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 71.16
Character \(\chi\) \(=\) 81.71
Dual form 81.9.f.a.8.16

$q$-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 \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

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

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 0.0901503 − 0.995928i \(-0.471265\pi\)
0.425341 + 0.905033i \(0.360154\pi\)
\(3\) 0 0
\(4\) −174.649 + 63.5671i −0.682223 + 0.248309i
\(5\) −723.153 + 861.820i −1.15704 + 1.37891i −0.244644 + 0.969613i \(0.578671\pi\)
−0.912400 + 0.409299i \(0.865773\pi\)
\(6\) 0 0
\(7\) 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
\(9\) 0 0
\(10\) −4711.10 + 8159.87i −0.471110 + 0.815987i
\(11\) −2798.09 − 3334.63i −0.191113 − 0.227760i 0.661976 − 0.749525i \(-0.269718\pi\)
−0.853089 + 0.521765i \(0.825274\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) 12706.1 − 10661.7i 0.193880 − 0.162685i
\(17\) −32378.2 − 18693.6i −0.387666 − 0.223819i 0.293482 − 0.955964i \(-0.405186\pi\)
−0.681148 + 0.732145i \(0.738519\pi\)
\(18\) 0 0
\(19\) 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
\(21\) 0 0
\(22\) −27927.9 − 23434.3i −0.119219 − 0.100037i
\(23\) −148858. − 408984.i −0.531938 − 1.46149i −0.856761 − 0.515713i \(-0.827527\pi\)
0.324823 − 0.945775i \(-0.394695\pi\)
\(24\) 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 −0.376743 − 0.926318i \(-0.622956\pi\)
−0.0372036 + 0.999308i \(0.511845\pi\)
\(30\) 0 0
\(31\) 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
\(33\) 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 0.385391 + 0.922753i \(0.374067\pi\)
−0.991823 + 0.127618i \(0.959267\pi\)
\(38\) 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 −0.0896896 − 0.995970i \(-0.528587\pi\)
−0.424922 + 0.905230i \(0.639699\pi\)
\(42\) 0 0
\(43\) 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
\(45\) 0 0
\(46\) −1.82256e6 − 3.15676e6i −0.407052 − 0.705034i
\(47\) 647951. − 1.78023e6i 0.132786 − 0.364825i −0.855425 − 0.517927i \(-0.826704\pi\)
0.988210 + 0.153102i \(0.0489262\pi\)
\(48\) 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 \(-0.864225\pi\)
0.910399 − 0.413731i \(-0.135775\pi\)
\(54\) 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 −0.497076 + 0.867707i \(0.665593\pi\)
−0.768208 + 0.640200i \(0.778851\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 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 −0.992623 − 0.121245i \(-0.961311\pi\)
0.974228 + 0.225564i \(0.0724223\pi\)
\(68\) 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 0.772135 − 0.635459i \(-0.219189\pi\)
−0.164256 + 0.986418i \(0.552522\pi\)
\(72\) 0 0
\(73\) −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
\(75\) 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 −0.813541 − 0.581508i \(-0.802463\pi\)
0.565591 − 0.824686i \(-0.308648\pi\)
\(80\) 1.86604e7i 0.455577i
\(81\) 0 0
\(82\) −1.23667e7 −0.273526
\(83\) 5.73546e7 − 1.01132e7i 1.20853 − 0.213096i 0.467145 − 0.884181i \(-0.345283\pi\)
0.741381 + 0.671085i \(0.234171\pi\)
\(84\) 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 −0.0921395 − 0.995746i \(-0.529371\pi\)
0.816272 + 0.577668i \(0.196037\pi\)
\(90\) 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 −0.949605 − 0.313449i \(-0.898515\pi\)
0.143790 + 0.989608i \(0.454071\pi\)
\(98\) −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.71.16 138
3.2 odd 2 27.9.f.a.14.8 yes 138
27.2 odd 18 inner 81.9.f.a.8.16 138
27.25 even 9 27.9.f.a.2.8 ✓ 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.8 ✓ 138 27.25 even 9
27.9.f.a.14.8 yes 138 3.2 odd 2
81.9.f.a.8.16 138 27.2 odd 18 inner
81.9.f.a.71.16 138 1.1 even 1 trivial