Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4225,2,Mod(1,4225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, 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("4225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4225 = 5^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4225.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: | \(33.7367948540\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1068321.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 8x^{3} - x^{2} + 12x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 325) |
| 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
\( x^{5} - 8x^{3} - x^{2} + 12x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_{2} + 3\beta _1 + 14 \)
|
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 |
|
−2.60481 | 0.135320 | 4.78505 | 0 | −0.352484 | 3.25233 | −7.25455 | −2.98169 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.09600 | 3.36467 | −0.798791 | 0 | −3.68767 | −1.59167 | 3.06747 | 8.32102 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.268864 | −0.602828 | −1.92771 | 0 | 0.162079 | 1.43094 | 1.05602 | −2.63660 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.80836 | −1.85803 | 1.27015 | 0 | −3.35998 | −4.16834 | −1.31983 | 0.452276 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.16132 | 1.96087 | 2.67130 | 0 | 4.23806 | 3.07674 | 1.45089 | 0.844995 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
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 | 4225.2.a.bo | 5 | |
| 5.b | even | 2 | 1 | 4225.2.a.bm | 5 | ||
| 13.b | even | 2 | 1 | 4225.2.a.bp | 5 | ||
| 13.e | even | 6 | 2 | 325.2.e.c | ✓ | 10 | |
| 65.d | even | 2 | 1 | 4225.2.a.bn | 5 | ||
| 65.l | even | 6 | 2 | 325.2.e.d | yes | 10 | |
| 65.r | odd | 12 | 4 | 325.2.o.c | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 325.2.e.c | ✓ | 10 | 13.e | even | 6 | 2 | |
| 325.2.e.d | yes | 10 | 65.l | even | 6 | 2 | |
| 325.2.o.c | 20 | 65.r | odd | 12 | 4 | ||
| 4225.2.a.bm | 5 | 5.b | even | 2 | 1 | ||
| 4225.2.a.bn | 5 | 65.d | even | 2 | 1 | ||
| 4225.2.a.bo | 5 | 1.a | even | 1 | 1 | trivial | |
| 4225.2.a.bp | 5 | 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):
|
\( T_{2}^{5} - 8T_{2}^{3} + T_{2}^{2} + 12T_{2} + 3 \)
|
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\( T_{3}^{5} - 3T_{3}^{4} - 5T_{3}^{3} + 11T_{3}^{2} + 6T_{3} - 1 \)
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\( T_{7}^{5} - 2T_{7}^{4} - 19T_{7}^{3} + 44T_{7}^{2} + 44T_{7} - 95 \)
|
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\( T_{11}^{5} + 3T_{11}^{4} - 17T_{11}^{3} - 56T_{11}^{2} - 9T_{11} + 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 8 T^{3} + \cdots + 3 \)
|
| $3$ |
\( T^{5} - 3 T^{4} + \cdots - 1 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 2 T^{4} + \cdots - 95 \)
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| $11$ |
\( T^{5} + 3 T^{4} + \cdots + 3 \)
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| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} - 4 T^{4} + \cdots - 489 \)
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| $19$ |
\( T^{5} - 4 T^{4} + \cdots - 15 \)
|
| $23$ |
\( T^{5} - 15 T^{4} + \cdots + 531 \)
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| $29$ |
\( T^{5} + T^{4} + \cdots + 939 \)
|
| $31$ |
\( T^{5} - 57 T^{3} + \cdots - 225 \)
|
| $37$ |
\( T^{5} + 17 T^{4} + \cdots - 909 \)
|
| $41$ |
\( T^{5} + 6 T^{4} + \cdots + 225 \)
|
| $43$ |
\( T^{5} - 12 T^{4} + \cdots - 3767 \)
|
| $47$ |
\( T^{5} - 12 T^{4} + \cdots - 2025 \)
|
| $53$ |
\( T^{5} - 8 T^{4} + \cdots + 6075 \)
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| $59$ |
\( T^{5} - 12 T^{4} + \cdots - 2187 \)
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| $61$ |
\( T^{5} - 5 T^{4} + \cdots - 635 \)
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| $67$ |
\( T^{5} - 16 T^{4} + \cdots - 50671 \)
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| $71$ |
\( T^{5} + 19 T^{4} + \cdots + 10089 \)
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| $73$ |
\( T^{5} - 8 T^{4} + \cdots - 10125 \)
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| $79$ |
\( T^{5} + 14 T^{4} + \cdots + 16875 \)
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| $83$ |
\( T^{5} - 7 T^{4} + \cdots + 53775 \)
|
| $89$ |
\( T^{5} - 10 T^{4} + \cdots - 1593 \)
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| $97$ |
\( T^{5} + 37 T^{4} + \cdots - 41029 \)
|
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