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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,9,Mod(2,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27.2"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 14.8
Character \(\chi\) \(=\) 27.14
Dual form 27.9.f.a.2.8

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-8.24786 + 1.45432i) q^{2} +(12.3580 + 80.0517i) q^{3} +(-174.649 + 63.5671i) q^{4} +(723.153 - 861.820i) q^{5} +(-218.348 - 642.283i) q^{6} +(2191.55 + 797.660i) q^{7} +(3204.82 - 1850.30i) q^{8} +(-6255.56 + 1978.56i) q^{9} +(-4711.10 + 8159.87i) q^{10} +(2798.09 + 3334.63i) q^{11} +(-7246.97 - 13195.4i) q^{12} +(-6274.91 + 35586.8i) q^{13} +(-19235.7 - 3391.77i) q^{14} +(77926.9 + 47239.2i) q^{15} +(12706.1 - 10661.7i) q^{16} +(32378.2 + 18693.6i) q^{17} +(48717.5 - 25416.5i) q^{18} +(34340.1 + 59478.9i) q^{19} +(-71514.6 + 196485. i) q^{20} +(-36770.8 + 185295. i) q^{21} +(-27927.9 - 23434.3i) q^{22} +(148858. + 408984. i) q^{23} +(187725. + 233685. i) q^{24} +(-151952. - 861764. i) q^{25} -302641. i q^{26} +(-235694. - 476317. i) q^{27} -433458. q^{28} +(292777. - 51624.5i) q^{29} +(-711431. - 276292. i) q^{30} +(764612. - 278296. i) q^{31} +(-698240. + 832130. i) q^{32} +(-232364. + 265201. i) q^{33} +(-294238. - 107094. i) q^{34} +(2.27227e6 - 1.31189e6i) q^{35} +(966756. - 743202. i) q^{36} +(-1.13655e6 + 1.96857e6i) q^{37} +(-369734. - 440632. i) q^{38} +(-2.92633e6 - 62535.2i) q^{39} +(722945. - 4.10002e6i) q^{40} +(1.45417e6 + 256409. i) q^{41} +(33802.1 - 1.58177e6i) q^{42} +(1.39504e6 - 1.17057e6i) q^{43} +(-700657. - 404524. i) q^{44} +(-2.81856e6 + 6.82197e6i) q^{45} +(-1.82256e6 - 3.15676e6i) q^{46} +(-647951. + 1.78023e6i) q^{47} +(1.01051e6 + 885390. i) q^{48} +(-249448. - 209312. i) q^{49} +(2.50656e6 + 6.88672e6i) q^{50} +(-1.09632e6 + 2.82295e6i) q^{51} +(-1.16624e6 - 6.61408e6i) q^{52} +6.52907e6i q^{53} +(2.63669e6 + 3.58582e6i) q^{54} +4.89730e6 q^{55} +(8.49944e6 - 1.49868e6i) q^{56} +(-4.33701e6 + 3.48403e6i) q^{57} +(-2.33970e6 + 851583. i) q^{58} +(1.53319e7 - 1.82719e7i) q^{59} +(-1.66127e7 - 3.29671e6i) q^{60} +(1.95580e7 + 7.11851e6i) q^{61} +(-5.90169e6 + 3.40734e6i) q^{62} +(-1.52876e7 - 653686. i) q^{63} +(2.42571e6 - 4.20146e6i) q^{64} +(2.61317e7 + 3.11425e7i) q^{65} +(1.53082e6 - 2.52528e6i) q^{66} +(-370662. + 2.10213e6i) q^{67} +(-6.84313e6 - 1.20663e6i) q^{68} +(-3.09003e7 + 1.69706e7i) q^{69} +(-1.68334e7 + 1.41249e7i) q^{70} +(-1.54472e7 - 8.91846e6i) q^{71} +(-1.63870e7 + 1.79156e7i) q^{72} +(-1.66329e7 - 2.88091e7i) q^{73} +(6.51120e6 - 1.78894e7i) q^{74} +(6.71079e7 - 2.28137e7i) q^{75} +(-9.77838e6 - 8.20503e6i) q^{76} +(3.47226e6 + 9.53995e6i) q^{77} +(2.42269e7 - 3.74004e6i) q^{78} +(-9.65767e6 - 5.47714e7i) q^{79} -1.86604e7i q^{80} +(3.52173e7 - 2.47540e7i) q^{81} -1.23667e7 q^{82} +(-5.73546e7 + 1.01132e7i) q^{83} +(-5.35668e6 - 3.46991e7i) q^{84} +(3.95249e7 - 1.43859e7i) q^{85} +(-9.80367e6 + 1.16836e7i) q^{86} +(7.75077e6 + 2.27993e7i) q^{87} +(1.51374e7 + 5.50958e6i) q^{88} +(-4.54337e7 + 2.62311e7i) q^{89} +(1.33258e7 - 6.03657e7i) q^{90} +(-4.21380e7 + 7.29851e7i) q^{91} +(-5.19959e7 - 6.19663e7i) q^{92} +(3.17272e7 + 5.77693e7i) q^{93} +(2.75518e6 - 1.56254e7i) q^{94} +(7.60932e7 + 1.34173e7i) q^{95} +(-7.52423e7 - 4.56119e7i) q^{96} +(-7.13382e7 + 5.98599e7i) q^{97} +(2.36182e6 + 1.36360e6i) q^{98} +(-2.41014e7 - 1.53238e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9} - 3 q^{10} + 28668 q^{11} + 77421 q^{12} - 6 q^{13} - 120975 q^{14} + 105507 q^{15} - 774 q^{16} - 9 q^{17}+ \cdots + 228876057 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.24786 + 1.45432i −0.515491 + 0.0908950i −0.425341 − 0.905033i \(-0.639846\pi\)
−0.0901503 + 0.995928i \(0.528735\pi\)
\(3\) 12.3580 + 80.0517i 0.152568 + 0.988293i
\(4\) −174.649 + 63.5671i −0.682223 + 0.248309i
\(5\) 723.153 − 861.820i 1.15704 − 1.37891i 0.244644 − 0.969613i \(-0.421329\pi\)
0.912400 − 0.409299i \(-0.134227\pi\)
\(6\) −218.348 − 642.283i −0.168479 − 0.495589i
\(7\) 2191.55 + 797.660i 0.912767 + 0.332220i 0.755357 − 0.655313i \(-0.227463\pi\)
0.157410 + 0.987533i \(0.449686\pi\)
\(8\) 3204.82 − 1850.30i 0.782426 − 0.451734i
\(9\) −6255.56 + 1978.56i −0.953446 + 0.301564i
\(10\) −4711.10 + 8159.87i −0.471110 + 0.815987i
\(11\) 2798.09 + 3334.63i 0.191113 + 0.227760i 0.853089 − 0.521765i \(-0.174726\pi\)
−0.661976 + 0.749525i \(0.730282\pi\)
\(12\) −7246.97 − 13195.4i −0.349487 − 0.636352i
\(13\) −6274.91 + 35586.8i −0.219702 + 1.24599i 0.652856 + 0.757482i \(0.273571\pi\)
−0.872558 + 0.488510i \(0.837541\pi\)
\(14\) −19235.7 − 3391.77i −0.500721 − 0.0882906i
\(15\) 77926.9 + 47239.2i 1.53930 + 0.933121i
\(16\) 12706.1 − 10661.7i 0.193880 − 0.162685i
\(17\) 32378.2 + 18693.6i 0.387666 + 0.223819i 0.681148 − 0.732145i \(-0.261481\pi\)
−0.293482 + 0.955964i \(0.594814\pi\)
\(18\) 48717.5 − 25416.5i 0.464082 − 0.242117i
\(19\) 34340.1 + 59478.9i 0.263504 + 0.456403i 0.967171 − 0.254128i \(-0.0817883\pi\)
−0.703666 + 0.710531i \(0.748455\pi\)
\(20\) −71514.6 + 196485.i −0.446966 + 1.22803i
\(21\) −36770.8 + 185295.i −0.189072 + 0.952767i
\(22\) −27927.9 − 23434.3i −0.119219 − 0.100037i
\(23\) 148858. + 408984.i 0.531938 + 1.46149i 0.856761 + 0.515713i \(0.172473\pi\)
−0.324823 + 0.945775i \(0.605305\pi\)
\(24\) 187725. + 233685.i 0.565818 + 0.704346i
\(25\) −151952. − 861764.i −0.388998 − 2.20612i
\(26\) − 302641.i − 0.662268i
\(27\) −235694. − 476317.i −0.443499 − 0.896275i
\(28\) −433458. −0.705204
\(29\) 292777. − 51624.5i 0.413947 − 0.0729900i 0.0372036 − 0.999308i \(-0.488155\pi\)
0.376743 + 0.926318i \(0.377044\pi\)
\(30\) −711431. − 276292.i −0.878310 − 0.341101i
\(31\) 764612. − 278296.i 0.827932 − 0.301342i 0.106922 − 0.994267i \(-0.465901\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(32\) −698240. + 832130.i −0.665894 + 0.793581i
\(33\) −232364. + 265201.i −0.195936 + 0.223625i
\(34\) −294238. − 107094.i −0.220182 − 0.0801399i
\(35\) 2.27227e6 − 1.31189e6i 1.51421 − 0.874232i
\(36\) 966756. − 743202.i 0.575582 − 0.442483i
\(37\) −1.13655e6 + 1.96857e6i −0.606432 + 1.05037i 0.385391 + 0.922753i \(0.374067\pi\)
−0.991823 + 0.127618i \(0.959267\pi\)
\(38\) −369734. − 440632.i −0.177319 − 0.211321i
\(39\) −2.92633e6 − 62535.2i −1.26492 − 0.0270312i
\(40\) 722945. − 4.10002e6i 0.282400 − 1.60157i
\(41\) 1.45417e6 + 256409.i 0.514612 + 0.0907400i 0.424922 − 0.905230i \(-0.360301\pi\)
0.0896896 + 0.995970i \(0.471413\pi\)
\(42\) 33802.1 − 1.58177e6i 0.0108629 − 0.508329i
\(43\) 1.39504e6 − 1.17057e6i 0.408048 − 0.342393i −0.415546 − 0.909572i \(-0.636410\pi\)
0.823595 + 0.567179i \(0.191965\pi\)
\(44\) −700657. − 404524.i −0.186937 − 0.107928i
\(45\) −2.81856e6 + 6.82197e6i −0.687349 + 1.66364i
\(46\) −1.82256e6 − 3.15676e6i −0.407052 − 0.705034i
\(47\) −647951. + 1.78023e6i −0.132786 + 0.364825i −0.988210 − 0.153102i \(-0.951074\pi\)
0.855425 + 0.517927i \(0.173296\pi\)
\(48\) 1.01051e6 + 885390.i 0.190360 + 0.166790i
\(49\) −249448. − 209312.i −0.0432709 − 0.0363086i
\(50\) 2.50656e6 + 6.88672e6i 0.401050 + 1.10188i
\(51\) −1.09632e6 + 2.82295e6i −0.162053 + 0.417275i
\(52\) −1.16624e6 − 6.61408e6i −0.159505 − 0.904599i
\(53\) 6.52907e6i 0.827462i 0.910399 + 0.413731i \(0.135775\pi\)
−0.910399 + 0.413731i \(0.864225\pi\)
\(54\) 2.63669e6 + 3.58582e6i 0.310087 + 0.421710i
\(55\) 4.89730e6 0.535187
\(56\) 8.49944e6 − 1.49868e6i 0.864247 − 0.152390i
\(57\) −4.33701e6 + 3.48403e6i −0.410857 + 0.330052i
\(58\) −2.33970e6 + 851583.i −0.206752 + 0.0752515i
\(59\) 1.53319e7 − 1.82719e7i 1.26528 − 1.50791i 0.497076 − 0.867707i \(-0.334407\pi\)
0.768208 − 0.640200i \(-0.221149\pi\)
\(60\) −1.66127e7 − 3.29671e6i −1.28185 − 0.254375i
\(61\) 1.95580e7 + 7.11851e6i 1.41255 + 0.514126i 0.931877 − 0.362773i \(-0.118170\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(62\) −5.90169e6 + 3.40734e6i −0.399401 + 0.230594i
\(63\) −1.52876e7 − 653686.i −0.970460 − 0.0414961i
\(64\) 2.42571e6 − 4.20146e6i 0.144584 − 0.250426i
\(65\) 2.61317e7 + 3.11425e7i 1.46391 + 1.74462i
\(66\) 1.53082e6 − 2.52528e6i 0.0806768 − 0.133086i
\(67\) −370662. + 2.10213e6i −0.0183941 + 0.104318i −0.992623 − 0.121245i \(-0.961311\pi\)
0.974228 + 0.225564i \(0.0724223\pi\)
\(68\) −6.84313e6 − 1.20663e6i −0.320051 − 0.0564336i
\(69\) −3.09003e7 + 1.69706e7i −1.36322 + 0.748687i
\(70\) −1.68334e7 + 1.41249e7i −0.701101 + 0.588294i
\(71\) −1.54472e7 − 8.91846e6i −0.607879 − 0.350959i 0.164256 − 0.986418i \(-0.447478\pi\)
−0.772135 + 0.635459i \(0.780811\pi\)
\(72\) −1.63870e7 + 1.79156e7i −0.609774 + 0.666655i
\(73\) −1.66329e7 − 2.88091e7i −0.585703 − 1.01447i −0.994787 − 0.101971i \(-0.967485\pi\)
0.409084 − 0.912497i \(-0.365848\pi\)
\(74\) 6.51120e6 − 1.78894e7i 0.217137 − 0.596579i
\(75\) 6.71079e7 − 2.28137e7i 2.12094 − 0.721027i
\(76\) −9.77838e6 − 8.20503e6i −0.293098 − 0.245938i
\(77\) 3.47226e6 + 9.53995e6i 0.0987755 + 0.271383i
\(78\) 2.42269e7 − 3.74004e6i 0.654515 − 0.101041i
\(79\) −9.65767e6 − 5.47714e7i −0.247950 − 1.40619i −0.813541 − 0.581508i \(-0.802463\pi\)
0.565591 − 0.824686i \(-0.308648\pi\)
\(80\) − 1.86604e7i − 0.455577i
\(81\) 3.52173e7 − 2.47540e7i 0.818118 − 0.575050i
\(82\) −1.23667e7 −0.273526
\(83\) −5.73546e7 + 1.01132e7i −1.20853 + 0.213096i −0.741381 − 0.671085i \(-0.765829\pi\)
−0.467145 + 0.884181i \(0.654717\pi\)
\(84\) −5.35668e6 − 3.46991e7i −0.107592 − 0.696948i
\(85\) 3.95249e7 − 1.43859e7i 0.757173 − 0.275588i
\(86\) −9.80367e6 + 1.16836e7i −0.179223 + 0.213590i
\(87\) 7.75077e6 + 2.27993e7i 0.135291 + 0.397965i
\(88\) 1.51374e7 + 5.50958e6i 0.252419 + 0.0918729i
\(89\) −4.54337e7 + 2.62311e7i −0.724132 + 0.418078i −0.816272 − 0.577668i \(-0.803963\pi\)
0.0921395 + 0.995746i \(0.470629\pi\)
\(90\) 1.33258e7 − 6.03657e7i 0.203106 − 0.920069i
\(91\) −4.21380e7 + 7.29851e7i −0.614480 + 1.06431i
\(92\) −5.19959e7 − 6.19663e7i −0.725801 − 0.864976i
\(93\) 3.17272e7 + 5.77693e7i 0.424131 + 0.772264i
\(94\) 2.75518e6 − 1.56254e7i 0.0352890 − 0.200134i
\(95\) 7.60932e7 + 1.34173e7i 0.934225 + 0.164729i
\(96\) −7.52423e7 − 4.56119e7i −0.885885 − 0.537023i
\(97\) −7.13382e7 + 5.98599e7i −0.805815 + 0.676159i −0.949605 − 0.313449i \(-0.898515\pi\)
0.143790 + 0.989608i \(0.454071\pi\)
\(98\) 2.36182e6 + 1.36360e6i 0.0256061 + 0.0147837i
\(99\) −2.41014e7 − 1.53238e7i −0.250900 − 0.159524i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 27.9.f.a.14.8 yes 138
3.2 odd 2 81.9.f.a.71.16 138
27.2 odd 18 inner 27.9.f.a.2.8 ✓ 138
27.25 even 9 81.9.f.a.8.16 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.8 ✓ 138 27.2 odd 18 inner
27.9.f.a.14.8 yes 138 1.1 even 1 trivial
81.9.f.a.8.16 138 27.25 even 9
81.9.f.a.71.16 138 3.2 odd 2