Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,6,Mod(1,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.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: | \(39.2940358542\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 43 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 52\nu - 48 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} - \beta _1 + 43 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{3} - 3\beta_{2} + 53\beta _1 + 5 \)
|
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 |
|
−6.05190 | −15.9755 | 4.62555 | −25.0000 | 96.6824 | 0 | 165.668 | 12.2177 | 151.298 | ||||||||||||||||||||||||||||||
| 1.2 | −2.73688 | 28.3358 | −24.5095 | −25.0000 | −77.5516 | 0 | 154.660 | 559.917 | 68.4220 | |||||||||||||||||||||||||||||||
| 1.3 | 5.86448 | −26.2268 | 2.39209 | −25.0000 | −153.807 | 0 | −173.635 | 444.847 | −146.612 | |||||||||||||||||||||||||||||||
| 1.4 | 9.92431 | −0.133419 | 66.4919 | −25.0000 | −1.32409 | 0 | 342.308 | −242.982 | −248.108 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
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 | 245.6.a.e | 4 | |
| 7.b | odd | 2 | 1 | 35.6.a.d | ✓ | 4 | |
| 21.c | even | 2 | 1 | 315.6.a.l | 4 | ||
| 28.d | even | 2 | 1 | 560.6.a.v | 4 | ||
| 35.c | odd | 2 | 1 | 175.6.a.f | 4 | ||
| 35.f | even | 4 | 2 | 175.6.b.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.6.a.d | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 175.6.a.f | 4 | 35.c | odd | 2 | 1 | ||
| 175.6.b.f | 8 | 35.f | even | 4 | 2 | ||
| 245.6.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 315.6.a.l | 4 | 21.c | even | 2 | 1 | ||
| 560.6.a.v | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):
|
\( T_{2}^{4} - 7T_{2}^{3} - 64T_{2}^{2} + 250T_{2} + 964 \)
|
|
\( T_{3}^{4} + 14T_{3}^{3} - 775T_{3}^{2} - 11976T_{3} - 1584 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 7 T^{3} + \cdots + 964 \)
|
| $3$ |
\( T^{4} + 14 T^{3} + \cdots - 1584 \)
|
| $5$ |
\( (T + 25)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 20510223536 \)
|
| $13$ |
\( T^{4} + \cdots + 430417376452 \)
|
| $17$ |
\( T^{4} + \cdots + 617989800868 \)
|
| $19$ |
\( T^{4} + \cdots + 5221683964480 \)
|
| $23$ |
\( T^{4} + \cdots - 19307649395456 \)
|
| $29$ |
\( T^{4} - 8066 T^{3} + \cdots + 831691300 \)
|
| $31$ |
\( T^{4} + \cdots - 738407035699200 \)
|
| $37$ |
\( T^{4} + \cdots - 57\!\cdots\!96 \)
|
| $41$ |
\( T^{4} + \cdots + 225655412958976 \)
|
| $43$ |
\( T^{4} + \cdots + 16\!\cdots\!68 \)
|
| $47$ |
\( T^{4} + \cdots - 16\!\cdots\!88 \)
|
| $53$ |
\( T^{4} + \cdots - 15\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots + 13\!\cdots\!40 \)
|
| $61$ |
\( T^{4} + \cdots - 25\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots - 74\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots - 17\!\cdots\!24 \)
|
| $73$ |
\( T^{4} + \cdots - 32\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 29\!\cdots\!64 \)
|
| $89$ |
\( T^{4} + \cdots + 21\!\cdots\!20 \)
|
| $97$ |
\( T^{4} + \cdots - 75\!\cdots\!24 \)
|
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