Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3850,2,Mod(1,3850)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3850, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("3850.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3850.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: | \(30.7424047782\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.117688.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 770) |
| 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
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 3\nu^{2} - \nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + \nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 9\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 5\nu^{2} - 7\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 8 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{4} + 3\beta_{3} + 5\beta_{2} - 3\beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{4} + 2\beta_{3} - \beta_{2} + 3\beta _1 + 14 ) / 2 \)
|
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 |
|
1.00000 | −3.27510 | 1.00000 | 0 | −3.27510 | −1.00000 | 1.00000 | 7.72630 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.63209 | 1.00000 | 0 | −1.63209 | −1.00000 | 1.00000 | −0.336288 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.426870 | 1.00000 | 0 | −0.426870 | −1.00000 | 1.00000 | −2.81778 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 2.34949 | 1.00000 | 0 | 2.34949 | −1.00000 | 1.00000 | 2.52013 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.98457 | 1.00000 | 0 | 2.98457 | −1.00000 | 1.00000 | 5.90764 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 3850.2.a.ca | 5 | |
| 5.b | even | 2 | 1 | 3850.2.a.bz | 5 | ||
| 5.c | odd | 4 | 2 | 770.2.c.f | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.c.f | ✓ | 10 | 5.c | odd | 4 | 2 | |
| 3850.2.a.bz | 5 | 5.b | even | 2 | 1 | ||
| 3850.2.a.ca | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3850))\):
|
\( T_{3}^{5} - 14T_{3}^{3} + 40T_{3} + 16 \)
|
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\( T_{13}^{5} - 10T_{13}^{4} + 14T_{13}^{3} + 68T_{13}^{2} - 72T_{13} + 16 \)
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\( T_{17}^{5} - 2T_{17}^{4} - 60T_{17}^{3} + 216T_{17}^{2} + 224T_{17} - 1088 \)
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\( T_{19}^{5} - 6T_{19}^{4} - 46T_{19}^{3} + 268T_{19}^{2} + 480T_{19} - 2768 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
|
| $3$ |
\( T^{5} - 14 T^{3} + \cdots + 16 \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( (T + 1)^{5} \)
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| $11$ |
\( (T - 1)^{5} \)
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| $13$ |
\( T^{5} - 10 T^{4} + \cdots + 16 \)
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| $17$ |
\( T^{5} - 2 T^{4} + \cdots - 1088 \)
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| $19$ |
\( T^{5} - 6 T^{4} + \cdots - 2768 \)
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| $23$ |
\( T^{5} + 8 T^{4} + \cdots + 544 \)
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| $29$ |
\( T^{5} - 8 T^{4} + \cdots - 32 \)
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| $31$ |
\( T^{5} - 12 T^{4} + \cdots - 256 \)
|
| $37$ |
\( T^{5} - 6 T^{4} + \cdots - 544 \)
|
| $41$ |
\( T^{5} - 20 T^{4} + \cdots + 512 \)
|
| $43$ |
\( T^{5} + 12 T^{4} + \cdots + 512 \)
|
| $47$ |
\( T^{5} - 10 T^{4} + \cdots - 2336 \)
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| $53$ |
\( T^{5} + 14 T^{4} + \cdots + 4832 \)
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| $59$ |
\( T^{5} - 36 T^{4} + \cdots + 26752 \)
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| $61$ |
\( T^{5} - 10 T^{4} + \cdots + 3664 \)
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| $67$ |
\( T^{5} + 2 T^{4} + \cdots + 2048 \)
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| $71$ |
\( T^{5} - 16 T^{4} + \cdots + 6400 \)
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| $73$ |
\( T^{5} - 22 T^{4} + \cdots + 2432 \)
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| $79$ |
\( T^{5} + 10 T^{4} + \cdots + 46112 \)
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| $83$ |
\( T^{5} + 12 T^{4} + \cdots + 2176 \)
|
| $89$ |
\( T^{5} - 14 T^{4} + \cdots + 14752 \)
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| $97$ |
\( T^{5} - 2 T^{4} + \cdots + 389408 \)
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