Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,9,Mod(8,81)] 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("81.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.f (of order \(18\), degree \(6\), not 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: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 8.16
Character \(\chi\) \(=\) 81.8
Dual form 81.9.f.a.71.16

$q$-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} +(-174.649 - 63.5671i) q^{4} +(-723.153 - 861.820i) q^{5} +(2191.55 - 797.660i) q^{7} +(-3204.82 - 1850.30i) q^{8} +(-4711.10 - 8159.87i) q^{10} +(-2798.09 + 3334.63i) q^{11} +(-6274.91 - 35586.8i) q^{13} +(19235.7 - 3391.77i) q^{14} +(12706.1 + 10661.7i) q^{16} +(-32378.2 + 18693.6i) q^{17} +(34340.1 - 59478.9i) q^{19} +(71514.6 + 196485. i) q^{20} +(-27927.9 + 23434.3i) q^{22} +(-148858. + 408984. i) q^{23} +(-151952. + 861764. i) q^{25} -302641. i q^{26} -433458. q^{28} +(-292777. - 51624.5i) q^{29} +(764612. + 278296. i) q^{31} +(698240. + 832130. i) q^{32} +(-294238. + 107094. i) q^{34} +(-2.27227e6 - 1.31189e6i) q^{35} +(-1.13655e6 - 1.96857e6i) q^{37} +(369734. - 440632. i) q^{38} +(722945. + 4.10002e6i) q^{40} +(-1.45417e6 + 256409. i) q^{41} +(1.39504e6 + 1.17057e6i) q^{43} +(700657. - 404524. i) q^{44} +(-1.82256e6 + 3.15676e6i) q^{46} +(647951. + 1.78023e6i) q^{47} +(-249448. + 209312. i) q^{49} +(-2.50656e6 + 6.88672e6i) q^{50} +(-1.16624e6 + 6.61408e6i) q^{52} +6.52907e6i q^{53} +4.89730e6 q^{55} +(-8.49944e6 - 1.49868e6i) q^{56} +(-2.33970e6 - 851583. i) q^{58} +(-1.53319e7 - 1.82719e7i) q^{59} +(1.95580e7 - 7.11851e6i) q^{61} +(5.90169e6 + 3.40734e6i) q^{62} +(2.42571e6 + 4.20146e6i) q^{64} +(-2.61317e7 + 3.11425e7i) q^{65} +(-370662. - 2.10213e6i) q^{67} +(6.84313e6 - 1.20663e6i) q^{68} +(-1.68334e7 - 1.41249e7i) q^{70} +(1.54472e7 - 8.91846e6i) q^{71} +(-1.66329e7 + 2.88091e7i) q^{73} +(-6.51120e6 - 1.78894e7i) q^{74} +(-9.77838e6 + 8.20503e6i) q^{76} +(-3.47226e6 + 9.53995e6i) q^{77} +(-9.65767e6 + 5.47714e7i) q^{79} -1.86604e7i q^{80} -1.23667e7 q^{82} +(5.73546e7 + 1.01132e7i) q^{83} +(3.95249e7 + 1.43859e7i) q^{85} +(9.80367e6 + 1.16836e7i) q^{86} +(1.51374e7 - 5.50958e6i) q^{88} +(4.54337e7 + 2.62311e7i) q^{89} +(-4.21380e7 - 7.29851e7i) q^{91} +(5.19959e7 - 6.19663e7i) q^{92} +(2.75518e6 + 1.56254e7i) q^{94} +(-7.60932e7 + 1.34173e7i) q^{95} +(-7.13382e7 - 5.98599e7i) q^{97} +(-2.36182e6 + 1.36360e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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 \(-0.360154\pi\)
0.0901503 + 0.995928i \(0.471265\pi\)
\(3\) 0 0
\(4\) −174.649 − 63.5671i −0.682223 − 0.248309i
\(5\) −723.153 − 861.820i −1.15704 − 1.37891i −0.912400 − 0.409299i \(-0.865773\pi\)
−0.244644 − 0.969613i \(-0.578671\pi\)
\(6\) 0 0
\(7\) 2191.55 − 797.660i 0.912767 − 0.332220i 0.157410 − 0.987533i \(-0.449686\pi\)
0.755357 + 0.655313i \(0.227463\pi\)
\(8\) −3204.82 − 1850.30i −0.782426 − 0.451734i
\(9\) 0 0
\(10\) −4711.10 − 8159.87i −0.471110 − 0.815987i
\(11\) −2798.09 + 3334.63i −0.191113 + 0.227760i −0.853089 − 0.521765i \(-0.825274\pi\)
0.661976 + 0.749525i \(0.269718\pi\)
\(12\) 0 0
\(13\) −6274.91 − 35586.8i −0.219702 − 1.24599i −0.872558 − 0.488510i \(-0.837541\pi\)
0.652856 − 0.757482i \(-0.273571\pi\)
\(14\) 19235.7 − 3391.77i 0.500721 − 0.0882906i
\(15\) 0 0
\(16\) 12706.1 + 10661.7i 0.193880 + 0.162685i
\(17\) −32378.2 + 18693.6i −0.387666 + 0.223819i −0.681148 − 0.732145i \(-0.738519\pi\)
0.293482 + 0.955964i \(0.405186\pi\)
\(18\) 0 0
\(19\) 34340.1 − 59478.9i 0.263504 − 0.456403i −0.703666 − 0.710531i \(-0.748455\pi\)
0.967171 + 0.254128i \(0.0817883\pi\)
\(20\) 71514.6 + 196485.i 0.446966 + 1.22803i
\(21\) 0 0
\(22\) −27927.9 + 23434.3i −0.119219 + 0.100037i
\(23\) −148858. + 408984.i −0.531938 + 1.46149i 0.324823 + 0.945775i \(0.394695\pi\)
−0.856761 + 0.515713i \(0.827527\pi\)
\(24\) 0 0
\(25\) −151952. + 861764.i −0.388998 + 2.20612i
\(26\) − 302641.i − 0.662268i
\(27\) 0 0
\(28\) −433458. −0.705204
\(29\) −292777. − 51624.5i −0.413947 − 0.0729900i −0.0372036 − 0.999308i \(-0.511845\pi\)
−0.376743 + 0.926318i \(0.622956\pi\)
\(30\) 0 0
\(31\) 764612. + 278296.i 0.827932 + 0.301342i 0.721010 − 0.692925i \(-0.243678\pi\)
0.106922 + 0.994267i \(0.465901\pi\)
\(32\) 698240. + 832130.i 0.665894 + 0.793581i
\(33\) 0 0
\(34\) −294238. + 107094.i −0.220182 + 0.0801399i
\(35\) −2.27227e6 − 1.31189e6i −1.51421 − 0.874232i
\(36\) 0 0
\(37\) −1.13655e6 − 1.96857e6i −0.606432 − 1.05037i −0.991823 − 0.127618i \(-0.959267\pi\)
0.385391 − 0.922753i \(-0.374067\pi\)
\(38\) 369734. − 440632.i 0.177319 − 0.211321i
\(39\) 0 0
\(40\) 722945. + 4.10002e6i 0.282400 + 1.60157i
\(41\) −1.45417e6 + 256409.i −0.514612 + 0.0907400i −0.424922 − 0.905230i \(-0.639699\pi\)
−0.0896896 + 0.995970i \(0.528587\pi\)
\(42\) 0 0
\(43\) 1.39504e6 + 1.17057e6i 0.408048 + 0.342393i 0.823595 − 0.567179i \(-0.191965\pi\)
−0.415546 + 0.909572i \(0.636410\pi\)
\(44\) 700657. − 404524.i 0.186937 − 0.107928i
\(45\) 0 0
\(46\) −1.82256e6 + 3.15676e6i −0.407052 + 0.705034i
\(47\) 647951. + 1.78023e6i 0.132786 + 0.364825i 0.988210 − 0.153102i \(-0.0489262\pi\)
−0.855425 + 0.517927i \(0.826704\pi\)
\(48\) 0 0
\(49\) −249448. + 209312.i −0.0432709 + 0.0363086i
\(50\) −2.50656e6 + 6.88672e6i −0.401050 + 1.10188i
\(51\) 0 0
\(52\) −1.16624e6 + 6.61408e6i −0.159505 + 0.904599i
\(53\) 6.52907e6i 0.827462i 0.910399 + 0.413731i \(0.135775\pi\)
−0.910399 + 0.413731i \(0.864225\pi\)
\(54\) 0 0
\(55\) 4.89730e6 0.535187
\(56\) −8.49944e6 − 1.49868e6i −0.864247 − 0.152390i
\(57\) 0 0
\(58\) −2.33970e6 − 851583.i −0.206752 − 0.0752515i
\(59\) −1.53319e7 − 1.82719e7i −1.26528 − 1.50791i −0.768208 − 0.640200i \(-0.778851\pi\)
−0.497076 − 0.867707i \(-0.665593\pi\)
\(60\) 0 0
\(61\) 1.95580e7 − 7.11851e6i 1.41255 − 0.514126i 0.480673 − 0.876900i \(-0.340392\pi\)
0.931877 + 0.362773i \(0.118170\pi\)
\(62\) 5.90169e6 + 3.40734e6i 0.399401 + 0.230594i
\(63\) 0 0
\(64\) 2.42571e6 + 4.20146e6i 0.144584 + 0.250426i
\(65\) −2.61317e7 + 3.11425e7i −1.46391 + 1.74462i
\(66\) 0 0
\(67\) −370662. − 2.10213e6i −0.0183941 − 0.104318i 0.974228 − 0.225564i \(-0.0724223\pi\)
−0.992623 + 0.121245i \(0.961311\pi\)
\(68\) 6.84313e6 − 1.20663e6i 0.320051 − 0.0564336i
\(69\) 0 0
\(70\) −1.68334e7 − 1.41249e7i −0.701101 − 0.588294i
\(71\) 1.54472e7 − 8.91846e6i 0.607879 − 0.350959i −0.164256 − 0.986418i \(-0.552522\pi\)
0.772135 + 0.635459i \(0.219189\pi\)
\(72\) 0 0
\(73\) −1.66329e7 + 2.88091e7i −0.585703 + 1.01447i 0.409084 + 0.912497i \(0.365848\pi\)
−0.994787 + 0.101971i \(0.967485\pi\)
\(74\) −6.51120e6 − 1.78894e7i −0.217137 − 0.596579i
\(75\) 0 0
\(76\) −9.77838e6 + 8.20503e6i −0.293098 + 0.245938i
\(77\) −3.47226e6 + 9.53995e6i −0.0987755 + 0.271383i
\(78\) 0 0
\(79\) −9.65767e6 + 5.47714e7i −0.247950 + 1.40619i 0.565591 + 0.824686i \(0.308648\pi\)
−0.813541 + 0.581508i \(0.802463\pi\)
\(80\) − 1.86604e7i − 0.455577i
\(81\) 0 0
\(82\) −1.23667e7 −0.273526
\(83\) 5.73546e7 + 1.01132e7i 1.20853 + 0.213096i 0.741381 − 0.671085i \(-0.234171\pi\)
0.467145 + 0.884181i \(0.345283\pi\)
\(84\) 0 0
\(85\) 3.95249e7 + 1.43859e7i 0.757173 + 0.275588i
\(86\) 9.80367e6 + 1.16836e7i 0.179223 + 0.213590i
\(87\) 0 0
\(88\) 1.51374e7 − 5.50958e6i 0.252419 − 0.0918729i
\(89\) 4.54337e7 + 2.62311e7i 0.724132 + 0.418078i 0.816272 − 0.577668i \(-0.196037\pi\)
−0.0921395 + 0.995746i \(0.529371\pi\)
\(90\) 0 0
\(91\) −4.21380e7 − 7.29851e7i −0.614480 − 1.06431i
\(92\) 5.19959e7 − 6.19663e7i 0.725801 − 0.864976i
\(93\) 0 0
\(94\) 2.75518e6 + 1.56254e7i 0.0352890 + 0.200134i
\(95\) −7.60932e7 + 1.34173e7i −0.934225 + 0.164729i
\(96\) 0 0
\(97\) −7.13382e7 − 5.98599e7i −0.805815 − 0.676159i 0.143790 − 0.989608i \(-0.454071\pi\)
−0.949605 + 0.313449i \(0.898515\pi\)
\(98\) −2.36182e6 + 1.36360e6i −0.0256061 + 0.0147837i
\(99\) 0 0
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 81.9.f.a.8.16 138
3.2 odd 2 27.9.f.a.2.8 ✓ 138
27.13 even 9 27.9.f.a.14.8 yes 138
27.14 odd 18 inner 81.9.f.a.71.16 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.8 ✓ 138 3.2 odd 2
27.9.f.a.14.8 yes 138 27.13 even 9
81.9.f.a.8.16 138 1.1 even 1 trivial
81.9.f.a.71.16 138 27.14 odd 18 inner