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Properties

Label 81.3.d.b
Level $81$
Weight $3$
Character orbit 81.d
Analytic conductor $2.207$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,3,Mod(26,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.20709014132\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (5 \beta_{2} - 5) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (5 \beta_{2} - 5) q^{7} + \beta_{3} q^{8} + 9 q^{10} - 5 \beta_1 q^{11} + 10 \beta_{2} q^{13} + (5 \beta_{3} - 5 \beta_1) q^{14} + ( - 11 \beta_{2} + 11) q^{16} - 6 \beta_{3} q^{17} - 16 q^{19} + 5 \beta_1 q^{20} - 45 \beta_{2} q^{22} + ( - 4 \beta_{3} + 4 \beta_1) q^{23} + (16 \beta_{2} - 16) q^{25} + 10 \beta_{3} q^{26} - 25 q^{28} + 10 \beta_1 q^{29} + \beta_{2} q^{31} + ( - 15 \beta_{3} + 15 \beta_1) q^{32} + ( - 54 \beta_{2} + 54) q^{34} + 5 \beta_{3} q^{35} + 20 q^{37} - 16 \beta_1 q^{38} + 9 \beta_{2} q^{40} + (20 \beta_{3} - 20 \beta_1) q^{41} + (50 \beta_{2} - 50) q^{43} - 25 \beta_{3} q^{44} + 36 q^{46} - 2 \beta_1 q^{47} + 24 \beta_{2} q^{49} + (16 \beta_{3} - 16 \beta_1) q^{50} + (50 \beta_{2} - 50) q^{52} + 9 \beta_{3} q^{53} - 45 q^{55} - 5 \beta_1 q^{56} + 90 \beta_{2} q^{58} + ( - 10 \beta_{3} + 10 \beta_1) q^{59} + ( - 76 \beta_{2} + 76) q^{61} + \beta_{3} q^{62} + 91 q^{64} + 10 \beta_1 q^{65} + 10 \beta_{2} q^{67} + ( - 30 \beta_{3} + 30 \beta_1) q^{68} + (45 \beta_{2} - 45) q^{70} + 30 \beta_{3} q^{71} + 65 q^{73} + 20 \beta_1 q^{74} - 80 \beta_{2} q^{76} + ( - 25 \beta_{3} + 25 \beta_1) q^{77} + (14 \beta_{2} - 14) q^{79} - 11 \beta_{3} q^{80} - 180 q^{82} + \beta_1 q^{83} - 54 \beta_{2} q^{85} + (50 \beta_{3} - 50 \beta_1) q^{86} + ( - 45 \beta_{2} + 45) q^{88} - 30 \beta_{3} q^{89} - 50 q^{91} + 20 \beta_1 q^{92} - 18 \beta_{2} q^{94} + (16 \beta_{3} - 16 \beta_1) q^{95} + ( - 85 \beta_{2} + 85) q^{97} + 24 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} - 10 q^{7} + 36 q^{10} + 20 q^{13} + 22 q^{16} - 64 q^{19} - 90 q^{22} - 32 q^{25} - 100 q^{28} + 2 q^{31} + 108 q^{34} + 80 q^{37} + 18 q^{40} - 100 q^{43} + 144 q^{46} + 48 q^{49} - 100 q^{52} - 180 q^{55} + 180 q^{58} + 152 q^{61} + 364 q^{64} + 20 q^{67} - 90 q^{70} + 260 q^{73} - 160 q^{76} - 28 q^{79} - 720 q^{82} - 108 q^{85} + 90 q^{88} - 200 q^{91} - 36 q^{94} + 170 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{12}^{3} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−2.59808 − 1.50000i 0 2.50000 + 4.33013i −2.59808 + 1.50000i 0 −2.50000 + 4.33013i − 3.00000i 0 9.00000
26.2 2.59808 + 1.50000i 0 2.50000 + 4.33013i 2.59808 − 1.50000i 0 −2.50000 + 4.33013i 3.00000i 0 9.00000
53.1 −2.59808 + 1.50000i 0 2.50000 − 4.33013i −2.59808 − 1.50000i 0 −2.50000 − 4.33013i 3.00000i 0 9.00000
53.2 2.59808 − 1.50000i 0 2.50000 − 4.33013i 2.59808 + 1.50000i 0 −2.50000 − 4.33013i − 3.00000i 0 9.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.3.d.b 4
3.b odd 2 1 inner 81.3.d.b 4
4.b odd 2 1 1296.3.q.j 4
9.c even 3 1 27.3.b.b ✓ 2
9.c even 3 1 inner 81.3.d.b 4
9.d odd 6 1 27.3.b.b ✓ 2
9.d odd 6 1 inner 81.3.d.b 4
12.b even 2 1 1296.3.q.j 4
36.f odd 6 1 432.3.e.c 2
36.f odd 6 1 1296.3.q.j 4
36.h even 6 1 432.3.e.c 2
36.h even 6 1 1296.3.q.j 4
45.h odd 6 1 675.3.c.h 2
45.j even 6 1 675.3.c.h 2
45.k odd 12 1 675.3.d.a 2
45.k odd 12 1 675.3.d.d 2
45.l even 12 1 675.3.d.a 2
45.l even 12 1 675.3.d.d 2
72.j odd 6 1 1728.3.e.m 2
72.l even 6 1 1728.3.e.g 2
72.n even 6 1 1728.3.e.m 2
72.p odd 6 1 1728.3.e.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.b.b ✓ 2 9.c even 3 1
27.3.b.b ✓ 2 9.d odd 6 1
81.3.d.b 4 1.a even 1 1 trivial
81.3.d.b 4 3.b odd 2 1 inner
81.3.d.b 4 9.c even 3 1 inner
81.3.d.b 4 9.d odd 6 1 inner
432.3.e.c 2 36.f odd 6 1
432.3.e.c 2 36.h even 6 1
675.3.c.h 2 45.h odd 6 1
675.3.c.h 2 45.j even 6 1
675.3.d.a 2 45.k odd 12 1
675.3.d.a 2 45.l even 12 1
675.3.d.d 2 45.k odd 12 1
675.3.d.d 2 45.l even 12 1
1296.3.q.j 4 4.b odd 2 1
1296.3.q.j 4 12.b even 2 1
1296.3.q.j 4 36.f odd 6 1
1296.3.q.j 4 36.h even 6 1
1728.3.e.g 2 72.l even 6 1
1728.3.e.g 2 72.p odd 6 1
1728.3.e.m 2 72.j odd 6 1
1728.3.e.m 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 9T_{2}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$7$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$13$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$19$ \( (T + 16)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$29$ \( T^{4} - 900 T^{2} + 810000 \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T - 20)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 3600 T^{2} + 12960000 \) Copy content Toggle raw display
$43$ \( (T^{2} + 50 T + 2500)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$53$ \( (T^{2} + 729)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 900 T^{2} + 810000 \) Copy content Toggle raw display
$61$ \( (T^{2} - 76 T + 5776)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8100)^{2} \) Copy content Toggle raw display
$73$ \( (T - 65)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$89$ \( (T^{2} + 8100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 85 T + 7225)^{2} \) Copy content Toggle raw display
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