Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3025,2,Mod(1,3025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, 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("3025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3025 = 5^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3025.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: | \(24.1547466114\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.27433728.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} + 15x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 605) |
| 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
\( x^{6} - 9x^{4} + 15x^{2} - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 8\nu^{2} + 7 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{3} + 9\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 10\nu^{3} - 19\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} + 8\beta_{2} + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 10\beta_{4} + 31\beta_1 \)
|
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 |
|
−2.62383 | −1.33988 | 4.88448 | 0 | 3.51561 | 2.62383 | −7.56839 | −1.20473 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.37268 | −3.26180 | −0.115749 | 0 | 4.47741 | 1.37268 | 2.90425 | 7.63935 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.480901 | 1.60168 | −1.76873 | 0 | −0.770249 | 0.480901 | 1.81239 | −0.434624 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.480901 | 1.60168 | −1.76873 | 0 | 0.770249 | −0.480901 | −1.81239 | −0.434624 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.37268 | −3.26180 | −0.115749 | 0 | −4.47741 | −1.37268 | −2.90425 | 7.63935 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.62383 | −1.33988 | 4.88448 | 0 | −3.51561 | −2.62383 | 7.56839 | −1.20473 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(11\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3025.2.a.bg | 6 | |
| 5.b | even | 2 | 1 | 605.2.a.m | ✓ | 6 | |
| 11.b | odd | 2 | 1 | inner | 3025.2.a.bg | 6 | |
| 15.d | odd | 2 | 1 | 5445.2.a.bx | 6 | ||
| 20.d | odd | 2 | 1 | 9680.2.a.cw | 6 | ||
| 55.d | odd | 2 | 1 | 605.2.a.m | ✓ | 6 | |
| 55.h | odd | 10 | 4 | 605.2.g.q | 24 | ||
| 55.j | even | 10 | 4 | 605.2.g.q | 24 | ||
| 165.d | even | 2 | 1 | 5445.2.a.bx | 6 | ||
| 220.g | even | 2 | 1 | 9680.2.a.cw | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 605.2.a.m | ✓ | 6 | 5.b | even | 2 | 1 | |
| 605.2.a.m | ✓ | 6 | 55.d | odd | 2 | 1 | |
| 605.2.g.q | 24 | 55.h | odd | 10 | 4 | ||
| 605.2.g.q | 24 | 55.j | even | 10 | 4 | ||
| 3025.2.a.bg | 6 | 1.a | even | 1 | 1 | trivial | |
| 3025.2.a.bg | 6 | 11.b | odd | 2 | 1 | inner | |
| 5445.2.a.bx | 6 | 15.d | odd | 2 | 1 | ||
| 5445.2.a.bx | 6 | 165.d | even | 2 | 1 | ||
| 9680.2.a.cw | 6 | 20.d | odd | 2 | 1 | ||
| 9680.2.a.cw | 6 | 220.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):
|
\( T_{2}^{6} - 9T_{2}^{4} + 15T_{2}^{2} - 3 \)
|
|
\( T_{3}^{3} + 3T_{3}^{2} - 3T_{3} - 7 \)
|
|
\( T_{19}^{6} - 36T_{19}^{4} + 96T_{19}^{2} - 48 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 9 T^{4} + \cdots - 3 \)
|
| $3$ |
\( (T^{3} + 3 T^{2} - 3 T - 7)^{2} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 9 T^{4} + \cdots - 3 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 72 T^{4} + \cdots - 3888 \)
|
| $17$ |
\( T^{6} - 48 T^{4} + \cdots - 768 \)
|
| $19$ |
\( T^{6} - 36 T^{4} + \cdots - 48 \)
|
| $23$ |
\( (T^{3} + 6 T^{2} - 12 T - 84)^{2} \)
|
| $29$ |
\( T^{6} - 144 T^{4} + \cdots - 6912 \)
|
| $31$ |
\( (T^{3} - 48 T - 124)^{2} \)
|
| $37$ |
\( (T^{3} - 24 T + 16)^{2} \)
|
| $41$ |
\( T^{6} - 105 T^{4} + \cdots - 7203 \)
|
| $43$ |
\( T^{6} - 129 T^{4} + \cdots - 43923 \)
|
| $47$ |
\( (T^{3} + 21 T^{2} + \cdots + 303)^{2} \)
|
| $53$ |
\( (T^{3} + 12 T^{2} + \cdots - 732)^{2} \)
|
| $59$ |
\( (T^{3} - 12 T + 12)^{2} \)
|
| $61$ |
\( T^{6} - 33 T^{4} + \cdots - 1083 \)
|
| $67$ |
\( (T^{3} + 15 T^{2} + \cdots + 97)^{2} \)
|
| $71$ |
\( (T^{3} - 12 T - 12)^{2} \)
|
| $73$ |
\( T^{6} - 192 T^{4} + \cdots - 49152 \)
|
| $79$ |
\( T^{6} - 276 T^{4} + \cdots - 101568 \)
|
| $83$ |
\( T^{6} - 312 T^{4} + \cdots - 40368 \)
|
| $89$ |
\( (T^{3} + 15 T^{2} + \cdots - 147)^{2} \)
|
| $97$ |
\( (T^{3} + 12 T^{2} + \cdots - 76)^{2} \)
|
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