Properties

Label 400.2.bi.d.303.2
Level $400$
Weight $2$
Character 400.303
Analytic conductor $3.194$
Analytic rank $0$
Dimension $80$
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] = [400,2,Mod(47,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.bi (of order \(20\), degree \(8\), 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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 303.2
Character \(\chi\) \(=\) 400.303
Dual form 400.2.bi.d.367.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.49574 + 0.395286i) q^{3} +(-0.616106 + 2.14951i) q^{5} +(-1.08050 - 1.08050i) q^{7} +(3.21930 - 1.04601i) q^{9} +O(q^{10})\) \(q+(-2.49574 + 0.395286i) q^{3} +(-0.616106 + 2.14951i) q^{5} +(-1.08050 - 1.08050i) q^{7} +(3.21930 - 1.04601i) q^{9} +(-0.665530 - 0.216244i) q^{11} +(-0.606168 + 1.18967i) q^{13} +(0.687965 - 5.60817i) q^{15} +(0.880206 - 5.55740i) q^{17} +(1.31160 - 0.952937i) q^{19} +(3.12376 + 2.26954i) q^{21} +(-3.79350 - 7.44517i) q^{23} +(-4.24083 - 2.64866i) q^{25} +(-0.866739 + 0.441626i) q^{27} +(4.72188 - 6.49912i) q^{29} +(2.81093 + 3.86891i) q^{31} +(1.74647 + 0.276613i) q^{33} +(2.98826 - 1.65685i) q^{35} +(-9.91663 - 5.05278i) q^{37} +(1.04258 - 3.20872i) q^{39} +(-0.0489618 - 0.150689i) q^{41} +(-5.11533 + 5.11533i) q^{43} +(0.264994 + 7.56439i) q^{45} +(1.04855 + 6.62028i) q^{47} -4.66503i q^{49} +14.2178i q^{51} +(0.570597 + 3.60261i) q^{53} +(0.874856 - 1.29734i) q^{55} +(-2.89674 + 2.89674i) q^{57} +(-4.70323 - 14.4751i) q^{59} +(-2.50953 + 7.72352i) q^{61} +(-4.60867 - 2.34824i) q^{63} +(-2.18375 - 2.03593i) q^{65} +(6.12258 + 0.969721i) q^{67} +(12.4106 + 17.0817i) q^{69} +(0.603305 - 0.830378i) q^{71} +(4.39246 - 2.23807i) q^{73} +(11.6310 + 4.93402i) q^{75} +(0.485454 + 0.952758i) q^{77} +(8.82257 + 6.40997i) q^{79} +(-6.22691 + 4.52411i) q^{81} +(-0.609010 + 3.84514i) q^{83} +(11.4034 + 5.31596i) q^{85} +(-9.21559 + 18.0866i) q^{87} +(-8.25547 - 2.68237i) q^{89} +(1.94041 - 0.630476i) q^{91} +(-8.54468 - 8.54468i) q^{93} +(1.24026 + 3.40642i) q^{95} +(3.75883 - 0.595340i) q^{97} -2.36873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 80 q + 4 q^{5} - 4 q^{13} - 24 q^{17} - 48 q^{25} - 40 q^{29} - 64 q^{33} - 20 q^{37} - 24 q^{45} + 28 q^{53} + 48 q^{57} + 112 q^{65} + 140 q^{69} + 108 q^{73} + 136 q^{77} - 20 q^{81} - 24 q^{85} + 80 q^{89} - 116 q^{93} - 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{20}\right)\) \(-1\)

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
\(3\) −2.49574 + 0.395286i −1.44092 + 0.228219i −0.827467 0.561515i \(-0.810219\pi\)
−0.613450 + 0.789734i \(0.710219\pi\)
\(4\) 0 0
\(5\) −0.616106 + 2.14951i −0.275531 + 0.961292i
\(6\) 0 0
\(7\) −1.08050 1.08050i −0.408391 0.408391i 0.472786 0.881177i \(-0.343248\pi\)
−0.881177 + 0.472786i \(0.843248\pi\)
\(8\) 0 0
\(9\) 3.21930 1.04601i 1.07310 0.348671i
\(10\) 0 0
\(11\) −0.665530 0.216244i −0.200665 0.0652000i 0.206960 0.978349i \(-0.433643\pi\)
−0.407625 + 0.913149i \(0.633643\pi\)
\(12\) 0 0
\(13\) −0.606168 + 1.18967i −0.168121 + 0.329955i −0.959660 0.281163i \(-0.909280\pi\)
0.791539 + 0.611118i \(0.209280\pi\)
\(14\) 0 0
\(15\) 0.687965 5.60817i 0.177632 1.44802i
\(16\) 0 0
\(17\) 0.880206 5.55740i 0.213481 1.34787i −0.615299 0.788294i \(-0.710965\pi\)
0.828780 0.559574i \(-0.189035\pi\)
\(18\) 0 0
\(19\) 1.31160 0.952937i 0.300903 0.218619i −0.427080 0.904214i \(-0.640458\pi\)
0.727983 + 0.685595i \(0.240458\pi\)
\(20\) 0 0
\(21\) 3.12376 + 2.26954i 0.681660 + 0.495255i
\(22\) 0 0
\(23\) −3.79350 7.44517i −0.791000 1.55243i −0.832978 0.553307i \(-0.813366\pi\)
0.0419773 0.999119i \(-0.486634\pi\)
\(24\) 0 0
\(25\) −4.24083 2.64866i −0.848166 0.529731i
\(26\) 0 0
\(27\) −0.866739 + 0.441626i −0.166804 + 0.0849909i
\(28\) 0 0
\(29\) 4.72188 6.49912i 0.876832 1.20686i −0.100456 0.994941i \(-0.532030\pi\)
0.977288 0.211914i \(-0.0679698\pi\)
\(30\) 0 0
\(31\) 2.81093 + 3.86891i 0.504858 + 0.694878i 0.983042 0.183382i \(-0.0587046\pi\)
−0.478184 + 0.878260i \(0.658705\pi\)
\(32\) 0 0
\(33\) 1.74647 + 0.276613i 0.304021 + 0.0481522i
\(34\) 0 0
\(35\) 2.98826 1.65685i 0.505107 0.280059i
\(36\) 0 0
\(37\) −9.91663 5.05278i −1.63028 0.830672i −0.998455 0.0555642i \(-0.982304\pi\)
−0.631830 0.775107i \(-0.717696\pi\)
\(38\) 0 0
\(39\) 1.04258 3.20872i 0.166946 0.513807i
\(40\) 0 0
\(41\) −0.0489618 0.150689i −0.00764656 0.0235337i 0.947160 0.320760i \(-0.103938\pi\)
−0.954807 + 0.297227i \(0.903938\pi\)
\(42\) 0 0
\(43\) −5.11533 + 5.11533i −0.780080 + 0.780080i −0.979844 0.199764i \(-0.935983\pi\)
0.199764 + 0.979844i \(0.435983\pi\)
\(44\) 0 0
\(45\) 0.264994 + 7.56439i 0.0395030 + 1.12763i
\(46\) 0 0
\(47\) 1.04855 + 6.62028i 0.152947 + 0.965667i 0.938100 + 0.346366i \(0.112584\pi\)
−0.785153 + 0.619302i \(0.787416\pi\)
\(48\) 0 0
\(49\) 4.66503i 0.666434i
\(50\) 0 0
\(51\) 14.2178i 1.99089i
\(52\) 0 0
\(53\) 0.570597 + 3.60261i 0.0783775 + 0.494856i 0.995383 + 0.0959818i \(0.0305991\pi\)
−0.917006 + 0.398874i \(0.869401\pi\)
\(54\) 0 0
\(55\) 0.874856 1.29734i 0.117966 0.174933i
\(56\) 0 0
\(57\) −2.89674 + 2.89674i −0.383683 + 0.383683i
\(58\) 0 0
\(59\) −4.70323 14.4751i −0.612308 1.88449i −0.435312 0.900280i \(-0.643362\pi\)
−0.176996 0.984211i \(-0.556638\pi\)
\(60\) 0 0
\(61\) −2.50953 + 7.72352i −0.321312 + 0.988896i 0.651766 + 0.758420i \(0.274028\pi\)
−0.973078 + 0.230476i \(0.925972\pi\)
\(62\) 0 0
\(63\) −4.60867 2.34824i −0.580638 0.295850i
\(64\) 0 0
\(65\) −2.18375 2.03593i −0.270861 0.252526i
\(66\) 0 0
\(67\) 6.12258 + 0.969721i 0.747992 + 0.118470i 0.518782 0.854907i \(-0.326386\pi\)
0.229210 + 0.973377i \(0.426386\pi\)
\(68\) 0 0
\(69\) 12.4106 + 17.0817i 1.49406 + 2.05639i
\(70\) 0 0
\(71\) 0.603305 0.830378i 0.0715991 0.0985477i −0.771716 0.635968i \(-0.780601\pi\)
0.843315 + 0.537420i \(0.180601\pi\)
\(72\) 0 0
\(73\) 4.39246 2.23807i 0.514098 0.261946i −0.177639 0.984096i \(-0.556846\pi\)
0.691737 + 0.722150i \(0.256846\pi\)
\(74\) 0 0
\(75\) 11.6310 + 4.93402i 1.34303 + 0.569731i
\(76\) 0 0
\(77\) 0.485454 + 0.952758i 0.0553226 + 0.108577i
\(78\) 0 0
\(79\) 8.82257 + 6.40997i 0.992617 + 0.721178i 0.960493 0.278305i \(-0.0897727\pi\)
0.0321244 + 0.999484i \(0.489773\pi\)
\(80\) 0 0
\(81\) −6.22691 + 4.52411i −0.691879 + 0.502679i
\(82\) 0 0
\(83\) −0.609010 + 3.84514i −0.0668476 + 0.422059i 0.931457 + 0.363852i \(0.118538\pi\)
−0.998305 + 0.0582073i \(0.981462\pi\)
\(84\) 0 0
\(85\) 11.4034 + 5.31596i 1.23687 + 0.576597i
\(86\) 0 0
\(87\) −9.21559 + 18.0866i −0.988015 + 1.93909i
\(88\) 0 0
\(89\) −8.25547 2.68237i −0.875078 0.284330i −0.163166 0.986599i \(-0.552171\pi\)
−0.711912 + 0.702268i \(0.752171\pi\)
\(90\) 0 0
\(91\) 1.94041 0.630476i 0.203410 0.0660919i
\(92\) 0 0
\(93\) −8.54468 8.54468i −0.886042 0.886042i
\(94\) 0 0
\(95\) 1.24026 + 3.40642i 0.127248 + 0.349492i
\(96\) 0 0
\(97\) 3.75883 0.595340i 0.381651 0.0604476i 0.0373386 0.999303i \(-0.488112\pi\)
0.344313 + 0.938855i \(0.388112\pi\)
\(98\) 0 0
\(99\) −2.36873 −0.238067
\(100\) 0 0
\(101\) −6.96453 −0.692996 −0.346498 0.938051i \(-0.612629\pi\)
−0.346498 + 0.938051i \(0.612629\pi\)
\(102\) 0 0
\(103\) −1.39178 + 0.220437i −0.137136 + 0.0217203i −0.224625 0.974445i \(-0.572116\pi\)
0.0874888 + 0.996165i \(0.472116\pi\)
\(104\) 0 0
\(105\) −6.80298 + 5.31629i −0.663903 + 0.518816i
\(106\) 0 0
\(107\) −10.0992 10.0992i −0.976324 0.976324i 0.0234025 0.999726i \(-0.492550\pi\)
−0.999726 + 0.0234025i \(0.992550\pi\)
\(108\) 0 0
\(109\) −18.4746 + 6.00276i −1.76954 + 0.574960i −0.998116 0.0613565i \(-0.980457\pi\)
−0.771428 + 0.636316i \(0.780457\pi\)
\(110\) 0 0
\(111\) 26.7466 + 8.69051i 2.53868 + 0.824867i
\(112\) 0 0
\(113\) 7.15358 14.0397i 0.672952 1.32074i −0.261689 0.965152i \(-0.584280\pi\)
0.934642 0.355591i \(-0.115720\pi\)
\(114\) 0 0
\(115\) 18.3407 3.56718i 1.71028 0.332641i
\(116\) 0 0
\(117\) −0.707023 + 4.46397i −0.0653643 + 0.412694i
\(118\) 0 0
\(119\) −6.95584 + 5.05372i −0.637641 + 0.463273i
\(120\) 0 0
\(121\) −8.50302 6.17780i −0.773002 0.561619i
\(122\) 0 0
\(123\) 0.181761 + 0.356727i 0.0163889 + 0.0321650i
\(124\) 0 0
\(125\) 8.30612 7.48387i 0.742922 0.669378i
\(126\) 0 0
\(127\) −12.1817 + 6.20688i −1.08095 + 0.550772i −0.901404 0.432978i \(-0.857463\pi\)
−0.179546 + 0.983750i \(0.557463\pi\)
\(128\) 0 0
\(129\) 10.7445 14.7886i 0.946001 1.30206i
\(130\) 0 0
\(131\) −5.87989 8.09298i −0.513728 0.707087i 0.470814 0.882232i \(-0.343960\pi\)
−0.984542 + 0.175146i \(0.943960\pi\)
\(132\) 0 0
\(133\) −2.44684 0.387541i −0.212168 0.0336041i
\(134\) 0 0
\(135\) −0.415278 2.13516i −0.0357415 0.183765i
\(136\) 0 0
\(137\) −9.50415 4.84261i −0.811994 0.413732i −0.00187601 0.999998i \(-0.500597\pi\)
−0.810118 + 0.586266i \(0.800597\pi\)
\(138\) 0 0
\(139\) 6.07838 18.7073i 0.515562 1.58674i −0.266696 0.963781i \(-0.585932\pi\)
0.782258 0.622955i \(-0.214068\pi\)
\(140\) 0 0
\(141\) −5.23382 16.1080i −0.440767 1.35654i
\(142\) 0 0
\(143\) 0.660682 0.660682i 0.0552490 0.0552490i
\(144\) 0 0
\(145\) 11.0608 + 14.1539i 0.918547 + 1.17542i
\(146\) 0 0
\(147\) 1.84402 + 11.6427i 0.152093 + 0.960275i
\(148\) 0 0
\(149\) 12.4064i 1.01637i 0.861247 + 0.508187i \(0.169684\pi\)
−0.861247 + 0.508187i \(0.830316\pi\)
\(150\) 0 0
\(151\) 6.27163i 0.510378i −0.966891 0.255189i \(-0.917862\pi\)
0.966891 0.255189i \(-0.0821377\pi\)
\(152\) 0 0
\(153\) −2.97947 18.8116i −0.240876 1.52083i
\(154\) 0 0
\(155\) −10.0481 + 3.65848i −0.807084 + 0.293856i
\(156\) 0 0
\(157\) 1.56883 1.56883i 0.125207 0.125207i −0.641727 0.766933i \(-0.721782\pi\)
0.766933 + 0.641727i \(0.221782\pi\)
\(158\) 0 0
\(159\) −2.84812 8.76562i −0.225871 0.695159i
\(160\) 0 0
\(161\) −3.94563 + 12.1434i −0.310959 + 0.957034i
\(162\) 0 0
\(163\) −1.15366 0.587821i −0.0903619 0.0460417i 0.408226 0.912881i \(-0.366148\pi\)
−0.498587 + 0.866839i \(0.666148\pi\)
\(164\) 0 0
\(165\) −1.67059 + 3.58364i −0.130056 + 0.278986i
\(166\) 0 0
\(167\) 6.57784 + 1.04183i 0.509009 + 0.0806191i 0.405656 0.914026i \(-0.367043\pi\)
0.103353 + 0.994645i \(0.467043\pi\)
\(168\) 0 0
\(169\) 6.59333 + 9.07494i 0.507179 + 0.698072i
\(170\) 0 0
\(171\) 3.22566 4.43974i 0.246673 0.339516i
\(172\) 0 0
\(173\) −0.526699 + 0.268367i −0.0400442 + 0.0204035i −0.473898 0.880580i \(-0.657153\pi\)
0.433854 + 0.900983i \(0.357153\pi\)
\(174\) 0 0
\(175\) 1.72034 + 7.44410i 0.130046 + 0.562721i
\(176\) 0 0
\(177\) 17.4598 + 34.2668i 1.31236 + 2.57565i
\(178\) 0 0
\(179\) −12.2871 8.92709i −0.918380 0.667242i 0.0247405 0.999694i \(-0.492124\pi\)
−0.943120 + 0.332452i \(0.892124\pi\)
\(180\) 0 0
\(181\) 10.9315 7.94221i 0.812533 0.590339i −0.102031 0.994781i \(-0.532534\pi\)
0.914564 + 0.404442i \(0.132534\pi\)
\(182\) 0 0
\(183\) 3.21012 20.2679i 0.237299 1.49825i
\(184\) 0 0
\(185\) 16.9707 18.2029i 1.24771 1.33830i
\(186\) 0 0
\(187\) −1.78756 + 3.50828i −0.130719 + 0.256551i
\(188\) 0 0
\(189\) 1.41369 + 0.459336i 0.102831 + 0.0334118i
\(190\) 0 0
\(191\) −12.7837 + 4.15367i −0.924994 + 0.300549i −0.732514 0.680752i \(-0.761653\pi\)
−0.192480 + 0.981301i \(0.561653\pi\)
\(192\) 0 0
\(193\) 6.62984 + 6.62984i 0.477227 + 0.477227i 0.904244 0.427017i \(-0.140436\pi\)
−0.427017 + 0.904244i \(0.640436\pi\)
\(194\) 0 0
\(195\) 6.25486 + 4.21794i 0.447919 + 0.302053i
\(196\) 0 0
\(197\) 14.3776 2.27719i 1.02436 0.162243i 0.378424 0.925633i \(-0.376466\pi\)
0.645938 + 0.763390i \(0.276466\pi\)
\(198\) 0 0
\(199\) −0.958761 −0.0679648 −0.0339824 0.999422i \(-0.510819\pi\)
−0.0339824 + 0.999422i \(0.510819\pi\)
\(200\) 0 0
\(201\) −15.6637 −1.10483
\(202\) 0 0
\(203\) −12.1243 + 1.92030i −0.850959 + 0.134779i
\(204\) 0 0
\(205\) 0.354074 0.0124038i 0.0247296 0.000866322i
\(206\) 0 0
\(207\) −20.0002 20.0002i −1.39011 1.39011i
\(208\) 0 0
\(209\) −1.07898 + 0.350582i −0.0746345 + 0.0242502i
\(210\) 0 0
\(211\) −11.0232 3.58166i −0.758869 0.246572i −0.0960759 0.995374i \(-0.530629\pi\)
−0.662793 + 0.748802i \(0.730629\pi\)
\(212\) 0 0
\(213\) −1.17746 + 2.31089i −0.0806779 + 0.158339i
\(214\) 0 0
\(215\) −7.84389 14.1471i −0.534949 0.964821i
\(216\) 0 0
\(217\) 1.14315 7.21758i 0.0776022 0.489961i
\(218\) 0 0
\(219\) −10.0778 + 7.32191i −0.680991 + 0.494769i
\(220\) 0 0
\(221\) 6.07793 + 4.41587i 0.408846 + 0.297044i
\(222\) 0 0
\(223\) 8.13525 + 15.9663i 0.544777 + 1.06918i 0.985202 + 0.171396i \(0.0548276\pi\)
−0.440425 + 0.897789i \(0.645172\pi\)
\(224\) 0 0
\(225\) −16.4230 4.09085i −1.09487 0.272723i
\(226\) 0 0
\(227\) 12.4217 6.32920i 0.824460 0.420084i 0.00975061 0.999952i \(-0.496896\pi\)
0.814710 + 0.579869i \(0.196896\pi\)
\(228\) 0 0
\(229\) 1.55986 2.14696i 0.103078 0.141875i −0.754362 0.656459i \(-0.772054\pi\)
0.857440 + 0.514584i \(0.172054\pi\)
\(230\) 0 0
\(231\) −1.58818 2.18594i −0.104495 0.143824i
\(232\) 0 0
\(233\) 14.0180 + 2.22024i 0.918351 + 0.145452i 0.597677 0.801737i \(-0.296091\pi\)
0.320674 + 0.947190i \(0.396091\pi\)
\(234\) 0 0
\(235\) −14.8764 1.82492i −0.970430 0.119045i
\(236\) 0 0
\(237\) −24.5526 12.5102i −1.59486 0.812624i
\(238\) 0 0
\(239\) 3.61203 11.1167i 0.233643 0.719078i −0.763656 0.645623i \(-0.776598\pi\)
0.997299 0.0734545i \(-0.0234024\pi\)
\(240\) 0 0
\(241\) 5.70865 + 17.5694i 0.367726 + 1.13175i 0.948256 + 0.317506i \(0.102845\pi\)
−0.580530 + 0.814239i \(0.697155\pi\)
\(242\) 0 0
\(243\) 15.8160 15.8160i 1.01460 1.01460i
\(244\) 0 0
\(245\) 10.0276 + 2.87415i 0.640637 + 0.183623i
\(246\) 0 0
\(247\) 0.338629 + 2.13802i 0.0215464 + 0.136039i
\(248\) 0 0
\(249\) 9.83720i 0.623407i
\(250\) 0 0
\(251\) 21.2264i 1.33980i 0.742452 + 0.669900i \(0.233663\pi\)
−0.742452 + 0.669900i \(0.766337\pi\)
\(252\) 0 0
\(253\) 0.914719 + 5.77531i 0.0575079 + 0.363090i
\(254\) 0 0
\(255\) −30.5613 8.75964i −1.91382 0.548550i
\(256\) 0 0
\(257\) 1.34040 1.34040i 0.0836121 0.0836121i −0.664064 0.747676i \(-0.731170\pi\)
0.747676 + 0.664064i \(0.231170\pi\)
\(258\) 0 0
\(259\) 5.25540 + 16.1745i 0.326555 + 1.00503i
\(260\) 0 0
\(261\) 8.40299 25.8618i 0.520132 1.60080i
\(262\) 0 0
\(263\) −12.6404 6.44061i −0.779441 0.397145i 0.0185397 0.999828i \(-0.494098\pi\)
−0.797981 + 0.602683i \(0.794098\pi\)
\(264\) 0 0
\(265\) −8.09541 0.993080i −0.497297 0.0610044i
\(266\) 0 0
\(267\) 21.6638 + 3.43121i 1.32580 + 0.209987i
\(268\) 0 0
\(269\) −4.34204 5.97631i −0.264739 0.364382i 0.655866 0.754877i \(-0.272304\pi\)
−0.920605 + 0.390496i \(0.872304\pi\)
\(270\) 0 0
\(271\) 14.6738 20.1968i 0.891371 1.22687i −0.0817692 0.996651i \(-0.526057\pi\)
0.973140 0.230215i \(-0.0739430\pi\)
\(272\) 0 0
\(273\) −4.59353 + 2.34052i −0.278013 + 0.141655i
\(274\) 0 0
\(275\) 2.24964 + 2.67981i 0.135659 + 0.161599i
\(276\) 0 0
\(277\) −8.52278 16.7269i −0.512084 1.00502i −0.991824 0.127611i \(-0.959269\pi\)
0.479740 0.877411i \(-0.340731\pi\)
\(278\) 0 0
\(279\) 13.0962 + 9.51492i 0.784047 + 0.569643i
\(280\) 0 0
\(281\) −0.766131 + 0.556627i −0.0457035 + 0.0332056i −0.610402 0.792091i \(-0.708992\pi\)
0.564699 + 0.825297i \(0.308992\pi\)
\(282\) 0 0
\(283\) −4.30567 + 27.1849i −0.255945 + 1.61598i 0.440080 + 0.897959i \(0.354950\pi\)
−0.696025 + 0.718017i \(0.745050\pi\)
\(284\) 0 0
\(285\) −4.44189 8.01129i −0.263115 0.474548i
\(286\) 0 0
\(287\) −0.109916 + 0.215723i −0.00648816 + 0.0127337i
\(288\) 0 0
\(289\) −13.9420 4.53003i −0.820118 0.266472i
\(290\) 0 0
\(291\) −9.14573 + 2.97163i −0.536132 + 0.174200i
\(292\) 0 0
\(293\) 11.5657 + 11.5657i 0.675677 + 0.675677i 0.959019 0.283342i \(-0.0914431\pi\)
−0.283342 + 0.959019i \(0.591443\pi\)
\(294\) 0 0
\(295\) 34.0120 1.19150i 1.98026 0.0693719i
\(296\) 0 0
\(297\) 0.672340 0.106488i 0.0390131 0.00617907i
\(298\) 0 0
\(299\) 11.1568 0.645215
\(300\) 0 0
\(301\) 11.0542 0.637156
\(302\) 0 0
\(303\) 17.3817 2.75298i 0.998550 0.158155i
\(304\) 0 0
\(305\) −15.0557 10.1528i −0.862087 0.581346i
\(306\) 0 0
\(307\) −16.2123 16.2123i −0.925284 0.925284i 0.0721127 0.997396i \(-0.477026\pi\)
−0.997396 + 0.0721127i \(0.977026\pi\)
\(308\) 0 0
\(309\) 3.38639 1.10031i 0.192645 0.0625942i
\(310\) 0 0
\(311\) −3.05745 0.993425i −0.173372 0.0563320i 0.221045 0.975264i \(-0.429053\pi\)
−0.394417 + 0.918932i \(0.629053\pi\)
\(312\) 0 0
\(313\) −9.38062 + 18.4105i −0.530224 + 1.04062i 0.458192 + 0.888853i \(0.348497\pi\)
−0.988416 + 0.151770i \(0.951503\pi\)
\(314\) 0 0
\(315\) 7.88700 8.45965i 0.444382 0.476647i
\(316\) 0 0
\(317\) 1.16618 7.36300i 0.0654994 0.413547i −0.933052 0.359742i \(-0.882865\pi\)
0.998551 0.0538056i \(-0.0171351\pi\)
\(318\) 0 0
\(319\) −4.54795 + 3.30428i −0.254636 + 0.185004i
\(320\) 0 0
\(321\) 29.1970 + 21.2128i 1.62962 + 1.18399i
\(322\) 0 0
\(323\) −4.14137 8.12790i −0.230432 0.452248i
\(324\) 0 0
\(325\) 5.72168 3.43966i 0.317382 0.190798i
\(326\) 0 0
\(327\) 43.7350 22.2841i 2.41855 1.23231i
\(328\) 0 0
\(329\) 6.02026 8.28618i 0.331908 0.456832i
\(330\) 0 0
\(331\) −13.8679 19.0875i −0.762247 1.04914i −0.997024 0.0770934i \(-0.975436\pi\)
0.234777 0.972049i \(-0.424564\pi\)
\(332\) 0 0
\(333\) −37.2099 5.89347i −2.03909 0.322960i
\(334\) 0 0
\(335\) −5.85658 + 12.5631i −0.319979 + 0.686397i
\(336\) 0 0
\(337\) 4.02128 + 2.04894i 0.219053 + 0.111613i 0.560074 0.828443i \(-0.310773\pi\)
−0.341021 + 0.940056i \(0.610773\pi\)
\(338\) 0 0
\(339\) −12.3038 + 37.8671i −0.668250 + 2.05666i
\(340\) 0 0
\(341\) −1.03413 3.18273i −0.0560013 0.172354i
\(342\) 0 0
\(343\) −12.6041 + 12.6041i −0.680557 + 0.680557i
\(344\) 0 0
\(345\) −44.3636 + 16.1526i −2.38845 + 0.869626i
\(346\) 0 0
\(347\) −2.09940 13.2551i −0.112702 0.711570i −0.977733 0.209853i \(-0.932702\pi\)
0.865031 0.501718i \(-0.167298\pi\)
\(348\) 0 0
\(349\) 5.64203i 0.302011i 0.988533 + 0.151005i \(0.0482511\pi\)
−0.988533 + 0.151005i \(0.951749\pi\)
\(350\) 0 0
\(351\) 1.29883i 0.0693266i
\(352\) 0 0
\(353\) 2.45754 + 15.5163i 0.130802 + 0.825848i 0.962631 + 0.270817i \(0.0872938\pi\)
−0.831829 + 0.555031i \(0.812706\pi\)
\(354\) 0 0
\(355\) 1.41321 + 1.80841i 0.0750054 + 0.0959806i
\(356\) 0 0
\(357\) 15.3623 15.3623i 0.813060 0.813060i
\(358\) 0 0
\(359\) 6.35999 + 19.5740i 0.335667 + 1.03308i 0.966392 + 0.257072i \(0.0827576\pi\)
−0.630725 + 0.776006i \(0.717242\pi\)
\(360\) 0 0
\(361\) −5.05910 + 15.5703i −0.266269 + 0.819491i
\(362\) 0 0
\(363\) 23.6633 + 12.0571i 1.24200 + 0.632832i
\(364\) 0 0
\(365\) 2.10454 + 10.8205i 0.110157 + 0.566373i
\(366\) 0 0
\(367\) 13.2423 + 2.09737i 0.691242 + 0.109482i 0.492165 0.870502i \(-0.336206\pi\)
0.199077 + 0.979984i \(0.436206\pi\)
\(368\) 0 0
\(369\) −0.315246 0.433898i −0.0164110 0.0225878i
\(370\) 0 0
\(371\) 3.27609 4.50915i 0.170086 0.234104i
\(372\) 0 0
\(373\) −14.7204 + 7.50041i −0.762193 + 0.388357i −0.791465 0.611215i \(-0.790681\pi\)
0.0292718 + 0.999571i \(0.490681\pi\)
\(374\) 0 0
\(375\) −17.7717 + 21.9611i −0.917724 + 1.13407i
\(376\) 0 0
\(377\) 4.86956 + 9.55705i 0.250795 + 0.492213i
\(378\) 0 0
\(379\) 27.4172 + 19.9197i 1.40833 + 1.02321i 0.993563 + 0.113279i \(0.0361353\pi\)
0.414763 + 0.909930i \(0.363865\pi\)
\(380\) 0 0
\(381\) 27.9488 20.3060i 1.43186 1.04031i
\(382\) 0 0
\(383\) −0.937636 + 5.92000i −0.0479109 + 0.302498i −0.999994 0.00334047i \(-0.998937\pi\)
0.952083 + 0.305838i \(0.0989367\pi\)
\(384\) 0 0
\(385\) −2.34706 + 0.456492i −0.119617 + 0.0232650i
\(386\) 0 0
\(387\) −11.1171 + 21.8185i −0.565112 + 1.10910i
\(388\) 0 0
\(389\) 4.85597 + 1.57780i 0.246208 + 0.0799977i 0.429521 0.903057i \(-0.358682\pi\)
−0.183314 + 0.983054i \(0.558682\pi\)
\(390\) 0 0
\(391\) −44.7149 + 14.5287i −2.26133 + 0.734750i
\(392\) 0 0
\(393\) 17.8737 + 17.8737i 0.901610 + 0.901610i
\(394\) 0 0
\(395\) −19.2140 + 15.0150i −0.966760 + 0.755488i
\(396\) 0 0
\(397\) −13.3182 + 2.10939i −0.668420 + 0.105867i −0.481417 0.876492i \(-0.659878\pi\)
−0.187004 + 0.982359i \(0.559878\pi\)
\(398\) 0 0
\(399\) 6.25987 0.313385
\(400\) 0 0
\(401\) −2.21166 −0.110445 −0.0552225 0.998474i \(-0.517587\pi\)
−0.0552225 + 0.998474i \(0.517587\pi\)
\(402\) 0 0
\(403\) −6.30663 + 0.998872i −0.314156 + 0.0497574i
\(404\) 0 0
\(405\) −5.88822 16.1722i −0.292588 0.803601i
\(406\) 0 0
\(407\) 5.50719 + 5.50719i 0.272981 + 0.272981i
\(408\) 0 0
\(409\) −11.4631 + 3.72457i −0.566812 + 0.184168i −0.578384 0.815765i \(-0.696316\pi\)
0.0115722 + 0.999933i \(0.496316\pi\)
\(410\) 0 0
\(411\) 25.6341 + 8.32903i 1.26444 + 0.410841i
\(412\) 0 0
\(413\) −10.5585 + 20.7222i −0.519548 + 1.01967i
\(414\) 0 0
\(415\) −7.88997 3.67809i −0.387303 0.180550i
\(416\) 0 0
\(417\) −7.77531 + 49.0914i −0.380758 + 2.40401i
\(418\) 0 0
\(419\) 1.24155 0.902039i 0.0606537 0.0440675i −0.557045 0.830482i \(-0.688065\pi\)
0.617699 + 0.786415i \(0.288065\pi\)
\(420\) 0 0
\(421\) 4.12933 + 3.00014i 0.201251 + 0.146218i 0.683847 0.729626i \(-0.260306\pi\)
−0.482595 + 0.875844i \(0.660306\pi\)
\(422\) 0 0
\(423\) 10.3005 + 20.2159i 0.500827 + 0.982929i
\(424\) 0 0
\(425\) −18.4524 + 21.2366i −0.895075 + 1.03013i
\(426\) 0 0
\(427\) 11.0568 5.63373i 0.535077 0.272635i
\(428\) 0 0
\(429\) −1.38773 + 1.91005i −0.0670003 + 0.0922180i
\(430\) 0 0
\(431\) −0.811616 1.11709i −0.0390942 0.0538085i 0.789022 0.614364i \(-0.210588\pi\)
−0.828117 + 0.560556i \(0.810588\pi\)
\(432\) 0 0
\(433\) 19.1398 + 3.03145i 0.919801 + 0.145682i 0.598341 0.801241i \(-0.295827\pi\)
0.321459 + 0.946923i \(0.395827\pi\)
\(434\) 0 0
\(435\) −33.1997 30.9523i −1.59180 1.48405i
\(436\) 0 0
\(437\) −12.0704 6.15015i −0.577403 0.294202i
\(438\) 0 0
\(439\) −6.79514 + 20.9133i −0.324314 + 0.998136i 0.647435 + 0.762121i \(0.275842\pi\)
−0.971749 + 0.236016i \(0.924158\pi\)
\(440\) 0 0
\(441\) −4.87969 15.0181i −0.232366 0.715150i
\(442\) 0 0
\(443\) 18.6542 18.6542i 0.886289 0.886289i −0.107875 0.994164i \(-0.534405\pi\)
0.994164 + 0.107875i \(0.0344047\pi\)
\(444\) 0 0
\(445\) 10.8520 16.0926i 0.514435 0.762864i
\(446\) 0 0
\(447\) −4.90409 30.9632i −0.231955 1.46451i
\(448\) 0 0
\(449\) 26.5049i 1.25084i −0.780287 0.625422i \(-0.784927\pi\)
0.780287 0.625422i \(-0.215073\pi\)
\(450\) 0 0
\(451\) 0.110876i 0.00522094i
\(452\) 0 0
\(453\) 2.47909 + 15.6524i 0.116478 + 0.735412i
\(454\) 0 0
\(455\) 0.159723 + 4.55937i 0.00748793 + 0.213747i
\(456\) 0 0
\(457\) 26.9003 26.9003i 1.25834 1.25834i 0.306461 0.951883i \(-0.400855\pi\)
0.951883 0.306461i \(-0.0991448\pi\)
\(458\) 0 0
\(459\) 1.69138 + 5.20554i 0.0789470 + 0.242974i
\(460\) 0 0
\(461\) −4.17045 + 12.8353i −0.194237 + 0.597800i 0.805748 + 0.592259i \(0.201764\pi\)
−0.999985 + 0.00554125i \(0.998236\pi\)
\(462\) 0 0
\(463\) 16.9581 + 8.64060i 0.788111 + 0.401563i 0.801234 0.598351i \(-0.204177\pi\)
−0.0131228 + 0.999914i \(0.504177\pi\)
\(464\) 0 0
\(465\) 23.6313 13.1025i 1.09588 0.607614i
\(466\) 0 0
\(467\) 26.4680 + 4.19212i 1.22479 + 0.193988i 0.735131 0.677925i \(-0.237120\pi\)
0.489661 + 0.871913i \(0.337120\pi\)
\(468\) 0 0
\(469\) −5.56767 7.66324i −0.257091 0.353855i
\(470\) 0 0
\(471\) −3.29526 + 4.53554i −0.151838 + 0.208987i
\(472\) 0 0
\(473\) 4.51056 2.29825i 0.207396 0.105673i
\(474\) 0 0
\(475\) −8.08629 + 0.567251i −0.371025 + 0.0260272i
\(476\) 0 0
\(477\) 5.60530 + 11.0010i 0.256649 + 0.503702i
\(478\) 0 0
\(479\) −22.6704 16.4710i −1.03584 0.752579i −0.0663679 0.997795i \(-0.521141\pi\)
−0.969468 + 0.245216i \(0.921141\pi\)
\(480\) 0 0
\(481\) 12.0223 8.73470i 0.548169 0.398268i
\(482\) 0 0
\(483\) 5.04715 31.8664i 0.229653 1.44997i
\(484\) 0 0
\(485\) −1.03614 + 8.44645i −0.0470488 + 0.383534i
\(486\) 0 0
\(487\) −7.14432 + 14.0215i −0.323740 + 0.635375i −0.994316 0.106465i \(-0.966047\pi\)
0.670577 + 0.741840i \(0.266047\pi\)
\(488\) 0 0
\(489\) 3.11160 + 1.01102i 0.140711 + 0.0457199i
\(490\) 0 0
\(491\) 24.2576 7.88179i 1.09473 0.355700i 0.294659 0.955602i \(-0.404794\pi\)
0.800073 + 0.599903i \(0.204794\pi\)
\(492\) 0 0
\(493\) −31.9620 31.9620i −1.43950 1.43950i
\(494\) 0 0
\(495\) 1.45939 5.09163i 0.0655947 0.228852i
\(496\) 0 0
\(497\) −1.54910 + 0.245353i −0.0694864 + 0.0110056i
\(498\) 0 0
\(499\) 7.96970 0.356773 0.178386 0.983961i \(-0.442912\pi\)
0.178386 + 0.983961i \(0.442912\pi\)
\(500\) 0 0
\(501\) −16.8284 −0.751838
\(502\) 0 0
\(503\) 36.5361 5.78675i 1.62906 0.258018i 0.726054 0.687638i \(-0.241352\pi\)
0.903010 + 0.429620i \(0.141352\pi\)
\(504\) 0 0
\(505\) 4.29088 14.9704i 0.190942 0.666172i
\(506\) 0 0
\(507\) −20.0424 20.0424i −0.890116 0.890116i
\(508\) 0 0
\(509\) 3.01087 0.978290i 0.133454 0.0433619i −0.241528 0.970394i \(-0.577649\pi\)
0.374983 + 0.927032i \(0.377649\pi\)
\(510\) 0 0
\(511\) −7.16429 2.32782i −0.316929 0.102977i
\(512\) 0 0
\(513\) −0.715978 + 1.40519i −0.0316112 + 0.0620405i
\(514\) 0 0
\(515\) 0.383653 3.12747i 0.0169058 0.137813i
\(516\) 0 0
\(517\) 0.733754 4.63274i 0.0322705 0.203748i
\(518\) 0 0
\(519\) 1.20842 0.877970i 0.0530438 0.0385386i
\(520\) 0 0
\(521\) −18.6034 13.5161i −0.815028 0.592152i 0.100256 0.994962i \(-0.468034\pi\)
−0.915284 + 0.402809i \(0.868034\pi\)
\(522\) 0 0
\(523\) −5.37688 10.5527i −0.235115 0.461438i 0.743060 0.669225i \(-0.233374\pi\)
−0.978174 + 0.207787i \(0.933374\pi\)
\(524\) 0 0
\(525\) −7.23608 17.8985i −0.315808 0.781155i
\(526\) 0 0
\(527\) 23.9753 12.2160i 1.04438 0.532139i
\(528\) 0 0
\(529\) −27.5208 + 37.8792i −1.19656 + 1.64692i
\(530\) 0 0
\(531\) −30.2822 41.6799i −1.31414 1.80875i
\(532\) 0 0
\(533\) 0.208950 + 0.0330944i 0.00905061 + 0.00143348i
\(534\) 0 0
\(535\) 27.9305 15.4862i 1.20754 0.669525i
\(536\) 0 0
\(537\) 34.1941 + 17.4228i 1.47559 + 0.751848i
\(538\) 0 0
\(539\) −1.00878 + 3.10472i −0.0434514 + 0.133730i
\(540\) 0 0
\(541\) −6.76204 20.8114i −0.290723 0.894753i −0.984625 0.174683i \(-0.944110\pi\)
0.693902 0.720069i \(-0.255890\pi\)
\(542\) 0 0
\(543\) −24.1428 + 24.1428i −1.03606 + 1.03606i
\(544\) 0 0
\(545\) −1.52072 43.4097i −0.0651405 1.85947i
\(546\) 0 0
\(547\) −0.461983 2.91684i −0.0197530 0.124715i 0.975841 0.218480i \(-0.0701098\pi\)
−0.995594 + 0.0937648i \(0.970110\pi\)
\(548\) 0 0
\(549\) 27.4893i 1.17322i
\(550\) 0 0
\(551\) 13.0239i 0.554838i
\(552\) 0 0
\(553\) −2.60682 16.4588i −0.110853 0.699899i
\(554\) 0 0
\(555\) −35.1591 + 52.1380i −1.49242 + 2.21314i
\(556\) 0 0
\(557\) −3.97120 + 3.97120i −0.168265 + 0.168265i −0.786216 0.617951i \(-0.787963\pi\)
0.617951 + 0.786216i \(0.287963\pi\)
\(558\) 0 0
\(559\) −2.98481 9.18631i −0.126244 0.388539i
\(560\) 0 0
\(561\) 3.07450 9.46235i 0.129806 0.399501i
\(562\) 0 0
\(563\) 10.2006 + 5.19747i 0.429904 + 0.219047i 0.655533 0.755166i \(-0.272444\pi\)
−0.225629 + 0.974213i \(0.572444\pi\)
\(564\) 0 0
\(565\) 25.7712 + 24.0267i 1.08420 + 1.01081i
\(566\) 0 0
\(567\) 11.6165 + 1.83987i 0.487847 + 0.0772674i
\(568\) 0 0
\(569\) −21.6548 29.8053i −0.907817 1.24950i −0.967907 0.251309i \(-0.919139\pi\)
0.0600903 0.998193i \(-0.480861\pi\)
\(570\) 0 0
\(571\) 20.2874 27.9232i 0.849002 1.16855i −0.135079 0.990835i \(-0.543129\pi\)
0.984082 0.177717i \(-0.0568710\pi\)
\(572\) 0 0
\(573\) 30.2628 15.4197i 1.26425 0.644167i
\(574\) 0 0
\(575\) −3.63210 + 41.6214i −0.151469 + 1.73573i
\(576\) 0 0
\(577\) 0.0627971 + 0.123246i 0.00261428 + 0.00513081i 0.892310 0.451423i \(-0.149084\pi\)
−0.889696 + 0.456554i \(0.849084\pi\)
\(578\) 0 0
\(579\) −19.1671 13.9257i −0.796556 0.578732i
\(580\) 0 0
\(581\) 4.81271 3.49664i 0.199665 0.145065i
\(582\) 0 0
\(583\) 0.399292 2.52103i 0.0165370 0.104410i
\(584\) 0 0
\(585\) −9.15976 4.27003i −0.378710 0.176544i
\(586\) 0 0
\(587\) −9.53903 + 18.7214i −0.393718 + 0.772715i −0.999742 0.0227315i \(-0.992764\pi\)
0.606024 + 0.795447i \(0.292764\pi\)
\(588\) 0 0
\(589\) 7.37366 + 2.39585i 0.303826 + 0.0987192i
\(590\) 0 0
\(591\) −34.9826 + 11.3665i −1.43899 + 0.467557i
\(592\) 0 0
\(593\) −8.53522 8.53522i −0.350500 0.350500i 0.509796 0.860295i \(-0.329721\pi\)
−0.860295 + 0.509796i \(0.829721\pi\)
\(594\) 0 0
\(595\) −6.57750 18.0653i −0.269651 0.740606i
\(596\) 0 0
\(597\) 2.39282 0.378985i 0.0979315 0.0155108i
\(598\) 0 0
\(599\) −43.2355 −1.76655 −0.883276 0.468852i \(-0.844668\pi\)
−0.883276 + 0.468852i \(0.844668\pi\)
\(600\) 0 0
\(601\) 3.78938 0.154572 0.0772861 0.997009i \(-0.475375\pi\)
0.0772861 + 0.997009i \(0.475375\pi\)
\(602\) 0 0
\(603\) 20.7247 3.28248i 0.843977 0.133673i
\(604\) 0 0
\(605\) 18.5180 14.4712i 0.752865 0.588337i
\(606\) 0 0
\(607\) −30.2283 30.2283i −1.22693 1.22693i −0.965119 0.261810i \(-0.915681\pi\)
−0.261810 0.965119i \(-0.584319\pi\)
\(608\) 0 0
\(609\) 29.5000 9.58515i 1.19540 0.388410i
\(610\) 0 0
\(611\) −8.51156 2.76557i −0.344341 0.111883i
\(612\) 0 0
\(613\) 16.8340 33.0385i 0.679917 1.33441i −0.250574 0.968098i \(-0.580619\pi\)
0.930491 0.366315i \(-0.119381\pi\)
\(614\) 0 0
\(615\) −0.878774 + 0.170917i −0.0354356 + 0.00689206i
\(616\) 0 0
\(617\) −3.69290 + 23.3161i −0.148671 + 0.938670i 0.794718 + 0.606979i \(0.207619\pi\)
−0.943388 + 0.331690i \(0.892381\pi\)
\(618\) 0 0
\(619\) 28.3479 20.5960i 1.13940 0.827821i 0.152363 0.988325i \(-0.451312\pi\)
0.987035 + 0.160503i \(0.0513117\pi\)
\(620\) 0 0
\(621\) 6.57596 + 4.77771i 0.263884 + 0.191723i
\(622\) 0 0
\(623\) 6.02175 + 11.8183i 0.241256 + 0.473492i
\(624\) 0 0
\(625\) 10.9692 + 22.4650i 0.438770 + 0.898600i
\(626\) 0 0
\(627\) 2.55427 1.30147i 0.102008 0.0519756i
\(628\) 0 0
\(629\) −36.8090 + 50.6632i −1.46767 + 2.02008i
\(630\) 0 0
\(631\) −25.9074 35.6585i −1.03136 1.41954i −0.903923 0.427695i \(-0.859326\pi\)
−0.127435 0.991847i \(-0.540674\pi\)
\(632\) 0 0
\(633\) 28.9269 + 4.58156i 1.14974 + 0.182101i
\(634\) 0 0
\(635\) −5.83658 30.0088i −0.231617 1.19086i
\(636\) 0 0
\(637\) 5.54986 + 2.82779i 0.219893 + 0.112041i
\(638\) 0 0
\(639\) 1.07363 3.30430i 0.0424722 0.130716i
\(640\) 0 0
\(641\) −1.59616 4.91247i −0.0630445 0.194031i 0.914573 0.404420i \(-0.132527\pi\)
−0.977618 + 0.210390i \(0.932527\pi\)
\(642\) 0 0
\(643\) −8.75915 + 8.75915i −0.345427 + 0.345427i −0.858403 0.512976i \(-0.828543\pi\)
0.512976 + 0.858403i \(0.328543\pi\)
\(644\) 0 0
\(645\) 25.1685 + 32.2068i 0.991007 + 1.26814i
\(646\) 0 0
\(647\) 0.169704 + 1.07147i 0.00667174 + 0.0421237i 0.990800 0.135337i \(-0.0432117\pi\)
−0.984128 + 0.177461i \(0.943212\pi\)
\(648\) 0 0
\(649\) 10.6506i 0.418074i
\(650\) 0 0
\(651\) 18.4651i 0.723704i
\(652\) 0 0
\(653\) 4.34961 + 27.4623i 0.170213 + 1.07468i 0.913836 + 0.406084i \(0.133106\pi\)
−0.743622 + 0.668600i \(0.766894\pi\)
\(654\) 0 0
\(655\) 21.0186 7.65279i 0.821265 0.299019i
\(656\) 0 0
\(657\) 11.7996 11.7996i 0.460345 0.460345i
\(658\) 0 0
\(659\) 7.33067 + 22.5615i 0.285562 + 0.878870i 0.986230 + 0.165382i \(0.0528858\pi\)
−0.700667 + 0.713488i \(0.747114\pi\)
\(660\) 0 0
\(661\) −0.722498 + 2.22362i −0.0281019 + 0.0864888i −0.964124 0.265453i \(-0.914478\pi\)
0.936022 + 0.351942i \(0.114478\pi\)
\(662\) 0 0
\(663\) −16.9145 8.61835i −0.656904 0.334709i
\(664\) 0 0
\(665\) 2.34054 5.02075i 0.0907621 0.194696i
\(666\) 0 0
\(667\) −66.2995 10.5008i −2.56713 0.406593i
\(668\) 0 0
\(669\) −26.6148 36.6321i −1.02899 1.41628i
\(670\) 0 0
\(671\) 3.34033 4.59757i 0.128952 0.177487i
\(672\) 0 0
\(673\) −30.7626 + 15.6743i −1.18581 + 0.604201i −0.931790 0.362998i \(-0.881753\pi\)
−0.254021 + 0.967199i \(0.581753\pi\)
\(674\) 0 0
\(675\) 4.84541 + 0.422835i 0.186500 + 0.0162749i
\(676\) 0 0
\(677\) 10.1098 + 19.8415i 0.388549 + 0.762571i 0.999578 0.0290371i \(-0.00924408\pi\)
−0.611029 + 0.791608i \(0.709244\pi\)
\(678\) 0 0
\(679\) −4.70468 3.41815i −0.180549 0.131177i
\(680\) 0 0
\(681\) −28.4996 + 20.7062i −1.09211 + 0.793463i
\(682\) 0 0
\(683\) −3.81619 + 24.0945i −0.146023 + 0.921950i 0.800505 + 0.599326i \(0.204565\pi\)
−0.946527 + 0.322624i \(0.895435\pi\)
\(684\) 0 0
\(685\) 16.2648 17.4458i 0.621447 0.666568i
\(686\) 0 0
\(687\) −3.04434 + 5.97485i −0.116149 + 0.227955i
\(688\) 0 0
\(689\) −4.63180 1.50496i −0.176457 0.0573345i
\(690\) 0 0
\(691\) 42.2488 13.7275i 1.60722 0.522217i 0.638341 0.769753i \(-0.279621\pi\)
0.968879 + 0.247536i \(0.0796208\pi\)
\(692\) 0 0
\(693\) 2.55942 + 2.55942i 0.0972243 + 0.0972243i
\(694\) 0 0
\(695\) 36.4668 + 24.5913i 1.38326 + 0.932800i
\(696\) 0 0
\(697\) −0.880536 + 0.139463i −0.0333527 + 0.00528255i
\(698\) 0 0
\(699\) −35.8630 −1.35646
\(700\) 0 0
\(701\) 14.5039 0.547805 0.273902 0.961757i \(-0.411685\pi\)
0.273902 + 0.961757i \(0.411685\pi\)
\(702\) 0 0
\(703\) −17.8217 + 2.82268i −0.672158 + 0.106459i
\(704\) 0 0
\(705\) 37.8490 1.32592i 1.42548 0.0499370i
\(706\) 0 0
\(707\) 7.52518 + 7.52518i 0.283014 + 0.283014i
\(708\) 0 0
\(709\) 10.6790 3.46982i 0.401059 0.130312i −0.101540 0.994831i \(-0.532377\pi\)
0.502599 + 0.864519i \(0.332377\pi\)
\(710\) 0 0
\(711\) 35.1074 + 11.4071i 1.31663 + 0.427799i
\(712\) 0 0
\(713\) 18.1415 35.6046i 0.679403 1.33340i
\(714\) 0 0
\(715\) 1.01310 + 1.82720i 0.0378876 + 0.0683332i
\(716\) 0 0
\(717\) −4.62041 + 29.1721i −0.172552 + 1.08945i
\(718\) 0 0
\(719\) 28.3054 20.5651i 1.05562 0.766949i 0.0823429 0.996604i \(-0.473760\pi\)
0.973272 + 0.229655i \(0.0737597\pi\)
\(720\) 0 0
\(721\) 1.74200 + 1.26564i 0.0648756 + 0.0471349i
\(722\) 0 0
\(723\) −21.1923 41.5921i −0.788148 1.54683i
\(724\) 0 0
\(725\) −37.2386 + 15.0550i −1.38301 + 0.559128i
\(726\) 0 0
\(727\) 2.73112 1.39158i 0.101292 0.0516107i −0.402611 0.915371i \(-0.631897\pi\)
0.503902 + 0.863761i \(0.331897\pi\)
\(728\) 0 0
\(729\) −19.6484 + 27.0437i −0.727718 + 1.00162i
\(730\) 0 0
\(731\) 23.9254 + 32.9305i 0.884913 + 1.21798i
\(732\) 0 0
\(733\) −19.7493 3.12799i −0.729458 0.115535i −0.219351 0.975646i \(-0.570394\pi\)
−0.510107 + 0.860111i \(0.670394\pi\)
\(734\) 0 0
\(735\) −26.1623 3.20938i −0.965011 0.118380i
\(736\) 0 0
\(737\) −3.86506 1.96935i −0.142371 0.0725419i
\(738\) 0 0
\(739\) −8.94023 + 27.5152i −0.328872 + 1.01216i 0.640791 + 0.767716i \(0.278607\pi\)
−0.969662 + 0.244448i \(0.921393\pi\)
\(740\) 0 0
\(741\) −1.69026 5.20208i −0.0620932 0.191103i
\(742\) 0 0
\(743\) 19.1037 19.1037i 0.700847 0.700847i −0.263746 0.964592i \(-0.584958\pi\)
0.964592 + 0.263746i \(0.0849580\pi\)
\(744\) 0 0
\(745\) −26.6678 7.64366i −0.977032 0.280042i
\(746\) 0 0
\(747\) 2.06148 + 13.0157i 0.0754257 + 0.476219i
\(748\) 0 0
\(749\) 21.8243i 0.797444i
\(750\) 0 0
\(751\) 28.6039i 1.04377i −0.853016 0.521885i \(-0.825229\pi\)
0.853016 0.521885i \(-0.174771\pi\)
\(752\) 0 0
\(753\) −8.39051 52.9756i −0.305767 1.93054i
\(754\) 0 0
\(755\) 13.4810 + 3.86399i 0.490623 + 0.140625i
\(756\) 0 0
\(757\) 28.8967 28.8967i 1.05027 1.05027i 0.0516009 0.998668i \(-0.483568\pi\)
0.998668 0.0516009i \(-0.0164324\pi\)
\(758\) 0 0
\(759\) −4.56580 14.0521i −0.165728 0.510059i
\(760\) 0 0
\(761\) −9.52096 + 29.3025i −0.345135 + 1.06222i 0.616377 + 0.787451i \(0.288600\pi\)
−0.961512 + 0.274764i \(0.911400\pi\)
\(762\) 0 0
\(763\) 26.4478 + 13.4758i 0.957474 + 0.487858i
\(764\) 0 0
\(765\) 42.2716 + 5.18554i 1.52833 + 0.187484i
\(766\) 0 0
\(767\) 20.0715 + 3.17901i 0.724740 + 0.114788i
\(768\) 0 0
\(769\) 1.90394 + 2.62054i 0.0686577 + 0.0944993i 0.841965 0.539532i \(-0.181399\pi\)
−0.773307 + 0.634032i \(0.781399\pi\)
\(770\) 0 0
\(771\) −2.81546 + 3.87514i −0.101396 + 0.139560i
\(772\) 0 0
\(773\) 19.9686 10.1745i 0.718220 0.365951i −0.0563745 0.998410i \(-0.517954\pi\)
0.774594 + 0.632458i \(0.217954\pi\)
\(774\) 0 0
\(775\) −1.67325 23.8526i −0.0601050 0.856810i
\(776\) 0 0
\(777\) −19.5097 38.2899i −0.699905 1.37364i
\(778\) 0 0
\(779\) −0.207816 0.150987i −0.00744577 0.00540967i
\(780\) 0 0
\(781\) −0.581082 + 0.422180i −0.0207927 + 0.0151068i
\(782\) 0 0
\(783\) −1.22247 + 7.71835i −0.0436874 + 0.275831i
\(784\) 0 0
\(785\) 2.40566 + 4.33880i 0.0858618 + 0.154858i
\(786\) 0 0
\(787\) −18.6369 + 36.5771i −0.664335 + 1.30383i 0.275203 + 0.961386i \(0.411255\pi\)
−0.939538 + 0.342445i \(0.888745\pi\)
\(788\) 0 0
\(789\) 34.0931 + 11.0775i 1.21375 + 0.394370i
\(790\) 0 0
\(791\) −22.8994 + 7.44045i −0.814207 + 0.264552i
\(792\) 0 0
\(793\) −7.66726 7.66726i −0.272272 0.272272i
\(794\) 0 0
\(795\) 20.5966 0.721535i 0.730485 0.0255902i
\(796\) 0 0
\(797\) −38.3233 + 6.06981i −1.35748 + 0.215004i −0.792385 0.610022i \(-0.791161\pi\)
−0.565095 + 0.825026i \(0.691161\pi\)
\(798\) 0 0
\(799\) 37.7145 1.33424
\(800\) 0 0
\(801\) −29.3826 −1.03818
\(802\) 0 0
\(803\) −3.40728 + 0.539660i −0.120240 + 0.0190442i
\(804\) 0 0
\(805\) −23.6715 15.9628i −0.834311 0.562615i
\(806\) 0 0
\(807\) 13.1990 + 13.1990i 0.464625 + 0.464625i
\(808\) 0 0
\(809\) 38.5178 12.5152i 1.35421 0.440010i 0.460106 0.887864i \(-0.347811\pi\)
0.894107 + 0.447854i \(0.147811\pi\)
\(810\) 0 0
\(811\) −22.7477 7.39116i −0.798779 0.259539i −0.118941 0.992901i \(-0.537950\pi\)
−0.679838 + 0.733362i \(0.737950\pi\)
\(812\) 0 0
\(813\) −28.6385 + 56.2063i −1.00440 + 1.97124i
\(814\) 0 0
\(815\) 1.97431 2.11766i 0.0691570 0.0741783i
\(816\) 0 0
\(817\) −1.83471 + 11.5839i −0.0641882 + 0.405268i
\(818\) 0 0
\(819\) 5.58726 4.05938i 0.195235 0.141846i
\(820\) 0 0
\(821\) −31.7877 23.0951i −1.10940 0.806025i −0.126829 0.991925i \(-0.540480\pi\)
−0.982569 + 0.185900i \(0.940480\pi\)
\(822\) 0 0
\(823\) −0.607127 1.19155i −0.0211631 0.0415350i 0.880183 0.474635i \(-0.157420\pi\)
−0.901346 + 0.433100i \(0.857420\pi\)
\(824\) 0 0
\(825\) −6.67382 5.79886i −0.232353 0.201891i
\(826\) 0 0
\(827\) 10.5547 5.37788i 0.367023 0.187007i −0.260746 0.965407i \(-0.583968\pi\)
0.627768 + 0.778400i \(0.283968\pi\)
\(828\) 0 0
\(829\) 7.15706 9.85084i 0.248575 0.342134i −0.666437 0.745562i \(-0.732181\pi\)
0.915012 + 0.403428i \(0.132181\pi\)
\(830\) 0 0
\(831\) 27.8826 + 38.3771i 0.967235 + 1.33129i
\(832\) 0 0
\(833\) −25.9255 4.10619i −0.898265 0.142271i
\(834\) 0 0
\(835\) −6.29207 + 13.4973i −0.217746 + 0.467093i
\(836\) 0 0
\(837\) −4.14496 2.11196i −0.143271 0.0730001i
\(838\) 0 0
\(839\) 6.77186 20.8416i 0.233791 0.719533i −0.763489 0.645821i \(-0.776515\pi\)
0.997280 0.0737124i \(-0.0234847\pi\)
\(840\) 0 0
\(841\) −10.9808 33.7955i −0.378649 1.16536i
\(842\) 0 0
\(843\) 1.69204 1.69204i 0.0582768 0.0582768i
\(844\) 0 0
\(845\) −23.5689 + 8.58134i −0.810795 + 0.295207i
\(846\) 0 0
\(847\) 2.51240 + 15.8626i 0.0863270 + 0.545047i
\(848\) 0 0
\(849\) 69.5485i 2.38690i
\(850\) 0 0
\(851\) 92.9988i 3.18796i
\(852\) 0 0
\(853\) 5.32987 + 33.6515i 0.182491 + 1.15221i 0.893514 + 0.449036i \(0.148232\pi\)
−0.711022 + 0.703169i \(0.751768\pi\)
\(854\) 0 0
\(855\) 7.55595 + 9.66896i 0.258408 + 0.330672i
\(856\) 0 0
\(857\) −3.05566 + 3.05566i −0.104380 + 0.104380i −0.757368 0.652988i \(-0.773515\pi\)
0.652988 + 0.757368i \(0.273515\pi\)
\(858\) 0 0
\(859\) 2.09476 + 6.44702i 0.0714724 + 0.219970i 0.980412 0.196959i \(-0.0631066\pi\)
−0.908939 + 0.416928i \(0.863107\pi\)
\(860\) 0 0
\(861\) 0.189050 0.581837i 0.00644282 0.0198290i
\(862\) 0 0
\(863\) −16.8674 8.59435i −0.574172 0.292555i 0.142694 0.989767i \(-0.454424\pi\)
−0.716865 + 0.697212i \(0.754424\pi\)
\(864\) 0 0
\(865\) −0.252356 1.29749i −0.00858035 0.0441160i
\(866\) 0 0
\(867\) 36.5863 + 5.79470i 1.24253 + 0.196798i
\(868\) 0 0
\(869\) −4.48557 6.17386i −0.152163 0.209434i
\(870\) 0 0
\(871\) −4.86496 + 6.69604i −0.164843 + 0.226887i
\(872\) 0 0
\(873\) 11.4781 5.84836i 0.388473 0.197937i
\(874\) 0 0
\(875\) −17.0611 0.888444i −0.576771 0.0300349i
\(876\) 0 0
\(877\) −5.08050 9.97105i −0.171556 0.336699i 0.789180 0.614163i \(-0.210506\pi\)
−0.960736 + 0.277464i \(0.910506\pi\)
\(878\) 0 0
\(879\) −33.4368 24.2933i −1.12780 0.819392i
\(880\) 0 0
\(881\) 10.2267 7.43016i 0.344548 0.250329i −0.402030 0.915626i \(-0.631695\pi\)
0.746578 + 0.665298i \(0.231695\pi\)
\(882\) 0 0
\(883\) −2.76358 + 17.4485i −0.0930018 + 0.587190i 0.896542 + 0.442959i \(0.146071\pi\)
−0.989544 + 0.144232i \(0.953929\pi\)
\(884\) 0 0
\(885\) −84.4142 + 16.4182i −2.83755 + 0.551891i
\(886\) 0 0
\(887\) −5.63298 + 11.0553i −0.189137 + 0.371202i −0.966030 0.258431i \(-0.916795\pi\)
0.776893 + 0.629633i \(0.216795\pi\)
\(888\) 0 0
\(889\) 19.8689 + 6.45579i 0.666380 + 0.216520i
\(890\) 0 0
\(891\) 5.12251 1.66440i 0.171610 0.0557596i
\(892\) 0 0
\(893\) 7.68399 + 7.68399i 0.257135 + 0.257135i
\(894\) 0 0
\(895\) 26.7590 20.9112i 0.894456 0.698986i
\(896\) 0 0
\(897\) −27.8445 + 4.41013i −0.929700 + 0.147250i
\(898\) 0 0
\(899\) 38.4174 1.28129
\(900\) 0 0
\(901\) 20.5234 0.683733
\(902\) 0 0
\(903\) −27.5885 + 4.36959i −0.918088 + 0.145411i
\(904\) 0 0
\(905\) 10.3369 + 28.3907i 0.343611 + 0.943738i
\(906\) 0 0
\(907\) 15.8504 + 15.8504i 0.526303 + 0.526303i 0.919468 0.393165i \(-0.128620\pi\)
−0.393165 + 0.919468i \(0.628620\pi\)
\(908\) 0 0
\(909\) −22.4209 + 7.28499i −0.743654 + 0.241628i
\(910\) 0 0
\(911\) 39.9966 + 12.9957i 1.32515 + 0.430566i 0.884259 0.466997i \(-0.154664\pi\)
0.440886 + 0.897563i \(0.354664\pi\)
\(912\) 0 0
\(913\) 1.23680 2.42736i 0.0409322 0.0803339i
\(914\) 0 0
\(915\) 41.5884 + 19.3874i 1.37487 + 0.640926i
\(916\) 0 0
\(917\) −2.39124 + 15.0977i −0.0789657 + 0.498570i
\(918\) 0 0
\(919\) −4.34396 + 3.15607i −0.143294 + 0.104109i −0.657123 0.753784i \(-0.728227\pi\)
0.513829 + 0.857893i \(0.328227\pi\)
\(920\) 0 0
\(921\) 46.8701 + 34.0531i 1.54442 + 1.12209i
\(922\) 0 0
\(923\) 0.622173 + 1.22108i 0.0204791 + 0.0401924i
\(924\) 0 0
\(925\) 28.6717 + 47.6937i 0.942719 + 1.56816i
\(926\) 0 0
\(927\) −4.24998 + 2.16547i −0.139588 + 0.0711235i
\(928\) 0 0
\(929\) 18.5366 25.5134i 0.608165 0.837067i −0.388260 0.921550i \(-0.626924\pi\)
0.996425 + 0.0844825i \(0.0269237\pi\)
\(930\) 0 0
\(931\) −4.44548 6.11868i −0.145695 0.200532i
\(932\) 0 0
\(933\) 8.02329 + 1.27076i 0.262671 + 0.0416029i
\(934\) 0 0
\(935\) −6.43977 6.00385i −0.210603 0.196347i
\(936\) 0 0
\(937\) −31.9461 16.2773i −1.04363 0.531758i −0.153828 0.988098i \(-0.549160\pi\)
−0.889806 + 0.456340i \(0.849160\pi\)
\(938\) 0 0
\(939\) 16.1342 49.6559i 0.526519 1.62046i
\(940\) 0 0
\(941\) 14.9733 + 46.0832i 0.488117 + 1.50227i 0.827415 + 0.561591i \(0.189811\pi\)
−0.339298 + 0.940679i \(0.610189\pi\)
\(942\) 0 0
\(943\) −0.936169 + 0.936169i −0.0304859 + 0.0304859i
\(944\) 0 0
\(945\) −1.85833 + 2.75575i −0.0604515 + 0.0896445i
\(946\) 0 0
\(947\) 0.711359 + 4.49135i 0.0231161 + 0.145949i 0.996547 0.0830323i \(-0.0264605\pi\)
−0.973431 + 0.228981i \(0.926460\pi\)
\(948\) 0 0
\(949\) 6.58222i 0.213668i
\(950\) 0 0
\(951\) 18.8371i 0.610835i
\(952\) 0 0
\(953\) −0.905342 5.71610i −0.0293269 0.185163i 0.968676 0.248327i \(-0.0798808\pi\)
−0.998003 + 0.0631644i \(0.979881\pi\)
\(954\) 0 0
\(955\) −1.05228 30.0378i −0.0340509 0.972000i
\(956\) 0 0
\(957\) 10.0444 10.0444i 0.324688 0.324688i
\(958\) 0 0
\(959\) 5.03680 + 15.5017i 0.162647 + 0.500576i
\(960\) 0 0
\(961\) 2.51236 7.73225i 0.0810439 0.249427i
\(962\) 0 0
\(963\) −43.0761 21.9484i −1.38811 0.707277i
\(964\) 0 0
\(965\) −18.3356 + 10.1663i −0.590245 + 0.327264i
\(966\) 0 0
\(967\) −34.2974 5.43217i −1.10293 0.174687i −0.421685 0.906742i \(-0.638561\pi\)
−0.681245 + 0.732055i \(0.738561\pi\)
\(968\) 0 0
\(969\) 13.5486 + 18.6481i 0.435245 + 0.599063i
\(970\) 0 0
\(971\) 14.5302 19.9991i 0.466296 0.641801i −0.509504 0.860469i \(-0.670171\pi\)
0.975799 + 0.218667i \(0.0701710\pi\)
\(972\) 0 0
\(973\) −26.7810 + 13.6456i −0.858559 + 0.437458i
\(974\) 0 0
\(975\) −12.9202 + 10.8462i −0.413777 + 0.347357i
\(976\) 0 0
\(977\) −16.9857 33.3364i −0.543422 1.06652i −0.985520 0.169558i \(-0.945766\pi\)
0.442099 0.896967i \(-0.354234\pi\)
\(978\) 0 0
\(979\) 4.91422 + 3.57039i 0.157059 + 0.114110i
\(980\) 0 0
\(981\) −53.1962 + 38.6493i −1.69843 + 1.23398i
\(982\) 0 0
\(983\) 2.11732 13.3682i 0.0675319 0.426380i −0.930640 0.365937i \(-0.880749\pi\)
0.998172 0.0604430i \(-0.0192513\pi\)
\(984\) 0 0
\(985\) −3.96327 + 32.3079i −0.126280 + 1.02941i
\(986\) 0 0
\(987\) −11.7496 + 23.0599i −0.373994 + 0.734004i
\(988\) 0 0
\(989\) 57.4895 + 18.6795i 1.82806 + 0.593973i
\(990\) 0 0
\(991\) −1.20979 + 0.393085i −0.0384302 + 0.0124867i −0.328169 0.944619i \(-0.606432\pi\)
0.289739 + 0.957106i \(0.406432\pi\)
\(992\) 0 0
\(993\) 42.1556 + 42.1556i 1.33777 + 1.33777i
\(994\) 0 0
\(995\) 0.590698 2.06087i 0.0187264 0.0653340i
\(996\) 0 0
\(997\) −27.7961 + 4.40246i −0.880310 + 0.139427i −0.580197 0.814476i \(-0.697024\pi\)
−0.300113 + 0.953904i \(0.597024\pi\)
\(998\) 0 0
\(999\) 10.8266 0.342538
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 400.2.bi.d.303.2 80
4.3 odd 2 inner 400.2.bi.d.303.9 yes 80
25.17 odd 20 inner 400.2.bi.d.367.9 yes 80
100.67 even 20 inner 400.2.bi.d.367.2 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.bi.d.303.2 80 1.1 even 1 trivial
400.2.bi.d.303.9 yes 80 4.3 odd 2 inner
400.2.bi.d.367.2 yes 80 100.67 even 20 inner
400.2.bi.d.367.9 yes 80 25.17 odd 20 inner