Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,36,Mod(1,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 36, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.1");
S:= CuspForms(chi, 36);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 36 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.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: | \(193.987826584\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2 x^{4} - 26764488186 x^{3} - 142880248812560 x^{2} + \cdots + 13\!\cdots\!66 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3^{6}\cdot 5^{4}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - 2 x^{4} - 26764488186 x^{3} - 142880248812560 x^{2} + \cdots + 13\!\cdots\!66 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 7 \nu^{4} + 6755835 \nu^{3} + 149101549293 \nu^{2} + \cdots - 94\!\cdots\!06 ) / 19315610707968 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7 \nu^{4} - 6755835 \nu^{3} + 19166509158675 \nu^{2} + \cdots - 20\!\cdots\!22 ) / 4828902676992 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 54685 \nu^{4} - 713907417 \nu^{3} + \cdots + 31\!\cdots\!42 ) / 19315610707968 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + 16019\beta _1 + 42823184303 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 742\beta_{4} + 216\beta_{3} + 5797474\beta_{2} + 16484922145\beta _1 + 171523828012161 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1432237020 \beta_{4} + 21717152859 \beta_{3} + 238216983840 \beta_{2} + 466065849284665 \beta _1 + 70\!\cdots\!09 ) / 4 \)
|
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 |
|
−195592. | −1.75923e8 | 3.89669e9 | 0 | 3.44092e13 | 4.08781e14 | 5.95834e15 | −1.90827e16 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −78064.7 | 3.02218e8 | −2.82656e10 | 0 | −2.35925e13 | −4.58916e14 | 4.88883e15 | 4.13040e16 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 111319. | 4.77965e7 | −2.19677e10 | 0 | 5.32068e12 | 4.16245e14 | −6.27034e15 | −4.77470e16 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 253960. | −1.81015e8 | 3.01358e10 | 0 | −4.59705e13 | −4.79980e14 | −1.07272e15 | −1.72651e16 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 367464. | 2.64714e8 | 1.00670e11 | 0 | 9.72731e13 | 7.92623e14 | 2.43667e16 | 2.00422e16 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
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 | 25.36.a.b | 5 | |
| 5.b | even | 2 | 1 | 5.36.a.a | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 25.36.b.b | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.36.a.a | ✓ | 5 | 5.b | even | 2 | 1 | |
| 25.36.a.b | 5 | 1.a | even | 1 | 1 | trivial | |
| 25.36.b.b | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 459086 T_{2}^{4} - 22753970592 T_{2}^{3} + \cdots - 15\!\cdots\!76 \)
acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots - 15\!\cdots\!76 \)
|
| $3$ |
\( T^{5} + \cdots - 12\!\cdots\!32 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 29\!\cdots\!76 \)
|
| $11$ |
\( T^{5} + \cdots + 10\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots + 24\!\cdots\!68 \)
|
| $17$ |
\( T^{5} + \cdots - 41\!\cdots\!76 \)
|
| $19$ |
\( T^{5} + \cdots + 28\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 10\!\cdots\!68 \)
|
| $29$ |
\( T^{5} + \cdots + 52\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 45\!\cdots\!32 \)
|
| $37$ |
\( T^{5} + \cdots - 20\!\cdots\!76 \)
|
| $41$ |
\( T^{5} + \cdots - 85\!\cdots\!32 \)
|
| $43$ |
\( T^{5} + \cdots - 32\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots + 37\!\cdots\!24 \)
|
| $53$ |
\( T^{5} + \cdots + 34\!\cdots\!68 \)
|
| $59$ |
\( T^{5} + \cdots - 25\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 12\!\cdots\!68 \)
|
| $67$ |
\( T^{5} + \cdots + 10\!\cdots\!24 \)
|
| $71$ |
\( T^{5} + \cdots - 42\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots + 14\!\cdots\!68 \)
|
| $79$ |
\( T^{5} + \cdots - 53\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 10\!\cdots\!32 \)
|
| $89$ |
\( T^{5} + \cdots + 42\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 34\!\cdots\!76 \)
|
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