Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1300,4,Mod(1,1300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1300, 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("1300.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1300.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: | \(76.7024830075\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2785296.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 30x^{2} + 2x + 65 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 260) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 30x^{2} + 2x + 65 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 49\nu + 5 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 25\nu - 17 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} + 17\nu - 43 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + \beta _1 + 32 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + 57\beta_{2} + 27\beta _1 + 182 ) / 4 \)
|
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 |
|
0 | −9.84775 | 0 | 0 | 0 | −1.86425 | 0 | 69.9781 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.14910 | 0 | 0 | 0 | −26.4482 | 0 | −22.3814 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.71495 | 0 | 0 | 0 | 31.3797 | 0 | −24.0590 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 6.28190 | 0 | 0 | 0 | 4.93274 | 0 | 12.4622 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(13\) | \(1\) |
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 | 1300.4.a.k | 4 | |
| 5.b | even | 2 | 1 | 260.4.a.c | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 1300.4.c.g | 8 | ||
| 15.d | odd | 2 | 1 | 2340.4.a.m | 4 | ||
| 20.d | odd | 2 | 1 | 1040.4.a.u | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.4.a.c | ✓ | 4 | 5.b | even | 2 | 1 | |
| 1040.4.a.u | 4 | 20.d | odd | 2 | 1 | ||
| 1300.4.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 1300.4.c.g | 8 | 5.c | odd | 4 | 2 | ||
| 2340.4.a.m | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4T_{3}^{3} - 64T_{3}^{2} - 40T_{3} + 228 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1300))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 228 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 8 T^{3} + \cdots + 7632 \)
|
| $11$ |
\( T^{4} - 28 T^{3} + \cdots + 2100852 \)
|
| $13$ |
\( (T + 13)^{4} \)
|
| $17$ |
\( T^{4} + 200 T^{3} + \cdots - 14951088 \)
|
| $19$ |
\( T^{4} - 132 T^{3} + \cdots - 8535020 \)
|
| $23$ |
\( T^{4} + 244 T^{3} + \cdots + 29171364 \)
|
| $29$ |
\( T^{4} + 8 T^{3} + \cdots + 293007168 \)
|
| $31$ |
\( T^{4} - 292 T^{3} + \cdots - 642276076 \)
|
| $37$ |
\( T^{4} + 168 T^{3} + \cdots + 805400832 \)
|
| $41$ |
\( T^{4} - 280 T^{3} + \cdots + 496536912 \)
|
| $43$ |
\( T^{4} + 452 T^{3} + \cdots - 45746748 \)
|
| $47$ |
\( T^{4} + \cdots + 1647798480 \)
|
| $53$ |
\( T^{4} + 584 T^{3} + \cdots - 168468912 \)
|
| $59$ |
\( T^{4} + 708 T^{3} + \cdots + 777261396 \)
|
| $61$ |
\( T^{4} + \cdots - 21509627072 \)
|
| $67$ |
\( T^{4} + \cdots + 33912154512 \)
|
| $71$ |
\( T^{4} + \cdots - 3387108492 \)
|
| $73$ |
\( T^{4} + \cdots + 67723906304 \)
|
| $79$ |
\( T^{4} + \cdots + 245418881600 \)
|
| $83$ |
\( T^{4} + 552 T^{3} + \cdots - 311448240 \)
|
| $89$ |
\( T^{4} + \cdots + 46519298064 \)
|
| $97$ |
\( T^{4} + \cdots - 226545046512 \)
|
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