Properties

Label 2268.2.l.j.109.3
Level $2268$
Weight $2$
Character 2268.109
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(109,2268)]
 
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, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.l (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 109.3
Root \(0.500000 - 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 2268.109
Dual form 2268.2.l.j.541.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.42107 q^{5} +(0.710533 - 2.54856i) q^{7} +O(q^{10})\) \(q+2.42107 q^{5} +(0.710533 - 2.54856i) q^{7} -4.70370 q^{11} +(1.71053 - 2.96273i) q^{13} +(0.851848 - 1.47544i) q^{17} +(-0.641315 - 1.11079i) q^{19} +1.12476 q^{23} +0.861564 q^{25} +(-2.35185 - 4.07352i) q^{29} +(1.71053 + 2.96273i) q^{31} +(1.72025 - 6.17023i) q^{35} +(-4.27292 - 7.40091i) q^{37} +(1.85868 - 3.21934i) q^{41} +(-2.77292 - 4.80283i) q^{43} +(-5.91423 + 10.2437i) q^{47} +(-5.99028 - 3.62167i) q^{49} +(5.13160 - 8.88819i) q^{53} -11.3880 q^{55} +(2.06238 + 3.57215i) q^{59} +(4.62476 - 8.01033i) q^{61} +(4.14132 - 7.17297i) q^{65} +(5.56238 + 9.63433i) q^{67} -14.9669 q^{71} +(-1.06922 + 1.85194i) q^{73} +(-3.34213 + 11.9876i) q^{77} +(7.26608 - 12.5852i) q^{79} +(4.21053 + 7.29286i) q^{83} +(2.06238 - 3.57215i) q^{85} +(8.04583 + 13.9358i) q^{89} +(-6.33530 - 6.46451i) q^{91} +(-1.55267 - 2.68930i) q^{95} +(-8.12476 - 14.0725i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} - 4 q^{7} - 10 q^{11} + 2 q^{13} - 4 q^{17} - 3 q^{19} - 28 q^{23} + 20 q^{25} - 5 q^{29} + 2 q^{31} + 26 q^{35} + 12 q^{41} + 9 q^{43} - 9 q^{47} - 12 q^{49} + 6 q^{53} + 16 q^{55} - 5 q^{59} - 7 q^{61} + 24 q^{65} + 16 q^{67} - 22 q^{71} + q^{73} + 13 q^{77} + 8 q^{79} + 17 q^{83} - 5 q^{85} - 3 q^{89} + 5 q^{91} + 32 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 2.42107 1.08273 0.541367 0.840786i \(-0.317907\pi\)
0.541367 + 0.840786i \(0.317907\pi\)
\(6\) 0 0
\(7\) 0.710533 2.54856i 0.268556 0.963264i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.70370 −1.41822 −0.709109 0.705099i \(-0.750903\pi\)
−0.709109 + 0.705099i \(0.750903\pi\)
\(12\) 0 0
\(13\) 1.71053 2.96273i 0.474417 0.821714i −0.525154 0.851007i \(-0.675992\pi\)
0.999571 + 0.0292934i \(0.00932572\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.851848 1.47544i 0.206604 0.357848i −0.744039 0.668136i \(-0.767092\pi\)
0.950642 + 0.310288i \(0.100426\pi\)
\(18\) 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
\(23\) 1.12476 0.234529 0.117265 0.993101i \(-0.462587\pi\)
0.117265 + 0.993101i \(0.462587\pi\)
\(24\) 0 0
\(25\) 0.861564 0.172313
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.35185 4.07352i −0.436727 0.756434i 0.560708 0.828014i \(-0.310529\pi\)
−0.997435 + 0.0715801i \(0.977196\pi\)
\(30\) 0 0
\(31\) 1.71053 + 2.96273i 0.307221 + 0.532122i 0.977753 0.209758i \(-0.0672676\pi\)
−0.670532 + 0.741880i \(0.733934\pi\)
\(32\) 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 \(-0.739586\pi\)
0.973875 + 0.227083i \(0.0729189\pi\)
\(42\) 0 0
\(43\) −2.77292 4.80283i −0.422866 0.732425i 0.573353 0.819309i \(-0.305642\pi\)
−0.996218 + 0.0868839i \(0.972309\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.91423 + 10.2437i −0.862679 + 1.49420i 0.00665422 + 0.999978i \(0.497882\pi\)
−0.869333 + 0.494226i \(0.835451\pi\)
\(48\) 0 0
\(49\) −5.99028 3.62167i −0.855755 0.517381i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.13160 8.88819i 0.704879 1.22089i −0.261856 0.965107i \(-0.584334\pi\)
0.966735 0.255780i \(-0.0823323\pi\)
\(54\) 0 0
\(55\) −11.3880 −1.53555
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.06238 + 3.57215i 0.268499 + 0.465054i 0.968474 0.249113i \(-0.0801390\pi\)
−0.699975 + 0.714167i \(0.746806\pi\)
\(60\) 0 0
\(61\) 4.62476 8.01033i 0.592140 1.02562i −0.401803 0.915726i \(-0.631616\pi\)
0.993944 0.109891i \(-0.0350502\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.14132 7.17297i 0.513667 0.889697i
\(66\) 0 0
\(67\) 5.56238 + 9.63433i 0.679553 + 1.17702i 0.975116 + 0.221697i \(0.0711596\pi\)
−0.295563 + 0.955323i \(0.595507\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.9669 −1.77624 −0.888122 0.459608i \(-0.847990\pi\)
−0.888122 + 0.459608i \(0.847990\pi\)
\(72\) 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
\(77\) −3.34213 + 11.9876i −0.380871 + 1.36612i
\(78\) 0 0
\(79\) 7.26608 12.5852i 0.817498 1.41595i −0.0900228 0.995940i \(-0.528694\pi\)
0.907520 0.420008i \(-0.137973\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.21053 + 7.29286i 0.462166 + 0.800495i 0.999069 0.0431491i \(-0.0137391\pi\)
−0.536903 + 0.843644i \(0.680406\pi\)
\(84\) 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 \(0.158465\pi\)
−0.0257633 + 0.999668i \(0.508202\pi\)
\(90\) 0 0
\(91\) −6.33530 6.46451i −0.664120 0.677665i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.55267 2.68930i −0.159300 0.275916i
\(96\) 0 0
\(97\) −8.12476 14.0725i −0.824945 1.42885i −0.901962 0.431816i \(-0.857873\pi\)
0.0770168 0.997030i \(-0.475460\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.9863 1.29219 0.646094 0.763258i \(-0.276401\pi\)
0.646094 + 0.763258i \(0.276401\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.35868 + 11.0136i 0.614717 + 1.06472i 0.990434 + 0.137987i \(0.0440632\pi\)
−0.375717 + 0.926735i \(0.622603\pi\)
\(108\) 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 \(-0.935137\pi\)
0.664909 + 0.746925i \(0.268470\pi\)
\(114\) 0 0
\(115\) 2.72313 0.253933
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.15499 3.21934i −0.289217 0.295116i
\(120\) 0 0
\(121\) 11.1248 1.01134
\(122\) 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
\(131\) 9.13844 0.798429 0.399214 0.916858i \(-0.369283\pi\)
0.399214 + 0.916858i \(0.369283\pi\)
\(132\) 0 0
\(133\) −3.28659 + 0.845174i −0.284983 + 0.0732859i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.24953 0.192190 0.0960950 0.995372i \(-0.469365\pi\)
0.0960950 + 0.995372i \(0.469365\pi\)
\(138\) 0 0
\(139\) 5.28947 9.16163i 0.448647 0.777079i −0.549652 0.835394i \(-0.685239\pi\)
0.998298 + 0.0583152i \(0.0185728\pi\)
\(140\) 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
\(149\) −24.3743 −1.99682 −0.998410 0.0563721i \(-0.982047\pi\)
−0.998410 + 0.0563721i \(0.982047\pi\)
\(150\) 0 0
\(151\) 11.1384 0.906433 0.453217 0.891400i \(-0.350276\pi\)
0.453217 + 0.891400i \(0.350276\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.14132 + 7.17297i 0.332638 + 0.576147i
\(156\) 0 0
\(157\) −1.63160 2.82601i −0.130216 0.225540i 0.793544 0.608513i \(-0.208234\pi\)
−0.923760 + 0.382973i \(0.874900\pi\)
\(158\) 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 0.167991 0.985789i \(-0.446272\pi\)
0.769723 0.638379i \(-0.220395\pi\)
\(168\) 0 0
\(169\) 0.648152 + 1.12263i 0.0498578 + 0.0863563i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.47661 6.02167i 0.264322 0.457819i −0.703064 0.711127i \(-0.748185\pi\)
0.967386 + 0.253308i \(0.0815185\pi\)
\(174\) 0 0
\(175\) 0.612170 2.19574i 0.0462757 0.165983i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.34897 12.7288i 0.549288 0.951394i −0.449036 0.893514i \(-0.648232\pi\)
0.998324 0.0578806i \(-0.0184343\pi\)
\(180\) 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
\(185\) −10.3450 17.9181i −0.760580 1.31736i
\(186\) 0 0
\(187\) −4.00684 + 6.94004i −0.293009 + 0.507506i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.4903 19.9018i 0.831408 1.44004i −0.0655141 0.997852i \(-0.520869\pi\)
0.896922 0.442189i \(-0.145798\pi\)
\(192\) 0 0
\(193\) −2.68715 4.65427i −0.193425 0.335022i 0.752958 0.658068i \(-0.228626\pi\)
−0.946383 + 0.323047i \(0.895293\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.57893 0.254988 0.127494 0.991839i \(-0.459307\pi\)
0.127494 + 0.991839i \(0.459307\pi\)
\(198\) 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
\(203\) −12.0527 + 3.09945i −0.845931 + 0.217539i
\(204\) 0 0
\(205\) 4.50000 7.79423i 0.314294 0.544373i
\(206\) 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 0.323345 + 0.946281i \(0.395193\pi\)
−0.981176 + 0.193115i \(0.938141\pi\)
\(212\) 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
\(221\) −2.91423 5.04759i −0.196032 0.339538i
\(222\) 0 0
\(223\) −12.1316 21.0125i −0.812392 1.40710i −0.911186 0.411996i \(-0.864832\pi\)
0.0987935 0.995108i \(-0.468502\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.84789 −0.255393 −0.127697 0.991813i \(-0.540758\pi\)
−0.127697 + 0.991813i \(0.540758\pi\)
\(228\) 0 0
\(229\) 3.53216 0.233412 0.116706 0.993167i \(-0.462767\pi\)
0.116706 + 0.993167i \(0.462767\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.6248 + 20.1347i 0.761564 + 1.31907i 0.942044 + 0.335488i \(0.108901\pi\)
−0.180481 + 0.983579i \(0.557765\pi\)
\(234\) 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 \(-0.747010\pi\)
0.968314 + 0.249736i \(0.0803438\pi\)
\(240\) 0 0
\(241\) 13.5264 0.871312 0.435656 0.900113i \(-0.356516\pi\)
0.435656 + 0.900113i \(0.356516\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −14.5029 8.76830i −0.926555 0.560186i
\(246\) 0 0
\(247\) −4.38796 −0.279199
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.11109 0.322609 0.161305 0.986905i \(-0.448430\pi\)
0.161305 + 0.986905i \(0.448430\pi\)
\(252\) 0 0
\(253\) −5.29055 −0.332614
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.0917 1.00377 0.501885 0.864934i \(-0.332640\pi\)
0.501885 + 0.864934i \(0.332640\pi\)
\(258\) 0 0
\(259\) −21.8977 + 5.63118i −1.36066 + 0.349904i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.7174 −0.784187 −0.392093 0.919925i \(-0.628249\pi\)
−0.392093 + 0.919925i \(0.628249\pi\)
\(264\) 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 \(0.409996\pi\)
−0.971137 + 0.238522i \(0.923337\pi\)
\(270\) 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
\(275\) −4.05253 −0.244377
\(276\) 0 0
\(277\) 23.5458 1.41473 0.707366 0.706848i \(-0.249883\pi\)
0.707366 + 0.706848i \(0.249883\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.77975 4.81467i −0.165826 0.287219i 0.771122 0.636687i \(-0.219696\pi\)
−0.936948 + 0.349468i \(0.886362\pi\)
\(282\) 0 0
\(283\) 6.97661 + 12.0838i 0.414717 + 0.718310i 0.995399 0.0958203i \(-0.0305474\pi\)
−0.580682 + 0.814130i \(0.697214\pi\)
\(284\) 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 \(-0.756800\pi\)
0.960176 + 0.279394i \(0.0901337\pi\)
\(294\) 0 0
\(295\) 4.99316 + 8.64841i 0.290713 + 0.503530i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.92395 3.33237i 0.111265 0.192716i
\(300\) 0 0
\(301\) −14.2105 + 3.65436i −0.819082 + 0.210634i
\(302\) 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
\(311\) 5.42107 + 9.38956i 0.307400 + 0.532433i 0.977793 0.209573i \(-0.0672075\pi\)
−0.670392 + 0.742007i \(0.733874\pi\)
\(312\) 0 0
\(313\) −16.1706 + 28.0083i −0.914016 + 1.58312i −0.105680 + 0.994400i \(0.533702\pi\)
−0.808336 + 0.588722i \(0.799631\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.9601 + 18.9834i −0.615578 + 1.06621i 0.374704 + 0.927144i \(0.377744\pi\)
−0.990283 + 0.139069i \(0.955589\pi\)
\(318\) 0 0
\(319\) 11.0624 + 19.1606i 0.619374 + 1.07279i
\(320\) 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
\(329\) 21.9045 + 22.3513i 1.20763 + 1.23227i
\(330\) 0 0
\(331\) −6.19686 + 10.7333i −0.340610 + 0.589954i −0.984546 0.175125i \(-0.943967\pi\)
0.643936 + 0.765079i \(0.277300\pi\)
\(332\) 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 −0.951893 0.306432i \(-0.900865\pi\)
0.741324 + 0.671147i \(0.234198\pi\)
\(338\) 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
\(347\) 14.1111 + 24.4411i 0.757523 + 1.31207i 0.944110 + 0.329630i \(0.106924\pi\)
−0.186587 + 0.982438i \(0.559743\pi\)
\(348\) 0 0
\(349\) 0.617927 + 1.07028i 0.0330769 + 0.0572908i 0.882090 0.471081i \(-0.156136\pi\)
−0.849013 + 0.528372i \(0.822803\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.9201 −1.00702 −0.503508 0.863990i \(-0.667958\pi\)
−0.503508 + 0.863990i \(0.667958\pi\)
\(354\) 0 0
\(355\) −36.2359 −1.92320
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.93078 + 10.2724i 0.313015 + 0.542157i 0.979013 0.203795i \(-0.0653278\pi\)
−0.665999 + 0.745953i \(0.731994\pi\)
\(360\) 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
\(367\) −14.2222 −0.742392 −0.371196 0.928555i \(-0.621052\pi\)
−0.371196 + 0.928555i \(0.621052\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −19.0059 19.3935i −0.986737 1.00686i
\(372\) 0 0
\(373\) 23.6843 1.22632 0.613162 0.789957i \(-0.289897\pi\)
0.613162 + 0.789957i \(0.289897\pi\)
\(374\) 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
\(383\) −4.40164 −0.224913 −0.112457 0.993657i \(-0.535872\pi\)
−0.112457 + 0.993657i \(0.535872\pi\)
\(384\) 0 0
\(385\) −8.09153 + 29.0229i −0.412382 + 1.47914i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.30998 0.167822 0.0839112 0.996473i \(-0.473259\pi\)
0.0839112 + 0.996473i \(0.473259\pi\)
\(390\) 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
\(401\) −0.586849 −0.0293058 −0.0146529 0.999893i \(-0.504664\pi\)
−0.0146529 + 0.999893i \(0.504664\pi\)
\(402\) 0 0
\(403\) 11.7037 0.583003
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0985 + 34.8116i 0.996245 + 1.72555i
\(408\) 0 0
\(409\) −14.5156 25.1418i −0.717750 1.24318i −0.961889 0.273440i \(-0.911839\pi\)
0.244139 0.969740i \(-0.421495\pi\)
\(410\) 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 0.430639 + 0.902524i \(0.358288\pi\)
−0.996928 + 0.0783181i \(0.975045\pi\)
\(420\) 0 0
\(421\) 14.5293 + 25.1654i 0.708114 + 1.22649i 0.965556 + 0.260195i \(0.0837869\pi\)
−0.257442 + 0.966294i \(0.582880\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.733922 1.27119i 0.0356004 0.0616617i
\(426\) 0 0
\(427\) −17.1287 17.4781i −0.828917 0.845823i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.990285 + 1.71522i −0.0477003 + 0.0826194i −0.888890 0.458121i \(-0.848523\pi\)
0.841189 + 0.540741i \(0.181856\pi\)
\(432\) 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
\(437\) −0.721328 1.24938i −0.0345058 0.0597658i
\(438\) 0 0
\(439\) −9.48633 + 16.4308i −0.452758 + 0.784199i −0.998556 0.0537171i \(-0.982893\pi\)
0.545799 + 0.837916i \(0.316226\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.6345 18.4195i 0.505259 0.875135i −0.494722 0.869051i \(-0.664730\pi\)
0.999981 0.00608363i \(-0.00193649\pi\)
\(444\) 0 0
\(445\) 19.4795 + 33.7395i 0.923416 + 1.59940i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.2690 1.00374 0.501872 0.864942i \(-0.332645\pi\)
0.501872 + 0.864942i \(0.332645\pi\)
\(450\) 0 0
\(451\) −8.74269 + 15.1428i −0.411677 + 0.713046i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −15.3382 15.6510i −0.719065 0.733731i
\(456\) 0 0
\(457\) 5.73392 9.93144i 0.268222 0.464573i −0.700181 0.713965i \(-0.746897\pi\)
0.968403 + 0.249392i \(0.0802307\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.70082 6.41001i −0.172364 0.298544i 0.766882 0.641788i \(-0.221807\pi\)
−0.939246 + 0.343245i \(0.888474\pi\)
\(462\) 0 0
\(463\) −2.84213 + 4.92272i −0.132085 + 0.228778i −0.924480 0.381230i \(-0.875501\pi\)
0.792395 + 0.610008i \(0.208834\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.6150 21.8499i −0.583755 1.01109i −0.995029 0.0995815i \(-0.968250\pi\)
0.411275 0.911511i \(-0.365084\pi\)
\(468\) 0 0
\(469\) 28.5059 7.33054i 1.31628 0.338493i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13.0430 + 22.5911i 0.599716 + 1.03874i
\(474\) 0 0
\(475\) −0.552534 0.957016i −0.0253520 0.0439109i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.2359 0.559070 0.279535 0.960135i \(-0.409820\pi\)
0.279535 + 0.960135i \(0.409820\pi\)
\(480\) 0 0
\(481\) −29.2359 −1.33304
\(482\) 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 \(-0.899917\pi\)
0.743319 + 0.668937i \(0.233251\pi\)
\(492\) 0 0
\(493\) −8.01367 −0.360918
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.6345 + 38.1440i −0.477022 + 1.71099i
\(498\) 0 0
\(499\) 19.6648 0.880319 0.440159 0.897920i \(-0.354922\pi\)
0.440159 + 0.897920i \(0.354922\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9.54583 −0.425628 −0.212814 0.977093i \(-0.568263\pi\)
−0.212814 + 0.977093i \(0.568263\pi\)
\(504\) 0 0
\(505\) 31.4408 1.39910
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.54583 −0.157166 −0.0785831 0.996908i \(-0.525040\pi\)
−0.0785831 + 0.996908i \(0.525040\pi\)
\(510\) 0 0
\(511\) 3.96006 + 4.04083i 0.175183 + 0.178756i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.9475 −0.746795
\(516\) 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 −0.984533 0.175201i \(-0.943942\pi\)
0.643995 + 0.765030i \(0.277276\pi\)
\(522\) 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
\(527\) 5.82846 0.253892
\(528\) 0 0
\(529\) −21.7349 −0.944996
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.35868 11.0136i −0.275425 0.477050i
\(534\) 0 0
\(535\) 15.3948 + 26.6646i 0.665575 + 1.15281i
\(536\) 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 0.997389 0.0722139i \(-0.0230064\pi\)
−0.561234 + 0.827657i \(0.689673\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.01655 + 5.22482i −0.128509 + 0.222585i
\(552\) 0 0
\(553\) −26.9114 27.4602i −1.14439 1.16773i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −17.7466 + 30.7381i −0.751950 + 1.30241i 0.194927 + 0.980818i \(0.437553\pi\)
−0.946877 + 0.321597i \(0.895780\pi\)
\(558\) 0 0
\(559\) −18.9727 −0.802458
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.64527 + 4.58175i 0.111485 + 0.193098i 0.916369 0.400334i \(-0.131106\pi\)
−0.804884 + 0.593432i \(0.797773\pi\)
\(564\) 0 0
\(565\) −8.09153 + 14.0149i −0.340413 + 0.589613i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.530225 + 0.918376i −0.0222282 + 0.0385003i −0.876926 0.480626i \(-0.840409\pi\)
0.854697 + 0.519127i \(0.173743\pi\)
\(570\) 0 0
\(571\) 14.5498 + 25.2010i 0.608890 + 1.05463i 0.991424 + 0.130686i \(0.0417181\pi\)
−0.382534 + 0.923941i \(0.624949\pi\)
\(572\) 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
\(581\) 21.5780 5.54897i 0.895206 0.230210i
\(582\) 0 0
\(583\) −24.1375 + 41.8074i −0.999673 + 1.73148i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.9863 22.4930i −0.536003 0.928385i −0.999114 0.0420843i \(-0.986600\pi\)
0.463111 0.886300i \(-0.346733\pi\)
\(588\) 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 0.964239 0.265035i \(-0.0853835\pi\)
−0.711646 + 0.702538i \(0.752050\pi\)
\(594\) 0 0
\(595\) −7.63844 7.79423i −0.313145 0.319532i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.8937 20.6005i −0.485964 0.841715i 0.513906 0.857847i \(-0.328198\pi\)
−0.999870 + 0.0161320i \(0.994865\pi\)
\(600\) 0 0
\(601\) 3.96006 + 6.85903i 0.161534 + 0.279785i 0.935419 0.353541i \(-0.115022\pi\)
−0.773885 + 0.633326i \(0.781689\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 26.9338 1.09501
\(606\) 0 0
\(607\) −0.0467764 −0.00189860 −0.000949298 1.00000i \(-0.500302\pi\)
−0.000949298 1.00000i \(0.500302\pi\)
\(608\) 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 −0.277708 0.960666i \(-0.589575\pi\)
0.970815 0.239831i \(-0.0770921\pi\)
\(618\) 0 0
\(619\) 47.5127 1.90970 0.954849 0.297092i \(-0.0960168\pi\)
0.954849 + 0.297092i \(0.0960168\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 41.2330 10.6034i 1.65196 0.424817i
\(624\) 0 0
\(625\) −28.5655 −1.14262
\(626\) 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
\(635\) −27.9201 −1.10798
\(636\) 0 0
\(637\) −20.9766 + 11.5526i −0.831124 + 0.457731i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.10533 0.122653 0.0613266 0.998118i \(-0.480467\pi\)
0.0613266 + 0.998118i \(0.480467\pi\)
\(642\) 0 0
\(643\) 8.86840 15.3605i 0.349736 0.605760i −0.636467 0.771304i \(-0.719605\pi\)
0.986202 + 0.165544i \(0.0529381\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.460060 + 0.796847i −0.0180868 + 0.0313273i −0.874927 0.484255i \(-0.839091\pi\)
0.856840 + 0.515582i \(0.172424\pi\)
\(648\) 0 0
\(649\) −9.70082 16.8023i −0.380790 0.659548i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.20275 −0.360131 −0.180066 0.983655i \(-0.557631\pi\)
−0.180066 + 0.983655i \(0.557631\pi\)
\(654\) 0 0
\(655\) 22.1248 0.864486
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −24.6999 42.7814i −0.962170 1.66653i −0.717032 0.697040i \(-0.754500\pi\)
−0.245138 0.969488i \(-0.578833\pi\)
\(660\) 0 0
\(661\) 24.9922 + 43.2878i 0.972085 + 1.68370i 0.689238 + 0.724535i \(0.257945\pi\)
0.282846 + 0.959165i \(0.408721\pi\)
\(662\) 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 0.760666 0.649143i \(-0.224872\pi\)
−0.942508 + 0.334185i \(0.891539\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −11.8937 + 20.6005i −0.457113 + 0.791743i −0.998807 0.0488331i \(-0.984450\pi\)
0.541694 + 0.840576i \(0.317783\pi\)
\(678\) 0 0
\(679\) −41.6375 + 10.7074i −1.59790 + 0.410914i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −17.3187 + 29.9969i −0.662683 + 1.14780i 0.317224 + 0.948350i \(0.397249\pi\)
−0.979908 + 0.199451i \(0.936084\pi\)
\(684\) 0 0
\(685\) 5.44625 0.208091
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17.5555 30.4071i −0.668813 1.15842i
\(690\) 0 0
\(691\) −17.9601 + 31.1077i −0.683233 + 1.18339i 0.290756 + 0.956797i \(0.406093\pi\)
−0.973989 + 0.226597i \(0.927240\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.8062 22.1809i 0.485765 0.841370i
\(696\) 0 0
\(697\) −3.16664 5.48477i −0.119945 0.207751i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.72529 0.367319 0.183659 0.982990i \(-0.441206\pi\)
0.183659 + 0.982990i \(0.441206\pi\)
\(702\) 0 0
\(703\) −5.48057 + 9.49263i −0.206704 + 0.358021i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.22722 33.0964i 0.347025 1.24472i
\(708\) 0 0
\(709\) 12.0555 20.8808i 0.452756 0.784196i −0.545801 0.837915i \(-0.683774\pi\)
0.998556 + 0.0537196i \(0.0171077\pi\)
\(710\) 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 0.999514 0.0311869i \(-0.00992872\pi\)
−0.526765 + 0.850011i \(0.676595\pi\)
\(720\) 0 0
\(721\) −4.97373 + 17.8399i −0.185232 + 0.664393i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.02627 3.50960i −0.0752537 0.130343i
\(726\) 0 0
\(727\) −23.7564 41.1472i −0.881075 1.52607i −0.850148 0.526545i \(-0.823487\pi\)
−0.0309272 0.999522i \(-0.509846\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −9.44841 −0.349462
\(732\) 0 0
\(733\) −30.8364 −1.13897 −0.569484 0.822003i \(-0.692857\pi\)
−0.569484 + 0.822003i \(0.692857\pi\)
\(734\) 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 −0.0638908 0.997957i \(-0.520351\pi\)
0.896201 0.443647i \(-0.146316\pi\)
\(744\) 0 0
\(745\) −59.0118 −2.16202
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 32.5868 8.37997i 1.19069 0.306197i
\(750\) 0 0
\(751\) 12.8421 0.468616 0.234308 0.972162i \(-0.424718\pi\)
0.234308 + 0.972162i \(0.424718\pi\)
\(752\) 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
\(761\) −41.4933 −1.50413 −0.752065 0.659088i \(-0.770942\pi\)
−0.752065 + 0.659088i \(0.770942\pi\)
\(762\) 0 0
\(763\) −13.7174 13.9971i −0.496602 0.506731i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.1111 0.509522
\(768\) 0 0
\(769\) 12.3450 21.3822i 0.445173 0.771061i −0.552892 0.833253i \(-0.686476\pi\)
0.998064 + 0.0621917i \(0.0198090\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 19.8256 34.3389i 0.713077 1.23508i −0.250620 0.968086i \(-0.580634\pi\)
0.963697 0.266999i \(-0.0860322\pi\)
\(774\) 0 0
\(775\) 1.47373 + 2.55258i 0.0529381 + 0.0916914i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.76801 −0.170832
\(780\) 0 0
\(781\) 70.3997 2.51910
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.95021 6.84197i −0.140989 0.244200i
\(786\) 0 0
\(787\) −9.01273 15.6105i −0.321269 0.556454i 0.659481 0.751721i \(-0.270776\pi\)
−0.980750 + 0.195267i \(0.937443\pi\)
\(788\) 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 −0.259927 0.965628i \(-0.583699\pi\)
0.966222 0.257711i \(-0.0829682\pi\)
\(798\) 0 0
\(799\) 10.0761 + 17.4522i 0.356465 + 0.617416i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.02928 8.71097i 0.177479 0.307403i
\(804\) 0 0
\(805\) 1.93487 6.94004i 0.0681953 0.244604i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −22.5848 + 39.1181i −0.794040 + 1.37532i 0.129407 + 0.991592i \(0.458693\pi\)
−0.923447 + 0.383726i \(0.874641\pi\)
\(810\) 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
\(815\) 0.184134 + 0.318929i 0.00644992 + 0.0111716i
\(816\) 0 0
\(817\) −3.55662 + 6.16025i −0.124431 + 0.215520i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −26.4572 + 45.8252i −0.923362 + 1.59931i −0.129187 + 0.991620i \(0.541237\pi\)
−0.794175 + 0.607690i \(0.792096\pi\)
\(822\) 0 0
\(823\) 15.6179 + 27.0510i 0.544407 + 0.942940i 0.998644 + 0.0520593i \(0.0165785\pi\)
−0.454237 + 0.890881i \(0.650088\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35.2243 −1.22487 −0.612435 0.790521i \(-0.709810\pi\)
−0.612435 + 0.790521i \(0.709810\pi\)
\(828\) 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
\(833\) −10.4464 + 5.75322i −0.361946 + 0.199337i
\(834\) 0 0
\(835\) 29.3383 50.8154i 1.01529 1.75854i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9.77004 16.9222i −0.337299 0.584219i 0.646625 0.762808i \(-0.276180\pi\)
−0.983924 + 0.178589i \(0.942847\pi\)
\(840\) 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
\(851\) −4.80602 8.32427i −0.164748 0.285352i
\(852\) 0 0
\(853\) −8.68715 15.0466i −0.297442 0.515185i 0.678108 0.734962i \(-0.262800\pi\)
−0.975550 + 0.219777i \(0.929467\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.5458 0.735992 0.367996 0.929827i \(-0.380044\pi\)
0.367996 + 0.929827i \(0.380044\pi\)
\(858\) 0 0
\(859\) 17.3743 0.592803 0.296402 0.955063i \(-0.404213\pi\)
0.296402 + 0.955063i \(0.404213\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.71053 15.0871i −0.296510 0.513570i 0.678825 0.734300i \(-0.262489\pi\)
−0.975335 + 0.220730i \(0.929156\pi\)
\(864\) 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
\(871\) 38.0586 1.28956
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −7.11914 + 25.5351i −0.240671 + 0.863244i
\(876\) 0 0
\(877\) 46.9981 1.58701 0.793507 0.608562i \(-0.208253\pi\)
0.793507 + 0.608562i \(0.208253\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 15.1715 0.511142 0.255571 0.966790i \(-0.417737\pi\)
0.255571 + 0.966790i \(0.417737\pi\)
\(882\) 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
\(887\) −2.86948 −0.0963477 −0.0481738 0.998839i \(-0.515340\pi\)
−0.0481738 + 0.998839i \(0.515340\pi\)
\(888\) 0 0
\(889\) −8.19398 + 29.3904i −0.274817 + 0.985721i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.1715 0.507696
\(894\) 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
\(905\) 41.4132 1.37662
\(906\) 0 0
\(907\) −32.4328 −1.07692 −0.538458 0.842653i \(-0.680993\pi\)
−0.538458 + 0.842653i \(0.680993\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 16.6219 + 28.7899i 0.550708 + 0.953854i 0.998224 + 0.0595777i \(0.0189754\pi\)
−0.447516 + 0.894276i \(0.647691\pi\)
\(912\) 0 0
\(913\) −19.8051 34.3034i −0.655452 1.13528i
\(914\) 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
\(929\) 27.2729 47.2381i 0.894795 1.54983i 0.0607376 0.998154i \(-0.480655\pi\)
0.834058 0.551677i \(-0.186012\pi\)
\(930\) 0 0
\(931\) −0.181255 + 8.97658i −0.00594039 + 0.294196i
\(932\) 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
\(941\) 9.64815 + 16.7111i 0.314521 + 0.544766i 0.979336 0.202242i \(-0.0648229\pi\)
−0.664815 + 0.747008i \(0.731490\pi\)
\(942\) 0 0
\(943\) 2.09058 3.62099i 0.0680787 0.117916i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.26608 + 9.12112i −0.171125 + 0.296396i −0.938813 0.344426i \(-0.888073\pi\)
0.767689 + 0.640823i \(0.221407\pi\)
\(948\) 0 0
\(949\) 3.65787 + 6.33561i 0.118739 + 0.205663i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 19.1970 0.621852 0.310926 0.950434i \(-0.399361\pi\)
0.310926 + 0.950434i \(0.399361\pi\)
\(954\) 0 0
\(955\) 27.8187 48.1835i 0.900193 1.55918i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.59836 5.73305i 0.0516139 0.185130i
\(960\) 0 0
\(961\) 9.64815 16.7111i 0.311231 0.539067i
\(962\) 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 −0.714257 0.699883i \(-0.753235\pi\)
0.963245 + 0.268623i \(0.0865686\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −0.904515 1.56667i −0.0290273 0.0502767i 0.851147 0.524928i \(-0.175908\pi\)
−0.880174 + 0.474651i \(0.842574\pi\)
\(972\) 0 0
\(973\) −19.5906 19.9901i −0.628045 0.640855i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.9493 + 32.8211i 0.606241 + 1.05004i 0.991854 + 0.127379i \(0.0406565\pi\)
−0.385613 + 0.922660i \(0.626010\pi\)
\(978\) 0 0
\(979\) −37.8451 65.5497i −1.20954 2.09498i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −38.1111 −1.21556 −0.607778 0.794107i \(-0.707939\pi\)
−0.607778 + 0.794107i \(0.707939\pi\)
\(984\) 0 0
\(985\) 8.66484 0.276085
\(986\) 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
\(997\) 14.8069 0.468938 0.234469 0.972124i \(-0.424665\pi\)
0.234469 + 0.972124i \(0.424665\pi\)
\(998\) 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.l.j.109.3 6
3.2 odd 2 2268.2.l.k.109.1 6
7.2 even 3 2268.2.i.k.2053.1 6
9.2 odd 6 2268.2.i.j.865.3 6
9.4 even 3 756.2.k.f.109.1 yes 6
9.5 odd 6 756.2.k.e.109.3 6
9.7 even 3 2268.2.i.k.865.1 6
21.2 odd 6 2268.2.i.j.2053.3 6
63.2 odd 6 2268.2.l.k.541.1 6
63.4 even 3 5292.2.a.u.1.3 3
63.16 even 3 inner 2268.2.l.j.541.3 6
63.23 odd 6 756.2.k.e.541.3 yes 6
63.31 odd 6 5292.2.a.w.1.1 3
63.32 odd 6 5292.2.a.x.1.1 3
63.58 even 3 756.2.k.f.541.1 yes 6
63.59 even 6 5292.2.a.v.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.k.e.109.3 6 9.5 odd 6
756.2.k.e.541.3 yes 6 63.23 odd 6
756.2.k.f.109.1 yes 6 9.4 even 3
756.2.k.f.541.1 yes 6 63.58 even 3
2268.2.i.j.865.3 6 9.2 odd 6
2268.2.i.j.2053.3 6 21.2 odd 6
2268.2.i.k.865.1 6 9.7 even 3
2268.2.i.k.2053.1 6 7.2 even 3
2268.2.l.j.109.3 6 1.1 even 1 trivial
2268.2.l.j.541.3 6 63.16 even 3 inner
2268.2.l.k.109.1 6 3.2 odd 2
2268.2.l.k.541.1 6 63.2 odd 6
5292.2.a.u.1.3 3 63.4 even 3
5292.2.a.v.1.3 3 63.59 even 6
5292.2.a.w.1.1 3 63.31 odd 6
5292.2.a.x.1.1 3 63.32 odd 6