Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,8,Mod(1,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.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: | \(98.4012830275\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{11}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 11 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = 2\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−14.6332 | 0 | 86.1320 | −125.000 | 0 | −343.000 | 612.665 | 0 | 1829.16 | ||||||||||||||||||||||||
| 1.2 | −1.36675 | 0 | −126.132 | −125.000 | 0 | −343.000 | 347.335 | 0 | 170.844 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 315.8.a.c | 2 | |
| 3.b | odd | 2 | 1 | 35.8.a.a | ✓ | 2 | |
| 12.b | even | 2 | 1 | 560.8.a.i | 2 | ||
| 15.d | odd | 2 | 1 | 175.8.a.b | 2 | ||
| 15.e | even | 4 | 2 | 175.8.b.c | 4 | ||
| 21.c | even | 2 | 1 | 245.8.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.8.a.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 175.8.a.b | 2 | 15.d | odd | 2 | 1 | ||
| 175.8.b.c | 4 | 15.e | even | 4 | 2 | ||
| 245.8.a.b | 2 | 21.c | even | 2 | 1 | ||
| 315.8.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 560.8.a.i | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 16T_{2} + 20 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 16T + 20 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T + 125)^{2} \)
|
| $7$ |
\( (T + 343)^{2} \)
|
| $11$ |
\( T^{2} - 7906 T + 9272609 \)
|
| $13$ |
\( T^{2} + 17818 T + 71682425 \)
|
| $17$ |
\( T^{2} - 2398 T - 213462799 \)
|
| $19$ |
\( T^{2} + \cdots - 1348143980 \)
|
| $23$ |
\( T^{2} + \cdots - 4980234316 \)
|
| $29$ |
\( T^{2} + \cdots - 20838975695 \)
|
| $31$ |
\( T^{2} + \cdots - 6409132848 \)
|
| $37$ |
\( T^{2} + \cdots - 66117717036 \)
|
| $41$ |
\( T^{2} + \cdots - 58262616304 \)
|
| $43$ |
\( T^{2} + \cdots + 197893738100 \)
|
| $47$ |
\( T^{2} + \cdots - 123267405047 \)
|
| $53$ |
\( T^{2} + \cdots + 214640651072 \)
|
| $59$ |
\( T^{2} + \cdots - 818710818800 \)
|
| $61$ |
\( T^{2} + \cdots + 816706565136 \)
|
| $67$ |
\( T^{2} + \cdots - 10082635080448 \)
|
| $71$ |
\( T^{2} + \cdots - 1610731398080 \)
|
| $73$ |
\( T^{2} + \cdots + 10866534315332 \)
|
| $79$ |
\( T^{2} + \cdots + 14225767990465 \)
|
| $83$ |
\( T^{2} + \cdots - 38809208931248 \)
|
| $89$ |
\( T^{2} + \cdots + 77796446568160 \)
|
| $97$ |
\( T^{2} + \cdots - 15859623239687 \)
|
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