Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,4,Mod(1,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1150.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1150.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: | \(67.8521965066\) |
| 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: |
\( x^{5} - 73x^{3} - 73x^{2} + 810x - 260 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{5} - 73x^{3} - 73x^{2} + 810x - 260 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 29 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 4\nu^{2} - 46\nu + 73 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 60\nu^{2} + 2\nu + 356 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 29 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 4\beta_{2} + 50\beta _1 + 43 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{4} + 6\beta_{3} + 68\beta_{2} + 158\beta _1 + 1470 \)
|
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 |
|
−2.00000 | −8.40587 | 4.00000 | 0 | 16.8117 | −13.1917 | −8.00000 | 43.6586 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −2.80523 | 4.00000 | 0 | 5.61046 | −28.6194 | −8.00000 | −19.1307 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −0.334433 | 4.00000 | 0 | 0.668867 | 14.7343 | −8.00000 | −26.8882 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 5.17669 | 4.00000 | 0 | −10.3534 | 22.9133 | −8.00000 | −0.201881 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 6.36884 | 4.00000 | 0 | −12.7377 | −15.8364 | −8.00000 | 13.5621 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 1150.4.a.r | ✓ | 5 |
| 5.b | even | 2 | 1 | 1150.4.a.u | yes | 5 | |
| 5.c | odd | 4 | 2 | 1150.4.b.q | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1150.4.a.r | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1150.4.a.u | yes | 5 | 5.b | even | 2 | 1 | |
| 1150.4.b.q | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1150))\):
|
\( T_{3}^{5} - 73T_{3}^{3} + 73T_{3}^{2} + 810T_{3} + 260 \)
|
|
\( T_{7}^{5} + 20T_{7}^{4} - 793T_{7}^{3} - 13700T_{7}^{2} + 125916T_{7} + 2018520 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( T^{5} - 73 T^{3} + \cdots + 260 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + 20 T^{4} + \cdots + 2018520 \)
|
| $11$ |
\( T^{5} - 16 T^{4} + \cdots + 3859464 \)
|
| $13$ |
\( T^{5} + 56 T^{4} + \cdots + 86273217 \)
|
| $17$ |
\( T^{5} + 70 T^{4} + \cdots - 177045120 \)
|
| $19$ |
\( T^{5} + \cdots + 3001283496 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} - 128 T^{4} + \cdots + 672972489 \)
|
| $31$ |
\( T^{5} + \cdots + 3684414060 \)
|
| $37$ |
\( T^{5} + \cdots - 24927292128 \)
|
| $41$ |
\( T^{5} + 102 T^{4} + \cdots - 974454795 \)
|
| $43$ |
\( T^{5} + \cdots + 18704792640 \)
|
| $47$ |
\( T^{5} + \cdots - 1763324403300 \)
|
| $53$ |
\( T^{5} + \cdots - 127529551872 \)
|
| $59$ |
\( T^{5} + \cdots + 7351600368780 \)
|
| $61$ |
\( T^{5} + \cdots + 3649011506400 \)
|
| $67$ |
\( T^{5} + \cdots - 20902706250240 \)
|
| $71$ |
\( T^{5} + \cdots - 1909230484860 \)
|
| $73$ |
\( T^{5} + \cdots - 16515313827759 \)
|
| $79$ |
\( T^{5} + \cdots + 1076509576512 \)
|
| $83$ |
\( T^{5} + \cdots + 56873063355048 \)
|
| $89$ |
\( T^{5} + \cdots - 29607769114848 \)
|
| $97$ |
\( T^{5} + \cdots + 649651401987168 \)
|
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