Properties

Label 2268.2.x.j.1889.3
Level $2268$
Weight $2$
Character 2268.1889
Analytic conductor $18.110$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(377,2268)]
 
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, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.377");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.x (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1889.3
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2268.1889
Dual form 2268.2.x.j.377.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 1.50000i) q^{5} +(-1.62132 - 2.09077i) q^{7} +O(q^{10})\) \(q+(0.866025 + 1.50000i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(3.67423 + 2.12132i) q^{11} +(-2.12132 + 1.22474i) q^{13} -1.73205 q^{17} +2.44949i q^{19} +(-7.34847 + 4.24264i) q^{23} +(1.00000 - 1.73205i) q^{25} +(3.67423 + 2.12132i) q^{29} +(-6.36396 + 3.67423i) q^{31} +(1.73205 - 4.24264i) q^{35} +1.00000 q^{37} +(-0.866025 - 1.50000i) q^{41} +(-3.50000 + 6.06218i) q^{43} +(6.06218 - 10.5000i) q^{47} +(-1.74264 + 6.77962i) q^{49} +7.34847i q^{55} +(4.33013 + 7.50000i) q^{59} +(-2.12132 - 1.22474i) q^{61} +(-3.67423 - 2.12132i) q^{65} +(5.00000 + 8.66025i) q^{67} -8.48528i q^{71} +9.79796i q^{73} +(-1.52192 - 11.1213i) q^{77} +(-2.50000 + 4.33013i) q^{79} +(-6.06218 + 10.5000i) q^{83} +(-1.50000 - 2.59808i) q^{85} -10.3923 q^{89} +(6.00000 + 2.44949i) q^{91} +(-3.67423 + 2.12132i) q^{95} +(2.12132 + 1.22474i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} + 8 q^{25} + 8 q^{37} - 28 q^{43} + 20 q^{49} + 40 q^{67} - 20 q^{79} - 12 q^{85} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) 0.866025 + 1.50000i 0.387298 + 0.670820i 0.992085 0.125567i \(-0.0400750\pi\)
−0.604787 + 0.796387i \(0.706742\pi\)
\(6\) 0 0
\(7\) −1.62132 2.09077i −0.612801 0.790237i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.67423 + 2.12132i 1.10782 + 0.639602i 0.938265 0.345918i \(-0.112432\pi\)
0.169559 + 0.985520i \(0.445766\pi\)
\(12\) 0 0
\(13\) −2.12132 + 1.22474i −0.588348 + 0.339683i −0.764444 0.644690i \(-0.776986\pi\)
0.176096 + 0.984373i \(0.443653\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.73205 −0.420084 −0.210042 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 0 0
\(19\) 2.44949i 0.561951i 0.959715 + 0.280976i \(0.0906580\pi\)
−0.959715 + 0.280976i \(0.909342\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.34847 + 4.24264i −1.53226 + 0.884652i −0.533005 + 0.846112i \(0.678937\pi\)
−0.999257 + 0.0385394i \(0.987729\pi\)
\(24\) 0 0
\(25\) 1.00000 1.73205i 0.200000 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.67423 + 2.12132i 0.682288 + 0.393919i 0.800717 0.599043i \(-0.204452\pi\)
−0.118428 + 0.992963i \(0.537786\pi\)
\(30\) 0 0
\(31\) −6.36396 + 3.67423i −1.14300 + 0.659912i −0.947172 0.320726i \(-0.896073\pi\)
−0.195829 + 0.980638i \(0.562740\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.73205 4.24264i 0.292770 0.717137i
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.866025 1.50000i −0.135250 0.234261i 0.790443 0.612536i \(-0.209851\pi\)
−0.925693 + 0.378275i \(0.876517\pi\)
\(42\) 0 0
\(43\) −3.50000 + 6.06218i −0.533745 + 0.924473i 0.465478 + 0.885059i \(0.345882\pi\)
−0.999223 + 0.0394140i \(0.987451\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.06218 10.5000i 0.884260 1.53158i 0.0376995 0.999289i \(-0.487997\pi\)
0.846560 0.532293i \(-0.178670\pi\)
\(48\) 0 0
\(49\) −1.74264 + 6.77962i −0.248949 + 0.968517i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 7.34847i 0.990867i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.33013 + 7.50000i 0.563735 + 0.976417i 0.997166 + 0.0752304i \(0.0239692\pi\)
−0.433432 + 0.901186i \(0.642697\pi\)
\(60\) 0 0
\(61\) −2.12132 1.22474i −0.271607 0.156813i 0.358011 0.933718i \(-0.383455\pi\)
−0.629618 + 0.776905i \(0.716789\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.67423 2.12132i −0.455733 0.263117i
\(66\) 0 0
\(67\) 5.00000 + 8.66025i 0.610847 + 1.05802i 0.991098 + 0.133135i \(0.0425044\pi\)
−0.380251 + 0.924883i \(0.624162\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.48528i 1.00702i −0.863990 0.503509i \(-0.832042\pi\)
0.863990 0.503509i \(-0.167958\pi\)
\(72\) 0 0
\(73\) 9.79796i 1.14676i 0.819288 + 0.573382i \(0.194369\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.52192 11.1213i −0.173439 1.26739i
\(78\) 0 0
\(79\) −2.50000 + 4.33013i −0.281272 + 0.487177i −0.971698 0.236225i \(-0.924090\pi\)
0.690426 + 0.723403i \(0.257423\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.06218 + 10.5000i −0.665410 + 1.15252i 0.313763 + 0.949501i \(0.398410\pi\)
−0.979174 + 0.203024i \(0.934923\pi\)
\(84\) 0 0
\(85\) −1.50000 2.59808i −0.162698 0.281801i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.3923 −1.10158 −0.550791 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) 0 0
\(91\) 6.00000 + 2.44949i 0.628971 + 0.256776i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.67423 + 2.12132i −0.376969 + 0.217643i
\(96\) 0 0
\(97\) 2.12132 + 1.22474i 0.215387 + 0.124354i 0.603813 0.797126i \(-0.293647\pi\)
−0.388425 + 0.921480i \(0.626981\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.19615 + 9.00000i −0.517036 + 0.895533i 0.482768 + 0.875748i \(0.339632\pi\)
−0.999804 + 0.0197851i \(0.993702\pi\)
\(102\) 0 0
\(103\) −16.9706 + 9.79796i −1.67216 + 0.965422i −0.705733 + 0.708478i \(0.749382\pi\)
−0.966426 + 0.256943i \(0.917285\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.48528i 0.820303i 0.912017 + 0.410152i \(0.134524\pi\)
−0.912017 + 0.410152i \(0.865476\pi\)
\(108\) 0 0
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.0227 6.36396i 1.03693 0.598671i 0.117967 0.993018i \(-0.462362\pi\)
0.918962 + 0.394346i \(0.129029\pi\)
\(114\) 0 0
\(115\) −12.7279 7.34847i −1.18688 0.685248i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.80821 + 3.62132i 0.257428 + 0.331966i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244 1.08444
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.19615 + 9.00000i 0.453990 + 0.786334i 0.998630 0.0523366i \(-0.0166669\pi\)
−0.544640 + 0.838670i \(0.683334\pi\)
\(132\) 0 0
\(133\) 5.12132 3.97141i 0.444075 0.344365i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.3712 10.6066i −1.56956 0.906183i −0.996220 0.0868620i \(-0.972316\pi\)
−0.573335 0.819321i \(-0.694351\pi\)
\(138\) 0 0
\(139\) −10.6066 + 6.12372i −0.899640 + 0.519408i −0.877083 0.480338i \(-0.840514\pi\)
−0.0225568 + 0.999746i \(0.507181\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.3923 −0.869048
\(144\) 0 0
\(145\) 7.34847i 0.610257i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.0227 6.36396i 0.903015 0.521356i 0.0248379 0.999691i \(-0.492093\pi\)
0.878177 + 0.478335i \(0.158760\pi\)
\(150\) 0 0
\(151\) 2.50000 4.33013i 0.203447 0.352381i −0.746190 0.665733i \(-0.768119\pi\)
0.949637 + 0.313353i \(0.101452\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −11.0227 6.36396i −0.885365 0.511166i
\(156\) 0 0
\(157\) −6.36396 + 3.67423i −0.507899 + 0.293236i −0.731970 0.681337i \(-0.761399\pi\)
0.224070 + 0.974573i \(0.428065\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.7846 + 8.48528i 1.63806 + 0.668734i
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.52628 16.5000i −0.737166 1.27681i −0.953767 0.300548i \(-0.902830\pi\)
0.216601 0.976260i \(-0.430503\pi\)
\(168\) 0 0
\(169\) −3.50000 + 6.06218i −0.269231 + 0.466321i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.3923 18.0000i 0.790112 1.36851i −0.135785 0.990738i \(-0.543356\pi\)
0.925897 0.377776i \(-0.123311\pi\)
\(174\) 0 0
\(175\) −5.24264 + 0.717439i −0.396306 + 0.0542333i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.24264i 0.317110i 0.987350 + 0.158555i \(0.0506835\pi\)
−0.987350 + 0.158555i \(0.949317\pi\)
\(180\) 0 0
\(181\) 4.89898i 0.364138i 0.983286 + 0.182069i \(0.0582795\pi\)
−0.983286 + 0.182069i \(0.941721\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.866025 + 1.50000i 0.0636715 + 0.110282i
\(186\) 0 0
\(187\) −6.36396 3.67423i −0.465379 0.268687i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.34847 + 4.24264i 0.531717 + 0.306987i 0.741715 0.670715i \(-0.234013\pi\)
−0.209999 + 0.977702i \(0.567346\pi\)
\(192\) 0 0
\(193\) −3.50000 6.06218i −0.251936 0.436365i 0.712123 0.702055i \(-0.247734\pi\)
−0.964059 + 0.265689i \(0.914400\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.2132i 1.51138i 0.654931 + 0.755689i \(0.272698\pi\)
−0.654931 + 0.755689i \(0.727302\pi\)
\(198\) 0 0
\(199\) 9.79796i 0.694559i −0.937762 0.347279i \(-0.887106\pi\)
0.937762 0.347279i \(-0.112894\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.52192 11.1213i −0.106818 0.780564i
\(204\) 0 0
\(205\) 1.50000 2.59808i 0.104765 0.181458i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.19615 + 9.00000i −0.359425 + 0.622543i
\(210\) 0 0
\(211\) 4.00000 + 6.92820i 0.275371 + 0.476957i 0.970229 0.242190i \(-0.0778659\pi\)
−0.694857 + 0.719148i \(0.744533\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −12.1244 −0.826874
\(216\) 0 0
\(217\) 18.0000 + 7.34847i 1.22192 + 0.498847i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.67423 2.12132i 0.247156 0.142695i
\(222\) 0 0
\(223\) −19.0919 11.0227i −1.27849 0.738135i −0.301917 0.953334i \(-0.597627\pi\)
−0.976570 + 0.215199i \(0.930960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 12.7279 7.34847i 0.841085 0.485601i −0.0165480 0.999863i \(-0.505268\pi\)
0.857633 + 0.514263i \(0.171934\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.24264i 0.277945i −0.990296 0.138972i \(-0.955620\pi\)
0.990296 0.138972i \(-0.0443799\pi\)
\(234\) 0 0
\(235\) 21.0000 1.36989
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −18.3712 + 10.6066i −1.18833 + 0.686084i −0.957927 0.287011i \(-0.907338\pi\)
−0.230405 + 0.973095i \(0.574005\pi\)
\(240\) 0 0
\(241\) 23.3345 + 13.4722i 1.50311 + 0.867820i 0.999994 + 0.00360098i \(0.00114623\pi\)
0.503115 + 0.864219i \(0.332187\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −11.6786 + 3.25736i −0.746118 + 0.208105i
\(246\) 0 0
\(247\) −3.00000 5.19615i −0.190885 0.330623i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 19.0526 1.20259 0.601293 0.799028i \(-0.294652\pi\)
0.601293 + 0.799028i \(0.294652\pi\)
\(252\) 0 0
\(253\) −36.0000 −2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.19615 + 9.00000i 0.324127 + 0.561405i 0.981335 0.192304i \(-0.0615961\pi\)
−0.657208 + 0.753709i \(0.728263\pi\)
\(258\) 0 0
\(259\) −1.62132 2.09077i −0.100744 0.129914i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.3712 + 10.6066i 1.13282 + 0.654031i 0.944641 0.328105i \(-0.106410\pi\)
0.188174 + 0.982136i \(0.439743\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.73205 0.105605 0.0528025 0.998605i \(-0.483185\pi\)
0.0528025 + 0.998605i \(0.483185\pi\)
\(270\) 0 0
\(271\) 4.89898i 0.297592i 0.988868 + 0.148796i \(0.0475397\pi\)
−0.988868 + 0.148796i \(0.952460\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.34847 4.24264i 0.443129 0.255841i
\(276\) 0 0
\(277\) 3.50000 6.06218i 0.210295 0.364241i −0.741512 0.670940i \(-0.765891\pi\)
0.951807 + 0.306699i \(0.0992243\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.34847 + 4.24264i 0.438373 + 0.253095i 0.702907 0.711282i \(-0.251885\pi\)
−0.264534 + 0.964376i \(0.585218\pi\)
\(282\) 0 0
\(283\) 6.36396 3.67423i 0.378298 0.218411i −0.298779 0.954322i \(-0.596579\pi\)
0.677078 + 0.735912i \(0.263246\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.73205 + 4.24264i −0.102240 + 0.250435i
\(288\) 0 0
\(289\) −14.0000 −0.823529
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.33013 + 7.50000i 0.252969 + 0.438155i 0.964342 0.264660i \(-0.0852597\pi\)
−0.711373 + 0.702815i \(0.751926\pi\)
\(294\) 0 0
\(295\) −7.50000 + 12.9904i −0.436667 + 0.756329i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.3923 18.0000i 0.601003 1.04097i
\(300\) 0 0
\(301\) 18.3492 2.51104i 1.05763 0.144734i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.24264i 0.242933i
\(306\) 0 0
\(307\) 29.3939i 1.67760i −0.544442 0.838799i \(-0.683259\pi\)
0.544442 0.838799i \(-0.316741\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.2583 + 19.5000i 0.638401 + 1.10574i 0.985784 + 0.168020i \(0.0537373\pi\)
−0.347382 + 0.937724i \(0.612929\pi\)
\(312\) 0 0
\(313\) 21.2132 + 12.2474i 1.19904 + 0.692267i 0.960341 0.278827i \(-0.0899455\pi\)
0.238700 + 0.971093i \(0.423279\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 9.00000 + 15.5885i 0.503903 + 0.872786i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.24264i 0.236067i
\(324\) 0 0
\(325\) 4.89898i 0.271746i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −31.7818 + 4.34924i −1.75219 + 0.239781i
\(330\) 0 0
\(331\) −6.50000 + 11.2583i −0.357272 + 0.618814i −0.987504 0.157593i \(-0.949627\pi\)
0.630232 + 0.776407i \(0.282960\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.66025 + 15.0000i −0.473160 + 0.819538i
\(336\) 0 0
\(337\) −12.5000 21.6506i −0.680918 1.17939i −0.974701 0.223513i \(-0.928247\pi\)
0.293783 0.955872i \(-0.405086\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −31.1769 −1.68832
\(342\) 0 0
\(343\) 17.0000 7.34847i 0.917914 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.3712 10.6066i 0.986216 0.569392i 0.0820751 0.996626i \(-0.473845\pi\)
0.904141 + 0.427234i \(0.140512\pi\)
\(348\) 0 0
\(349\) 16.9706 + 9.79796i 0.908413 + 0.524473i 0.879920 0.475121i \(-0.157596\pi\)
0.0284931 + 0.999594i \(0.490929\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.866025 1.50000i 0.0460939 0.0798369i −0.842058 0.539387i \(-0.818656\pi\)
0.888152 + 0.459550i \(0.151989\pi\)
\(354\) 0 0
\(355\) 12.7279 7.34847i 0.675528 0.390016i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.9411i 1.79134i −0.444715 0.895672i \(-0.646695\pi\)
0.444715 0.895672i \(-0.353305\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −14.6969 + 8.48528i −0.769273 + 0.444140i
\(366\) 0 0
\(367\) −4.24264 2.44949i −0.221464 0.127862i 0.385164 0.922848i \(-0.374145\pi\)
−0.606628 + 0.794986i \(0.707478\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5.50000 9.52628i −0.284779 0.493252i 0.687776 0.725923i \(-0.258587\pi\)
−0.972556 + 0.232671i \(0.925254\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.3923 −0.535231
\(378\) 0 0
\(379\) −35.0000 −1.79783 −0.898915 0.438124i \(-0.855643\pi\)
−0.898915 + 0.438124i \(0.855643\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.52628 + 16.5000i 0.486770 + 0.843111i 0.999884 0.0152097i \(-0.00484160\pi\)
−0.513114 + 0.858320i \(0.671508\pi\)
\(384\) 0 0
\(385\) 15.3640 11.9142i 0.783020 0.607205i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 25.7196 + 14.8492i 1.30404 + 0.752886i 0.981094 0.193532i \(-0.0619942\pi\)
0.322944 + 0.946418i \(0.395328\pi\)
\(390\) 0 0
\(391\) 12.7279 7.34847i 0.643679 0.371628i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.66025 −0.435745
\(396\) 0 0
\(397\) 29.3939i 1.47524i −0.675218 0.737618i \(-0.735950\pi\)
0.675218 0.737618i \(-0.264050\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.34847 + 4.24264i −0.366965 + 0.211867i −0.672132 0.740432i \(-0.734621\pi\)
0.305167 + 0.952299i \(0.401288\pi\)
\(402\) 0 0
\(403\) 9.00000 15.5885i 0.448322 0.776516i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.67423 + 2.12132i 0.182125 + 0.105150i
\(408\) 0 0
\(409\) 33.9411 19.5959i 1.67828 0.968956i 0.715523 0.698589i \(-0.246188\pi\)
0.962757 0.270367i \(-0.0871450\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.66025 21.2132i 0.426143 1.04383i
\(414\) 0 0
\(415\) −21.0000 −1.03085
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.7224 25.5000i −0.719238 1.24576i −0.961302 0.275496i \(-0.911158\pi\)
0.242064 0.970260i \(-0.422176\pi\)
\(420\) 0 0
\(421\) −10.0000 + 17.3205i −0.487370 + 0.844150i −0.999895 0.0145228i \(-0.995377\pi\)
0.512524 + 0.858673i \(0.328710\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.73205 + 3.00000i −0.0840168 + 0.145521i
\(426\) 0 0
\(427\) 0.878680 + 6.42090i 0.0425223 + 0.310729i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.9706i 0.817443i −0.912659 0.408722i \(-0.865975\pi\)
0.912659 0.408722i \(-0.134025\pi\)
\(432\) 0 0
\(433\) 22.0454i 1.05943i 0.848174 + 0.529717i \(0.177702\pi\)
−0.848174 + 0.529717i \(0.822298\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.3923 18.0000i −0.497131 0.861057i
\(438\) 0 0
\(439\) −25.4558 14.6969i −1.21494 0.701447i −0.251110 0.967959i \(-0.580795\pi\)
−0.963832 + 0.266512i \(0.914129\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11.0227 6.36396i −0.523704 0.302361i 0.214745 0.976670i \(-0.431108\pi\)
−0.738449 + 0.674309i \(0.764441\pi\)
\(444\) 0 0
\(445\) −9.00000 15.5885i −0.426641 0.738964i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.2132i 1.00111i −0.865704 0.500556i \(-0.833129\pi\)
0.865704 0.500556i \(-0.166871\pi\)
\(450\) 0 0
\(451\) 7.34847i 0.346026i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.52192 + 11.1213i 0.0713486 + 0.521376i
\(456\) 0 0
\(457\) 4.00000 6.92820i 0.187112 0.324088i −0.757174 0.653213i \(-0.773421\pi\)
0.944286 + 0.329125i \(0.106754\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.7224 25.5000i 0.685692 1.18765i −0.287527 0.957773i \(-0.592833\pi\)
0.973219 0.229881i \(-0.0738336\pi\)
\(462\) 0 0
\(463\) −14.5000 25.1147i −0.673872 1.16718i −0.976797 0.214166i \(-0.931297\pi\)
0.302925 0.953014i \(-0.402037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) 10.0000 24.4949i 0.461757 1.13107i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −25.7196 + 14.8492i −1.18259 + 0.682769i
\(474\) 0 0
\(475\) 4.24264 + 2.44949i 0.194666 + 0.112390i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19.9186 34.5000i 0.910103 1.57635i 0.0961869 0.995363i \(-0.469335\pi\)
0.813916 0.580982i \(-0.197331\pi\)
\(480\) 0 0
\(481\) −2.12132 + 1.22474i −0.0967239 + 0.0558436i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.24264i 0.192648i
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.0227 6.36396i 0.497448 0.287202i −0.230211 0.973141i \(-0.573942\pi\)
0.727659 + 0.685939i \(0.240608\pi\)
\(492\) 0 0
\(493\) −6.36396 3.67423i −0.286618 0.165479i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −17.7408 + 13.7574i −0.795782 + 0.617102i
\(498\) 0 0
\(499\) −9.50000 16.4545i −0.425278 0.736604i 0.571168 0.820833i \(-0.306490\pi\)
−0.996446 + 0.0842294i \(0.973157\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.73205 0.0772283 0.0386142 0.999254i \(-0.487706\pi\)
0.0386142 + 0.999254i \(0.487706\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.52628 + 16.5000i 0.422245 + 0.731350i 0.996159 0.0875661i \(-0.0279089\pi\)
−0.573914 + 0.818916i \(0.694576\pi\)
\(510\) 0 0
\(511\) 20.4853 15.8856i 0.906215 0.702739i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −29.3939 16.9706i −1.29525 0.747812i
\(516\) 0 0
\(517\) 44.5477 25.7196i 1.95921 1.13115i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.73205 0.0758825 0.0379413 0.999280i \(-0.487920\pi\)
0.0379413 + 0.999280i \(0.487920\pi\)
\(522\) 0 0
\(523\) 31.8434i 1.39241i −0.717841 0.696207i \(-0.754870\pi\)
0.717841 0.696207i \(-0.245130\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.0227 6.36396i 0.480157 0.277218i
\(528\) 0 0
\(529\) 24.5000 42.4352i 1.06522 1.84501i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.67423 + 2.12132i 0.159149 + 0.0918846i
\(534\) 0 0
\(535\) −12.7279 + 7.34847i −0.550276 + 0.317702i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −20.7846 + 21.2132i −0.895257 + 0.913717i
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 11.2583 + 19.5000i 0.482254 + 0.835288i
\(546\) 0 0
\(547\) −20.5000 + 35.5070i −0.876517 + 1.51817i −0.0213785 + 0.999771i \(0.506805\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.19615 + 9.00000i −0.221364 + 0.383413i
\(552\) 0 0
\(553\) 13.1066 1.79360i 0.557349 0.0762715i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.48528i 0.359533i 0.983709 + 0.179766i \(0.0575342\pi\)
−0.983709 + 0.179766i \(0.942466\pi\)
\(558\) 0 0
\(559\) 17.1464i 0.725217i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.3923 18.0000i −0.437983 0.758610i 0.559550 0.828796i \(-0.310974\pi\)
−0.997534 + 0.0701867i \(0.977640\pi\)
\(564\) 0 0
\(565\) 19.0919 + 11.0227i 0.803202 + 0.463729i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 29.3939 + 16.9706i 1.23226 + 0.711443i 0.967500 0.252872i \(-0.0813753\pi\)
0.264756 + 0.964315i \(0.414709\pi\)
\(570\) 0 0
\(571\) 12.5000 + 21.6506i 0.523109 + 0.906051i 0.999638 + 0.0268925i \(0.00856117\pi\)
−0.476530 + 0.879158i \(0.658105\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.9706i 0.707721i
\(576\) 0 0
\(577\) 19.5959i 0.815789i −0.913029 0.407894i \(-0.866263\pi\)
0.913029 0.407894i \(-0.133737\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 31.7818 4.34924i 1.31853 0.180437i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.7846 + 36.0000i −0.857873 + 1.48588i 0.0160815 + 0.999871i \(0.494881\pi\)
−0.873954 + 0.486008i \(0.838452\pi\)
\(588\) 0 0
\(589\) −9.00000 15.5885i −0.370839 0.642311i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −8.66025 −0.355634 −0.177817 0.984064i \(-0.556904\pi\)
−0.177817 + 0.984064i \(0.556904\pi\)
\(594\) 0 0
\(595\) −3.00000 + 7.34847i −0.122988 + 0.301258i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.67423 + 2.12132i −0.150125 + 0.0866748i −0.573181 0.819429i \(-0.694291\pi\)
0.423056 + 0.906104i \(0.360957\pi\)
\(600\) 0 0
\(601\) 19.0919 + 11.0227i 0.778774 + 0.449625i 0.835996 0.548736i \(-0.184891\pi\)
−0.0572215 + 0.998362i \(0.518224\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.06218 + 10.5000i −0.246463 + 0.426886i
\(606\) 0 0
\(607\) 23.3345 13.4722i 0.947119 0.546819i 0.0549343 0.998490i \(-0.482505\pi\)
0.892185 + 0.451671i \(0.149172\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 29.6985i 1.20147i
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0454 12.7279i 0.887515 0.512407i 0.0143859 0.999897i \(-0.495421\pi\)
0.873129 + 0.487490i \(0.162087\pi\)
\(618\) 0 0
\(619\) 2.12132 + 1.22474i 0.0852631 + 0.0492267i 0.542025 0.840362i \(-0.317658\pi\)
−0.456762 + 0.889589i \(0.650991\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 16.8493 + 21.7279i 0.675051 + 0.870511i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.73205 −0.0690614
\(630\) 0 0
\(631\) −37.0000 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.2583 19.5000i −0.446773 0.773834i
\(636\) 0 0
\(637\) −4.60660 16.5160i −0.182520 0.654389i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.3712 10.6066i −0.725618 0.418936i 0.0911991 0.995833i \(-0.470930\pi\)
−0.816817 + 0.576897i \(0.804263\pi\)
\(642\) 0 0
\(643\) −12.7279 + 7.34847i −0.501940 + 0.289795i −0.729514 0.683965i \(-0.760254\pi\)
0.227574 + 0.973761i \(0.426921\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.7846 0.817127 0.408564 0.912730i \(-0.366030\pi\)
0.408564 + 0.912730i \(0.366030\pi\)
\(648\) 0 0
\(649\) 36.7423i 1.44226i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −25.7196 + 14.8492i −1.00649 + 0.581096i −0.910161 0.414254i \(-0.864042\pi\)
−0.0963261 + 0.995350i \(0.530709\pi\)
\(654\) 0 0
\(655\) −9.00000 + 15.5885i −0.351659 + 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −14.6969 8.48528i −0.572511 0.330540i 0.185640 0.982618i \(-0.440564\pi\)
−0.758152 + 0.652078i \(0.773897\pi\)
\(660\) 0 0
\(661\) 10.6066 6.12372i 0.412549 0.238185i −0.279335 0.960194i \(-0.590114\pi\)
0.691884 + 0.722008i \(0.256781\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10.3923 + 4.24264i 0.402996 + 0.164523i
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.19615 9.00000i −0.200595 0.347441i
\(672\) 0 0
\(673\) −8.00000 + 13.8564i −0.308377 + 0.534125i −0.978008 0.208569i \(-0.933119\pi\)
0.669630 + 0.742695i \(0.266453\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15.5885 + 27.0000i −0.599113 + 1.03769i 0.393839 + 0.919179i \(0.371147\pi\)
−0.992952 + 0.118515i \(0.962187\pi\)
\(678\) 0 0
\(679\) −0.878680 6.42090i −0.0337206 0.246411i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 38.1838i 1.46106i 0.682880 + 0.730531i \(0.260727\pi\)
−0.682880 + 0.730531i \(0.739273\pi\)
\(684\) 0 0
\(685\) 36.7423i 1.40385i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 19.0919 + 11.0227i 0.726289 + 0.419323i 0.817063 0.576548i \(-0.195601\pi\)
−0.0907737 + 0.995872i \(0.528934\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −18.3712 10.6066i −0.696858 0.402331i
\(696\) 0 0
\(697\) 1.50000 + 2.59808i 0.0568166 + 0.0984092i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.7279i 0.480727i −0.970683 0.240363i \(-0.922733\pi\)
0.970683 0.240363i \(-0.0772666\pi\)
\(702\) 0 0
\(703\) 2.44949i 0.0923843i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.2416 3.72792i 1.02452 0.140203i
\(708\) 0 0
\(709\) −24.5000 + 42.4352i −0.920117 + 1.59369i −0.120885 + 0.992667i \(0.538573\pi\)
−0.799232 + 0.601023i \(0.794760\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 31.1769 54.0000i 1.16758 2.02232i
\(714\) 0 0
\(715\) −9.00000 15.5885i −0.336581 0.582975i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32.9090 1.22730 0.613649 0.789579i \(-0.289701\pi\)
0.613649 + 0.789579i \(0.289701\pi\)
\(720\) 0 0
\(721\) 48.0000 + 19.5959i 1.78761 + 0.729790i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.34847 4.24264i 0.272915 0.157568i
\(726\) 0 0
\(727\) 6.36396 + 3.67423i 0.236026 + 0.136270i 0.613349 0.789812i \(-0.289822\pi\)
−0.377323 + 0.926082i \(0.623155\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.06218 10.5000i 0.224218 0.388357i
\(732\) 0 0
\(733\) 25.4558 14.6969i 0.940233 0.542844i 0.0501997 0.998739i \(-0.484014\pi\)
0.890033 + 0.455895i \(0.150681\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.4264i 1.56280i
\(738\) 0 0
\(739\) 38.0000 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.3712 10.6066i 0.673973 0.389118i −0.123607 0.992331i \(-0.539446\pi\)
0.797580 + 0.603213i \(0.206113\pi\)
\(744\) 0 0
\(745\) 19.0919 + 11.0227i 0.699472 + 0.403841i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 17.7408 13.7574i 0.648234 0.502683i
\(750\) 0 0
\(751\) −5.00000 8.66025i −0.182453 0.316017i 0.760263 0.649616i \(-0.225070\pi\)
−0.942715 + 0.333599i \(0.891737\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.66025 0.315179
\(756\) 0 0
\(757\) −53.0000 −1.92632 −0.963159 0.268933i \(-0.913329\pi\)
−0.963159 + 0.268933i \(0.913329\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −11.2583 19.5000i −0.408114 0.706874i 0.586564 0.809903i \(-0.300480\pi\)
−0.994678 + 0.103028i \(0.967147\pi\)
\(762\) 0 0
\(763\) −21.0772 27.1800i −0.763045 0.983983i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.3712 10.6066i −0.663345 0.382982i
\(768\) 0 0
\(769\) −4.24264 + 2.44949i −0.152994 + 0.0883309i −0.574542 0.818475i \(-0.694820\pi\)
0.421549 + 0.906806i \(0.361487\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.8372 1.43284 0.716422 0.697668i \(-0.245779\pi\)
0.716422 + 0.697668i \(0.245779\pi\)
\(774\) 0 0
\(775\) 14.6969i 0.527930i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.67423 2.12132i 0.131643 0.0760042i
\(780\) 0 0
\(781\) 18.0000 31.1769i 0.644091 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −11.0227 6.36396i −0.393417 0.227140i
\(786\) 0 0
\(787\) 4.24264 2.44949i 0.151234 0.0873149i −0.422473 0.906375i \(-0.638838\pi\)
0.573707 + 0.819060i \(0.305505\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −31.1769 12.7279i −1.10852 0.452553i
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.19615 + 9.00000i 0.184057 + 0.318796i 0.943258 0.332060i \(-0.107744\pi\)
−0.759201 + 0.650856i \(0.774410\pi\)
\(798\) 0 0
\(799\) −10.5000 + 18.1865i −0.371463 + 0.643393i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −20.7846 + 36.0000i −0.733473 + 1.27041i
\(804\) 0 0
\(805\) 5.27208 + 38.5254i 0.185816 + 1.35784i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 25.4558i 0.894980i −0.894289 0.447490i \(-0.852318\pi\)
0.894289 0.447490i \(-0.147682\pi\)
\(810\) 0 0
\(811\) 51.4393i 1.80628i 0.429349 + 0.903139i \(0.358743\pi\)
−0.429349 + 0.903139i \(0.641257\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.866025 1.50000i −0.0303355 0.0525427i
\(816\) 0 0
\(817\) −14.8492 8.57321i −0.519509 0.299939i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −22.0454 12.7279i −0.769390 0.444208i 0.0632669 0.997997i \(-0.479848\pi\)
−0.832657 + 0.553789i \(0.813181\pi\)
\(822\) 0 0
\(823\) 12.5000 + 21.6506i 0.435723 + 0.754694i 0.997354 0.0726937i \(-0.0231595\pi\)
−0.561632 + 0.827387i \(0.689826\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.1838i 1.32778i 0.747830 + 0.663890i \(0.231096\pi\)
−0.747830 + 0.663890i \(0.768904\pi\)
\(828\) 0 0
\(829\) 31.8434i 1.10597i −0.833193 0.552983i \(-0.813489\pi\)
0.833193 0.552983i \(-0.186511\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.01834 11.7426i 0.104579 0.406858i
\(834\) 0 0
\(835\) 16.5000 28.5788i 0.571006 0.989011i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0.866025 1.50000i 0.0298985 0.0517858i −0.850689 0.525669i \(-0.823815\pi\)
0.880587 + 0.473884i \(0.157148\pi\)
\(840\) 0 0
\(841\) −5.50000 9.52628i −0.189655 0.328492i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.1244 −0.417091
\(846\) 0 0
\(847\) 7.00000 17.1464i 0.240523 0.589158i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −7.34847 + 4.24264i −0.251902 + 0.145436i
\(852\) 0 0
\(853\) −16.9706 9.79796i −0.581061 0.335476i 0.180494 0.983576i \(-0.442230\pi\)
−0.761555 + 0.648100i \(0.775564\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.06218 10.5000i 0.207080 0.358673i −0.743713 0.668499i \(-0.766937\pi\)
0.950793 + 0.309825i \(0.100271\pi\)
\(858\) 0 0
\(859\) −33.9411 + 19.5959i −1.15806 + 0.668604i −0.950837 0.309691i \(-0.899775\pi\)
−0.207219 + 0.978295i \(0.566441\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.9706i 0.577685i 0.957377 + 0.288842i \(0.0932703\pi\)
−0.957377 + 0.288842i \(0.906730\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.3712 + 10.6066i −0.623199 + 0.359804i
\(870\) 0 0
\(871\) −21.2132 12.2474i −0.718782 0.414989i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −19.6575 25.3492i −0.664544 0.856961i
\(876\) 0 0
\(877\) 3.50000 + 6.06218i 0.118187 + 0.204705i 0.919049 0.394143i \(-0.128959\pi\)
−0.800862 + 0.598848i \(0.795625\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −31.1769 −1.05038 −0.525188 0.850986i \(-0.676005\pi\)
−0.525188 + 0.850986i \(0.676005\pi\)
\(882\) 0 0
\(883\) −17.0000 −0.572096 −0.286048 0.958215i \(-0.592342\pi\)
−0.286048 + 0.958215i \(0.592342\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −9.52628 16.5000i −0.319861 0.554016i 0.660598 0.750740i \(-0.270303\pi\)
−0.980459 + 0.196724i \(0.936970\pi\)
\(888\) 0 0
\(889\) 21.0772 + 27.1800i 0.706905 + 0.911588i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.7196 + 14.8492i 0.860675 + 0.496911i
\(894\) 0 0
\(895\) −6.36396 + 3.67423i −0.212724 + 0.122816i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −31.1769 −1.03981
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.34847 + 4.24264i −0.244271 + 0.141030i
\(906\) 0 0
\(907\) −6.50000 + 11.2583i −0.215829 + 0.373827i −0.953529 0.301302i \(-0.902579\pi\)
0.737700 + 0.675129i \(0.235912\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 25.7196 + 14.8492i 0.852130 + 0.491977i 0.861369 0.507980i \(-0.169608\pi\)
−0.00923912 + 0.999957i \(0.502941\pi\)
\(912\) 0 0
\(913\) −44.5477 + 25.7196i −1.47431 + 0.851196i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.3923 25.4558i 0.343184 0.840626i
\(918\) 0 0
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 10.3923 + 18.0000i 0.342067 + 0.592477i
\(924\) 0 0
\(925\) 1.00000 1.73205i 0.0328798 0.0569495i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19.9186 + 34.5000i −0.653508 + 1.13191i 0.328758 + 0.944414i \(0.393370\pi\)
−0.982266 + 0.187494i \(0.939963\pi\)
\(930\) 0 0
\(931\) −16.6066 4.26858i −0.544259 0.139897i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12.7279i 0.416248i
\(936\) 0 0
\(937\) 2.44949i 0.0800213i 0.999199 + 0.0400107i \(0.0127392\pi\)
−0.999199 + 0.0400107i \(0.987261\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −26.8468 46.5000i −0.875180 1.51586i −0.856571 0.516030i \(-0.827409\pi\)
−0.0186097 0.999827i \(-0.505924\pi\)
\(942\) 0 0
\(943\) 12.7279 + 7.34847i 0.414478 + 0.239299i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.6969 8.48528i −0.477586 0.275735i 0.241824 0.970320i \(-0.422254\pi\)
−0.719410 + 0.694586i \(0.755588\pi\)
\(948\) 0 0
\(949\) −12.0000 20.7846i −0.389536 0.674697i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 50.9117i 1.64919i −0.565723 0.824596i \(-0.691403\pi\)
0.565723 0.824596i \(-0.308597\pi\)
\(954\) 0 0
\(955\) 14.6969i 0.475582i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 7.60959 + 55.6066i 0.245726 + 1.79563i
\(960\) 0 0
\(961\) 11.5000 19.9186i 0.370968 0.642535i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.06218 10.5000i 0.195148 0.338007i
\(966\) 0 0
\(967\) 23.0000 + 39.8372i 0.739630 + 1.28108i 0.952662 + 0.304032i \(0.0983329\pi\)
−0.213032 + 0.977045i \(0.568334\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 32.9090 1.05610 0.528049 0.849214i \(-0.322924\pi\)
0.528049 + 0.849214i \(0.322924\pi\)
\(972\) 0 0
\(973\) 30.0000 + 12.2474i 0.961756 + 0.392635i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36.7423 21.2132i 1.17549 0.678671i 0.220524 0.975381i \(-0.429223\pi\)
0.954967 + 0.296711i \(0.0958898\pi\)
\(978\) 0 0
\(979\) −38.1838 22.0454i −1.22036 0.704574i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −0.866025 + 1.50000i −0.0276219 + 0.0478426i −0.879506 0.475888i \(-0.842127\pi\)
0.851884 + 0.523731i \(0.175460\pi\)
\(984\) 0 0
\(985\) −31.8198 + 18.3712i −1.01386 + 0.585354i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 59.3970i 1.88871i
\(990\) 0 0
\(991\) 55.0000 1.74713 0.873566 0.486705i \(-0.161801\pi\)
0.873566 + 0.486705i \(0.161801\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.6969 8.48528i 0.465924 0.269002i
\(996\) 0 0
\(997\) −31.8198 18.3712i −1.00774 0.581821i −0.0972132 0.995264i \(-0.530993\pi\)
−0.910530 + 0.413443i \(0.864326\pi\)
\(998\) 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.x.j.1889.3 8
3.2 odd 2 inner 2268.2.x.j.1889.1 8
7.6 odd 2 inner 2268.2.x.j.1889.2 8
9.2 odd 6 756.2.f.d.377.4 yes 4
9.4 even 3 inner 2268.2.x.j.377.4 8
9.5 odd 6 inner 2268.2.x.j.377.2 8
9.7 even 3 756.2.f.d.377.2 yes 4
21.20 even 2 inner 2268.2.x.j.1889.4 8
36.7 odd 6 3024.2.k.h.1889.1 4
36.11 even 6 3024.2.k.h.1889.3 4
63.13 odd 6 inner 2268.2.x.j.377.1 8
63.20 even 6 756.2.f.d.377.1 4
63.34 odd 6 756.2.f.d.377.3 yes 4
63.41 even 6 inner 2268.2.x.j.377.3 8
252.83 odd 6 3024.2.k.h.1889.2 4
252.223 even 6 3024.2.k.h.1889.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.f.d.377.1 4 63.20 even 6
756.2.f.d.377.2 yes 4 9.7 even 3
756.2.f.d.377.3 yes 4 63.34 odd 6
756.2.f.d.377.4 yes 4 9.2 odd 6
2268.2.x.j.377.1 8 63.13 odd 6 inner
2268.2.x.j.377.2 8 9.5 odd 6 inner
2268.2.x.j.377.3 8 63.41 even 6 inner
2268.2.x.j.377.4 8 9.4 even 3 inner
2268.2.x.j.1889.1 8 3.2 odd 2 inner
2268.2.x.j.1889.2 8 7.6 odd 2 inner
2268.2.x.j.1889.3 8 1.1 even 1 trivial
2268.2.x.j.1889.4 8 21.20 even 2 inner
3024.2.k.h.1889.1 4 36.7 odd 6
3024.2.k.h.1889.2 4 252.83 odd 6
3024.2.k.h.1889.3 4 36.11 even 6
3024.2.k.h.1889.4 4 252.223 even 6