Properties

Label 3456.2.i.f.2305.1
Level $3456$
Weight $2$
Character 3456.2305
Analytic conductor $27.596$
Analytic rank $0$
Dimension $10$
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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.8528759163648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2305.1
Root \(1.06839 + 1.36328i\) of defining polynomial
Character \(\chi\) \(=\) 3456.2305
Dual form 3456.2.i.f.1153.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.34011 + 2.32114i) q^{5} +(-2.48656 - 4.30684i) q^{7} +O(q^{10})\) \(q+(-1.34011 + 2.32114i) q^{5} +(-2.48656 - 4.30684i) q^{7} +(1.26947 + 2.19879i) q^{11} +(-2.21483 + 3.83620i) q^{13} +2.43417 q^{17} +4.18361 q^{19} +(-0.570641 + 0.988379i) q^{23} +(-1.09180 - 1.89106i) q^{25} +(3.00434 + 5.20366i) q^{29} +(2.65303 - 4.59518i) q^{31} +13.3291 q^{35} -0.241556 q^{37} +(3.21105 - 5.56170i) q^{41} +(-5.57273 - 9.65224i) q^{43} +(-2.37728 - 4.11757i) q^{47} +(-8.86592 + 15.3562i) q^{49} -9.38345 q^{53} -6.80494 q^{55} +(-5.40906 + 9.36876i) q^{59} +(-4.16044 - 7.20610i) q^{61} +(-5.93625 - 10.2819i) q^{65} +(-1.13269 + 1.96188i) q^{67} -4.52639 q^{71} -3.34728 q^{73} +(6.31322 - 10.9348i) q^{77} +(3.14159 + 5.44139i) q^{79} +(0.738248 + 1.27868i) q^{83} +(-3.26206 + 5.65005i) q^{85} -14.8823 q^{89} +22.0292 q^{91} +(-5.60651 + 9.71076i) q^{95} +(5.89884 + 10.2171i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{7} + q^{11} - 6 q^{13} + 6 q^{17} + 18 q^{19} - 4 q^{23} + q^{25} - 4 q^{29} - 8 q^{31} + 24 q^{35} + 20 q^{37} + 5 q^{41} - 13 q^{43} + 6 q^{47} + 3 q^{49} + 12 q^{55} + 13 q^{59} - 10 q^{61} - 17 q^{67} - 8 q^{71} - 34 q^{73} + 8 q^{77} - 6 q^{79} - 12 q^{83} - 18 q^{85} - 44 q^{89} + 36 q^{91} + 6 q^{95} + 27 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −1.34011 + 2.32114i −0.599317 + 1.03805i 0.393605 + 0.919279i \(0.371228\pi\)
−0.992922 + 0.118767i \(0.962106\pi\)
\(6\) 0 0
\(7\) −2.48656 4.30684i −0.939830 1.62783i −0.765786 0.643095i \(-0.777650\pi\)
−0.174043 0.984738i \(-0.555683\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.26947 + 2.19879i 0.382760 + 0.662960i 0.991456 0.130444i \(-0.0416402\pi\)
−0.608696 + 0.793404i \(0.708307\pi\)
\(12\) 0 0
\(13\) −2.21483 + 3.83620i −0.614284 + 1.06397i 0.376225 + 0.926528i \(0.377222\pi\)
−0.990510 + 0.137443i \(0.956111\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.43417 0.590373 0.295186 0.955440i \(-0.404618\pi\)
0.295186 + 0.955440i \(0.404618\pi\)
\(18\) 0 0
\(19\) 4.18361 0.959786 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.570641 + 0.988379i −0.118987 + 0.206091i −0.919366 0.393402i \(-0.871298\pi\)
0.800380 + 0.599494i \(0.204631\pi\)
\(24\) 0 0
\(25\) −1.09180 1.89106i −0.218361 0.378212i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00434 + 5.20366i 0.557891 + 0.966296i 0.997672 + 0.0681908i \(0.0217227\pi\)
−0.439781 + 0.898105i \(0.644944\pi\)
\(30\) 0 0
\(31\) 2.65303 4.59518i 0.476498 0.825319i −0.523139 0.852247i \(-0.675239\pi\)
0.999637 + 0.0269282i \(0.00857256\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 13.3291 2.25302
\(36\) 0 0
\(37\) −0.241556 −0.0397115 −0.0198558 0.999803i \(-0.506321\pi\)
−0.0198558 + 0.999803i \(0.506321\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.21105 5.56170i 0.501482 0.868592i −0.498517 0.866880i \(-0.666122\pi\)
0.999999 0.00171211i \(-0.000544983\pi\)
\(42\) 0 0
\(43\) −5.57273 9.65224i −0.849833 1.47195i −0.881357 0.472451i \(-0.843370\pi\)
0.0315245 0.999503i \(-0.489964\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.37728 4.11757i −0.346762 0.600609i 0.638910 0.769281i \(-0.279385\pi\)
−0.985672 + 0.168672i \(0.946052\pi\)
\(48\) 0 0
\(49\) −8.86592 + 15.3562i −1.26656 + 2.19375i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.38345 −1.28892 −0.644458 0.764640i \(-0.722917\pi\)
−0.644458 + 0.764640i \(0.722917\pi\)
\(54\) 0 0
\(55\) −6.80494 −0.917578
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.40906 + 9.36876i −0.704199 + 1.21971i 0.262781 + 0.964856i \(0.415360\pi\)
−0.966980 + 0.254853i \(0.917973\pi\)
\(60\) 0 0
\(61\) −4.16044 7.20610i −0.532690 0.922646i −0.999271 0.0381677i \(-0.987848\pi\)
0.466581 0.884478i \(-0.345485\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.93625 10.2819i −0.736302 1.27531i
\(66\) 0 0
\(67\) −1.13269 + 1.96188i −0.138380 + 0.239682i −0.926884 0.375349i \(-0.877523\pi\)
0.788503 + 0.615030i \(0.210856\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.52639 −0.537183 −0.268592 0.963254i \(-0.586558\pi\)
−0.268592 + 0.963254i \(0.586558\pi\)
\(72\) 0 0
\(73\) −3.34728 −0.391769 −0.195885 0.980627i \(-0.562758\pi\)
−0.195885 + 0.980627i \(0.562758\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.31322 10.9348i 0.719459 1.24614i
\(78\) 0 0
\(79\) 3.14159 + 5.44139i 0.353457 + 0.612205i 0.986853 0.161623i \(-0.0516729\pi\)
−0.633396 + 0.773828i \(0.718340\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.738248 + 1.27868i 0.0810332 + 0.140354i 0.903694 0.428179i \(-0.140845\pi\)
−0.822661 + 0.568533i \(0.807511\pi\)
\(84\) 0 0
\(85\) −3.26206 + 5.65005i −0.353820 + 0.612834i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.8823 −1.57752 −0.788762 0.614698i \(-0.789278\pi\)
−0.788762 + 0.614698i \(0.789278\pi\)
\(90\) 0 0
\(91\) 22.0292 2.30929
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.60651 + 9.71076i −0.575216 + 0.996303i
\(96\) 0 0
\(97\) 5.89884 + 10.2171i 0.598937 + 1.03739i 0.992978 + 0.118296i \(0.0377432\pi\)
−0.394042 + 0.919093i \(0.628923\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.81211 10.0669i −0.578326 1.00169i −0.995671 0.0929424i \(-0.970373\pi\)
0.417345 0.908748i \(-0.362961\pi\)
\(102\) 0 0
\(103\) −9.54962 + 16.5404i −0.940952 + 1.62978i −0.177292 + 0.984158i \(0.556734\pi\)
−0.763660 + 0.645618i \(0.776600\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.99694 −0.483072 −0.241536 0.970392i \(-0.577651\pi\)
−0.241536 + 0.970392i \(0.577651\pi\)
\(108\) 0 0
\(109\) 2.15690 0.206594 0.103297 0.994651i \(-0.467061\pi\)
0.103297 + 0.994651i \(0.467061\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.2451 + 17.7451i −0.963782 + 1.66932i −0.250929 + 0.968005i \(0.580736\pi\)
−0.712853 + 0.701314i \(0.752597\pi\)
\(114\) 0 0
\(115\) −1.52945 2.64908i −0.142622 0.247028i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.05270 10.4836i −0.554850 0.961028i
\(120\) 0 0
\(121\) 2.27688 3.94368i 0.206989 0.358516i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.54856 −0.675164
\(126\) 0 0
\(127\) −9.71867 −0.862392 −0.431196 0.902258i \(-0.641908\pi\)
−0.431196 + 0.902258i \(0.641908\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.49219 + 6.04865i −0.305114 + 0.528473i −0.977287 0.211921i \(-0.932028\pi\)
0.672173 + 0.740394i \(0.265361\pi\)
\(132\) 0 0
\(133\) −10.4028 18.0181i −0.902035 1.56237i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.2072 19.4114i −0.957493 1.65843i −0.728558 0.684984i \(-0.759809\pi\)
−0.228935 0.973442i \(-0.573524\pi\)
\(138\) 0 0
\(139\) 2.56822 4.44830i 0.217834 0.377299i −0.736312 0.676643i \(-0.763434\pi\)
0.954146 + 0.299343i \(0.0967675\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −11.2467 −0.940494
\(144\) 0 0
\(145\) −16.1046 −1.33741
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.08562 + 1.88034i −0.0889372 + 0.154044i −0.907062 0.420997i \(-0.861680\pi\)
0.818125 + 0.575040i \(0.195014\pi\)
\(150\) 0 0
\(151\) 3.10508 + 5.37816i 0.252688 + 0.437669i 0.964265 0.264940i \(-0.0853520\pi\)
−0.711577 + 0.702608i \(0.752019\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.11072 + 12.3161i 0.571147 + 0.989255i
\(156\) 0 0
\(157\) −7.97578 + 13.8144i −0.636536 + 1.10251i 0.349651 + 0.936880i \(0.386300\pi\)
−0.986187 + 0.165633i \(0.947033\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.67572 0.447310
\(162\) 0 0
\(163\) −4.72991 −0.370475 −0.185237 0.982694i \(-0.559305\pi\)
−0.185237 + 0.982694i \(0.559305\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.18611 2.05440i 0.0917836 0.158974i −0.816478 0.577376i \(-0.804077\pi\)
0.908262 + 0.418403i \(0.137410\pi\)
\(168\) 0 0
\(169\) −3.31097 5.73477i −0.254690 0.441137i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.19528 + 2.07028i 0.0908752 + 0.157400i 0.907880 0.419231i \(-0.137700\pi\)
−0.817004 + 0.576631i \(0.804367\pi\)
\(174\) 0 0
\(175\) −5.42967 + 9.40446i −0.410444 + 0.710910i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.19567 0.0893689 0.0446845 0.999001i \(-0.485772\pi\)
0.0446845 + 0.999001i \(0.485772\pi\)
\(180\) 0 0
\(181\) −5.17390 −0.384573 −0.192287 0.981339i \(-0.561590\pi\)
−0.192287 + 0.981339i \(0.561590\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.323712 0.560686i 0.0237998 0.0412224i
\(186\) 0 0
\(187\) 3.09011 + 5.35222i 0.225971 + 0.391393i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.654726 + 1.13402i 0.0473743 + 0.0820548i 0.888740 0.458411i \(-0.151581\pi\)
−0.841366 + 0.540466i \(0.818248\pi\)
\(192\) 0 0
\(193\) 10.0726 17.4463i 0.725043 1.25581i −0.233913 0.972257i \(-0.575153\pi\)
0.958956 0.283554i \(-0.0915135\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.46023 −0.531520 −0.265760 0.964039i \(-0.585623\pi\)
−0.265760 + 0.964039i \(0.585623\pi\)
\(198\) 0 0
\(199\) 3.47361 0.246238 0.123119 0.992392i \(-0.460710\pi\)
0.123119 + 0.992392i \(0.460710\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 14.9409 25.8784i 1.04865 1.81631i
\(204\) 0 0
\(205\) 8.60634 + 14.9066i 0.601093 + 1.04112i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.31097 + 9.19888i 0.367368 + 0.636300i
\(210\) 0 0
\(211\) 4.38878 7.60159i 0.302136 0.523315i −0.674483 0.738290i \(-0.735634\pi\)
0.976620 + 0.214975i \(0.0689670\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 29.8723 2.03728
\(216\) 0 0
\(217\) −26.3876 −1.79131
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.39128 + 9.33797i −0.362657 + 0.628140i
\(222\) 0 0
\(223\) −7.42411 12.8589i −0.497155 0.861098i 0.502840 0.864380i \(-0.332289\pi\)
−0.999995 + 0.00328197i \(0.998955\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.49314 16.4426i −0.630082 1.09133i −0.987535 0.157403i \(-0.949688\pi\)
0.357453 0.933931i \(-0.383645\pi\)
\(228\) 0 0
\(229\) 1.17194 2.02986i 0.0774441 0.134137i −0.824702 0.565567i \(-0.808657\pi\)
0.902147 + 0.431430i \(0.141991\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0912 −0.792120 −0.396060 0.918225i \(-0.629623\pi\)
−0.396060 + 0.918225i \(0.629623\pi\)
\(234\) 0 0
\(235\) 12.7433 0.831281
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.54375 13.0662i 0.487965 0.845180i −0.511939 0.859022i \(-0.671073\pi\)
0.999904 + 0.0138417i \(0.00440610\pi\)
\(240\) 0 0
\(241\) 1.01095 + 1.75101i 0.0651208 + 0.112793i 0.896748 0.442542i \(-0.145923\pi\)
−0.831627 + 0.555335i \(0.812590\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −23.7627 41.1581i −1.51814 2.62950i
\(246\) 0 0
\(247\) −9.26600 + 16.0492i −0.589581 + 1.02118i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.7796 1.18536 0.592679 0.805438i \(-0.298070\pi\)
0.592679 + 0.805438i \(0.298070\pi\)
\(252\) 0 0
\(253\) −2.89765 −0.182174
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.5006 19.9197i 0.717389 1.24255i −0.244642 0.969613i \(-0.578671\pi\)
0.962031 0.272940i \(-0.0879961\pi\)
\(258\) 0 0
\(259\) 0.600642 + 1.04034i 0.0373221 + 0.0646437i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.77774 16.9356i −0.602922 1.04429i −0.992376 0.123245i \(-0.960670\pi\)
0.389455 0.921046i \(-0.372663\pi\)
\(264\) 0 0
\(265\) 12.5749 21.7803i 0.772469 1.33796i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −13.2107 −0.805468 −0.402734 0.915317i \(-0.631940\pi\)
−0.402734 + 0.915317i \(0.631940\pi\)
\(270\) 0 0
\(271\) 8.63055 0.524268 0.262134 0.965031i \(-0.415574\pi\)
0.262134 + 0.965031i \(0.415574\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.77203 4.80130i 0.167160 0.289529i
\(276\) 0 0
\(277\) 1.21650 + 2.10705i 0.0730927 + 0.126600i 0.900255 0.435362i \(-0.143380\pi\)
−0.827163 + 0.561963i \(0.810046\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.75715 + 8.23962i 0.283787 + 0.491534i 0.972314 0.233676i \(-0.0750756\pi\)
−0.688527 + 0.725211i \(0.741742\pi\)
\(282\) 0 0
\(283\) −5.27042 + 9.12864i −0.313294 + 0.542641i −0.979073 0.203507i \(-0.934766\pi\)
0.665779 + 0.746149i \(0.268099\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −31.9378 −1.88523
\(288\) 0 0
\(289\) −11.0748 −0.651460
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.10888 5.38474i 0.181623 0.314580i −0.760811 0.648974i \(-0.775198\pi\)
0.942433 + 0.334394i \(0.108532\pi\)
\(294\) 0 0
\(295\) −14.4975 25.1104i −0.844077 1.46198i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.52775 4.37819i −0.146184 0.253197i
\(300\) 0 0
\(301\) −27.7138 + 48.0017i −1.59740 + 2.76677i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 22.3018 1.27700
\(306\) 0 0
\(307\) 2.45594 0.140168 0.0700839 0.997541i \(-0.477673\pi\)
0.0700839 + 0.997541i \(0.477673\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.06337 3.57386i 0.117003 0.202655i −0.801576 0.597893i \(-0.796005\pi\)
0.918579 + 0.395238i \(0.129338\pi\)
\(312\) 0 0
\(313\) 2.66061 + 4.60831i 0.150386 + 0.260477i 0.931370 0.364075i \(-0.118615\pi\)
−0.780983 + 0.624552i \(0.785282\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.1291 + 17.5440i 0.568904 + 0.985371i 0.996675 + 0.0814836i \(0.0259658\pi\)
−0.427770 + 0.903887i \(0.640701\pi\)
\(318\) 0 0
\(319\) −7.62784 + 13.2118i −0.427077 + 0.739719i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.1836 0.566631
\(324\) 0 0
\(325\) 9.67266 0.536543
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.8225 + 20.4771i −0.651794 + 1.12894i
\(330\) 0 0
\(331\) −4.88990 8.46955i −0.268773 0.465529i 0.699772 0.714366i \(-0.253285\pi\)
−0.968545 + 0.248837i \(0.919952\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.03587 5.25828i −0.165867 0.287290i
\(336\) 0 0
\(337\) −7.61501 + 13.1896i −0.414816 + 0.718482i −0.995409 0.0957115i \(-0.969487\pi\)
0.580593 + 0.814194i \(0.302821\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 13.4718 0.729538
\(342\) 0 0
\(343\) 53.3706 2.88174
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.12881 + 5.41925i −0.167963 + 0.290921i −0.937704 0.347436i \(-0.887052\pi\)
0.769741 + 0.638357i \(0.220386\pi\)
\(348\) 0 0
\(349\) 10.0277 + 17.3685i 0.536772 + 0.929716i 0.999075 + 0.0429941i \(0.0136897\pi\)
−0.462304 + 0.886722i \(0.652977\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.26961 + 16.0554i 0.493372 + 0.854545i 0.999971 0.00763681i \(-0.00243089\pi\)
−0.506599 + 0.862182i \(0.669098\pi\)
\(354\) 0 0
\(355\) 6.06587 10.5064i 0.321943 0.557621i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −25.2838 −1.33443 −0.667213 0.744867i \(-0.732513\pi\)
−0.667213 + 0.744867i \(0.732513\pi\)
\(360\) 0 0
\(361\) −1.49741 −0.0788112
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.48573 7.76951i 0.234794 0.406675i
\(366\) 0 0
\(367\) −6.32458 10.9545i −0.330141 0.571820i 0.652399 0.757876i \(-0.273763\pi\)
−0.982539 + 0.186056i \(0.940429\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 23.3325 + 40.4130i 1.21136 + 2.09814i
\(372\) 0 0
\(373\) 9.39906 16.2796i 0.486665 0.842928i −0.513218 0.858258i \(-0.671547\pi\)
0.999882 + 0.0153303i \(0.00487998\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −26.6164 −1.37081
\(378\) 0 0
\(379\) −2.15057 −0.110467 −0.0552337 0.998473i \(-0.517590\pi\)
−0.0552337 + 0.998473i \(0.517590\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.2847 28.2060i 0.832111 1.44126i −0.0642505 0.997934i \(-0.520466\pi\)
0.896361 0.443324i \(-0.146201\pi\)
\(384\) 0 0
\(385\) 16.9209 + 29.3078i 0.862367 + 1.49366i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.9569 + 20.7100i 0.606241 + 1.05004i 0.991854 + 0.127380i \(0.0406567\pi\)
−0.385613 + 0.922661i \(0.626010\pi\)
\(390\) 0 0
\(391\) −1.38904 + 2.40588i −0.0702466 + 0.121671i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.8403 −0.847329
\(396\) 0 0
\(397\) −4.42557 −0.222113 −0.111056 0.993814i \(-0.535423\pi\)
−0.111056 + 0.993814i \(0.535423\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.04462 5.27344i 0.152041 0.263343i −0.779936 0.625859i \(-0.784749\pi\)
0.931978 + 0.362515i \(0.118082\pi\)
\(402\) 0 0
\(403\) 11.7520 + 20.3551i 0.585411 + 1.01396i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.306648 0.531130i −0.0152000 0.0263272i
\(408\) 0 0
\(409\) −1.23061 + 2.13148i −0.0608497 + 0.105395i −0.894845 0.446376i \(-0.852714\pi\)
0.833996 + 0.551771i \(0.186048\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 53.7997 2.64731
\(414\) 0 0
\(415\) −3.95734 −0.194258
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9.75667 + 16.8991i −0.476645 + 0.825573i −0.999642 0.0267617i \(-0.991480\pi\)
0.522997 + 0.852334i \(0.324814\pi\)
\(420\) 0 0
\(421\) −9.52846 16.5038i −0.464389 0.804345i 0.534785 0.844988i \(-0.320393\pi\)
−0.999174 + 0.0406435i \(0.987059\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.65764 4.60316i −0.128914 0.223286i
\(426\) 0 0
\(427\) −20.6903 + 35.8367i −1.00128 + 1.73426i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 23.6111 1.13731 0.568654 0.822577i \(-0.307464\pi\)
0.568654 + 0.822577i \(0.307464\pi\)
\(432\) 0 0
\(433\) −2.09660 −0.100756 −0.0503780 0.998730i \(-0.516043\pi\)
−0.0503780 + 0.998730i \(0.516043\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.38734 + 4.13499i −0.114202 + 0.197804i
\(438\) 0 0
\(439\) −4.38026 7.58684i −0.209059 0.362100i 0.742360 0.670002i \(-0.233707\pi\)
−0.951418 + 0.307901i \(0.900373\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.51251 + 7.81590i 0.214396 + 0.371345i 0.953086 0.302701i \(-0.0978884\pi\)
−0.738690 + 0.674046i \(0.764555\pi\)
\(444\) 0 0
\(445\) 19.9440 34.5440i 0.945437 1.63754i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.54295 0.120009 0.0600047 0.998198i \(-0.480888\pi\)
0.0600047 + 0.998198i \(0.480888\pi\)
\(450\) 0 0
\(451\) 16.3054 0.767789
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −29.5216 + 51.1330i −1.38400 + 2.39715i
\(456\) 0 0
\(457\) 12.9090 + 22.3590i 0.603855 + 1.04591i 0.992231 + 0.124407i \(0.0397029\pi\)
−0.388376 + 0.921501i \(0.626964\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.4517 21.5669i −0.579932 1.00447i −0.995486 0.0949039i \(-0.969746\pi\)
0.415554 0.909568i \(-0.363588\pi\)
\(462\) 0 0
\(463\) −11.0655 + 19.1660i −0.514258 + 0.890721i 0.485605 + 0.874178i \(0.338599\pi\)
−0.999863 + 0.0165428i \(0.994734\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −24.7135 −1.14361 −0.571803 0.820391i \(-0.693756\pi\)
−0.571803 + 0.820391i \(0.693756\pi\)
\(468\) 0 0
\(469\) 11.2660 0.520215
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14.1488 24.5065i 0.650564 1.12681i
\(474\) 0 0
\(475\) −4.56768 7.91146i −0.209580 0.363003i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19.6023 + 33.9522i 0.895653 + 1.55132i 0.832995 + 0.553281i \(0.186624\pi\)
0.0626578 + 0.998035i \(0.480042\pi\)
\(480\) 0 0
\(481\) 0.535006 0.926657i 0.0243942 0.0422519i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −31.6204 −1.43581
\(486\) 0 0
\(487\) −29.3140 −1.32835 −0.664173 0.747579i \(-0.731216\pi\)
−0.664173 + 0.747579i \(0.731216\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.57211 4.45503i 0.116078 0.201052i −0.802132 0.597146i \(-0.796301\pi\)
0.918210 + 0.396094i \(0.129635\pi\)
\(492\) 0 0
\(493\) 7.31306 + 12.6666i 0.329364 + 0.570474i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.2551 + 19.4944i 0.504861 + 0.874445i
\(498\) 0 0
\(499\) −17.5877 + 30.4627i −0.787332 + 1.36370i 0.140263 + 0.990114i \(0.455205\pi\)
−0.927596 + 0.373585i \(0.878128\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −13.2366 −0.590192 −0.295096 0.955468i \(-0.595352\pi\)
−0.295096 + 0.955468i \(0.595352\pi\)
\(504\) 0 0
\(505\) 31.1555 1.38640
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −15.8738 + 27.4942i −0.703593 + 1.21866i 0.263604 + 0.964631i \(0.415089\pi\)
−0.967197 + 0.254028i \(0.918245\pi\)
\(510\) 0 0
\(511\) 8.32319 + 14.4162i 0.368196 + 0.637735i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −25.5951 44.3321i −1.12786 1.95350i
\(516\) 0 0
\(517\) 6.03578 10.4543i 0.265453 0.459778i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.83339 0.124133 0.0620666 0.998072i \(-0.480231\pi\)
0.0620666 + 0.998072i \(0.480231\pi\)
\(522\) 0 0
\(523\) −32.0242 −1.40032 −0.700160 0.713986i \(-0.746888\pi\)
−0.700160 + 0.713986i \(0.746888\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.45792 11.1854i 0.281311 0.487246i
\(528\) 0 0
\(529\) 10.8487 + 18.7906i 0.471684 + 0.816981i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.2239 + 24.6365i 0.616105 + 1.06713i
\(534\) 0 0
\(535\) 6.69646 11.5986i 0.289513 0.501452i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −45.0201 −1.93915
\(540\) 0 0
\(541\) 3.84546 0.165329 0.0826646 0.996577i \(-0.473657\pi\)
0.0826646 + 0.996577i \(0.473657\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.89049 + 5.00648i −0.123815 + 0.214454i
\(546\) 0 0
\(547\) −22.4749 38.9277i −0.960958 1.66443i −0.720102 0.693868i \(-0.755905\pi\)
−0.240856 0.970561i \(-0.577428\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.5690 + 21.7701i 0.535456 + 0.927437i
\(552\) 0 0
\(553\) 15.6235 27.0607i 0.664378 1.15074i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 35.8629 1.51956 0.759780 0.650180i \(-0.225307\pi\)
0.759780 + 0.650180i \(0.225307\pi\)
\(558\) 0 0
\(559\) 49.3706 2.08816
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.11092 8.85237i 0.215400 0.373083i −0.737997 0.674805i \(-0.764228\pi\)
0.953396 + 0.301721i \(0.0975613\pi\)
\(564\) 0 0
\(565\) −27.4593 47.5609i −1.15522 2.00090i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.4319 19.8006i −0.479250 0.830085i 0.520467 0.853882i \(-0.325758\pi\)
−0.999717 + 0.0237967i \(0.992425\pi\)
\(570\) 0 0
\(571\) 1.63442 2.83091i 0.0683985 0.118470i −0.829798 0.558064i \(-0.811544\pi\)
0.898197 + 0.439594i \(0.144878\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.49211 0.103928
\(576\) 0 0
\(577\) −40.2432 −1.67535 −0.837673 0.546172i \(-0.816084\pi\)
−0.837673 + 0.546172i \(0.816084\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.67139 6.35903i 0.152315 0.263817i
\(582\) 0 0
\(583\) −11.9120 20.6322i −0.493346 0.854500i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.3312 + 23.0903i 0.550236 + 0.953037i 0.998257 + 0.0590141i \(0.0187957\pi\)
−0.448021 + 0.894023i \(0.647871\pi\)
\(588\) 0 0
\(589\) 11.0992 19.2244i 0.457336 0.792129i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −36.4349 −1.49620 −0.748101 0.663585i \(-0.769034\pi\)
−0.748101 + 0.663585i \(0.769034\pi\)
\(594\) 0 0
\(595\) 32.4452 1.33012
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.51357 11.2818i 0.266137 0.460963i −0.701724 0.712449i \(-0.747586\pi\)
0.967861 + 0.251486i \(0.0809193\pi\)
\(600\) 0 0
\(601\) −23.1094 40.0267i −0.942653 1.63272i −0.760384 0.649473i \(-0.774989\pi\)
−0.182268 0.983249i \(-0.558344\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.10256 + 10.5699i 0.248104 + 0.429729i
\(606\) 0 0
\(607\) 6.32515 10.9555i 0.256730 0.444669i −0.708634 0.705576i \(-0.750688\pi\)
0.965364 + 0.260907i \(0.0840215\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.0611 0.852041
\(612\) 0 0
\(613\) 16.2013 0.654365 0.327182 0.944961i \(-0.393901\pi\)
0.327182 + 0.944961i \(0.393901\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.54706 4.41164i 0.102541 0.177606i −0.810190 0.586167i \(-0.800636\pi\)
0.912731 + 0.408561i \(0.133969\pi\)
\(618\) 0 0
\(619\) −12.5926 21.8111i −0.506140 0.876660i −0.999975 0.00710457i \(-0.997739\pi\)
0.493835 0.869556i \(-0.335595\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 37.0058 + 64.0959i 1.48260 + 2.56795i
\(624\) 0 0
\(625\) 15.5749 26.9766i 0.622998 1.07906i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.587987 −0.0234446
\(630\) 0 0
\(631\) 18.4509 0.734518 0.367259 0.930119i \(-0.380296\pi\)
0.367259 + 0.930119i \(0.380296\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 13.0241 22.5584i 0.516846 0.895204i
\(636\) 0 0
\(637\) −39.2731 68.0230i −1.55606 2.69517i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.2431 + 29.8659i 0.681062 + 1.17963i 0.974657 + 0.223704i \(0.0718149\pi\)
−0.293595 + 0.955930i \(0.594852\pi\)
\(642\) 0 0
\(643\) −5.28665 + 9.15675i −0.208485 + 0.361107i −0.951238 0.308459i \(-0.900187\pi\)
0.742752 + 0.669566i \(0.233520\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.5011 0.570098 0.285049 0.958513i \(-0.407990\pi\)
0.285049 + 0.958513i \(0.407990\pi\)
\(648\) 0 0
\(649\) −27.4666 −1.07816
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.29761 + 5.71162i −0.129045 + 0.223513i −0.923307 0.384063i \(-0.874525\pi\)
0.794262 + 0.607576i \(0.207858\pi\)
\(654\) 0 0
\(655\) −9.35986 16.2117i −0.365720 0.633445i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −15.5531 26.9387i −0.605861 1.04938i −0.991915 0.126906i \(-0.959495\pi\)
0.386053 0.922476i \(-0.373838\pi\)
\(660\) 0 0
\(661\) −17.1695 + 29.7384i −0.667816 + 1.15669i 0.310698 + 0.950509i \(0.399437\pi\)
−0.978514 + 0.206182i \(0.933896\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 55.7636 2.16242
\(666\) 0 0
\(667\) −6.85759 −0.265527
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.5631 18.2959i 0.407785 0.706304i
\(672\) 0 0
\(673\) 6.32809 + 10.9606i 0.243930 + 0.422499i 0.961830 0.273647i \(-0.0882299\pi\)
−0.717900 + 0.696146i \(0.754897\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.78910 + 11.7591i 0.260926 + 0.451938i 0.966488 0.256710i \(-0.0826387\pi\)
−0.705562 + 0.708648i \(0.749305\pi\)
\(678\) 0 0
\(679\) 29.3356 50.8107i 1.12580 1.94994i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.85174 0.147383 0.0736914 0.997281i \(-0.476522\pi\)
0.0736914 + 0.997281i \(0.476522\pi\)
\(684\) 0 0
\(685\) 60.0755 2.29537
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 20.7828 35.9968i 0.791761 1.37137i
\(690\) 0 0
\(691\) −20.3658 35.2746i −0.774752 1.34191i −0.934934 0.354822i \(-0.884542\pi\)
0.160182 0.987087i \(-0.448792\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.88342 + 11.9224i 0.261103 + 0.452244i
\(696\) 0 0
\(697\) 7.81624 13.5381i 0.296061 0.512793i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17.5260 −0.661948 −0.330974 0.943640i \(-0.607377\pi\)
−0.330974 + 0.943640i \(0.607377\pi\)
\(702\) 0 0
\(703\) −1.01057 −0.0381146
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −28.9043 + 50.0636i −1.08706 + 1.88284i
\(708\) 0 0
\(709\) 11.5020 + 19.9220i 0.431965 + 0.748185i 0.997042 0.0768527i \(-0.0244871\pi\)
−0.565078 + 0.825038i \(0.691154\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.02786 + 5.24440i 0.113394 + 0.196404i
\(714\) 0 0
\(715\) 15.0718 26.1051i 0.563654 0.976277i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 11.6172 0.433250 0.216625 0.976255i \(-0.430495\pi\)
0.216625 + 0.976255i \(0.430495\pi\)
\(720\) 0 0
\(721\) 94.9826 3.53734
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.56030 11.3628i 0.243643 0.422003i
\(726\) 0 0
\(727\) 23.5416 + 40.7752i 0.873109 + 1.51227i 0.858763 + 0.512373i \(0.171233\pi\)
0.0143459 + 0.999897i \(0.495433\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.5650 23.4952i −0.501718 0.869001i
\(732\) 0 0
\(733\) −12.5946 + 21.8145i −0.465193 + 0.805737i −0.999210 0.0397362i \(-0.987348\pi\)
0.534018 + 0.845473i \(0.320682\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.75168 −0.211866
\(738\) 0 0
\(739\) 9.73735 0.358194 0.179097 0.983831i \(-0.442682\pi\)
0.179097 + 0.983831i \(0.442682\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −18.0084 + 31.1915i −0.660666 + 1.14431i 0.319775 + 0.947493i \(0.396393\pi\)
−0.980441 + 0.196813i \(0.936941\pi\)
\(744\) 0 0
\(745\) −2.90970 5.03974i −0.106603 0.184642i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.4252 + 21.5210i 0.454006 + 0.786361i
\(750\) 0 0
\(751\) 14.3734 24.8954i 0.524493 0.908448i −0.475101 0.879931i \(-0.657588\pi\)
0.999593 0.0285164i \(-0.00907828\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.6446 −0.605761
\(756\) 0 0
\(757\) 53.3478 1.93896 0.969479 0.245174i \(-0.0788450\pi\)
0.969479 + 0.245174i \(0.0788450\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −13.4580 + 23.3100i −0.487853 + 0.844986i −0.999902 0.0139702i \(-0.995553\pi\)
0.512050 + 0.858956i \(0.328886\pi\)
\(762\) 0 0
\(763\) −5.36326 9.28943i −0.194163 0.336300i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −23.9603 41.5005i −0.865157 1.49850i
\(768\) 0 0
\(769\) −4.56541 + 7.90752i −0.164633 + 0.285152i −0.936525 0.350601i \(-0.885977\pi\)
0.771892 + 0.635754i \(0.219311\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.5628 −0.739591 −0.369796 0.929113i \(-0.620572\pi\)
−0.369796 + 0.929113i \(0.620572\pi\)
\(774\) 0 0
\(775\) −11.5864 −0.416194
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.4338 23.2680i 0.481315 0.833663i
\(780\) 0 0
\(781\) −5.74612 9.95257i −0.205612 0.356131i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −21.3769 37.0258i −0.762974 1.32151i
\(786\) 0 0
\(787\) 3.17998 5.50789i 0.113354 0.196335i −0.803767 0.594945i \(-0.797174\pi\)
0.917121 + 0.398610i \(0.130507\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 101.900 3.62316
\(792\) 0 0
\(793\) 36.8587 1.30889
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.3241 + 17.8819i −0.365699 + 0.633409i −0.988888 0.148662i \(-0.952503\pi\)
0.623189 + 0.782071i \(0.285837\pi\)
\(798\) 0 0
\(799\) −5.78670 10.0229i −0.204719 0.354583i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4.24928 7.35996i −0.149954 0.259727i
\(804\) 0 0
\(805\) −7.60611 + 13.1742i −0.268080 + 0.464328i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 45.2805 1.59198 0.795989 0.605311i \(-0.206951\pi\)
0.795989 + 0.605311i \(0.206951\pi\)
\(810\) 0 0
\(811\) −4.29363 −0.150770 −0.0753848 0.997155i \(-0.524019\pi\)
−0.0753848 + 0.997155i \(0.524019\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.33861 10.9788i 0.222032 0.384570i
\(816\) 0 0
\(817\) −23.3141 40.3812i −0.815657 1.41276i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.49228 14.7091i −0.296383 0.513350i 0.678923 0.734210i \(-0.262447\pi\)
−0.975306 + 0.220860i \(0.929114\pi\)
\(822\) 0 0
\(823\) −2.98907 + 5.17721i −0.104192 + 0.180466i −0.913408 0.407045i \(-0.866559\pi\)
0.809216 + 0.587512i \(0.199892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.4590 −0.398469 −0.199234 0.979952i \(-0.563845\pi\)
−0.199234 + 0.979952i \(0.563845\pi\)
\(828\) 0 0
\(829\) 34.5859 1.20122 0.600609 0.799543i \(-0.294925\pi\)
0.600609 + 0.799543i \(0.294925\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −21.5811 + 37.3796i −0.747742 + 1.29513i
\(834\) 0 0
\(835\) 3.17903 + 5.50625i 0.110015 + 0.190551i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12.1093 20.9740i −0.418060 0.724102i 0.577684 0.816261i \(-0.303957\pi\)
−0.995744 + 0.0921587i \(0.970623\pi\)
\(840\) 0 0
\(841\) −3.55207 + 6.15236i −0.122485 + 0.212150i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 17.7483 0.610561
\(846\) 0 0
\(847\) −22.6464 −0.778139
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.137842 0.238749i 0.00472515 0.00818420i
\(852\) 0 0
\(853\) 26.7241 + 46.2875i 0.915016 + 1.58485i 0.806878 + 0.590719i \(0.201156\pi\)
0.108138 + 0.994136i \(0.465511\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.79836 3.11486i −0.0614310 0.106402i 0.833674 0.552256i \(-0.186233\pi\)
−0.895105 + 0.445855i \(0.852900\pi\)
\(858\) 0 0
\(859\) −15.9691 + 27.6594i −0.544860 + 0.943726i 0.453755 + 0.891126i \(0.350084\pi\)
−0.998616 + 0.0525994i \(0.983249\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.20625 0.0751016 0.0375508 0.999295i \(-0.488044\pi\)
0.0375508 + 0.999295i \(0.488044\pi\)
\(864\) 0 0
\(865\) −6.40722 −0.217852
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7.97632 + 13.8154i −0.270578 + 0.468655i
\(870\) 0 0
\(871\) −5.01744 8.69047i −0.170010 0.294465i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 18.7699 + 32.5105i 0.634539 + 1.09905i
\(876\) 0 0
\(877\) −3.07129 + 5.31962i −0.103710 + 0.179631i −0.913210 0.407488i \(-0.866405\pi\)
0.809500 + 0.587119i \(0.199738\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 38.6422 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(882\) 0 0
\(883\) −38.4651 −1.29446 −0.647228 0.762297i \(-0.724072\pi\)
−0.647228 + 0.762297i \(0.724072\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.1015 43.4770i 0.842825 1.45982i −0.0446713 0.999002i \(-0.514224\pi\)
0.887497 0.460814i \(-0.152443\pi\)
\(888\) 0 0
\(889\) 24.1660 + 41.8568i 0.810502 + 1.40383i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.94561 17.2263i −0.332817 0.576456i
\(894\) 0 0
\(895\) −1.60234 + 2.77533i −0.0535603 + 0.0927691i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 31.8824 1.06334
\(900\) 0 0
\(901\) −22.8409 −0.760941
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.93361 12.0094i 0.230481 0.399205i
\(906\) 0 0
\(907\) 28.5437 + 49.4391i 0.947777 + 1.64160i 0.750093 + 0.661332i \(0.230009\pi\)
0.197684 + 0.980266i \(0.436658\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.15893 14.1317i −0.270317 0.468204i 0.698626 0.715487i \(-0.253795\pi\)
−0.968943 + 0.247284i \(0.920462\pi\)
\(912\) 0 0
\(913\) −1.87437 + 3.24650i −0.0620326 + 0.107444i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 34.7341 1.14702
\(918\) 0 0
\(919\) 25.5992 0.844441 0.422220 0.906493i \(-0.361251\pi\)
0.422220 + 0.906493i \(0.361251\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 10.0252 17.3641i 0.329983 0.571548i
\(924\) 0 0
\(925\) 0.263732 + 0.456797i 0.00867145 + 0.0150194i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 12.3480 + 21.3874i 0.405126 + 0.701699i 0.994336 0.106281i \(-0.0338944\pi\)
−0.589210 + 0.807980i \(0.700561\pi\)
\(930\) 0 0
\(931\) −37.0915 + 64.2444i −1.21563 + 2.10553i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −16.5644 −0.541713
\(936\) 0 0
\(937\) −27.3574 −0.893728 −0.446864 0.894602i \(-0.647459\pi\)
−0.446864 + 0.894602i \(0.647459\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.2953 21.2962i 0.400817 0.694235i −0.593008 0.805197i \(-0.702060\pi\)
0.993825 + 0.110962i \(0.0353931\pi\)
\(942\) 0 0
\(943\) 3.66472 + 6.34747i 0.119340 + 0.206702i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −10.0107 17.3391i −0.325305 0.563445i 0.656269 0.754527i \(-0.272134\pi\)
−0.981574 + 0.191082i \(0.938800\pi\)
\(948\) 0 0
\(949\) 7.41366 12.8408i 0.240658 0.416831i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 7.59573 0.246050 0.123025 0.992404i \(-0.460741\pi\)
0.123025 + 0.992404i \(0.460741\pi\)
\(954\) 0 0
\(955\) −3.50963 −0.113569
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −55.7345 + 96.5350i −1.79976 + 3.11728i
\(960\) 0 0
\(961\) 1.42287 + 2.46448i 0.0458990 + 0.0794994i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 26.9969 + 46.7600i 0.869061 + 1.50526i
\(966\) 0 0
\(967\) −10.2203 + 17.7020i −0.328662 + 0.569259i −0.982247 0.187594i \(-0.939931\pi\)
0.653585 + 0.756853i \(0.273264\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −47.8189 −1.53458 −0.767290 0.641300i \(-0.778395\pi\)
−0.767290 + 0.641300i \(0.778395\pi\)
\(972\) 0 0
\(973\) −25.5441 −0.818907
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.33054 9.23277i 0.170539 0.295382i −0.768069 0.640367i \(-0.778782\pi\)
0.938608 + 0.344984i \(0.112116\pi\)
\(978\) 0 0
\(979\) −18.8927 32.7231i −0.603813 1.04584i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −11.9147 20.6369i −0.380020 0.658214i 0.611045 0.791596i \(-0.290750\pi\)
−0.991065 + 0.133382i \(0.957416\pi\)
\(984\) 0 0
\(985\) 9.99755 17.3163i 0.318549 0.551742i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.7201 0.404476
\(990\) 0 0
\(991\) 11.3401 0.360231 0.180116 0.983645i \(-0.442353\pi\)
0.180116 + 0.983645i \(0.442353\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.65503 + 8.06276i −0.147575 + 0.255607i
\(996\) 0 0
\(997\) −16.0454 27.7914i −0.508163 0.880164i −0.999955 0.00945137i \(-0.996991\pi\)
0.491793 0.870712i \(-0.336342\pi\)
\(998\) 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.f.2305.1 10
3.2 odd 2 1152.2.i.g.769.5 yes 10
4.3 odd 2 3456.2.i.g.2305.1 10
8.3 odd 2 3456.2.i.h.2305.5 10
8.5 even 2 3456.2.i.e.2305.5 10
9.2 odd 6 1152.2.i.g.385.5 yes 10
9.7 even 3 inner 3456.2.i.f.1153.1 10
12.11 even 2 1152.2.i.f.769.1 yes 10
24.5 odd 2 1152.2.i.e.769.1 yes 10
24.11 even 2 1152.2.i.h.769.5 yes 10
36.7 odd 6 3456.2.i.g.1153.1 10
36.11 even 6 1152.2.i.f.385.1 yes 10
72.11 even 6 1152.2.i.h.385.5 yes 10
72.29 odd 6 1152.2.i.e.385.1 10
72.43 odd 6 3456.2.i.h.1153.5 10
72.61 even 6 3456.2.i.e.1153.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.i.e.385.1 10 72.29 odd 6
1152.2.i.e.769.1 yes 10 24.5 odd 2
1152.2.i.f.385.1 yes 10 36.11 even 6
1152.2.i.f.769.1 yes 10 12.11 even 2
1152.2.i.g.385.5 yes 10 9.2 odd 6
1152.2.i.g.769.5 yes 10 3.2 odd 2
1152.2.i.h.385.5 yes 10 72.11 even 6
1152.2.i.h.769.5 yes 10 24.11 even 2
3456.2.i.e.1153.5 10 72.61 even 6
3456.2.i.e.2305.5 10 8.5 even 2
3456.2.i.f.1153.1 10 9.7 even 3 inner
3456.2.i.f.2305.1 10 1.1 even 1 trivial
3456.2.i.g.1153.1 10 36.7 odd 6
3456.2.i.g.2305.1 10 4.3 odd 2
3456.2.i.h.1153.5 10 72.43 odd 6
3456.2.i.h.2305.5 10 8.3 odd 2