Properties

Label 3456.2.i.d.2305.1
Level $3456$
Weight $2$
Character 3456.2305
Analytic conductor $27.596$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2305.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3456.2305
Dual form 3456.2.i.d.1153.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(1.00000 - 1.73205i) q^{5} +(1.00000 + 1.73205i) q^{7} +(2.50000 + 4.33013i) q^{11} +(2.00000 - 3.46410i) q^{13} -1.00000 q^{17} -5.00000 q^{19} +(-2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +(3.00000 + 5.19615i) q^{29} +4.00000 q^{35} +10.0000 q^{37} +(-1.50000 + 2.59808i) q^{41} +(4.50000 + 7.79423i) q^{43} +(4.00000 + 6.92820i) q^{47} +(1.50000 - 2.59808i) q^{49} -12.0000 q^{53} +10.0000 q^{55} +(3.50000 - 6.06218i) q^{59} +(2.00000 + 3.46410i) q^{61} +(-4.00000 - 6.92820i) q^{65} +(3.50000 - 6.06218i) q^{67} -6.00000 q^{71} -13.0000 q^{73} +(-5.00000 + 8.66025i) q^{77} +(-1.00000 - 1.73205i) q^{79} +(6.00000 + 10.3923i) q^{83} +(-1.00000 + 1.73205i) q^{85} -10.0000 q^{89} +8.00000 q^{91} +(-5.00000 + 8.66025i) q^{95} +(-6.50000 - 11.2583i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} + 5 q^{11} + 4 q^{13} - 2 q^{17} - 10 q^{19} - 4 q^{23} + q^{25} + 6 q^{29} + 8 q^{35} + 20 q^{37} - 3 q^{41} + 9 q^{43} + 8 q^{47} + 3 q^{49} - 24 q^{53} + 20 q^{55} + 7 q^{59} + 4 q^{61} - 8 q^{65} + 7 q^{67} - 12 q^{71} - 26 q^{73} - 10 q^{77} - 2 q^{79} + 12 q^{83} - 2 q^{85} - 20 q^{89} + 16 q^{91} - 10 q^{95} - 13 q^{97}+O(q^{100}) \) 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
\(5\) 1.00000 1.73205i 0.447214 0.774597i −0.550990 0.834512i \(-0.685750\pi\)
0.998203 + 0.0599153i \(0.0190830\pi\)
\(6\) 0 0
\(7\) 1.00000 + 1.73205i 0.377964 + 0.654654i 0.990766 0.135583i \(-0.0432908\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.50000 + 4.33013i 0.753778 + 1.30558i 0.945979 + 0.324227i \(0.105104\pi\)
−0.192201 + 0.981356i \(0.561563\pi\)
\(12\) 0 0
\(13\) 2.00000 3.46410i 0.554700 0.960769i −0.443227 0.896410i \(-0.646166\pi\)
0.997927 0.0643593i \(-0.0205004\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 + 3.46410i −0.417029 + 0.722315i −0.995639 0.0932891i \(-0.970262\pi\)
0.578610 + 0.815604i \(0.303595\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.50000 + 2.59808i −0.234261 + 0.405751i −0.959058 0.283211i \(-0.908600\pi\)
0.724797 + 0.688963i \(0.241934\pi\)
\(42\) 0 0
\(43\) 4.50000 + 7.79423i 0.686244 + 1.18861i 0.973044 + 0.230618i \(0.0740749\pi\)
−0.286801 + 0.957990i \(0.592592\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 10.0000 1.34840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.50000 6.06218i 0.455661 0.789228i −0.543065 0.839691i \(-0.682736\pi\)
0.998726 + 0.0504625i \(0.0160695\pi\)
\(60\) 0 0
\(61\) 2.00000 + 3.46410i 0.256074 + 0.443533i 0.965187 0.261562i \(-0.0842377\pi\)
−0.709113 + 0.705095i \(0.750904\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 6.92820i −0.496139 0.859338i
\(66\) 0 0
\(67\) 3.50000 6.06218i 0.427593 0.740613i −0.569066 0.822292i \(-0.692695\pi\)
0.996659 + 0.0816792i \(0.0260283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −13.0000 −1.52153 −0.760767 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 + 8.66025i −0.569803 + 0.986928i
\(78\) 0 0
\(79\) −1.00000 1.73205i −0.112509 0.194871i 0.804272 0.594261i \(-0.202555\pi\)
−0.916781 + 0.399390i \(0.869222\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 0 0
\(85\) −1.00000 + 1.73205i −0.108465 + 0.187867i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.00000 + 8.66025i −0.512989 + 0.888523i
\(96\) 0 0
\(97\) −6.50000 11.2583i −0.659975 1.14311i −0.980622 0.195911i \(-0.937234\pi\)
0.320647 0.947199i \(-0.396100\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 + 5.19615i 0.298511 + 0.517036i 0.975796 0.218685i \(-0.0701767\pi\)
−0.677284 + 0.735721i \(0.736843\pi\)
\(102\) 0 0
\(103\) 4.00000 6.92820i 0.394132 0.682656i −0.598858 0.800855i \(-0.704379\pi\)
0.992990 + 0.118199i \(0.0377120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.0000 −1.06341 −0.531705 0.846930i \(-0.678449\pi\)
−0.531705 + 0.846930i \(0.678449\pi\)
\(108\) 0 0
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.00000 12.1244i 0.658505 1.14056i −0.322498 0.946570i \(-0.604523\pi\)
0.981003 0.193993i \(-0.0621440\pi\)
\(114\) 0 0
\(115\) 4.00000 + 6.92820i 0.373002 + 0.646058i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.00000 1.73205i −0.0916698 0.158777i
\(120\) 0 0
\(121\) −7.00000 + 12.1244i −0.636364 + 1.10221i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 + 10.3923i −0.524222 + 0.907980i 0.475380 + 0.879781i \(0.342311\pi\)
−0.999602 + 0.0281993i \(0.991023\pi\)
\(132\) 0 0
\(133\) −5.00000 8.66025i −0.433555 0.750939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.500000 + 0.866025i 0.0427179 + 0.0739895i 0.886594 0.462549i \(-0.153065\pi\)
−0.843876 + 0.536538i \(0.819732\pi\)
\(138\) 0 0
\(139\) −0.500000 + 0.866025i −0.0424094 + 0.0734553i −0.886451 0.462822i \(-0.846837\pi\)
0.844042 + 0.536278i \(0.180170\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.0000 1.67248
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(150\) 0 0
\(151\) 8.00000 + 13.8564i 0.651031 + 1.12762i 0.982873 + 0.184284i \(0.0589965\pi\)
−0.331842 + 0.943335i \(0.607670\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.0000 + 20.7846i −0.957704 + 1.65879i −0.229650 + 0.973273i \(0.573758\pi\)
−0.728055 + 0.685519i \(0.759575\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 1.73205i 0.0773823 0.134030i −0.824737 0.565516i \(-0.808677\pi\)
0.902120 + 0.431486i \(0.142010\pi\)
\(168\) 0 0
\(169\) −1.50000 2.59808i −0.115385 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.00000 + 15.5885i 0.684257 + 1.18517i 0.973670 + 0.227964i \(0.0732068\pi\)
−0.289412 + 0.957205i \(0.593460\pi\)
\(174\) 0 0
\(175\) −1.00000 + 1.73205i −0.0755929 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.0000 17.3205i 0.735215 1.27343i
\(186\) 0 0
\(187\) −2.50000 4.33013i −0.182818 0.316650i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.00000 1.73205i −0.0723575 0.125327i 0.827577 0.561353i \(-0.189719\pi\)
−0.899934 + 0.436026i \(0.856386\pi\)
\(192\) 0 0
\(193\) −0.500000 + 0.866025i −0.0359908 + 0.0623379i −0.883460 0.468507i \(-0.844792\pi\)
0.847469 + 0.530845i \(0.178125\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.0000 1.42494 0.712470 0.701702i \(-0.247576\pi\)
0.712470 + 0.701702i \(0.247576\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.00000 + 10.3923i −0.421117 + 0.729397i
\(204\) 0 0
\(205\) 3.00000 + 5.19615i 0.209529 + 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.5000 21.6506i −0.864643 1.49761i
\(210\) 0 0
\(211\) 10.0000 17.3205i 0.688428 1.19239i −0.283918 0.958849i \(-0.591634\pi\)
0.972346 0.233544i \(-0.0750324\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 18.0000 1.22759
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 + 3.46410i −0.134535 + 0.233021i
\(222\) 0 0
\(223\) −12.0000 20.7846i −0.803579 1.39184i −0.917246 0.398321i \(-0.869593\pi\)
0.113666 0.993519i \(-0.463740\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.50000 2.59808i −0.0995585 0.172440i 0.811943 0.583736i \(-0.198410\pi\)
−0.911502 + 0.411296i \(0.865076\pi\)
\(228\) 0 0
\(229\) 7.00000 12.1244i 0.462573 0.801200i −0.536515 0.843891i \(-0.680260\pi\)
0.999088 + 0.0426906i \(0.0135930\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 + 10.3923i −0.388108 + 0.672222i −0.992195 0.124696i \(-0.960204\pi\)
0.604087 + 0.796918i \(0.293538\pi\)
\(240\) 0 0
\(241\) −12.5000 21.6506i −0.805196 1.39464i −0.916159 0.400815i \(-0.868727\pi\)
0.110963 0.993825i \(-0.464606\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.00000 5.19615i −0.191663 0.331970i
\(246\) 0 0
\(247\) −10.0000 + 17.3205i −0.636285 + 1.10208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.50000 2.59808i 0.0935674 0.162064i −0.815442 0.578838i \(-0.803506\pi\)
0.909010 + 0.416775i \(0.136840\pi\)
\(258\) 0 0
\(259\) 10.0000 + 17.3205i 0.621370 + 1.07624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.00000 + 15.5885i 0.554964 + 0.961225i 0.997906 + 0.0646755i \(0.0206012\pi\)
−0.442943 + 0.896550i \(0.646065\pi\)
\(264\) 0 0
\(265\) −12.0000 + 20.7846i −0.737154 + 1.27679i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.50000 + 4.33013i −0.150756 + 0.261116i
\(276\) 0 0
\(277\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.0000 + 25.9808i 0.894825 + 1.54988i 0.834021 + 0.551733i \(0.186033\pi\)
0.0608039 + 0.998150i \(0.480634\pi\)
\(282\) 0 0
\(283\) 12.0000 20.7846i 0.713326 1.23552i −0.250276 0.968175i \(-0.580521\pi\)
0.963602 0.267342i \(-0.0861454\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.00000 8.66025i 0.292103 0.505937i −0.682204 0.731162i \(-0.738978\pi\)
0.974307 + 0.225225i \(0.0723116\pi\)
\(294\) 0 0
\(295\) −7.00000 12.1244i −0.407556 0.705907i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.00000 + 13.8564i 0.462652 + 0.801337i
\(300\) 0 0
\(301\) −9.00000 + 15.5885i −0.518751 + 0.898504i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) −3.00000 −0.171219 −0.0856095 0.996329i \(-0.527284\pi\)
−0.0856095 + 0.996329i \(0.527284\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.00000 8.66025i 0.283524 0.491078i −0.688726 0.725022i \(-0.741830\pi\)
0.972250 + 0.233944i \(0.0751631\pi\)
\(312\) 0 0
\(313\) −12.5000 21.6506i −0.706542 1.22377i −0.966132 0.258047i \(-0.916921\pi\)
0.259590 0.965719i \(-0.416412\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 + 10.3923i 0.336994 + 0.583690i 0.983866 0.178908i \(-0.0572566\pi\)
−0.646872 + 0.762598i \(0.723923\pi\)
\(318\) 0 0
\(319\) −15.0000 + 25.9808i −0.839839 + 1.45464i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.00000 + 13.8564i −0.441054 + 0.763928i
\(330\) 0 0
\(331\) 10.0000 + 17.3205i 0.549650 + 0.952021i 0.998298 + 0.0583130i \(0.0185721\pi\)
−0.448649 + 0.893708i \(0.648095\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.00000 12.1244i −0.382451 0.662424i
\(336\) 0 0
\(337\) −9.50000 + 16.4545i −0.517498 + 0.896333i 0.482295 + 0.876009i \(0.339803\pi\)
−0.999793 + 0.0203242i \(0.993530\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.50000 + 2.59808i −0.0805242 + 0.139472i −0.903475 0.428640i \(-0.858993\pi\)
0.822951 + 0.568112i \(0.192326\pi\)
\(348\) 0 0
\(349\) −8.00000 13.8564i −0.428230 0.741716i 0.568486 0.822693i \(-0.307529\pi\)
−0.996716 + 0.0809766i \(0.974196\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.50000 7.79423i −0.239511 0.414845i 0.721063 0.692869i \(-0.243654\pi\)
−0.960574 + 0.278024i \(0.910320\pi\)
\(354\) 0 0
\(355\) −6.00000 + 10.3923i −0.318447 + 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −34.0000 −1.79445 −0.897226 0.441572i \(-0.854421\pi\)
−0.897226 + 0.441572i \(0.854421\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.0000 + 22.5167i −0.680451 + 1.17858i
\(366\) 0 0
\(367\) 11.0000 + 19.0526i 0.574195 + 0.994535i 0.996129 + 0.0879086i \(0.0280183\pi\)
−0.421933 + 0.906627i \(0.638648\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 20.7846i −0.623009 1.07908i
\(372\) 0 0
\(373\) 16.0000 27.7128i 0.828449 1.43492i −0.0708063 0.997490i \(-0.522557\pi\)
0.899255 0.437425i \(-0.144109\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.00000 5.19615i 0.153293 0.265511i −0.779143 0.626846i \(-0.784346\pi\)
0.932436 + 0.361335i \(0.117679\pi\)
\(384\) 0 0
\(385\) 10.0000 + 17.3205i 0.509647 + 0.882735i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −19.0000 32.9090i −0.963338 1.66855i −0.714015 0.700130i \(-0.753125\pi\)
−0.249323 0.968420i \(-0.580208\pi\)
\(390\) 0 0
\(391\) 2.00000 3.46410i 0.101144 0.175187i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.50000 + 2.59808i −0.0749064 + 0.129742i −0.901046 0.433724i \(-0.857199\pi\)
0.826139 + 0.563466i \(0.190532\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 25.0000 + 43.3013i 1.23920 + 2.14636i
\(408\) 0 0
\(409\) −12.5000 + 21.6506i −0.618085 + 1.07056i 0.371750 + 0.928333i \(0.378758\pi\)
−0.989835 + 0.142222i \(0.954575\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.0000 0.688895
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10.0000 + 17.3205i −0.488532 + 0.846162i −0.999913 0.0131919i \(-0.995801\pi\)
0.511381 + 0.859354i \(0.329134\pi\)
\(420\) 0 0
\(421\) 1.00000 + 1.73205i 0.0487370 + 0.0844150i 0.889365 0.457198i \(-0.151147\pi\)
−0.840628 + 0.541613i \(0.817814\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.500000 0.866025i −0.0242536 0.0420084i
\(426\) 0 0
\(427\) −4.00000 + 6.92820i −0.193574 + 0.335279i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 0 0
\(433\) 33.0000 1.58588 0.792939 0.609301i \(-0.208550\pi\)
0.792939 + 0.609301i \(0.208550\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.0000 17.3205i 0.478365 0.828552i
\(438\) 0 0
\(439\) −12.0000 20.7846i −0.572729 0.991995i −0.996284 0.0861252i \(-0.972552\pi\)
0.423556 0.905870i \(-0.360782\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.50000 + 12.9904i 0.356336 + 0.617192i 0.987346 0.158583i \(-0.0506926\pi\)
−0.631010 + 0.775775i \(0.717359\pi\)
\(444\) 0 0
\(445\) −10.0000 + 17.3205i −0.474045 + 0.821071i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17.0000 −0.802280 −0.401140 0.916017i \(-0.631386\pi\)
−0.401140 + 0.916017i \(0.631386\pi\)
\(450\) 0 0
\(451\) −15.0000 −0.706322
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.00000 13.8564i 0.375046 0.649598i
\(456\) 0 0
\(457\) 1.50000 + 2.59808i 0.0701670 + 0.121533i 0.898974 0.438001i \(-0.144313\pi\)
−0.828807 + 0.559534i \(0.810980\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −10.0000 17.3205i −0.465746 0.806696i 0.533488 0.845807i \(-0.320881\pi\)
−0.999235 + 0.0391109i \(0.987547\pi\)
\(462\) 0 0
\(463\) −1.00000 + 1.73205i −0.0464739 + 0.0804952i −0.888327 0.459212i \(-0.848132\pi\)
0.841853 + 0.539707i \(0.181465\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −29.0000 −1.34196 −0.670980 0.741475i \(-0.734126\pi\)
−0.670980 + 0.741475i \(0.734126\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −22.5000 + 38.9711i −1.03455 + 1.79190i
\(474\) 0 0
\(475\) −2.50000 4.33013i −0.114708 0.198680i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.00000 15.5885i −0.411220 0.712255i 0.583803 0.811895i \(-0.301564\pi\)
−0.995023 + 0.0996406i \(0.968231\pi\)
\(480\) 0 0
\(481\) 20.0000 34.6410i 0.911922 1.57949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −26.0000 −1.18060
\(486\) 0 0
\(487\) −10.0000 −0.453143 −0.226572 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.5000 21.6506i 0.564117 0.977079i −0.433014 0.901387i \(-0.642550\pi\)
0.997131 0.0756923i \(-0.0241167\pi\)
\(492\) 0 0
\(493\) −3.00000 5.19615i −0.135113 0.234023i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 10.3923i −0.269137 0.466159i
\(498\) 0 0
\(499\) 2.50000 4.33013i 0.111915 0.193843i −0.804627 0.593780i \(-0.797635\pi\)
0.916542 + 0.399937i \(0.130968\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000 24.2487i 0.620539 1.07481i −0.368846 0.929490i \(-0.620247\pi\)
0.989385 0.145315i \(-0.0464195\pi\)
\(510\) 0 0
\(511\) −13.0000 22.5167i −0.575086 0.996078i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.00000 13.8564i −0.352522 0.610586i
\(516\) 0 0
\(517\) −20.0000 + 34.6410i −0.879599 + 1.52351i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.00000 0.306676 0.153338 0.988174i \(-0.450998\pi\)
0.153338 + 0.988174i \(0.450998\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.50000 + 6.06218i 0.152174 + 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 + 10.3923i 0.259889 + 0.450141i
\(534\) 0 0
\(535\) −11.0000 + 19.0526i −0.475571 + 0.823714i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 15.0000 0.646096
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.0000 20.7846i 0.514024 0.890315i
\(546\) 0 0
\(547\) 0.500000 + 0.866025i 0.0213785 + 0.0370286i 0.876517 0.481371i \(-0.159861\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −15.0000 25.9808i −0.639021 1.10682i
\(552\) 0 0
\(553\) 2.00000 3.46410i 0.0850487 0.147309i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −26.0000 −1.10166 −0.550828 0.834619i \(-0.685688\pi\)
−0.550828 + 0.834619i \(0.685688\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.5000 28.5788i 0.695392 1.20445i −0.274656 0.961542i \(-0.588564\pi\)
0.970048 0.242912i \(-0.0781026\pi\)
\(564\) 0 0
\(565\) −14.0000 24.2487i −0.588984 1.02015i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.50000 16.4545i −0.398261 0.689808i 0.595251 0.803540i \(-0.297053\pi\)
−0.993511 + 0.113732i \(0.963719\pi\)
\(570\) 0 0
\(571\) −9.50000 + 16.4545i −0.397563 + 0.688599i −0.993425 0.114488i \(-0.963477\pi\)
0.595862 + 0.803087i \(0.296811\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 23.0000 0.957503 0.478751 0.877951i \(-0.341090\pi\)
0.478751 + 0.877951i \(0.341090\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 + 20.7846i −0.497844 + 0.862291i
\(582\) 0 0
\(583\) −30.0000 51.9615i −1.24247 2.15203i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.50000 4.33013i −0.103186 0.178723i 0.809810 0.586693i \(-0.199570\pi\)
−0.912996 + 0.407969i \(0.866237\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21.0000 36.3731i 0.858037 1.48616i −0.0157622 0.999876i \(-0.505017\pi\)
0.873799 0.486287i \(-0.161649\pi\)
\(600\) 0 0
\(601\) 4.50000 + 7.79423i 0.183559 + 0.317933i 0.943090 0.332538i \(-0.107905\pi\)
−0.759531 + 0.650471i \(0.774572\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.0000 + 24.2487i 0.569181 + 0.985850i
\(606\) 0 0
\(607\) 4.00000 6.92820i 0.162355 0.281207i −0.773358 0.633970i \(-0.781424\pi\)
0.935713 + 0.352763i \(0.114758\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.50000 + 14.7224i −0.342197 + 0.592703i −0.984840 0.173463i \(-0.944504\pi\)
0.642643 + 0.766165i \(0.277838\pi\)
\(618\) 0 0
\(619\) −17.5000 30.3109i −0.703384 1.21830i −0.967271 0.253744i \(-0.918338\pi\)
0.263887 0.964554i \(-0.414995\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.0000 17.3205i −0.400642 0.693932i
\(624\) 0 0
\(625\) 9.50000 16.4545i 0.380000 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.0000 34.6410i 0.793676 1.37469i
\(636\) 0 0
\(637\) −6.00000 10.3923i −0.237729 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5000 + 28.5788i 0.651711 + 1.12880i 0.982708 + 0.185164i \(0.0592817\pi\)
−0.330997 + 0.943632i \(0.607385\pi\)
\(642\) 0 0
\(643\) 10.5000 18.1865i 0.414080 0.717207i −0.581252 0.813724i \(-0.697437\pi\)
0.995331 + 0.0965169i \(0.0307702\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 0 0
\(649\) 35.0000 1.37387
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.00000 + 15.5885i −0.352197 + 0.610023i −0.986634 0.162951i \(-0.947899\pi\)
0.634437 + 0.772975i \(0.281232\pi\)
\(654\) 0 0
\(655\) 12.0000 + 20.7846i 0.468879 + 0.812122i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 20.7846i −0.467454 0.809653i 0.531855 0.846836i \(-0.321495\pi\)
−0.999309 + 0.0371821i \(0.988162\pi\)
\(660\) 0 0
\(661\) 1.00000 1.73205i 0.0388955 0.0673690i −0.845922 0.533306i \(-0.820949\pi\)
0.884818 + 0.465937i \(0.154283\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −20.0000 −0.775567
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 + 17.3205i −0.386046 + 0.668651i
\(672\) 0 0
\(673\) −5.00000 8.66025i −0.192736 0.333828i 0.753420 0.657539i \(-0.228403\pi\)
−0.946156 + 0.323711i \(0.895069\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −14.0000 24.2487i −0.538064 0.931954i −0.999008 0.0445248i \(-0.985823\pi\)
0.460945 0.887429i \(-0.347511\pi\)
\(678\) 0 0
\(679\) 13.0000 22.5167i 0.498894 0.864110i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −15.0000 −0.573959 −0.286980 0.957937i \(-0.592651\pi\)
−0.286980 + 0.957937i \(0.592651\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −24.0000 + 41.5692i −0.914327 + 1.58366i
\(690\) 0 0
\(691\) 6.00000 + 10.3923i 0.228251 + 0.395342i 0.957290 0.289130i \(-0.0933661\pi\)
−0.729039 + 0.684472i \(0.760033\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.00000 + 1.73205i 0.0379322 + 0.0657004i
\(696\) 0 0
\(697\) 1.50000 2.59808i 0.0568166 0.0984092i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.0000 0.604312 0.302156 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) 0 0
\(703\) −50.0000 −1.88579
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 + 10.3923i −0.225653 + 0.390843i
\(708\) 0 0
\(709\) −10.0000 17.3205i −0.375558 0.650485i 0.614852 0.788642i \(-0.289216\pi\)
−0.990410 + 0.138157i \(0.955882\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 20.0000 34.6410i 0.747958 1.29550i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.00000 + 5.19615i −0.111417 + 0.192980i
\(726\) 0 0
\(727\) 24.0000 + 41.5692i 0.890111 + 1.54172i 0.839742 + 0.542986i \(0.182706\pi\)
0.0503692 + 0.998731i \(0.483960\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.50000 7.79423i −0.166439 0.288280i
\(732\) 0 0
\(733\) −9.00000 + 15.5885i −0.332423 + 0.575773i −0.982986 0.183679i \(-0.941199\pi\)
0.650564 + 0.759452i \(0.274533\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 35.0000 1.28924
\(738\) 0 0
\(739\) −21.0000 −0.772497 −0.386249 0.922395i \(-0.626229\pi\)
−0.386249 + 0.922395i \(0.626229\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −19.0000 + 32.9090i −0.697042 + 1.20731i 0.272445 + 0.962171i \(0.412168\pi\)
−0.969487 + 0.245141i \(0.921166\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.0000 19.0526i −0.401931 0.696165i
\(750\) 0 0
\(751\) 1.00000 1.73205i 0.0364905 0.0632034i −0.847203 0.531269i \(-0.821715\pi\)
0.883694 + 0.468065i \(0.155049\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.00000 8.66025i 0.181250 0.313934i −0.761057 0.648686i \(-0.775319\pi\)
0.942306 + 0.334752i \(0.108652\pi\)
\(762\) 0 0
\(763\) 12.0000 + 20.7846i 0.434429 + 0.752453i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14.0000 24.2487i −0.505511 0.875570i
\(768\) 0 0
\(769\) −1.00000 + 1.73205i −0.0360609 + 0.0624593i −0.883493 0.468445i \(-0.844814\pi\)
0.847432 + 0.530904i \(0.178148\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.50000 12.9904i 0.268715 0.465429i
\(780\) 0 0
\(781\) −15.0000 25.9808i −0.536742 0.929665i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.0000 + 41.5692i 0.856597 + 1.48367i
\(786\) 0 0
\(787\) −14.0000 + 24.2487i −0.499046 + 0.864373i −0.999999 0.00110111i \(-0.999650\pi\)
0.500953 + 0.865474i \(0.332983\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 28.0000 0.995565
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.0000 25.9808i 0.531327 0.920286i −0.468004 0.883726i \(-0.655027\pi\)
0.999331 0.0365596i \(-0.0116399\pi\)
\(798\) 0 0
\(799\) −4.00000 6.92820i −0.141510 0.245102i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −32.5000 56.2917i −1.14690 1.98649i
\(804\) 0 0
\(805\) −8.00000 + 13.8564i −0.281963 + 0.488374i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.00000 0.105474 0.0527372 0.998608i \(-0.483205\pi\)
0.0527372 + 0.998608i \(0.483205\pi\)
\(810\) 0 0
\(811\) 9.00000 0.316033 0.158016 0.987436i \(-0.449490\pi\)
0.158016 + 0.987436i \(0.449490\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.00000 6.92820i 0.140114 0.242684i
\(816\) 0 0
\(817\) −22.5000 38.9711i −0.787175 1.36343i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.00000 12.1244i −0.244302 0.423143i 0.717633 0.696421i \(-0.245225\pi\)
−0.961935 + 0.273278i \(0.911892\pi\)
\(822\) 0 0
\(823\) −17.0000 + 29.4449i −0.592583 + 1.02638i 0.401300 + 0.915947i \(0.368558\pi\)
−0.993883 + 0.110437i \(0.964775\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.50000 + 2.59808i −0.0519719 + 0.0900180i
\(834\) 0 0
\(835\) −2.00000 3.46410i −0.0692129 0.119880i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.0000 41.5692i −0.828572 1.43513i −0.899158 0.437623i \(-0.855820\pi\)
0.0705865 0.997506i \(-0.477513\pi\)
\(840\) 0 0
\(841\) −3.50000 + 6.06218i −0.120690 + 0.209041i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.00000 −0.206406
\(846\) 0 0
\(847\) −28.0000 −0.962091
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −20.0000 + 34.6410i −0.685591 + 1.18748i
\(852\) 0 0
\(853\) −4.00000 6.92820i −0.136957 0.237217i 0.789386 0.613897i \(-0.210399\pi\)
−0.926343 + 0.376680i \(0.877066\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.00000 + 15.5885i 0.307434 + 0.532492i 0.977800 0.209539i \(-0.0671963\pi\)
−0.670366 + 0.742030i \(0.733863\pi\)
\(858\) 0 0
\(859\) 6.50000 11.2583i 0.221777 0.384129i −0.733571 0.679613i \(-0.762148\pi\)
0.955348 + 0.295484i \(0.0954809\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22.0000 0.748889 0.374444 0.927249i \(-0.377833\pi\)
0.374444 + 0.927249i \(0.377833\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 5.00000 8.66025i 0.169613 0.293779i
\(870\) 0 0
\(871\) −14.0000 24.2487i −0.474372 0.821636i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0000 + 20.7846i 0.405674 + 0.702648i
\(876\) 0 0
\(877\) 8.00000 13.8564i 0.270141 0.467898i −0.698757 0.715359i \(-0.746263\pi\)
0.968898 + 0.247462i \(0.0795964\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) −7.00000 −0.235569 −0.117784 0.993039i \(-0.537579\pi\)
−0.117784 + 0.993039i \(0.537579\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 15.0000 25.9808i 0.503651 0.872349i −0.496340 0.868128i \(-0.665323\pi\)
0.999991 0.00422062i \(-0.00134347\pi\)
\(888\) 0 0
\(889\) 20.0000 + 34.6410i 0.670778 + 1.16182i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −20.0000 34.6410i −0.669274 1.15922i
\(894\) 0 0
\(895\) 12.0000 20.7846i 0.401116 0.694753i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.0000 34.6410i 0.664822 1.15151i
\(906\) 0 0
\(907\) 5.50000 + 9.52628i 0.182625 + 0.316315i 0.942773 0.333434i \(-0.108207\pi\)
−0.760149 + 0.649749i \(0.774874\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.00000 10.3923i −0.198789 0.344312i 0.749347 0.662177i \(-0.230367\pi\)
−0.948136 + 0.317865i \(0.897034\pi\)
\(912\) 0 0
\(913\) −30.0000 + 51.9615i −0.992855 + 1.71968i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.0000 + 20.7846i −0.394985 + 0.684134i
\(924\) 0 0
\(925\) 5.00000 + 8.66025i 0.164399 + 0.284747i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.00000 5.19615i −0.0984268 0.170480i 0.812607 0.582812i \(-0.198048\pi\)
−0.911034 + 0.412332i \(0.864714\pi\)
\(930\) 0 0
\(931\) −7.50000 + 12.9904i −0.245803 + 0.425743i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −10.0000 −0.327035
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 14.0000 24.2487i 0.456387 0.790485i −0.542380 0.840133i \(-0.682477\pi\)
0.998767 + 0.0496480i \(0.0158099\pi\)
\(942\) 0 0
\(943\) −6.00000 10.3923i −0.195387 0.338420i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.5000 + 19.9186i 0.373700 + 0.647267i 0.990132 0.140141i \(-0.0447557\pi\)
−0.616432 + 0.787408i \(0.711422\pi\)
\(948\) 0 0
\(949\) −26.0000 + 45.0333i −0.843996 + 1.46184i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.00000 −0.0323932 −0.0161966 0.999869i \(-0.505156\pi\)
−0.0161966 + 0.999869i \(0.505156\pi\)
\(954\) 0 0
\(955\) −4.00000 −0.129437
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.00000 + 1.73205i −0.0322917 + 0.0559308i
\(960\) 0 0
\(961\) 15.5000 + 26.8468i 0.500000 + 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.00000 + 1.73205i 0.0321911 + 0.0557567i
\(966\) 0 0
\(967\) −7.00000 + 12.1244i −0.225105 + 0.389893i −0.956351 0.292221i \(-0.905606\pi\)
0.731246 + 0.682114i \(0.238939\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) −2.00000 −0.0641171
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.5000 49.3634i 0.911796 1.57928i 0.100270 0.994960i \(-0.468029\pi\)
0.811526 0.584316i \(-0.198637\pi\)
\(978\) 0 0
\(979\) −25.0000 43.3013i −0.799003 1.38391i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 23.0000 + 39.8372i 0.733586 + 1.27061i 0.955341 + 0.295506i \(0.0954882\pi\)
−0.221755 + 0.975102i \(0.571178\pi\)
\(984\) 0 0
\(985\) 20.0000 34.6410i 0.637253 1.10375i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.00000 + 3.46410i −0.0634043 + 0.109819i
\(996\) 0 0
\(997\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) 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.d.2305.1 2
3.2 odd 2 1152.2.i.a.769.1 yes 2
4.3 odd 2 3456.2.i.c.2305.1 2
8.3 odd 2 3456.2.i.a.2305.1 2
8.5 even 2 3456.2.i.b.2305.1 2
9.2 odd 6 1152.2.i.a.385.1 2
9.7 even 3 inner 3456.2.i.d.1153.1 2
12.11 even 2 1152.2.i.c.769.1 yes 2
24.5 odd 2 1152.2.i.d.769.1 yes 2
24.11 even 2 1152.2.i.b.769.1 yes 2
36.7 odd 6 3456.2.i.c.1153.1 2
36.11 even 6 1152.2.i.c.385.1 yes 2
72.11 even 6 1152.2.i.b.385.1 yes 2
72.29 odd 6 1152.2.i.d.385.1 yes 2
72.43 odd 6 3456.2.i.a.1153.1 2
72.61 even 6 3456.2.i.b.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.i.a.385.1 2 9.2 odd 6
1152.2.i.a.769.1 yes 2 3.2 odd 2
1152.2.i.b.385.1 yes 2 72.11 even 6
1152.2.i.b.769.1 yes 2 24.11 even 2
1152.2.i.c.385.1 yes 2 36.11 even 6
1152.2.i.c.769.1 yes 2 12.11 even 2
1152.2.i.d.385.1 yes 2 72.29 odd 6
1152.2.i.d.769.1 yes 2 24.5 odd 2
3456.2.i.a.1153.1 2 72.43 odd 6
3456.2.i.a.2305.1 2 8.3 odd 2
3456.2.i.b.1153.1 2 72.61 even 6
3456.2.i.b.2305.1 2 8.5 even 2
3456.2.i.c.1153.1 2 36.7 odd 6
3456.2.i.c.2305.1 2 4.3 odd 2
3456.2.i.d.1153.1 2 9.7 even 3 inner
3456.2.i.d.2305.1 2 1.1 even 1 trivial