Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [725,4,Mod(1,725)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(725, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("725.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 725.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: | \(42.7763847542\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 29) |
| 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 \(\beta = \sqrt{2}\). 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 |
|
−0.414214 | 9.24264 | −7.82843 | 0 | −3.82843 | −6.14214 | 6.55635 | 58.4264 | 0 | ||||||||||||||||||||||||
| 1.2 | 2.41421 | 0.757359 | −2.17157 | 0 | 1.82843 | 22.1421 | −24.5563 | −26.4264 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(29\) | \(-1\) |
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 | 725.4.a.b | 2 | |
| 5.b | even | 2 | 1 | 29.4.a.a | ✓ | 2 | |
| 15.d | odd | 2 | 1 | 261.4.a.b | 2 | ||
| 20.d | odd | 2 | 1 | 464.4.a.f | 2 | ||
| 35.c | odd | 2 | 1 | 1421.4.a.c | 2 | ||
| 40.e | odd | 2 | 1 | 1856.4.a.h | 2 | ||
| 40.f | even | 2 | 1 | 1856.4.a.n | 2 | ||
| 145.d | even | 2 | 1 | 841.4.a.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.4.a.a | ✓ | 2 | 5.b | even | 2 | 1 | |
| 261.4.a.b | 2 | 15.d | odd | 2 | 1 | ||
| 464.4.a.f | 2 | 20.d | odd | 2 | 1 | ||
| 725.4.a.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 841.4.a.a | 2 | 145.d | even | 2 | 1 | ||
| 1421.4.a.c | 2 | 35.c | odd | 2 | 1 | ||
| 1856.4.a.h | 2 | 40.e | odd | 2 | 1 | ||
| 1856.4.a.n | 2 | 40.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2T_{2} - 1 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(725))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 2T - 1 \)
|
| $3$ |
\( T^{2} - 10T + 7 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 16T - 136 \)
|
| $11$ |
\( T^{2} + 26T - 2569 \)
|
| $13$ |
\( T^{2} - 26T - 1183 \)
|
| $17$ |
\( T^{2} + 60T + 252 \)
|
| $19$ |
\( T^{2} + 220T + 10052 \)
|
| $23$ |
\( T^{2} + 52T - 3932 \)
|
| $29$ |
\( (T - 29)^{2} \)
|
| $31$ |
\( T^{2} + 294T + 13671 \)
|
| $37$ |
\( T^{2} + 312T + 18064 \)
|
| $41$ |
\( T^{2} - 40T - 37688 \)
|
| $43$ |
\( T^{2} - 322T - 32561 \)
|
| $47$ |
\( T^{2} - 130T - 81473 \)
|
| $53$ |
\( T^{2} + 1002 T + 221233 \)
|
| $59$ |
\( T^{2} + 900T + 79492 \)
|
| $61$ |
\( T^{2} + 948T + 161308 \)
|
| $67$ |
\( T^{2} + 320T - 442912 \)
|
| $71$ |
\( T^{2} + 660T + 106588 \)
|
| $73$ |
\( T^{2} + 648T - 714224 \)
|
| $79$ |
\( T^{2} - 258T - 215921 \)
|
| $83$ |
\( T^{2} + 1212 T + 359044 \)
|
| $89$ |
\( T^{2} - 760T - 400568 \)
|
| $97$ |
\( T^{2} + 24T - 668024 \)
|
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