Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [975,6,Mod(1,975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("975.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 975.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: | \(156.374224318\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{14}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 39) |
| 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 \(\beta = 2\sqrt{14}\). 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 |
|
−5.48331 | 9.00000 | −1.93326 | 0 | −49.3498 | 46.3165 | 186.067 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 9.48331 | 9.00000 | 57.9333 | 0 | 85.3498 | −118.316 | 245.933 | 81.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 975.6.a.c | 2 | |
| 5.b | even | 2 | 1 | 39.6.a.b | ✓ | 2 | |
| 15.d | odd | 2 | 1 | 117.6.a.b | 2 | ||
| 20.d | odd | 2 | 1 | 624.6.a.k | 2 | ||
| 65.d | even | 2 | 1 | 507.6.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.6.a.b | ✓ | 2 | 5.b | even | 2 | 1 | |
| 117.6.a.b | 2 | 15.d | odd | 2 | 1 | ||
| 507.6.a.c | 2 | 65.d | even | 2 | 1 | ||
| 624.6.a.k | 2 | 20.d | odd | 2 | 1 | ||
| 975.6.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(975))\):
|
\( T_{2}^{2} - 4T_{2} - 52 \)
|
|
\( T_{7}^{2} + 72T_{7} - 5480 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 4T - 52 \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 72T - 5480 \)
|
| $11$ |
\( T^{2} + 596T - 9980 \)
|
| $13$ |
\( (T - 169)^{2} \)
|
| $17$ |
\( T^{2} - 268T - 611260 \)
|
| $19$ |
\( T^{2} - 1128T + 19672 \)
|
| $23$ |
\( T^{2} - 1768 T - 799088 \)
|
| $29$ |
\( T^{2} + 7612 T + 14090500 \)
|
| $31$ |
\( T^{2} + 4160 T - 33052984 \)
|
| $37$ |
\( T^{2} + 17468 T + 45451172 \)
|
| $41$ |
\( T^{2} + 28000 T + 180799304 \)
|
| $43$ |
\( T^{2} - 24328 T + 73902896 \)
|
| $47$ |
\( T^{2} - 18108 T - 180267260 \)
|
| $53$ |
\( T^{2} + 1420 T - 35695196 \)
|
| $59$ |
\( T^{2} + \cdots - 1582962940 \)
|
| $61$ |
\( T^{2} + 37148 T + 199475012 \)
|
| $67$ |
\( T^{2} + \cdots + 2672473736 \)
|
| $71$ |
\( T^{2} + \cdots - 1074465500 \)
|
| $73$ |
\( T^{2} + \cdots - 2379218876 \)
|
| $79$ |
\( T^{2} + \cdots - 2524694336 \)
|
| $83$ |
\( T^{2} + \cdots + 10699975748 \)
|
| $89$ |
\( T^{2} + \cdots - 5742473000 \)
|
| $97$ |
\( T^{2} + \cdots + 3235033508 \)
|
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