Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7605,2,Mod(1,7605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7605, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("7605.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7605.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: | \(60.7262307372\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.3352656.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 585) |
| 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
\( x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \)
|
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 |
|
−2.12283 | 0 | 2.50640 | 1.00000 | 0 | −1.46707 | −1.07500 | 0 | −2.12283 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.946366 | 0 | −1.10439 | 1.00000 | 0 | 1.56306 | 2.93789 | 0 | −0.946366 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.626791 | 0 | −1.60713 | 1.00000 | 0 | −4.43127 | −2.26092 | 0 | 0.626791 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.81031 | 0 | 1.27724 | 1.00000 | 0 | 4.42503 | −1.30843 | 0 | 1.81031 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.63209 | 0 | 4.92789 | 1.00000 | 0 | −1.08975 | 7.70645 | 0 | 2.63209 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
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 | 7605.2.a.cn | 5 | |
| 3.b | odd | 2 | 1 | 7605.2.a.cl | 5 | ||
| 13.b | even | 2 | 1 | 7605.2.a.cm | 5 | ||
| 13.e | even | 6 | 2 | 585.2.j.i | yes | 10 | |
| 39.d | odd | 2 | 1 | 7605.2.a.co | 5 | ||
| 39.h | odd | 6 | 2 | 585.2.j.h | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 585.2.j.h | ✓ | 10 | 39.h | odd | 6 | 2 | |
| 585.2.j.i | yes | 10 | 13.e | even | 6 | 2 | |
| 7605.2.a.cl | 5 | 3.b | odd | 2 | 1 | ||
| 7605.2.a.cm | 5 | 13.b | even | 2 | 1 | ||
| 7605.2.a.cn | 5 | 1.a | even | 1 | 1 | trivial | |
| 7605.2.a.co | 5 | 39.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):
|
\( T_{2}^{5} - 2T_{2}^{4} - 6T_{2}^{3} + 10T_{2}^{2} + 6T_{2} - 6 \)
|
|
\( T_{7}^{5} + T_{7}^{4} - 22T_{7}^{3} - 22T_{7}^{2} + 47T_{7} + 49 \)
|
|
\( T_{11}^{5} - 8T_{11}^{4} + 64T_{11}^{2} + 48T_{11} - 24 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 6 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} + T^{4} + \cdots + 49 \)
|
| $11$ |
\( T^{5} - 8 T^{4} + \cdots - 24 \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} - 50 T^{3} + \cdots - 642 \)
|
| $19$ |
\( T^{5} - 4 T^{4} + \cdots - 12 \)
|
| $23$ |
\( T^{5} - 6 T^{4} + \cdots + 72 \)
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| $29$ |
\( T^{5} + 16 T^{4} + \cdots - 258 \)
|
| $31$ |
\( T^{5} + 9 T^{4} + \cdots - 603 \)
|
| $37$ |
\( T^{5} - 4 T^{4} + \cdots + 144 \)
|
| $41$ |
\( T^{5} - 6 T^{4} + \cdots - 11898 \)
|
| $43$ |
\( T^{5} - 15 T^{4} + \cdots + 2059 \)
|
| $47$ |
\( T^{5} - 10 T^{4} + \cdots - 7434 \)
|
| $53$ |
\( T^{5} - 20 T^{4} + \cdots + 15552 \)
|
| $59$ |
\( T^{5} - 12 T^{4} + \cdots - 2754 \)
|
| $61$ |
\( T^{5} - 11 T^{4} + \cdots - 3023 \)
|
| $67$ |
\( T^{5} + 5 T^{4} + \cdots + 17 \)
|
| $71$ |
\( T^{5} - 10 T^{4} + \cdots + 162 \)
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| $73$ |
\( T^{5} + T^{4} + \cdots - 9693 \)
|
| $79$ |
\( T^{5} + 17 T^{4} + \cdots + 2349 \)
|
| $83$ |
\( T^{5} - 16 T^{4} + \cdots - 6264 \)
|
| $89$ |
\( T^{5} - 4 T^{4} + \cdots - 54 \)
|
| $97$ |
\( T^{5} - 11 T^{4} + \cdots + 7679 \)
|
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