Properties

Label 108.5.d.a.55.11
Level $108$
Weight $5$
Character 108.55
Analytic conductor $11.164$
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] = [108,5,Mod(55,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.55");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.d (of order \(2\), degree \(1\), 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: \(11.1639560131\)
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: \( x^{16} - 2 x^{15} + 6 x^{14} - 22 x^{13} + 19 x^{12} + 18 x^{11} + 1423 x^{10} + 660 x^{9} - 7353 x^{8} - 22934 x^{7} - 36353 x^{6} - 16248 x^{5} + 360646 x^{4} + \cdots + 2924100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{26} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 55.11
Root \(-1.98442 - 2.21794i\) of defining polynomial
Character \(\chi\) \(=\) 108.55
Dual form 108.5.d.a.55.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.59266 - 3.66925i) q^{2} +(-10.9268 - 11.6878i) q^{4} +29.4580 q^{5} -32.9098i q^{7} +(-60.2882 + 21.4787i) q^{8} +O(q^{10})\) \(q+(1.59266 - 3.66925i) q^{2} +(-10.9268 - 11.6878i) q^{4} +29.4580 q^{5} -32.9098i q^{7} +(-60.2882 + 21.4787i) q^{8} +(46.9167 - 108.089i) q^{10} -136.639i q^{11} +164.522 q^{13} +(-120.754 - 52.4142i) q^{14} +(-17.2082 + 255.421i) q^{16} -380.756 q^{17} -439.574i q^{19} +(-321.883 - 344.298i) q^{20} +(-501.364 - 217.620i) q^{22} -171.733i q^{23} +242.773 q^{25} +(262.028 - 603.673i) q^{26} +(-384.642 + 359.600i) q^{28} -1041.21 q^{29} +1181.17i q^{31} +(909.798 + 469.941i) q^{32} +(-606.417 + 1397.09i) q^{34} -969.456i q^{35} +2700.79 q^{37} +(-1612.91 - 700.094i) q^{38} +(-1775.97 + 632.718i) q^{40} +1556.42 q^{41} -2218.02i q^{43} +(-1597.01 + 1493.04i) q^{44} +(-630.130 - 273.512i) q^{46} +1292.12i q^{47} +1317.95 q^{49} +(386.655 - 890.794i) q^{50} +(-1797.71 - 1922.90i) q^{52} +1015.00 q^{53} -4025.12i q^{55} +(706.858 + 1984.07i) q^{56} +(-1658.29 + 3820.45i) q^{58} +2434.15i q^{59} +3839.98 q^{61} +(4334.02 + 1881.21i) q^{62} +(3173.33 - 2589.82i) q^{64} +4846.48 q^{65} -2352.60i q^{67} +(4160.46 + 4450.19i) q^{68} +(-3557.18 - 1544.02i) q^{70} -884.064i q^{71} +6921.45 q^{73} +(4301.45 - 9909.89i) q^{74} +(-5137.65 + 4803.16i) q^{76} -4496.77 q^{77} +10308.0i q^{79} +(-506.918 + 7524.19i) q^{80} +(2478.85 - 5710.89i) q^{82} +12322.0i q^{83} -11216.3 q^{85} +(-8138.47 - 3532.56i) q^{86} +(2934.83 + 8237.73i) q^{88} -5751.10 q^{89} -5414.38i q^{91} +(-2007.17 + 1876.49i) q^{92} +(4741.11 + 2057.91i) q^{94} -12949.0i q^{95} -8159.93 q^{97} +(2099.04 - 4835.88i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 14 q^{4} - 202 q^{10} - 352 q^{13} - 206 q^{16} + 738 q^{22} + 1632 q^{25} + 342 q^{28} - 2536 q^{34} + 3200 q^{37} - 2854 q^{40} + 36 q^{46} - 896 q^{49} + 2288 q^{52} + 2492 q^{58} - 2752 q^{61} + 682 q^{64} - 14166 q^{70} + 8240 q^{73} - 33084 q^{76} + 68 q^{82} + 8800 q^{85} + 48294 q^{88} + 52596 q^{94} - 6928 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.59266 3.66925i 0.398166 0.917313i
\(3\) 0 0
\(4\) −10.9268 11.6878i −0.682928 0.730486i
\(5\) 29.4580 1.17832 0.589160 0.808017i \(-0.299459\pi\)
0.589160 + 0.808017i \(0.299459\pi\)
\(6\) 0 0
\(7\) 32.9098i 0.671628i −0.941928 0.335814i \(-0.890989\pi\)
0.941928 0.335814i \(-0.109011\pi\)
\(8\) −60.2882 + 21.4787i −0.942003 + 0.335604i
\(9\) 0 0
\(10\) 46.9167 108.089i 0.469167 1.08089i
\(11\) 136.639i 1.12925i −0.825348 0.564625i \(-0.809021\pi\)
0.825348 0.564625i \(-0.190979\pi\)
\(12\) 0 0
\(13\) 164.522 0.973503 0.486751 0.873541i \(-0.338182\pi\)
0.486751 + 0.873541i \(0.338182\pi\)
\(14\) −120.754 52.4142i −0.616094 0.267420i
\(15\) 0 0
\(16\) −17.2082 + 255.421i −0.0672195 + 0.997738i
\(17\) −380.756 −1.31750 −0.658748 0.752364i \(-0.728914\pi\)
−0.658748 + 0.752364i \(0.728914\pi\)
\(18\) 0 0
\(19\) 439.574i 1.21766i −0.793302 0.608829i \(-0.791640\pi\)
0.793302 0.608829i \(-0.208360\pi\)
\(20\) −321.883 344.298i −0.804707 0.860746i
\(21\) 0 0
\(22\) −501.364 217.620i −1.03588 0.449629i
\(23\) 171.733i 0.324636i −0.986738 0.162318i \(-0.948103\pi\)
0.986738 0.162318i \(-0.0518971\pi\)
\(24\) 0 0
\(25\) 242.773 0.388436
\(26\) 262.028 603.673i 0.387616 0.893007i
\(27\) 0 0
\(28\) −384.642 + 359.600i −0.490615 + 0.458674i
\(29\) −1041.21 −1.23806 −0.619029 0.785368i \(-0.712474\pi\)
−0.619029 + 0.785368i \(0.712474\pi\)
\(30\) 0 0
\(31\) 1181.17i 1.22911i 0.788876 + 0.614553i \(0.210664\pi\)
−0.788876 + 0.614553i \(0.789336\pi\)
\(32\) 909.798 + 469.941i 0.888474 + 0.458927i
\(33\) 0 0
\(34\) −606.417 + 1397.09i −0.524582 + 1.20856i
\(35\) 969.456i 0.791393i
\(36\) 0 0
\(37\) 2700.79 1.97282 0.986410 0.164301i \(-0.0525370\pi\)
0.986410 + 0.164301i \(0.0525370\pi\)
\(38\) −1612.91 700.094i −1.11697 0.484830i
\(39\) 0 0
\(40\) −1775.97 + 632.718i −1.10998 + 0.395449i
\(41\) 1556.42 0.925887 0.462944 0.886388i \(-0.346793\pi\)
0.462944 + 0.886388i \(0.346793\pi\)
\(42\) 0 0
\(43\) 2218.02i 1.19958i −0.800158 0.599789i \(-0.795251\pi\)
0.800158 0.599789i \(-0.204749\pi\)
\(44\) −1597.01 + 1493.04i −0.824901 + 0.771196i
\(45\) 0 0
\(46\) −630.130 273.512i −0.297793 0.129259i
\(47\) 1292.12i 0.584934i 0.956276 + 0.292467i \(0.0944761\pi\)
−0.956276 + 0.292467i \(0.905524\pi\)
\(48\) 0 0
\(49\) 1317.95 0.548915
\(50\) 386.655 890.794i 0.154662 0.356318i
\(51\) 0 0
\(52\) −1797.71 1922.90i −0.664832 0.711130i
\(53\) 1015.00 0.361337 0.180669 0.983544i \(-0.442174\pi\)
0.180669 + 0.983544i \(0.442174\pi\)
\(54\) 0 0
\(55\) 4025.12i 1.33062i
\(56\) 706.858 + 1984.07i 0.225401 + 0.632676i
\(57\) 0 0
\(58\) −1658.29 + 3820.45i −0.492952 + 1.13569i
\(59\) 2434.15i 0.699268i 0.936886 + 0.349634i \(0.113694\pi\)
−0.936886 + 0.349634i \(0.886306\pi\)
\(60\) 0 0
\(61\) 3839.98 1.03198 0.515988 0.856596i \(-0.327425\pi\)
0.515988 + 0.856596i \(0.327425\pi\)
\(62\) 4334.02 + 1881.21i 1.12748 + 0.489388i
\(63\) 0 0
\(64\) 3173.33 2589.82i 0.774740 0.632280i
\(65\) 4846.48 1.14710
\(66\) 0 0
\(67\) 2352.60i 0.524081i −0.965057 0.262040i \(-0.915605\pi\)
0.965057 0.262040i \(-0.0843954\pi\)
\(68\) 4160.46 + 4450.19i 0.899754 + 0.962412i
\(69\) 0 0
\(70\) −3557.18 1544.02i −0.725955 0.315106i
\(71\) 884.064i 0.175375i −0.996148 0.0876874i \(-0.972052\pi\)
0.996148 0.0876874i \(-0.0279476\pi\)
\(72\) 0 0
\(73\) 6921.45 1.29883 0.649413 0.760436i \(-0.275014\pi\)
0.649413 + 0.760436i \(0.275014\pi\)
\(74\) 4301.45 9909.89i 0.785510 1.80969i
\(75\) 0 0
\(76\) −5137.65 + 4803.16i −0.889482 + 0.831572i
\(77\) −4496.77 −0.758436
\(78\) 0 0
\(79\) 10308.0i 1.65165i 0.563924 + 0.825826i \(0.309291\pi\)
−0.563924 + 0.825826i \(0.690709\pi\)
\(80\) −506.918 + 7524.19i −0.0792060 + 1.17565i
\(81\) 0 0
\(82\) 2478.85 5710.89i 0.368657 0.849329i
\(83\) 12322.0i 1.78865i 0.447417 + 0.894326i \(0.352344\pi\)
−0.447417 + 0.894326i \(0.647656\pi\)
\(84\) 0 0
\(85\) −11216.3 −1.55243
\(86\) −8138.47 3532.56i −1.10039 0.477631i
\(87\) 0 0
\(88\) 2934.83 + 8237.73i 0.378981 + 1.06376i
\(89\) −5751.10 −0.726057 −0.363028 0.931778i \(-0.618257\pi\)
−0.363028 + 0.931778i \(0.618257\pi\)
\(90\) 0 0
\(91\) 5414.38i 0.653832i
\(92\) −2007.17 + 1876.49i −0.237142 + 0.221703i
\(93\) 0 0
\(94\) 4741.11 + 2057.91i 0.536568 + 0.232901i
\(95\) 12949.0i 1.43479i
\(96\) 0 0
\(97\) −8159.93 −0.867247 −0.433624 0.901094i \(-0.642765\pi\)
−0.433624 + 0.901094i \(0.642765\pi\)
\(98\) 2099.04 4835.88i 0.218559 0.503527i
\(99\) 0 0
\(100\) −2652.74 2837.47i −0.265274 0.283747i
\(101\) 11440.4 1.12150 0.560749 0.827986i \(-0.310513\pi\)
0.560749 + 0.827986i \(0.310513\pi\)
\(102\) 0 0
\(103\) 11724.0i 1.10510i 0.833479 + 0.552552i \(0.186346\pi\)
−0.833479 + 0.552552i \(0.813654\pi\)
\(104\) −9918.73 + 3533.71i −0.917043 + 0.326711i
\(105\) 0 0
\(106\) 1616.55 3724.28i 0.143872 0.331460i
\(107\) 21224.4i 1.85382i −0.375286 0.926909i \(-0.622456\pi\)
0.375286 0.926909i \(-0.377544\pi\)
\(108\) 0 0
\(109\) −5752.64 −0.484188 −0.242094 0.970253i \(-0.577834\pi\)
−0.242094 + 0.970253i \(0.577834\pi\)
\(110\) −14769.2 6410.66i −1.22059 0.529806i
\(111\) 0 0
\(112\) 8405.85 + 566.318i 0.670109 + 0.0451465i
\(113\) −6018.68 −0.471351 −0.235675 0.971832i \(-0.575730\pi\)
−0.235675 + 0.971832i \(0.575730\pi\)
\(114\) 0 0
\(115\) 5058.90i 0.382525i
\(116\) 11377.1 + 12169.4i 0.845504 + 0.904384i
\(117\) 0 0
\(118\) 8931.52 + 3876.79i 0.641448 + 0.278425i
\(119\) 12530.6i 0.884868i
\(120\) 0 0
\(121\) −4029.28 −0.275205
\(122\) 6115.80 14089.9i 0.410898 0.946646i
\(123\) 0 0
\(124\) 13805.3 12906.5i 0.897845 0.839391i
\(125\) −11259.6 −0.720617
\(126\) 0 0
\(127\) 7591.93i 0.470700i 0.971911 + 0.235350i \(0.0756237\pi\)
−0.971911 + 0.235350i \(0.924376\pi\)
\(128\) −4448.65 15768.5i −0.271524 0.962432i
\(129\) 0 0
\(130\) 7718.82 17783.0i 0.456735 1.05225i
\(131\) 3334.49i 0.194306i 0.995269 + 0.0971532i \(0.0309737\pi\)
−0.995269 + 0.0971532i \(0.969026\pi\)
\(132\) 0 0
\(133\) −14466.3 −0.817813
\(134\) −8632.28 3746.90i −0.480746 0.208671i
\(135\) 0 0
\(136\) 22955.1 8178.14i 1.24109 0.442157i
\(137\) −11823.7 −0.629960 −0.314980 0.949098i \(-0.601998\pi\)
−0.314980 + 0.949098i \(0.601998\pi\)
\(138\) 0 0
\(139\) 5544.34i 0.286959i 0.989653 + 0.143480i \(0.0458292\pi\)
−0.989653 + 0.143480i \(0.954171\pi\)
\(140\) −11330.8 + 10593.1i −0.578101 + 0.540464i
\(141\) 0 0
\(142\) −3243.85 1408.02i −0.160874 0.0698282i
\(143\) 22480.2i 1.09933i
\(144\) 0 0
\(145\) −30671.8 −1.45883
\(146\) 11023.5 25396.5i 0.517149 1.19143i
\(147\) 0 0
\(148\) −29511.1 31566.2i −1.34729 1.44112i
\(149\) −11073.6 −0.498790 −0.249395 0.968402i \(-0.580232\pi\)
−0.249395 + 0.968402i \(0.580232\pi\)
\(150\) 0 0
\(151\) 27450.4i 1.20391i −0.798529 0.601956i \(-0.794388\pi\)
0.798529 0.601956i \(-0.205612\pi\)
\(152\) 9441.47 + 26501.1i 0.408651 + 1.14704i
\(153\) 0 0
\(154\) −7161.84 + 16499.8i −0.301984 + 0.695724i
\(155\) 34794.9i 1.44828i
\(156\) 0 0
\(157\) 14606.6 0.592583 0.296291 0.955098i \(-0.404250\pi\)
0.296291 + 0.955098i \(0.404250\pi\)
\(158\) 37822.5 + 16417.1i 1.51508 + 0.657632i
\(159\) 0 0
\(160\) 26800.8 + 13843.5i 1.04691 + 0.540762i
\(161\) −5651.68 −0.218035
\(162\) 0 0
\(163\) 39065.5i 1.47034i 0.677883 + 0.735170i \(0.262898\pi\)
−0.677883 + 0.735170i \(0.737102\pi\)
\(164\) −17006.7 18191.1i −0.632314 0.676348i
\(165\) 0 0
\(166\) 45212.6 + 19624.8i 1.64075 + 0.712180i
\(167\) 3123.15i 0.111985i −0.998431 0.0559926i \(-0.982168\pi\)
0.998431 0.0559926i \(-0.0178323\pi\)
\(168\) 0 0
\(169\) −1493.53 −0.0522925
\(170\) −17863.8 + 41155.5i −0.618125 + 1.42407i
\(171\) 0 0
\(172\) −25923.7 + 24235.9i −0.876274 + 0.819225i
\(173\) 50749.6 1.69567 0.847834 0.530262i \(-0.177907\pi\)
0.847834 + 0.530262i \(0.177907\pi\)
\(174\) 0 0
\(175\) 7989.60i 0.260885i
\(176\) 34900.5 + 2351.31i 1.12670 + 0.0759076i
\(177\) 0 0
\(178\) −9159.56 + 21102.2i −0.289091 + 0.666022i
\(179\) 4456.66i 0.139092i 0.997579 + 0.0695462i \(0.0221551\pi\)
−0.997579 + 0.0695462i \(0.977845\pi\)
\(180\) 0 0
\(181\) −8462.68 −0.258316 −0.129158 0.991624i \(-0.541227\pi\)
−0.129158 + 0.991624i \(0.541227\pi\)
\(182\) −19866.7 8623.29i −0.599769 0.260334i
\(183\) 0 0
\(184\) 3688.59 + 10353.4i 0.108949 + 0.305808i
\(185\) 79559.9 2.32461
\(186\) 0 0
\(187\) 52026.2i 1.48778i
\(188\) 15102.0 14118.8i 0.427286 0.399467i
\(189\) 0 0
\(190\) −47513.1 20623.4i −1.31615 0.571284i
\(191\) 17699.2i 0.485163i 0.970131 + 0.242582i \(0.0779942\pi\)
−0.970131 + 0.242582i \(0.922006\pi\)
\(192\) 0 0
\(193\) 10944.5 0.293820 0.146910 0.989150i \(-0.453067\pi\)
0.146910 + 0.989150i \(0.453067\pi\)
\(194\) −12996.0 + 29940.9i −0.345308 + 0.795538i
\(195\) 0 0
\(196\) −14401.0 15403.9i −0.374869 0.400975i
\(197\) 10101.3 0.260284 0.130142 0.991495i \(-0.458457\pi\)
0.130142 + 0.991495i \(0.458457\pi\)
\(198\) 0 0
\(199\) 31394.8i 0.792777i −0.918083 0.396388i \(-0.870263\pi\)
0.918083 0.396388i \(-0.129737\pi\)
\(200\) −14636.3 + 5214.43i −0.365908 + 0.130361i
\(201\) 0 0
\(202\) 18220.7 41977.8i 0.446543 1.02877i
\(203\) 34265.9i 0.831515i
\(204\) 0 0
\(205\) 45848.9 1.09099
\(206\) 43018.5 + 18672.5i 1.01373 + 0.440015i
\(207\) 0 0
\(208\) −2831.12 + 42022.4i −0.0654383 + 0.971301i
\(209\) −60063.1 −1.37504
\(210\) 0 0
\(211\) 83361.8i 1.87241i −0.351448 0.936207i \(-0.614311\pi\)
0.351448 0.936207i \(-0.385689\pi\)
\(212\) −11090.7 11863.1i −0.246767 0.263952i
\(213\) 0 0
\(214\) −77877.6 33803.3i −1.70053 0.738127i
\(215\) 65338.3i 1.41349i
\(216\) 0 0
\(217\) 38872.1 0.825503
\(218\) −9162.02 + 21107.9i −0.192787 + 0.444152i
\(219\) 0 0
\(220\) −47044.7 + 43981.8i −0.971997 + 0.908715i
\(221\) −62642.8 −1.28259
\(222\) 0 0
\(223\) 13301.4i 0.267478i −0.991017 0.133739i \(-0.957302\pi\)
0.991017 0.133739i \(-0.0426983\pi\)
\(224\) 15465.7 29941.2i 0.308228 0.596724i
\(225\) 0 0
\(226\) −9585.73 + 22084.1i −0.187676 + 0.432377i
\(227\) 50075.5i 0.971792i 0.874017 + 0.485896i \(0.161507\pi\)
−0.874017 + 0.485896i \(0.838493\pi\)
\(228\) 0 0
\(229\) 60039.8 1.14490 0.572451 0.819939i \(-0.305993\pi\)
0.572451 + 0.819939i \(0.305993\pi\)
\(230\) −18562.4 8057.12i −0.350895 0.152308i
\(231\) 0 0
\(232\) 62772.5 22363.7i 1.16625 0.415497i
\(233\) −31252.0 −0.575659 −0.287830 0.957682i \(-0.592934\pi\)
−0.287830 + 0.957682i \(0.592934\pi\)
\(234\) 0 0
\(235\) 38063.2i 0.689239i
\(236\) 28449.8 26597.6i 0.510805 0.477549i
\(237\) 0 0
\(238\) 45978.0 + 19957.0i 0.811701 + 0.352324i
\(239\) 62883.1i 1.10087i −0.834876 0.550437i \(-0.814461\pi\)
0.834876 0.550437i \(-0.185539\pi\)
\(240\) 0 0
\(241\) 64613.0 1.11246 0.556232 0.831027i \(-0.312247\pi\)
0.556232 + 0.831027i \(0.312247\pi\)
\(242\) −6417.29 + 14784.5i −0.109577 + 0.252450i
\(243\) 0 0
\(244\) −41958.9 44880.9i −0.704765 0.753844i
\(245\) 38824.0 0.646797
\(246\) 0 0
\(247\) 72319.6i 1.18539i
\(248\) −25370.0 71210.7i −0.412493 1.15782i
\(249\) 0 0
\(250\) −17932.8 + 41314.5i −0.286925 + 0.661032i
\(251\) 78853.1i 1.25162i 0.779977 + 0.625808i \(0.215231\pi\)
−0.779977 + 0.625808i \(0.784769\pi\)
\(252\) 0 0
\(253\) −23465.4 −0.366595
\(254\) 27856.7 + 12091.4i 0.431780 + 0.187417i
\(255\) 0 0
\(256\) −64943.8 8790.66i −0.990963 0.134135i
\(257\) −25571.1 −0.387153 −0.193576 0.981085i \(-0.562009\pi\)
−0.193576 + 0.981085i \(0.562009\pi\)
\(258\) 0 0
\(259\) 88882.5i 1.32500i
\(260\) −52956.8 56644.6i −0.783384 0.837938i
\(261\) 0 0
\(262\) 12235.1 + 5310.73i 0.178240 + 0.0773662i
\(263\) 110011.i 1.59046i −0.606305 0.795232i \(-0.707349\pi\)
0.606305 0.795232i \(-0.292651\pi\)
\(264\) 0 0
\(265\) 29899.8 0.425771
\(266\) −23040.0 + 53080.5i −0.325625 + 0.750191i
\(267\) 0 0
\(268\) −27496.6 + 25706.5i −0.382834 + 0.357909i
\(269\) −39818.1 −0.550270 −0.275135 0.961406i \(-0.588723\pi\)
−0.275135 + 0.961406i \(0.588723\pi\)
\(270\) 0 0
\(271\) 14054.7i 0.191375i 0.995411 + 0.0956873i \(0.0305049\pi\)
−0.995411 + 0.0956873i \(0.969495\pi\)
\(272\) 6552.12 97253.1i 0.0885614 1.31452i
\(273\) 0 0
\(274\) −18831.2 + 43384.2i −0.250829 + 0.577871i
\(275\) 33172.3i 0.438642i
\(276\) 0 0
\(277\) −31583.7 −0.411627 −0.205813 0.978591i \(-0.565984\pi\)
−0.205813 + 0.978591i \(0.565984\pi\)
\(278\) 20343.6 + 8830.27i 0.263232 + 0.114257i
\(279\) 0 0
\(280\) 20822.6 + 58446.8i 0.265595 + 0.745494i
\(281\) 66899.3 0.847244 0.423622 0.905839i \(-0.360758\pi\)
0.423622 + 0.905839i \(0.360758\pi\)
\(282\) 0 0
\(283\) 160099.i 1.99901i 0.0314612 + 0.999505i \(0.489984\pi\)
−0.0314612 + 0.999505i \(0.510016\pi\)
\(284\) −10332.7 + 9660.03i −0.128109 + 0.119768i
\(285\) 0 0
\(286\) −82485.4 35803.3i −1.00843 0.437715i
\(287\) 51221.4i 0.621852i
\(288\) 0 0
\(289\) 61454.3 0.735795
\(290\) −48849.9 + 112543.i −0.580855 + 1.33820i
\(291\) 0 0
\(292\) −75629.6 80896.3i −0.887005 0.948775i
\(293\) −92671.2 −1.07947 −0.539734 0.841836i \(-0.681475\pi\)
−0.539734 + 0.841836i \(0.681475\pi\)
\(294\) 0 0
\(295\) 71705.2i 0.823961i
\(296\) −162826. + 58009.4i −1.85840 + 0.662087i
\(297\) 0 0
\(298\) −17636.6 + 40631.9i −0.198601 + 0.457546i
\(299\) 28253.8i 0.316034i
\(300\) 0 0
\(301\) −72994.5 −0.805670
\(302\) −100722. 43719.3i −1.10436 0.479357i
\(303\) 0 0
\(304\) 112277. + 7564.28i 1.21490 + 0.0818503i
\(305\) 113118. 1.21600
\(306\) 0 0
\(307\) 86852.2i 0.921518i −0.887525 0.460759i \(-0.847577\pi\)
0.887525 0.460759i \(-0.152423\pi\)
\(308\) 49135.5 + 52557.2i 0.517957 + 0.554027i
\(309\) 0 0
\(310\) 127671. + 55416.6i 1.32853 + 0.576656i
\(311\) 43940.9i 0.454306i 0.973859 + 0.227153i \(0.0729417\pi\)
−0.973859 + 0.227153i \(0.927058\pi\)
\(312\) 0 0
\(313\) −100801. −1.02890 −0.514452 0.857519i \(-0.672005\pi\)
−0.514452 + 0.857519i \(0.672005\pi\)
\(314\) 23263.4 53595.2i 0.235946 0.543584i
\(315\) 0 0
\(316\) 120477. 112634.i 1.20651 1.12796i
\(317\) −101986. −1.01490 −0.507449 0.861682i \(-0.669411\pi\)
−0.507449 + 0.861682i \(0.669411\pi\)
\(318\) 0 0
\(319\) 142270.i 1.39808i
\(320\) 93480.0 76290.9i 0.912891 0.745028i
\(321\) 0 0
\(322\) −9001.23 + 20737.5i −0.0868141 + 0.200006i
\(323\) 167371.i 1.60426i
\(324\) 0 0
\(325\) 39941.4 0.378144
\(326\) 143341. + 62218.2i 1.34876 + 0.585439i
\(327\) 0 0
\(328\) −93833.6 + 33429.8i −0.872189 + 0.310732i
\(329\) 42523.4 0.392858
\(330\) 0 0
\(331\) 39677.8i 0.362153i −0.983469 0.181076i \(-0.942042\pi\)
0.983469 0.181076i \(-0.0579581\pi\)
\(332\) 144017. 134641.i 1.30658 1.22152i
\(333\) 0 0
\(334\) −11459.6 4974.13i −0.102725 0.0445887i
\(335\) 69302.8i 0.617534i
\(336\) 0 0
\(337\) −129324. −1.13873 −0.569364 0.822086i \(-0.692810\pi\)
−0.569364 + 0.822086i \(0.692810\pi\)
\(338\) −2378.68 + 5480.12i −0.0208211 + 0.0479686i
\(339\) 0 0
\(340\) 122559. + 131094.i 1.06020 + 1.13403i
\(341\) 161394. 1.38797
\(342\) 0 0
\(343\) 122390.i 1.04030i
\(344\) 47640.1 + 133720.i 0.402583 + 1.13001i
\(345\) 0 0
\(346\) 80827.1 186213.i 0.675157 1.55546i
\(347\) 174482.i 1.44908i 0.689233 + 0.724540i \(0.257948\pi\)
−0.689233 + 0.724540i \(0.742052\pi\)
\(348\) 0 0
\(349\) −39421.3 −0.323654 −0.161827 0.986819i \(-0.551739\pi\)
−0.161827 + 0.986819i \(0.551739\pi\)
\(350\) −29315.9 12724.7i −0.239313 0.103875i
\(351\) 0 0
\(352\) 64212.4 124314.i 0.518243 1.00331i
\(353\) −123080. −0.987733 −0.493866 0.869538i \(-0.664417\pi\)
−0.493866 + 0.869538i \(0.664417\pi\)
\(354\) 0 0
\(355\) 26042.7i 0.206647i
\(356\) 62841.3 + 67217.5i 0.495844 + 0.530374i
\(357\) 0 0
\(358\) 16352.6 + 7097.96i 0.127591 + 0.0553818i
\(359\) 20967.0i 0.162685i 0.996686 + 0.0813424i \(0.0259207\pi\)
−0.996686 + 0.0813424i \(0.974079\pi\)
\(360\) 0 0
\(361\) −62904.6 −0.482690
\(362\) −13478.2 + 31051.7i −0.102852 + 0.236956i
\(363\) 0 0
\(364\) −63282.1 + 59162.1i −0.477615 + 0.446520i
\(365\) 203892. 1.53043
\(366\) 0 0
\(367\) 58300.8i 0.432855i 0.976299 + 0.216427i \(0.0694405\pi\)
−0.976299 + 0.216427i \(0.930560\pi\)
\(368\) 43864.1 + 2955.21i 0.323902 + 0.0218219i
\(369\) 0 0
\(370\) 126712. 291925.i 0.925581 2.13240i
\(371\) 33403.3i 0.242684i
\(372\) 0 0
\(373\) −79775.3 −0.573391 −0.286695 0.958022i \(-0.592557\pi\)
−0.286695 + 0.958022i \(0.592557\pi\)
\(374\) 190897. + 82860.3i 1.36476 + 0.592384i
\(375\) 0 0
\(376\) −27753.0 77899.5i −0.196306 0.551009i
\(377\) −171301. −1.20525
\(378\) 0 0
\(379\) 80878.2i 0.563058i −0.959553 0.281529i \(-0.909159\pi\)
0.959553 0.281529i \(-0.0908415\pi\)
\(380\) −151345. + 141491.i −1.04809 + 0.979857i
\(381\) 0 0
\(382\) 64943.0 + 28188.9i 0.445047 + 0.193176i
\(383\) 99819.8i 0.680486i −0.940338 0.340243i \(-0.889491\pi\)
0.940338 0.340243i \(-0.110509\pi\)
\(384\) 0 0
\(385\) −132466. −0.893680
\(386\) 17430.9 40158.1i 0.116989 0.269525i
\(387\) 0 0
\(388\) 89162.3 + 95371.4i 0.592267 + 0.633512i
\(389\) 15583.1 0.102981 0.0514903 0.998673i \(-0.483603\pi\)
0.0514903 + 0.998673i \(0.483603\pi\)
\(390\) 0 0
\(391\) 65388.3i 0.427707i
\(392\) −79456.6 + 28307.7i −0.517080 + 0.184218i
\(393\) 0 0
\(394\) 16088.0 37064.4i 0.103636 0.238762i
\(395\) 303652.i 1.94617i
\(396\) 0 0
\(397\) −67256.6 −0.426730 −0.213365 0.976973i \(-0.568442\pi\)
−0.213365 + 0.976973i \(0.568442\pi\)
\(398\) −115195. 50001.3i −0.727225 0.315657i
\(399\) 0 0
\(400\) −4177.68 + 62009.2i −0.0261105 + 0.387558i
\(401\) 151843. 0.944288 0.472144 0.881521i \(-0.343480\pi\)
0.472144 + 0.881521i \(0.343480\pi\)
\(402\) 0 0
\(403\) 194329.i 1.19654i
\(404\) −125008. 133713.i −0.765903 0.819239i
\(405\) 0 0
\(406\) 125730. + 54574.0i 0.762760 + 0.331081i
\(407\) 369034.i 2.22781i
\(408\) 0 0
\(409\) −64170.7 −0.383610 −0.191805 0.981433i \(-0.561434\pi\)
−0.191805 + 0.981433i \(0.561434\pi\)
\(410\) 73021.9 168231.i 0.434396 1.00078i
\(411\) 0 0
\(412\) 137028. 128107.i 0.807262 0.754706i
\(413\) 80107.4 0.469648
\(414\) 0 0
\(415\) 362982.i 2.10760i
\(416\) 149682. + 77315.6i 0.864932 + 0.446766i
\(417\) 0 0
\(418\) −95660.3 + 220387.i −0.547494 + 1.26134i
\(419\) 85171.9i 0.485142i 0.970134 + 0.242571i \(0.0779907\pi\)
−0.970134 + 0.242571i \(0.922009\pi\)
\(420\) 0 0
\(421\) 214896. 1.21245 0.606224 0.795294i \(-0.292683\pi\)
0.606224 + 0.795294i \(0.292683\pi\)
\(422\) −305875. 132767.i −1.71759 0.745532i
\(423\) 0 0
\(424\) −61192.3 + 21800.8i −0.340381 + 0.121266i
\(425\) −92437.2 −0.511763
\(426\) 0 0
\(427\) 126373.i 0.693105i
\(428\) −248066. + 231915.i −1.35419 + 1.26602i
\(429\) 0 0
\(430\) −239743. 104062.i −1.29661 0.562802i
\(431\) 19885.9i 0.107051i −0.998566 0.0535254i \(-0.982954\pi\)
0.998566 0.0535254i \(-0.0170458\pi\)
\(432\) 0 0
\(433\) 103996. 0.554679 0.277339 0.960772i \(-0.410547\pi\)
0.277339 + 0.960772i \(0.410547\pi\)
\(434\) 61910.2 142632.i 0.328687 0.757244i
\(435\) 0 0
\(436\) 62858.2 + 67235.6i 0.330666 + 0.353693i
\(437\) −75489.2 −0.395296
\(438\) 0 0
\(439\) 55956.4i 0.290349i −0.989406 0.145175i \(-0.953626\pi\)
0.989406 0.145175i \(-0.0463744\pi\)
\(440\) 86454.1 + 242667.i 0.446561 + 1.25345i
\(441\) 0 0
\(442\) −99768.9 + 229852.i −0.510682 + 1.17653i
\(443\) 22607.4i 0.115198i −0.998340 0.0575988i \(-0.981656\pi\)
0.998340 0.0575988i \(-0.0183444\pi\)
\(444\) 0 0
\(445\) −169416. −0.855527
\(446\) −48806.2 21184.7i −0.245361 0.106501i
\(447\) 0 0
\(448\) −85230.4 104434.i −0.424657 0.520337i
\(449\) 77577.5 0.384807 0.192404 0.981316i \(-0.438372\pi\)
0.192404 + 0.981316i \(0.438372\pi\)
\(450\) 0 0
\(451\) 212668.i 1.04556i
\(452\) 65765.2 + 70345.0i 0.321899 + 0.344315i
\(453\) 0 0
\(454\) 183740. + 79753.4i 0.891438 + 0.386934i
\(455\) 159497.i 0.770423i
\(456\) 0 0
\(457\) −242860. −1.16285 −0.581425 0.813600i \(-0.697505\pi\)
−0.581425 + 0.813600i \(0.697505\pi\)
\(458\) 95623.2 220301.i 0.455861 1.05023i
\(459\) 0 0
\(460\) −59127.2 + 55277.8i −0.279429 + 0.261237i
\(461\) −199640. −0.939391 −0.469696 0.882828i \(-0.655636\pi\)
−0.469696 + 0.882828i \(0.655636\pi\)
\(462\) 0 0
\(463\) 63490.8i 0.296175i −0.988974 0.148088i \(-0.952688\pi\)
0.988974 0.148088i \(-0.0473118\pi\)
\(464\) 17917.3 265946.i 0.0832216 1.23526i
\(465\) 0 0
\(466\) −49773.9 + 114671.i −0.229208 + 0.528060i
\(467\) 112982.i 0.518053i −0.965870 0.259026i \(-0.916598\pi\)
0.965870 0.259026i \(-0.0834017\pi\)
\(468\) 0 0
\(469\) −77423.5 −0.351987
\(470\) 139664. + 60621.9i 0.632248 + 0.274431i
\(471\) 0 0
\(472\) −52282.3 146751.i −0.234677 0.658713i
\(473\) −303068. −1.35462
\(474\) 0 0
\(475\) 106717.i 0.472982i
\(476\) 146455. 136920.i 0.646383 0.604301i
\(477\) 0 0
\(478\) −230734. 100152.i −1.00985 0.438331i
\(479\) 313437.i 1.36609i −0.730376 0.683045i \(-0.760655\pi\)
0.730376 0.683045i \(-0.239345\pi\)
\(480\) 0 0
\(481\) 444339. 1.92055
\(482\) 102907. 237081.i 0.442945 1.02048i
\(483\) 0 0
\(484\) 44027.3 + 47093.4i 0.187945 + 0.201034i
\(485\) −240375. −1.02189
\(486\) 0 0
\(487\) 64950.6i 0.273858i 0.990581 + 0.136929i \(0.0437232\pi\)
−0.990581 + 0.136929i \(0.956277\pi\)
\(488\) −231506. + 82477.7i −0.972125 + 0.346336i
\(489\) 0 0
\(490\) 61833.6 142455.i 0.257533 0.593316i
\(491\) 358585.i 1.48741i 0.668510 + 0.743703i \(0.266932\pi\)
−0.668510 + 0.743703i \(0.733068\pi\)
\(492\) 0 0
\(493\) 396446. 1.63114
\(494\) −265359. 115181.i −1.08738 0.471983i
\(495\) 0 0
\(496\) −301696. 20325.8i −1.22633 0.0826198i
\(497\) −29094.4 −0.117787
\(498\) 0 0
\(499\) 12696.3i 0.0509890i 0.999675 + 0.0254945i \(0.00811603\pi\)
−0.999675 + 0.0254945i \(0.991884\pi\)
\(500\) 123032. + 131600.i 0.492130 + 0.526401i
\(501\) 0 0
\(502\) 289332. + 125586.i 1.14812 + 0.498351i
\(503\) 114739.i 0.453498i 0.973953 + 0.226749i \(0.0728097\pi\)
−0.973953 + 0.226749i \(0.927190\pi\)
\(504\) 0 0
\(505\) 337011. 1.32148
\(506\) −37372.5 + 86100.5i −0.145966 + 0.336283i
\(507\) 0 0
\(508\) 88732.8 82955.8i 0.343840 0.321454i
\(509\) −44847.2 −0.173101 −0.0865505 0.996247i \(-0.527584\pi\)
−0.0865505 + 0.996247i \(0.527584\pi\)
\(510\) 0 0
\(511\) 227783.i 0.872329i
\(512\) −135689. + 224295.i −0.517611 + 0.855616i
\(513\) 0 0
\(514\) −40726.1 + 93826.7i −0.154151 + 0.355141i
\(515\) 345367.i 1.30216i
\(516\) 0 0
\(517\) 176554. 0.660536
\(518\) −326132. 141560.i −1.21544 0.527571i
\(519\) 0 0
\(520\) −292186. + 104096.i −1.08057 + 0.384970i
\(521\) 285591. 1.05213 0.526065 0.850444i \(-0.323667\pi\)
0.526065 + 0.850444i \(0.323667\pi\)
\(522\) 0 0
\(523\) 164178.i 0.600220i 0.953905 + 0.300110i \(0.0970234\pi\)
−0.953905 + 0.300110i \(0.902977\pi\)
\(524\) 38972.8 36435.5i 0.141938 0.132697i
\(525\) 0 0
\(526\) −403658. 175210.i −1.45895 0.633269i
\(527\) 449738.i 1.61934i
\(528\) 0 0
\(529\) 250349. 0.894611
\(530\) 47620.3 109710.i 0.169527 0.390565i
\(531\) 0 0
\(532\) 158071. + 169079.i 0.558507 + 0.597401i
\(533\) 256065. 0.901354
\(534\) 0 0
\(535\) 625227.i 2.18439i
\(536\) 50530.7 + 141834.i 0.175884 + 0.493686i
\(537\) 0 0
\(538\) −63416.9 + 146103.i −0.219099 + 0.504770i
\(539\) 180083.i 0.619863i
\(540\) 0 0
\(541\) −300448. −1.02654 −0.513269 0.858228i \(-0.671566\pi\)
−0.513269 + 0.858228i \(0.671566\pi\)
\(542\) 51570.4 + 22384.5i 0.175551 + 0.0761989i
\(543\) 0 0
\(544\) −346411. 178933.i −1.17056 0.604634i
\(545\) −169461. −0.570528
\(546\) 0 0
\(547\) 74251.6i 0.248160i 0.992272 + 0.124080i \(0.0395979\pi\)
−0.992272 + 0.124080i \(0.960402\pi\)
\(548\) 129196. + 138193.i 0.430217 + 0.460177i
\(549\) 0 0
\(550\) −121717. 52832.3i −0.402372 0.174652i
\(551\) 457688.i 1.50753i
\(552\) 0 0
\(553\) 339233. 1.10930
\(554\) −50302.2 + 115889.i −0.163896 + 0.377591i
\(555\) 0 0
\(556\) 64801.0 60582.1i 0.209620 0.195972i
\(557\) 193413. 0.623411 0.311705 0.950179i \(-0.399100\pi\)
0.311705 + 0.950179i \(0.399100\pi\)
\(558\) 0 0
\(559\) 364913.i 1.16779i
\(560\) 247619. + 16682.6i 0.789603 + 0.0531970i
\(561\) 0 0
\(562\) 106548. 245470.i 0.337344 0.777188i
\(563\) 177354.i 0.559530i −0.960068 0.279765i \(-0.909743\pi\)
0.960068 0.279765i \(-0.0902566\pi\)
\(564\) 0 0
\(565\) −177298. −0.555402
\(566\) 587443. + 254983.i 1.83372 + 0.795938i
\(567\) 0 0
\(568\) 18988.5 + 53298.6i 0.0588565 + 0.165204i
\(569\) 590091. 1.82261 0.911306 0.411730i \(-0.135075\pi\)
0.911306 + 0.411730i \(0.135075\pi\)
\(570\) 0 0
\(571\) 524012.i 1.60720i 0.595172 + 0.803598i \(0.297084\pi\)
−0.595172 + 0.803598i \(0.702916\pi\)
\(572\) −262743. + 245637.i −0.803044 + 0.750761i
\(573\) 0 0
\(574\) −187944. 81578.4i −0.570433 0.247600i
\(575\) 41692.0i 0.126100i
\(576\) 0 0
\(577\) −337273. −1.01305 −0.506523 0.862226i \(-0.669070\pi\)
−0.506523 + 0.862226i \(0.669070\pi\)
\(578\) 97876.1 225492.i 0.292969 0.674955i
\(579\) 0 0
\(580\) 335146. + 358486.i 0.996273 + 1.06565i
\(581\) 405515. 1.20131
\(582\) 0 0
\(583\) 138688.i 0.408040i
\(584\) −417282. + 148663.i −1.22350 + 0.435892i
\(585\) 0 0
\(586\) −147594. + 340034.i −0.429807 + 0.990210i
\(587\) 4289.71i 0.0124495i 0.999981 + 0.00622475i \(0.00198141\pi\)
−0.999981 + 0.00622475i \(0.998019\pi\)
\(588\) 0 0
\(589\) 519212. 1.49663
\(590\) 263105. + 114202.i 0.755830 + 0.328073i
\(591\) 0 0
\(592\) −46475.7 + 689839.i −0.132612 + 1.96836i
\(593\) 150227. 0.427207 0.213604 0.976920i \(-0.431480\pi\)
0.213604 + 0.976920i \(0.431480\pi\)
\(594\) 0 0
\(595\) 369126.i 1.04266i
\(596\) 121000. + 129426.i 0.340637 + 0.364359i
\(597\) 0 0
\(598\) −103670. 44998.8i −0.289902 0.125834i
\(599\) 473108.i 1.31858i 0.751888 + 0.659290i \(0.229143\pi\)
−0.751888 + 0.659290i \(0.770857\pi\)
\(600\) 0 0
\(601\) −83320.1 −0.230675 −0.115338 0.993326i \(-0.536795\pi\)
−0.115338 + 0.993326i \(0.536795\pi\)
\(602\) −116256. + 267835.i −0.320790 + 0.739052i
\(603\) 0 0
\(604\) −320834. + 299946.i −0.879441 + 0.822185i
\(605\) −118695. −0.324280
\(606\) 0 0
\(607\) 143905.i 0.390570i 0.980747 + 0.195285i \(0.0625632\pi\)
−0.980747 + 0.195285i \(0.937437\pi\)
\(608\) 206574. 399924.i 0.558816 1.08186i
\(609\) 0 0
\(610\) 180159. 415059.i 0.484169 1.11545i
\(611\) 212582.i 0.569435i
\(612\) 0 0
\(613\) −5563.04 −0.0148044 −0.00740221 0.999973i \(-0.502356\pi\)
−0.00740221 + 0.999973i \(0.502356\pi\)
\(614\) −318683. 138326.i −0.845321 0.366917i
\(615\) 0 0
\(616\) 271102. 96584.6i 0.714449 0.254534i
\(617\) −300749. −0.790012 −0.395006 0.918679i \(-0.629257\pi\)
−0.395006 + 0.918679i \(0.629257\pi\)
\(618\) 0 0
\(619\) 158539.i 0.413767i −0.978366 0.206883i \(-0.933668\pi\)
0.978366 0.206883i \(-0.0663321\pi\)
\(620\) 406675. 380199.i 1.05795 0.989070i
\(621\) 0 0
\(622\) 161230. + 69983.1i 0.416741 + 0.180889i
\(623\) 189267.i 0.487640i
\(624\) 0 0
\(625\) −483419. −1.23755
\(626\) −160542. + 369863.i −0.409675 + 0.943828i
\(627\) 0 0
\(628\) −159604. 170718.i −0.404691 0.432873i
\(629\) −1.02834e6 −2.59918
\(630\) 0 0
\(631\) 557374.i 1.39987i 0.714206 + 0.699936i \(0.246788\pi\)
−0.714206 + 0.699936i \(0.753212\pi\)
\(632\) −221401. 621449.i −0.554301 1.55586i
\(633\) 0 0
\(634\) −162429. + 374213.i −0.404098 + 0.930979i
\(635\) 223643.i 0.554635i
\(636\) 0 0
\(637\) 216831. 0.534370
\(638\) 522023. + 226588.i 1.28247 + 0.556667i
\(639\) 0 0
\(640\) −131048. 464508.i −0.319942 1.13405i
\(641\) 101106. 0.246071 0.123036 0.992402i \(-0.460737\pi\)
0.123036 + 0.992402i \(0.460737\pi\)
\(642\) 0 0
\(643\) 406950.i 0.984281i −0.870516 0.492141i \(-0.836215\pi\)
0.870516 0.492141i \(-0.163785\pi\)
\(644\) 61755.1 + 66055.6i 0.148902 + 0.159271i
\(645\) 0 0
\(646\) 614126. + 266565.i 1.47161 + 0.638761i
\(647\) 427706.i 1.02173i −0.859660 0.510866i \(-0.829325\pi\)
0.859660 0.510866i \(-0.170675\pi\)
\(648\) 0 0
\(649\) 332601. 0.789648
\(650\) 63613.3 146555.i 0.150564 0.346876i
\(651\) 0 0
\(652\) 456588. 426862.i 1.07406 1.00414i
\(653\) 430055. 1.00855 0.504275 0.863543i \(-0.331760\pi\)
0.504275 + 0.863543i \(0.331760\pi\)
\(654\) 0 0
\(655\) 98227.4i 0.228955i
\(656\) −26783.1 + 397542.i −0.0622377 + 0.923793i
\(657\) 0 0
\(658\) 67725.4 156029.i 0.156423 0.360374i
\(659\) 179686.i 0.413755i 0.978367 + 0.206878i \(0.0663302\pi\)
−0.978367 + 0.206878i \(0.933670\pi\)
\(660\) 0 0
\(661\) −539348. −1.23443 −0.617214 0.786795i \(-0.711739\pi\)
−0.617214 + 0.786795i \(0.711739\pi\)
\(662\) −145588. 63193.4i −0.332208 0.144197i
\(663\) 0 0
\(664\) −264660. 742872.i −0.600279 1.68492i
\(665\) −426148. −0.963645
\(666\) 0 0
\(667\) 178809.i 0.401918i
\(668\) −36502.7 + 34126.2i −0.0818036 + 0.0764777i
\(669\) 0 0
\(670\) −254290. 110376.i −0.566473 0.245881i
\(671\) 524693.i 1.16536i
\(672\) 0 0
\(673\) −758520. −1.67470 −0.837350 0.546667i \(-0.815896\pi\)
−0.837350 + 0.546667i \(0.815896\pi\)
\(674\) −205970. + 474523.i −0.453403 + 1.04457i
\(675\) 0 0
\(676\) 16319.5 + 17456.0i 0.0357120 + 0.0381989i
\(677\) −394155. −0.859982 −0.429991 0.902833i \(-0.641483\pi\)
−0.429991 + 0.902833i \(0.641483\pi\)
\(678\) 0 0
\(679\) 268542.i 0.582468i
\(680\) 676211. 240911.i 1.46239 0.521002i
\(681\) 0 0
\(682\) 257047. 592197.i 0.552642 1.27320i
\(683\) 483705.i 1.03691i −0.855106 0.518453i \(-0.826508\pi\)
0.855106 0.518453i \(-0.173492\pi\)
\(684\) 0 0
\(685\) −348303. −0.742294
\(686\) −449079. 194926.i −0.954277 0.414210i
\(687\) 0 0
\(688\) 566528. + 38168.1i 1.19686 + 0.0806349i
\(689\) 166989. 0.351763
\(690\) 0 0
\(691\) 136607.i 0.286098i 0.989716 + 0.143049i \(0.0456907\pi\)
−0.989716 + 0.143049i \(0.954309\pi\)
\(692\) −554533. 593150.i −1.15802 1.23866i
\(693\) 0 0
\(694\) 640219. + 277891.i 1.32926 + 0.576974i
\(695\) 163325.i 0.338130i
\(696\) 0 0
\(697\) −592615. −1.21985
\(698\) −62784.9 + 144647.i −0.128868 + 0.296892i
\(699\) 0 0
\(700\) −93380.6 + 87301.1i −0.190573 + 0.178165i
\(701\) 878847. 1.78845 0.894226 0.447617i \(-0.147727\pi\)
0.894226 + 0.447617i \(0.147727\pi\)
\(702\) 0 0
\(703\) 1.18720e6i 2.40222i
\(704\) −353871. 433602.i −0.714002 0.874875i
\(705\) 0 0
\(706\) −196026. + 451613.i −0.393282 + 0.906061i
\(707\) 376502.i 0.753231i
\(708\) 0 0
\(709\) 445304. 0.885857 0.442929 0.896557i \(-0.353939\pi\)
0.442929 + 0.896557i \(0.353939\pi\)
\(710\) −95557.4 41477.3i −0.189560 0.0822800i
\(711\) 0 0
\(712\) 346723. 123526.i 0.683948 0.243668i
\(713\) 202846. 0.399012
\(714\) 0 0
\(715\) 662220.i 1.29536i
\(716\) 52088.4 48697.2i 0.101605 0.0949900i
\(717\) 0 0
\(718\) 76933.2 + 33393.4i 0.149233 + 0.0647756i
\(719\) 731078.i 1.41418i −0.707121 0.707092i \(-0.750007\pi\)
0.707121 0.707092i \(-0.249993\pi\)
\(720\) 0 0
\(721\) 385836. 0.742219
\(722\) −100186. + 230813.i −0.192191 + 0.442778i
\(723\) 0 0
\(724\) 92470.3 + 98909.8i 0.176411 + 0.188696i
\(725\) −252776. −0.480906
\(726\) 0 0
\(727\) 992100.i 1.87710i 0.345147 + 0.938549i \(0.387829\pi\)
−0.345147 + 0.938549i \(0.612171\pi\)
\(728\) 116294. + 326423.i 0.219429 + 0.615912i
\(729\) 0 0
\(730\) 324731. 748131.i 0.609366 1.40389i
\(731\) 844524.i 1.58044i
\(732\) 0 0
\(733\) −121286. −0.225737 −0.112868 0.993610i \(-0.536004\pi\)
−0.112868 + 0.993610i \(0.536004\pi\)
\(734\) 213920. + 92853.6i 0.397064 + 0.172348i
\(735\) 0 0
\(736\) 80704.2 156242.i 0.148984 0.288431i
\(737\) −321457. −0.591818
\(738\) 0 0
\(739\) 273800.i 0.501355i 0.968071 + 0.250677i \(0.0806534\pi\)
−0.968071 + 0.250677i \(0.919347\pi\)
\(740\) −869338. 929878.i −1.58754 1.69810i
\(741\) 0 0
\(742\) −122565. 53200.3i −0.222618 0.0966287i
\(743\) 389402.i 0.705376i 0.935741 + 0.352688i \(0.114732\pi\)
−0.935741 + 0.352688i \(0.885268\pi\)
\(744\) 0 0
\(745\) −326207. −0.587733
\(746\) −127055. + 292716.i −0.228305 + 0.525979i
\(747\) 0 0
\(748\) 608071. 568483.i 1.08680 1.01605i
\(749\) −698489. −1.24508
\(750\) 0 0
\(751\) 430569.i 0.763419i 0.924282 + 0.381710i \(0.124665\pi\)
−0.924282 + 0.381710i \(0.875335\pi\)
\(752\) −330034. 22235.0i −0.583611 0.0393189i
\(753\) 0 0
\(754\) −272825. + 628548.i −0.479891 + 1.10559i
\(755\) 808633.i 1.41859i
\(756\) 0 0
\(757\) −1.04226e6 −1.81880 −0.909401 0.415920i \(-0.863460\pi\)
−0.909401 + 0.415920i \(0.863460\pi\)
\(758\) −296762. 128812.i −0.516500 0.224190i
\(759\) 0 0
\(760\) 278127. + 780670.i 0.481521 + 1.35158i
\(761\) −117139. −0.202271 −0.101135 0.994873i \(-0.532248\pi\)
−0.101135 + 0.994873i \(0.532248\pi\)
\(762\) 0 0
\(763\) 189318.i 0.325195i
\(764\) 206865. 193397.i 0.354405 0.331331i
\(765\) 0 0
\(766\) −366264. 158979.i −0.624219 0.270946i
\(767\) 400471.i 0.680739i
\(768\) 0 0
\(769\) −104458. −0.176639 −0.0883197 0.996092i \(-0.528150\pi\)
−0.0883197 + 0.996092i \(0.528150\pi\)
\(770\) −210973. + 486050.i −0.355833 + 0.819785i
\(771\) 0 0
\(772\) −119589. 127917.i −0.200658 0.214631i
\(773\) 34677.2 0.0580343 0.0290172 0.999579i \(-0.490762\pi\)
0.0290172 + 0.999579i \(0.490762\pi\)
\(774\) 0 0
\(775\) 286756.i 0.477429i
\(776\) 491948. 175264.i 0.816950 0.291052i
\(777\) 0 0
\(778\) 24818.7 57178.4i 0.0410034 0.0944654i
\(779\) 684161.i 1.12741i
\(780\) 0 0
\(781\) −120798. −0.198042
\(782\) 239926. + 104142.i 0.392341 + 0.170298i
\(783\) 0 0
\(784\) −22679.4 + 336631.i −0.0368978 + 0.547674i
\(785\) 430280. 0.698252
\(786\) 0 0
\(787\) 1.08172e6i 1.74649i 0.487285 + 0.873243i \(0.337987\pi\)
−0.487285 + 0.873243i \(0.662013\pi\)
\(788\) −110376. 118062.i −0.177755 0.190133i
\(789\) 0 0
\(790\) 1.11418e6 + 483615.i 1.78525 + 0.774900i
\(791\) 198074.i 0.316573i
\(792\) 0 0
\(793\) 631762. 1.00463
\(794\) −107117. + 246781.i −0.169910 + 0.391446i
\(795\) 0 0
\(796\) −366935. + 343046.i −0.579112 + 0.541409i
\(797\) −370822. −0.583779 −0.291889 0.956452i \(-0.594284\pi\)
−0.291889 + 0.956452i \(0.594284\pi\)
\(798\) 0 0
\(799\) 491982.i 0.770648i
\(800\) 220874. + 114089.i 0.345116 + 0.178264i
\(801\) 0 0
\(802\) 241834. 557149.i 0.375983 0.866208i
\(803\) 945741.i 1.46670i
\(804\) 0 0
\(805\) −166487. −0.256915
\(806\) 713041. + 309500.i 1.09760 + 0.476421i
\(807\) 0 0
\(808\) −689722. + 245725.i −1.05646 + 0.376380i
\(809\) −763253. −1.16620 −0.583098 0.812402i \(-0.698160\pi\)
−0.583098 + 0.812402i \(0.698160\pi\)
\(810\) 0 0
\(811\) 976424.i 1.48456i −0.670091 0.742279i \(-0.733745\pi\)
0.670091 0.742279i \(-0.266255\pi\)
\(812\) 400492. 374418.i 0.607410 0.567864i
\(813\) 0 0
\(814\) −1.35408e6 587747.i −2.04360 0.887037i
\(815\) 1.15079e6i 1.73253i
\(816\) 0 0
\(817\) −974984. −1.46067
\(818\) −102202. + 235459.i −0.152741 + 0.351891i
\(819\) 0 0
\(820\) −500984. 535872.i −0.745068 0.796954i
\(821\) 1.05933e6 1.57162 0.785808 0.618470i \(-0.212247\pi\)
0.785808 + 0.618470i \(0.212247\pi\)
\(822\) 0 0
\(823\) 649256.i 0.958553i 0.877664 + 0.479277i \(0.159101\pi\)
−0.877664 + 0.479277i \(0.840899\pi\)
\(824\) −251817. 706821.i −0.370877 1.04101i
\(825\) 0 0
\(826\) 127584. 293934.i 0.186998 0.430815i
\(827\) 1.23981e6i 1.81278i 0.422439 + 0.906391i \(0.361174\pi\)
−0.422439 + 0.906391i \(0.638826\pi\)
\(828\) 0 0
\(829\) 656712. 0.955577 0.477788 0.878475i \(-0.341439\pi\)
0.477788 + 0.878475i \(0.341439\pi\)
\(830\) 1.33187e6 + 578108.i 1.93333 + 0.839175i
\(831\) 0 0
\(832\) 522083. 426082.i 0.754211 0.615526i
\(833\) −501816. −0.723194
\(834\) 0 0
\(835\) 92001.8i 0.131954i
\(836\) 656300. + 702004.i 0.939053 + 1.00445i
\(837\) 0 0
\(838\) 312517. + 135650.i 0.445027 + 0.193167i
\(839\) 895246.i 1.27180i 0.771772 + 0.635899i \(0.219371\pi\)
−0.771772 + 0.635899i \(0.780629\pi\)
\(840\) 0 0
\(841\) 376830. 0.532787
\(842\) 342257. 788507.i 0.482756 1.11220i
\(843\) 0 0
\(844\) −974314. + 910881.i −1.36777 + 1.27872i
\(845\) −43996.2 −0.0616172
\(846\) 0 0
\(847\) 132603.i 0.184836i
\(848\) −17466.2 + 259251.i −0.0242889 + 0.360520i
\(849\) 0 0
\(850\) −147221. + 339176.i −0.203767 + 0.469447i
\(851\) 463814.i 0.640449i
\(852\) 0 0
\(853\) 1.15283e6 1.58441 0.792206 0.610254i \(-0.208933\pi\)
0.792206 + 0.610254i \(0.208933\pi\)
\(854\) −463695. 201270.i −0.635794 0.275971i
\(855\) 0 0
\(856\) 455871. + 1.27958e6i 0.622149 + 1.74630i
\(857\) 576402. 0.784809 0.392404 0.919793i \(-0.371643\pi\)
0.392404 + 0.919793i \(0.371643\pi\)
\(858\) 0 0
\(859\) 365493.i 0.495327i −0.968846 0.247664i \(-0.920337\pi\)
0.968846 0.247664i \(-0.0796628\pi\)
\(860\) −763660. + 713942.i −1.03253 + 0.965308i
\(861\) 0 0
\(862\) −72966.3 31671.5i −0.0981992 0.0426240i
\(863\) 233229.i 0.313156i 0.987666 + 0.156578i \(0.0500463\pi\)
−0.987666 + 0.156578i \(0.949954\pi\)
\(864\) 0 0
\(865\) 1.49498e6 1.99804
\(866\) 165631. 381588.i 0.220854 0.508814i
\(867\) 0 0
\(868\) −424749. 454328.i −0.563759 0.603018i
\(869\) 1.40847e6 1.86513
\(870\) 0 0
\(871\) 387054.i 0.510194i
\(872\) 346816. 123559.i 0.456107 0.162496i
\(873\) 0 0
\(874\) −120229. + 276989.i −0.157393 + 0.362610i
\(875\) 370553.i 0.483987i
\(876\) 0 0
\(877\) −939899. −1.22203 −0.611015 0.791619i \(-0.709238\pi\)
−0.611015 + 0.791619i \(0.709238\pi\)
\(878\) −205318. 89119.8i −0.266341 0.115607i
\(879\) 0 0
\(880\) 1.02810e6 + 69264.9i 1.32761 + 0.0894434i
\(881\) 918880. 1.18388 0.591939 0.805983i \(-0.298363\pi\)
0.591939 + 0.805983i \(0.298363\pi\)
\(882\) 0 0
\(883\) 842123.i 1.08008i −0.841641 0.540038i \(-0.818410\pi\)
0.841641 0.540038i \(-0.181590\pi\)
\(884\) 684488. + 732155.i 0.875913 + 0.936911i
\(885\) 0 0
\(886\) −82952.3 36006.0i −0.105672 0.0458677i
\(887\) 255495.i 0.324740i −0.986730 0.162370i \(-0.948086\pi\)
0.986730 0.162370i \(-0.0519138\pi\)
\(888\) 0 0
\(889\) 249849. 0.316136
\(890\) −269822. + 621629.i −0.340642 + 0.784786i
\(891\) 0 0
\(892\) −155464. + 145342.i −0.195389 + 0.182668i
\(893\) 567982. 0.712249
\(894\) 0 0
\(895\) 131284.i 0.163895i
\(896\) −518937. + 146404.i −0.646396 + 0.182363i
\(897\) 0 0
\(898\) 123555. 284651.i 0.153217 0.352989i
\(899\) 1.22984e6i 1.52170i
\(900\) 0 0
\(901\) −386466. −0.476060
\(902\) −780331. 338708.i −0.959105 0.416306i
\(903\) 0 0
\(904\) 362855. 129273.i 0.444014 0.158187i
\(905\) −249293. −0.304378
\(906\) 0 0
\(907\) 1.42137e6i 1.72780i −0.503663 0.863900i \(-0.668014\pi\)
0.503663 0.863900i \(-0.331986\pi\)
\(908\) 585271. 547167.i 0.709880 0.663664i
\(909\) 0 0
\(910\) −585234. 254025.i −0.706719 0.306756i
\(911\) 42728.5i 0.0514851i −0.999669 0.0257425i \(-0.991805\pi\)
0.999669 0.0257425i \(-0.00819501\pi\)
\(912\) 0 0
\(913\) 1.68367e6 2.01983
\(914\) −386794. + 891115.i −0.463007 + 1.06670i
\(915\) 0 0
\(916\) −656045. 701731.i −0.781885 0.836334i
\(917\) 109737. 0.130502
\(918\) 0 0
\(919\) 32211.9i 0.0381404i −0.999818 0.0190702i \(-0.993929\pi\)
0.999818 0.0190702i \(-0.00607060\pi\)
\(920\) 108658. + 304992.i 0.128377 + 0.360340i
\(921\) 0 0
\(922\) −317960. + 732531.i −0.374034 + 0.861716i
\(923\) 145448.i 0.170728i
\(924\) 0 0
\(925\) 655678. 0.766315
\(926\) −232964. 101120.i −0.271686 0.117927i
\(927\) 0 0
\(928\) −947287. 489306.i −1.09998 0.568178i
\(929\) −1.39589e6 −1.61741 −0.808706 0.588214i \(-0.799831\pi\)
−0.808706 + 0.588214i \(0.799831\pi\)
\(930\) 0 0
\(931\) 579335.i 0.668391i
\(932\) 341485. + 365266.i 0.393134 + 0.420511i
\(933\) 0 0
\(934\) −414558. 179942.i −0.475217 0.206271i
\(935\) 1.53259e6i 1.75308i
\(936\) 0 0
\(937\) −525528. −0.598572 −0.299286 0.954163i \(-0.596749\pi\)
−0.299286 + 0.954163i \(0.596749\pi\)
\(938\) −123310. + 284087.i −0.140149 + 0.322883i
\(939\) 0 0
\(940\) 444874. 415911.i 0.503479 0.470700i
\(941\) −1.40325e6 −1.58473 −0.792364 0.610048i \(-0.791150\pi\)
−0.792364 + 0.610048i \(0.791150\pi\)
\(942\) 0 0
\(943\) 267287.i 0.300577i
\(944\) −621733. 41887.3i −0.697686 0.0470044i
\(945\) 0 0
\(946\) −482686. + 1.11203e6i −0.539365 + 1.24261i
\(947\) 1.40028e6i 1.56140i −0.624905 0.780701i \(-0.714862\pi\)
0.624905 0.780701i \(-0.285138\pi\)
\(948\) 0 0
\(949\) 1.13873e6 1.26441
\(950\) −391570. 169964.i −0.433873 0.188325i
\(951\) 0 0
\(952\) −269141. 755448.i −0.296965 0.833548i
\(953\) 537716. 0.592062 0.296031 0.955178i \(-0.404337\pi\)
0.296031 + 0.955178i \(0.404337\pi\)
\(954\) 0 0
\(955\) 521384.i 0.571677i
\(956\) −734963. + 687113.i −0.804174 + 0.751818i
\(957\) 0 0
\(958\) −1.15008e6 499200.i −1.25313 0.543931i
\(959\) 389116.i 0.423099i
\(960\) 0 0
\(961\) −471644. −0.510701
\(962\) 707683. 1.63039e6i 0.764696 1.76174i
\(963\) 0 0
\(964\) −706016. 755182.i −0.759732 0.812639i
\(965\) 322403. 0.346213
\(966\) 0 0
\(967\) 153678.i 0.164346i −0.996618 0.0821732i \(-0.973814\pi\)
0.996618 0.0821732i \(-0.0261861\pi\)
\(968\) 242918. 86543.6i 0.259244 0.0923601i
\(969\) 0 0
\(970\) −382837. + 881997.i −0.406884 + 0.937397i
\(971\) 36471.1i 0.0386821i −0.999813 0.0193411i \(-0.993843\pi\)
0.999813 0.0193411i \(-0.00615683\pi\)
\(972\) 0 0
\(973\) 182463. 0.192730
\(974\) 238320. + 103444.i 0.251214 + 0.109041i
\(975\) 0 0
\(976\) −66079.2 + 980813.i −0.0693689 + 1.02964i
\(977\) 1.18898e6 1.24562 0.622811 0.782373i \(-0.285991\pi\)
0.622811 + 0.782373i \(0.285991\pi\)
\(978\) 0 0
\(979\) 785825.i 0.819900i
\(980\) −424224. 453766.i −0.441716 0.472476i
\(981\) 0 0
\(982\) 1.31574e6 + 571106.i 1.36442 + 0.592235i
\(983\) 890220.i 0.921277i 0.887588 + 0.460639i \(0.152380\pi\)
−0.887588 + 0.460639i \(0.847620\pi\)
\(984\) 0 0
\(985\) 297565. 0.306697
\(986\) 631405. 1.45466e6i 0.649463 1.49626i
\(987\) 0 0
\(988\) −845256. + 790225.i −0.865913 + 0.809538i
\(989\) −380906. −0.389426
\(990\) 0 0
\(991\) 844659.i 0.860070i 0.902812 + 0.430035i \(0.141499\pi\)
−0.902812 + 0.430035i \(0.858501\pi\)
\(992\) −555081. + 1.07463e6i −0.564070 + 1.09203i
\(993\) 0 0
\(994\) −46337.5 + 106755.i −0.0468986 + 0.108047i
\(995\) 924826.i 0.934144i
\(996\) 0 0
\(997\) 1.30909e6 1.31698 0.658492 0.752587i \(-0.271194\pi\)
0.658492 + 0.752587i \(0.271194\pi\)
\(998\) 46586.0 + 20221.0i 0.0467729 + 0.0203021i
\(999\) 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 108.5.d.a.55.11 yes 16
3.2 odd 2 inner 108.5.d.a.55.6 yes 16
4.3 odd 2 inner 108.5.d.a.55.12 yes 16
12.11 even 2 inner 108.5.d.a.55.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.5.d.a.55.5 16 12.11 even 2 inner
108.5.d.a.55.6 yes 16 3.2 odd 2 inner
108.5.d.a.55.11 yes 16 1.1 even 1 trivial
108.5.d.a.55.12 yes 16 4.3 odd 2 inner