Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [312,2,Mod(205,312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(312, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("312.205");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 312 = 2^{3} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 312.bk (of order \(6\), 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: | \(2.49133254306\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 5 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 13\nu ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{6} + 8\nu^{4} - 16\nu^{2} + 1 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + 8\nu^{5} - 24\nu^{3} + 9\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} + 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} + 3\beta_{4} + 3\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{7} - 5\beta_{6} + 3\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{2} - 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{3} - 13\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(157\) | \(209\) |
| \(\chi(n)\) | \(1\) | \(\beta_{5}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 205.1 |
|
−1.40126 | − | 0.190983i | 0.866025 | − | 0.500000i | 1.92705 | + | 0.535233i | −2.23607 | −1.30902 | + | 0.535233i | −0.436492 | − | 0.252009i | −2.59808 | − | 1.11803i | 0.500000 | − | 0.866025i | 3.13331 | + | 0.427051i | ||||||||||||||||||||||||||
| 205.2 | −0.535233 | + | 1.30902i | −0.866025 | + | 0.500000i | −1.42705 | − | 1.40126i | 2.23607 | −0.190983 | − | 1.40126i | −0.436492 | − | 0.252009i | 2.59808 | − | 1.11803i | 0.500000 | − | 0.866025i | −1.19682 | + | 2.92705i | |||||||||||||||||||||||||||
| 205.3 | 0.535233 | − | 1.30902i | 0.866025 | − | 0.500000i | −1.42705 | − | 1.40126i | 2.23607 | −0.190983 | − | 1.40126i | 3.43649 | + | 1.98406i | −2.59808 | + | 1.11803i | 0.500000 | − | 0.866025i | 1.19682 | − | 2.92705i | |||||||||||||||||||||||||||
| 205.4 | 1.40126 | + | 0.190983i | −0.866025 | + | 0.500000i | 1.92705 | + | 0.535233i | −2.23607 | −1.30902 | + | 0.535233i | 3.43649 | + | 1.98406i | 2.59808 | + | 1.11803i | 0.500000 | − | 0.866025i | −3.13331 | − | 0.427051i | |||||||||||||||||||||||||||
| 277.1 | −1.40126 | + | 0.190983i | 0.866025 | + | 0.500000i | 1.92705 | − | 0.535233i | −2.23607 | −1.30902 | − | 0.535233i | −0.436492 | + | 0.252009i | −2.59808 | + | 1.11803i | 0.500000 | + | 0.866025i | 3.13331 | − | 0.427051i | |||||||||||||||||||||||||||
| 277.2 | −0.535233 | − | 1.30902i | −0.866025 | − | 0.500000i | −1.42705 | + | 1.40126i | 2.23607 | −0.190983 | + | 1.40126i | −0.436492 | + | 0.252009i | 2.59808 | + | 1.11803i | 0.500000 | + | 0.866025i | −1.19682 | − | 2.92705i | |||||||||||||||||||||||||||
| 277.3 | 0.535233 | + | 1.30902i | 0.866025 | + | 0.500000i | −1.42705 | + | 1.40126i | 2.23607 | −0.190983 | + | 1.40126i | 3.43649 | − | 1.98406i | −2.59808 | − | 1.11803i | 0.500000 | + | 0.866025i | 1.19682 | + | 2.92705i | |||||||||||||||||||||||||||
| 277.4 | 1.40126 | − | 0.190983i | −0.866025 | − | 0.500000i | 1.92705 | − | 0.535233i | −2.23607 | −1.30902 | − | 0.535233i | 3.43649 | − | 1.98406i | 2.59808 | − | 1.11803i | 0.500000 | + | 0.866025i | −3.13331 | + | 0.427051i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
| 104.s | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 312.2.bk.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 936.2.dg.c | 8 | ||
| 4.b | odd | 2 | 1 | 1248.2.ca.a | 8 | ||
| 8.b | even | 2 | 1 | inner | 312.2.bk.a | ✓ | 8 |
| 8.d | odd | 2 | 1 | 1248.2.ca.a | 8 | ||
| 13.e | even | 6 | 1 | inner | 312.2.bk.a | ✓ | 8 |
| 24.h | odd | 2 | 1 | 936.2.dg.c | 8 | ||
| 39.h | odd | 6 | 1 | 936.2.dg.c | 8 | ||
| 52.i | odd | 6 | 1 | 1248.2.ca.a | 8 | ||
| 104.p | odd | 6 | 1 | 1248.2.ca.a | 8 | ||
| 104.s | even | 6 | 1 | inner | 312.2.bk.a | ✓ | 8 |
| 312.bg | odd | 6 | 1 | 936.2.dg.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.2.bk.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 312.2.bk.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 312.2.bk.a | ✓ | 8 | 13.e | even | 6 | 1 | inner |
| 312.2.bk.a | ✓ | 8 | 104.s | even | 6 | 1 | inner |
| 936.2.dg.c | 8 | 3.b | odd | 2 | 1 | ||
| 936.2.dg.c | 8 | 24.h | odd | 2 | 1 | ||
| 936.2.dg.c | 8 | 39.h | odd | 6 | 1 | ||
| 936.2.dg.c | 8 | 312.bg | odd | 6 | 1 | ||
| 1248.2.ca.a | 8 | 4.b | odd | 2 | 1 | ||
| 1248.2.ca.a | 8 | 8.d | odd | 2 | 1 | ||
| 1248.2.ca.a | 8 | 52.i | odd | 6 | 1 | ||
| 1248.2.ca.a | 8 | 104.p | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{6} + \cdots + 16 \)
|
| $3$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $5$ |
\( (T^{2} - 5)^{4} \)
|
| $7$ |
\( (T^{4} - 6 T^{3} + 10 T^{2} + \cdots + 4)^{2} \)
|
| $11$ |
\( T^{8} + 16 T^{6} + \cdots + 16 \)
|
| $13$ |
\( (T^{4} - T^{2} + 169)^{2} \)
|
| $17$ |
\( (T^{4} + 4 T^{3} + \cdots + 121)^{2} \)
|
| $19$ |
\( T^{8} + 16 T^{6} + \cdots + 16 \)
|
| $23$ |
\( (T^{4} + 10 T^{3} + \cdots + 100)^{2} \)
|
| $29$ |
\( T^{8} - 102 T^{6} + \cdots + 194481 \)
|
| $31$ |
\( (T^{2} + 20)^{4} \)
|
| $37$ |
\( T^{8} + 166 T^{6} + \cdots + 35153041 \)
|
| $41$ |
\( (T^{4} - 12 T^{3} + \cdots + 49)^{2} \)
|
| $43$ |
\( T^{8} - 80 T^{6} + \cdots + 10000 \)
|
| $47$ |
\( (T^{4} + 16 T^{2} + 4)^{2} \)
|
| $53$ |
\( (T^{4} + 62 T^{2} + 1)^{2} \)
|
| $59$ |
\( T^{8} + 136 T^{6} + \cdots + 614656 \)
|
| $61$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $67$ |
\( T^{8} + 240 T^{6} + \cdots + 810000 \)
|
| $71$ |
\( (T^{4} - 6 T^{3} + \cdots + 1764)^{2} \)
|
| $73$ |
\( (T^{4} + 190 T^{2} + 3025)^{2} \)
|
| $79$ |
\( (T - 14)^{8} \)
|
| $83$ |
\( (T^{4} - 64 T^{2} + 484)^{2} \)
|
| $89$ |
\( (T^{4} - 24 T^{3} + \cdots + 1024)^{2} \)
|
| $97$ |
\( (T^{2} + 30 T + 300)^{4} \)
|
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