Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2025,2,Mod(1,2025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2025 = 3^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2025.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: | \(16.1697064093\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.11661.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 5x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 225) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 4x^{2} + 5x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.63372 | 0 | 4.93650 | 0 | 0 | −1.79743 | −7.73393 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.47325 | 0 | 0.170479 | 0 | 0 | 3.86583 | 2.69535 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0.473255 | 0 | −1.77603 | 0 | 0 | −2.56305 | −1.78702 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.63372 | 0 | 0.669052 | 0 | 0 | −0.505348 | −2.17440 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 2025.2.a.p | 4 | |
| 3.b | odd | 2 | 1 | 2025.2.a.y | 4 | ||
| 5.b | even | 2 | 1 | 2025.2.a.z | 4 | ||
| 5.c | odd | 4 | 2 | 2025.2.b.o | 8 | ||
| 9.c | even | 3 | 2 | 675.2.e.e | 8 | ||
| 9.d | odd | 6 | 2 | 225.2.e.c | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 2025.2.a.q | 4 | ||
| 15.e | even | 4 | 2 | 2025.2.b.n | 8 | ||
| 45.h | odd | 6 | 2 | 225.2.e.e | yes | 8 | |
| 45.j | even | 6 | 2 | 675.2.e.c | 8 | ||
| 45.k | odd | 12 | 4 | 675.2.k.c | 16 | ||
| 45.l | even | 12 | 4 | 225.2.k.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 225.2.e.c | ✓ | 8 | 9.d | odd | 6 | 2 | |
| 225.2.e.e | yes | 8 | 45.h | odd | 6 | 2 | |
| 225.2.k.c | 16 | 45.l | even | 12 | 4 | ||
| 675.2.e.c | 8 | 45.j | even | 6 | 2 | ||
| 675.2.e.e | 8 | 9.c | even | 3 | 2 | ||
| 675.2.k.c | 16 | 45.k | odd | 12 | 4 | ||
| 2025.2.a.p | 4 | 1.a | even | 1 | 1 | trivial | |
| 2025.2.a.q | 4 | 15.d | odd | 2 | 1 | ||
| 2025.2.a.y | 4 | 3.b | odd | 2 | 1 | ||
| 2025.2.a.z | 4 | 5.b | even | 2 | 1 | ||
| 2025.2.b.n | 8 | 15.e | even | 4 | 2 | ||
| 2025.2.b.o | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2025))\):
|
\( T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 5T_{2} + 3 \)
|
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\( T_{7}^{4} + T_{7}^{3} - 12T_{7}^{2} - 24T_{7} - 9 \)
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\( T_{11}^{4} - T_{11}^{3} - 25T_{11}^{2} - 41T_{11} - 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 3 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + T^{3} - 12 T^{2} + \cdots - 9 \)
|
| $11$ |
\( T^{4} - T^{3} - 25 T^{2} + \cdots - 9 \)
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| $13$ |
\( T^{4} - 2 T^{3} + \cdots + 107 \)
|
| $17$ |
\( T^{4} + 11 T^{3} + \cdots - 303 \)
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| $19$ |
\( T^{4} - 2 T^{3} + \cdots - 25 \)
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| $23$ |
\( T^{4} + 15 T^{3} + \cdots - 243 \)
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| $29$ |
\( T^{4} + T^{3} + \cdots - 129 \)
|
| $31$ |
\( T^{4} + 4 T^{3} + \cdots + 243 \)
|
| $37$ |
\( T^{4} + T^{3} + \cdots - 647 \)
|
| $41$ |
\( T^{4} - 5 T^{3} + \cdots - 207 \)
|
| $43$ |
\( T^{4} + 10 T^{3} + \cdots - 673 \)
|
| $47$ |
\( T^{4} + 20 T^{3} + \cdots - 381 \)
|
| $53$ |
\( T^{4} + 20 T^{3} + \cdots - 471 \)
|
| $59$ |
\( T^{4} + 17 T^{3} + \cdots - 2313 \)
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| $61$ |
\( T^{4} + 13 T^{3} + \cdots - 1 \)
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| $67$ |
\( T^{4} - 17 T^{3} + \cdots + 243 \)
|
| $71$ |
\( T^{4} - 8 T^{3} + \cdots + 381 \)
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| $73$ |
\( T^{4} - 2 T^{3} + \cdots + 113 \)
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| $79$ |
\( T^{4} + 7 T^{3} + \cdots + 207 \)
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| $83$ |
\( T^{4} + 30 T^{3} + \cdots + 729 \)
|
| $89$ |
\( T^{4} - 9 T^{3} + \cdots + 2025 \)
|
| $97$ |
\( T^{4} + 19 T^{3} + \cdots - 953 \)
|
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