Properties

Label 2240.2.k.f.1791.12
Level $2240$
Weight $2$
Character 2240.1791
Analytic conductor $17.886$
Analytic rank $0$
Dimension $16$
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] = [2240,2,Mod(1791,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1791");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.k (of order \(2\), degree \(1\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} - 179 x^{8} + 3992 x^{7} - 5596 x^{6} - 488 x^{5} + 16080 x^{4} - 33776 x^{3} + \cdots + 5956 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 1120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1791.12
Root \(0.715495 + 0.510550i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1791
Dual form 2240.2.k.f.1791.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.02110 q^{3} +1.00000i q^{5} +(1.49445 - 2.18326i) q^{7} -1.95735 q^{9} +O(q^{10})\) \(q+1.02110 q^{3} +1.00000i q^{5} +(1.49445 - 2.18326i) q^{7} -1.95735 q^{9} -2.58102i q^{11} -5.79751i q^{13} +1.02110i q^{15} +4.41988i q^{17} -3.62638 q^{19} +(1.52598 - 2.22933i) q^{21} -6.61528i q^{23} -1.00000 q^{25} -5.06196 q^{27} -10.1146 q^{29} -1.36590 q^{31} -2.63548i q^{33} +(2.18326 + 1.49445i) q^{35} +1.44165 q^{37} -5.91984i q^{39} +2.41261i q^{41} +8.71716i q^{43} -1.95735i q^{45} +7.09441 q^{47} +(-2.53326 - 6.52554i) q^{49} +4.51315i q^{51} -9.51402 q^{53} +2.58102 q^{55} -3.70290 q^{57} -10.4752 q^{59} -9.00793i q^{61} +(-2.92516 + 4.27341i) q^{63} +5.79751 q^{65} -8.04086i q^{67} -6.75486i q^{69} +6.09463i q^{71} -5.74076i q^{73} -1.02110 q^{75} +(-5.63503 - 3.85719i) q^{77} +6.97180i q^{79} +0.703293 q^{81} +8.92102 q^{83} -4.41988 q^{85} -10.3280 q^{87} -0.519989i q^{89} +(-12.6575 - 8.66407i) q^{91} -1.39472 q^{93} -3.62638i q^{95} -9.08985i q^{97} +5.05196i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} + 16 q^{9} - 8 q^{19} - 4 q^{21} - 16 q^{25} + 48 q^{27} - 8 q^{29} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{53} + 8 q^{55} + 16 q^{57} - 8 q^{59} - 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} + 64 q^{83} - 16 q^{87} - 64 q^{91} + 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 1.02110 0.589533 0.294766 0.955569i \(-0.404758\pi\)
0.294766 + 0.955569i \(0.404758\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 1.49445 2.18326i 0.564848 0.825195i
\(8\) 0 0
\(9\) −1.95735 −0.652451
\(10\) 0 0
\(11\) 2.58102i 0.778206i −0.921194 0.389103i \(-0.872785\pi\)
0.921194 0.389103i \(-0.127215\pi\)
\(12\) 0 0
\(13\) 5.79751i 1.60794i −0.594670 0.803970i \(-0.702717\pi\)
0.594670 0.803970i \(-0.297283\pi\)
\(14\) 0 0
\(15\) 1.02110i 0.263647i
\(16\) 0 0
\(17\) 4.41988i 1.07198i 0.844225 + 0.535990i \(0.180061\pi\)
−0.844225 + 0.535990i \(0.819939\pi\)
\(18\) 0 0
\(19\) −3.62638 −0.831949 −0.415975 0.909376i \(-0.636560\pi\)
−0.415975 + 0.909376i \(0.636560\pi\)
\(20\) 0 0
\(21\) 1.52598 2.22933i 0.332996 0.486479i
\(22\) 0 0
\(23\) 6.61528i 1.37938i −0.724104 0.689690i \(-0.757747\pi\)
0.724104 0.689690i \(-0.242253\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.06196 −0.974174
\(28\) 0 0
\(29\) −10.1146 −1.87823 −0.939117 0.343598i \(-0.888354\pi\)
−0.939117 + 0.343598i \(0.888354\pi\)
\(30\) 0 0
\(31\) −1.36590 −0.245322 −0.122661 0.992449i \(-0.539143\pi\)
−0.122661 + 0.992449i \(0.539143\pi\)
\(32\) 0 0
\(33\) 2.63548i 0.458778i
\(34\) 0 0
\(35\) 2.18326 + 1.49445i 0.369038 + 0.252608i
\(36\) 0 0
\(37\) 1.44165 0.237006 0.118503 0.992954i \(-0.462191\pi\)
0.118503 + 0.992954i \(0.462191\pi\)
\(38\) 0 0
\(39\) 5.91984i 0.947933i
\(40\) 0 0
\(41\) 2.41261i 0.376786i 0.982094 + 0.188393i \(0.0603279\pi\)
−0.982094 + 0.188393i \(0.939672\pi\)
\(42\) 0 0
\(43\) 8.71716i 1.32935i 0.747131 + 0.664677i \(0.231431\pi\)
−0.747131 + 0.664677i \(0.768569\pi\)
\(44\) 0 0
\(45\) 1.95735i 0.291785i
\(46\) 0 0
\(47\) 7.09441 1.03483 0.517413 0.855736i \(-0.326895\pi\)
0.517413 + 0.855736i \(0.326895\pi\)
\(48\) 0 0
\(49\) −2.53326 6.52554i −0.361894 0.932219i
\(50\) 0 0
\(51\) 4.51315i 0.631967i
\(52\) 0 0
\(53\) −9.51402 −1.30685 −0.653426 0.756991i \(-0.726669\pi\)
−0.653426 + 0.756991i \(0.726669\pi\)
\(54\) 0 0
\(55\) 2.58102 0.348024
\(56\) 0 0
\(57\) −3.70290 −0.490461
\(58\) 0 0
\(59\) −10.4752 −1.36376 −0.681881 0.731464i \(-0.738838\pi\)
−0.681881 + 0.731464i \(0.738838\pi\)
\(60\) 0 0
\(61\) 9.00793i 1.15335i −0.816975 0.576674i \(-0.804350\pi\)
0.816975 0.576674i \(-0.195650\pi\)
\(62\) 0 0
\(63\) −2.92516 + 4.27341i −0.368536 + 0.538399i
\(64\) 0 0
\(65\) 5.79751 0.719093
\(66\) 0 0
\(67\) 8.04086i 0.982347i −0.871062 0.491173i \(-0.836568\pi\)
0.871062 0.491173i \(-0.163432\pi\)
\(68\) 0 0
\(69\) 6.75486i 0.813190i
\(70\) 0 0
\(71\) 6.09463i 0.723300i 0.932314 + 0.361650i \(0.117786\pi\)
−0.932314 + 0.361650i \(0.882214\pi\)
\(72\) 0 0
\(73\) 5.74076i 0.671905i −0.941879 0.335953i \(-0.890942\pi\)
0.941879 0.335953i \(-0.109058\pi\)
\(74\) 0 0
\(75\) −1.02110 −0.117907
\(76\) 0 0
\(77\) −5.63503 3.85719i −0.642172 0.439568i
\(78\) 0 0
\(79\) 6.97180i 0.784389i 0.919882 + 0.392195i \(0.128284\pi\)
−0.919882 + 0.392195i \(0.871716\pi\)
\(80\) 0 0
\(81\) 0.703293 0.0781436
\(82\) 0 0
\(83\) 8.92102 0.979210 0.489605 0.871944i \(-0.337141\pi\)
0.489605 + 0.871944i \(0.337141\pi\)
\(84\) 0 0
\(85\) −4.41988 −0.479404
\(86\) 0 0
\(87\) −10.3280 −1.10728
\(88\) 0 0
\(89\) 0.519989i 0.0551187i −0.999620 0.0275594i \(-0.991226\pi\)
0.999620 0.0275594i \(-0.00877353\pi\)
\(90\) 0 0
\(91\) −12.6575 8.66407i −1.32686 0.908242i
\(92\) 0 0
\(93\) −1.39472 −0.144626
\(94\) 0 0
\(95\) 3.62638i 0.372059i
\(96\) 0 0
\(97\) 9.08985i 0.922934i −0.887157 0.461467i \(-0.847323\pi\)
0.887157 0.461467i \(-0.152677\pi\)
\(98\) 0 0
\(99\) 5.05196i 0.507741i
\(100\) 0 0
\(101\) 18.8656i 1.87720i −0.345012 0.938598i \(-0.612125\pi\)
0.345012 0.938598i \(-0.387875\pi\)
\(102\) 0 0
\(103\) 16.0722 1.58364 0.791819 0.610756i \(-0.209134\pi\)
0.791819 + 0.610756i \(0.209134\pi\)
\(104\) 0 0
\(105\) 2.22933 + 1.52598i 0.217560 + 0.148920i
\(106\) 0 0
\(107\) 5.93693i 0.573945i 0.957939 + 0.286972i \(0.0926488\pi\)
−0.957939 + 0.286972i \(0.907351\pi\)
\(108\) 0 0
\(109\) 11.4574 1.09742 0.548712 0.836011i \(-0.315118\pi\)
0.548712 + 0.836011i \(0.315118\pi\)
\(110\) 0 0
\(111\) 1.47207 0.139723
\(112\) 0 0
\(113\) 10.6475 1.00163 0.500816 0.865554i \(-0.333033\pi\)
0.500816 + 0.865554i \(0.333033\pi\)
\(114\) 0 0
\(115\) 6.61528 0.616878
\(116\) 0 0
\(117\) 11.3478i 1.04910i
\(118\) 0 0
\(119\) 9.64976 + 6.60528i 0.884592 + 0.605505i
\(120\) 0 0
\(121\) 4.33835 0.394396
\(122\) 0 0
\(123\) 2.46352i 0.222128i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 11.8640i 1.05276i −0.850249 0.526380i \(-0.823549\pi\)
0.850249 0.526380i \(-0.176451\pi\)
\(128\) 0 0
\(129\) 8.90110i 0.783698i
\(130\) 0 0
\(131\) −2.91333 −0.254539 −0.127269 0.991868i \(-0.540621\pi\)
−0.127269 + 0.991868i \(0.540621\pi\)
\(132\) 0 0
\(133\) −5.41944 + 7.91734i −0.469925 + 0.686521i
\(134\) 0 0
\(135\) 5.06196i 0.435664i
\(136\) 0 0
\(137\) −19.3320 −1.65164 −0.825822 0.563931i \(-0.809288\pi\)
−0.825822 + 0.563931i \(0.809288\pi\)
\(138\) 0 0
\(139\) 17.9727 1.52443 0.762214 0.647326i \(-0.224113\pi\)
0.762214 + 0.647326i \(0.224113\pi\)
\(140\) 0 0
\(141\) 7.24411 0.610064
\(142\) 0 0
\(143\) −14.9635 −1.25131
\(144\) 0 0
\(145\) 10.1146i 0.839972i
\(146\) 0 0
\(147\) −2.58671 6.66323i −0.213348 0.549574i
\(148\) 0 0
\(149\) −2.37426 −0.194507 −0.0972533 0.995260i \(-0.531006\pi\)
−0.0972533 + 0.995260i \(0.531006\pi\)
\(150\) 0 0
\(151\) 2.19781i 0.178855i 0.995993 + 0.0894276i \(0.0285038\pi\)
−0.995993 + 0.0894276i \(0.971496\pi\)
\(152\) 0 0
\(153\) 8.65128i 0.699414i
\(154\) 0 0
\(155\) 1.36590i 0.109711i
\(156\) 0 0
\(157\) 16.0690i 1.28244i −0.767356 0.641221i \(-0.778428\pi\)
0.767356 0.641221i \(-0.221572\pi\)
\(158\) 0 0
\(159\) −9.71477 −0.770432
\(160\) 0 0
\(161\) −14.4429 9.88618i −1.13826 0.779140i
\(162\) 0 0
\(163\) 0.289542i 0.0226787i 0.999936 + 0.0113393i \(0.00360950\pi\)
−0.999936 + 0.0113393i \(0.996390\pi\)
\(164\) 0 0
\(165\) 2.63548 0.205172
\(166\) 0 0
\(167\) 23.9468 1.85306 0.926530 0.376220i \(-0.122776\pi\)
0.926530 + 0.376220i \(0.122776\pi\)
\(168\) 0 0
\(169\) −20.6111 −1.58547
\(170\) 0 0
\(171\) 7.09811 0.542806
\(172\) 0 0
\(173\) 12.3976i 0.942573i 0.881980 + 0.471287i \(0.156210\pi\)
−0.881980 + 0.471287i \(0.843790\pi\)
\(174\) 0 0
\(175\) −1.49445 + 2.18326i −0.112970 + 0.165039i
\(176\) 0 0
\(177\) −10.6963 −0.803982
\(178\) 0 0
\(179\) 2.91386i 0.217792i 0.994053 + 0.108896i \(0.0347316\pi\)
−0.994053 + 0.108896i \(0.965268\pi\)
\(180\) 0 0
\(181\) 2.06385i 0.153405i −0.997054 0.0767023i \(-0.975561\pi\)
0.997054 0.0767023i \(-0.0244391\pi\)
\(182\) 0 0
\(183\) 9.19800i 0.679936i
\(184\) 0 0
\(185\) 1.44165i 0.105992i
\(186\) 0 0
\(187\) 11.4078 0.834221
\(188\) 0 0
\(189\) −7.56483 + 11.0516i −0.550260 + 0.803884i
\(190\) 0 0
\(191\) 15.3564i 1.11115i 0.831467 + 0.555574i \(0.187501\pi\)
−0.831467 + 0.555574i \(0.812499\pi\)
\(192\) 0 0
\(193\) 6.09771 0.438923 0.219462 0.975621i \(-0.429570\pi\)
0.219462 + 0.975621i \(0.429570\pi\)
\(194\) 0 0
\(195\) 5.91984 0.423929
\(196\) 0 0
\(197\) −11.5165 −0.820515 −0.410258 0.911970i \(-0.634561\pi\)
−0.410258 + 0.911970i \(0.634561\pi\)
\(198\) 0 0
\(199\) −17.5402 −1.24339 −0.621695 0.783259i \(-0.713556\pi\)
−0.621695 + 0.783259i \(0.713556\pi\)
\(200\) 0 0
\(201\) 8.21052i 0.579126i
\(202\) 0 0
\(203\) −15.1157 + 22.0828i −1.06092 + 1.54991i
\(204\) 0 0
\(205\) −2.41261 −0.168504
\(206\) 0 0
\(207\) 12.9484i 0.899979i
\(208\) 0 0
\(209\) 9.35976i 0.647428i
\(210\) 0 0
\(211\) 24.0289i 1.65422i −0.562041 0.827109i \(-0.689984\pi\)
0.562041 0.827109i \(-0.310016\pi\)
\(212\) 0 0
\(213\) 6.22323i 0.426409i
\(214\) 0 0
\(215\) −8.71716 −0.594505
\(216\) 0 0
\(217\) −2.04126 + 2.98211i −0.138570 + 0.202439i
\(218\) 0 0
\(219\) 5.86190i 0.396110i
\(220\) 0 0
\(221\) 25.6243 1.72368
\(222\) 0 0
\(223\) −25.2618 −1.69165 −0.845827 0.533458i \(-0.820892\pi\)
−0.845827 + 0.533458i \(0.820892\pi\)
\(224\) 0 0
\(225\) 1.95735 0.130490
\(226\) 0 0
\(227\) 8.16769 0.542109 0.271054 0.962564i \(-0.412628\pi\)
0.271054 + 0.962564i \(0.412628\pi\)
\(228\) 0 0
\(229\) 2.20416i 0.145655i −0.997345 0.0728274i \(-0.976798\pi\)
0.997345 0.0728274i \(-0.0232022\pi\)
\(230\) 0 0
\(231\) −5.75394 3.93858i −0.378581 0.259140i
\(232\) 0 0
\(233\) 6.92417 0.453618 0.226809 0.973939i \(-0.427171\pi\)
0.226809 + 0.973939i \(0.427171\pi\)
\(234\) 0 0
\(235\) 7.09441i 0.462788i
\(236\) 0 0
\(237\) 7.11891i 0.462423i
\(238\) 0 0
\(239\) 4.36537i 0.282372i −0.989983 0.141186i \(-0.954908\pi\)
0.989983 0.141186i \(-0.0450916\pi\)
\(240\) 0 0
\(241\) 4.62318i 0.297805i −0.988852 0.148903i \(-0.952426\pi\)
0.988852 0.148903i \(-0.0475741\pi\)
\(242\) 0 0
\(243\) 15.9040 1.02024
\(244\) 0 0
\(245\) 6.52554 2.53326i 0.416901 0.161844i
\(246\) 0 0
\(247\) 21.0240i 1.33773i
\(248\) 0 0
\(249\) 9.10926 0.577276
\(250\) 0 0
\(251\) 5.78316 0.365030 0.182515 0.983203i \(-0.441576\pi\)
0.182515 + 0.983203i \(0.441576\pi\)
\(252\) 0 0
\(253\) −17.0741 −1.07344
\(254\) 0 0
\(255\) −4.51315 −0.282624
\(256\) 0 0
\(257\) 5.26190i 0.328228i −0.986441 0.164114i \(-0.947523\pi\)
0.986441 0.164114i \(-0.0524766\pi\)
\(258\) 0 0
\(259\) 2.15447 3.14750i 0.133872 0.195576i
\(260\) 0 0
\(261\) 19.7978 1.22546
\(262\) 0 0
\(263\) 0.921675i 0.0568329i −0.999596 0.0284165i \(-0.990954\pi\)
0.999596 0.0284165i \(-0.00904646\pi\)
\(264\) 0 0
\(265\) 9.51402i 0.584442i
\(266\) 0 0
\(267\) 0.530961i 0.0324943i
\(268\) 0 0
\(269\) 3.23020i 0.196949i −0.995140 0.0984743i \(-0.968604\pi\)
0.995140 0.0984743i \(-0.0313962\pi\)
\(270\) 0 0
\(271\) 15.5410 0.944051 0.472025 0.881585i \(-0.343523\pi\)
0.472025 + 0.881585i \(0.343523\pi\)
\(272\) 0 0
\(273\) −12.9246 8.84689i −0.782230 0.535438i
\(274\) 0 0
\(275\) 2.58102i 0.155641i
\(276\) 0 0
\(277\) −26.9121 −1.61699 −0.808497 0.588501i \(-0.799718\pi\)
−0.808497 + 0.588501i \(0.799718\pi\)
\(278\) 0 0
\(279\) 2.67354 0.160061
\(280\) 0 0
\(281\) 16.1430 0.963012 0.481506 0.876443i \(-0.340090\pi\)
0.481506 + 0.876443i \(0.340090\pi\)
\(282\) 0 0
\(283\) 10.3611 0.615905 0.307953 0.951402i \(-0.400356\pi\)
0.307953 + 0.951402i \(0.400356\pi\)
\(284\) 0 0
\(285\) 3.70290i 0.219341i
\(286\) 0 0
\(287\) 5.26736 + 3.60552i 0.310922 + 0.212827i
\(288\) 0 0
\(289\) −2.53537 −0.149140
\(290\) 0 0
\(291\) 9.28165i 0.544100i
\(292\) 0 0
\(293\) 8.52467i 0.498017i 0.968501 + 0.249008i \(0.0801047\pi\)
−0.968501 + 0.249008i \(0.919895\pi\)
\(294\) 0 0
\(295\) 10.4752i 0.609893i
\(296\) 0 0
\(297\) 13.0650i 0.758108i
\(298\) 0 0
\(299\) −38.3521 −2.21796
\(300\) 0 0
\(301\) 19.0318 + 13.0273i 1.09698 + 0.750883i
\(302\) 0 0
\(303\) 19.2637i 1.10667i
\(304\) 0 0
\(305\) 9.00793 0.515793
\(306\) 0 0
\(307\) 11.5082 0.656806 0.328403 0.944538i \(-0.393490\pi\)
0.328403 + 0.944538i \(0.393490\pi\)
\(308\) 0 0
\(309\) 16.4113 0.933607
\(310\) 0 0
\(311\) 1.26053 0.0714782 0.0357391 0.999361i \(-0.488621\pi\)
0.0357391 + 0.999361i \(0.488621\pi\)
\(312\) 0 0
\(313\) 16.5665i 0.936394i −0.883624 0.468197i \(-0.844904\pi\)
0.883624 0.468197i \(-0.155096\pi\)
\(314\) 0 0
\(315\) −4.27341 2.92516i −0.240780 0.164814i
\(316\) 0 0
\(317\) 17.8458 1.00232 0.501161 0.865354i \(-0.332906\pi\)
0.501161 + 0.865354i \(0.332906\pi\)
\(318\) 0 0
\(319\) 26.1059i 1.46165i
\(320\) 0 0
\(321\) 6.06221i 0.338359i
\(322\) 0 0
\(323\) 16.0282i 0.891833i
\(324\) 0 0
\(325\) 5.79751i 0.321588i
\(326\) 0 0
\(327\) 11.6992 0.646968
\(328\) 0 0
\(329\) 10.6022 15.4890i 0.584519 0.853934i
\(330\) 0 0
\(331\) 4.38744i 0.241156i −0.992704 0.120578i \(-0.961525\pi\)
0.992704 0.120578i \(-0.0384747\pi\)
\(332\) 0 0
\(333\) −2.82182 −0.154635
\(334\) 0 0
\(335\) 8.04086 0.439319
\(336\) 0 0
\(337\) −36.2982 −1.97729 −0.988643 0.150281i \(-0.951982\pi\)
−0.988643 + 0.150281i \(0.951982\pi\)
\(338\) 0 0
\(339\) 10.8722 0.590494
\(340\) 0 0
\(341\) 3.52540i 0.190911i
\(342\) 0 0
\(343\) −18.0328 4.22131i −0.973678 0.227929i
\(344\) 0 0
\(345\) 6.75486 0.363670
\(346\) 0 0
\(347\) 7.01135i 0.376389i −0.982132 0.188194i \(-0.939736\pi\)
0.982132 0.188194i \(-0.0602635\pi\)
\(348\) 0 0
\(349\) 23.7131i 1.26933i 0.772786 + 0.634667i \(0.218863\pi\)
−0.772786 + 0.634667i \(0.781137\pi\)
\(350\) 0 0
\(351\) 29.3468i 1.56641i
\(352\) 0 0
\(353\) 27.7599i 1.47751i 0.673973 + 0.738756i \(0.264586\pi\)
−0.673973 + 0.738756i \(0.735414\pi\)
\(354\) 0 0
\(355\) −6.09463 −0.323469
\(356\) 0 0
\(357\) 9.85338 + 6.74466i 0.521496 + 0.356965i
\(358\) 0 0
\(359\) 19.6167i 1.03533i −0.855584 0.517664i \(-0.826802\pi\)
0.855584 0.517664i \(-0.173198\pi\)
\(360\) 0 0
\(361\) −5.84934 −0.307860
\(362\) 0 0
\(363\) 4.42989 0.232509
\(364\) 0 0
\(365\) 5.74076 0.300485
\(366\) 0 0
\(367\) −32.7026 −1.70706 −0.853531 0.521041i \(-0.825544\pi\)
−0.853531 + 0.521041i \(0.825544\pi\)
\(368\) 0 0
\(369\) 4.72233i 0.245835i
\(370\) 0 0
\(371\) −14.2182 + 20.7716i −0.738172 + 1.07841i
\(372\) 0 0
\(373\) −4.67865 −0.242251 −0.121126 0.992637i \(-0.538650\pi\)
−0.121126 + 0.992637i \(0.538650\pi\)
\(374\) 0 0
\(375\) 1.02110i 0.0527294i
\(376\) 0 0
\(377\) 58.6395i 3.02009i
\(378\) 0 0
\(379\) 16.3039i 0.837478i 0.908107 + 0.418739i \(0.137528\pi\)
−0.908107 + 0.418739i \(0.862472\pi\)
\(380\) 0 0
\(381\) 12.1143i 0.620637i
\(382\) 0 0
\(383\) −5.99798 −0.306482 −0.153241 0.988189i \(-0.548971\pi\)
−0.153241 + 0.988189i \(0.548971\pi\)
\(384\) 0 0
\(385\) 3.85719 5.63503i 0.196581 0.287188i
\(386\) 0 0
\(387\) 17.0626i 0.867339i
\(388\) 0 0
\(389\) −3.81109 −0.193230 −0.0966149 0.995322i \(-0.530802\pi\)
−0.0966149 + 0.995322i \(0.530802\pi\)
\(390\) 0 0
\(391\) 29.2388 1.47867
\(392\) 0 0
\(393\) −2.97480 −0.150059
\(394\) 0 0
\(395\) −6.97180 −0.350790
\(396\) 0 0
\(397\) 29.3567i 1.47337i 0.676234 + 0.736686i \(0.263611\pi\)
−0.676234 + 0.736686i \(0.736389\pi\)
\(398\) 0 0
\(399\) −5.53379 + 8.08440i −0.277036 + 0.404726i
\(400\) 0 0
\(401\) 25.9716 1.29696 0.648479 0.761232i \(-0.275405\pi\)
0.648479 + 0.761232i \(0.275405\pi\)
\(402\) 0 0
\(403\) 7.91880i 0.394464i
\(404\) 0 0
\(405\) 0.703293i 0.0349469i
\(406\) 0 0
\(407\) 3.72092i 0.184439i
\(408\) 0 0
\(409\) 18.6606i 0.922709i 0.887216 + 0.461355i \(0.152636\pi\)
−0.887216 + 0.461355i \(0.847364\pi\)
\(410\) 0 0
\(411\) −19.7399 −0.973698
\(412\) 0 0
\(413\) −15.6547 + 22.8702i −0.770318 + 1.12537i
\(414\) 0 0
\(415\) 8.92102i 0.437916i
\(416\) 0 0
\(417\) 18.3520 0.898700
\(418\) 0 0
\(419\) 28.2356 1.37940 0.689699 0.724096i \(-0.257743\pi\)
0.689699 + 0.724096i \(0.257743\pi\)
\(420\) 0 0
\(421\) 7.85966 0.383056 0.191528 0.981487i \(-0.438656\pi\)
0.191528 + 0.981487i \(0.438656\pi\)
\(422\) 0 0
\(423\) −13.8863 −0.675174
\(424\) 0 0
\(425\) 4.41988i 0.214396i
\(426\) 0 0
\(427\) −19.6667 13.4619i −0.951736 0.651466i
\(428\) 0 0
\(429\) −15.2792 −0.737687
\(430\) 0 0
\(431\) 20.1789i 0.971983i −0.873963 0.485992i \(-0.838459\pi\)
0.873963 0.485992i \(-0.161541\pi\)
\(432\) 0 0
\(433\) 3.74851i 0.180142i −0.995935 0.0900709i \(-0.971291\pi\)
0.995935 0.0900709i \(-0.0287094\pi\)
\(434\) 0 0
\(435\) 10.3280i 0.495191i
\(436\) 0 0
\(437\) 23.9895i 1.14758i
\(438\) 0 0
\(439\) 26.6858 1.27365 0.636823 0.771010i \(-0.280248\pi\)
0.636823 + 0.771010i \(0.280248\pi\)
\(440\) 0 0
\(441\) 4.95848 + 12.7728i 0.236118 + 0.608228i
\(442\) 0 0
\(443\) 12.3567i 0.587084i 0.955946 + 0.293542i \(0.0948341\pi\)
−0.955946 + 0.293542i \(0.905166\pi\)
\(444\) 0 0
\(445\) 0.519989 0.0246498
\(446\) 0 0
\(447\) −2.42435 −0.114668
\(448\) 0 0
\(449\) 8.59414 0.405582 0.202791 0.979222i \(-0.434999\pi\)
0.202791 + 0.979222i \(0.434999\pi\)
\(450\) 0 0
\(451\) 6.22699 0.293217
\(452\) 0 0
\(453\) 2.24419i 0.105441i
\(454\) 0 0
\(455\) 8.66407 12.6575i 0.406178 0.593392i
\(456\) 0 0
\(457\) 9.34157 0.436980 0.218490 0.975839i \(-0.429887\pi\)
0.218490 + 0.975839i \(0.429887\pi\)
\(458\) 0 0
\(459\) 22.3733i 1.04429i
\(460\) 0 0
\(461\) 36.9266i 1.71984i −0.510425 0.859922i \(-0.670512\pi\)
0.510425 0.859922i \(-0.329488\pi\)
\(462\) 0 0
\(463\) 20.1423i 0.936092i −0.883704 0.468046i \(-0.844958\pi\)
0.883704 0.468046i \(-0.155042\pi\)
\(464\) 0 0
\(465\) 1.39472i 0.0646785i
\(466\) 0 0
\(467\) 2.86546 0.132598 0.0662988 0.997800i \(-0.478881\pi\)
0.0662988 + 0.997800i \(0.478881\pi\)
\(468\) 0 0
\(469\) −17.5553 12.0166i −0.810628 0.554877i
\(470\) 0 0
\(471\) 16.4080i 0.756042i
\(472\) 0 0
\(473\) 22.4991 1.03451
\(474\) 0 0
\(475\) 3.62638 0.166390
\(476\) 0 0
\(477\) 18.6223 0.852657
\(478\) 0 0
\(479\) −23.9569 −1.09462 −0.547309 0.836931i \(-0.684348\pi\)
−0.547309 + 0.836931i \(0.684348\pi\)
\(480\) 0 0
\(481\) 8.35798i 0.381091i
\(482\) 0 0
\(483\) −14.7476 10.0948i −0.671040 0.459329i
\(484\) 0 0
\(485\) 9.08985 0.412749
\(486\) 0 0
\(487\) 19.7786i 0.896255i −0.893970 0.448128i \(-0.852091\pi\)
0.893970 0.448128i \(-0.147909\pi\)
\(488\) 0 0
\(489\) 0.295652i 0.0133698i
\(490\) 0 0
\(491\) 19.6113i 0.885047i 0.896757 + 0.442524i \(0.145917\pi\)
−0.896757 + 0.442524i \(0.854083\pi\)
\(492\) 0 0
\(493\) 44.7054i 2.01343i
\(494\) 0 0
\(495\) −5.05196 −0.227069
\(496\) 0 0
\(497\) 13.3062 + 9.10811i 0.596863 + 0.408554i
\(498\) 0 0
\(499\) 17.2166i 0.770722i −0.922766 0.385361i \(-0.874077\pi\)
0.922766 0.385361i \(-0.125923\pi\)
\(500\) 0 0
\(501\) 24.4521 1.09244
\(502\) 0 0
\(503\) 27.8339 1.24105 0.620526 0.784186i \(-0.286919\pi\)
0.620526 + 0.784186i \(0.286919\pi\)
\(504\) 0 0
\(505\) 18.8656 0.839508
\(506\) 0 0
\(507\) −21.0460 −0.934688
\(508\) 0 0
\(509\) 6.15369i 0.272757i −0.990657 0.136379i \(-0.956454\pi\)
0.990657 0.136379i \(-0.0435464\pi\)
\(510\) 0 0
\(511\) −12.5336 8.57927i −0.554453 0.379524i
\(512\) 0 0
\(513\) 18.3566 0.810464
\(514\) 0 0
\(515\) 16.0722i 0.708225i
\(516\) 0 0
\(517\) 18.3108i 0.805308i
\(518\) 0 0
\(519\) 12.6592i 0.555678i
\(520\) 0 0
\(521\) 11.9506i 0.523565i 0.965127 + 0.261782i \(0.0843103\pi\)
−0.965127 + 0.261782i \(0.915690\pi\)
\(522\) 0 0
\(523\) −1.76482 −0.0771700 −0.0385850 0.999255i \(-0.512285\pi\)
−0.0385850 + 0.999255i \(0.512285\pi\)
\(524\) 0 0
\(525\) −1.52598 + 2.22933i −0.0665993 + 0.0972959i
\(526\) 0 0
\(527\) 6.03710i 0.262980i
\(528\) 0 0
\(529\) −20.7619 −0.902691
\(530\) 0 0
\(531\) 20.5038 0.889788
\(532\) 0 0
\(533\) 13.9871 0.605850
\(534\) 0 0
\(535\) −5.93693 −0.256676
\(536\) 0 0
\(537\) 2.97534i 0.128396i
\(538\) 0 0
\(539\) −16.8425 + 6.53837i −0.725459 + 0.281628i
\(540\) 0 0
\(541\) 27.6583 1.18912 0.594561 0.804050i \(-0.297326\pi\)
0.594561 + 0.804050i \(0.297326\pi\)
\(542\) 0 0
\(543\) 2.10740i 0.0904370i
\(544\) 0 0
\(545\) 11.4574i 0.490783i
\(546\) 0 0
\(547\) 16.1575i 0.690845i 0.938447 + 0.345422i \(0.112264\pi\)
−0.938447 + 0.345422i \(0.887736\pi\)
\(548\) 0 0
\(549\) 17.6317i 0.752503i
\(550\) 0 0
\(551\) 36.6794 1.56260
\(552\) 0 0
\(553\) 15.2213 + 10.4190i 0.647274 + 0.443061i
\(554\) 0 0
\(555\) 1.47207i 0.0624859i
\(556\) 0 0
\(557\) −29.9564 −1.26930 −0.634648 0.772802i \(-0.718855\pi\)
−0.634648 + 0.772802i \(0.718855\pi\)
\(558\) 0 0
\(559\) 50.5378 2.13752
\(560\) 0 0
\(561\) 11.6485 0.491800
\(562\) 0 0
\(563\) −12.4054 −0.522827 −0.261413 0.965227i \(-0.584189\pi\)
−0.261413 + 0.965227i \(0.584189\pi\)
\(564\) 0 0
\(565\) 10.6475i 0.447943i
\(566\) 0 0
\(567\) 1.05103 1.53547i 0.0441393 0.0644837i
\(568\) 0 0
\(569\) −17.8500 −0.748310 −0.374155 0.927366i \(-0.622067\pi\)
−0.374155 + 0.927366i \(0.622067\pi\)
\(570\) 0 0
\(571\) 32.7029i 1.36857i 0.729214 + 0.684286i \(0.239886\pi\)
−0.729214 + 0.684286i \(0.760114\pi\)
\(572\) 0 0
\(573\) 15.6804i 0.655058i
\(574\) 0 0
\(575\) 6.61528i 0.275876i
\(576\) 0 0
\(577\) 24.1299i 1.00454i 0.864710 + 0.502271i \(0.167502\pi\)
−0.864710 + 0.502271i \(0.832498\pi\)
\(578\) 0 0
\(579\) 6.22638 0.258760
\(580\) 0 0
\(581\) 13.3320 19.4769i 0.553105 0.808039i
\(582\) 0 0
\(583\) 24.5558i 1.01700i
\(584\) 0 0
\(585\) −11.3478 −0.469173
\(586\) 0 0
\(587\) 25.2089 1.04048 0.520241 0.854020i \(-0.325842\pi\)
0.520241 + 0.854020i \(0.325842\pi\)
\(588\) 0 0
\(589\) 4.95326 0.204096
\(590\) 0 0
\(591\) −11.7595 −0.483721
\(592\) 0 0
\(593\) 47.1605i 1.93665i 0.249692 + 0.968325i \(0.419671\pi\)
−0.249692 + 0.968325i \(0.580329\pi\)
\(594\) 0 0
\(595\) −6.60528 + 9.64976i −0.270790 + 0.395602i
\(596\) 0 0
\(597\) −17.9103 −0.733019
\(598\) 0 0
\(599\) 40.0947i 1.63823i −0.573632 0.819113i \(-0.694466\pi\)
0.573632 0.819113i \(-0.305534\pi\)
\(600\) 0 0
\(601\) 34.9606i 1.42607i −0.701127 0.713036i \(-0.747320\pi\)
0.701127 0.713036i \(-0.252680\pi\)
\(602\) 0 0
\(603\) 15.7388i 0.640933i
\(604\) 0 0
\(605\) 4.33835i 0.176379i
\(606\) 0 0
\(607\) −20.1706 −0.818698 −0.409349 0.912378i \(-0.634244\pi\)
−0.409349 + 0.912378i \(0.634244\pi\)
\(608\) 0 0
\(609\) −15.4347 + 22.5488i −0.625445 + 0.913722i
\(610\) 0 0
\(611\) 41.1299i 1.66394i
\(612\) 0 0
\(613\) 25.7382 1.03956 0.519779 0.854301i \(-0.326014\pi\)
0.519779 + 0.854301i \(0.326014\pi\)
\(614\) 0 0
\(615\) −2.46352 −0.0993386
\(616\) 0 0
\(617\) −12.7826 −0.514607 −0.257303 0.966331i \(-0.582834\pi\)
−0.257303 + 0.966331i \(0.582834\pi\)
\(618\) 0 0
\(619\) −16.8997 −0.679258 −0.339629 0.940560i \(-0.610301\pi\)
−0.339629 + 0.940560i \(0.610301\pi\)
\(620\) 0 0
\(621\) 33.4863i 1.34376i
\(622\) 0 0
\(623\) −1.13527 0.777096i −0.0454837 0.0311337i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 9.55725i 0.381680i
\(628\) 0 0
\(629\) 6.37193i 0.254065i
\(630\) 0 0
\(631\) 45.5707i 1.81414i 0.420980 + 0.907070i \(0.361686\pi\)
−0.420980 + 0.907070i \(0.638314\pi\)
\(632\) 0 0
\(633\) 24.5359i 0.975216i
\(634\) 0 0
\(635\) 11.8640 0.470809
\(636\) 0 0
\(637\) −37.8319 + 14.6866i −1.49895 + 0.581903i
\(638\) 0 0
\(639\) 11.9293i 0.471918i
\(640\) 0 0
\(641\) 11.3043 0.446493 0.223246 0.974762i \(-0.428335\pi\)
0.223246 + 0.974762i \(0.428335\pi\)
\(642\) 0 0
\(643\) 3.74609 0.147732 0.0738658 0.997268i \(-0.476466\pi\)
0.0738658 + 0.997268i \(0.476466\pi\)
\(644\) 0 0
\(645\) −8.90110 −0.350480
\(646\) 0 0
\(647\) 34.4780 1.35547 0.677735 0.735306i \(-0.262962\pi\)
0.677735 + 0.735306i \(0.262962\pi\)
\(648\) 0 0
\(649\) 27.0368i 1.06129i
\(650\) 0 0
\(651\) −2.08433 + 3.04503i −0.0816914 + 0.119344i
\(652\) 0 0
\(653\) −7.06377 −0.276427 −0.138213 0.990402i \(-0.544136\pi\)
−0.138213 + 0.990402i \(0.544136\pi\)
\(654\) 0 0
\(655\) 2.91333i 0.113833i
\(656\) 0 0
\(657\) 11.2367i 0.438386i
\(658\) 0 0
\(659\) 11.0259i 0.429508i 0.976668 + 0.214754i \(0.0688949\pi\)
−0.976668 + 0.214754i \(0.931105\pi\)
\(660\) 0 0
\(661\) 23.1116i 0.898935i −0.893297 0.449468i \(-0.851614\pi\)
0.893297 0.449468i \(-0.148386\pi\)
\(662\) 0 0
\(663\) 26.1650 1.01617
\(664\) 0 0
\(665\) −7.91734 5.41944i −0.307021 0.210157i
\(666\) 0 0
\(667\) 66.9109i 2.59080i
\(668\) 0 0
\(669\) −25.7948 −0.997285
\(670\) 0 0
\(671\) −23.2496 −0.897541
\(672\) 0 0
\(673\) −16.8939 −0.651214 −0.325607 0.945505i \(-0.605569\pi\)
−0.325607 + 0.945505i \(0.605569\pi\)
\(674\) 0 0
\(675\) 5.06196 0.194835
\(676\) 0 0
\(677\) 11.2595i 0.432738i −0.976312 0.216369i \(-0.930579\pi\)
0.976312 0.216369i \(-0.0694215\pi\)
\(678\) 0 0
\(679\) −19.8455 13.5843i −0.761601 0.521317i
\(680\) 0 0
\(681\) 8.34004 0.319591
\(682\) 0 0
\(683\) 14.6641i 0.561106i 0.959839 + 0.280553i \(0.0905178\pi\)
−0.959839 + 0.280553i \(0.909482\pi\)
\(684\) 0 0
\(685\) 19.3320i 0.738638i
\(686\) 0 0
\(687\) 2.25067i 0.0858682i
\(688\) 0 0
\(689\) 55.1576i 2.10134i
\(690\) 0 0
\(691\) 15.3976 0.585754 0.292877 0.956150i \(-0.405387\pi\)
0.292877 + 0.956150i \(0.405387\pi\)
\(692\) 0 0
\(693\) 11.0298 + 7.54989i 0.418986 + 0.286797i
\(694\) 0 0
\(695\) 17.9727i 0.681745i
\(696\) 0 0
\(697\) −10.6635 −0.403907
\(698\) 0 0
\(699\) 7.07028 0.267422
\(700\) 0 0
\(701\) −33.8811 −1.27967 −0.639836 0.768512i \(-0.720998\pi\)
−0.639836 + 0.768512i \(0.720998\pi\)
\(702\) 0 0
\(703\) −5.22798 −0.197177
\(704\) 0 0
\(705\) 7.24411i 0.272829i
\(706\) 0 0
\(707\) −41.1885 28.1936i −1.54905 1.06033i
\(708\) 0 0
\(709\) −2.75297 −0.103390 −0.0516949 0.998663i \(-0.516462\pi\)
−0.0516949 + 0.998663i \(0.516462\pi\)
\(710\) 0 0
\(711\) 13.6463i 0.511776i
\(712\) 0 0
\(713\) 9.03579i 0.338393i
\(714\) 0 0
\(715\) 14.9635i 0.559602i
\(716\) 0 0
\(717\) 4.45748i 0.166468i
\(718\) 0 0
\(719\) 23.2515 0.867135 0.433568 0.901121i \(-0.357255\pi\)
0.433568 + 0.901121i \(0.357255\pi\)
\(720\) 0 0
\(721\) 24.0190 35.0897i 0.894515 1.30681i
\(722\) 0 0
\(723\) 4.72073i 0.175566i
\(724\) 0 0
\(725\) 10.1146 0.375647
\(726\) 0 0
\(727\) −12.7237 −0.471894 −0.235947 0.971766i \(-0.575819\pi\)
−0.235947 + 0.971766i \(0.575819\pi\)
\(728\) 0 0
\(729\) 14.1297 0.523323
\(730\) 0 0
\(731\) −38.5288 −1.42504
\(732\) 0 0
\(733\) 21.3314i 0.787894i 0.919133 + 0.393947i \(0.128891\pi\)
−0.919133 + 0.393947i \(0.871109\pi\)
\(734\) 0 0
\(735\) 6.66323 2.58671i 0.245777 0.0954122i
\(736\) 0 0
\(737\) −20.7536 −0.764468
\(738\) 0 0
\(739\) 12.0391i 0.442864i 0.975176 + 0.221432i \(0.0710730\pi\)
−0.975176 + 0.221432i \(0.928927\pi\)
\(740\) 0 0
\(741\) 21.4676i 0.788633i
\(742\) 0 0
\(743\) 13.2484i 0.486036i 0.970022 + 0.243018i \(0.0781374\pi\)
−0.970022 + 0.243018i \(0.921863\pi\)
\(744\) 0 0
\(745\) 2.37426i 0.0869860i
\(746\) 0 0
\(747\) −17.4616 −0.638886
\(748\) 0 0
\(749\) 12.9619 + 8.87243i 0.473616 + 0.324192i
\(750\) 0 0
\(751\) 6.61180i 0.241268i 0.992697 + 0.120634i \(0.0384928\pi\)
−0.992697 + 0.120634i \(0.961507\pi\)
\(752\) 0 0
\(753\) 5.90518 0.215197
\(754\) 0 0
\(755\) −2.19781 −0.0799865
\(756\) 0 0
\(757\) 3.89444 0.141546 0.0707730 0.997492i \(-0.477453\pi\)
0.0707730 + 0.997492i \(0.477453\pi\)
\(758\) 0 0
\(759\) −17.4344 −0.632829
\(760\) 0 0
\(761\) 6.62391i 0.240117i 0.992767 + 0.120058i \(0.0383082\pi\)
−0.992767 + 0.120058i \(0.961692\pi\)
\(762\) 0 0
\(763\) 17.1225 25.0146i 0.619878 0.905589i
\(764\) 0 0
\(765\) 8.65128 0.312788
\(766\) 0 0
\(767\) 60.7304i 2.19285i
\(768\) 0 0
\(769\) 16.8942i 0.609220i −0.952477 0.304610i \(-0.901474\pi\)
0.952477 0.304610i \(-0.0985261\pi\)
\(770\) 0 0
\(771\) 5.37293i 0.193501i
\(772\) 0 0
\(773\) 4.87500i 0.175342i −0.996150 0.0876708i \(-0.972058\pi\)
0.996150 0.0876708i \(-0.0279424\pi\)
\(774\) 0 0
\(775\) 1.36590 0.0490645
\(776\) 0 0
\(777\) 2.19993 3.21391i 0.0789221 0.115298i
\(778\) 0 0
\(779\) 8.74905i 0.313467i
\(780\) 0 0
\(781\) 15.7303 0.562876
\(782\) 0 0
\(783\) 51.1997 1.82973
\(784\) 0 0
\(785\) 16.0690 0.573526
\(786\) 0 0
\(787\) 2.84166 0.101294 0.0506471 0.998717i \(-0.483872\pi\)
0.0506471 + 0.998717i \(0.483872\pi\)
\(788\) 0 0
\(789\) 0.941123i 0.0335049i
\(790\) 0 0
\(791\) 15.9121 23.2462i 0.565769 0.826541i
\(792\) 0 0
\(793\) −52.2236 −1.85451
\(794\) 0 0
\(795\) 9.71477i 0.344547i
\(796\) 0 0
\(797\) 5.67920i 0.201168i −0.994929 0.100584i \(-0.967929\pi\)
0.994929 0.100584i \(-0.0320711\pi\)
\(798\) 0 0
\(799\) 31.3565i 1.10931i
\(800\) 0 0
\(801\) 1.01780i 0.0359623i
\(802\) 0 0
\(803\) −14.8170 −0.522881
\(804\) 0 0
\(805\) 9.88618 14.4429i 0.348442 0.509045i
\(806\) 0 0
\(807\) 3.29836i 0.116108i
\(808\) 0 0
\(809\) 40.2852 1.41635 0.708176 0.706036i \(-0.249518\pi\)
0.708176 + 0.706036i \(0.249518\pi\)
\(810\) 0 0
\(811\) −4.14029 −0.145385 −0.0726926 0.997354i \(-0.523159\pi\)
−0.0726926 + 0.997354i \(0.523159\pi\)
\(812\) 0 0
\(813\) 15.8690 0.556549
\(814\) 0 0
\(815\) −0.289542 −0.0101422
\(816\) 0 0
\(817\) 31.6118i 1.10596i
\(818\) 0 0
\(819\) 24.7752 + 16.9587i 0.865714 + 0.592583i
\(820\) 0 0
\(821\) −25.7883 −0.900018 −0.450009 0.893024i \(-0.648579\pi\)
−0.450009 + 0.893024i \(0.648579\pi\)
\(822\) 0 0
\(823\) 27.7051i 0.965740i 0.875692 + 0.482870i \(0.160406\pi\)
−0.875692 + 0.482870i \(0.839594\pi\)
\(824\) 0 0
\(825\) 2.63548i 0.0917556i
\(826\) 0 0
\(827\) 6.36134i 0.221205i 0.993865 + 0.110603i \(0.0352781\pi\)
−0.993865 + 0.110603i \(0.964722\pi\)
\(828\) 0 0
\(829\) 18.3234i 0.636399i −0.948024 0.318199i \(-0.896922\pi\)
0.948024 0.318199i \(-0.103078\pi\)
\(830\) 0 0
\(831\) −27.4800 −0.953271
\(832\) 0 0
\(833\) 28.8421 11.1967i 0.999320 0.387943i
\(834\) 0 0
\(835\) 23.9468i 0.828714i
\(836\) 0 0
\(837\) 6.91411 0.238987
\(838\) 0 0
\(839\) −14.4848 −0.500071 −0.250036 0.968237i \(-0.580442\pi\)
−0.250036 + 0.968237i \(0.580442\pi\)
\(840\) 0 0
\(841\) 73.3051 2.52776
\(842\) 0 0
\(843\) 16.4836 0.567727
\(844\) 0 0
\(845\) 20.6111i 0.709045i
\(846\) 0 0
\(847\) 6.48344 9.47176i 0.222774 0.325453i
\(848\) 0 0
\(849\) 10.5798 0.363096
\(850\) 0 0
\(851\) 9.53692i 0.326921i
\(852\) 0 0
\(853\) 0.520567i 0.0178239i −0.999960 0.00891193i \(-0.997163\pi\)
0.999960 0.00891193i \(-0.00283679\pi\)
\(854\) 0 0
\(855\) 7.09811i 0.242750i
\(856\) 0 0
\(857\) 47.8352i 1.63402i −0.576624 0.817010i \(-0.695630\pi\)
0.576624 0.817010i \(-0.304370\pi\)
\(858\) 0 0
\(859\) 16.1624 0.551456 0.275728 0.961236i \(-0.411081\pi\)
0.275728 + 0.961236i \(0.411081\pi\)
\(860\) 0 0
\(861\) 5.37850 + 3.68160i 0.183299 + 0.125468i
\(862\) 0 0
\(863\) 17.3447i 0.590419i −0.955433 0.295209i \(-0.904611\pi\)
0.955433 0.295209i \(-0.0953894\pi\)
\(864\) 0 0
\(865\) −12.3976 −0.421532
\(866\) 0 0
\(867\) −2.58887 −0.0879227
\(868\) 0 0
\(869\) 17.9943 0.610416
\(870\) 0 0
\(871\) −46.6170 −1.57956
\(872\) 0 0
\(873\) 17.7920i 0.602169i
\(874\) 0 0
\(875\) −2.18326 1.49445i −0.0738077 0.0505215i
\(876\) 0 0
\(877\) −35.1021 −1.18531 −0.592657 0.805455i \(-0.701921\pi\)
−0.592657 + 0.805455i \(0.701921\pi\)
\(878\) 0 0
\(879\) 8.70455i 0.293597i
\(880\) 0 0
\(881\) 3.12523i 0.105292i 0.998613 + 0.0526458i \(0.0167654\pi\)
−0.998613 + 0.0526458i \(0.983235\pi\)
\(882\) 0 0
\(883\) 28.1541i 0.947461i −0.880670 0.473731i \(-0.842907\pi\)
0.880670 0.473731i \(-0.157093\pi\)
\(884\) 0 0
\(885\) 10.6963i 0.359552i
\(886\) 0 0
\(887\) −30.7626 −1.03291 −0.516454 0.856315i \(-0.672748\pi\)
−0.516454 + 0.856315i \(0.672748\pi\)
\(888\) 0 0
\(889\) −25.9022 17.7301i −0.868733 0.594649i
\(890\) 0 0
\(891\) 1.81521i 0.0608118i
\(892\) 0 0
\(893\) −25.7271 −0.860923
\(894\) 0 0
\(895\) −2.91386 −0.0973996
\(896\) 0 0
\(897\) −39.1614 −1.30756
\(898\) 0 0
\(899\) 13.8155 0.460773
\(900\) 0 0
\(901\) 42.0509i 1.40092i
\(902\) 0 0
\(903\) 19.4334 + 13.3022i 0.646704 + 0.442670i
\(904\) 0 0
\(905\) 2.06385 0.0686046
\(906\) 0 0
\(907\) 25.4398i 0.844716i −0.906429 0.422358i \(-0.861202\pi\)
0.906429 0.422358i \(-0.138798\pi\)
\(908\) 0 0
\(909\) 36.9266i 1.22478i
\(910\) 0 0
\(911\) 28.6451i 0.949054i −0.880241 0.474527i \(-0.842619\pi\)
0.880241 0.474527i \(-0.157381\pi\)
\(912\) 0 0
\(913\) 23.0253i 0.762027i
\(914\) 0 0
\(915\) 9.19800 0.304077
\(916\) 0 0
\(917\) −4.35382 + 6.36056i −0.143776 + 0.210044i
\(918\) 0 0
\(919\) 36.3761i 1.19994i −0.800023 0.599969i \(-0.795180\pi\)
0.800023 0.599969i \(-0.204820\pi\)
\(920\) 0 0
\(921\) 11.7510 0.387209
\(922\) 0 0
\(923\) 35.3337 1.16302
\(924\) 0 0
\(925\) −1.44165 −0.0474012
\(926\) 0 0
\(927\) −31.4589 −1.03325
\(928\) 0 0
\(929\) 53.2086i 1.74572i −0.487974 0.872858i \(-0.662264\pi\)
0.487974 0.872858i \(-0.337736\pi\)
\(930\) 0 0
\(931\) 9.18656 + 23.6641i 0.301077 + 0.775559i
\(932\) 0 0
\(933\) 1.28713 0.0421388
\(934\) 0 0
\(935\) 11.4078i 0.373075i
\(936\) 0 0
\(937\) 37.6232i 1.22910i −0.788879 0.614549i \(-0.789338\pi\)
0.788879 0.614549i \(-0.210662\pi\)
\(938\) 0 0
\(939\) 16.9161i 0.552035i
\(940\) 0 0
\(941\) 31.0252i 1.01139i 0.862711 + 0.505697i \(0.168765\pi\)
−0.862711 + 0.505697i \(0.831235\pi\)
\(942\) 0 0
\(943\) 15.9601 0.519732
\(944\) 0 0
\(945\) −11.0516 7.56483i −0.359508 0.246084i
\(946\) 0 0
\(947\) 34.2651i 1.11347i 0.830691 + 0.556734i \(0.187946\pi\)
−0.830691 + 0.556734i \(0.812054\pi\)
\(948\) 0 0
\(949\) −33.2821 −1.08038
\(950\) 0 0
\(951\) 18.2224 0.590902
\(952\) 0 0
\(953\) −49.5311 −1.60447 −0.802236 0.597007i \(-0.796356\pi\)
−0.802236 + 0.597007i \(0.796356\pi\)
\(954\) 0 0
\(955\) −15.3564 −0.496921
\(956\) 0 0
\(957\) 26.6568i 0.861692i
\(958\) 0 0
\(959\) −28.8906 + 42.2068i −0.932928 + 1.36293i
\(960\) 0 0
\(961\) −29.1343 −0.939817
\(962\) 0 0
\(963\) 11.6207i 0.374471i
\(964\) 0 0
\(965\) 6.09771i 0.196292i
\(966\) 0 0
\(967\) 59.9908i 1.92917i −0.263767 0.964587i \(-0.584965\pi\)
0.263767 0.964587i \(-0.415035\pi\)
\(968\) 0 0
\(969\) 16.3664i 0.525765i
\(970\) 0 0
\(971\) 46.5169 1.49280 0.746400 0.665498i \(-0.231781\pi\)
0.746400 + 0.665498i \(0.231781\pi\)
\(972\) 0 0
\(973\) 26.8593 39.2392i 0.861070 1.25795i
\(974\) 0 0
\(975\) 5.91984i 0.189587i
\(976\) 0 0
\(977\) 0.576069 0.0184301 0.00921504 0.999958i \(-0.497067\pi\)
0.00921504 + 0.999958i \(0.497067\pi\)
\(978\) 0 0
\(979\) −1.34210 −0.0428937
\(980\) 0 0
\(981\) −22.4263 −0.716016
\(982\) 0 0
\(983\) −7.50742 −0.239449 −0.119725 0.992807i \(-0.538201\pi\)
−0.119725 + 0.992807i \(0.538201\pi\)
\(984\) 0 0
\(985\) 11.5165i 0.366946i
\(986\) 0 0
\(987\) 10.8259 15.8158i 0.344593 0.503422i
\(988\) 0 0
\(989\) 57.6664 1.83369
\(990\) 0 0
\(991\) 40.7914i 1.29578i −0.761734 0.647890i \(-0.775652\pi\)
0.761734 0.647890i \(-0.224348\pi\)
\(992\) 0 0
\(993\) 4.48002i 0.142169i
\(994\) 0 0
\(995\) 17.5402i 0.556061i
\(996\) 0 0
\(997\) 38.4148i 1.21661i 0.793704 + 0.608304i \(0.208150\pi\)
−0.793704 + 0.608304i \(0.791850\pi\)
\(998\) 0 0
\(999\) −7.29757 −0.230885
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 2240.2.k.f.1791.12 16
4.3 odd 2 2240.2.k.g.1791.6 16
7.6 odd 2 2240.2.k.g.1791.5 16
8.3 odd 2 1120.2.k.b.671.11 yes 16
8.5 even 2 1120.2.k.a.671.5 16
28.27 even 2 inner 2240.2.k.f.1791.11 16
56.13 odd 2 1120.2.k.b.671.12 yes 16
56.27 even 2 1120.2.k.a.671.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1120.2.k.a.671.5 16 8.5 even 2
1120.2.k.a.671.6 yes 16 56.27 even 2
1120.2.k.b.671.11 yes 16 8.3 odd 2
1120.2.k.b.671.12 yes 16 56.13 odd 2
2240.2.k.f.1791.11 16 28.27 even 2 inner
2240.2.k.f.1791.12 16 1.1 even 1 trivial
2240.2.k.g.1791.5 16 7.6 odd 2
2240.2.k.g.1791.6 16 4.3 odd 2