Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,2,Mod(10,81)] 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("81.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 81.e (of order \(9\), 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: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.1952986685049.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 64.2
Root \(0.500000 - 1.27297i\) of defining polynomial
Character \(\chi\) \(=\) 81.64
Dual form 81.2.e.a.19.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.57954 + 0.574906i) q^{2} +(0.632343 + 0.530599i) q^{4} +(0.196143 - 1.11238i) q^{5} +(-2.99441 + 2.51261i) q^{7} +(-0.987144 - 1.70978i) q^{8} +(0.949332 - 1.64429i) q^{10} +(0.324801 + 1.84204i) q^{11} +(0.688417 - 0.250563i) q^{13} +(-6.17430 + 2.24726i) q^{14} +(-0.862951 - 4.89404i) q^{16} +(0.944822 - 1.63648i) q^{17} +(-1.37143 - 2.37538i) q^{19} +(0.714260 - 0.599335i) q^{20} +(-0.545962 + 3.09631i) q^{22} +(4.46428 + 3.74597i) q^{23} +(3.49954 + 1.27373i) q^{25} +1.23143 q^{26} -3.22668 q^{28} +(-4.99910 - 1.81953i) q^{29} +(1.02696 + 0.861722i) q^{31} +(0.764882 - 4.33786i) q^{32} +(2.43321 - 2.04170i) q^{34} +(2.20765 + 3.82376i) q^{35} +(-1.69806 + 2.94112i) q^{37} +(-0.800605 - 4.54046i) q^{38} +(-2.09556 + 0.762721i) q^{40} +(1.68800 - 0.614382i) q^{41} +(0.873477 + 4.95373i) q^{43} +(-0.771999 + 1.33714i) q^{44} +(4.89793 + 8.48346i) q^{46} +(-1.30892 + 1.09832i) q^{47} +(1.43775 - 8.15389i) q^{49} +(4.79539 + 4.02381i) q^{50} +(0.568265 + 0.206831i) q^{52} -2.84494 q^{53} +2.11276 q^{55} +(7.25193 + 2.63949i) q^{56} +(-6.85023 - 5.74803i) q^{58} +(1.95529 - 11.0890i) q^{59} +(4.00710 - 3.36235i) q^{61} +(1.12672 + 1.95153i) q^{62} +(-1.26751 + 2.19540i) q^{64} +(-0.143694 - 0.814930i) q^{65} +(1.77511 - 0.646086i) q^{67} +(1.46577 - 0.533495i) q^{68} +(1.28877 + 7.30898i) q^{70} +(-6.09193 + 10.5515i) q^{71} +(-4.94384 - 8.56298i) q^{73} +(-4.37302 + 3.66940i) q^{74} +(0.393163 - 2.22974i) q^{76} +(-5.60091 - 4.69972i) q^{77} +(-11.6079 - 4.22493i) q^{79} -5.61331 q^{80} +3.01948 q^{82} +(-10.9786 - 3.99588i) q^{83} +(-1.63507 - 1.37199i) q^{85} +(-1.46824 + 8.32679i) q^{86} +(2.82886 - 2.37370i) q^{88} +(-2.86437 - 4.96123i) q^{89} +(-1.43183 + 2.48001i) q^{91} +(0.835346 + 4.73748i) q^{92} +(-2.69893 + 0.982329i) q^{94} +(-2.91133 + 1.05964i) q^{95} +(-0.0596270 - 0.338162i) q^{97} +(6.95870 - 12.0528i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 6 q^{4} + 3 q^{5} - 6 q^{7} - 6 q^{8} - 3 q^{10} - 3 q^{11} - 6 q^{13} - 15 q^{14} - 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 12 q^{23} + 3 q^{25} + 30 q^{26} - 12 q^{28} + 6 q^{29}+ \cdots + 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.57954 + 0.574906i 1.11690 + 0.406520i 0.833521 − 0.552487i \(-0.186321\pi\)
0.283383 + 0.959007i \(0.408543\pi\)
\(3\) 0 0
\(4\) 0.632343 + 0.530599i 0.316172 + 0.265300i
\(5\) 0.196143 − 1.11238i 0.0877179 − 0.497473i −0.909019 − 0.416755i \(-0.863167\pi\)
0.996737 − 0.0807185i \(-0.0257215\pi\)
\(6\) 0 0
\(7\) −2.99441 + 2.51261i −1.13178 + 0.949676i −0.999139 − 0.0414879i \(-0.986790\pi\)
−0.132641 + 0.991164i \(0.542346\pi\)
\(8\) −0.987144 − 1.70978i −0.349008 − 0.604500i
\(9\) 0 0
\(10\) 0.949332 − 1.64429i 0.300205 − 0.519971i
\(11\) 0.324801 + 1.84204i 0.0979313 + 0.555396i 0.993810 + 0.111096i \(0.0354362\pi\)
−0.895878 + 0.444299i \(0.853453\pi\)
\(12\) 0 0
\(13\) 0.688417 − 0.250563i 0.190932 − 0.0694937i −0.244784 − 0.969578i \(-0.578717\pi\)
0.435717 + 0.900084i \(0.356495\pi\)
\(14\) −6.17430 + 2.24726i −1.65015 + 0.600606i
\(15\) 0 0
\(16\) −0.862951 − 4.89404i −0.215738 − 1.22351i
\(17\) 0.944822 − 1.63648i 0.229153 − 0.396905i −0.728404 − 0.685147i \(-0.759738\pi\)
0.957557 + 0.288243i \(0.0930711\pi\)
\(18\) 0 0
\(19\) −1.37143 − 2.37538i −0.314627 − 0.544950i 0.664731 − 0.747083i \(-0.268546\pi\)
−0.979358 + 0.202133i \(0.935213\pi\)
\(20\) 0.714260 − 0.599335i 0.159713 − 0.134015i
\(21\) 0 0
\(22\) −0.545962 + 3.09631i −0.116400 + 0.660135i
\(23\) 4.46428 + 3.74597i 0.930866 + 0.781089i 0.975973 − 0.217893i \(-0.0699183\pi\)
−0.0451066 + 0.998982i \(0.514363\pi\)
\(24\) 0 0
\(25\) 3.49954 + 1.27373i 0.699908 + 0.254746i
\(26\) 1.23143 0.241504
\(27\) 0 0
\(28\) −3.22668 −0.609786
\(29\) −4.99910 − 1.81953i −0.928310 − 0.337877i −0.166771 − 0.985996i \(-0.553334\pi\)
−0.761540 + 0.648118i \(0.775556\pi\)
\(30\) 0 0
\(31\) 1.02696 + 0.861722i 0.184447 + 0.154770i 0.730336 − 0.683088i \(-0.239363\pi\)
−0.545889 + 0.837858i \(0.683808\pi\)
\(32\) 0.764882 − 4.33786i 0.135213 − 0.766833i
\(33\) 0 0
\(34\) 2.43321 − 2.04170i 0.417291 − 0.350149i
\(35\) 2.20765 + 3.82376i 0.373161 + 0.646334i
\(36\) 0 0
\(37\) −1.69806 + 2.94112i −0.279159 + 0.483517i −0.971176 − 0.238364i \(-0.923389\pi\)
0.692017 + 0.721881i \(0.256722\pi\)
\(38\) −0.800605 − 4.54046i −0.129875 − 0.736559i
\(39\) 0 0
\(40\) −2.09556 + 0.762721i −0.331337 + 0.120597i
\(41\) 1.68800 − 0.614382i 0.263621 − 0.0959503i −0.206828 − 0.978377i \(-0.566314\pi\)
0.470449 + 0.882427i \(0.344092\pi\)
\(42\) 0 0
\(43\) 0.873477 + 4.95373i 0.133204 + 0.755437i 0.976094 + 0.217351i \(0.0697415\pi\)
−0.842890 + 0.538087i \(0.819147\pi\)
\(44\) −0.771999 + 1.33714i −0.116383 + 0.201582i
\(45\) 0 0
\(46\) 4.89793 + 8.48346i 0.722160 + 1.25082i
\(47\) −1.30892 + 1.09832i −0.190926 + 0.160206i −0.733240 − 0.679970i \(-0.761993\pi\)
0.542314 + 0.840176i \(0.317548\pi\)
\(48\) 0 0
\(49\) 1.43775 − 8.15389i 0.205393 − 1.16484i
\(50\) 4.79539 + 4.02381i 0.678170 + 0.569053i
\(51\) 0 0
\(52\) 0.568265 + 0.206831i 0.0788041 + 0.0286824i
\(53\) −2.84494 −0.390783 −0.195391 − 0.980725i \(-0.562598\pi\)
−0.195391 + 0.980725i \(0.562598\pi\)
\(54\) 0 0
\(55\) 2.11276 0.284885
\(56\) 7.25193 + 2.63949i 0.969080 + 0.352716i
\(57\) 0 0
\(58\) −6.85023 − 5.74803i −0.899480 − 0.754753i
\(59\) 1.95529 − 11.0890i 0.254557 − 1.44366i −0.542652 − 0.839958i \(-0.682580\pi\)
0.797208 − 0.603704i \(-0.206309\pi\)
\(60\) 0 0
\(61\) 4.00710 − 3.36235i 0.513056 − 0.430505i −0.349147 − 0.937068i \(-0.613529\pi\)
0.862203 + 0.506563i \(0.169084\pi\)
\(62\) 1.12672 + 1.95153i 0.143093 + 0.247844i
\(63\) 0 0
\(64\) −1.26751 + 2.19540i −0.158439 + 0.274425i
\(65\) −0.143694 − 0.814930i −0.0178231 − 0.101080i
\(66\) 0 0
\(67\) 1.77511 − 0.646086i 0.216864 − 0.0789319i −0.231304 − 0.972882i \(-0.574299\pi\)
0.448167 + 0.893950i \(0.352077\pi\)
\(68\) 1.46577 − 0.533495i 0.177750 − 0.0646958i
\(69\) 0 0
\(70\) 1.28877 + 7.30898i 0.154038 + 0.873590i
\(71\) −6.09193 + 10.5515i −0.722980 + 1.25224i 0.236821 + 0.971553i \(0.423895\pi\)
−0.959800 + 0.280684i \(0.909439\pi\)
\(72\) 0 0
\(73\) −4.94384 − 8.56298i −0.578633 − 1.00222i −0.995637 − 0.0933164i \(-0.970253\pi\)
0.417004 − 0.908905i \(-0.363080\pi\)
\(74\) −4.37302 + 3.66940i −0.508353 + 0.426559i
\(75\) 0 0
\(76\) 0.393163 − 2.22974i 0.0450989 − 0.255768i
\(77\) −5.60091 − 4.69972i −0.638283 − 0.535583i
\(78\) 0 0
\(79\) −11.6079 − 4.22493i −1.30599 − 0.475342i −0.407048 − 0.913407i \(-0.633442\pi\)
−0.898943 + 0.438065i \(0.855664\pi\)
\(80\) −5.61331 −0.627587
\(81\) 0 0
\(82\) 3.01948 0.333445
\(83\) −10.9786 − 3.99588i −1.20506 − 0.438605i −0.340070 − 0.940400i \(-0.610451\pi\)
−0.864986 + 0.501796i \(0.832673\pi\)
\(84\) 0 0
\(85\) −1.63507 − 1.37199i −0.177349 − 0.148813i
\(86\) −1.46824 + 8.32679i −0.158324 + 0.897901i
\(87\) 0 0
\(88\) 2.82886 − 2.37370i 0.301558 − 0.253037i
\(89\) −2.86437 − 4.96123i −0.303622 − 0.525889i 0.673331 − 0.739341i \(-0.264863\pi\)
−0.976954 + 0.213452i \(0.931529\pi\)
\(90\) 0 0
\(91\) −1.43183 + 2.48001i −0.150097 + 0.259976i
\(92\) 0.835346 + 4.73748i 0.0870908 + 0.493917i
\(93\) 0 0
\(94\) −2.69893 + 0.982329i −0.278373 + 0.101319i
\(95\) −2.91133 + 1.05964i −0.298697 + 0.108717i
\(96\) 0 0
\(97\) −0.0596270 − 0.338162i −0.00605421 − 0.0343351i 0.981631 − 0.190789i \(-0.0611047\pi\)
−0.987685 + 0.156454i \(0.949994\pi\)
\(98\) 6.95870 − 12.0528i 0.702935 − 1.21752i
\(99\) 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 81.2.e.a.64.2 12
3.2 odd 2 27.2.e.a.4.1 ✓ 12
9.2 odd 6 243.2.e.c.28.2 12
9.4 even 3 243.2.e.a.109.1 12
9.5 odd 6 243.2.e.d.109.2 12
9.7 even 3 243.2.e.b.28.1 12
12.11 even 2 432.2.u.c.193.1 12
15.2 even 4 675.2.u.b.274.4 24
15.8 even 4 675.2.u.b.274.1 24
15.14 odd 2 675.2.l.c.301.2 12
27.2 odd 18 243.2.e.d.136.2 12
27.4 even 9 729.2.c.b.244.6 12
27.5 odd 18 729.2.c.e.487.1 12
27.7 even 9 inner 81.2.e.a.19.2 12
27.11 odd 18 243.2.e.c.217.2 12
27.13 even 9 729.2.a.d.1.1 6
27.14 odd 18 729.2.a.a.1.6 6
27.16 even 9 243.2.e.b.217.1 12
27.20 odd 18 27.2.e.a.7.1 yes 12
27.22 even 9 729.2.c.b.487.6 12
27.23 odd 18 729.2.c.e.244.1 12
27.25 even 9 243.2.e.a.136.1 12
108.47 even 18 432.2.u.c.385.1 12
135.47 even 36 675.2.u.b.574.1 24
135.74 odd 18 675.2.l.c.601.2 12
135.128 even 36 675.2.u.b.574.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.1 ✓ 12 3.2 odd 2
27.2.e.a.7.1 yes 12 27.20 odd 18
81.2.e.a.19.2 12 27.7 even 9 inner
81.2.e.a.64.2 12 1.1 even 1 trivial
243.2.e.a.109.1 12 9.4 even 3
243.2.e.a.136.1 12 27.25 even 9
243.2.e.b.28.1 12 9.7 even 3
243.2.e.b.217.1 12 27.16 even 9
243.2.e.c.28.2 12 9.2 odd 6
243.2.e.c.217.2 12 27.11 odd 18
243.2.e.d.109.2 12 9.5 odd 6
243.2.e.d.136.2 12 27.2 odd 18
432.2.u.c.193.1 12 12.11 even 2
432.2.u.c.385.1 12 108.47 even 18
675.2.l.c.301.2 12 15.14 odd 2
675.2.l.c.601.2 12 135.74 odd 18
675.2.u.b.274.1 24 15.8 even 4
675.2.u.b.274.4 24 15.2 even 4
675.2.u.b.574.1 24 135.47 even 36
675.2.u.b.574.4 24 135.128 even 36
729.2.a.a.1.6 6 27.14 odd 18
729.2.a.d.1.1 6 27.13 even 9
729.2.c.b.244.6 12 27.4 even 9
729.2.c.b.487.6 12 27.22 even 9
729.2.c.e.244.1 12 27.23 odd 18
729.2.c.e.487.1 12 27.5 odd 18