Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,5,Mod(26,81)] 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("81.26"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.37296700979\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 26.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 81.26
Dual form 81.5.d.a.53.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-8.00000 - 13.8564i) q^{4} +(-35.5000 + 61.4878i) q^{7} +(168.500 + 291.851i) q^{13} +(-128.000 + 221.703i) q^{16} -601.000 q^{19} +(-312.500 + 541.266i) q^{25} +1136.00 q^{28} +(-97.0000 - 168.009i) q^{31} -529.000 q^{37} +(1607.00 - 2783.41i) q^{43} +(-1320.00 - 2286.31i) q^{49} +(2696.00 - 4669.61i) q^{52} +(-3599.50 + 6234.52i) q^{61} +4096.00 q^{64} +(-1451.50 - 2514.07i) q^{67} -1249.00 q^{73} +(4808.00 + 8327.70i) q^{76} +(-2339.50 + 4052.13i) q^{79} -23927.0 q^{91} +(-4535.50 + 7855.72i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 71 q^{7} + 337 q^{13} - 256 q^{16} - 1202 q^{19} - 625 q^{25} + 2272 q^{28} - 194 q^{31} - 1058 q^{37} + 3214 q^{43} - 2640 q^{49} + 5392 q^{52} - 7199 q^{61} + 8192 q^{64} - 2903 q^{67}+ \cdots - 9071 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0
\(4\) −8.00000 − 13.8564i −0.500000 − 0.866025i
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) −35.5000 + 61.4878i −0.724490 + 1.25485i 0.234694 + 0.972069i \(0.424591\pi\)
−0.959184 + 0.282784i \(0.908742\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 168.500 + 291.851i 0.997041 + 1.72693i 0.565089 + 0.825030i \(0.308842\pi\)
0.431953 + 0.901896i \(0.357825\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −128.000 + 221.703i −0.500000 + 0.866025i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −601.000 −1.66482 −0.832410 − 0.554160i \(-0.813039\pi\)
−0.832410 + 0.554160i \(0.813039\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −312.500 + 541.266i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 1136.00 1.44898
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −97.0000 − 168.009i −0.100937 − 0.174827i 0.811134 − 0.584860i \(-0.198851\pi\)
−0.912071 + 0.410033i \(0.865517\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −529.000 −0.386413 −0.193207 − 0.981158i \(-0.561889\pi\)
−0.193207 + 0.981158i \(0.561889\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) 1607.00 − 2783.41i 0.869118 − 1.50536i 0.00621958 − 0.999981i \(-0.498020\pi\)
0.862899 − 0.505377i \(-0.168646\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) −1320.00 − 2286.31i −0.549771 − 0.952231i
\(50\) 0 0
\(51\) 0 0
\(52\) 2696.00 − 4669.61i 0.997041 − 1.72693i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −3599.50 + 6234.52i −0.967347 + 1.67549i −0.264176 + 0.964474i \(0.585100\pi\)
−0.703171 + 0.711021i \(0.748233\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4096.00 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1451.50 − 2514.07i −0.323346 − 0.560052i 0.657830 − 0.753166i \(-0.271474\pi\)
−0.981176 + 0.193115i \(0.938141\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1249.00 −0.234378 −0.117189 − 0.993110i \(-0.537388\pi\)
−0.117189 + 0.993110i \(0.537388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 4808.00 + 8327.70i 0.832410 + 1.44178i
\(77\) 0 0
\(78\) 0 0
\(79\) −2339.50 + 4052.13i −0.374860 + 0.649276i −0.990306 − 0.138903i \(-0.955642\pi\)
0.615446 + 0.788179i \(0.288976\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −23927.0 −2.88939
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4535.50 + 7855.72i −0.482038 + 0.834915i −0.999787 − 0.0206175i \(-0.993437\pi\)
0.517749 + 0.855533i \(0.326770\pi\)
\(98\) 0 0
\(99\) 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 81.5.d.a.26.1 2
3.2 odd 2 CM 81.5.d.a.26.1 2
9.2 odd 6 27.5.b.a.26.1 ✓ 1
9.4 even 3 inner 81.5.d.a.53.1 2
9.5 odd 6 inner 81.5.d.a.53.1 2
9.7 even 3 27.5.b.a.26.1 ✓ 1
36.7 odd 6 432.5.e.a.161.1 1
36.11 even 6 432.5.e.a.161.1 1
45.2 even 12 675.5.d.c.674.2 2
45.7 odd 12 675.5.d.c.674.2 2
45.29 odd 6 675.5.c.b.26.1 1
45.34 even 6 675.5.c.b.26.1 1
45.38 even 12 675.5.d.c.674.1 2
45.43 odd 12 675.5.d.c.674.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.a.26.1 ✓ 1 9.2 odd 6
27.5.b.a.26.1 ✓ 1 9.7 even 3
81.5.d.a.26.1 2 1.1 even 1 trivial
81.5.d.a.26.1 2 3.2 odd 2 CM
81.5.d.a.53.1 2 9.4 even 3 inner
81.5.d.a.53.1 2 9.5 odd 6 inner
432.5.e.a.161.1 1 36.7 odd 6
432.5.e.a.161.1 1 36.11 even 6
675.5.c.b.26.1 1 45.29 odd 6
675.5.c.b.26.1 1 45.34 even 6
675.5.d.c.674.1 2 45.38 even 12
675.5.d.c.674.1 2 45.43 odd 12
675.5.d.c.674.2 2 45.2 even 12
675.5.d.c.674.2 2 45.7 odd 12