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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| 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 \(\beta = \sqrt{2}\). 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 |
|
−5.41421 | 0 | 21.3137 | 5.00000 | 0 | −7.00000 | −72.0833 | 0 | −27.0711 | ||||||||||||||||||||||||
| 1.2 | −2.58579 | 0 | −1.31371 | 5.00000 | 0 | −7.00000 | 24.0833 | 0 | −12.9289 | |||||||||||||||||||||||||
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.f | 2 | |
| 3.b | odd | 2 | 1 | 35.4.a.b | ✓ | 2 | |
| 5.b | even | 2 | 1 | 1575.4.a.z | 2 | ||
| 7.b | odd | 2 | 1 | 2205.4.a.u | 2 | ||
| 12.b | even | 2 | 1 | 560.4.a.r | 2 | ||
| 15.d | odd | 2 | 1 | 175.4.a.c | 2 | ||
| 15.e | even | 4 | 2 | 175.4.b.c | 4 | ||
| 21.c | even | 2 | 1 | 245.4.a.k | 2 | ||
| 21.g | even | 6 | 2 | 245.4.e.i | 4 | ||
| 21.h | odd | 6 | 2 | 245.4.e.h | 4 | ||
| 24.f | even | 2 | 1 | 2240.4.a.bo | 2 | ||
| 24.h | odd | 2 | 1 | 2240.4.a.bn | 2 | ||
| 105.g | even | 2 | 1 | 1225.4.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.a.b | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 175.4.a.c | 2 | 15.d | odd | 2 | 1 | ||
| 175.4.b.c | 4 | 15.e | even | 4 | 2 | ||
| 245.4.a.k | 2 | 21.c | even | 2 | 1 | ||
| 245.4.e.h | 4 | 21.h | odd | 6 | 2 | ||
| 245.4.e.i | 4 | 21.g | even | 6 | 2 | ||
| 315.4.a.f | 2 | 1.a | even | 1 | 1 | trivial | |
| 560.4.a.r | 2 | 12.b | even | 2 | 1 | ||
| 1225.4.a.m | 2 | 105.g | even | 2 | 1 | ||
| 1575.4.a.z | 2 | 5.b | even | 2 | 1 | ||
| 2205.4.a.u | 2 | 7.b | odd | 2 | 1 | ||
| 2240.4.a.bn | 2 | 24.h | odd | 2 | 1 | ||
| 2240.4.a.bo | 2 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 8T_{2} + 14 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 8T + 14 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T - 5)^{2} \)
|
| $7$ |
\( (T + 7)^{2} \)
|
| $11$ |
\( T^{2} - 14T - 1999 \)
|
| $13$ |
\( T^{2} - 50T + 593 \)
|
| $17$ |
\( T^{2} - 50T - 3247 \)
|
| $19$ |
\( T^{2} - 36T - 3548 \)
|
| $23$ |
\( T^{2} + 244T + 5636 \)
|
| $29$ |
\( T^{2} - 26T - 983 \)
|
| $31$ |
\( T^{2} + 120T - 61200 \)
|
| $37$ |
\( T^{2} - 564T + 72324 \)
|
| $41$ |
\( T^{2} - 328T - 3856 \)
|
| $43$ |
\( T^{2} + 260T + 7652 \)
|
| $47$ |
\( T^{2} - 350T - 4223 \)
|
| $53$ |
\( T^{2} - 56T - 31984 \)
|
| $59$ |
\( (T - 616)^{2} \)
|
| $61$ |
\( T^{2} - 336T + 4896 \)
|
| $67$ |
\( T^{2} + 152T - 2416 \)
|
| $71$ |
\( (T - 952)^{2} \)
|
| $73$ |
\( T^{2} - 676T - 122428 \)
|
| $79$ |
\( T^{2} - 1014 T + 134041 \)
|
| $83$ |
\( T^{2} - 376T - 684656 \)
|
| $89$ |
\( T^{2} - 216T + 7792 \)
|
| $97$ |
\( T^{2} - 2742 T + 1782841 \)
|
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