Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,4,Mod(324,425)]
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([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.324");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.b (of order \(2\), degree \(1\), not 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: | \(6\) |
| Coefficient field: | 6.0.5161984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 85) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{5} + 25\nu^{4} + 10\nu^{3} - 4\nu^{2} + 323 ) / 121 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} - 27\nu^{4} - 35\nu^{3} + 14\nu^{2} - 223 ) / 121 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 75\nu^{5} + 30\nu^{4} + 12\nu^{3} - 392\nu^{2} + 1815\nu - 774 ) / 242 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - 3\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{5} + 5\beta_{4} + 4\beta_{3} - 5\beta_{2} - 5\beta _1 + 4 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{2} + 7\beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -29\beta_{5} - 25\beta_{4} - 32\beta_{3} - 25\beta_{2} - 29\beta _1 + 32 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 324.1 |
|
− | 4.50466i | 5.08998i | −12.2920 | 0 | 22.9287 | 0.616696i | 19.3340i | 1.09206 | 0 | |||||||||||||||||||||||||||||||||||
| 324.2 | − | 1.64002i | 5.37466i | 5.31032 | 0 | 8.81456 | 2.20103i | − | 21.8292i | −1.88693 | 0 | |||||||||||||||||||||||||||||||||||
| 324.3 | − | 0.135359i | 9.28467i | 7.98168 | 0 | 1.25676 | − | 32.4157i | − | 2.16327i | −59.2051 | 0 | ||||||||||||||||||||||||||||||||||
| 324.4 | 0.135359i | − | 9.28467i | 7.98168 | 0 | 1.25676 | 32.4157i | 2.16327i | −59.2051 | 0 | ||||||||||||||||||||||||||||||||||||
| 324.5 | 1.64002i | − | 5.37466i | 5.31032 | 0 | 8.81456 | − | 2.20103i | 21.8292i | −1.88693 | 0 | |||||||||||||||||||||||||||||||||||
| 324.6 | 4.50466i | − | 5.08998i | −12.2920 | 0 | 22.9287 | − | 0.616696i | − | 19.3340i | 1.09206 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.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.b.h | 6 | |
| 5.b | even | 2 | 1 | inner | 425.4.b.h | 6 | |
| 5.c | odd | 4 | 1 | 85.4.a.f | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 425.4.a.f | 3 | ||
| 15.e | even | 4 | 1 | 765.4.a.k | 3 | ||
| 20.e | even | 4 | 1 | 1360.4.a.p | 3 | ||
| 85.g | odd | 4 | 1 | 1445.4.a.k | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.4.a.f | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 425.4.a.f | 3 | 5.c | odd | 4 | 1 | ||
| 425.4.b.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 425.4.b.h | 6 | 5.b | even | 2 | 1 | inner | |
| 765.4.a.k | 3 | 15.e | even | 4 | 1 | ||
| 1360.4.a.p | 3 | 20.e | even | 4 | 1 | ||
| 1445.4.a.k | 3 | 85.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):
|
\( T_{2}^{6} + 23T_{2}^{4} + 55T_{2}^{2} + 1 \)
|
|
\( T_{3}^{6} + 141T_{3}^{4} + 5472T_{3}^{2} + 64516 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 23 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} + 141 T^{4} + \cdots + 64516 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 1056 T^{4} + \cdots + 1936 \)
|
| $11$ |
\( (T^{3} - 52 T^{2} + \cdots + 6784)^{2} \)
|
| $13$ |
\( T^{6} + 6537 T^{4} + \cdots + 416486464 \)
|
| $17$ |
\( (T^{2} + 289)^{3} \)
|
| $19$ |
\( (T^{3} - 153 T^{2} + \cdots + 43112)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 344874307600 \)
|
| $29$ |
\( (T^{3} + 45 T^{2} + \cdots + 1563820)^{2} \)
|
| $31$ |
\( (T^{3} + 67 T^{2} + \cdots + 634)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 773217731584 \)
|
| $41$ |
\( (T^{3} - 498 T^{2} + \cdots + 948856)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 65024354357824 \)
|
| $47$ |
\( T^{6} + \cdots + 14\!\cdots\!76 \)
|
| $53$ |
\( T^{6} + \cdots + 533738813372944 \)
|
| $59$ |
\( (T^{3} + 355 T^{2} + \cdots - 23789032)^{2} \)
|
| $61$ |
\( (T^{3} - 1019 T^{2} + \cdots - 32261500)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 5029254760000 \)
|
| $71$ |
\( (T^{3} - 313 T^{2} + \cdots - 104660798)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 555271333126416 \)
|
| $79$ |
\( (T^{3} - 92 T^{2} + \cdots - 310900832)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 56\!\cdots\!04 \)
|
| $89$ |
\( (T^{3} + 1623 T^{2} + \cdots - 25616512)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 22\!\cdots\!24 \)
|
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