Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,2,Mod(347,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("348.347");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 348.b (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: | \(2.77879399034\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.12960000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| 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
\( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{5} - 4\nu^{3} - 7\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{4} + 6\nu^{2} + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 4\nu^{4} - 12\nu^{2} + 7 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{5} + 10\nu^{3} - 7\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + 3\nu^{4} - 7\nu^{2} + 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} + 20\nu^{5} - 52\nu^{3} + 17\nu ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{7} - 5\nu^{5} + 14\nu^{3} + \nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 2\beta_{6} + 2\beta_{4} - \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 2\beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{6} + \beta_{4} + 2\beta_1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{5} - 2\beta_{3} + 5\beta_{2} - 5 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5\beta_{7} - 2\beta_{6} - 14\beta_{4} + 13\beta_1 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4\beta_{3} + 4\beta_{2} - 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -13\beta_{7} - 34\beta_{6} - 50\beta_{4} + 5\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/348\mathbb{Z}\right)^\times\).
| \(n\) | \(175\) | \(205\) | \(233\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 347.1 |
|
−1.11803 | − | 0.866025i | −1.73205 | 0.500000 | + | 1.93649i | 3.00000i | 1.93649 | + | 1.50000i | − | 3.87298i | 1.11803 | − | 2.59808i | 3.00000 | 2.59808 | − | 3.35410i | |||||||||||||||||||||||||||||||
| 347.2 | −1.11803 | − | 0.866025i | 1.73205 | 0.500000 | + | 1.93649i | − | 3.00000i | −1.93649 | − | 1.50000i | − | 3.87298i | 1.11803 | − | 2.59808i | 3.00000 | −2.59808 | + | 3.35410i | |||||||||||||||||||||||||||||||
| 347.3 | −1.11803 | + | 0.866025i | −1.73205 | 0.500000 | − | 1.93649i | − | 3.00000i | 1.93649 | − | 1.50000i | 3.87298i | 1.11803 | + | 2.59808i | 3.00000 | 2.59808 | + | 3.35410i | ||||||||||||||||||||||||||||||||
| 347.4 | −1.11803 | + | 0.866025i | 1.73205 | 0.500000 | − | 1.93649i | 3.00000i | −1.93649 | + | 1.50000i | 3.87298i | 1.11803 | + | 2.59808i | 3.00000 | −2.59808 | − | 3.35410i | |||||||||||||||||||||||||||||||||
| 347.5 | 1.11803 | − | 0.866025i | −1.73205 | 0.500000 | − | 1.93649i | 3.00000i | −1.93649 | + | 1.50000i | 3.87298i | −1.11803 | − | 2.59808i | 3.00000 | 2.59808 | + | 3.35410i | |||||||||||||||||||||||||||||||||
| 347.6 | 1.11803 | − | 0.866025i | 1.73205 | 0.500000 | − | 1.93649i | − | 3.00000i | 1.93649 | − | 1.50000i | 3.87298i | −1.11803 | − | 2.59808i | 3.00000 | −2.59808 | − | 3.35410i | ||||||||||||||||||||||||||||||||
| 347.7 | 1.11803 | + | 0.866025i | −1.73205 | 0.500000 | + | 1.93649i | − | 3.00000i | −1.93649 | − | 1.50000i | − | 3.87298i | −1.11803 | + | 2.59808i | 3.00000 | 2.59808 | − | 3.35410i | |||||||||||||||||||||||||||||||
| 347.8 | 1.11803 | + | 0.866025i | 1.73205 | 0.500000 | + | 1.93649i | 3.00000i | 1.93649 | + | 1.50000i | − | 3.87298i | −1.11803 | + | 2.59808i | 3.00000 | −2.59808 | + | 3.35410i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
| 29.b | even | 2 | 1 | inner |
| 87.d | odd | 2 | 1 | inner |
| 116.d | odd | 2 | 1 | inner |
| 348.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 348.2.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 348.2.b.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 348.2.b.a | ✓ | 8 |
| 12.b | even | 2 | 1 | inner | 348.2.b.a | ✓ | 8 |
| 29.b | even | 2 | 1 | inner | 348.2.b.a | ✓ | 8 |
| 87.d | odd | 2 | 1 | inner | 348.2.b.a | ✓ | 8 |
| 116.d | odd | 2 | 1 | inner | 348.2.b.a | ✓ | 8 |
| 348.b | even | 2 | 1 | inner | 348.2.b.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 348.2.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 348.2.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 348.2.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 348.2.b.a | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 348.2.b.a | ✓ | 8 | 29.b | even | 2 | 1 | inner |
| 348.2.b.a | ✓ | 8 | 87.d | odd | 2 | 1 | inner |
| 348.2.b.a | ✓ | 8 | 116.d | odd | 2 | 1 | inner |
| 348.2.b.a | ✓ | 8 | 348.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(348, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 4)^{2} \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( (T^{2} + 9)^{4} \)
|
| $7$ |
\( (T^{2} + 15)^{4} \)
|
| $11$ |
\( (T^{2} + 12)^{4} \)
|
| $13$ |
\( (T + 4)^{8} \)
|
| $17$ |
\( (T^{2} - 5)^{4} \)
|
| $19$ |
\( (T^{2} - 3)^{4} \)
|
| $23$ |
\( (T^{2} - 60)^{4} \)
|
| $29$ |
\( (T^{4} - 22 T^{2} + 841)^{2} \)
|
| $31$ |
\( (T^{2} - 12)^{4} \)
|
| $37$ |
\( (T^{2} + 45)^{4} \)
|
| $41$ |
\( (T^{2} - 5)^{4} \)
|
| $43$ |
\( (T^{2} - 75)^{4} \)
|
| $47$ |
\( (T^{2} + 3)^{4} \)
|
| $53$ |
\( (T^{2} + 36)^{4} \)
|
| $59$ |
\( (T^{2} - 15)^{4} \)
|
| $61$ |
\( (T^{2} + 180)^{4} \)
|
| $67$ |
\( (T^{2} + 60)^{4} \)
|
| $71$ |
\( (T^{2} - 60)^{4} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( (T^{2} - 108)^{4} \)
|
| $83$ |
\( (T^{2} - 60)^{4} \)
|
| $89$ |
\( (T^{2} - 20)^{4} \)
|
| $97$ |
\( (T^{2} + 180)^{4} \)
|
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