Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8325,2,Mod(1,8325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8325, 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("8325.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8325.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: | \(66.4754596827\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.528933.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 7x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 2775) |
| 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
\( x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 7x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - \nu + 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 5\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 6\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 2\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - 2\beta_{3} + 3\beta_{2} + \beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{4} - 3\beta_{3} + 5\beta_{2} + 3\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9\beta_{4} - 22\beta_{3} + 33\beta_{2} + 15\beta _1 + 44 ) / 2 \)
|
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 |
|
−2.74664 | 0 | 5.54401 | 0 | 0 | −2.38589 | −9.73410 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.52233 | 0 | 0.317493 | 0 | 0 | 4.77701 | 2.56133 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.15875 | 0 | −0.657292 | 0 | 0 | −4.20154 | 3.07914 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.03520 | 0 | −0.928359 | 0 | 0 | −1.11914 | −3.03144 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.39252 | 0 | 3.72415 | 0 | 0 | 2.92955 | 4.12506 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(37\) | \( -1 \) |
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 | 8325.2.a.ca | 5 | |
| 3.b | odd | 2 | 1 | 2775.2.a.bc | yes | 5 | |
| 5.b | even | 2 | 1 | 8325.2.a.cg | 5 | ||
| 15.d | odd | 2 | 1 | 2775.2.a.z | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2775.2.a.z | ✓ | 5 | 15.d | odd | 2 | 1 | |
| 2775.2.a.bc | yes | 5 | 3.b | odd | 2 | 1 | |
| 8325.2.a.ca | 5 | 1.a | even | 1 | 1 | trivial | |
| 8325.2.a.cg | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):
|
\( T_{2}^{5} + 2T_{2}^{4} - 7T_{2}^{3} - 13T_{2}^{2} + 6T_{2} + 12 \)
|
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\( T_{7}^{5} - 28T_{7}^{3} - 15T_{7}^{2} + 157T_{7} + 157 \)
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\( T_{11}^{5} - T_{11}^{4} - 18T_{11}^{3} + 31T_{11}^{2} + 18T_{11} - 12 \)
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\( T_{13}^{5} - 14T_{13}^{4} + 64T_{13}^{3} - 91T_{13}^{2} - 63T_{13} + 167 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 2 T^{4} + \cdots + 12 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} - 28 T^{3} + \cdots + 157 \)
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| $11$ |
\( T^{5} - T^{4} + \cdots - 12 \)
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| $13$ |
\( T^{5} - 14 T^{4} + \cdots + 167 \)
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| $17$ |
\( T^{5} + 11 T^{4} + \cdots - 48 \)
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| $19$ |
\( T^{5} - 10 T^{4} + \cdots - 349 \)
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| $23$ |
\( T^{5} + 4 T^{4} + \cdots + 516 \)
|
| $29$ |
\( T^{5} + 5 T^{4} + \cdots - 228 \)
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| $31$ |
\( T^{5} + 3 T^{4} + \cdots + 2395 \)
|
| $37$ |
\( (T - 1)^{5} \)
|
| $41$ |
\( T^{5} - 8 T^{4} + \cdots - 60 \)
|
| $43$ |
\( T^{5} - T^{4} + \cdots + 1161 \)
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| $47$ |
\( T^{5} + T^{4} + \cdots - 48 \)
|
| $53$ |
\( T^{5} - T^{4} + \cdots + 48 \)
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| $59$ |
\( T^{5} - 21 T^{4} + \cdots - 108 \)
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| $61$ |
\( T^{5} - 26 T^{4} + \cdots + 30839 \)
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| $67$ |
\( T^{5} - 21 T^{4} + \cdots + 911 \)
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| $71$ |
\( T^{5} - 17 T^{4} + \cdots - 684 \)
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| $73$ |
\( T^{5} - 25 T^{4} + \cdots + 4 \)
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| $79$ |
\( T^{5} - 10 T^{4} + \cdots - 2384 \)
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| $83$ |
\( T^{5} - 3 T^{4} + \cdots - 17196 \)
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| $89$ |
\( T^{5} + 33 T^{4} + \cdots - 7284 \)
|
| $97$ |
\( T^{5} - 16 T^{4} + \cdots + 5464 \)
|
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