Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,4,Mod(101,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.101");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.d (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(25.0758117524\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-19}, \sqrt{-1131})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 575x^{2} + 77284 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 575x^{2} + 77284 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 297\nu ) / 278 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 278\nu^{2} + 575\nu + 79786 ) / 556 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} - \beta _1 - 287 \)
|
| \(\nu^{3}\) | \(=\) |
\( -278\beta_{2} - 297\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
−1.00000 | − | 4.35890i | −7.00000 | 0 | 4.35890i | − | 32.0713i | 15.0000 | 8.00000 | 0 | ||||||||||||||||||||||||||||
| 101.2 | −1.00000 | − | 4.35890i | −7.00000 | 0 | 4.35890i | 1.55903i | 15.0000 | 8.00000 | 0 | ||||||||||||||||||||||||||||||
| 101.3 | −1.00000 | 4.35890i | −7.00000 | 0 | − | 4.35890i | − | 1.55903i | 15.0000 | 8.00000 | 0 | |||||||||||||||||||||||||||||
| 101.4 | −1.00000 | 4.35890i | −7.00000 | 0 | − | 4.35890i | 32.0713i | 15.0000 | 8.00000 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 425.4.d.b | ✓ | 4 |
| 5.b | even | 2 | 1 | 425.4.d.d | yes | 4 | |
| 5.c | odd | 4 | 2 | 425.4.c.d | 8 | ||
| 17.b | even | 2 | 1 | inner | 425.4.d.b | ✓ | 4 |
| 85.c | even | 2 | 1 | 425.4.d.d | yes | 4 | |
| 85.g | odd | 4 | 2 | 425.4.c.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 425.4.c.d | 8 | 5.c | odd | 4 | 2 | ||
| 425.4.c.d | 8 | 85.g | odd | 4 | 2 | ||
| 425.4.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 425.4.d.b | ✓ | 4 | 17.b | even | 2 | 1 | inner |
| 425.4.d.d | yes | 4 | 5.b | even | 2 | 1 | |
| 425.4.d.d | yes | 4 | 85.c | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):
|
\( T_{2} + 1 \)
|
|
\( T_{3}^{2} + 19 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( (T^{2} + 19)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 1031 T^{2} + 2500 \)
|
| $11$ |
\( T^{4} + 1031 T^{2} + 2500 \)
|
| $13$ |
\( (T^{2} + 7 T - 5360)^{2} \)
|
| $17$ |
\( T^{4} - 35 T^{3} + \cdots + 24137569 \)
|
| $19$ |
\( (T^{2} - 7 T - 5360)^{2} \)
|
| $23$ |
\( T^{4} + 43640 T^{2} + 206554384 \)
|
| $29$ |
\( (T^{2} + 18096)^{2} \)
|
| $31$ |
\( T^{4} + 86999 T^{2} + 249324100 \)
|
| $37$ |
\( T^{4} + \cdots + 1255993600 \)
|
| $41$ |
\( T^{4} + 106379 T^{2} + 649230400 \)
|
| $43$ |
\( (T - 148)^{4} \)
|
| $47$ |
\( (T - 224)^{4} \)
|
| $53$ |
\( (T^{2} - 315 T - 238434)^{2} \)
|
| $59$ |
\( (T^{2} - 504 T - 280320)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 247844665600 \)
|
| $67$ |
\( (T^{2} + 1041 T + 7680)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 366146010000 \)
|
| $73$ |
\( T^{4} + \cdots + 4517452944 \)
|
| $79$ |
\( T^{4} + 151895 T^{2} + 385022884 \)
|
| $83$ |
\( (T^{2} + 1071 T + 281388)^{2} \)
|
| $89$ |
\( (T^{2} - 931 T - 46550)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 33609155584 \)
|
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