Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,9,Mod(7,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.7");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.f (of order \(4\), degree \(2\), 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: | \(30.5533957546\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.1485512441856.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 119x^{4} + 625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 119x^{4} + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 79\nu ) / 65 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 144\nu^{2} ) / 325 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{7} + 827\nu^{3} ) / 1625 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13\nu^{6} + 50\nu^{4} - 1222\nu^{2} + 2975 ) / 325 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -13\nu^{6} - 50\nu^{4} - 1222\nu^{2} - 2975 ) / 325 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 54\nu^{7} + 75\nu^{5} + 6801\nu^{3} + 15675\nu ) / 1625 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 54\nu^{7} - 75\nu^{5} + 6801\nu^{3} - 15675\nu ) / 1625 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - 6\beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 26\beta_{2} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{6} - 27\beta_{3} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -13\beta_{5} + 13\beta_{4} - 238 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 79\beta_{7} - 79\beta_{6} + 1254\beta_1 ) / 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( -36\beta_{5} - 36\beta_{4} - 611\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -827\beta_{7} - 827\beta_{6} + 13602\beta_{3} ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−10.7561 | − | 10.7561i | −33.0681 | + | 33.0681i | − | 24.6120i | 0 | 711.369 | 986.622 | + | 986.622i | −3018.29 | + | 3018.29i | − | 2187.00i | 0 | ||||||||||||||||||||||||||||||||
| 7.2 | −5.85713 | − | 5.85713i | 33.0681 | − | 33.0681i | − | 187.388i | 0 | −387.369 | 1106.65 | + | 1106.65i | −2596.98 | + | 2596.98i | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||
| 7.3 | 5.85713 | + | 5.85713i | −33.0681 | + | 33.0681i | − | 187.388i | 0 | −387.369 | −1106.65 | − | 1106.65i | 2596.98 | − | 2596.98i | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||
| 7.4 | 10.7561 | + | 10.7561i | 33.0681 | − | 33.0681i | − | 24.6120i | 0 | 711.369 | −986.622 | − | 986.622i | 3018.29 | − | 3018.29i | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||
| 43.1 | −10.7561 | + | 10.7561i | −33.0681 | − | 33.0681i | 24.6120i | 0 | 711.369 | 986.622 | − | 986.622i | −3018.29 | − | 3018.29i | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||
| 43.2 | −5.85713 | + | 5.85713i | 33.0681 | + | 33.0681i | 187.388i | 0 | −387.369 | 1106.65 | − | 1106.65i | −2596.98 | − | 2596.98i | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||
| 43.3 | 5.85713 | − | 5.85713i | −33.0681 | − | 33.0681i | 187.388i | 0 | −387.369 | −1106.65 | + | 1106.65i | 2596.98 | + | 2596.98i | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||
| 43.4 | 10.7561 | − | 10.7561i | 33.0681 | + | 33.0681i | 24.6120i | 0 | 711.369 | −986.622 | + | 986.622i | 3018.29 | + | 3018.29i | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.9.f.b | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 75.9.f.b | ✓ | 8 |
| 5.c | odd | 4 | 2 | inner | 75.9.f.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.9.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 75.9.f.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 75.9.f.b | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 58248T_{2}^{4} + 252047376 \)
acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 58248 T^{4} + 252047376 \)
|
| $3$ |
\( (T^{4} + 4782969)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 22\!\cdots\!25 \)
|
| $11$ |
\( (T^{2} - 13188 T - 144589428)^{4} \)
|
| $13$ |
\( T^{8} + \cdots + 72\!\cdots\!81 \)
|
| $17$ |
\( T^{8} + \cdots + 71\!\cdots\!56 \)
|
| $19$ |
\( (T^{4} + \cdots + 16\!\cdots\!25)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 30\!\cdots\!00 \)
|
| $29$ |
\( (T^{4} + \cdots + 35\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{2} + 1710538 T + 669683339425)^{4} \)
|
| $37$ |
\( T^{8} + \cdots + 65\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots + 12090158821080)^{4} \)
|
| $43$ |
\( T^{8} + \cdots + 27\!\cdots\!01 \)
|
| $47$ |
\( T^{8} + \cdots + 33\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 36\!\cdots\!00 \)
|
| $59$ |
\( (T^{4} + \cdots + 38\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 221887828921799)^{4} \)
|
| $67$ |
\( T^{8} + \cdots + 40\!\cdots\!41 \)
|
| $71$ |
\( (T^{2} + \cdots - 13\!\cdots\!56)^{4} \)
|
| $73$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $79$ |
\( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 55\!\cdots\!76 \)
|
| $89$ |
\( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 31\!\cdots\!61 \)
|
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