Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,10,Mod(1,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.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: | \(115.883063137\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 652x + 4000 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 25) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 652x + 4000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 3\nu - 436 ) / 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 33\nu - 445 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 4\beta _1 + 3 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{2} + 132\beta _1 + 4351 ) / 10 \)
|
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 |
|
−31.7828 | 0 | 498.143 | 0 | 0 | −637.237 | 440.406 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | −21.4187 | 0 | −53.2406 | 0 | 0 | 9905.49 | 12106.7 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 20.2014 | 0 | −103.903 | 0 | 0 | −4010.25 | −12442.1 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 225.10.a.m | 3 | |
| 3.b | odd | 2 | 1 | 25.10.a.d | yes | 3 | |
| 5.b | even | 2 | 1 | 225.10.a.p | 3 | ||
| 5.c | odd | 4 | 2 | 225.10.b.m | 6 | ||
| 12.b | even | 2 | 1 | 400.10.a.u | 3 | ||
| 15.d | odd | 2 | 1 | 25.10.a.c | ✓ | 3 | |
| 15.e | even | 4 | 2 | 25.10.b.c | 6 | ||
| 60.h | even | 2 | 1 | 400.10.a.y | 3 | ||
| 60.l | odd | 4 | 2 | 400.10.c.q | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.10.a.c | ✓ | 3 | 15.d | odd | 2 | 1 | |
| 25.10.a.d | yes | 3 | 3.b | odd | 2 | 1 | |
| 25.10.b.c | 6 | 15.e | even | 4 | 2 | ||
| 225.10.a.m | 3 | 1.a | even | 1 | 1 | trivial | |
| 225.10.a.p | 3 | 5.b | even | 2 | 1 | ||
| 225.10.b.m | 6 | 5.c | odd | 4 | 2 | ||
| 400.10.a.u | 3 | 12.b | even | 2 | 1 | ||
| 400.10.a.y | 3 | 60.h | even | 2 | 1 | ||
| 400.10.c.q | 6 | 60.l | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\):
|
\( T_{2}^{3} + 33T_{2}^{2} - 394T_{2} - 13752 \)
|
|
\( T_{7}^{3} - 5258T_{7}^{2} - 43480164T_{7} - 25313297688 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 33 T^{2} + \cdots - 13752 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + \cdots - 25313297688 \)
|
| $11$ |
\( T^{3} + \cdots + 75283351667163 \)
|
| $13$ |
\( T^{3} + \cdots + 14\!\cdots\!84 \)
|
| $17$ |
\( T^{3} + \cdots - 17\!\cdots\!07 \)
|
| $19$ |
\( T^{3} + \cdots + 22\!\cdots\!25 \)
|
| $23$ |
\( T^{3} + \cdots + 38\!\cdots\!16 \)
|
| $29$ |
\( T^{3} + \cdots - 39\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 81\!\cdots\!48 \)
|
| $37$ |
\( T^{3} + \cdots - 17\!\cdots\!28 \)
|
| $41$ |
\( T^{3} + \cdots - 15\!\cdots\!47 \)
|
| $43$ |
\( T^{3} + \cdots + 33\!\cdots\!84 \)
|
| $47$ |
\( T^{3} + \cdots + 14\!\cdots\!08 \)
|
| $53$ |
\( T^{3} + \cdots - 40\!\cdots\!84 \)
|
| $59$ |
\( T^{3} + \cdots + 49\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 62\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots - 27\!\cdots\!93 \)
|
| $71$ |
\( T^{3} + \cdots - 11\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots + 13\!\cdots\!09 \)
|
| $79$ |
\( T^{3} + \cdots - 26\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 83\!\cdots\!41 \)
|
| $89$ |
\( T^{3} + \cdots + 19\!\cdots\!75 \)
|
| $97$ |
\( T^{3} + \cdots - 34\!\cdots\!08 \)
|
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