Properties

Label 3456.2.i.l.1153.1
Level $3456$
Weight $2$
Character 3456.1153
Analytic conductor $27.596$
Analytic rank $0$
Dimension $12$
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] = [3456,2,Mod(1153,3456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3456.1153");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3456 = 2^{7} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3456.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: \(27.5962989386\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1153.1
Root \(-0.433633 + 1.67689i\) of defining polynomial
Character \(\chi\) \(=\) 3456.1153
Dual form 3456.2.i.l.2305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.22043 - 3.84590i) q^{5} +(1.45488 - 2.51992i) q^{7} +O(q^{10})\) \(q+(-2.22043 - 3.84590i) q^{5} +(1.45488 - 2.51992i) q^{7} +(-1.08263 + 1.87517i) q^{11} +(1.96377 + 3.40135i) q^{13} -1.79720 q^{17} -1.76882 q^{19} +(3.44197 + 5.96166i) q^{23} +(-7.36062 + 12.7490i) q^{25} +(-2.87353 + 4.97710i) q^{29} +(3.27671 + 5.67542i) q^{31} -12.9218 q^{35} +2.51332 q^{37} +(3.68420 + 6.38122i) q^{41} +(-2.53640 + 4.39317i) q^{43} +(4.98598 - 8.63597i) q^{47} +(-0.733339 - 1.27018i) q^{49} -3.30620 q^{53} +9.61562 q^{55} +(-2.30090 - 3.98528i) q^{59} +(-1.87353 + 3.24505i) q^{61} +(8.72084 - 15.1049i) q^{65} +(-2.36045 - 4.08841i) q^{67} -0.907539 q^{71} -1.87740 q^{73} +(3.15019 + 5.45629i) q^{77} +(1.23661 - 2.14187i) q^{79} +(-1.09251 + 1.89227i) q^{83} +(3.99056 + 6.91185i) q^{85} +5.30620 q^{89} +11.4282 q^{91} +(3.92754 + 6.80271i) q^{95} +(4.45302 - 7.71286i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{5} + 6 q^{7} - 4 q^{11} + 10 q^{13} - 4 q^{17} + 4 q^{19} - 8 q^{23} - 14 q^{25} + 2 q^{29} + 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} + 14 q^{47} - 18 q^{49} - 24 q^{53} - 16 q^{55} - 6 q^{59} + 14 q^{61} + 8 q^{65} + 4 q^{67} + 28 q^{71} + 60 q^{73} - 2 q^{77} + 16 q^{79} - 24 q^{83} + 16 q^{85} + 48 q^{89} - 52 q^{91} + 20 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(2431\) \(2945\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −2.22043 3.84590i −0.993006 1.71994i −0.598746 0.800939i \(-0.704334\pi\)
−0.394260 0.918999i \(-0.628999\pi\)
\(6\) 0 0
\(7\) 1.45488 2.51992i 0.549892 0.952441i −0.448389 0.893838i \(-0.648002\pi\)
0.998281 0.0586028i \(-0.0186645\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.08263 + 1.87517i −0.326425 + 0.565385i −0.981800 0.189919i \(-0.939178\pi\)
0.655374 + 0.755304i \(0.272511\pi\)
\(12\) 0 0
\(13\) 1.96377 + 3.40135i 0.544652 + 0.943366i 0.998629 + 0.0523518i \(0.0166717\pi\)
−0.453976 + 0.891014i \(0.649995\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.79720 −0.435885 −0.217943 0.975962i \(-0.569935\pi\)
−0.217943 + 0.975962i \(0.569935\pi\)
\(18\) 0 0
\(19\) −1.76882 −0.405795 −0.202898 0.979200i \(-0.565036\pi\)
−0.202898 + 0.979200i \(0.565036\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.44197 + 5.96166i 0.717700 + 1.24309i 0.961909 + 0.273370i \(0.0881384\pi\)
−0.244209 + 0.969723i \(0.578528\pi\)
\(24\) 0 0
\(25\) −7.36062 + 12.7490i −1.47212 + 2.54979i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.87353 + 4.97710i −0.533601 + 0.924224i 0.465629 + 0.884980i \(0.345828\pi\)
−0.999230 + 0.0392435i \(0.987505\pi\)
\(30\) 0 0
\(31\) 3.27671 + 5.67542i 0.588514 + 1.01934i 0.994427 + 0.105425i \(0.0336201\pi\)
−0.405913 + 0.913912i \(0.633047\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −12.9218 −2.18419
\(36\) 0 0
\(37\) 2.51332 0.413187 0.206593 0.978427i \(-0.433762\pi\)
0.206593 + 0.978427i \(0.433762\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.68420 + 6.38122i 0.575376 + 0.996580i 0.996001 + 0.0893453i \(0.0284775\pi\)
−0.420625 + 0.907235i \(0.638189\pi\)
\(42\) 0 0
\(43\) −2.53640 + 4.39317i −0.386797 + 0.669953i −0.992017 0.126106i \(-0.959752\pi\)
0.605219 + 0.796059i \(0.293085\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.98598 8.63597i 0.727280 1.25969i −0.230748 0.973013i \(-0.574117\pi\)
0.958029 0.286673i \(-0.0925493\pi\)
\(48\) 0 0
\(49\) −0.733339 1.27018i −0.104763 0.181454i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.30620 −0.454141 −0.227070 0.973878i \(-0.572915\pi\)
−0.227070 + 0.973878i \(0.572915\pi\)
\(54\) 0 0
\(55\) 9.61562 1.29657
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.30090 3.98528i −0.299552 0.518839i 0.676482 0.736459i \(-0.263504\pi\)
−0.976033 + 0.217621i \(0.930170\pi\)
\(60\) 0 0
\(61\) −1.87353 + 3.24505i −0.239881 + 0.415485i −0.960680 0.277658i \(-0.910442\pi\)
0.720799 + 0.693144i \(0.243775\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.72084 15.1049i 1.08169 1.87354i
\(66\) 0 0
\(67\) −2.36045 4.08841i −0.288374 0.499479i 0.685047 0.728498i \(-0.259781\pi\)
−0.973422 + 0.229019i \(0.926448\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.907539 −0.107705 −0.0538525 0.998549i \(-0.517150\pi\)
−0.0538525 + 0.998549i \(0.517150\pi\)
\(72\) 0 0
\(73\) −1.87740 −0.219733 −0.109866 0.993946i \(-0.535042\pi\)
−0.109866 + 0.993946i \(0.535042\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.15019 + 5.45629i 0.358998 + 0.621802i
\(78\) 0 0
\(79\) 1.23661 2.14187i 0.139129 0.240979i −0.788038 0.615627i \(-0.788903\pi\)
0.927167 + 0.374648i \(0.122236\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.09251 + 1.89227i −0.119918 + 0.207704i −0.919735 0.392540i \(-0.871597\pi\)
0.799817 + 0.600244i \(0.204930\pi\)
\(84\) 0 0
\(85\) 3.99056 + 6.91185i 0.432837 + 0.749696i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.30620 0.562456 0.281228 0.959641i \(-0.409258\pi\)
0.281228 + 0.959641i \(0.409258\pi\)
\(90\) 0 0
\(91\) 11.4282 1.19800
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.92754 + 6.80271i 0.402958 + 0.697943i
\(96\) 0 0
\(97\) 4.45302 7.71286i 0.452136 0.783123i −0.546382 0.837536i \(-0.683995\pi\)
0.998519 + 0.0544132i \(0.0173288\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.689326 1.19395i 0.0685905 0.118802i −0.829691 0.558224i \(-0.811483\pi\)
0.898281 + 0.439421i \(0.144816\pi\)
\(102\) 0 0
\(103\) 2.54512 + 4.40828i 0.250778 + 0.434361i 0.963740 0.266842i \(-0.0859801\pi\)
−0.712962 + 0.701203i \(0.752647\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.2062 −1.66338 −0.831692 0.555238i \(-0.812627\pi\)
−0.831692 + 0.555238i \(0.812627\pi\)
\(108\) 0 0
\(109\) 6.59351 0.631544 0.315772 0.948835i \(-0.397737\pi\)
0.315772 + 0.948835i \(0.397737\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.90072 + 15.4165i 0.837309 + 1.45026i 0.892137 + 0.451766i \(0.149206\pi\)
−0.0548276 + 0.998496i \(0.517461\pi\)
\(114\) 0 0
\(115\) 15.2853 26.4749i 1.42536 2.46880i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.61471 + 4.52881i −0.239690 + 0.415155i
\(120\) 0 0
\(121\) 3.15582 + 5.46604i 0.286893 + 0.496913i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 43.1706 3.86130
\(126\) 0 0
\(127\) −18.2258 −1.61728 −0.808639 0.588305i \(-0.799795\pi\)
−0.808639 + 0.588305i \(0.799795\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.33057 + 7.50076i 0.378363 + 0.655345i 0.990824 0.135156i \(-0.0431537\pi\)
−0.612461 + 0.790501i \(0.709820\pi\)
\(132\) 0 0
\(133\) −2.57342 + 4.45729i −0.223144 + 0.386496i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.774446 1.34138i 0.0661654 0.114602i −0.831045 0.556205i \(-0.812257\pi\)
0.897210 + 0.441603i \(0.145590\pi\)
\(138\) 0 0
\(139\) −9.78618 16.9502i −0.830053 1.43769i −0.897996 0.440005i \(-0.854977\pi\)
0.0679426 0.997689i \(-0.478357\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.50416 −0.711154
\(144\) 0 0
\(145\) 25.5219 2.11948
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.945984 + 1.63849i 0.0774980 + 0.134230i 0.902170 0.431381i \(-0.141974\pi\)
−0.824672 + 0.565612i \(0.808640\pi\)
\(150\) 0 0
\(151\) −4.27927 + 7.41191i −0.348242 + 0.603173i −0.985937 0.167116i \(-0.946555\pi\)
0.637695 + 0.770289i \(0.279888\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 14.5514 25.2038i 1.16880 2.02441i
\(156\) 0 0
\(157\) 2.22265 + 3.84974i 0.177387 + 0.307242i 0.940985 0.338449i \(-0.109902\pi\)
−0.763598 + 0.645692i \(0.776569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.0306 1.57863
\(162\) 0 0
\(163\) −18.8817 −1.47893 −0.739465 0.673195i \(-0.764922\pi\)
−0.739465 + 0.673195i \(0.764922\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.31394 7.47197i −0.333823 0.578198i 0.649435 0.760417i \(-0.275005\pi\)
−0.983258 + 0.182219i \(0.941672\pi\)
\(168\) 0 0
\(169\) −1.21280 + 2.10063i −0.0932924 + 0.161587i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.91423 6.77965i 0.297594 0.515447i −0.677991 0.735070i \(-0.737149\pi\)
0.975585 + 0.219623i \(0.0704826\pi\)
\(174\) 0 0
\(175\) 21.4176 + 37.0964i 1.61902 + 2.80422i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.6390 1.01943 0.509714 0.860344i \(-0.329751\pi\)
0.509714 + 0.860344i \(0.329751\pi\)
\(180\) 0 0
\(181\) −0.504672 −0.0375120 −0.0187560 0.999824i \(-0.505971\pi\)
−0.0187560 + 0.999824i \(0.505971\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.58064 9.66596i −0.410297 0.710655i
\(186\) 0 0
\(187\) 1.94571 3.37006i 0.142284 0.246443i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0083 17.3349i 0.724175 1.25431i −0.235138 0.971962i \(-0.575554\pi\)
0.959313 0.282345i \(-0.0911124\pi\)
\(192\) 0 0
\(193\) −1.08462 1.87862i −0.0780726 0.135226i 0.824346 0.566087i \(-0.191543\pi\)
−0.902418 + 0.430861i \(0.858210\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.67460 0.404298 0.202149 0.979355i \(-0.435207\pi\)
0.202149 + 0.979355i \(0.435207\pi\)
\(198\) 0 0
\(199\) −11.5032 −0.815439 −0.407719 0.913107i \(-0.633676\pi\)
−0.407719 + 0.913107i \(0.633676\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.36126 + 14.4821i 0.586846 + 1.01645i
\(204\) 0 0
\(205\) 16.3610 28.3381i 1.14270 1.97922i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.91498 3.31684i 0.132462 0.229431i
\(210\) 0 0
\(211\) 10.3177 + 17.8707i 0.710297 + 1.23027i 0.964746 + 0.263184i \(0.0847726\pi\)
−0.254449 + 0.967086i \(0.581894\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 22.5276 1.53637
\(216\) 0 0
\(217\) 19.0688 1.29448
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.52929 6.11292i −0.237406 0.411199i
\(222\) 0 0
\(223\) 2.54291 4.40444i 0.170286 0.294943i −0.768234 0.640169i \(-0.778864\pi\)
0.938520 + 0.345226i \(0.112198\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.14484 + 15.8393i −0.606964 + 1.05129i 0.384773 + 0.923011i \(0.374280\pi\)
−0.991738 + 0.128282i \(0.959054\pi\)
\(228\) 0 0
\(229\) 9.62341 + 16.6682i 0.635933 + 1.10147i 0.986317 + 0.164862i \(0.0527178\pi\)
−0.350384 + 0.936606i \(0.613949\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.4263 −1.07612 −0.538061 0.842906i \(-0.680843\pi\)
−0.538061 + 0.842906i \(0.680843\pi\)
\(234\) 0 0
\(235\) −44.2841 −2.88878
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.08563 + 15.7368i 0.587700 + 1.01793i 0.994533 + 0.104424i \(0.0332999\pi\)
−0.406833 + 0.913503i \(0.633367\pi\)
\(240\) 0 0
\(241\) −11.4344 + 19.8050i −0.736556 + 1.27575i 0.217481 + 0.976065i \(0.430216\pi\)
−0.954037 + 0.299688i \(0.903117\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.25666 + 5.64070i −0.208060 + 0.360371i
\(246\) 0 0
\(247\) −3.47356 6.01639i −0.221017 0.382813i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.139530 0.00880707 0.00440353 0.999990i \(-0.498598\pi\)
0.00440353 + 0.999990i \(0.498598\pi\)
\(252\) 0 0
\(253\) −14.9055 −0.937102
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.17682 + 12.4306i 0.447678 + 0.775400i 0.998234 0.0593974i \(-0.0189179\pi\)
−0.550557 + 0.834798i \(0.685585\pi\)
\(258\) 0 0
\(259\) 3.65657 6.33336i 0.227208 0.393536i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.968751 1.67793i 0.0597357 0.103465i −0.834611 0.550840i \(-0.814308\pi\)
0.894347 + 0.447374i \(0.147641\pi\)
\(264\) 0 0
\(265\) 7.34118 + 12.7153i 0.450965 + 0.781094i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.91415 −0.604477 −0.302238 0.953232i \(-0.597734\pi\)
−0.302238 + 0.953232i \(0.597734\pi\)
\(270\) 0 0
\(271\) −4.56777 −0.277472 −0.138736 0.990329i \(-0.544304\pi\)
−0.138736 + 0.990329i \(0.544304\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.9377 27.6048i −0.961077 1.66463i
\(276\) 0 0
\(277\) −14.4728 + 25.0676i −0.869585 + 1.50616i −0.00716263 + 0.999974i \(0.502280\pi\)
−0.862422 + 0.506190i \(0.831053\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.1351 + 19.2865i −0.664262 + 1.15054i 0.315223 + 0.949018i \(0.397921\pi\)
−0.979485 + 0.201518i \(0.935412\pi\)
\(282\) 0 0
\(283\) 6.79946 + 11.7770i 0.404186 + 0.700071i 0.994226 0.107303i \(-0.0342215\pi\)
−0.590040 + 0.807374i \(0.700888\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 21.4403 1.26558
\(288\) 0 0
\(289\) −13.7701 −0.810004
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.21821 + 12.5023i 0.421693 + 0.730393i 0.996105 0.0881730i \(-0.0281028\pi\)
−0.574413 + 0.818566i \(0.694770\pi\)
\(294\) 0 0
\(295\) −10.2180 + 17.6980i −0.594913 + 1.03042i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −13.5185 + 23.4147i −0.781794 + 1.35411i
\(300\) 0 0
\(301\) 7.38030 + 12.7831i 0.425394 + 0.736804i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16.6401 0.952812
\(306\) 0 0
\(307\) −16.5451 −0.944280 −0.472140 0.881524i \(-0.656518\pi\)
−0.472140 + 0.881524i \(0.656518\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.19366 + 8.99568i 0.294505 + 0.510098i 0.974870 0.222776i \(-0.0715118\pi\)
−0.680364 + 0.732874i \(0.738178\pi\)
\(312\) 0 0
\(313\) −6.76501 + 11.7173i −0.382381 + 0.662303i −0.991402 0.130851i \(-0.958229\pi\)
0.609021 + 0.793154i \(0.291562\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.9869 20.7619i 0.673251 1.16611i −0.303726 0.952760i \(-0.598231\pi\)
0.976977 0.213346i \(-0.0684360\pi\)
\(318\) 0 0
\(319\) −6.22194 10.7767i −0.348362 0.603380i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.17893 0.176880
\(324\) 0 0
\(325\) −57.8183 −3.20718
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −14.5080 25.1286i −0.799851 1.38538i
\(330\) 0 0
\(331\) −1.29103 + 2.23612i −0.0709612 + 0.122908i −0.899323 0.437285i \(-0.855940\pi\)
0.828362 + 0.560194i \(0.189273\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −10.4824 + 18.1561i −0.572715 + 0.991972i
\(336\) 0 0
\(337\) −1.79736 3.11313i −0.0979087 0.169583i 0.812910 0.582389i \(-0.197882\pi\)
−0.910819 + 0.412806i \(0.864549\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.1899 −0.768424
\(342\) 0 0
\(343\) 16.1006 0.869351
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.85180 10.1356i −0.314141 0.544108i 0.665114 0.746742i \(-0.268383\pi\)
−0.979255 + 0.202634i \(0.935050\pi\)
\(348\) 0 0
\(349\) 9.34856 16.1922i 0.500417 0.866747i −0.499583 0.866266i \(-0.666513\pi\)
1.00000 0.000481224i \(-0.000153178\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −14.3410 + 24.8394i −0.763295 + 1.32207i 0.177848 + 0.984058i \(0.443086\pi\)
−0.941143 + 0.338008i \(0.890247\pi\)
\(354\) 0 0
\(355\) 2.01513 + 3.49030i 0.106952 + 0.185246i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.8202 0.834958 0.417479 0.908687i \(-0.362914\pi\)
0.417479 + 0.908687i \(0.362914\pi\)
\(360\) 0 0
\(361\) −15.8713 −0.835330
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.16863 + 7.22028i 0.218196 + 0.377927i
\(366\) 0 0
\(367\) 13.1383 22.7563i 0.685815 1.18787i −0.287364 0.957821i \(-0.592779\pi\)
0.973180 0.230046i \(-0.0738876\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.81011 + 8.33136i −0.249729 + 0.432543i
\(372\) 0 0
\(373\) 10.8735 + 18.8335i 0.563010 + 0.975162i 0.997232 + 0.0743558i \(0.0236901\pi\)
−0.434222 + 0.900806i \(0.642977\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −22.5718 −1.16251
\(378\) 0 0
\(379\) 32.8861 1.68925 0.844623 0.535362i \(-0.179825\pi\)
0.844623 + 0.535362i \(0.179825\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.81269 10.0679i −0.297015 0.514444i 0.678437 0.734659i \(-0.262658\pi\)
−0.975452 + 0.220214i \(0.929324\pi\)
\(384\) 0 0
\(385\) 13.9896 24.2306i 0.712974 1.23491i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.61687 6.26460i 0.183383 0.317628i −0.759648 0.650335i \(-0.774629\pi\)
0.943030 + 0.332707i \(0.107962\pi\)
\(390\) 0 0
\(391\) −6.18591 10.7143i −0.312835 0.541846i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.9832 −0.552625
\(396\) 0 0
\(397\) 29.8911 1.50019 0.750095 0.661330i \(-0.230008\pi\)
0.750095 + 0.661330i \(0.230008\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.03226 5.25202i −0.151424 0.262273i 0.780327 0.625371i \(-0.215052\pi\)
−0.931751 + 0.363098i \(0.881719\pi\)
\(402\) 0 0
\(403\) −12.8694 + 22.2905i −0.641071 + 1.11037i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.72099 + 4.71290i −0.134875 + 0.233610i
\(408\) 0 0
\(409\) −14.4396 25.0101i −0.713993 1.23667i −0.963347 0.268259i \(-0.913552\pi\)
0.249354 0.968412i \(-0.419782\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −13.3901 −0.658884
\(414\) 0 0
\(415\) 9.70332 0.476317
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.63281 9.75631i −0.275181 0.476627i 0.695000 0.719010i \(-0.255404\pi\)
−0.970181 + 0.242383i \(0.922071\pi\)
\(420\) 0 0
\(421\) −6.03050 + 10.4451i −0.293909 + 0.509065i −0.974730 0.223384i \(-0.928289\pi\)
0.680822 + 0.732449i \(0.261623\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.2285 22.9125i 0.641677 1.11142i
\(426\) 0 0
\(427\) 5.45151 + 9.44229i 0.263817 + 0.456944i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.5079 1.22867 0.614336 0.789045i \(-0.289424\pi\)
0.614336 + 0.789045i \(0.289424\pi\)
\(432\) 0 0
\(433\) 29.4513 1.41534 0.707670 0.706543i \(-0.249746\pi\)
0.707670 + 0.706543i \(0.249746\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.08823 10.5451i −0.291239 0.504441i
\(438\) 0 0
\(439\) 17.8086 30.8454i 0.849959 1.47217i −0.0312845 0.999511i \(-0.509960\pi\)
0.881244 0.472662i \(-0.156707\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.60886 11.4469i 0.313996 0.543857i −0.665227 0.746641i \(-0.731665\pi\)
0.979224 + 0.202783i \(0.0649987\pi\)
\(444\) 0 0
\(445\) −11.7820 20.4071i −0.558522 0.967389i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.83739 0.275483 0.137742 0.990468i \(-0.456016\pi\)
0.137742 + 0.990468i \(0.456016\pi\)
\(450\) 0 0
\(451\) −15.9545 −0.751269
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −25.3755 43.9517i −1.18962 2.06049i
\(456\) 0 0
\(457\) −13.5037 + 23.3891i −0.631677 + 1.09410i 0.355532 + 0.934664i \(0.384300\pi\)
−0.987209 + 0.159433i \(0.949034\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.78550 + 3.09258i −0.0831591 + 0.144036i −0.904605 0.426250i \(-0.859834\pi\)
0.821446 + 0.570286i \(0.193168\pi\)
\(462\) 0 0
\(463\) −19.8396 34.3631i −0.922023 1.59699i −0.796281 0.604927i \(-0.793202\pi\)
−0.125742 0.992063i \(-0.540131\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.8522 0.872376 0.436188 0.899855i \(-0.356328\pi\)
0.436188 + 0.899855i \(0.356328\pi\)
\(468\) 0 0
\(469\) −13.7366 −0.634299
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.49197 9.51237i −0.252521 0.437379i
\(474\) 0 0
\(475\) 13.0196 22.5506i 0.597381 1.03469i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −11.0879 + 19.2049i −0.506621 + 0.877492i 0.493350 + 0.869831i \(0.335772\pi\)
−0.999971 + 0.00766167i \(0.997561\pi\)
\(480\) 0 0
\(481\) 4.93558 + 8.54867i 0.225043 + 0.389786i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −39.5505 −1.79590
\(486\) 0 0
\(487\) −17.9432 −0.813086 −0.406543 0.913632i \(-0.633266\pi\)
−0.406543 + 0.913632i \(0.633266\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.71919 2.97773i −0.0775861 0.134383i 0.824622 0.565684i \(-0.191388\pi\)
−0.902208 + 0.431301i \(0.858055\pi\)
\(492\) 0 0
\(493\) 5.16431 8.94485i 0.232589 0.402856i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.32036 + 2.28693i −0.0592262 + 0.102583i
\(498\) 0 0
\(499\) 5.41124 + 9.37254i 0.242240 + 0.419572i 0.961352 0.275322i \(-0.0887845\pi\)
−0.719112 + 0.694894i \(0.755451\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9.71510 −0.433175 −0.216587 0.976263i \(-0.569493\pi\)
−0.216587 + 0.976263i \(0.569493\pi\)
\(504\) 0 0
\(505\) −6.12240 −0.272443
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.5991 + 30.4825i 0.780066 + 1.35111i 0.931903 + 0.362708i \(0.118148\pi\)
−0.151837 + 0.988406i \(0.548519\pi\)
\(510\) 0 0
\(511\) −2.73138 + 4.73090i −0.120829 + 0.209283i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11.3025 19.5766i 0.498049 0.862646i
\(516\) 0 0
\(517\) 10.7960 + 18.6991i 0.474806 + 0.822387i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.57440 0.331840 0.165920 0.986139i \(-0.446941\pi\)
0.165920 + 0.986139i \(0.446941\pi\)
\(522\) 0 0
\(523\) 10.0630 0.440025 0.220013 0.975497i \(-0.429390\pi\)
0.220013 + 0.975497i \(0.429390\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.88890 10.1999i −0.256525 0.444314i
\(528\) 0 0
\(529\) −12.1943 + 21.1211i −0.530187 + 0.918310i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14.4699 + 25.0625i −0.626759 + 1.08558i
\(534\) 0 0
\(535\) 38.2051 + 66.1732i 1.65175 + 2.86092i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.17574 0.136789
\(540\) 0 0
\(541\) −26.2133 −1.12700 −0.563498 0.826117i \(-0.690545\pi\)
−0.563498 + 0.826117i \(0.690545\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −14.6404 25.3580i −0.627127 1.08622i
\(546\) 0 0
\(547\) 9.57620 16.5865i 0.409449 0.709186i −0.585379 0.810760i \(-0.699054\pi\)
0.994828 + 0.101574i \(0.0323878\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.08276 8.80359i 0.216533 0.375046i
\(552\) 0 0
\(553\) −3.59823 6.23232i −0.153012 0.265025i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −22.5019 −0.953435 −0.476717 0.879057i \(-0.658173\pi\)
−0.476717 + 0.879057i \(0.658173\pi\)
\(558\) 0 0
\(559\) −19.9236 −0.842680
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.7085 22.0118i −0.535599 0.927685i −0.999134 0.0416066i \(-0.986752\pi\)
0.463535 0.886079i \(-0.346581\pi\)
\(564\) 0 0
\(565\) 39.5268 68.4625i 1.66291 2.88024i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.14798 + 15.8448i −0.383503 + 0.664247i −0.991560 0.129646i \(-0.958616\pi\)
0.608057 + 0.793893i \(0.291949\pi\)
\(570\) 0 0
\(571\) 1.27484 + 2.20808i 0.0533503 + 0.0924054i 0.891467 0.453085i \(-0.149677\pi\)
−0.838117 + 0.545491i \(0.816343\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −101.340 −4.22617
\(576\) 0 0
\(577\) 2.22842 0.0927702 0.0463851 0.998924i \(-0.485230\pi\)
0.0463851 + 0.998924i \(0.485230\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.17892 + 5.50606i 0.131884 + 0.228430i
\(582\) 0 0
\(583\) 3.57939 6.19968i 0.148243 0.256765i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.2694 + 26.4473i −0.630234 + 1.09160i 0.357270 + 0.934001i \(0.383708\pi\)
−0.987504 + 0.157596i \(0.949626\pi\)
\(588\) 0 0
\(589\) −5.79591 10.0388i −0.238816 0.413642i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.96281 0.244863 0.122432 0.992477i \(-0.460931\pi\)
0.122432 + 0.992477i \(0.460931\pi\)
\(594\) 0 0
\(595\) 23.2231 0.952055
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.29265 + 7.43508i 0.175393 + 0.303789i 0.940297 0.340355i \(-0.110547\pi\)
−0.764904 + 0.644144i \(0.777214\pi\)
\(600\) 0 0
\(601\) 1.44648 2.50538i 0.0590033 0.102197i −0.835015 0.550227i \(-0.814541\pi\)
0.894018 + 0.448031i \(0.147874\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.0146 24.2739i 0.569773 0.986876i
\(606\) 0 0
\(607\) 9.96773 + 17.2646i 0.404577 + 0.700749i 0.994272 0.106877i \(-0.0340853\pi\)
−0.589695 + 0.807626i \(0.700752\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 39.1653 1.58446
\(612\) 0 0
\(613\) 35.4941 1.43359 0.716797 0.697282i \(-0.245607\pi\)
0.716797 + 0.697282i \(0.245607\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 15.6891 + 27.1743i 0.631618 + 1.09399i 0.987221 + 0.159357i \(0.0509422\pi\)
−0.355603 + 0.934637i \(0.615724\pi\)
\(618\) 0 0
\(619\) 16.7289 28.9752i 0.672389 1.16461i −0.304835 0.952405i \(-0.598601\pi\)
0.977225 0.212207i \(-0.0680652\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.71987 13.3712i 0.309290 0.535706i
\(624\) 0 0
\(625\) −59.0543 102.285i −2.36217 4.09140i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.51694 −0.180102
\(630\) 0 0
\(631\) −8.12216 −0.323338 −0.161669 0.986845i \(-0.551688\pi\)
−0.161669 + 0.986845i \(0.551688\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 40.4691 + 70.0945i 1.60597 + 2.78162i
\(636\) 0 0
\(637\) 2.88022 4.98869i 0.114119 0.197659i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.4782 18.1488i 0.413865 0.716836i −0.581443 0.813587i \(-0.697512\pi\)
0.995309 + 0.0967511i \(0.0308451\pi\)
\(642\) 0 0
\(643\) 16.3547 + 28.3272i 0.644967 + 1.11712i 0.984309 + 0.176453i \(0.0564623\pi\)
−0.339342 + 0.940663i \(0.610204\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.7820 −0.738395 −0.369198 0.929351i \(-0.620367\pi\)
−0.369198 + 0.929351i \(0.620367\pi\)
\(648\) 0 0
\(649\) 9.96410 0.391125
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.85977 8.41736i −0.190177 0.329397i 0.755132 0.655573i \(-0.227573\pi\)
−0.945309 + 0.326176i \(0.894240\pi\)
\(654\) 0 0
\(655\) 19.2314 33.3098i 0.751435 1.30152i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −16.1773 + 28.0198i −0.630177 + 1.09150i 0.357338 + 0.933975i \(0.383684\pi\)
−0.987515 + 0.157523i \(0.949649\pi\)
\(660\) 0 0
\(661\) −13.0319 22.5719i −0.506883 0.877946i −0.999968 0.00796563i \(-0.997464\pi\)
0.493086 0.869981i \(-0.335869\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 22.8564 0.886333
\(666\) 0 0
\(667\) −39.5624 −1.53186
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.05668 7.02637i −0.156606 0.271250i
\(672\) 0 0
\(673\) 16.6951 28.9167i 0.643549 1.11466i −0.341086 0.940032i \(-0.610795\pi\)
0.984635 0.174627i \(-0.0558719\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.6991 21.9955i 0.488065 0.845354i −0.511840 0.859081i \(-0.671036\pi\)
0.999906 + 0.0137265i \(0.00436941\pi\)
\(678\) 0 0
\(679\) −12.9572 22.4425i −0.497252 0.861266i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 37.2800 1.42648 0.713241 0.700919i \(-0.247227\pi\)
0.713241 + 0.700919i \(0.247227\pi\)
\(684\) 0 0
\(685\) −6.87841 −0.262811
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.49261 11.2455i −0.247349 0.428421i
\(690\) 0 0
\(691\) −6.41730 + 11.1151i −0.244126 + 0.422838i −0.961885 0.273453i \(-0.911834\pi\)
0.717760 + 0.696291i \(0.245168\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −43.4591 + 75.2733i −1.64850 + 2.85528i
\(696\) 0 0
\(697\) −6.62125 11.4683i −0.250798 0.434395i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.89156 0.260290 0.130145 0.991495i \(-0.458456\pi\)
0.130145 + 0.991495i \(0.458456\pi\)
\(702\) 0 0
\(703\) −4.44561 −0.167669
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.00577 3.47410i −0.0754347 0.130657i
\(708\) 0 0
\(709\) 10.1178 17.5246i 0.379983 0.658150i −0.611076 0.791572i \(-0.709263\pi\)
0.991059 + 0.133422i \(0.0425964\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −22.5566 + 39.0693i −0.844753 + 1.46316i
\(714\) 0 0
\(715\) 18.8829 + 32.7061i 0.706180 + 1.22314i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −44.1706 −1.64729 −0.823643 0.567108i \(-0.808062\pi\)
−0.823643 + 0.567108i \(0.808062\pi\)
\(720\) 0 0
\(721\) 14.8114 0.551604
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −42.3019 73.2690i −1.57105 2.72114i
\(726\) 0 0
\(727\) 7.29193 12.6300i 0.270443 0.468421i −0.698532 0.715578i \(-0.746163\pi\)
0.968975 + 0.247158i \(0.0794965\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.55842 7.89542i 0.168599 0.292023i
\(732\) 0 0
\(733\) 16.4444 + 28.4826i 0.607388 + 1.05203i 0.991669 + 0.128811i \(0.0411161\pi\)
−0.384281 + 0.923216i \(0.625551\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.2220 0.376531
\(738\) 0 0
\(739\) 35.3966 1.30208 0.651042 0.759041i \(-0.274332\pi\)
0.651042 + 0.759041i \(0.274332\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −18.8177 32.5932i −0.690353 1.19573i −0.971722 0.236127i \(-0.924122\pi\)
0.281369 0.959600i \(-0.409212\pi\)
\(744\) 0 0
\(745\) 4.20098 7.27631i 0.153912 0.266584i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −25.0329 + 43.3582i −0.914682 + 1.58427i
\(750\) 0 0
\(751\) 8.38950 + 14.5310i 0.306137 + 0.530245i 0.977514 0.210871i \(-0.0676300\pi\)
−0.671377 + 0.741116i \(0.734297\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 38.0073 1.38323
\(756\) 0 0
\(757\) 19.4825 0.708103 0.354051 0.935226i \(-0.384804\pi\)
0.354051 + 0.935226i \(0.384804\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.49573 + 16.4471i 0.344220 + 0.596206i 0.985212 0.171341i \(-0.0548101\pi\)
−0.640992 + 0.767548i \(0.721477\pi\)
\(762\) 0 0
\(763\) 9.59276 16.6151i 0.347281 0.601508i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.03688 15.6523i 0.326303 0.565173i
\(768\) 0 0
\(769\) 21.2098 + 36.7365i 0.764846 + 1.32475i 0.940328 + 0.340270i \(0.110518\pi\)
−0.175482 + 0.984483i \(0.556148\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.55333 −0.0918368 −0.0459184 0.998945i \(-0.514621\pi\)
−0.0459184 + 0.998945i \(0.514621\pi\)
\(774\) 0 0
\(775\) −96.4744 −3.46546
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.51670 11.2872i −0.233485 0.404408i
\(780\) 0 0
\(781\) 0.982529 1.70179i 0.0351577 0.0608949i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.87046 17.0961i 0.352292 0.610188i
\(786\) 0 0
\(787\) −6.70128 11.6069i −0.238875 0.413743i 0.721517 0.692397i \(-0.243445\pi\)
−0.960392 + 0.278654i \(0.910112\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 51.7978 1.84172
\(792\) 0 0
\(793\) −14.7167 −0.522606
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16.0873 + 27.8640i 0.569840 + 0.986992i 0.996581 + 0.0826182i \(0.0263282\pi\)
−0.426741 + 0.904374i \(0.640338\pi\)
\(798\) 0 0
\(799\) −8.96082 + 15.5206i −0.317011 + 0.549079i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.03253 3.52044i 0.0717264 0.124234i
\(804\) 0 0
\(805\) −44.4765 77.0355i −1.56759 2.71515i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −34.7417 −1.22145 −0.610727 0.791841i \(-0.709123\pi\)
−0.610727 + 0.791841i \(0.709123\pi\)
\(810\) 0 0
\(811\) 40.7570 1.43117 0.715587 0.698524i \(-0.246159\pi\)
0.715587 + 0.698524i \(0.246159\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 41.9255 + 72.6172i 1.46859 + 2.54367i
\(816\) 0 0
\(817\) 4.48644 7.77074i 0.156961 0.271864i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.25420 + 14.2967i −0.288073 + 0.498958i −0.973350 0.229325i \(-0.926348\pi\)
0.685276 + 0.728283i \(0.259681\pi\)
\(822\) 0 0
\(823\) −2.28675 3.96078i −0.0797113 0.138064i 0.823414 0.567441i \(-0.192066\pi\)
−0.903125 + 0.429377i \(0.858733\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −47.2992 −1.64475 −0.822377 0.568943i \(-0.807352\pi\)
−0.822377 + 0.568943i \(0.807352\pi\)
\(828\) 0 0
\(829\) 2.10329 0.0730501 0.0365251 0.999333i \(-0.488371\pi\)
0.0365251 + 0.999333i \(0.488371\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.31796 + 2.28277i 0.0456646 + 0.0790934i
\(834\) 0 0
\(835\) −19.1576 + 33.1820i −0.662977 + 1.14831i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.60255 + 7.97185i −0.158898 + 0.275219i −0.934471 0.356038i \(-0.884127\pi\)
0.775574 + 0.631257i \(0.217461\pi\)
\(840\) 0 0
\(841\) −2.01432 3.48891i −0.0694595 0.120307i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10.7718 0.370560
\(846\) 0 0
\(847\) 18.3653 0.631041
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.65075 + 14.9835i 0.296544 + 0.513629i
\(852\) 0 0
\(853\) −7.43348 + 12.8752i −0.254518 + 0.440837i −0.964764 0.263115i \(-0.915250\pi\)
0.710247 + 0.703953i \(0.248583\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22.9611 39.7698i 0.784337 1.35851i −0.145058 0.989423i \(-0.546337\pi\)
0.929394 0.369088i \(-0.120330\pi\)
\(858\) 0 0
\(859\) 14.6542 + 25.3818i 0.499994 + 0.866015i 1.00000 6.84699e-6i \(-2.17947e-6\pi\)
−0.500006 + 0.866022i \(0.666669\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 23.5606 0.802012 0.401006 0.916075i \(-0.368661\pi\)
0.401006 + 0.916075i \(0.368661\pi\)
\(864\) 0 0
\(865\) −34.7651 −1.18205
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.67758 + 4.63771i 0.0908307 + 0.157323i
\(870\) 0 0
\(871\) 9.27076 16.0574i 0.314128 0.544085i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 62.8080 108.787i 2.12330 3.67766i
\(876\) 0 0
\(877\) −9.29438 16.0983i −0.313849 0.543602i 0.665343 0.746538i \(-0.268285\pi\)
−0.979192 + 0.202935i \(0.934952\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −24.6693 −0.831129 −0.415564 0.909564i \(-0.636416\pi\)
−0.415564 + 0.909564i \(0.636416\pi\)
\(882\) 0 0
\(883\) 4.42122 0.148786 0.0743930 0.997229i \(-0.476298\pi\)
0.0743930 + 0.997229i \(0.476298\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.1558 38.3750i −0.743919 1.28851i −0.950698 0.310118i \(-0.899631\pi\)
0.206779 0.978388i \(-0.433702\pi\)
\(888\) 0 0
\(889\) −26.5163 + 45.9276i −0.889328 + 1.54036i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.81931 + 15.2755i −0.295127 + 0.511175i
\(894\) 0 0
\(895\) −30.2845 52.4543i −1.01230 1.75335i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −37.6628 −1.25613
\(900\) 0 0
\(901\) 5.94190 0.197953
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.12059 + 1.94092i 0.0372497 + 0.0645183i
\(906\) 0 0
\(907\) −23.2939 + 40.3462i −0.773461 + 1.33967i 0.162195 + 0.986759i \(0.448143\pi\)
−0.935656 + 0.352914i \(0.885191\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.29458 7.43844i 0.142286 0.246446i −0.786071 0.618136i \(-0.787888\pi\)
0.928357 + 0.371690i \(0.121221\pi\)
\(912\) 0 0
\(913\) −2.36556 4.09727i −0.0782886 0.135600i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 25.2018 0.832236
\(918\) 0 0
\(919\) 41.7394 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.78220 3.08686i −0.0586618 0.101605i
\(924\) 0 0
\(925\) −18.4996 + 32.0422i −0.608262 + 1.05354i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −20.5014 + 35.5094i −0.672628 + 1.16503i 0.304528 + 0.952503i \(0.401501\pi\)
−0.977156 + 0.212523i \(0.931832\pi\)
\(930\) 0 0
\(931\) 1.29715 + 2.24672i 0.0425123 + 0.0736334i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −17.2812 −0.565156
\(936\) 0 0
\(937\) −10.1921 −0.332962 −0.166481 0.986045i \(-0.553240\pi\)
−0.166481 + 0.986045i \(0.553240\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −20.7130 35.8761i −0.675226 1.16953i −0.976403 0.215958i \(-0.930713\pi\)
0.301177 0.953568i \(-0.402621\pi\)
\(942\) 0 0
\(943\) −25.3618 + 43.9279i −0.825894 + 1.43049i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.04880 + 1.81658i −0.0340815 + 0.0590309i −0.882563 0.470194i \(-0.844184\pi\)
0.848482 + 0.529225i \(0.177517\pi\)
\(948\) 0 0
\(949\) −3.68678 6.38569i −0.119678 0.207288i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −31.7663 −1.02901 −0.514506 0.857487i \(-0.672025\pi\)
−0.514506 + 0.857487i \(0.672025\pi\)
\(954\) 0 0
\(955\) −88.8908 −2.87644
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.25345 3.90309i −0.0727677 0.126037i
\(960\) 0 0
\(961\) −5.97362 + 10.3466i −0.192697 + 0.333762i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.81664 + 8.34267i −0.155053 + 0.268560i
\(966\) 0 0
\(967\) −1.76817 3.06256i −0.0568605 0.0984853i 0.836194 0.548434i \(-0.184776\pi\)
−0.893055 + 0.449948i \(0.851442\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21.4727 −0.689093 −0.344546 0.938769i \(-0.611967\pi\)
−0.344546 + 0.938769i \(0.611967\pi\)
\(972\) 0 0
\(973\) −56.9508 −1.82576
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −23.6237 40.9174i −0.755788 1.30906i −0.944982 0.327124i \(-0.893921\pi\)
0.189193 0.981940i \(-0.439413\pi\)
\(978\) 0 0
\(979\) −5.74465 + 9.95003i −0.183600 + 0.318004i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 21.4576 37.1656i 0.684390 1.18540i −0.289238 0.957257i \(-0.593402\pi\)
0.973628 0.228141i \(-0.0732648\pi\)
\(984\) 0 0
\(985\) −12.6001 21.8239i −0.401471 0.695368i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −34.9208 −1.11042
\(990\) 0 0
\(991\) −11.5080 −0.365563 −0.182782 0.983154i \(-0.558510\pi\)
−0.182782 + 0.983154i \(0.558510\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 25.5420 + 44.2401i 0.809736 + 1.40250i
\(996\) 0 0
\(997\) −8.46934 + 14.6693i −0.268227 + 0.464582i −0.968404 0.249387i \(-0.919771\pi\)
0.700177 + 0.713969i \(0.253104\pi\)
\(998\) 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 3456.2.i.l.1153.1 12
3.2 odd 2 1152.2.i.k.385.5 yes 12
4.3 odd 2 3456.2.i.k.1153.1 12
8.3 odd 2 3456.2.i.i.1153.6 12
8.5 even 2 3456.2.i.j.1153.6 12
9.4 even 3 inner 3456.2.i.l.2305.1 12
9.5 odd 6 1152.2.i.k.769.5 yes 12
12.11 even 2 1152.2.i.i.385.2 12
24.5 odd 2 1152.2.i.j.385.2 yes 12
24.11 even 2 1152.2.i.l.385.5 yes 12
36.23 even 6 1152.2.i.i.769.2 yes 12
36.31 odd 6 3456.2.i.k.2305.1 12
72.5 odd 6 1152.2.i.j.769.2 yes 12
72.13 even 6 3456.2.i.j.2305.6 12
72.59 even 6 1152.2.i.l.769.5 yes 12
72.67 odd 6 3456.2.i.i.2305.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.i.i.385.2 12 12.11 even 2
1152.2.i.i.769.2 yes 12 36.23 even 6
1152.2.i.j.385.2 yes 12 24.5 odd 2
1152.2.i.j.769.2 yes 12 72.5 odd 6
1152.2.i.k.385.5 yes 12 3.2 odd 2
1152.2.i.k.769.5 yes 12 9.5 odd 6
1152.2.i.l.385.5 yes 12 24.11 even 2
1152.2.i.l.769.5 yes 12 72.59 even 6
3456.2.i.i.1153.6 12 8.3 odd 2
3456.2.i.i.2305.6 12 72.67 odd 6
3456.2.i.j.1153.6 12 8.5 even 2
3456.2.i.j.2305.6 12 72.13 even 6
3456.2.i.k.1153.1 12 4.3 odd 2
3456.2.i.k.2305.1 12 36.31 odd 6
3456.2.i.l.1153.1 12 1.1 even 1 trivial
3456.2.i.l.2305.1 12 9.4 even 3 inner