Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [750,6,Mod(499,750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(750, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("750.499");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 750 = 2 \cdot 3 \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 750.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: | \(120.287864860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.289444000000.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 39x^{6} + 541x^{4} + 3084x^{2} + 5776 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 39x^{6} + 541x^{4} + 3084x^{2} + 5776 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -5\nu^{7} - 119\nu^{5} - 805\nu^{3} - 1360\nu ) / 456 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 35\nu^{4} + 317\nu^{2} + 568 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} + 138\nu^{5} + 1090\nu^{3} + 2025\nu ) / 114 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13\nu^{7} - 355\nu^{5} - 3005\nu^{3} - 8096\nu ) / 114 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{6} - 135\nu^{4} - 1105\nu^{2} - 2556 ) / 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{6} - 50\nu^{4} - 364\nu^{2} - 700 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -129\nu^{7} - 3511\nu^{5} - 29205\nu^{3} - 69668\nu ) / 456 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - 2\beta_{4} + \beta_{3} - \beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - 2\beta_{5} - 2\beta_{2} - 98 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13\beta_{7} + 21\beta_{4} - 25\beta_{3} + 17\beta_1 ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -12\beta_{6} + 27\beta_{5} + 39\beta_{2} + 1105 ) / 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 32\beta_{7} - 49\beta_{4} + 80\beta_{3} + 4\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 103\beta_{6} - 311\beta_{5} - 611\beta_{2} - 13289 ) / 10 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -1987\beta_{7} + 2994\beta_{4} - 5767\beta_{3} - 3853\beta_1 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(251\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 499.1 |
|
− | 4.00000i | 9.00000i | −16.0000 | 0 | 36.0000 | − | 124.927i | 64.0000i | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 499.2 | − | 4.00000i | 9.00000i | −16.0000 | 0 | 36.0000 | − | 89.0490i | 64.0000i | −81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 499.3 | − | 4.00000i | 9.00000i | −16.0000 | 0 | 36.0000 | 89.9244i | 64.0000i | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 499.4 | − | 4.00000i | 9.00000i | −16.0000 | 0 | 36.0000 | 166.052i | 64.0000i | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 499.5 | 4.00000i | − | 9.00000i | −16.0000 | 0 | 36.0000 | − | 166.052i | − | 64.0000i | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 499.6 | 4.00000i | − | 9.00000i | −16.0000 | 0 | 36.0000 | − | 89.9244i | − | 64.0000i | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 499.7 | 4.00000i | − | 9.00000i | −16.0000 | 0 | 36.0000 | 89.0490i | − | 64.0000i | −81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 499.8 | 4.00000i | − | 9.00000i | −16.0000 | 0 | 36.0000 | 124.927i | − | 64.0000i | −81.0000 | 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 | 750.6.c.c | 8 | |
| 5.b | even | 2 | 1 | inner | 750.6.c.c | 8 | |
| 5.c | odd | 4 | 1 | 750.6.a.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 750.6.a.h | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 750.6.a.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 750.6.a.h | yes | 4 | 5.c | odd | 4 | 1 | |
| 750.6.c.c | 8 | 1.a | even | 1 | 1 | trivial | |
| 750.6.c.c | 8 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 59196T_{7}^{6} + 1186024016T_{7}^{4} + 9660987768576T_{7}^{2} + 27593813155188736 \)
acting on \(S_{6}^{\mathrm{new}}(750, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{4} \)
|
| $3$ |
\( (T^{2} + 81)^{4} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 27\!\cdots\!36 \)
|
| $11$ |
\( (T^{4} + 402 T^{3} + \cdots - 3668594624)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 12\!\cdots\!16 \)
|
| $17$ |
\( T^{8} + \cdots + 24\!\cdots\!81 \)
|
| $19$ |
\( (T^{4} + \cdots + 10103065458775)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 73\!\cdots\!61 \)
|
| $29$ |
\( (T^{4} + \cdots - 142429670315600)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 62975346422249)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 15\!\cdots\!56 \)
|
| $41$ |
\( (T^{4} + \cdots - 18\!\cdots\!44)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 21\!\cdots\!76 \)
|
| $47$ |
\( T^{8} + \cdots + 14\!\cdots\!61 \)
|
| $53$ |
\( T^{8} + \cdots + 11\!\cdots\!01 \)
|
| $59$ |
\( (T^{4} + \cdots - 74\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 10\!\cdots\!21)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 88\!\cdots\!56 \)
|
| $71$ |
\( (T^{4} + \cdots - 23\!\cdots\!84)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 48\!\cdots\!16 \)
|
| $79$ |
\( (T^{4} + \cdots - 18\!\cdots\!25)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 22\!\cdots\!61 \)
|
| $89$ |
\( (T^{4} + \cdots - 55\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 28\!\cdots\!76 \)
|
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