Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6525,2,Mod(1,6525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6525, 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("6525.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6525.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: | \(52.1023873189\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.331312.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 7x^{3} - 4x^{2} + 7x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 435) |
| 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
\( x^{5} - 7x^{3} - 4x^{2} + 7x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 2\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{4} + 7\beta_{3} + \beta_{2} + 9\beta _1 + 16 \)
|
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 |
|
−2.68032 | 0 | 5.18413 | 0 | 0 | −3.20454 | −8.53449 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.45441 | 0 | 4.02413 | 0 | 0 | 2.54244 | −4.96805 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.146109 | 0 | −1.97865 | 0 | 0 | 2.71260 | −0.581318 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.10415 | 0 | −0.780857 | 0 | 0 | −2.02596 | −3.07048 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.88448 | 0 | 1.55125 | 0 | 0 | 3.97545 | −0.845662 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 6525.2.a.bn | 5 | |
| 3.b | odd | 2 | 1 | 2175.2.a.y | 5 | ||
| 5.b | even | 2 | 1 | 6525.2.a.br | 5 | ||
| 5.c | odd | 4 | 2 | 1305.2.c.i | 10 | ||
| 15.d | odd | 2 | 1 | 2175.2.a.x | 5 | ||
| 15.e | even | 4 | 2 | 435.2.c.d | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.2.c.d | ✓ | 10 | 15.e | even | 4 | 2 | |
| 1305.2.c.i | 10 | 5.c | odd | 4 | 2 | ||
| 2175.2.a.x | 5 | 15.d | odd | 2 | 1 | ||
| 2175.2.a.y | 5 | 3.b | odd | 2 | 1 | ||
| 6525.2.a.bn | 5 | 1.a | even | 1 | 1 | trivial | |
| 6525.2.a.br | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):
|
\( T_{2}^{5} + 2T_{2}^{4} - 7T_{2}^{3} - 8T_{2}^{2} + 15T_{2} - 2 \)
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\( T_{7}^{5} - 4T_{7}^{4} - 14T_{7}^{3} + 58T_{7}^{2} + 37T_{7} - 178 \)
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\( T_{11}^{5} - 5T_{11}^{4} - 16T_{11}^{3} + 88T_{11}^{2} - 5T_{11} - 191 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 2 T^{4} + \cdots - 2 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} - 4 T^{4} + \cdots - 178 \)
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| $11$ |
\( T^{5} - 5 T^{4} + \cdots - 191 \)
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| $13$ |
\( T^{5} - 6 T^{4} + \cdots + 16 \)
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| $17$ |
\( T^{5} + 14 T^{4} + \cdots + 4 \)
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| $19$ |
\( T^{5} - 2 T^{4} + \cdots + 352 \)
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| $23$ |
\( T^{5} + 13 T^{4} + \cdots - 356 \)
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| $29$ |
\( (T - 1)^{5} \)
|
| $31$ |
\( T^{5} - 2 T^{4} + \cdots + 2672 \)
|
| $37$ |
\( T^{5} - 17 T^{4} + \cdots + 3532 \)
|
| $41$ |
\( T^{5} - 19 T^{4} + \cdots - 656 \)
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| $43$ |
\( T^{5} - 7 T^{4} + \cdots - 15772 \)
|
| $47$ |
\( T^{5} + 18 T^{4} + \cdots + 2924 \)
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| $53$ |
\( T^{5} + 3 T^{4} + \cdots + 2948 \)
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| $59$ |
\( T^{5} - 22 T^{4} + \cdots + 8744 \)
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| $61$ |
\( T^{5} + 8 T^{4} + \cdots + 200 \)
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| $67$ |
\( T^{5} - 10 T^{4} + \cdots - 2032 \)
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| $71$ |
\( T^{5} - 18 T^{4} + \cdots - 176 \)
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| $73$ |
\( T^{5} - 19 T^{4} + \cdots - 1132 \)
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| $79$ |
\( T^{5} - 16 T^{4} + \cdots + 14488 \)
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| $83$ |
\( T^{5} - 3 T^{4} + \cdots - 41796 \)
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| $89$ |
\( T^{5} + 2 T^{4} + \cdots + 165482 \)
|
| $97$ |
\( T^{5} - 5 T^{4} + \cdots - 12164 \)
|
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