Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4235,2,Mod(1,4235)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, 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("4235.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4235 = 5 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4235.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: | \(33.8166452560\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.580017.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 7x^{3} - 4x^{2} + 6x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{5} - 7x^{3} - 4x^{2} + 6x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + \nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{4} + 7\beta_{3} + \beta_{2} + 3\beta _1 + 10 \)
|
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 |
|
−2.74137 | −3.04102 | 5.51511 | −1.00000 | 8.33658 | 1.00000 | −9.63623 | 6.24783 | 2.74137 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.941569 | 1.53997 | −1.11345 | −1.00000 | −1.44999 | 1.00000 | 2.93153 | −0.628495 | 0.941569 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.470042 | −0.577925 | −1.77906 | −1.00000 | −0.271649 | 1.00000 | −1.77632 | −2.66600 | −0.470042 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.27779 | 2.68935 | −0.367260 | −1.00000 | 3.43642 | 1.00000 | −3.02485 | 4.23262 | −1.27779 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.93511 | −2.61037 | 1.74465 | −1.00000 | −5.05136 | 1.00000 | −0.494123 | 3.81405 | −1.93511 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 4235.2.a.bb | yes | 5 |
| 11.b | odd | 2 | 1 | 4235.2.a.ba | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4235.2.a.ba | ✓ | 5 | 11.b | odd | 2 | 1 | |
| 4235.2.a.bb | yes | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):
|
\( T_{2}^{5} - 7T_{2}^{3} + 4T_{2}^{2} + 6T_{2} - 3 \)
|
|
\( T_{3}^{5} + 2T_{3}^{4} - 11T_{3}^{3} - 17T_{3}^{2} + 27T_{3} + 19 \)
|
|
\( T_{13}^{5} + 10T_{13}^{4} + 8T_{13}^{3} - 99T_{13}^{2} - 4T_{13} + 133 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 7 T^{3} + \cdots - 3 \)
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| $3$ |
\( T^{5} + 2 T^{4} + \cdots + 19 \)
|
| $5$ |
\( (T + 1)^{5} \)
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| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} + 10 T^{4} + \cdots + 133 \)
|
| $17$ |
\( T^{5} + 2 T^{4} + \cdots - 9 \)
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| $19$ |
\( T^{5} - 3 T^{4} + \cdots + 269 \)
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| $23$ |
\( T^{5} - 3 T^{4} + \cdots - 3573 \)
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| $29$ |
\( T^{5} + T^{4} + \cdots + 147 \)
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| $31$ |
\( T^{5} + 12 T^{4} + \cdots - 5161 \)
|
| $37$ |
\( T^{5} - 145 T^{3} + \cdots + 11545 \)
|
| $41$ |
\( T^{5} - 11 T^{4} + \cdots + 1425 \)
|
| $43$ |
\( T^{5} + 10 T^{4} + \cdots - 761 \)
|
| $47$ |
\( T^{5} - 16 T^{4} + \cdots - 1335 \)
|
| $53$ |
\( T^{5} - 4 T^{4} + \cdots - 2307 \)
|
| $59$ |
\( T^{5} + 32 T^{4} + \cdots + 3267 \)
|
| $61$ |
\( T^{5} + 40 T^{4} + \cdots + 13937 \)
|
| $67$ |
\( T^{5} - 7 T^{4} + \cdots - 5 \)
|
| $71$ |
\( T^{5} - 10 T^{4} + \cdots - 1155 \)
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| $73$ |
\( T^{5} + 11 T^{4} + \cdots - 121 \)
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| $79$ |
\( T^{5} + 13 T^{4} + \cdots + 59 \)
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| $83$ |
\( T^{5} + 26 T^{4} + \cdots - 23445 \)
|
| $89$ |
\( T^{5} - 5 T^{4} + \cdots - 927 \)
|
| $97$ |
\( T^{5} + 5 T^{4} + \cdots + 49 \)
|
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