Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(239,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.239");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.p (of order \(4\), 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: | \(4.35983195036\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.45474709504.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 38x^{6} + 481x^{4} + 2112x^{2} + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 38x^{6} + 481x^{4} + 2112x^{2} + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 6\nu^{5} + 223\nu^{3} + 1504\nu ) / 1024 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 6\nu^{6} - 82\nu^{5} - 164\nu^{4} - 611\nu^{3} - 1094\nu^{2} - 800\nu - 192 ) / 512 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{7} - 6\nu^{6} + 82\nu^{5} - 164\nu^{4} + 611\nu^{3} - 1094\nu^{2} + 800\nu - 192 ) / 512 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 22\nu^{6} - 82\nu^{5} + 516\nu^{4} - 611\nu^{3} + 2902\nu^{2} - 800\nu + 704 ) / 512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} - 52\nu^{6} - 170\nu^{5} - 1336\nu^{4} - 743\nu^{3} - 8884\nu^{2} + 2720\nu - 6784 ) / 1024 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\nu^{7} - 64\nu^{6} + 334\nu^{5} - 1664\nu^{4} + 1965\nu^{3} - 11072\nu^{2} - 1120\nu - 7168 ) / 1024 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{6} - \beta_{5} + 3\beta_{4} + 3\beta_{3} - 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} + 2\beta_{6} + 4\beta_{4} - 2\beta_{3} - 4\beta_{2} - 11\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 13\beta_{7} + 13\beta_{6} + 7\beta_{5} - 53\beta_{4} - 47\beta_{3} + 130 \)
|
| \(\nu^{5}\) | \(=\) |
\( 40\beta_{7} - 40\beta_{6} - 76\beta_{4} + 36\beta_{3} + 128\beta_{2} + 137\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -173\beta_{7} - 173\beta_{6} - 9\beta_{5} + 859\beta_{4} + 695\beta_{3} - 1762 \)
|
| \(\nu^{7}\) | \(=\) |
\( -686\beta_{7} + 686\beta_{6} + 1348\beta_{4} - 662\beta_{3} - 2684\beta_{2} - 1771\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239.1 |
|
−0.707107 | − | 0.707107i | −1.41421 | − | 1.00000i | 1.00000i | −0.524897 | − | 0.524897i | 0.292893 | + | 1.70711i | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000 | + | 2.82843i | 0.742317i | ||||||||||||||||||||||||||||
| 239.2 | −0.707107 | − | 0.707107i | −1.41421 | − | 1.00000i | 1.00000i | 2.23200 | + | 2.23200i | 0.292893 | + | 1.70711i | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000 | + | 2.82843i | − | 3.15653i | ||||||||||||||||||||||||||||
| 239.3 | 0.707107 | + | 0.707107i | 1.41421 | − | 1.00000i | 1.00000i | −2.47078 | − | 2.47078i | 1.70711 | + | 0.292893i | −0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000 | − | 2.82843i | − | 3.49421i | ||||||||||||||||||||||||||||
| 239.4 | 0.707107 | + | 0.707107i | 1.41421 | − | 1.00000i | 1.00000i | 2.76367 | + | 2.76367i | 1.70711 | + | 0.292893i | −0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000 | − | 2.82843i | 3.90842i | |||||||||||||||||||||||||||||
| 281.1 | −0.707107 | + | 0.707107i | −1.41421 | + | 1.00000i | − | 1.00000i | −0.524897 | + | 0.524897i | 0.292893 | − | 1.70711i | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | 1.00000 | − | 2.82843i | − | 0.742317i | |||||||||||||||||||||||||||
| 281.2 | −0.707107 | + | 0.707107i | −1.41421 | + | 1.00000i | − | 1.00000i | 2.23200 | − | 2.23200i | 0.292893 | − | 1.70711i | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | 1.00000 | − | 2.82843i | 3.15653i | ||||||||||||||||||||||||||||
| 281.3 | 0.707107 | − | 0.707107i | 1.41421 | + | 1.00000i | − | 1.00000i | −2.47078 | + | 2.47078i | 1.70711 | − | 0.292893i | −0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | 1.00000 | + | 2.82843i | 3.49421i | ||||||||||||||||||||||||||||
| 281.4 | 0.707107 | − | 0.707107i | 1.41421 | + | 1.00000i | − | 1.00000i | 2.76367 | − | 2.76367i | 1.70711 | − | 0.292893i | −0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | 1.00000 | + | 2.82843i | − | 3.90842i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 39.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 546.2.p.b | yes | 8 |
| 3.b | odd | 2 | 1 | 546.2.p.a | ✓ | 8 | |
| 13.d | odd | 4 | 1 | 546.2.p.a | ✓ | 8 | |
| 39.f | even | 4 | 1 | inner | 546.2.p.b | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.2.p.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 546.2.p.a | ✓ | 8 | 13.d | odd | 4 | 1 | |
| 546.2.p.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 546.2.p.b | yes | 8 | 39.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 12T_{5}^{5} + 161T_{5}^{4} - 592T_{5}^{3} + 1152T_{5}^{2} + 1536T_{5} + 1024 \)
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( (T^{4} - 2 T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots + 1024 \)
|
| $7$ |
\( (T^{4} + 1)^{2} \)
|
| $11$ |
\( T^{8} - 8 T^{7} + \cdots + 18496 \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( (T^{4} + 10 T^{3} + \cdots - 68)^{2} \)
|
| $19$ |
\( T^{8} + 20 T^{7} + \cdots + 3136 \)
|
| $23$ |
\( (T^{4} - 6 T^{3} + \cdots + 2254)^{2} \)
|
| $29$ |
\( T^{8} + 50 T^{6} + \cdots + 2116 \)
|
| $31$ |
\( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \)
|
| $37$ |
\( T^{8} + 24 T^{7} + \cdots + 16 \)
|
| $41$ |
\( T^{8} - 20 T^{7} + \cdots + 61504 \)
|
| $43$ |
\( T^{8} + 230 T^{6} + \cdots + 3268864 \)
|
| $47$ |
\( (T^{4} + 1296)^{2} \)
|
| $53$ |
\( T^{8} + 404 T^{6} + \cdots + 16192576 \)
|
| $59$ |
\( T^{8} - 448 T^{5} + \cdots + 7772944 \)
|
| $61$ |
\( (T^{4} - 10 T^{3} + \cdots - 4738)^{2} \)
|
| $67$ |
\( T^{8} - 24 T^{7} + \cdots + 44782864 \)
|
| $71$ |
\( T^{8} + 28 T^{7} + \cdots + 107584 \)
|
| $73$ |
\( T^{8} - 16 T^{7} + \cdots + 28601104 \)
|
| $79$ |
\( (T^{4} - 12 T^{3} + \cdots - 496)^{2} \)
|
| $83$ |
\( T^{8} - 28 T^{7} + \cdots + 107584 \)
|
| $89$ |
\( T^{8} - 1792 T^{5} + \cdots + 2347024 \)
|
| $97$ |
\( (T^{4} - 16 T^{3} + \cdots + 784)^{2} \)
|
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