Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [670,2,Mod(269,670)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(670, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("670.269");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 670 = 2 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 670.c (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: | \(5.34997693543\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.0.1016580161536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 12x^{8} + 48x^{6} + 72x^{4} + 36x^{2} + 4 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 12x^{8} + 48x^{6} + 72x^{4} + 36x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 6\nu^{3} + 6\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 6\nu^{4} - 6\nu^{2} ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 8\nu^{5} + 18\nu^{3} + 12\nu ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} + 11\nu^{6} + 38\nu^{4} + 42\nu^{2} + 10 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{9} + 11\nu^{7} + 38\nu^{5} + 42\nu^{3} + 10\nu ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{9} - 12\nu^{7} - 47\nu^{5} - 64\nu^{3} - 20\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 5\beta_{2} + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -6\beta_{9} - 6\beta_{8} - 6\beta_{6} - 4\beta_{3} + 18\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -6\beta_{5} - 8\beta_{4} + 24\beta_{2} - 36 \)
|
| \(\nu^{7}\) | \(=\) |
\( 30\beta_{9} + 30\beta_{8} + 32\beta_{6} + 14\beta_{3} - 84\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2\beta_{7} + 28\beta_{5} + 50\beta_{4} - 116\beta_{2} + 166 \)
|
| \(\nu^{9}\) | \(=\) |
\( -144\beta_{9} - 142\beta_{8} - 166\beta_{6} - 44\beta_{3} + 398\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/670\mathbb{Z}\right)^\times\).
| \(n\) | \(471\) | \(537\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 269.1 |
|
− | 1.00000i | − | 2.39276i | −1.00000 | −1.79565 | + | 1.33253i | −2.39276 | 1.57128i | 1.00000i | −2.72528 | 1.33253 | + | 1.79565i | ||||||||||||||||||||||||||||||||||||||||||
| 269.2 | − | 1.00000i | − | 1.69775i | −1.00000 | 2.08209 | − | 0.815403i | −1.69775 | − | 3.41721i | 1.00000i | 0.117657 | −0.815403 | − | 2.08209i | ||||||||||||||||||||||||||||||||||||||||||
| 269.3 | − | 1.00000i | − | 0.359470i | −1.00000 | 0.161179 | − | 2.23025i | −0.359470 | 2.81284i | 1.00000i | 2.87078 | −2.23025 | − | 0.161179i | |||||||||||||||||||||||||||||||||||||||||||
| 269.4 | − | 1.00000i | 0.863067i | −1.00000 | −2.20143 | − | 0.392048i | 0.863067 | 0.355133i | 1.00000i | 2.25512 | −0.392048 | + | 2.20143i | ||||||||||||||||||||||||||||||||||||||||||||
| 269.5 | − | 1.00000i | 1.58691i | −1.00000 | 0.753811 | + | 2.10518i | 1.58691 | 1.67796i | 1.00000i | 0.481729 | 2.10518 | − | 0.753811i | ||||||||||||||||||||||||||||||||||||||||||||
| 269.6 | 1.00000i | − | 1.58691i | −1.00000 | 0.753811 | − | 2.10518i | 1.58691 | − | 1.67796i | − | 1.00000i | 0.481729 | 2.10518 | + | 0.753811i | ||||||||||||||||||||||||||||||||||||||||||
| 269.7 | 1.00000i | − | 0.863067i | −1.00000 | −2.20143 | + | 0.392048i | 0.863067 | − | 0.355133i | − | 1.00000i | 2.25512 | −0.392048 | − | 2.20143i | ||||||||||||||||||||||||||||||||||||||||||
| 269.8 | 1.00000i | 0.359470i | −1.00000 | 0.161179 | + | 2.23025i | −0.359470 | − | 2.81284i | − | 1.00000i | 2.87078 | −2.23025 | + | 0.161179i | |||||||||||||||||||||||||||||||||||||||||||
| 269.9 | 1.00000i | 1.69775i | −1.00000 | 2.08209 | + | 0.815403i | −1.69775 | 3.41721i | − | 1.00000i | 0.117657 | −0.815403 | + | 2.08209i | ||||||||||||||||||||||||||||||||||||||||||||
| 269.10 | 1.00000i | 2.39276i | −1.00000 | −1.79565 | − | 1.33253i | −2.39276 | − | 1.57128i | − | 1.00000i | −2.72528 | 1.33253 | − | 1.79565i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 670.2.c.a | ✓ | 10 |
| 5.b | even | 2 | 1 | inner | 670.2.c.a | ✓ | 10 |
| 5.c | odd | 4 | 1 | 3350.2.a.p | 5 | ||
| 5.c | odd | 4 | 1 | 3350.2.a.u | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 670.2.c.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 670.2.c.a | ✓ | 10 | 5.b | even | 2 | 1 | inner |
| 3350.2.a.p | 5 | 5.c | odd | 4 | 1 | ||
| 3350.2.a.u | 5 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 12T_{3}^{8} + 48T_{3}^{6} + 76T_{3}^{4} + 40T_{3}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{5} \)
|
| $3$ |
\( T^{10} + 12 T^{8} + \cdots + 4 \)
|
| $5$ |
\( T^{10} + 2 T^{9} + \cdots + 3125 \)
|
| $7$ |
\( T^{10} + 25 T^{8} + \cdots + 81 \)
|
| $11$ |
\( (T^{5} + 5 T^{4} - 14 T^{3} + \cdots + 9)^{2} \)
|
| $13$ |
\( T^{10} + 36 T^{8} + \cdots + 4 \)
|
| $17$ |
\( T^{10} + 88 T^{8} + \cdots + 45796 \)
|
| $19$ |
\( (T^{5} - 16 T^{4} + \cdots - 24)^{2} \)
|
| $23$ |
\( T^{10} + 168 T^{8} + \cdots + 118336 \)
|
| $29$ |
\( (T^{5} - 6 T^{4} + \cdots - 498)^{2} \)
|
| $31$ |
\( (T^{5} + 22 T^{4} + \cdots - 3326)^{2} \)
|
| $37$ |
\( T^{10} + 57 T^{8} + \cdots + 7569 \)
|
| $41$ |
\( (T^{5} + 14 T^{4} + \cdots + 5718)^{2} \)
|
| $43$ |
\( T^{10} + 76 T^{8} + \cdots + 31684 \)
|
| $47$ |
\( T^{10} + 208 T^{8} + \cdots + 36 \)
|
| $53$ |
\( T^{10} + 360 T^{8} + \cdots + 4990756 \)
|
| $59$ |
\( (T^{5} - 14 T^{4} + \cdots + 1678)^{2} \)
|
| $61$ |
\( (T^{5} + 17 T^{4} + \cdots + 547)^{2} \)
|
| $67$ |
\( (T^{2} + 1)^{5} \)
|
| $71$ |
\( (T^{5} - 21 T^{4} + \cdots + 3827)^{2} \)
|
| $73$ |
\( T^{10} + 168 T^{8} + \cdots + 27556 \)
|
| $79$ |
\( (T^{5} - 16 T^{4} + \cdots + 12248)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 65286893169 \)
|
| $89$ |
\( (T^{5} - 11 T^{4} + \cdots + 11369)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 314317441 \)
|
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