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: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 128x^{3} + 288x^{2} + 3551x - 6510 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 128x^{3} + 288x^{2} + 3551x - 6510 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 9\nu^{3} - 89\nu^{2} - 571\nu + 1350 ) / 60 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + \nu - 52 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{4} - 3\nu^{3} + 178\nu^{2} + 107\nu - 2025 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta _1 + 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 8\beta_{2} + 69\beta _1 - 45 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{4} + 89\beta_{3} - 12\beta_{2} - 139\beta _1 + 3683 \)
|
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 |
|
−10.6532 | 22.7173 | 81.4900 | 25.0000 | −242.012 | 0 | −527.226 | 273.078 | −266.329 | |||||||||||||||||||||||||||||||||
| 1.2 | −7.03043 | −10.0622 | 17.4270 | 25.0000 | 70.7416 | 0 | 102.455 | −141.752 | −175.761 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.780243 | −4.65090 | −31.3912 | 25.0000 | −3.62883 | 0 | −49.4606 | −221.369 | 19.5061 | ||||||||||||||||||||||||||||||||||
| 1.4 | 6.31036 | 11.8302 | 7.82066 | 25.0000 | 74.6531 | 0 | −152.580 | −103.046 | 157.759 | ||||||||||||||||||||||||||||||||||
| 1.5 | 7.59300 | −26.8345 | 25.6536 | 25.0000 | −203.754 | 0 | −48.1883 | 477.089 | 189.825 | ||||||||||||||||||||||||||||||||||
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.f | ✓ | 5 |
| 7.b | odd | 2 | 1 | 245.6.a.g | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 245.6.a.f | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 245.6.a.g | yes | 5 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):
|
\( T_{2}^{5} + 3T_{2}^{4} - 126T_{2}^{3} - 98T_{2}^{2} + 3740T_{2} - 2800 \)
|
|
\( T_{3}^{5} + 7T_{3}^{4} - 725T_{3}^{3} - 2835T_{3}^{2} + 75300T_{3} + 337500 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 3 T^{4} + \cdots - 2800 \)
|
| $3$ |
\( T^{5} + 7 T^{4} + \cdots + 337500 \)
|
| $5$ |
\( (T - 25)^{5} \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + \cdots - 7633598722544 \)
|
| $13$ |
\( T^{5} + \cdots - 7076320312500 \)
|
| $17$ |
\( T^{5} + \cdots - 32\!\cdots\!00 \)
|
| $19$ |
\( T^{5} + \cdots + 28\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 26\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots - 23\!\cdots\!76 \)
|
| $31$ |
\( T^{5} + \cdots - 31\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 11\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 30\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 69\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots + 31\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots - 29\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 14\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots + 26\!\cdots\!00 \)
|
| $71$ |
\( T^{5} + \cdots + 77\!\cdots\!52 \)
|
| $73$ |
\( T^{5} + \cdots - 12\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots + 20\!\cdots\!44 \)
|
| $83$ |
\( T^{5} + \cdots + 88\!\cdots\!00 \)
|
| $89$ |
\( T^{5} + \cdots + 29\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 57\!\cdots\!00 \)
|
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