Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(18.5856016518\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.14360.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 17x - 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 17x - 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
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 |
|
−3.48565 | 0 | 4.14976 | −5.00000 | 0 | 7.00000 | 13.4206 | 0 | 17.4283 | |||||||||||||||||||||||||||
| 1.2 | 1.86109 | 0 | −4.53636 | −5.00000 | 0 | 7.00000 | −23.3312 | 0 | −9.30543 | ||||||||||||||||||||||||||||
| 1.3 | 4.62456 | 0 | 13.3866 | −5.00000 | 0 | 7.00000 | 24.9107 | 0 | −23.1228 | ||||||||||||||||||||||||||||
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 | 315.4.a.p | 3 | |
| 3.b | odd | 2 | 1 | 35.4.a.c | ✓ | 3 | |
| 5.b | even | 2 | 1 | 1575.4.a.ba | 3 | ||
| 7.b | odd | 2 | 1 | 2205.4.a.bm | 3 | ||
| 12.b | even | 2 | 1 | 560.4.a.u | 3 | ||
| 15.d | odd | 2 | 1 | 175.4.a.f | 3 | ||
| 15.e | even | 4 | 2 | 175.4.b.e | 6 | ||
| 21.c | even | 2 | 1 | 245.4.a.l | 3 | ||
| 21.g | even | 6 | 2 | 245.4.e.n | 6 | ||
| 21.h | odd | 6 | 2 | 245.4.e.m | 6 | ||
| 24.f | even | 2 | 1 | 2240.4.a.bv | 3 | ||
| 24.h | odd | 2 | 1 | 2240.4.a.bt | 3 | ||
| 105.g | even | 2 | 1 | 1225.4.a.y | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.a.c | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 175.4.a.f | 3 | 15.d | odd | 2 | 1 | ||
| 175.4.b.e | 6 | 15.e | even | 4 | 2 | ||
| 245.4.a.l | 3 | 21.c | even | 2 | 1 | ||
| 245.4.e.m | 6 | 21.h | odd | 6 | 2 | ||
| 245.4.e.n | 6 | 21.g | even | 6 | 2 | ||
| 315.4.a.p | 3 | 1.a | even | 1 | 1 | trivial | |
| 560.4.a.u | 3 | 12.b | even | 2 | 1 | ||
| 1225.4.a.y | 3 | 105.g | even | 2 | 1 | ||
| 1575.4.a.ba | 3 | 5.b | even | 2 | 1 | ||
| 2205.4.a.bm | 3 | 7.b | odd | 2 | 1 | ||
| 2240.4.a.bt | 3 | 24.h | odd | 2 | 1 | ||
| 2240.4.a.bv | 3 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3T_{2}^{2} - 14T_{2} + 30 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 3 T^{2} + \cdots + 30 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} - 74 T^{2} + \cdots - 7692 \)
|
| $13$ |
\( T^{3} - 44 T^{2} + \cdots - 44870 \)
|
| $17$ |
\( T^{3} - 52 T^{2} + \cdots + 56706 \)
|
| $19$ |
\( T^{3} - 168 T^{2} + \cdots - 28720 \)
|
| $23$ |
\( T^{3} - 124 T^{2} + \cdots + 94368 \)
|
| $29$ |
\( T^{3} + 332 T^{2} + \cdots - 2565450 \)
|
| $31$ |
\( T^{3} - 320 T^{2} + \cdots + 50176 \)
|
| $37$ |
\( T^{3} + 54 T^{2} + \cdots + 25736 \)
|
| $41$ |
\( T^{3} + 362 T^{2} + \cdots + 1536192 \)
|
| $43$ |
\( T^{3} + 16 T^{2} + \cdots - 1524560 \)
|
| $47$ |
\( T^{3} - 730 T^{2} + \cdots - 4968912 \)
|
| $53$ |
\( T^{3} + 110 T^{2} + \cdots - 90318336 \)
|
| $59$ |
\( T^{3} - 180 T^{2} + \cdots + 202459200 \)
|
| $61$ |
\( T^{3} - 1222 T^{2} + \cdots - 38393792 \)
|
| $67$ |
\( T^{3} - 204 T^{2} + \cdots + 324944128 \)
|
| $71$ |
\( T^{3} - 136 T^{2} + \cdots - 15575040 \)
|
| $73$ |
\( T^{3} - 310 T^{2} + \cdots + 48718616 \)
|
| $79$ |
\( T^{3} + 1034 T^{2} + \cdots - 343615600 \)
|
| $83$ |
\( T^{3} - 1660 T^{2} + \cdots + 42727104 \)
|
| $89$ |
\( T^{3} + 242 T^{2} + \cdots + 6359520 \)
|
| $97$ |
\( T^{3} - 100 T^{2} + \cdots - 1978018 \)
|
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