Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,4,Mod(1,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.a (trivial) |
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: | yes |
| Analytic conductor: | \(4.77915471046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{57}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.27492 | 0 | 19.8248 | −1.54983 | 0 | −17.6495 | −62.3746 | 0 | 8.17525 | ||||||||||||||||||||||||
| 1.2 | 2.27492 | 0 | −2.82475 | 13.5498 | 0 | 27.6495 | −24.6254 | 0 | 30.8248 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.4.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 81.4.a.e | yes | 2 | |
| 4.b | odd | 2 | 1 | 1296.4.a.t | 2 | ||
| 5.b | even | 2 | 1 | 2025.4.a.o | 2 | ||
| 9.c | even | 3 | 2 | 81.4.c.g | 4 | ||
| 9.d | odd | 6 | 2 | 81.4.c.d | 4 | ||
| 12.b | even | 2 | 1 | 1296.4.a.k | 2 | ||
| 15.d | odd | 2 | 1 | 2025.4.a.h | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 81.4.a.e | yes | 2 | 3.b | odd | 2 | 1 | |
| 81.4.c.d | 4 | 9.d | odd | 6 | 2 | ||
| 81.4.c.g | 4 | 9.c | even | 3 | 2 | ||
| 1296.4.a.k | 2 | 12.b | even | 2 | 1 | ||
| 1296.4.a.t | 2 | 4.b | odd | 2 | 1 | ||
| 2025.4.a.h | 2 | 15.d | odd | 2 | 1 | ||
| 2025.4.a.o | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 3T_{2} - 12 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 3T - 12 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 12T - 21 \)
|
| $7$ |
\( T^{2} - 10T - 488 \)
|
| $11$ |
\( T^{2} - 42T + 384 \)
|
| $13$ |
\( T^{2} - 64T + 511 \)
|
| $17$ |
\( T^{2} + 18T - 8127 \)
|
| $19$ |
\( T^{2} - 130T - 392 \)
|
| $23$ |
\( T^{2} + 114T - 3648 \)
|
| $29$ |
\( T^{2} - 240T + 4767 \)
|
| $31$ |
\( T^{2} + 8T - 32816 \)
|
| $37$ |
\( T^{2} - 184T + 3847 \)
|
| $41$ |
\( T^{2} + 672T + 101724 \)
|
| $43$ |
\( T^{2} - 370T + 33712 \)
|
| $47$ |
\( T^{2} - 276T + 18816 \)
|
| $53$ |
\( (T + 162)^{2} \)
|
| $59$ |
\( T^{2} + 204T - 110208 \)
|
| $61$ |
\( T^{2} - 208T - 260561 \)
|
| $67$ |
\( T^{2} - 802T + 74104 \)
|
| $71$ |
\( T^{2} + 126T - 181224 \)
|
| $73$ |
\( T^{2} + 374T - 131243 \)
|
| $79$ |
\( T^{2} - 1378 T + 474208 \)
|
| $83$ |
\( T^{2} + 1164 T + 196224 \)
|
| $89$ |
\( T^{2} + 954T - 593271 \)
|
| $97$ |
\( T^{2} + 872T + 138796 \)
|
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