Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [750,2,Mod(151,750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(750, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("750.151");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 750 = 2 \cdot 3 \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 750.g (of order \(5\), degree \(4\), 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: | \(5.98878015160\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | no (minimal twist has level 150) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{20}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{20}^{3} + \zeta_{20} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{20}^{4} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{20}^{6} \)
|
| \(\beta_{5}\) | \(=\) |
\( \zeta_{20}^{7} + \zeta_{20} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{20}^{5} + \zeta_{20} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + 2\zeta_{20} \)
|
| \(\zeta_{20}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} ) / 5 \)
|
| \(\zeta_{20}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{20}^{3}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - \beta_{5} + 4\beta_{2} ) / 5 \)
|
| \(\zeta_{20}^{4}\) | \(=\) |
\( \beta_{3} \)
|
| \(\zeta_{20}^{5}\) | \(=\) |
\( ( \beta_{7} - 4\beta_{6} + \beta_{5} + \beta_{2} ) / 5 \)
|
| \(\zeta_{20}^{6}\) | \(=\) |
\( \beta_{4} \)
|
| \(\zeta_{20}^{7}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + 4\beta_{5} - \beta_{2} ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(251\) |
| \(\chi(n)\) | \(-1 + \beta_{1} - \beta_{3} + \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 151.1 |
|
−0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | + | 0.587785i | 0 | 0.809017 | + | 0.587785i | −2.07768 | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0 | ||||||||||||||||||||||||||||||
| 151.2 | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | + | 0.587785i | 0 | 0.809017 | + | 0.587785i | 4.07768 | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||
| 301.1 | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | − | 0.951057i | 0 | −0.309017 | − | 0.951057i | 0.273457 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||
| 301.2 | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | − | 0.951057i | 0 | −0.309017 | − | 0.951057i | 1.72654 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||
| 451.1 | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | + | 0.951057i | 0 | −0.309017 | + | 0.951057i | 0.273457 | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||
| 451.2 | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | + | 0.951057i | 0 | −0.309017 | + | 0.951057i | 1.72654 | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||
| 601.1 | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | − | 0.587785i | 0 | 0.809017 | − | 0.587785i | −2.07768 | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||
| 601.2 | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | − | 0.587785i | 0 | 0.809017 | − | 0.587785i | 4.07768 | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 750.2.g.e | 8 | |
| 5.b | even | 2 | 1 | 750.2.g.c | 8 | ||
| 5.c | odd | 4 | 1 | 150.2.h.a | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 750.2.h.c | 8 | ||
| 15.e | even | 4 | 1 | 450.2.l.a | 8 | ||
| 25.d | even | 5 | 1 | inner | 750.2.g.e | 8 | |
| 25.d | even | 5 | 1 | 3750.2.a.m | 4 | ||
| 25.e | even | 10 | 1 | 750.2.g.c | 8 | ||
| 25.e | even | 10 | 1 | 3750.2.a.o | 4 | ||
| 25.f | odd | 20 | 1 | 150.2.h.a | ✓ | 8 | |
| 25.f | odd | 20 | 1 | 750.2.h.c | 8 | ||
| 25.f | odd | 20 | 2 | 3750.2.c.e | 8 | ||
| 75.l | even | 20 | 1 | 450.2.l.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 150.2.h.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 150.2.h.a | ✓ | 8 | 25.f | odd | 20 | 1 | |
| 450.2.l.a | 8 | 15.e | even | 4 | 1 | ||
| 450.2.l.a | 8 | 75.l | even | 20 | 1 | ||
| 750.2.g.c | 8 | 5.b | even | 2 | 1 | ||
| 750.2.g.c | 8 | 25.e | even | 10 | 1 | ||
| 750.2.g.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 750.2.g.e | 8 | 25.d | even | 5 | 1 | inner | |
| 750.2.h.c | 8 | 5.c | odd | 4 | 1 | ||
| 750.2.h.c | 8 | 25.f | odd | 20 | 1 | ||
| 3750.2.a.m | 4 | 25.d | even | 5 | 1 | ||
| 3750.2.a.o | 4 | 25.e | even | 10 | 1 | ||
| 3750.2.c.e | 8 | 25.f | odd | 20 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 4T_{7}^{3} - 4T_{7}^{2} + 16T_{7} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} - 4 T^{3} - 4 T^{2} + \cdots - 4)^{2} \)
|
| $11$ |
\( T^{8} - 10 T^{7} + \cdots + 400 \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots + 3721 \)
|
| $17$ |
\( T^{8} + 2 T^{7} + \cdots + 3481 \)
|
| $19$ |
\( T^{8} - 8 T^{7} + \cdots + 5776 \)
|
| $23$ |
\( T^{8} - 10 T^{7} + \cdots + 10000 \)
|
| $29$ |
\( T^{8} - 22 T^{7} + \cdots + 12952801 \)
|
| $31$ |
\( T^{8} - 24 T^{7} + \cdots + 2085136 \)
|
| $37$ |
\( T^{8} + 14 T^{7} + \cdots + 143641 \)
|
| $41$ |
\( T^{8} - 22 T^{7} + \cdots + 192721 \)
|
| $43$ |
\( (T^{4} - 14 T^{3} + \cdots - 1684)^{2} \)
|
| $47$ |
\( T^{8} - 30 T^{7} + \cdots + 144400 \)
|
| $53$ |
\( T^{8} - 12 T^{7} + \cdots + 1739761 \)
|
| $59$ |
\( T^{8} - 20 T^{7} + \cdots + 102400 \)
|
| $61$ |
\( T^{8} + 240 T^{6} + \cdots + 17682025 \)
|
| $67$ |
\( T^{8} - 18 T^{7} + \cdots + 29637136 \)
|
| $71$ |
\( T^{8} - 20 T^{7} + \cdots + 1392400 \)
|
| $73$ |
\( T^{8} - 16 T^{7} + \cdots + 175561 \)
|
| $79$ |
\( T^{8} + 16 T^{7} + \cdots + 6885376 \)
|
| $83$ |
\( T^{8} - 6 T^{7} + \cdots + 167857936 \)
|
| $89$ |
\( T^{8} - 34 T^{7} + \cdots + 32478601 \)
|
| $97$ |
\( T^{8} + 4 T^{7} + \cdots + 85396081 \)
|
| show more | |
| show less |