Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,3,Mod(7,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 11]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.l (of order \(12\), degree \(4\), 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: | \(2.12534606201\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{12}\) |
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 |
|
−1.36603 | + | 0.366025i | 0.866025 | − | 1.50000i | 1.73205 | − | 1.00000i | −4.73205 | − | 4.73205i | −0.633975 | + | 2.36603i | −2.50000 | − | 0.669873i | −2.00000 | + | 2.00000i | −1.50000 | − | 2.59808i | 8.19615 | + | 4.73205i | ||||||||||||
| 19.1 | 0.366025 | + | 1.36603i | −0.866025 | + | 1.50000i | −1.73205 | + | 1.00000i | −1.26795 | + | 1.26795i | −2.36603 | − | 0.633975i | −2.50000 | + | 9.33013i | −2.00000 | − | 2.00000i | −1.50000 | − | 2.59808i | −2.19615 | − | 1.26795i | |||||||||||||
| 37.1 | 0.366025 | − | 1.36603i | −0.866025 | − | 1.50000i | −1.73205 | − | 1.00000i | −1.26795 | − | 1.26795i | −2.36603 | + | 0.633975i | −2.50000 | − | 9.33013i | −2.00000 | + | 2.00000i | −1.50000 | + | 2.59808i | −2.19615 | + | 1.26795i | |||||||||||||
| 67.1 | −1.36603 | − | 0.366025i | 0.866025 | + | 1.50000i | 1.73205 | + | 1.00000i | −4.73205 | + | 4.73205i | −0.633975 | − | 2.36603i | −2.50000 | + | 0.669873i | −2.00000 | − | 2.00000i | −1.50000 | + | 2.59808i | 8.19615 | − | 4.73205i | |||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.3.l.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 234.3.bb.c | 4 | ||
| 13.c | even | 3 | 1 | 1014.3.f.e | 4 | ||
| 13.e | even | 6 | 1 | 1014.3.f.d | 4 | ||
| 13.f | odd | 12 | 1 | inner | 78.3.l.a | ✓ | 4 |
| 13.f | odd | 12 | 1 | 1014.3.f.d | 4 | ||
| 13.f | odd | 12 | 1 | 1014.3.f.e | 4 | ||
| 39.k | even | 12 | 1 | 234.3.bb.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.3.l.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 78.3.l.a | ✓ | 4 | 13.f | odd | 12 | 1 | inner |
| 234.3.bb.c | 4 | 3.b | odd | 2 | 1 | ||
| 234.3.bb.c | 4 | 39.k | even | 12 | 1 | ||
| 1014.3.f.d | 4 | 13.e | even | 6 | 1 | ||
| 1014.3.f.d | 4 | 13.f | odd | 12 | 1 | ||
| 1014.3.f.e | 4 | 13.c | even | 3 | 1 | ||
| 1014.3.f.e | 4 | 13.f | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 12T_{5}^{3} + 72T_{5}^{2} + 144T_{5} + 144 \)
acting on \(S_{3}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $5$ |
\( T^{4} + 12 T^{3} + \cdots + 144 \)
|
| $7$ |
\( T^{4} + 10 T^{3} + \cdots + 625 \)
|
| $11$ |
\( T^{4} - 12 T^{3} + \cdots + 24336 \)
|
| $13$ |
\( T^{4} - 169 T^{2} + 28561 \)
|
| $17$ |
\( T^{4} - 24 T^{3} + \cdots + 9216 \)
|
| $19$ |
\( T^{4} - 62 T^{3} + \cdots + 14884 \)
|
| $23$ |
\( T^{4} - 48 T^{3} + \cdots + 17424 \)
|
| $29$ |
\( T^{4} + 300 T^{2} + 90000 \)
|
| $31$ |
\( T^{4} - 106 T^{3} + \cdots + 1868689 \)
|
| $37$ |
\( T^{4} + 98 T^{3} + \cdots + 3587236 \)
|
| $41$ |
\( T^{4} + 96 T^{3} + \cdots + 1218816 \)
|
| $43$ |
\( T^{4} - 30 T^{3} + \cdots + 251001 \)
|
| $47$ |
\( T^{4} - 132 T^{3} + \cdots + 685584 \)
|
| $53$ |
\( (T^{2} - 36 T - 6024)^{2} \)
|
| $59$ |
\( T^{4} - 84 T^{3} + \cdots + 16353936 \)
|
| $61$ |
\( T^{4} + 72 T^{3} + \cdots + 335241 \)
|
| $67$ |
\( T^{4} + 148 T^{3} + \cdots + 6723649 \)
|
| $71$ |
\( T^{4} - 180 T^{3} + \cdots + 33315984 \)
|
| $73$ |
\( T^{4} + 190 T^{3} + \cdots + 13830961 \)
|
| $79$ |
\( (T^{2} - 48 T + 429)^{2} \)
|
| $83$ |
\( T^{4} - 264 T^{3} + \cdots + 21678336 \)
|
| $89$ |
\( T^{4} - 288 T^{3} + \cdots + 137170944 \)
|
| $97$ |
\( T^{4} - 310 T^{3} + \cdots + 8162449 \)
|
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