Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,2,Mod(75,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.75");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 76.d (of order \(2\), degree \(1\), 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: | \(0.606863055362\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.14453810176.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 6x^{4} + 12x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} + 3x^{6} + 6x^{4} + 12x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + \nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} - 2\nu^{3} - 4\nu ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + \nu^{4} - 2\nu^{2} - 4 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 3\nu^{4} + 6\nu^{2} + 8 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{5} + 6\nu^{3} + 8\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 2\beta_{4} - \beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} - 3\beta_{5} - 3\beta_{2} - 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} - 6\beta_{4} - 3\beta_{3} - 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(39\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 75.1 |
|
−1.06789 | − | 0.927153i | 1.19935 | 0.280776 | + | 1.98019i | 1.56155 | −1.28078 | − | 1.11198i | − | 0.868210i | 1.53610 | − | 2.37495i | −1.56155 | −1.66757 | − | 1.44780i | |||||||||||||||||||||||||||||||
| 75.2 | −1.06789 | + | 0.927153i | 1.19935 | 0.280776 | − | 1.98019i | 1.56155 | −1.28078 | + | 1.11198i | 0.868210i | 1.53610 | + | 2.37495i | −1.56155 | −1.66757 | + | 1.44780i | |||||||||||||||||||||||||||||||||
| 75.3 | −0.331077 | − | 1.37491i | −2.35829 | −1.78078 | + | 0.910404i | −2.56155 | 0.780776 | + | 3.24245i | − | 4.15286i | 1.84130 | + | 2.14700i | 2.56155 | 0.848071 | + | 3.52191i | ||||||||||||||||||||||||||||||||
| 75.4 | −0.331077 | + | 1.37491i | −2.35829 | −1.78078 | − | 0.910404i | −2.56155 | 0.780776 | − | 3.24245i | 4.15286i | 1.84130 | − | 2.14700i | 2.56155 | 0.848071 | − | 3.52191i | |||||||||||||||||||||||||||||||||
| 75.5 | 0.331077 | − | 1.37491i | 2.35829 | −1.78078 | − | 0.910404i | −2.56155 | 0.780776 | − | 3.24245i | 4.15286i | −1.84130 | + | 2.14700i | 2.56155 | −0.848071 | + | 3.52191i | |||||||||||||||||||||||||||||||||
| 75.6 | 0.331077 | + | 1.37491i | 2.35829 | −1.78078 | + | 0.910404i | −2.56155 | 0.780776 | + | 3.24245i | − | 4.15286i | −1.84130 | − | 2.14700i | 2.56155 | −0.848071 | − | 3.52191i | ||||||||||||||||||||||||||||||||
| 75.7 | 1.06789 | − | 0.927153i | −1.19935 | 0.280776 | − | 1.98019i | 1.56155 | −1.28078 | + | 1.11198i | 0.868210i | −1.53610 | − | 2.37495i | −1.56155 | 1.66757 | − | 1.44780i | |||||||||||||||||||||||||||||||||
| 75.8 | 1.06789 | + | 0.927153i | −1.19935 | 0.280776 | + | 1.98019i | 1.56155 | −1.28078 | − | 1.11198i | − | 0.868210i | −1.53610 | + | 2.37495i | −1.56155 | 1.66757 | + | 1.44780i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 19.b | odd | 2 | 1 | inner |
| 76.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 76.2.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 684.2.f.b | 8 | ||
| 4.b | odd | 2 | 1 | inner | 76.2.d.a | ✓ | 8 |
| 8.b | even | 2 | 1 | 1216.2.h.d | 8 | ||
| 8.d | odd | 2 | 1 | 1216.2.h.d | 8 | ||
| 12.b | even | 2 | 1 | 684.2.f.b | 8 | ||
| 19.b | odd | 2 | 1 | inner | 76.2.d.a | ✓ | 8 |
| 57.d | even | 2 | 1 | 684.2.f.b | 8 | ||
| 76.d | even | 2 | 1 | inner | 76.2.d.a | ✓ | 8 |
| 152.b | even | 2 | 1 | 1216.2.h.d | 8 | ||
| 152.g | odd | 2 | 1 | 1216.2.h.d | 8 | ||
| 228.b | odd | 2 | 1 | 684.2.f.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.2.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 76.2.d.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 76.2.d.a | ✓ | 8 | 19.b | odd | 2 | 1 | inner |
| 76.2.d.a | ✓ | 8 | 76.d | even | 2 | 1 | inner |
| 684.2.f.b | 8 | 3.b | odd | 2 | 1 | ||
| 684.2.f.b | 8 | 12.b | even | 2 | 1 | ||
| 684.2.f.b | 8 | 57.d | even | 2 | 1 | ||
| 684.2.f.b | 8 | 228.b | odd | 2 | 1 | ||
| 1216.2.h.d | 8 | 8.b | even | 2 | 1 | ||
| 1216.2.h.d | 8 | 8.d | odd | 2 | 1 | ||
| 1216.2.h.d | 8 | 152.b | even | 2 | 1 | ||
| 1216.2.h.d | 8 | 152.g | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(76, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 3 T^{6} + \cdots + 16 \)
|
| $3$ |
\( (T^{4} - 7 T^{2} + 8)^{2} \)
|
| $5$ |
\( (T^{2} + T - 4)^{4} \)
|
| $7$ |
\( (T^{4} + 18 T^{2} + 13)^{2} \)
|
| $11$ |
\( (T^{4} + 15 T^{2} + 52)^{2} \)
|
| $13$ |
\( (T^{4} + 41 T^{2} + 416)^{2} \)
|
| $17$ |
\( (T + 1)^{8} \)
|
| $19$ |
\( T^{8} - 16 T^{6} + \cdots + 130321 \)
|
| $23$ |
\( (T^{4} + 19 T^{2} + 52)^{2} \)
|
| $29$ |
\( (T^{4} + 73 T^{2} + 104)^{2} \)
|
| $31$ |
\( (T^{4} - 20 T^{2} + 32)^{2} \)
|
| $37$ |
\( (T^{4} + 44 T^{2} + 416)^{2} \)
|
| $41$ |
\( (T^{4} + 44 T^{2} + 416)^{2} \)
|
| $43$ |
\( (T^{4} + 123 T^{2} + 208)^{2} \)
|
| $47$ |
\( (T^{4} + 135 T^{2} + 4212)^{2} \)
|
| $53$ |
\( (T^{4} + 97 T^{2} + 104)^{2} \)
|
| $59$ |
\( (T^{4} - 175 T^{2} + 5408)^{2} \)
|
| $61$ |
\( (T^{2} - 11 T - 8)^{4} \)
|
| $67$ |
\( (T^{4} - 95 T^{2} + 8)^{2} \)
|
| $71$ |
\( (T^{4} - 124 T^{2} + 512)^{2} \)
|
| $73$ |
\( (T^{2} - 8 T - 1)^{4} \)
|
| $79$ |
\( (T^{4} - 244 T^{2} + 11552)^{2} \)
|
| $83$ |
\( (T^{4} + 236 T^{2} + 13312)^{2} \)
|
| $89$ |
\( (T^{4} + 232 T^{2} + 6656)^{2} \)
|
| $97$ |
\( (T^{4} + 292 T^{2} + 1664)^{2} \)
|
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