Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2325,2,Mod(1,2325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2325, 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("2325.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2325 = 3 \cdot 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2325.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: | \(18.5652184699\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.136751504.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 7x^{3} + 20x^{2} - 8x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 9x^{4} + 7x^{3} + 20x^{2} - 8x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 7\nu^{2} - 5\nu - 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 6\nu^{2} + 12\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 7\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 8\beta_{4} + 9\beta_{3} + \beta_{2} + 28\beta _1 - 1 \)
|
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 |
|
−2.41669 | −1.00000 | 3.84041 | 0 | 2.41669 | −1.22754 | −4.44772 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.28017 | −1.00000 | −0.361175 | 0 | 1.28017 | 0.786026 | 3.02270 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.759793 | −1.00000 | −1.42271 | 0 | 0.759793 | −4.29226 | 2.60056 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.11920 | −1.00000 | −0.747403 | 0 | −1.11920 | −0.800818 | −3.07488 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.79183 | −1.00000 | 1.21066 | 0 | −1.79183 | 4.16628 | −1.41436 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.54563 | −1.00000 | 4.48022 | 0 | −2.54563 | −4.63170 | 6.31371 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(31\) | \(-1\) |
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 | 2325.2.a.ba | yes | 6 |
| 3.b | odd | 2 | 1 | 6975.2.a.bz | 6 | ||
| 5.b | even | 2 | 1 | 2325.2.a.z | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 2325.2.c.q | 12 | ||
| 15.d | odd | 2 | 1 | 6975.2.a.cd | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2325.2.a.z | ✓ | 6 | 5.b | even | 2 | 1 | |
| 2325.2.a.ba | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 2325.2.c.q | 12 | 5.c | odd | 4 | 2 | ||
| 6975.2.a.bz | 6 | 3.b | odd | 2 | 1 | ||
| 6975.2.a.cd | 6 | 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2325))\):
|
\( T_{2}^{6} - T_{2}^{5} - 9T_{2}^{4} + 7T_{2}^{3} + 20T_{2}^{2} - 8T_{2} - 12 \)
|
|
\( T_{7}^{6} + 6T_{7}^{5} - 12T_{7}^{4} - 108T_{7}^{3} - 96T_{7}^{2} + 64T_{7} + 64 \)
|
|
\( T_{11}^{6} - 9T_{11}^{5} - 5T_{11}^{4} + 217T_{11}^{3} - 432T_{11}^{2} - 332T_{11} + 852 \)
|
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\( T_{13}^{6} + 4T_{13}^{5} - 24T_{13}^{4} - 108T_{13}^{3} - 48T_{13}^{2} + 96T_{13} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + \cdots - 12 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 64 \)
|
| $11$ |
\( T^{6} - 9 T^{5} + \cdots + 852 \)
|
| $13$ |
\( T^{6} + 4 T^{5} + \cdots + 64 \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 27 \)
|
| $19$ |
\( T^{6} - 17 T^{5} + \cdots + 540 \)
|
| $23$ |
\( T^{6} - T^{5} + \cdots - 1080 \)
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| $29$ |
\( T^{6} - 10 T^{5} + \cdots + 447 \)
|
| $31$ |
\( (T - 1)^{6} \)
|
| $37$ |
\( T^{6} + 8 T^{5} + \cdots + 80 \)
|
| $41$ |
\( T^{6} - 6 T^{5} + \cdots + 78576 \)
|
| $43$ |
\( T^{6} + T^{5} - 35 T^{4} + \cdots - 8 \)
|
| $47$ |
\( T^{6} - 7 T^{5} + \cdots + 360 \)
|
| $53$ |
\( T^{6} + T^{5} + \cdots - 26868 \)
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| $59$ |
\( T^{6} - 22 T^{5} + \cdots + 432 \)
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| $61$ |
\( T^{6} - 14 T^{5} + \cdots + 160 \)
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| $67$ |
\( T^{6} + 7 T^{5} + \cdots - 23612 \)
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| $71$ |
\( T^{6} - 5 T^{5} + \cdots - 63048 \)
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| $73$ |
\( T^{6} + 4 T^{5} + \cdots - 58304 \)
|
| $79$ |
\( T^{6} - 19 T^{5} + \cdots + 2556 \)
|
| $83$ |
\( T^{6} - 19 T^{5} + \cdots - 57600 \)
|
| $89$ |
\( T^{6} - 14 T^{5} + \cdots + 578769 \)
|
| $97$ |
\( T^{6} + 34 T^{5} + \cdots - 131247 \)
|
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