Properties

Label 2268.2.k.g.1621.6
Level $2268$
Weight $2$
Character 2268.1621
Analytic conductor $18.110$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(1297,2268)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1297");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.k (of order \(3\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 9x^{14} + 31x^{12} - 282x^{10} + 1695x^{8} - 3318x^{6} + 4606x^{4} - 4116x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1621.6
Root \(-1.04556 - 0.339889i\) of defining polynomial
Character \(\chi\) \(=\) 2268.1621
Dual form 2268.2.k.g.1297.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.515559 + 0.892975i) q^{5} +(1.55575 + 2.14001i) q^{7} +O(q^{10})\) \(q+(0.515559 + 0.892975i) q^{5} +(1.55575 + 2.14001i) q^{7} +(-0.792879 + 1.37331i) q^{11} +5.04830 q^{13} +(2.58242 - 4.47288i) q^{17} +(-0.392975 - 0.680652i) q^{19} +(-2.93289 - 5.07991i) q^{23} +(1.96840 - 3.40936i) q^{25} +8.89021 q^{29} +(0.575423 - 0.996661i) q^{31} +(-1.10889 + 2.49255i) q^{35} +(4.07991 + 7.06661i) q^{37} -7.74411 q^{41} -2.53256 q^{43} +(4.24595 + 7.35420i) q^{47} +(-2.15926 + 6.65865i) q^{49} +(-2.41270 + 4.17892i) q^{53} -1.63510 q^{55} +(1.93622 - 3.35364i) q^{59} +(4.82204 + 8.35201i) q^{61} +(2.60270 + 4.50801i) q^{65} +(0.837220 - 1.45011i) q^{67} -14.2795 q^{71} +(3.04382 - 5.27205i) q^{73} +(-4.17241 + 0.439762i) q^{77} +(4.15533 + 7.19724i) q^{79} +14.2419 q^{83} +5.32556 q^{85} +(-6.69272 - 11.5921i) q^{89} +(7.85392 + 10.8034i) q^{91} +(0.405203 - 0.701833i) q^{95} +5.34999 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{7} - 20 q^{13} + 8 q^{19} - 8 q^{31} - 4 q^{37} + 20 q^{43} + 10 q^{49} - 32 q^{55} + 28 q^{61} + 18 q^{67} - 20 q^{79} + 76 q^{85} - 2 q^{91} - 84 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 0.515559 + 0.892975i 0.230565 + 0.399350i 0.957975 0.286853i \(-0.0926092\pi\)
−0.727409 + 0.686204i \(0.759276\pi\)
\(6\) 0 0
\(7\) 1.55575 + 2.14001i 0.588020 + 0.808846i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.792879 + 1.37331i −0.239062 + 0.414067i −0.960445 0.278468i \(-0.910173\pi\)
0.721383 + 0.692536i \(0.243507\pi\)
\(12\) 0 0
\(13\) 5.04830 1.40015 0.700074 0.714071i \(-0.253150\pi\)
0.700074 + 0.714071i \(0.253150\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.58242 4.47288i 0.626329 1.08483i −0.361954 0.932196i \(-0.617890\pi\)
0.988282 0.152637i \(-0.0487765\pi\)
\(18\) 0 0
\(19\) −0.392975 0.680652i −0.0901546 0.156152i 0.817421 0.576040i \(-0.195403\pi\)
−0.907576 + 0.419888i \(0.862069\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.93289 5.07991i −0.611549 1.05923i −0.990980 0.134014i \(-0.957213\pi\)
0.379431 0.925220i \(-0.376120\pi\)
\(24\) 0 0
\(25\) 1.96840 3.40936i 0.393679 0.681873i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.89021 1.65087 0.825435 0.564496i \(-0.190930\pi\)
0.825435 + 0.564496i \(0.190930\pi\)
\(30\) 0 0
\(31\) 0.575423 0.996661i 0.103349 0.179006i −0.809713 0.586825i \(-0.800377\pi\)
0.913062 + 0.407820i \(0.133711\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.10889 + 2.49255i −0.187436 + 0.421318i
\(36\) 0 0
\(37\) 4.07991 + 7.06661i 0.670732 + 1.16174i 0.977697 + 0.210022i \(0.0673535\pi\)
−0.306964 + 0.951721i \(0.599313\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.74411 −1.20943 −0.604714 0.796443i \(-0.706712\pi\)
−0.604714 + 0.796443i \(0.706712\pi\)
\(42\) 0 0
\(43\) −2.53256 −0.386212 −0.193106 0.981178i \(-0.561856\pi\)
−0.193106 + 0.981178i \(0.561856\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.24595 + 7.35420i 0.619335 + 1.07272i 0.989607 + 0.143796i \(0.0459310\pi\)
−0.370272 + 0.928923i \(0.620736\pi\)
\(48\) 0 0
\(49\) −2.15926 + 6.65865i −0.308465 + 0.951236i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.41270 + 4.17892i −0.331410 + 0.574019i −0.982789 0.184734i \(-0.940858\pi\)
0.651378 + 0.758753i \(0.274191\pi\)
\(54\) 0 0
\(55\) −1.63510 −0.220477
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.93622 3.35364i 0.252075 0.436607i −0.712022 0.702157i \(-0.752220\pi\)
0.964097 + 0.265551i \(0.0855537\pi\)
\(60\) 0 0
\(61\) 4.82204 + 8.35201i 0.617398 + 1.06937i 0.989959 + 0.141357i \(0.0451467\pi\)
−0.372560 + 0.928008i \(0.621520\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.60270 + 4.50801i 0.322825 + 0.559150i
\(66\) 0 0
\(67\) 0.837220 1.45011i 0.102283 0.177159i −0.810342 0.585957i \(-0.800719\pi\)
0.912625 + 0.408798i \(0.134052\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.2795 −1.69467 −0.847333 0.531062i \(-0.821793\pi\)
−0.847333 + 0.531062i \(0.821793\pi\)
\(72\) 0 0
\(73\) 3.04382 5.27205i 0.356252 0.617047i −0.631079 0.775718i \(-0.717388\pi\)
0.987331 + 0.158671i \(0.0507211\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.17241 + 0.439762i −0.475490 + 0.0501155i
\(78\) 0 0
\(79\) 4.15533 + 7.19724i 0.467511 + 0.809753i 0.999311 0.0371172i \(-0.0118175\pi\)
−0.531800 + 0.846870i \(0.678484\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.2419 1.56325 0.781626 0.623747i \(-0.214391\pi\)
0.781626 + 0.623747i \(0.214391\pi\)
\(84\) 0 0
\(85\) 5.32556 0.577638
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.69272 11.5921i −0.709426 1.22876i −0.965070 0.261992i \(-0.915621\pi\)
0.255644 0.966771i \(-0.417713\pi\)
\(90\) 0 0
\(91\) 7.85392 + 10.8034i 0.823315 + 1.13250i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.405203 0.701833i 0.0415730 0.0720065i
\(96\) 0 0
\(97\) 5.34999 0.543210 0.271605 0.962409i \(-0.412446\pi\)
0.271605 + 0.962409i \(0.412446\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.59038 + 2.75462i −0.158249 + 0.274095i −0.934237 0.356652i \(-0.883918\pi\)
0.775988 + 0.630747i \(0.217251\pi\)
\(102\) 0 0
\(103\) 5.70660 + 9.88412i 0.562288 + 0.973911i 0.997296 + 0.0734850i \(0.0234121\pi\)
−0.435008 + 0.900426i \(0.643255\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.311386 + 0.539337i 0.0301028 + 0.0521396i 0.880684 0.473704i \(-0.157083\pi\)
−0.850582 + 0.525843i \(0.823750\pi\)
\(108\) 0 0
\(109\) 0.971921 1.68342i 0.0930932 0.161242i −0.815718 0.578450i \(-0.803658\pi\)
0.908811 + 0.417208i \(0.136991\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.58833 −0.337561 −0.168781 0.985654i \(-0.553983\pi\)
−0.168781 + 0.985654i \(0.553983\pi\)
\(114\) 0 0
\(115\) 3.02415 5.23798i 0.282004 0.488445i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 13.5896 1.43231i 1.24576 0.131300i
\(120\) 0 0
\(121\) 4.24269 + 7.34855i 0.385699 + 0.668050i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.21489 0.824205
\(126\) 0 0
\(127\) 11.6202 1.03113 0.515563 0.856852i \(-0.327583\pi\)
0.515563 + 0.856852i \(0.327583\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.07898 7.06501i −0.356382 0.617272i 0.630971 0.775806i \(-0.282657\pi\)
−0.987354 + 0.158534i \(0.949323\pi\)
\(132\) 0 0
\(133\) 0.845228 1.89990i 0.0732906 0.164742i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.28181 + 12.6125i −0.622127 + 1.07756i 0.366962 + 0.930236i \(0.380398\pi\)
−0.989089 + 0.147320i \(0.952935\pi\)
\(138\) 0 0
\(139\) −11.8343 −1.00377 −0.501884 0.864935i \(-0.667360\pi\)
−0.501884 + 0.864935i \(0.667360\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00269 + 6.93287i −0.334722 + 0.579756i
\(144\) 0 0
\(145\) 4.58343 + 7.93873i 0.380633 + 0.659276i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.02477 12.1673i −0.575492 0.996781i −0.995988 0.0894868i \(-0.971477\pi\)
0.420496 0.907294i \(-0.361856\pi\)
\(150\) 0 0
\(151\) −1.24269 + 2.15240i −0.101128 + 0.175159i −0.912150 0.409857i \(-0.865579\pi\)
0.811021 + 0.585016i \(0.198912\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.18666 0.0953147
\(156\) 0 0
\(157\) 1.51967 2.63214i 0.121283 0.210068i −0.798991 0.601343i \(-0.794633\pi\)
0.920274 + 0.391275i \(0.127966\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.30818 14.1795i 0.497154 1.11750i
\(162\) 0 0
\(163\) −8.75883 15.1707i −0.686045 1.18826i −0.973107 0.230352i \(-0.926012\pi\)
0.287063 0.957912i \(-0.407321\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −24.5476 −1.89955 −0.949775 0.312935i \(-0.898688\pi\)
−0.949775 + 0.312935i \(0.898688\pi\)
\(168\) 0 0
\(169\) 12.4854 0.960413
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.0344569 0.0596811i −0.00261971 0.00453747i 0.864713 0.502267i \(-0.167501\pi\)
−0.867332 + 0.497730i \(0.834167\pi\)
\(174\) 0 0
\(175\) 10.3584 1.09175i 0.783022 0.0825286i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.96086 + 6.86041i −0.296049 + 0.512771i −0.975228 0.221201i \(-0.929002\pi\)
0.679180 + 0.733972i \(0.262336\pi\)
\(180\) 0 0
\(181\) 3.59688 0.267354 0.133677 0.991025i \(-0.457322\pi\)
0.133677 + 0.991025i \(0.457322\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.20687 + 7.28651i −0.309295 + 0.535715i
\(186\) 0 0
\(187\) 4.09509 + 7.09291i 0.299463 + 0.518685i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.02458 + 13.8990i 0.580638 + 1.00569i 0.995404 + 0.0957664i \(0.0305302\pi\)
−0.414766 + 0.909928i \(0.636136\pi\)
\(192\) 0 0
\(193\) 10.7865 18.6828i 0.776430 1.34482i −0.157558 0.987510i \(-0.550362\pi\)
0.933987 0.357306i \(-0.116305\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.4441 1.45659 0.728293 0.685266i \(-0.240314\pi\)
0.728293 + 0.685266i \(0.240314\pi\)
\(198\) 0 0
\(199\) −6.50056 + 11.2593i −0.460812 + 0.798150i −0.999002 0.0446737i \(-0.985775\pi\)
0.538189 + 0.842824i \(0.319109\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 13.8310 + 19.0251i 0.970745 + 1.33530i
\(204\) 0 0
\(205\) −3.99255 6.91530i −0.278852 0.482985i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.24633 0.0862101
\(210\) 0 0
\(211\) 4.23849 0.291789 0.145895 0.989300i \(-0.453394\pi\)
0.145895 + 0.989300i \(0.453394\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.30569 2.26151i −0.0890470 0.154234i
\(216\) 0 0
\(217\) 3.02808 0.319152i 0.205559 0.0216655i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.0368 22.5805i 0.876953 1.51893i
\(222\) 0 0
\(223\) −11.6666 −0.781252 −0.390626 0.920549i \(-0.627742\pi\)
−0.390626 + 0.920549i \(0.627742\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.08989 3.61980i 0.138711 0.240255i −0.788298 0.615294i \(-0.789037\pi\)
0.927009 + 0.375039i \(0.122371\pi\)
\(228\) 0 0
\(229\) 4.19086 + 7.25878i 0.276940 + 0.479674i 0.970623 0.240607i \(-0.0773464\pi\)
−0.693683 + 0.720280i \(0.744013\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.499512 0.865180i −0.0327241 0.0566798i 0.849200 0.528072i \(-0.177085\pi\)
−0.881924 + 0.471392i \(0.843752\pi\)
\(234\) 0 0
\(235\) −4.37807 + 7.58305i −0.285594 + 0.494663i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.3082 0.796154 0.398077 0.917352i \(-0.369678\pi\)
0.398077 + 0.917352i \(0.369678\pi\)
\(240\) 0 0
\(241\) −3.23916 + 5.61039i −0.208653 + 0.361397i −0.951290 0.308296i \(-0.900241\pi\)
0.742638 + 0.669694i \(0.233575\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.05923 + 1.50477i −0.450998 + 0.0961360i
\(246\) 0 0
\(247\) −1.98386 3.43614i −0.126230 0.218636i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.2236 −0.771544 −0.385772 0.922594i \(-0.626065\pi\)
−0.385772 + 0.922594i \(0.626065\pi\)
\(252\) 0 0
\(253\) 9.30169 0.584792
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.96355 + 13.7933i 0.496753 + 0.860401i 0.999993 0.00374541i \(-0.00119220\pi\)
−0.503240 + 0.864147i \(0.667859\pi\)
\(258\) 0 0
\(259\) −8.77525 + 19.7249i −0.545267 + 1.22565i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 11.5305 19.9714i 0.711002 1.23149i −0.253480 0.967341i \(-0.581575\pi\)
0.964481 0.264150i \(-0.0850916\pi\)
\(264\) 0 0
\(265\) −4.97556 −0.305647
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.51107 + 9.54546i −0.336016 + 0.581997i −0.983679 0.179930i \(-0.942413\pi\)
0.647663 + 0.761927i \(0.275746\pi\)
\(270\) 0 0
\(271\) 2.74213 + 4.74951i 0.166572 + 0.288512i 0.937213 0.348759i \(-0.113397\pi\)
−0.770640 + 0.637271i \(0.780063\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.12140 + 5.40643i 0.188228 + 0.326020i
\(276\) 0 0
\(277\) 15.0331 26.0381i 0.903253 1.56448i 0.0800068 0.996794i \(-0.474506\pi\)
0.823246 0.567685i \(-0.192161\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.8182 −0.705013 −0.352506 0.935809i \(-0.614671\pi\)
−0.352506 + 0.935809i \(0.614671\pi\)
\(282\) 0 0
\(283\) −10.5605 + 18.2914i −0.627758 + 1.08731i 0.360243 + 0.932859i \(0.382694\pi\)
−0.988001 + 0.154450i \(0.950639\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0479 16.5725i −0.711167 0.978241i
\(288\) 0 0
\(289\) −4.83778 8.37928i −0.284575 0.492899i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −12.3020 −0.718688 −0.359344 0.933205i \(-0.617000\pi\)
−0.359344 + 0.933205i \(0.617000\pi\)
\(294\) 0 0
\(295\) 3.99295 0.232479
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −14.8061 25.6449i −0.856259 1.48308i
\(300\) 0 0
\(301\) −3.94005 5.41970i −0.227100 0.312386i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.97209 + 8.61191i −0.284701 + 0.493117i
\(306\) 0 0
\(307\) −33.5033 −1.91213 −0.956067 0.293147i \(-0.905297\pi\)
−0.956067 + 0.293147i \(0.905297\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 14.8057 25.6442i 0.839555 1.45415i −0.0507130 0.998713i \(-0.516149\pi\)
0.890268 0.455438i \(-0.150517\pi\)
\(312\) 0 0
\(313\) 5.85745 + 10.1454i 0.331082 + 0.573452i 0.982724 0.185075i \(-0.0592529\pi\)
−0.651642 + 0.758527i \(0.725920\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.08084 + 1.87207i 0.0607061 + 0.105146i 0.894781 0.446505i \(-0.147331\pi\)
−0.834075 + 0.551651i \(0.813998\pi\)
\(318\) 0 0
\(319\) −7.04886 + 12.2090i −0.394660 + 0.683572i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.05930 −0.225866
\(324\) 0 0
\(325\) 9.93707 17.2115i 0.551209 0.954723i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.13238 + 20.5277i −0.503484 + 1.13173i
\(330\) 0 0
\(331\) −15.6001 27.0202i −0.857461 1.48517i −0.874343 0.485308i \(-0.838707\pi\)
0.0168824 0.999857i \(-0.494626\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.72655 0.0943313
\(336\) 0 0
\(337\) 3.88849 0.211820 0.105910 0.994376i \(-0.466225\pi\)
0.105910 + 0.994376i \(0.466225\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.912481 + 1.58046i 0.0494136 + 0.0855869i
\(342\) 0 0
\(343\) −17.6088 + 5.73840i −0.950787 + 0.309845i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.37831 14.5117i 0.449771 0.779027i −0.548600 0.836085i \(-0.684839\pi\)
0.998371 + 0.0570585i \(0.0181722\pi\)
\(348\) 0 0
\(349\) −31.8194 −1.70325 −0.851625 0.524151i \(-0.824383\pi\)
−0.851625 + 0.524151i \(0.824383\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.66867 6.35431i 0.195263 0.338206i −0.751723 0.659478i \(-0.770777\pi\)
0.946987 + 0.321272i \(0.104111\pi\)
\(354\) 0 0
\(355\) −7.36193 12.7512i −0.390731 0.676765i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.16644 + 12.4126i 0.378230 + 0.655114i 0.990805 0.135299i \(-0.0431994\pi\)
−0.612574 + 0.790413i \(0.709866\pi\)
\(360\) 0 0
\(361\) 9.19114 15.9195i 0.483744 0.837870i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.27708 0.328557
\(366\) 0 0
\(367\) −12.7865 + 22.1469i −0.667450 + 1.15606i 0.311165 + 0.950356i \(0.399281\pi\)
−0.978615 + 0.205702i \(0.934052\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.6965 + 1.33818i −0.659169 + 0.0694748i
\(372\) 0 0
\(373\) −15.5734 26.9739i −0.806361 1.39666i −0.915368 0.402617i \(-0.868101\pi\)
0.109008 0.994041i \(-0.465233\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 44.8805 2.31146
\(378\) 0 0
\(379\) −6.72979 −0.345686 −0.172843 0.984949i \(-0.555295\pi\)
−0.172843 + 0.984949i \(0.555295\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 18.7673 + 32.5060i 0.958967 + 1.66098i 0.725018 + 0.688730i \(0.241832\pi\)
0.233949 + 0.972249i \(0.424835\pi\)
\(384\) 0 0
\(385\) −2.54382 3.49913i −0.129645 0.178332i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.3213 + 26.5372i −0.776819 + 1.34549i 0.156947 + 0.987607i \(0.449835\pi\)
−0.933766 + 0.357883i \(0.883499\pi\)
\(390\) 0 0
\(391\) −30.2958 −1.53212
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.28464 + 7.42121i −0.215583 + 0.373401i
\(396\) 0 0
\(397\) −2.65885 4.60527i −0.133444 0.231132i 0.791558 0.611094i \(-0.209270\pi\)
−0.925002 + 0.379962i \(0.875937\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.328399 + 0.568803i 0.0163994 + 0.0284047i 0.874109 0.485730i \(-0.161446\pi\)
−0.857709 + 0.514135i \(0.828113\pi\)
\(402\) 0 0
\(403\) 2.90491 5.03145i 0.144704 0.250634i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.9395 −0.641387
\(408\) 0 0
\(409\) 15.7994 27.3654i 0.781230 1.35313i −0.149995 0.988687i \(-0.547926\pi\)
0.931226 0.364443i \(-0.118741\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.1891 1.07391i 0.501373 0.0528435i
\(414\) 0 0
\(415\) 7.34254 + 12.7177i 0.360431 + 0.624285i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −20.8848 −1.02029 −0.510146 0.860088i \(-0.670408\pi\)
−0.510146 + 0.860088i \(0.670408\pi\)
\(420\) 0 0
\(421\) 13.5958 0.662617 0.331309 0.943522i \(-0.392510\pi\)
0.331309 + 0.943522i \(0.392510\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −10.1665 17.6088i −0.493145 0.854153i
\(426\) 0 0
\(427\) −10.3715 + 23.3129i −0.501910 + 1.12819i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13.9416 + 24.1475i −0.671541 + 1.16314i 0.305926 + 0.952055i \(0.401034\pi\)
−0.977467 + 0.211088i \(0.932299\pi\)
\(432\) 0 0
\(433\) −22.7059 −1.09118 −0.545589 0.838053i \(-0.683694\pi\)
−0.545589 + 0.838053i \(0.683694\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.30510 + 3.99255i −0.110268 + 0.190990i
\(438\) 0 0
\(439\) −15.9508 27.6276i −0.761290 1.31859i −0.942186 0.335091i \(-0.891233\pi\)
0.180895 0.983502i \(-0.442100\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −17.5629 30.4198i −0.834436 1.44529i −0.894488 0.447091i \(-0.852460\pi\)
0.0600520 0.998195i \(-0.480873\pi\)
\(444\) 0 0
\(445\) 6.90098 11.9529i 0.327138 0.566620i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.0966532 0.00456135 0.00228067 0.999997i \(-0.499274\pi\)
0.00228067 + 0.999997i \(0.499274\pi\)
\(450\) 0 0
\(451\) 6.14014 10.6350i 0.289128 0.500785i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.59800 + 12.5831i −0.262438 + 0.589907i
\(456\) 0 0
\(457\) 8.50925 + 14.7385i 0.398046 + 0.689436i 0.993485 0.113965i \(-0.0363551\pi\)
−0.595439 + 0.803401i \(0.703022\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.1763 −0.753407 −0.376704 0.926334i \(-0.622942\pi\)
−0.376704 + 0.926334i \(0.622942\pi\)
\(462\) 0 0
\(463\) −12.4859 −0.580271 −0.290135 0.956986i \(-0.593700\pi\)
−0.290135 + 0.956986i \(0.593700\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.72717 11.6518i −0.311296 0.539181i 0.667347 0.744747i \(-0.267430\pi\)
−0.978643 + 0.205566i \(0.934097\pi\)
\(468\) 0 0
\(469\) 4.40575 0.464355i 0.203439 0.0214419i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.00802 3.47798i 0.0923286 0.159918i
\(474\) 0 0
\(475\) −3.09412 −0.141968
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.98107 + 12.0916i −0.318973 + 0.552478i −0.980274 0.197643i \(-0.936671\pi\)
0.661301 + 0.750121i \(0.270005\pi\)
\(480\) 0 0
\(481\) 20.5966 + 35.6744i 0.939124 + 1.62661i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.75824 + 4.77741i 0.125245 + 0.216931i
\(486\) 0 0
\(487\) 14.4858 25.0901i 0.656413 1.13694i −0.325124 0.945671i \(-0.605406\pi\)
0.981538 0.191270i \(-0.0612605\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 41.7550 1.88438 0.942189 0.335081i \(-0.108764\pi\)
0.942189 + 0.335081i \(0.108764\pi\)
\(492\) 0 0
\(493\) 22.9583 39.7649i 1.03399 1.79092i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −22.2154 30.5582i −0.996497 1.37072i
\(498\) 0 0
\(499\) −10.7230 18.5728i −0.480028 0.831433i 0.519709 0.854343i \(-0.326040\pi\)
−0.999738 + 0.0229099i \(0.992707\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 28.3098 1.26227 0.631135 0.775673i \(-0.282589\pi\)
0.631135 + 0.775673i \(0.282589\pi\)
\(504\) 0 0
\(505\) −3.27974 −0.145947
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.8142 + 27.3910i 0.700951 + 1.21408i 0.968133 + 0.250437i \(0.0805743\pi\)
−0.267182 + 0.963646i \(0.586092\pi\)
\(510\) 0 0
\(511\) 16.0177 1.68822i 0.708580 0.0746826i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.88418 + 10.1917i −0.259288 + 0.449100i
\(516\) 0 0
\(517\) −13.4661 −0.592238
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.0364 26.0439i 0.658759 1.14100i −0.322179 0.946679i \(-0.604415\pi\)
0.980937 0.194325i \(-0.0622515\pi\)
\(522\) 0 0
\(523\) 7.73793 + 13.4025i 0.338356 + 0.586050i 0.984124 0.177484i \(-0.0567959\pi\)
−0.645768 + 0.763534i \(0.723463\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.97197 5.14760i −0.129461 0.224233i
\(528\) 0 0
\(529\) −5.70363 + 9.87898i −0.247984 + 0.429521i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −39.0946 −1.69338
\(534\) 0 0
\(535\) −0.321076 + 0.556120i −0.0138813 + 0.0240432i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.43234 8.24482i −0.320133 0.355130i
\(540\) 0 0
\(541\) −8.11884 14.0622i −0.349056 0.604583i 0.637026 0.770842i \(-0.280164\pi\)
−0.986082 + 0.166259i \(0.946831\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.00433 0.0858562
\(546\) 0 0
\(547\) −38.9081 −1.66359 −0.831795 0.555083i \(-0.812687\pi\)
−0.831795 + 0.555083i \(0.812687\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.49363 6.05114i −0.148834 0.257787i
\(552\) 0 0
\(553\) −8.93747 + 20.0896i −0.380060 + 0.854296i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.86210 4.95730i 0.121271 0.210048i −0.798998 0.601334i \(-0.794636\pi\)
0.920269 + 0.391286i \(0.127970\pi\)
\(558\) 0 0
\(559\) −12.7851 −0.540754
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.47334 + 4.28394i −0.104239 + 0.180547i −0.913427 0.407003i \(-0.866574\pi\)
0.809188 + 0.587549i \(0.199907\pi\)
\(564\) 0 0
\(565\) −1.84999 3.20429i −0.0778299 0.134805i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.10972 14.0465i −0.339977 0.588858i 0.644451 0.764646i \(-0.277086\pi\)
−0.984428 + 0.175788i \(0.943753\pi\)
\(570\) 0 0
\(571\) 18.3029 31.7016i 0.765954 1.32667i −0.173787 0.984783i \(-0.555601\pi\)
0.939741 0.341887i \(-0.111066\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −23.0923 −0.963017
\(576\) 0 0
\(577\) −18.2684 + 31.6417i −0.760522 + 1.31726i 0.182060 + 0.983287i \(0.441723\pi\)
−0.942582 + 0.333975i \(0.891610\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 22.1569 + 30.4778i 0.919223 + 1.26443i
\(582\) 0 0
\(583\) −3.82596 6.62676i −0.158455 0.274452i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.8474 0.984289 0.492144 0.870514i \(-0.336213\pi\)
0.492144 + 0.870514i \(0.336213\pi\)
\(588\) 0 0
\(589\) −0.904506 −0.0372695
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.54751 11.3406i −0.268874 0.465704i 0.699697 0.714439i \(-0.253318\pi\)
−0.968571 + 0.248736i \(0.919985\pi\)
\(594\) 0 0
\(595\) 8.28526 + 11.3967i 0.339663 + 0.467220i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 18.5770 32.1762i 0.759034 1.31469i −0.184310 0.982868i \(-0.559005\pi\)
0.943344 0.331817i \(-0.107662\pi\)
\(600\) 0 0
\(601\) −17.0627 −0.696000 −0.348000 0.937494i \(-0.613139\pi\)
−0.348000 + 0.937494i \(0.613139\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.37471 + 7.57722i −0.177857 + 0.308058i
\(606\) 0 0
\(607\) −11.5973 20.0871i −0.470719 0.815310i 0.528720 0.848796i \(-0.322672\pi\)
−0.999439 + 0.0334867i \(0.989339\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.4348 + 37.1262i 0.867160 + 1.50197i
\(612\) 0 0
\(613\) 22.2875 38.6030i 0.900182 1.55916i 0.0729255 0.997337i \(-0.476766\pi\)
0.827257 0.561824i \(-0.189900\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.25406 0.251779 0.125890 0.992044i \(-0.459822\pi\)
0.125890 + 0.992044i \(0.459822\pi\)
\(618\) 0 0
\(619\) 0.770208 1.33404i 0.0309573 0.0536196i −0.850132 0.526570i \(-0.823478\pi\)
0.881089 + 0.472951i \(0.156811\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.3950 32.3570i 0.576723 1.29635i
\(624\) 0 0
\(625\) −5.09116 8.81816i −0.203647 0.352726i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 42.1441 1.68040
\(630\) 0 0
\(631\) −44.5148 −1.77210 −0.886052 0.463585i \(-0.846563\pi\)
−0.886052 + 0.463585i \(0.846563\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5.99090 + 10.3765i 0.237742 + 0.411781i
\(636\) 0 0
\(637\) −10.9006 + 33.6149i −0.431897 + 1.33187i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.87520 + 3.24794i −0.0740660 + 0.128286i −0.900680 0.434484i \(-0.856931\pi\)
0.826614 + 0.562770i \(0.190264\pi\)
\(642\) 0 0
\(643\) 1.63678 0.0645485 0.0322742 0.999479i \(-0.489725\pi\)
0.0322742 + 0.999479i \(0.489725\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.0748 + 17.4501i −0.396082 + 0.686034i −0.993239 0.116091i \(-0.962964\pi\)
0.597157 + 0.802125i \(0.296297\pi\)
\(648\) 0 0
\(649\) 3.07038 + 5.31806i 0.120523 + 0.208752i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21.4570 37.1646i −0.839677 1.45436i −0.890165 0.455638i \(-0.849411\pi\)
0.0504888 0.998725i \(-0.483922\pi\)
\(654\) 0 0
\(655\) 4.20591 7.28486i 0.164339 0.284643i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 21.7052 0.845515 0.422758 0.906243i \(-0.361062\pi\)
0.422758 + 0.906243i \(0.361062\pi\)
\(660\) 0 0
\(661\) 12.6907 21.9810i 0.493613 0.854962i −0.506360 0.862322i \(-0.669009\pi\)
0.999973 + 0.00735996i \(0.00234277\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.13232 0.224742i 0.0826880 0.00871511i
\(666\) 0 0
\(667\) −26.0740 45.1614i −1.00959 1.74866i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.2932 −0.590386
\(672\) 0 0
\(673\) −15.7735 −0.608024 −0.304012 0.952668i \(-0.598326\pi\)
−0.304012 + 0.952668i \(0.598326\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.32531 + 9.22370i 0.204668 + 0.354496i 0.950027 0.312168i \(-0.101055\pi\)
−0.745359 + 0.666664i \(0.767722\pi\)
\(678\) 0 0
\(679\) 8.32328 + 11.4490i 0.319418 + 0.439373i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −14.3656 + 24.8819i −0.549683 + 0.952078i 0.448613 + 0.893726i \(0.351918\pi\)
−0.998296 + 0.0583524i \(0.981415\pi\)
\(684\) 0 0
\(685\) −15.0168 −0.573763
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.1801 + 21.0965i −0.464023 + 0.803712i
\(690\) 0 0
\(691\) 19.6136 + 33.9717i 0.746136 + 1.29235i 0.949662 + 0.313276i \(0.101426\pi\)
−0.203526 + 0.979069i \(0.565240\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.10126 10.5677i −0.231434 0.400855i
\(696\) 0 0
\(697\) −19.9985 + 34.6385i −0.757499 + 1.31203i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −34.1643 −1.29037 −0.645184 0.764028i \(-0.723219\pi\)
−0.645184 + 0.764028i \(0.723219\pi\)
\(702\) 0 0
\(703\) 3.20660 5.55399i 0.120939 0.209473i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.36915 + 0.882088i −0.314754 + 0.0331743i
\(708\) 0 0
\(709\) −9.85352 17.0668i −0.370057 0.640957i 0.619517 0.784983i \(-0.287328\pi\)
−0.989574 + 0.144026i \(0.953995\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.75060 −0.252812
\(714\) 0 0
\(715\) −8.25450 −0.308701
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5.87829 10.1815i −0.219223 0.379705i 0.735348 0.677690i \(-0.237019\pi\)
−0.954571 + 0.297985i \(0.903686\pi\)
\(720\) 0 0
\(721\) −12.2740 + 27.5894i −0.457108 + 1.02748i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 17.4995 30.3100i 0.649914 1.12568i
\(726\) 0 0
\(727\) −45.6322 −1.69240 −0.846202 0.532862i \(-0.821117\pi\)
−0.846202 + 0.532862i \(0.821117\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.54014 + 11.3279i −0.241896 + 0.418976i
\(732\) 0 0
\(733\) −19.4901 33.7579i −0.719885 1.24688i −0.961045 0.276392i \(-0.910861\pi\)
0.241160 0.970485i \(-0.422472\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.32763 + 2.29952i 0.0489038 + 0.0847039i
\(738\) 0 0
\(739\) −11.7719 + 20.3895i −0.433036 + 0.750040i −0.997133 0.0756686i \(-0.975891\pi\)
0.564097 + 0.825708i \(0.309224\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.19224 0.300544 0.150272 0.988645i \(-0.451985\pi\)
0.150272 + 0.988645i \(0.451985\pi\)
\(744\) 0 0
\(745\) 7.24337 12.5459i 0.265377 0.459646i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.669743 + 1.50544i −0.0244719 + 0.0550077i
\(750\) 0 0
\(751\) 14.0936 + 24.4109i 0.514284 + 0.890766i 0.999863 + 0.0165733i \(0.00527570\pi\)
−0.485578 + 0.874193i \(0.661391\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.56271 −0.0932667
\(756\) 0 0
\(757\) 7.42352 0.269812 0.134906 0.990858i \(-0.456927\pi\)
0.134906 + 0.990858i \(0.456927\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.3534 + 26.5928i 0.556559 + 0.963988i 0.997780 + 0.0665900i \(0.0212120\pi\)
−0.441222 + 0.897398i \(0.645455\pi\)
\(762\) 0 0
\(763\) 5.11460 0.539066i 0.185161 0.0195155i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.77465 16.9302i 0.352942 0.611314i
\(768\) 0 0
\(769\) −40.1946 −1.44945 −0.724727 0.689037i \(-0.758034\pi\)
−0.724727 + 0.689037i \(0.758034\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.69611 2.93775i 0.0610050 0.105664i −0.833910 0.551901i \(-0.813903\pi\)
0.894915 + 0.446237i \(0.147236\pi\)
\(774\) 0 0
\(775\) −2.26532 3.92365i −0.0813727 0.140942i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.04324 + 5.27105i 0.109035 + 0.188855i
\(780\) 0 0
\(781\) 11.3219 19.6101i 0.405130 0.701706i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.13392 0.111854
\(786\) 0 0
\(787\) −5.41657 + 9.38177i −0.193080 + 0.334424i −0.946269 0.323379i \(-0.895181\pi\)
0.753190 + 0.657804i \(0.228514\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.58256 7.67904i −0.198493 0.273035i
\(792\) 0 0
\(793\) 24.3431 + 42.1635i 0.864449 + 1.49727i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −46.9965 −1.66470 −0.832350 0.554250i \(-0.813005\pi\)
−0.832350 + 0.554250i \(0.813005\pi\)
\(798\) 0 0
\(799\) 43.8593 1.55163
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.82676 + 8.36020i 0.170333 + 0.295025i
\(804\) 0 0
\(805\) 15.9142 1.67731i 0.560900 0.0591175i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.5052 25.1238i 0.509977 0.883306i −0.489957 0.871747i \(-0.662987\pi\)
0.999933 0.0115587i \(-0.00367932\pi\)
\(810\) 0 0
\(811\) 34.6805 1.21780 0.608899 0.793248i \(-0.291612\pi\)
0.608899 + 0.793248i \(0.291612\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.03139 15.6428i 0.316356 0.547944i
\(816\) 0 0
\(817\) 0.995233 + 1.72379i 0.0348188 + 0.0603079i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.6606 + 34.0532i 0.686161 + 1.18847i 0.973070 + 0.230508i \(0.0740388\pi\)
−0.286909 + 0.957958i \(0.592628\pi\)
\(822\) 0 0
\(823\) 21.3432 36.9675i 0.743978 1.28861i −0.206693 0.978406i \(-0.566270\pi\)
0.950671 0.310202i \(-0.100397\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14.5672 −0.506551 −0.253276 0.967394i \(-0.581508\pi\)
−0.253276 + 0.967394i \(0.581508\pi\)
\(828\) 0 0
\(829\) 16.6920 28.9113i 0.579736 1.00413i −0.415773 0.909468i \(-0.636489\pi\)
0.995509 0.0946641i \(-0.0301777\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 24.2072 + 26.8535i 0.838731 + 0.930419i
\(834\) 0 0
\(835\) −12.6557 21.9204i −0.437970 0.758586i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.97458 −0.0681701 −0.0340850 0.999419i \(-0.510852\pi\)
−0.0340850 + 0.999419i \(0.510852\pi\)
\(840\) 0 0
\(841\) 50.0359 1.72537
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.43695 + 11.1491i 0.221438 + 0.383541i
\(846\) 0 0
\(847\) −9.12536 + 20.5119i −0.313551 + 0.704798i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 23.9318 41.4511i 0.820371 1.42092i
\(852\) 0 0
\(853\) −13.3727 −0.457874 −0.228937 0.973441i \(-0.573525\pi\)
−0.228937 + 0.973441i \(0.573525\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.4324 23.2656i 0.458842 0.794738i −0.540058 0.841628i \(-0.681598\pi\)
0.998900 + 0.0468899i \(0.0149310\pi\)
\(858\) 0 0
\(859\) 22.9212 + 39.7007i 0.782062 + 1.35457i 0.930739 + 0.365685i \(0.119165\pi\)
−0.148677 + 0.988886i \(0.547502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.90612 10.2297i −0.201047 0.348223i 0.747819 0.663902i \(-0.231101\pi\)
−0.948866 + 0.315679i \(0.897768\pi\)
\(864\) 0 0
\(865\) 0.0355291 0.0615383i 0.00120803 0.00209236i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13.1787 −0.447056
\(870\) 0 0
\(871\) 4.22654 7.32059i 0.143211 0.248049i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 14.3361 + 19.7199i 0.484649 + 0.666655i
\(876\) 0 0
\(877\) 12.6046 + 21.8318i 0.425628 + 0.737209i 0.996479 0.0838449i \(-0.0267200\pi\)
−0.570851 + 0.821054i \(0.693387\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.268589 0.00904898 0.00452449 0.999990i \(-0.498560\pi\)
0.00452449 + 0.999990i \(0.498560\pi\)
\(882\) 0 0
\(883\) 25.8915 0.871319 0.435660 0.900112i \(-0.356515\pi\)
0.435660 + 0.900112i \(0.356515\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21.8396 + 37.8274i 0.733303 + 1.27012i 0.955464 + 0.295109i \(0.0953558\pi\)
−0.222160 + 0.975010i \(0.571311\pi\)
\(888\) 0 0
\(889\) 18.0782 + 24.8673i 0.606323 + 0.834023i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.33710 5.78003i 0.111672 0.193421i
\(894\) 0 0
\(895\) −8.16823 −0.273034
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.11563 8.86053i 0.170616 0.295515i
\(900\) 0 0
\(901\) 12.4612 + 21.5835i 0.415143 + 0.719050i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.85441 + 3.21192i 0.0616425 + 0.106768i
\(906\) 0 0
\(907\) 0.0146274 0.0253355i 0.000485696 0.000841251i −0.865782 0.500421i \(-0.833179\pi\)
0.866268 + 0.499579i \(0.166512\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28.0095 −0.927996 −0.463998 0.885836i \(-0.653586\pi\)
−0.463998 + 0.885836i \(0.653586\pi\)
\(912\) 0 0
\(913\) −11.2921 + 19.5585i −0.373714 + 0.647292i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.77326 19.7205i 0.289719 0.651227i
\(918\) 0 0
\(919\) −6.11476 10.5911i −0.201707 0.349367i 0.747371 0.664407i \(-0.231316\pi\)
−0.949079 + 0.315039i \(0.897982\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −72.0873 −2.37278
\(924\) 0 0
\(925\) 32.1235 1.05621
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.98202 6.89707i −0.130646 0.226285i 0.793280 0.608857i \(-0.208372\pi\)
−0.923926 + 0.382572i \(0.875038\pi\)
\(930\) 0 0
\(931\) 5.38076 1.14698i 0.176347 0.0375907i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.22252 + 7.31363i −0.138091 + 0.239181i
\(936\) 0 0
\(937\) −15.0407 −0.491358 −0.245679 0.969351i \(-0.579011\pi\)
−0.245679 + 0.969351i \(0.579011\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 28.9559 50.1531i 0.943936 1.63495i 0.186069 0.982537i \(-0.440425\pi\)
0.757867 0.652409i \(-0.226242\pi\)
\(942\) 0 0
\(943\) 22.7126 + 39.3394i 0.739624 + 1.28107i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.8442 + 20.5147i 0.384883 + 0.666638i 0.991753 0.128164i \(-0.0409083\pi\)
−0.606870 + 0.794801i \(0.707575\pi\)
\(948\) 0 0
\(949\) 15.3661 26.6149i 0.498806 0.863957i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −14.2226 −0.460716 −0.230358 0.973106i \(-0.573990\pi\)
−0.230358 + 0.973106i \(0.573990\pi\)
\(954\) 0 0
\(955\) −8.27429 + 14.3315i −0.267750 + 0.463756i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −38.3195 + 4.03878i −1.23740 + 0.130419i
\(960\) 0 0
\(961\) 14.8378 + 25.6998i 0.478638 + 0.829025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 22.2443 0.716070
\(966\) 0 0
\(967\) −20.7331 −0.666730 −0.333365 0.942798i \(-0.608184\pi\)
−0.333365 + 0.942798i \(0.608184\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −8.39847 14.5466i −0.269520 0.466822i 0.699218 0.714908i \(-0.253532\pi\)
−0.968738 + 0.248087i \(0.920198\pi\)
\(972\) 0 0
\(973\) −18.4112 25.3254i −0.590236 0.811894i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.7163 + 32.4176i −0.598788 + 1.03713i 0.394212 + 0.919020i \(0.371018\pi\)
−0.993000 + 0.118112i \(0.962316\pi\)
\(978\) 0 0
\(979\) 21.2261 0.678388
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −25.8389 + 44.7543i −0.824132 + 1.42744i 0.0784482 + 0.996918i \(0.475003\pi\)
−0.902581 + 0.430521i \(0.858330\pi\)
\(984\) 0 0
\(985\) 10.5402 + 18.2561i 0.335838 + 0.581688i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.42771 + 12.8652i 0.236188 + 0.409089i
\(990\) 0 0
\(991\) 28.5452 49.4418i 0.906769 1.57057i 0.0882435 0.996099i \(-0.471875\pi\)
0.818525 0.574471i \(-0.194792\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −13.4057 −0.424989
\(996\) 0 0
\(997\) 1.00745 1.74496i 0.0319063 0.0552633i −0.849631 0.527377i \(-0.823176\pi\)
0.881538 + 0.472114i \(0.156509\pi\)
\(998\) 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.k.g.1621.6 yes 16
3.2 odd 2 inner 2268.2.k.g.1621.3 yes 16
7.2 even 3 inner 2268.2.k.g.1297.6 yes 16
9.2 odd 6 2268.2.l.n.109.6 16
9.4 even 3 2268.2.i.n.865.6 16
9.5 odd 6 2268.2.i.n.865.3 16
9.7 even 3 2268.2.l.n.109.3 16
21.2 odd 6 inner 2268.2.k.g.1297.3 16
63.2 odd 6 2268.2.i.n.2053.3 16
63.16 even 3 2268.2.i.n.2053.6 16
63.23 odd 6 2268.2.l.n.541.6 16
63.58 even 3 2268.2.l.n.541.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2268.2.i.n.865.3 16 9.5 odd 6
2268.2.i.n.865.6 16 9.4 even 3
2268.2.i.n.2053.3 16 63.2 odd 6
2268.2.i.n.2053.6 16 63.16 even 3
2268.2.k.g.1297.3 16 21.2 odd 6 inner
2268.2.k.g.1297.6 yes 16 7.2 even 3 inner
2268.2.k.g.1621.3 yes 16 3.2 odd 2 inner
2268.2.k.g.1621.6 yes 16 1.1 even 1 trivial
2268.2.l.n.109.3 16 9.7 even 3
2268.2.l.n.109.6 16 9.2 odd 6
2268.2.l.n.541.3 16 63.58 even 3
2268.2.l.n.541.6 16 63.23 odd 6