 + 0.637387i 0.989252 0.146220i \(-0.0467108\pi\)
−0.621256 + 0.783607i \(0.713377\pi\)
\(858\) 0 0
\(859\) −8.68715 + 15.0466i −0.296402 + 0.513383i −0.975310 0.220840i \(-0.929120\pi\)
0.678908 + 0.734223i \(0.262453\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.71053 + 15.0871i 0.296510 + 0.513570i 0.975335 0.220730i \(-0.0708439\pi\)
−0.678825 + 0.734300i \(0.737511\pi\)
\(864\) 0 0
\(865\) 8.41711 14.5789i 0.286190 0.495696i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 34.1774 + 59.1970i 1.15939 + 2.00812i
\(870\) 0 0
\(871\) −19.0293 32.9597i −0.644782 1.11680i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −25.6736 + 6.60219i −0.867926 + 0.223195i
\(876\) 0 0
\(877\) −23.4991 + 40.7016i −0.793507 + 1.37439i 0.130276 + 0.991478i \(0.458414\pi\)
−0.923783 + 0.382916i \(0.874920\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −15.1715 −0.511142 −0.255571 0.966790i \(-0.582263\pi\)
−0.255571 + 0.966790i \(0.582263\pi\)
\(882\) 0 0
\(883\) −32.4660 −1.09257 −0.546283 0.837601i \(-0.683958\pi\)
−0.546283 + 0.837601i \(0.683958\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.43474 + 2.48504i −0.0481738 + 0.0834395i −0.889107 0.457700i \(-0.848673\pi\)
0.840933 + 0.541139i \(0.182007\pi\)
\(888\) 0 0
\(889\) 29.5498 7.59898i 0.991068 0.254862i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.58577 + 13.1389i 0.253848 + 0.439678i
\(894\) 0 0
\(895\) 17.7923 + 30.8172i 0.594733 + 1.03011i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8.04583 + 13.9358i −0.268343 + 0.464784i
\(900\) 0 0
\(901\) −8.74269 15.1428i −0.291261 0.504479i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.7066 35.8648i 0.688310 1.19219i
\(906\) 0 0
\(907\) 16.2164 + 28.0877i 0.538458 + 0.932636i 0.998987 + 0.0449915i \(0.0143261\pi\)
−0.460530 + 0.887644i \(0.652341\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −16.6219 + 28.7899i −0.550708 + 0.953854i 0.447516 + 0.894276i \(0.352309\pi\)
−0.998224 + 0.0595777i \(0.981025\pi\)
\(912\) 0 0
\(913\) 39.6101 1.31090
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6.49316 + 23.2898i −0.214423 + 0.769098i
\(918\) 0 0
\(919\) 14.3752 + 24.8986i 0.474195 + 0.821330i 0.999563 0.0295447i \(-0.00940573\pi\)
−0.525368 + 0.850875i \(0.676072\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 25.6014 + 44.3429i 0.842680 + 1.45956i
\(924\) 0 0
\(925\) −3.68139 + 6.37635i −0.121043 + 0.209653i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 54.5458 1.78959 0.894795 0.446477i \(-0.147321\pi\)
0.894795 + 0.446477i \(0.147321\pi\)
\(930\) 0 0
\(931\) −7.68332 4.64526i −0.251811 0.152242i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9.70082 16.8023i 0.317251 0.549494i
\(936\) 0 0
\(937\) 28.6979 0.937521 0.468760 0.883325i \(-0.344701\pi\)
0.468760 + 0.883325i \(0.344701\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.2963 0.629042 0.314521 0.949251i \(-0.398156\pi\)
0.314521 + 0.949251i \(0.398156\pi\)
\(942\) 0 0
\(943\) −4.18116 −0.136157
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −10.5322 −0.342249 −0.171125 0.985249i \(-0.554740\pi\)
−0.171125 + 0.985249i \(0.554740\pi\)
\(948\) 0 0
\(949\) −7.31573 −0.237479
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.1970 −0.621852 −0.310926 0.950434i \(-0.600639\pi\)
−0.310926 + 0.950434i \(0.600639\pi\)
\(954\) 0 0
\(955\) 27.8187 48.1835i 0.900193 1.55918i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.16578 4.25075i −0.134520 0.137264i
\(960\) 0 0
\(961\) −19.2963 −0.622461
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.50576 11.2683i 0.209428 0.362739i
\(966\) 0 0
\(967\) 7.74269 + 13.4107i 0.248988 + 0.431260i 0.963245 0.268623i \(-0.0865686\pi\)
−0.714257 + 0.699883i \(0.753235\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.904515 + 1.56667i 0.0290273 + 0.0502767i 0.880174 0.474651i \(-0.157426\pi\)
−0.851147 + 0.524928i \(0.824092\pi\)
\(972\) 0 0
\(973\) −19.5906 19.9901i −0.628045 0.640855i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37.8985 1.21248 0.606241 0.795281i \(-0.292677\pi\)
0.606241 + 0.795281i \(0.292677\pi\)
\(978\) 0 0
\(979\) −37.8451 + 65.5497i −1.20954 + 2.09498i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −19.0555 33.0052i −0.607778 1.05270i −0.991606 0.129297i \(-0.958728\pi\)
0.383828 0.923404i \(-0.374605\pi\)
\(984\) 0 0
\(985\) −4.33242 + 7.50397i −0.138042 + 0.239096i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.11887 + 5.40205i 0.0991744 + 0.171775i
\(990\) 0 0
\(991\) −5.89480 + 10.2101i −0.187254 + 0.324334i −0.944334 0.328989i \(-0.893292\pi\)
0.757079 + 0.653323i \(0.226626\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.47373 14.6769i −0.268635 0.465290i
\(996\) 0 0
\(997\) −7.40344 12.8231i −0.234469 0.406112i 0.724649 0.689118i \(-0.242002\pi\)
−0.959118 + 0.283006i \(0.908669\pi\)
\(998\) 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 2268.2.i.j.865.3 6
3.2 odd 2 2268.2.i.k.865.1 6
7.2 even 3 2268.2.l.k.541.1 6
9.2 odd 6 756.2.k.f.109.1 yes 6
9.4 even 3 2268.2.l.k.109.1 6
9.5 odd 6 2268.2.l.j.109.3 6
9.7 even 3 756.2.k.e.109.3 6
21.2 odd 6 2268.2.l.j.541.3 6
63.2 odd 6 756.2.k.f.541.1 yes 6
63.11 odd 6 5292.2.a.u.1.3 3
63.16 even 3 756.2.k.e.541.3 yes 6
63.23 odd 6 2268.2.i.k.2053.1 6
63.25 even 3 5292.2.a.x.1.1 3
63.38 even 6 5292.2.a.w.1.1 3
63.52 odd 6 5292.2.a.v.1.3 3
63.58 even 3 inner 2268.2.i.j.2053.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.k.e.109.3 6 9.7 even 3
756.2.k.e.541.3 yes 6 63.16 even 3
756.2.k.f.109.1 yes 6 9.2 odd 6
756.2.k.f.541.1 yes 6 63.2 odd 6
2268.2.i.j.865.3 6 1.1 even 1 trivial
2268.2.i.j.2053.3 6 63.58 even 3 inner
2268.2.i.k.865.1 6 3.2 odd 2
2268.2.i.k.2053.1 6 63.23 odd 6
2268.2.l.j.109.3 6 9.5 odd 6
2268.2.l.j.541.3 6 21.2 odd 6
2268.2.l.k.109.1 6 9.4 even 3
2268.2.l.k.541.1 6 7.2 even 3
5292.2.a.u.1.3 3 63.11 odd 6
5292.2.a.v.1.3 3 63.52 odd 6
5292.2.a.w.1.1 3 63.38 even 6
5292.2.a.x.1.1 3 63.25 even 3