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 71.2
Character \(\chi\) \(=\) 81.71
Dual form 81.9.f.a.8.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-26.8116 + 4.72761i) q^{2} +(455.951 - 165.953i) q^{4} +(-232.968 + 277.641i) q^{5} +(45.0827 + 16.4088i) q^{7} +(-5404.32 + 3120.19i) q^{8} +(4933.68 - 8545.38i) q^{10} +(735.313 + 876.312i) q^{11} +(6621.57 - 37552.8i) q^{13} +(-1286.31 - 226.812i) q^{14} +(34993.7 - 29363.2i) q^{16} +(-67597.9 - 39027.7i) q^{17} +(113236. + 196131. i) q^{19} +(-60146.9 + 165252. i) q^{20} +(-23857.8 - 20019.1i) q^{22} +(82558.0 + 226826. i) q^{23} +(45021.1 + 255328. i) q^{25} +1.03816e6i q^{26} +23278.6 q^{28} +(-572834. + 101006. i) q^{29} +(-24300.0 + 8844.49i) q^{31} +(227456. - 271071. i) q^{32} +(1.99692e6 + 726818. i) q^{34} +(-15058.6 + 8694.08i) q^{35} +(-1.45625e6 + 2.52229e6i) q^{37} +(-3.96328e6 - 4.72325e6i) q^{38} +(392744. - 2.22736e6i) q^{40} +(-535074. - 94347.9i) q^{41} +(4.54620e6 - 3.81472e6i) q^{43} +(480693. + 277528. i) q^{44} +(-3.28586e6 - 5.69128e6i) q^{46} +(3.01327e6 - 8.27890e6i) q^{47} +(-4.41433e6 - 3.70406e6i) q^{49} +(-2.41418e6 - 6.63290e6i) q^{50} +(-3.21287e6 - 1.82211e7i) q^{52} +3.89255e6i q^{53} -414605. q^{55} +(-294840. + 51988.2i) q^{56} +(1.48811e7 - 5.41627e6i) q^{58} +(-2.64580e6 + 3.15314e6i) q^{59} +(-1.75097e7 - 6.37302e6i) q^{61} +(609710. - 352016. i) q^{62} +(-1.06641e7 + 1.84708e7i) q^{64} +(8.88358e6 + 1.05870e7i) q^{65} +(-493265. + 2.79744e6i) q^{67} +(-3.72981e7 - 6.57666e6i) q^{68} +(362643. - 304293. i) q^{70} +(-893484. - 515853. i) q^{71} +(-1.26668e7 - 2.19395e7i) q^{73} +(2.71199e7 - 7.45114e7i) q^{74} +(8.41786e7 + 7.06343e7i) q^{76} +(18770.7 + 51572.1i) q^{77} +(-1.00312e7 - 5.68895e7i) q^{79} +1.65564e7i q^{80} +1.47922e7 q^{82} +(-1.09503e7 + 1.93083e6i) q^{83} +(2.65838e7 - 9.67572e6i) q^{85} +(-1.03856e8 + 1.23771e8i) q^{86} +(-6.70813e6 - 2.44156e6i) q^{88} +(2.45007e7 - 1.41455e7i) q^{89} +(914714. - 1.58433e6i) q^{91} +(7.52848e7 + 8.97209e7i) q^{92} +(-4.16513e7 + 2.36216e8i) q^{94} +(-8.08344e7 - 1.42533e7i) q^{95} +(-4.04609e7 + 3.39507e7i) q^{97} +(1.35867e8 + 7.84427e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −26.8116 + 4.72761i −1.67573 + 0.295476i −0.929117 − 0.369787i \(-0.879431\pi\)
−0.746609 + 0.665263i \(0.768320\pi\)
\(3\) 0 0
\(4\) 455.951 − 165.953i 1.78106 − 0.648252i
\(5\) −232.968 + 277.641i −0.372749 + 0.444225i −0.919512 − 0.393063i \(-0.871416\pi\)
0.546763 + 0.837288i \(0.315860\pi\)
\(6\) 0 0
\(7\) 45.0827 + 16.4088i 0.0187766 + 0.00683414i 0.351391 − 0.936229i \(-0.385709\pi\)
−0.332615 + 0.943063i \(0.607931\pi\)
\(8\) −5404.32 + 3120.19i −1.31941 + 0.761764i
\(9\) 0 0
\(10\) 4933.68 − 8545.38i 0.493368 − 0.854538i
\(11\) 735.313 + 876.312i 0.0502229 + 0.0598533i 0.790571 − 0.612370i \(-0.209784\pi\)
−0.740348 + 0.672224i \(0.765339\pi\)
\(12\) 0 0
\(13\) 6621.57 − 37552.8i 0.231840 − 1.31483i −0.617328 − 0.786706i \(-0.711785\pi\)
0.849168 − 0.528123i \(-0.177104\pi\)
\(14\) −1286.31 − 226.812i −0.0334838 − 0.00590410i
\(15\) 0 0
\(16\) 34993.7 − 29363.2i 0.533962 − 0.448047i
\(17\) −67597.9 − 39027.7i −0.809352 − 0.467280i 0.0373788 − 0.999301i \(-0.488099\pi\)
−0.846731 + 0.532022i \(0.821433\pi\)
\(18\) 0 0
\(19\) 113236. + 196131.i 0.868903 + 1.50498i 0.863119 + 0.505000i \(0.168508\pi\)
0.00578360 + 0.999983i \(0.498159\pi\)
\(20\) −60146.9 + 165252.i −0.375918 + 1.03283i
\(21\) 0 0
\(22\) −23857.8 − 20019.1i −0.101845 − 0.0854581i
\(23\) 82558.0 + 226826.i 0.295018 + 0.810554i 0.995313 + 0.0967014i \(0.0308292\pi\)
−0.700296 + 0.713853i \(0.746949\pi\)
\(24\) 0 0
\(25\) 45021.1 + 255328.i 0.115254 + 0.653639i
\(26\) 1.03816e6i 2.27179i
\(27\) 0 0
\(28\) 23278.6 0.0378725
\(29\) −572834. + 101006.i −0.809910 + 0.142809i −0.563243 − 0.826291i \(-0.690447\pi\)
−0.246667 + 0.969100i \(0.579335\pi\)
\(30\) 0 0
\(31\) −24300.0 + 8844.49i −0.0263124 + 0.00957693i −0.355143 − 0.934812i \(-0.615568\pi\)
0.328830 + 0.944389i \(0.393346\pi\)
\(32\) 227456. − 271071.i 0.216919 − 0.258514i
\(33\) 0 0
\(34\) 1.99692e6 + 726818.i 1.49432 + 0.543889i
\(35\) −15058.6 + 8694.08i −0.0100349 + 0.00579364i
\(36\) 0 0
\(37\) −1.45625e6 + 2.52229e6i −0.777013 + 1.34583i 0.156643 + 0.987655i \(0.449933\pi\)
−0.933656 + 0.358171i \(0.883401\pi\)
\(38\) −3.96328e6 − 4.72325e6i −1.90073 − 2.26520i
\(39\) 0 0
\(40\) 392744. − 2.22736e6i 0.153416 − 0.870064i
\(41\) −535074. − 94347.9i −0.189356 − 0.0333885i 0.0781660 − 0.996940i \(-0.475094\pi\)
−0.267522 + 0.963552i \(0.586205\pi\)
\(42\) 0 0
\(43\) 4.54620e6 − 3.81472e6i 1.32976 − 1.11581i 0.345632 − 0.938370i \(-0.387665\pi\)
0.984133 − 0.177435i \(-0.0567799\pi\)
\(44\) 480693. + 277528.i 0.128250 + 0.0740451i
\(45\) 0 0
\(46\) −3.28586e6 − 5.69128e6i −0.733868 − 1.27110i
\(47\) 3.01327e6 − 8.27890e6i 0.617515 − 1.69661i −0.0954743 − 0.995432i \(-0.530437\pi\)
0.712989 − 0.701175i \(-0.247341\pi\)
\(48\) 0 0
\(49\) −4.41433e6 − 3.70406e6i −0.765739 − 0.642531i
\(50\) −2.41418e6 − 6.63290e6i −0.386269 − 1.06126i
\(51\) 0 0
\(52\) −3.21287e6 − 1.82211e7i −0.439420 − 2.49208i
\(53\) 3.89255e6i 0.493322i 0.969102 + 0.246661i \(0.0793335\pi\)
−0.969102 + 0.246661i \(0.920667\pi\)
\(54\) 0 0
\(55\) −414605. −0.0453089
\(56\) −294840. + 51988.2i −0.0299802 + 0.00528631i
\(57\) 0 0
\(58\) 1.48811e7 − 5.41627e6i 1.31499 − 0.478618i
\(59\) −2.64580e6 + 3.15314e6i −0.218348 + 0.260217i −0.864088 − 0.503340i \(-0.832104\pi\)
0.645741 + 0.763557i \(0.276549\pi\)
\(60\) 0 0
\(61\) −1.75097e7 − 6.37302e6i −1.26462 − 0.460284i −0.379303 − 0.925273i \(-0.623836\pi\)
−0.885317 + 0.464989i \(0.846058\pi\)
\(62\) 609710. − 352016.i 0.0412626 − 0.0238230i
\(63\) 0 0
\(64\) −1.06641e7 + 1.84708e7i −0.635631 + 1.10094i
\(65\) 8.88358e6 + 1.05870e7i 0.497662 + 0.593090i
\(66\) 0 0
\(67\) −493265. + 2.79744e6i −0.0244783 + 0.138823i −0.994598 − 0.103805i \(-0.966898\pi\)
0.970119 + 0.242628i \(0.0780094\pi\)
\(68\) −3.72981e7 − 6.57666e6i −1.74442 − 0.307588i
\(69\) 0 0
\(70\) 362643. − 304293.i 0.0151038 − 0.0126736i
\(71\) −893484. − 515853.i −0.0351603 − 0.0202998i 0.482317 − 0.875997i \(-0.339795\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(72\) 0 0
\(73\) −1.26668e7 − 2.19395e7i −0.446042 − 0.772567i 0.552082 − 0.833790i \(-0.313833\pi\)
−0.998124 + 0.0612227i \(0.980500\pi\)
\(74\) 2.71199e7 − 7.45114e7i 0.904402 − 2.48482i
\(75\) 0 0
\(76\) 8.41786e7 + 7.06343e7i 2.52318 + 2.11720i
\(77\) 18770.7 + 51572.1i 0.000533971 + 0.00146707i
\(78\) 0 0
\(79\) −1.00312e7 − 5.68895e7i −0.257539 − 1.46057i −0.789471 − 0.613788i \(-0.789645\pi\)
0.531932 − 0.846787i \(-0.321466\pi\)
\(80\) 1.65564e7i 0.404209i
\(81\) 0 0
\(82\) 1.47922e7 0.327174
\(83\) −1.09503e7 + 1.93083e6i −0.230735 + 0.0406848i −0.287820 − 0.957684i \(-0.592931\pi\)
0.0570850 + 0.998369i \(0.481819\pi\)
\(84\) 0 0
\(85\) 2.65838e7 − 9.67572e6i 0.509263 − 0.185356i
\(86\) −1.03856e8 + 1.23771e8i −1.89863 + 2.26270i
\(87\) 0 0
\(88\) −6.70813e6 − 2.44156e6i −0.111859 − 0.0407133i
\(89\) 2.45007e7 − 1.41455e7i 0.390497 − 0.225454i −0.291878 − 0.956455i \(-0.594280\pi\)
0.682376 + 0.731002i \(0.260947\pi\)
\(90\) 0 0
\(91\) 914714. − 1.58433e6i 0.0133389 − 0.0231036i
\(92\) 7.52848e7 + 8.97209e7i 1.05089 + 1.25240i
\(93\) 0 0
\(94\) −4.16513e7 + 2.36216e8i −0.533479 + 3.02551i
\(95\) −8.08344e7 − 1.42533e7i −0.992434 − 0.174993i
\(96\) 0 0
\(97\) −4.04609e7 + 3.39507e7i −0.457034 + 0.383497i −0.842038 − 0.539418i \(-0.818644\pi\)
0.385005 + 0.922915i \(0.374200\pi\)
\(98\) 1.35867e8 + 7.84427e7i 1.47302 + 0.850449i
\(99\) 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.71.2 138
3.2 odd 2 27.9.f.a.14.22 yes 138
27.2 odd 18 inner 81.9.f.a.8.2 138
27.25 even 9 27.9.f.a.2.22 ✓ 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.22 ✓ 138 27.25 even 9
27.9.f.a.14.22 yes 138 3.2 odd 2
81.9.f.a.8.2 138 27.2 odd 18 inner
81.9.f.a.71.2 138 1.1 even 1 trivial