Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3800,2,Mod(1,3800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3800.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3800.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: | \(30.3431527681\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 9\nu^{2} + 6\nu + 5 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 15\nu^{2} + 6\nu - 8 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 15\nu^{2} + 15\nu - 7 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} + 2\beta_{4} + 9\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 12\beta _1 + 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( -22\beta_{5} + 24\beta_{4} + 4\beta_{3} + 3\beta_{2} + 84\beta _1 + 21 \)
|
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 | −2.77008 | 0 | 0 | 0 | −2.31077 | 0 | 4.67334 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.08999 | 0 | 0 | 0 | 4.19727 | 0 | −1.81192 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.185519 | 0 | 0 | 0 | −4.45651 | 0 | −2.96558 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.848258 | 0 | 0 | 0 | 1.74484 | 0 | −2.28046 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.93590 | 0 | 0 | 0 | −1.24708 | 0 | 0.747704 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.26143 | 0 | 0 | 0 | 4.07225 | 0 | 7.63693 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(1\) |
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 | 3800.2.a.bc | yes | 6 |
| 4.b | odd | 2 | 1 | 7600.2.a.ch | 6 | ||
| 5.b | even | 2 | 1 | 3800.2.a.ba | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 3800.2.d.q | 12 | ||
| 20.d | odd | 2 | 1 | 7600.2.a.cl | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3800.2.a.ba | ✓ | 6 | 5.b | even | 2 | 1 | |
| 3800.2.a.bc | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 3800.2.d.q | 12 | 5.c | odd | 4 | 2 | ||
| 7600.2.a.ch | 6 | 4.b | odd | 2 | 1 | ||
| 7600.2.a.cl | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 16T_{3}^{3} + 15T_{3}^{2} - 14T_{3} - 3 \)
|
|
\( T_{7}^{6} - 2T_{7}^{5} - 30T_{7}^{4} + 48T_{7}^{3} + 223T_{7}^{2} - 154T_{7} - 383 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots - 3 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 383 \)
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| $11$ |
\( T^{6} - 3 T^{5} + \cdots - 88 \)
|
| $13$ |
\( T^{6} + T^{5} + \cdots + 825 \)
|
| $17$ |
\( T^{6} + 14 T^{5} + \cdots + 3147 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} - 12 T^{5} + \cdots - 2487 \)
|
| $29$ |
\( T^{6} - 9 T^{5} + \cdots + 21951 \)
|
| $31$ |
\( T^{6} - 5 T^{5} + \cdots + 11000 \)
|
| $37$ |
\( T^{6} + 8 T^{5} + \cdots - 7365 \)
|
| $41$ |
\( T^{6} - 3 T^{5} + \cdots - 113472 \)
|
| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 6120 \)
|
| $47$ |
\( T^{6} - 4 T^{5} + \cdots + 1791 \)
|
| $53$ |
\( T^{6} - 13 T^{5} + \cdots + 9285 \)
|
| $59$ |
\( T^{6} - 193 T^{4} + \cdots - 32328 \)
|
| $61$ |
\( T^{6} - 9 T^{5} + \cdots + 12424 \)
|
| $67$ |
\( T^{6} + 3 T^{5} + \cdots - 136033 \)
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| $71$ |
\( T^{6} - 19 T^{5} + \cdots - 27576 \)
|
| $73$ |
\( T^{6} - 3 T^{5} + \cdots - 741033 \)
|
| $79$ |
\( T^{6} + 16 T^{5} + \cdots - 553536 \)
|
| $83$ |
\( T^{6} - 31 T^{5} + \cdots - 71160 \)
|
| $89$ |
\( T^{6} - 14 T^{5} + \cdots + 3240 \)
|
| $97$ |
\( T^{6} + 11 T^{5} + \cdots - 288792 \)
|
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