Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3750,2,Mod(1249,3750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3750, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3750.1249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3750 = 2 \cdot 3 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3750.c (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: | \(29.9439007580\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{20})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 150) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{20}^{5} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{20}^{6} + \zeta_{20}^{4} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{20}^{7} + \zeta_{20}^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( -\zeta_{20}^{7} + \zeta_{20}^{3} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{20}^{6} + \zeta_{20}^{4} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + 2\zeta_{20} \)
|
| \(\beta_{7}\) | \(=\) |
\( \zeta_{20}^{6} - \zeta_{20}^{4} + 2\zeta_{20}^{2} - 1 \)
|
| \(\zeta_{20}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} - \beta_1 ) / 2 \)
|
| \(\zeta_{20}^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} + 1 ) / 2 \)
|
| \(\zeta_{20}^{3}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\zeta_{20}^{4}\) | \(=\) |
\( ( \beta_{5} + \beta_{2} ) / 2 \)
|
| \(\zeta_{20}^{5}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{20}^{6}\) | \(=\) |
\( ( -\beta_{5} + \beta_{2} ) / 2 \)
|
| \(\zeta_{20}^{7}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times\).
| \(n\) | \(2501\) | \(3127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1249.1 |
|
− | 1.00000i | − | 1.00000i | −1.00000 | 0 | −1.00000 | − | 2.07768i | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1249.2 | − | 1.00000i | − | 1.00000i | −1.00000 | 0 | −1.00000 | 0.273457i | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.3 | − | 1.00000i | − | 1.00000i | −1.00000 | 0 | −1.00000 | 1.72654i | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.4 | − | 1.00000i | − | 1.00000i | −1.00000 | 0 | −1.00000 | 4.07768i | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.5 | 1.00000i | 1.00000i | −1.00000 | 0 | −1.00000 | − | 4.07768i | − | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.6 | 1.00000i | 1.00000i | −1.00000 | 0 | −1.00000 | − | 1.72654i | − | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.7 | 1.00000i | 1.00000i | −1.00000 | 0 | −1.00000 | − | 0.273457i | − | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.8 | 1.00000i | 1.00000i | −1.00000 | 0 | −1.00000 | 2.07768i | − | 1.00000i | −1.00000 | 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 | 3750.2.c.e | 8 | |
| 5.b | even | 2 | 1 | inner | 3750.2.c.e | 8 | |
| 5.c | odd | 4 | 1 | 3750.2.a.m | 4 | ||
| 5.c | odd | 4 | 1 | 3750.2.a.o | 4 | ||
| 25.d | even | 5 | 1 | 150.2.h.a | ✓ | 8 | |
| 25.d | even | 5 | 1 | 750.2.h.c | 8 | ||
| 25.e | even | 10 | 1 | 150.2.h.a | ✓ | 8 | |
| 25.e | even | 10 | 1 | 750.2.h.c | 8 | ||
| 25.f | odd | 20 | 2 | 750.2.g.c | 8 | ||
| 25.f | odd | 20 | 2 | 750.2.g.e | 8 | ||
| 75.h | odd | 10 | 1 | 450.2.l.a | 8 | ||
| 75.j | odd | 10 | 1 | 450.2.l.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 150.2.h.a | ✓ | 8 | 25.d | even | 5 | 1 | |
| 150.2.h.a | ✓ | 8 | 25.e | even | 10 | 1 | |
| 450.2.l.a | 8 | 75.h | odd | 10 | 1 | ||
| 450.2.l.a | 8 | 75.j | odd | 10 | 1 | ||
| 750.2.g.c | 8 | 25.f | odd | 20 | 2 | ||
| 750.2.g.e | 8 | 25.f | odd | 20 | 2 | ||
| 750.2.h.c | 8 | 25.d | even | 5 | 1 | ||
| 750.2.h.c | 8 | 25.e | even | 10 | 1 | ||
| 3750.2.a.m | 4 | 5.c | odd | 4 | 1 | ||
| 3750.2.a.o | 4 | 5.c | odd | 4 | 1 | ||
| 3750.2.c.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 3750.2.c.e | 8 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 24T_{7}^{6} + 136T_{7}^{4} + 224T_{7}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(3750, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 24 T^{6} + \cdots + 16 \)
|
| $11$ |
\( (T^{4} + 10 T^{3} + \cdots - 20)^{2} \)
|
| $13$ |
\( T^{8} + 46 T^{6} + \cdots + 3721 \)
|
| $17$ |
\( T^{8} + 34 T^{6} + \cdots + 3481 \)
|
| $19$ |
\( (T^{4} - 6 T^{3} - 14 T^{2} + \cdots + 76)^{2} \)
|
| $23$ |
\( T^{8} + 120 T^{6} + \cdots + 10000 \)
|
| $29$ |
\( (T^{4} - 14 T^{3} + \cdots - 3599)^{2} \)
|
| $31$ |
\( (T^{4} + 18 T^{3} + \cdots - 1444)^{2} \)
|
| $37$ |
\( T^{8} + 126 T^{6} + \cdots + 143641 \)
|
| $41$ |
\( (T^{4} + 14 T^{3} + \cdots - 439)^{2} \)
|
| $43$ |
\( T^{8} + 224 T^{6} + \cdots + 2835856 \)
|
| $47$ |
\( T^{8} + 160 T^{6} + \cdots + 144400 \)
|
| $53$ |
\( T^{8} + 234 T^{6} + \cdots + 1739761 \)
|
| $59$ |
\( (T^{4} - 20 T^{3} + \cdots - 320)^{2} \)
|
| $61$ |
\( (T^{4} - 185 T^{2} + 4205)^{2} \)
|
| $67$ |
\( T^{8} + 344 T^{6} + \cdots + 29637136 \)
|
| $71$ |
\( (T^{4} + 30 T^{3} + \cdots + 1180)^{2} \)
|
| $73$ |
\( T^{8} + 146 T^{6} + \cdots + 175561 \)
|
| $79$ |
\( (T^{4} - 28 T^{3} + \cdots - 2624)^{2} \)
|
| $83$ |
\( T^{8} + 536 T^{6} + \cdots + 167857936 \)
|
| $89$ |
\( (T^{4} - 28 T^{3} + \cdots - 5699)^{2} \)
|
| $97$ |
\( T^{8} + 466 T^{6} + \cdots + 85396081 \)
|
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