Properties

Label 1900.2.i.g.201.8
Level $1900$
Weight $2$
Character 1900.201
Analytic conductor $15.172$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,2,Mod(201,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, 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("1900.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.i (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 20 x^{18} + 261 x^{16} + 1994 x^{14} + 11074 x^{12} + 39211 x^{10} + 99376 x^{8} + 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 201.8
Root \(-1.00667 - 1.74361i\) of defining polynomial
Character \(\chi\) \(=\) 1900.201
Dual form 1900.2.i.g.501.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00667 - 1.74361i) q^{3} +1.34403 q^{7} +(-0.526784 - 0.912416i) q^{9} +5.25594 q^{11} +(1.21773 + 2.10918i) q^{13} +(-0.679914 + 1.17765i) q^{17} +(-2.89815 - 3.25587i) q^{19} +(1.35300 - 2.34346i) q^{21} +(4.07329 + 7.05514i) q^{23} +3.91884 q^{27} +(1.03597 + 1.79435i) q^{29} -0.513207 q^{31} +(5.29102 - 9.16431i) q^{33} +5.57175 q^{37} +4.90344 q^{39} +(2.70353 - 4.68265i) q^{41} +(-6.36221 + 11.0197i) q^{43} +(1.63266 + 2.82785i) q^{47} -5.19359 q^{49} +(1.36890 + 2.37101i) q^{51} +(-5.88276 - 10.1892i) q^{53} +(-8.59447 + 1.77564i) q^{57} +(-0.0175979 + 0.0304805i) q^{59} +(0.518372 + 0.897846i) q^{61} +(-0.708011 - 1.22631i) q^{63} +(0.383377 + 0.664028i) q^{67} +16.4019 q^{69} +(5.68450 - 9.84583i) q^{71} +(-1.07635 + 1.86429i) q^{73} +7.06413 q^{77} +(6.48576 - 11.2337i) q^{79} +(5.52535 - 9.57019i) q^{81} -4.20304 q^{83} +4.17153 q^{87} +(3.65426 + 6.32937i) q^{89} +(1.63667 + 2.83479i) q^{91} +(-0.516632 + 0.894833i) q^{93} +(-0.416683 + 0.721716i) q^{97} +(-2.76875 - 4.79561i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 1.00667 1.74361i 0.581203 1.00667i −0.414134 0.910216i \(-0.635915\pi\)
0.995337 0.0964577i \(-0.0307512\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.34403 0.507994 0.253997 0.967205i \(-0.418255\pi\)
0.253997 + 0.967205i \(0.418255\pi\)
\(8\) 0 0
\(9\) −0.526784 0.912416i −0.175595 0.304139i
\(10\) 0 0
\(11\) 5.25594 1.58473 0.792363 0.610050i \(-0.208851\pi\)
0.792363 + 0.610050i \(0.208851\pi\)
\(12\) 0 0
\(13\) 1.21773 + 2.10918i 0.337738 + 0.584980i 0.984007 0.178130i \(-0.0570048\pi\)
−0.646269 + 0.763110i \(0.723671\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.679914 + 1.17765i −0.164903 + 0.285621i −0.936621 0.350344i \(-0.886065\pi\)
0.771718 + 0.635965i \(0.219398\pi\)
\(18\) 0 0
\(19\) −2.89815 3.25587i −0.664882 0.746949i
\(20\) 0 0
\(21\) 1.35300 2.34346i 0.295248 0.511384i
\(22\) 0 0
\(23\) 4.07329 + 7.05514i 0.849339 + 1.47110i 0.881799 + 0.471625i \(0.156332\pi\)
−0.0324603 + 0.999473i \(0.510334\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.91884 0.754182
\(28\) 0 0
\(29\) 1.03597 + 1.79435i 0.192375 + 0.333203i 0.946037 0.324059i \(-0.105048\pi\)
−0.753662 + 0.657262i \(0.771714\pi\)
\(30\) 0 0
\(31\) −0.513207 −0.0921747 −0.0460873 0.998937i \(-0.514675\pi\)
−0.0460873 + 0.998937i \(0.514675\pi\)
\(32\) 0 0
\(33\) 5.29102 9.16431i 0.921048 1.59530i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.57175 0.915991 0.457995 0.888955i \(-0.348568\pi\)
0.457995 + 0.888955i \(0.348568\pi\)
\(38\) 0 0
\(39\) 4.90344 0.785179
\(40\) 0 0
\(41\) 2.70353 4.68265i 0.422220 0.731307i −0.573936 0.818900i \(-0.694584\pi\)
0.996156 + 0.0875933i \(0.0279176\pi\)
\(42\) 0 0
\(43\) −6.36221 + 11.0197i −0.970228 + 1.68048i −0.275369 + 0.961339i \(0.588800\pi\)
−0.694859 + 0.719146i \(0.744533\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.63266 + 2.82785i 0.238148 + 0.412485i 0.960183 0.279372i \(-0.0901262\pi\)
−0.722035 + 0.691857i \(0.756793\pi\)
\(48\) 0 0
\(49\) −5.19359 −0.741942
\(50\) 0 0
\(51\) 1.36890 + 2.37101i 0.191685 + 0.332008i
\(52\) 0 0
\(53\) −5.88276 10.1892i −0.808060 1.39960i −0.914206 0.405250i \(-0.867185\pi\)
0.106146 0.994351i \(-0.466149\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.59447 + 1.77564i −1.13837 + 0.235190i
\(58\) 0 0
\(59\) −0.0175979 + 0.0304805i −0.00229105 + 0.00396822i −0.867169 0.498015i \(-0.834063\pi\)
0.864878 + 0.501983i \(0.167396\pi\)
\(60\) 0 0
\(61\) 0.518372 + 0.897846i 0.0663707 + 0.114957i 0.897301 0.441419i \(-0.145525\pi\)
−0.830930 + 0.556376i \(0.812191\pi\)
\(62\) 0 0
\(63\) −0.708011 1.22631i −0.0892010 0.154501i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.383377 + 0.664028i 0.0468369 + 0.0811239i 0.888493 0.458889i \(-0.151753\pi\)
−0.841657 + 0.540013i \(0.818419\pi\)
\(68\) 0 0
\(69\) 16.4019 1.97455
\(70\) 0 0
\(71\) 5.68450 9.84583i 0.674625 1.16849i −0.301953 0.953323i \(-0.597638\pi\)
0.976578 0.215163i \(-0.0690282\pi\)
\(72\) 0 0
\(73\) −1.07635 + 1.86429i −0.125977 + 0.218199i −0.922115 0.386917i \(-0.873540\pi\)
0.796137 + 0.605116i \(0.206873\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.06413 0.805032
\(78\) 0 0
\(79\) 6.48576 11.2337i 0.729705 1.26389i −0.227302 0.973824i \(-0.572991\pi\)
0.957008 0.290062i \(-0.0936761\pi\)
\(80\) 0 0
\(81\) 5.52535 9.57019i 0.613928 1.06335i
\(82\) 0 0
\(83\) −4.20304 −0.461343 −0.230672 0.973032i \(-0.574092\pi\)
−0.230672 + 0.973032i \(0.574092\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.17153 0.447235
\(88\) 0 0
\(89\) 3.65426 + 6.32937i 0.387351 + 0.670912i 0.992092 0.125510i \(-0.0400568\pi\)
−0.604741 + 0.796422i \(0.706723\pi\)
\(90\) 0 0
\(91\) 1.63667 + 2.83479i 0.171569 + 0.297166i
\(92\) 0 0
\(93\) −0.516632 + 0.894833i −0.0535722 + 0.0927898i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.416683 + 0.721716i −0.0423077 + 0.0732791i −0.886404 0.462913i \(-0.846804\pi\)
0.844096 + 0.536192i \(0.180138\pi\)
\(98\) 0 0
\(99\) −2.76875 4.79561i −0.278269 0.481977i
\(100\) 0 0
\(101\) −7.40992 12.8344i −0.737315 1.27707i −0.953700 0.300759i \(-0.902760\pi\)
0.216385 0.976308i \(-0.430573\pi\)
\(102\) 0 0
\(103\) 9.40773 0.926971 0.463486 0.886104i \(-0.346599\pi\)
0.463486 + 0.886104i \(0.346599\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.8130 −1.23868 −0.619338 0.785124i \(-0.712599\pi\)
−0.619338 + 0.785124i \(0.712599\pi\)
\(108\) 0 0
\(109\) −0.996875 + 1.72664i −0.0954833 + 0.165382i −0.909810 0.415025i \(-0.863773\pi\)
0.814327 + 0.580406i \(0.197106\pi\)
\(110\) 0 0
\(111\) 5.60894 9.71497i 0.532377 0.922104i
\(112\) 0 0
\(113\) −8.34647 −0.785170 −0.392585 0.919716i \(-0.628419\pi\)
−0.392585 + 0.919716i \(0.628419\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.28296 2.22216i 0.118610 0.205439i
\(118\) 0 0
\(119\) −0.913823 + 1.58279i −0.0837700 + 0.145094i
\(120\) 0 0
\(121\) 16.6249 1.51136
\(122\) 0 0
\(123\) −5.44314 9.42780i −0.490791 0.850076i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −10.2907 17.8240i −0.913153 1.58163i −0.809583 0.587005i \(-0.800307\pi\)
−0.103569 0.994622i \(-0.533026\pi\)
\(128\) 0 0
\(129\) 12.8093 + 22.1864i 1.12780 + 1.95341i
\(130\) 0 0
\(131\) −5.36554 + 9.29339i −0.468790 + 0.811967i −0.999364 0.0356712i \(-0.988643\pi\)
0.530574 + 0.847639i \(0.321976\pi\)
\(132\) 0 0
\(133\) −3.89519 4.37598i −0.337756 0.379446i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.58931 14.8771i −0.733834 1.27104i −0.955233 0.295855i \(-0.904396\pi\)
0.221399 0.975183i \(-0.428938\pi\)
\(138\) 0 0
\(139\) −3.66394 6.34613i −0.310771 0.538272i 0.667758 0.744378i \(-0.267254\pi\)
−0.978530 + 0.206106i \(0.933921\pi\)
\(140\) 0 0
\(141\) 6.57423 0.553650
\(142\) 0 0
\(143\) 6.40033 + 11.0857i 0.535223 + 0.927033i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.22825 + 9.05560i −0.431219 + 0.746893i
\(148\) 0 0
\(149\) 6.12292 10.6052i 0.501609 0.868812i −0.498389 0.866953i \(-0.666075\pi\)
0.999998 0.00185904i \(-0.000591751\pi\)
\(150\) 0 0
\(151\) −11.5577 −0.940549 −0.470274 0.882520i \(-0.655845\pi\)
−0.470274 + 0.882520i \(0.655845\pi\)
\(152\) 0 0
\(153\) 1.43267 0.115825
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.06996 + 3.58528i −0.165201 + 0.286137i −0.936727 0.350062i \(-0.886161\pi\)
0.771526 + 0.636198i \(0.219494\pi\)
\(158\) 0 0
\(159\) −23.6881 −1.87859
\(160\) 0 0
\(161\) 5.47460 + 9.48229i 0.431459 + 0.747309i
\(162\) 0 0
\(163\) 9.41672 0.737575 0.368787 0.929514i \(-0.379773\pi\)
0.368787 + 0.929514i \(0.379773\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.83551 + 11.8395i 0.528948 + 0.916164i 0.999430 + 0.0337550i \(0.0107466\pi\)
−0.470482 + 0.882409i \(0.655920\pi\)
\(168\) 0 0
\(169\) 3.53425 6.12151i 0.271866 0.470885i
\(170\) 0 0
\(171\) −1.44401 + 4.35946i −0.110426 + 0.333376i
\(172\) 0 0
\(173\) −5.88891 + 10.1999i −0.447726 + 0.775484i −0.998238 0.0593437i \(-0.981099\pi\)
0.550512 + 0.834827i \(0.314433\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.0354307 + 0.0613678i 0.00266314 + 0.00461269i
\(178\) 0 0
\(179\) −16.5727 −1.23870 −0.619350 0.785115i \(-0.712604\pi\)
−0.619350 + 0.785115i \(0.712604\pi\)
\(180\) 0 0
\(181\) 7.19552 + 12.4630i 0.534839 + 0.926368i 0.999171 + 0.0407069i \(0.0129610\pi\)
−0.464332 + 0.885661i \(0.653706\pi\)
\(182\) 0 0
\(183\) 2.08733 0.154300
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.57359 + 6.18964i −0.261327 + 0.452631i
\(188\) 0 0
\(189\) 5.26703 0.383120
\(190\) 0 0
\(191\) −5.97170 −0.432097 −0.216049 0.976383i \(-0.569317\pi\)
−0.216049 + 0.976383i \(0.569317\pi\)
\(192\) 0 0
\(193\) 8.14331 14.1046i 0.586168 1.01527i −0.408560 0.912731i \(-0.633969\pi\)
0.994729 0.102542i \(-0.0326977\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.5233 −0.820998 −0.410499 0.911861i \(-0.634645\pi\)
−0.410499 + 0.911861i \(0.634645\pi\)
\(198\) 0 0
\(199\) −4.79943 8.31285i −0.340222 0.589283i 0.644251 0.764814i \(-0.277169\pi\)
−0.984474 + 0.175531i \(0.943836\pi\)
\(200\) 0 0
\(201\) 1.54374 0.108887
\(202\) 0 0
\(203\) 1.39237 + 2.41166i 0.0977253 + 0.169265i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.29148 7.43307i 0.298279 0.516634i
\(208\) 0 0
\(209\) −15.2325 17.1127i −1.05366 1.18371i
\(210\) 0 0
\(211\) 7.28207 12.6129i 0.501318 0.868308i −0.498681 0.866786i \(-0.666182\pi\)
0.999999 0.00152265i \(-0.000484675\pi\)
\(212\) 0 0
\(213\) −11.4449 19.8231i −0.784189 1.35826i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.689764 −0.0468242
\(218\) 0 0
\(219\) 2.16707 + 3.75347i 0.146437 + 0.253636i
\(220\) 0 0
\(221\) −3.31182 −0.222777
\(222\) 0 0
\(223\) −3.78635 + 6.55816i −0.253553 + 0.439167i −0.964501 0.264077i \(-0.914933\pi\)
0.710949 + 0.703244i \(0.248266\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.86640 −0.455739 −0.227869 0.973692i \(-0.573176\pi\)
−0.227869 + 0.973692i \(0.573176\pi\)
\(228\) 0 0
\(229\) 22.1011 1.46048 0.730240 0.683191i \(-0.239408\pi\)
0.730240 + 0.683191i \(0.239408\pi\)
\(230\) 0 0
\(231\) 7.11127 12.3171i 0.467887 0.810404i
\(232\) 0 0
\(233\) 1.48844 2.57806i 0.0975111 0.168894i −0.813143 0.582064i \(-0.802245\pi\)
0.910654 + 0.413170i \(0.135579\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −13.0581 22.6173i −0.848214 1.46915i
\(238\) 0 0
\(239\) −9.71289 −0.628275 −0.314137 0.949378i \(-0.601715\pi\)
−0.314137 + 0.949378i \(0.601715\pi\)
\(240\) 0 0
\(241\) 9.34287 + 16.1823i 0.601827 + 1.04239i 0.992544 + 0.121884i \(0.0388937\pi\)
−0.390717 + 0.920511i \(0.627773\pi\)
\(242\) 0 0
\(243\) −5.24618 9.08665i −0.336543 0.582909i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.33803 10.0775i 0.212394 0.641216i
\(248\) 0 0
\(249\) −4.23109 + 7.32846i −0.268134 + 0.464422i
\(250\) 0 0
\(251\) 2.10091 + 3.63888i 0.132608 + 0.229684i 0.924681 0.380742i \(-0.124332\pi\)
−0.792073 + 0.610426i \(0.790998\pi\)
\(252\) 0 0
\(253\) 21.4090 + 37.0814i 1.34597 + 2.33129i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.0842 + 24.3946i 0.878548 + 1.52169i 0.852934 + 0.522018i \(0.174821\pi\)
0.0256140 + 0.999672i \(0.491846\pi\)
\(258\) 0 0
\(259\) 7.48859 0.465318
\(260\) 0 0
\(261\) 1.09146 1.89047i 0.0675599 0.117017i
\(262\) 0 0
\(263\) 1.35425 2.34563i 0.0835065 0.144637i −0.821247 0.570572i \(-0.806721\pi\)
0.904754 + 0.425935i \(0.140055\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 14.7146 0.900519
\(268\) 0 0
\(269\) −5.64101 + 9.77052i −0.343938 + 0.595719i −0.985160 0.171637i \(-0.945094\pi\)
0.641222 + 0.767356i \(0.278428\pi\)
\(270\) 0 0
\(271\) −3.16690 + 5.48523i −0.192375 + 0.333204i −0.946037 0.324059i \(-0.894952\pi\)
0.753662 + 0.657263i \(0.228286\pi\)
\(272\) 0 0
\(273\) 6.59035 0.398866
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.1435 0.669549 0.334774 0.942298i \(-0.391340\pi\)
0.334774 + 0.942298i \(0.391340\pi\)
\(278\) 0 0
\(279\) 0.270349 + 0.468258i 0.0161854 + 0.0280339i
\(280\) 0 0
\(281\) −12.9061 22.3541i −0.769916 1.33353i −0.937608 0.347694i \(-0.886965\pi\)
0.167693 0.985839i \(-0.446368\pi\)
\(282\) 0 0
\(283\) −14.4304 + 24.9942i −0.857799 + 1.48575i 0.0162249 + 0.999868i \(0.494835\pi\)
−0.874024 + 0.485883i \(0.838498\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.63361 6.29360i 0.214485 0.371500i
\(288\) 0 0
\(289\) 7.57543 + 13.1210i 0.445614 + 0.771826i
\(290\) 0 0
\(291\) 0.838927 + 1.45306i 0.0491788 + 0.0851801i
\(292\) 0 0
\(293\) 22.7742 1.33048 0.665242 0.746628i \(-0.268328\pi\)
0.665242 + 0.746628i \(0.268328\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 20.5972 1.19517
\(298\) 0 0
\(299\) −9.92035 + 17.1825i −0.573709 + 0.993692i
\(300\) 0 0
\(301\) −8.55098 + 14.8107i −0.492870 + 0.853676i
\(302\) 0 0
\(303\) −29.8375 −1.71412
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −8.09880 + 14.0275i −0.462223 + 0.800593i −0.999071 0.0430854i \(-0.986281\pi\)
0.536849 + 0.843679i \(0.319615\pi\)
\(308\) 0 0
\(309\) 9.47052 16.4034i 0.538759 0.933158i
\(310\) 0 0
\(311\) −28.3483 −1.60749 −0.803743 0.594977i \(-0.797161\pi\)
−0.803743 + 0.594977i \(0.797161\pi\)
\(312\) 0 0
\(313\) −1.54464 2.67539i −0.0873081 0.151222i 0.819064 0.573702i \(-0.194493\pi\)
−0.906372 + 0.422480i \(0.861160\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0225 + 20.8236i 0.675251 + 1.16957i 0.976395 + 0.215991i \(0.0692982\pi\)
−0.301144 + 0.953579i \(0.597368\pi\)
\(318\) 0 0
\(319\) 5.44500 + 9.43101i 0.304861 + 0.528035i
\(320\) 0 0
\(321\) −12.8985 + 22.3408i −0.719923 + 1.24694i
\(322\) 0 0
\(323\) 5.80476 1.19928i 0.322986 0.0667298i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.00706 + 3.47632i 0.110990 + 0.192241i
\(328\) 0 0
\(329\) 2.19434 + 3.80071i 0.120978 + 0.209540i
\(330\) 0 0
\(331\) −20.7717 −1.14171 −0.570857 0.821049i \(-0.693389\pi\)
−0.570857 + 0.821049i \(0.693389\pi\)
\(332\) 0 0
\(333\) −2.93511 5.08376i −0.160843 0.278588i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.21324 2.10139i 0.0660892 0.114470i −0.831087 0.556142i \(-0.812281\pi\)
0.897177 + 0.441672i \(0.145614\pi\)
\(338\) 0 0
\(339\) −8.40217 + 14.5530i −0.456343 + 0.790410i
\(340\) 0 0
\(341\) −2.69739 −0.146072
\(342\) 0 0
\(343\) −16.3885 −0.884896
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.43664 + 2.48834i −0.0771230 + 0.133581i −0.902008 0.431720i \(-0.857907\pi\)
0.824885 + 0.565301i \(0.191240\pi\)
\(348\) 0 0
\(349\) −5.89385 −0.315490 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(350\) 0 0
\(351\) 4.77211 + 8.26553i 0.254716 + 0.441181i
\(352\) 0 0
\(353\) 12.0238 0.639962 0.319981 0.947424i \(-0.396324\pi\)
0.319981 + 0.947424i \(0.396324\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.83984 + 3.18670i 0.0973748 + 0.168658i
\(358\) 0 0
\(359\) 2.26590 3.92466i 0.119590 0.207136i −0.800015 0.599979i \(-0.795175\pi\)
0.919605 + 0.392844i \(0.128509\pi\)
\(360\) 0 0
\(361\) −2.20143 + 18.8720i −0.115865 + 0.993265i
\(362\) 0 0
\(363\) 16.7359 28.9874i 0.878406 1.52144i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 17.2710 + 29.9143i 0.901539 + 1.56151i 0.825496 + 0.564407i \(0.190895\pi\)
0.0760429 + 0.997105i \(0.475771\pi\)
\(368\) 0 0
\(369\) −5.69670 −0.296558
\(370\) 0 0
\(371\) −7.90659 13.6946i −0.410490 0.710989i
\(372\) 0 0
\(373\) −24.0801 −1.24682 −0.623411 0.781894i \(-0.714254\pi\)
−0.623411 + 0.781894i \(0.714254\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.52307 + 4.37008i −0.129945 + 0.225071i
\(378\) 0 0
\(379\) 30.1565 1.54904 0.774518 0.632552i \(-0.217992\pi\)
0.774518 + 0.632552i \(0.217992\pi\)
\(380\) 0 0
\(381\) −41.4375 −2.12291
\(382\) 0 0
\(383\) 19.4101 33.6192i 0.991809 1.71786i 0.385286 0.922797i \(-0.374103\pi\)
0.606522 0.795066i \(-0.292564\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 13.4060 0.681467
\(388\) 0 0
\(389\) −18.6935 32.3781i −0.947799 1.64164i −0.750048 0.661383i \(-0.769970\pi\)
−0.197750 0.980252i \(-0.563364\pi\)
\(390\) 0 0
\(391\) −11.0779 −0.560236
\(392\) 0 0
\(393\) 10.8027 + 18.7108i 0.544924 + 0.943836i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.78211 13.4790i 0.390573 0.676492i −0.601952 0.798532i \(-0.705610\pi\)
0.992525 + 0.122040i \(0.0389436\pi\)
\(398\) 0 0
\(399\) −11.5512 + 2.38651i −0.578283 + 0.119475i
\(400\) 0 0
\(401\) −11.3113 + 19.5918i −0.564860 + 0.978366i 0.432203 + 0.901777i \(0.357737\pi\)
−0.997063 + 0.0765898i \(0.975597\pi\)
\(402\) 0 0
\(403\) −0.624949 1.08244i −0.0311309 0.0539203i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 29.2848 1.45159
\(408\) 0 0
\(409\) 18.1239 + 31.3915i 0.896169 + 1.55221i 0.832351 + 0.554249i \(0.186995\pi\)
0.0638187 + 0.997962i \(0.479672\pi\)
\(410\) 0 0
\(411\) −34.5865 −1.70603
\(412\) 0 0
\(413\) −0.0236521 + 0.0409666i −0.00116384 + 0.00201583i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −14.7536 −0.722486
\(418\) 0 0
\(419\) −14.5598 −0.711293 −0.355647 0.934621i \(-0.615739\pi\)
−0.355647 + 0.934621i \(0.615739\pi\)
\(420\) 0 0
\(421\) −0.784161 + 1.35821i −0.0382177 + 0.0661950i −0.884502 0.466537i \(-0.845501\pi\)
0.846284 + 0.532732i \(0.178835\pi\)
\(422\) 0 0
\(423\) 1.72012 2.97934i 0.0836351 0.144860i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.696705 + 1.20673i 0.0337159 + 0.0583977i
\(428\) 0 0
\(429\) 25.7722 1.24429
\(430\) 0 0
\(431\) 12.7303 + 22.0495i 0.613197 + 1.06209i 0.990698 + 0.136080i \(0.0434503\pi\)
−0.377500 + 0.926009i \(0.623216\pi\)
\(432\) 0 0
\(433\) −8.44155 14.6212i −0.405675 0.702650i 0.588725 0.808334i \(-0.299630\pi\)
−0.994400 + 0.105684i \(0.966297\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11.1656 33.7090i 0.534125 1.61252i
\(438\) 0 0
\(439\) −9.93240 + 17.2034i −0.474048 + 0.821075i −0.999558 0.0297121i \(-0.990541\pi\)
0.525511 + 0.850787i \(0.323874\pi\)
\(440\) 0 0
\(441\) 2.73590 + 4.73872i 0.130281 + 0.225653i
\(442\) 0 0
\(443\) −2.57742 4.46422i −0.122457 0.212101i 0.798279 0.602288i \(-0.205744\pi\)
−0.920736 + 0.390186i \(0.872411\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −12.3276 21.3520i −0.583074 1.00991i
\(448\) 0 0
\(449\) −33.2207 −1.56778 −0.783892 0.620897i \(-0.786768\pi\)
−0.783892 + 0.620897i \(0.786768\pi\)
\(450\) 0 0
\(451\) 14.2096 24.6117i 0.669103 1.15892i
\(452\) 0 0
\(453\) −11.6348 + 20.1520i −0.546650 + 0.946826i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.7126 −0.547894 −0.273947 0.961745i \(-0.588329\pi\)
−0.273947 + 0.961745i \(0.588329\pi\)
\(458\) 0 0
\(459\) −2.66448 + 4.61501i −0.124367 + 0.215410i
\(460\) 0 0
\(461\) 3.68501 6.38263i 0.171628 0.297269i −0.767361 0.641215i \(-0.778431\pi\)
0.938989 + 0.343947i \(0.111764\pi\)
\(462\) 0 0
\(463\) 28.8020 1.33854 0.669271 0.743019i \(-0.266607\pi\)
0.669271 + 0.743019i \(0.266607\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −34.1251 −1.57912 −0.789561 0.613673i \(-0.789691\pi\)
−0.789561 + 0.613673i \(0.789691\pi\)
\(468\) 0 0
\(469\) 0.515268 + 0.892471i 0.0237929 + 0.0412105i
\(470\) 0 0
\(471\) 4.16756 + 7.21842i 0.192031 + 0.332607i
\(472\) 0 0
\(473\) −33.4394 + 57.9188i −1.53755 + 2.66311i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.19789 + 10.7351i −0.283782 + 0.491525i
\(478\) 0 0
\(479\) −14.4130 24.9640i −0.658546 1.14064i −0.980992 0.194048i \(-0.937838\pi\)
0.322446 0.946588i \(-0.395495\pi\)
\(480\) 0 0
\(481\) 6.78491 + 11.7518i 0.309365 + 0.535836i
\(482\) 0 0
\(483\) 22.0446 1.00306
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −16.5796 −0.751294 −0.375647 0.926763i \(-0.622579\pi\)
−0.375647 + 0.926763i \(0.622579\pi\)
\(488\) 0 0
\(489\) 9.47957 16.4191i 0.428681 0.742497i
\(490\) 0 0
\(491\) −4.94615 + 8.56698i −0.223217 + 0.386623i −0.955783 0.294073i \(-0.904989\pi\)
0.732566 + 0.680696i \(0.238322\pi\)
\(492\) 0 0
\(493\) −2.81748 −0.126893
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.64011 13.2331i 0.342706 0.593584i
\(498\) 0 0
\(499\) 15.4949 26.8380i 0.693649 1.20144i −0.276985 0.960874i \(-0.589335\pi\)
0.970634 0.240561i \(-0.0773315\pi\)
\(500\) 0 0
\(501\) 27.5245 1.22970
\(502\) 0 0
\(503\) −6.43203 11.1406i −0.286790 0.496735i 0.686252 0.727364i \(-0.259255\pi\)
−0.973042 + 0.230629i \(0.925922\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −7.11568 12.3247i −0.316018 0.547360i
\(508\) 0 0
\(509\) 7.35312 + 12.7360i 0.325921 + 0.564512i 0.981698 0.190442i \(-0.0609922\pi\)
−0.655777 + 0.754955i \(0.727659\pi\)
\(510\) 0 0
\(511\) −1.44664 + 2.50566i −0.0639957 + 0.110844i
\(512\) 0 0
\(513\) −11.3574 12.7593i −0.501442 0.563335i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.58118 + 14.8630i 0.377400 + 0.653676i
\(518\) 0 0
\(519\) 11.8564 + 20.5359i 0.520439 + 0.901427i
\(520\) 0 0
\(521\) 5.35528 0.234619 0.117310 0.993095i \(-0.462573\pi\)
0.117310 + 0.993095i \(0.462573\pi\)
\(522\) 0 0
\(523\) 7.98981 + 13.8388i 0.349370 + 0.605127i 0.986138 0.165929i \(-0.0530623\pi\)
−0.636768 + 0.771056i \(0.719729\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.348937 0.604376i 0.0151999 0.0263270i
\(528\) 0 0
\(529\) −21.6833 + 37.5566i −0.942753 + 1.63290i
\(530\) 0 0
\(531\) 0.0370812 0.00160919
\(532\) 0 0
\(533\) 13.1687 0.570400
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −16.6833 + 28.8963i −0.719937 + 1.24697i
\(538\) 0 0
\(539\) −27.2972 −1.17577
\(540\) 0 0
\(541\) −17.9500 31.0904i −0.771732 1.33668i −0.936613 0.350366i \(-0.886057\pi\)
0.164881 0.986313i \(-0.447276\pi\)
\(542\) 0 0
\(543\) 28.9742 1.24340
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −13.0754 22.6473i −0.559064 0.968327i −0.997575 0.0696011i \(-0.977827\pi\)
0.438511 0.898726i \(-0.355506\pi\)
\(548\) 0 0
\(549\) 0.546140 0.945942i 0.0233087 0.0403718i
\(550\) 0 0
\(551\) 2.83979 8.57329i 0.120979 0.365235i
\(552\) 0 0
\(553\) 8.71704 15.0983i 0.370686 0.642047i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.10412 3.64444i −0.0891543 0.154420i 0.818000 0.575219i \(-0.195083\pi\)
−0.907154 + 0.420799i \(0.861750\pi\)
\(558\) 0 0
\(559\) −30.9899 −1.31073
\(560\) 0 0
\(561\) 7.19488 + 12.4619i 0.303768 + 0.526142i
\(562\) 0 0
\(563\) −40.5225 −1.70782 −0.853909 0.520422i \(-0.825775\pi\)
−0.853909 + 0.520422i \(0.825775\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.42622 12.8626i 0.311872 0.540178i
\(568\) 0 0
\(569\) 23.9522 1.00413 0.502064 0.864831i \(-0.332574\pi\)
0.502064 + 0.864831i \(0.332574\pi\)
\(570\) 0 0
\(571\) −7.78949 −0.325980 −0.162990 0.986628i \(-0.552114\pi\)
−0.162990 + 0.986628i \(0.552114\pi\)
\(572\) 0 0
\(573\) −6.01155 + 10.4123i −0.251136 + 0.434981i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −34.1385 −1.42121 −0.710603 0.703593i \(-0.751578\pi\)
−0.710603 + 0.703593i \(0.751578\pi\)
\(578\) 0 0
\(579\) −16.3953 28.3975i −0.681366 1.18016i
\(580\) 0 0
\(581\) −5.64899 −0.234360
\(582\) 0 0
\(583\) −30.9195 53.5541i −1.28055 2.21798i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11.0700 + 19.1737i −0.456906 + 0.791384i −0.998796 0.0490654i \(-0.984376\pi\)
0.541890 + 0.840450i \(0.317709\pi\)
\(588\) 0 0
\(589\) 1.48735 + 1.67094i 0.0612853 + 0.0688498i
\(590\) 0 0
\(591\) −11.6002 + 20.0921i −0.477167 + 0.826477i
\(592\) 0 0
\(593\) 20.6767 + 35.8131i 0.849089 + 1.47067i 0.882022 + 0.471208i \(0.156182\pi\)
−0.0329325 + 0.999458i \(0.510485\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −19.3258 −0.790954
\(598\) 0 0
\(599\) −16.8243 29.1406i −0.687423 1.19065i −0.972669 0.232197i \(-0.925409\pi\)
0.285246 0.958454i \(-0.407925\pi\)
\(600\) 0 0
\(601\) 38.4939 1.57020 0.785100 0.619369i \(-0.212611\pi\)
0.785100 + 0.619369i \(0.212611\pi\)
\(602\) 0 0
\(603\) 0.403913 0.699598i 0.0164486 0.0284899i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 22.1827 0.900367 0.450183 0.892936i \(-0.351359\pi\)
0.450183 + 0.892936i \(0.351359\pi\)
\(608\) 0 0
\(609\) 5.60665 0.227193
\(610\) 0 0
\(611\) −3.97629 + 6.88714i −0.160864 + 0.278624i
\(612\) 0 0
\(613\) 2.38703 4.13445i 0.0964111 0.166989i −0.813786 0.581165i \(-0.802597\pi\)
0.910197 + 0.414176i \(0.135930\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.09317 14.0178i −0.325819 0.564335i 0.655859 0.754883i \(-0.272307\pi\)
−0.981678 + 0.190549i \(0.938973\pi\)
\(618\) 0 0
\(619\) 13.4892 0.542176 0.271088 0.962555i \(-0.412617\pi\)
0.271088 + 0.962555i \(0.412617\pi\)
\(620\) 0 0
\(621\) 15.9626 + 27.6480i 0.640556 + 1.10948i
\(622\) 0 0
\(623\) 4.91143 + 8.50684i 0.196772 + 0.340819i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −45.1720 + 9.33268i −1.80400 + 0.372711i
\(628\) 0 0
\(629\) −3.78832 + 6.56156i −0.151050 + 0.261626i
\(630\) 0 0
\(631\) 13.2207 + 22.8989i 0.526308 + 0.911592i 0.999530 + 0.0306488i \(0.00975735\pi\)
−0.473222 + 0.880943i \(0.656909\pi\)
\(632\) 0 0
\(633\) −14.6613 25.3942i −0.582735 1.00933i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.32441 10.9542i −0.250582 0.434021i
\(638\) 0 0
\(639\) −11.9780 −0.473842
\(640\) 0 0
\(641\) −4.27817 + 7.41000i −0.168977 + 0.292677i −0.938061 0.346471i \(-0.887380\pi\)
0.769083 + 0.639149i \(0.220713\pi\)
\(642\) 0 0
\(643\) −14.0112 + 24.2681i −0.552548 + 0.957042i 0.445541 + 0.895261i \(0.353011\pi\)
−0.998090 + 0.0617804i \(0.980322\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.57376 0.376383 0.188192 0.982132i \(-0.439737\pi\)
0.188192 + 0.982132i \(0.439737\pi\)
\(648\) 0 0
\(649\) −0.0924936 + 0.160204i −0.00363069 + 0.00628854i
\(650\) 0 0
\(651\) −0.694367 + 1.20268i −0.0272144 + 0.0471367i
\(652\) 0 0
\(653\) −16.4168 −0.642439 −0.321219 0.947005i \(-0.604093\pi\)
−0.321219 + 0.947005i \(0.604093\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.26801 0.0884837
\(658\) 0 0
\(659\) −7.08162 12.2657i −0.275861 0.477805i 0.694491 0.719501i \(-0.255629\pi\)
−0.970352 + 0.241697i \(0.922296\pi\)
\(660\) 0 0
\(661\) −18.5170 32.0724i −0.720229 1.24747i −0.960908 0.276868i \(-0.910704\pi\)
0.240679 0.970605i \(-0.422630\pi\)
\(662\) 0 0
\(663\) −3.33392 + 5.77452i −0.129479 + 0.224264i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.43960 + 14.6178i −0.326783 + 0.566004i
\(668\) 0 0
\(669\) 7.62324 + 13.2038i 0.294732 + 0.510490i
\(670\) 0 0
\(671\) 2.72453 + 4.71903i 0.105179 + 0.182176i
\(672\) 0 0
\(673\) 42.3293 1.63167 0.815837 0.578282i \(-0.196277\pi\)
0.815837 + 0.578282i \(0.196277\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.9856 0.537510 0.268755 0.963209i \(-0.413388\pi\)
0.268755 + 0.963209i \(0.413388\pi\)
\(678\) 0 0
\(679\) −0.560033 + 0.970005i −0.0214921 + 0.0372254i
\(680\) 0 0
\(681\) −6.91222 + 11.9723i −0.264877 + 0.458780i
\(682\) 0 0
\(683\) −11.6668 −0.446416 −0.223208 0.974771i \(-0.571653\pi\)
−0.223208 + 0.974771i \(0.571653\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 22.2486 38.5356i 0.848835 1.47023i
\(688\) 0 0
\(689\) 14.3273 24.8156i 0.545825 0.945397i
\(690\) 0 0
\(691\) −15.7886 −0.600627 −0.300313 0.953841i \(-0.597091\pi\)
−0.300313 + 0.953841i \(0.597091\pi\)
\(692\) 0 0
\(693\) −3.72127 6.44542i −0.141359 0.244841i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.67633 + 6.36760i 0.139251 + 0.241190i
\(698\) 0 0
\(699\) −2.99675 5.19053i −0.113348 0.196324i
\(700\) 0 0
\(701\) 2.64450 4.58042i 0.0998816 0.173000i −0.811754 0.584000i \(-0.801487\pi\)
0.911635 + 0.411000i \(0.134820\pi\)
\(702\) 0 0
\(703\) −16.1478 18.1409i −0.609025 0.684198i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.95913 17.2497i −0.374552 0.648743i
\(708\) 0 0
\(709\) 12.2529 + 21.2226i 0.460166 + 0.797031i 0.998969 0.0454011i \(-0.0144566\pi\)
−0.538803 + 0.842432i \(0.681123\pi\)
\(710\) 0 0
\(711\) −13.6664 −0.512529
\(712\) 0 0
\(713\) −2.09044 3.62075i −0.0782875 0.135598i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −9.77771 + 16.9355i −0.365155 + 0.632467i
\(718\) 0 0
\(719\) −22.4239 + 38.8393i −0.836269 + 1.44846i 0.0567236 + 0.998390i \(0.481935\pi\)
−0.892993 + 0.450071i \(0.851399\pi\)
\(720\) 0 0
\(721\) 12.6442 0.470896
\(722\) 0 0
\(723\) 37.6209 1.39914
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 16.2453 28.1376i 0.602504 1.04357i −0.389937 0.920842i \(-0.627503\pi\)
0.992441 0.122726i \(-0.0391635\pi\)
\(728\) 0 0
\(729\) 12.0273 0.445457
\(730\) 0 0
\(731\) −8.65152 14.9849i −0.319988 0.554235i
\(732\) 0 0
\(733\) 11.1969 0.413568 0.206784 0.978387i \(-0.433700\pi\)
0.206784 + 0.978387i \(0.433700\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.01501 + 3.49009i 0.0742237 + 0.128559i
\(738\) 0 0
\(739\) −0.466361 + 0.807761i −0.0171554 + 0.0297140i −0.874476 0.485069i \(-0.838794\pi\)
0.857320 + 0.514783i \(0.172128\pi\)
\(740\) 0 0
\(741\) −14.2109 15.9650i −0.522051 0.586488i
\(742\) 0 0
\(743\) −13.4736 + 23.3370i −0.494300 + 0.856153i −0.999978 0.00656939i \(-0.997909\pi\)
0.505678 + 0.862722i \(0.331242\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.21409 + 3.83492i 0.0810094 + 0.140312i
\(748\) 0 0
\(749\) −17.2210 −0.629240
\(750\) 0 0
\(751\) 2.33645 + 4.04686i 0.0852584 + 0.147672i 0.905501 0.424343i \(-0.139495\pi\)
−0.820243 + 0.572015i \(0.806162\pi\)
\(752\) 0 0
\(753\) 8.45971 0.308289
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −21.4140 + 37.0902i −0.778306 + 1.34807i 0.154612 + 0.987975i \(0.450587\pi\)
−0.932918 + 0.360090i \(0.882746\pi\)
\(758\) 0 0
\(759\) 86.2073 3.12913
\(760\) 0 0
\(761\) −16.7169 −0.605987 −0.302994 0.952993i \(-0.597986\pi\)
−0.302994 + 0.952993i \(0.597986\pi\)
\(762\) 0 0
\(763\) −1.33983 + 2.32065i −0.0485050 + 0.0840131i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.0857182 −0.00309511
\(768\) 0 0
\(769\) 7.70852 + 13.3516i 0.277976 + 0.481469i 0.970882 0.239559i \(-0.0770029\pi\)
−0.692905 + 0.721029i \(0.743670\pi\)
\(770\) 0 0
\(771\) 56.7128 2.04246
\(772\) 0 0
\(773\) −16.5897 28.7343i −0.596691 1.03350i −0.993306 0.115514i \(-0.963148\pi\)
0.396615 0.917985i \(-0.370185\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.53856 13.0572i 0.270444 0.468423i
\(778\) 0 0
\(779\) −23.0813 + 4.76868i −0.826975 + 0.170856i
\(780\) 0 0
\(781\) 29.8774 51.7491i 1.06910 1.85173i
\(782\) 0 0
\(783\) 4.05980 + 7.03179i 0.145086 + 0.251296i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.8318 0.457405 0.228702 0.973496i \(-0.426552\pi\)
0.228702 + 0.973496i \(0.426552\pi\)
\(788\) 0 0
\(789\) −2.72657 4.72256i −0.0970685 0.168128i
\(790\) 0 0
\(791\) −11.2179 −0.398862
\(792\) 0 0
\(793\) −1.26248 + 2.18667i −0.0448319 + 0.0776511i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.28485 0.0809336 0.0404668 0.999181i \(-0.487115\pi\)
0.0404668 + 0.999181i \(0.487115\pi\)
\(798\) 0 0
\(799\) −4.44028 −0.157086
\(800\) 0 0
\(801\) 3.85001 6.66842i 0.136034 0.235617i
\(802\) 0 0
\(803\) −5.65723 + 9.79861i −0.199639 + 0.345786i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 11.3573 + 19.6714i 0.399796 + 0.692468i
\(808\) 0 0
\(809\) 17.3304 0.609305 0.304652 0.952464i \(-0.401460\pi\)
0.304652 + 0.952464i \(0.401460\pi\)
\(810\) 0 0
\(811\) −24.7926 42.9420i −0.870586 1.50790i −0.861392 0.507941i \(-0.830407\pi\)
−0.00919378 0.999958i \(-0.502927\pi\)
\(812\) 0 0
\(813\) 6.37606 + 11.0437i 0.223618 + 0.387318i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 54.3173 11.2221i 1.90032 0.392612i
\(818\) 0 0
\(819\) 1.72434 2.98664i 0.0602532 0.104362i
\(820\) 0 0
\(821\) −11.1029 19.2308i −0.387495 0.671160i 0.604617 0.796516i \(-0.293326\pi\)
−0.992112 + 0.125356i \(0.959993\pi\)
\(822\) 0 0
\(823\) 9.56519 + 16.5674i 0.333422 + 0.577503i 0.983180 0.182637i \(-0.0584634\pi\)
−0.649759 + 0.760141i \(0.725130\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.52461 + 13.0330i 0.261656 + 0.453202i 0.966682 0.255980i \(-0.0823980\pi\)
−0.705026 + 0.709182i \(0.749065\pi\)
\(828\) 0 0
\(829\) 3.62995 0.126074 0.0630368 0.998011i \(-0.479921\pi\)
0.0630368 + 0.998011i \(0.479921\pi\)
\(830\) 0 0
\(831\) 11.2179 19.4299i 0.389144 0.674017i
\(832\) 0 0
\(833\) 3.53120 6.11621i 0.122349 0.211914i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.01118 −0.0695165
\(838\) 0 0
\(839\) 9.47453 16.4104i 0.327097 0.566549i −0.654838 0.755770i \(-0.727263\pi\)
0.981935 + 0.189221i \(0.0605963\pi\)
\(840\) 0 0
\(841\) 12.3535 21.3969i 0.425984 0.737826i
\(842\) 0 0
\(843\) −51.9691 −1.78991
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 22.3443 0.767761
\(848\) 0 0
\(849\) 29.0534 + 50.3220i 0.997111 + 1.72705i
\(850\) 0 0
\(851\) 22.6954 + 39.3095i 0.777987 + 1.34751i
\(852\) 0 0
\(853\) 21.3785 37.0287i 0.731987 1.26784i −0.224045 0.974579i \(-0.571926\pi\)
0.956033 0.293260i \(-0.0947403\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.57024 2.71973i 0.0536383 0.0929043i −0.837960 0.545732i \(-0.816252\pi\)
0.891598 + 0.452828i \(0.149585\pi\)
\(858\) 0 0
\(859\) −3.53437 6.12170i −0.120591 0.208870i 0.799410 0.600786i \(-0.205146\pi\)
−0.920001 + 0.391916i \(0.871812\pi\)
\(860\) 0 0
\(861\) −7.31572 12.6712i −0.249319 0.431834i
\(862\) 0 0
\(863\) 0.464328 0.0158059 0.00790296 0.999969i \(-0.497484\pi\)
0.00790296 + 0.999969i \(0.497484\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 30.5040 1.03597
\(868\) 0 0
\(869\) 34.0888 59.0435i 1.15638 2.00291i
\(870\) 0 0
\(871\) −0.933701 + 1.61722i −0.0316373 + 0.0547973i
\(872\) 0 0
\(873\) 0.878007 0.0297160
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.463552 + 0.802896i −0.0156530 + 0.0271119i −0.873746 0.486383i \(-0.838316\pi\)
0.858093 + 0.513495i \(0.171649\pi\)
\(878\) 0 0
\(879\) 22.9262 39.7094i 0.773282 1.33936i
\(880\) 0 0
\(881\) 4.50850 0.151895 0.0759477 0.997112i \(-0.475802\pi\)
0.0759477 + 0.997112i \(0.475802\pi\)
\(882\) 0 0
\(883\) −17.1245 29.6605i −0.576286 0.998156i −0.995901 0.0904542i \(-0.971168\pi\)
0.419615 0.907702i \(-0.362165\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −14.2791 24.7321i −0.479445 0.830423i 0.520277 0.853997i \(-0.325829\pi\)
−0.999722 + 0.0235747i \(0.992495\pi\)
\(888\) 0 0
\(889\) −13.8310 23.9560i −0.463876 0.803457i
\(890\) 0 0
\(891\) 29.0409 50.3003i 0.972907 1.68512i
\(892\) 0 0
\(893\) 4.47544 13.5113i 0.149765 0.452138i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 19.9731 + 34.5944i 0.666883 + 1.15507i
\(898\) 0 0
\(899\) −0.531667 0.920874i −0.0177321 0.0307129i
\(900\) 0 0
\(901\) 15.9991 0.533007
\(902\) 0 0
\(903\) 17.2161 + 29.8191i 0.572916 + 0.992319i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −7.85765 + 13.6098i −0.260909 + 0.451907i −0.966484 0.256728i \(-0.917356\pi\)
0.705575 + 0.708635i \(0.250689\pi\)
\(908\) 0 0
\(909\) −7.80686 + 13.5219i −0.258937 + 0.448492i
\(910\) 0 0
\(911\) 20.7125 0.686237 0.343119 0.939292i \(-0.388517\pi\)
0.343119 + 0.939292i \(0.388517\pi\)
\(912\) 0 0
\(913\) −22.0909 −0.731103
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.21143 + 12.4906i −0.238142 + 0.412475i
\(918\) 0 0
\(919\) 30.6628 1.01147 0.505737 0.862688i \(-0.331221\pi\)
0.505737 + 0.862688i \(0.331221\pi\)
\(920\) 0 0
\(921\) 16.3057 + 28.2423i 0.537291 + 0.930615i
\(922\) 0 0
\(923\) 27.6888 0.911388
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.95584 8.58377i −0.162771 0.281928i
\(928\) 0 0
\(929\) 7.65011 13.2504i 0.250992 0.434731i −0.712807 0.701360i \(-0.752577\pi\)
0.963799 + 0.266629i \(0.0859099\pi\)
\(930\) 0 0
\(931\) 15.0518 + 16.9097i 0.493303 + 0.554193i
\(932\) 0 0
\(933\) −28.5375 + 49.4284i −0.934276 + 1.61821i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.21623 + 3.83862i 0.0724011 + 0.125402i 0.899953 0.435987i \(-0.143601\pi\)
−0.827552 + 0.561389i \(0.810267\pi\)
\(938\) 0 0
\(939\) −6.21978 −0.202975
\(940\) 0 0
\(941\) −17.3400 30.0338i −0.565268 0.979073i −0.997025 0.0770825i \(-0.975440\pi\)
0.431757 0.901990i \(-0.357894\pi\)
\(942\) 0 0
\(943\) 44.0490 1.43443
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.603945 1.04606i 0.0196256 0.0339925i −0.856046 0.516900i \(-0.827086\pi\)
0.875671 + 0.482907i \(0.160419\pi\)
\(948\) 0 0
\(949\) −5.24283 −0.170189
\(950\) 0 0
\(951\) 48.4110 1.56983
\(952\) 0 0
\(953\) 29.6627 51.3773i 0.960868 1.66427i 0.240540 0.970639i \(-0.422676\pi\)
0.720329 0.693633i \(-0.243991\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 21.9253 0.708746
\(958\) 0 0
\(959\) −11.5443 19.9952i −0.372784 0.645680i
\(960\) 0 0
\(961\) −30.7366 −0.991504
\(962\) 0 0
\(963\) 6.74966 + 11.6908i 0.217505 + 0.376729i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −22.7736 + 39.4450i −0.732350 + 1.26847i 0.223527 + 0.974698i \(0.428243\pi\)
−0.955876 + 0.293769i \(0.905090\pi\)
\(968\) 0 0
\(969\) 3.75242 11.3285i 0.120545 0.363925i
\(970\) 0 0
\(971\) −0.738715 + 1.27949i −0.0237065 + 0.0410608i −0.877635 0.479329i \(-0.840880\pi\)
0.853929 + 0.520390i \(0.174213\pi\)
\(972\) 0 0
\(973\) −4.92443 8.52937i −0.157870 0.273439i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.8443 1.49868 0.749341 0.662184i \(-0.230370\pi\)
0.749341 + 0.662184i \(0.230370\pi\)
\(978\) 0 0
\(979\) 19.2066 + 33.2668i 0.613846 + 1.06321i
\(980\) 0 0
\(981\) 2.10055 0.0670654
\(982\) 0 0
\(983\) 10.0466 17.4012i 0.320436 0.555011i −0.660142 0.751141i \(-0.729504\pi\)
0.980578 + 0.196130i \(0.0628373\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8.83595 0.281251
\(988\) 0 0
\(989\) −103.660 −3.29621
\(990\) 0 0
\(991\) −8.95274 + 15.5066i −0.284393 + 0.492583i −0.972462 0.233062i \(-0.925125\pi\)
0.688069 + 0.725646i \(0.258459\pi\)
\(992\) 0 0
\(993\) −20.9103 + 36.2177i −0.663568 + 1.14933i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.989838 + 1.71445i 0.0313485 + 0.0542972i 0.881274 0.472606i \(-0.156686\pi\)
−0.849926 + 0.526903i \(0.823353\pi\)
\(998\) 0 0
\(999\) 21.8348 0.690824
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 1900.2.i.g.201.8 20
5.2 odd 4 380.2.r.a.49.3 20
5.3 odd 4 380.2.r.a.49.8 yes 20
5.4 even 2 inner 1900.2.i.g.201.3 20
15.2 even 4 3420.2.bj.c.1189.8 20
15.8 even 4 3420.2.bj.c.1189.6 20
19.7 even 3 inner 1900.2.i.g.501.8 20
95.7 odd 12 380.2.r.a.349.8 yes 20
95.64 even 6 inner 1900.2.i.g.501.3 20
95.83 odd 12 380.2.r.a.349.3 yes 20
285.83 even 12 3420.2.bj.c.2629.8 20
285.197 even 12 3420.2.bj.c.2629.6 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.3 20 5.2 odd 4
380.2.r.a.49.8 yes 20 5.3 odd 4
380.2.r.a.349.3 yes 20 95.83 odd 12
380.2.r.a.349.8 yes 20 95.7 odd 12
1900.2.i.g.201.3 20 5.4 even 2 inner
1900.2.i.g.201.8 20 1.1 even 1 trivial
1900.2.i.g.501.3 20 95.64 even 6 inner
1900.2.i.g.501.8 20 19.7 even 3 inner
3420.2.bj.c.1189.6 20 15.8 even 4
3420.2.bj.c.1189.8 20 15.2 even 4
3420.2.bj.c.2629.6 20 285.197 even 12
3420.2.bj.c.2629.8 20 285.83 even 12