Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [237,2,Mod(1,237)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(237, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("237.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 237 = 3 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 237.a (trivial) |
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: | yes |
| Analytic conductor: | \(1.89245452790\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 11x^{5} + 22x^{4} + 30x^{3} - 65x^{2} - 2x + 23 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 11x^{5} + 22x^{4} + 30x^{3} - 65x^{2} - 2x + 23 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 8\nu^{3} + 24\nu^{2} - 17\nu - 5 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 11\nu^{4} - \nu^{3} + 30\nu^{2} + \nu - 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 12\nu^{4} + \nu^{3} + 36\nu^{2} - 9\nu - 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{6} - \nu^{5} - 34\nu^{4} + 8\nu^{3} + 98\nu^{2} - 23\nu - 39 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 8\beta_{2} + 2\beta _1 + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{6} + 10\beta_{5} + 2\beta_{4} + 9\beta_{3} + 11\beta_{2} + 38\beta _1 + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( -23\beta_{6} + 12\beta_{5} + 12\beta_{4} + 23\beta_{3} + 59\beta_{2} + 27\beta _1 + 133 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.43464 | 1.00000 | 3.92748 | 2.56086 | −2.43464 | 2.52280 | −4.69271 | 1.00000 | −6.23477 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.20621 | 1.00000 | 2.86737 | −4.22897 | −2.20621 | 0.978483 | −1.91359 | 1.00000 | 9.33001 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.574496 | 1.00000 | −1.66995 | 0.0786605 | −0.574496 | 2.32841 | 2.10837 | 1.00000 | −0.0451902 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.809104 | 1.00000 | −1.34535 | 3.39246 | 0.809104 | −1.45542 | −2.70674 | 1.00000 | 2.74485 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.78428 | 1.00000 | 1.18367 | −2.05764 | 1.78428 | 4.97154 | −1.45656 | 1.00000 | −3.67141 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.88935 | 1.00000 | 1.56966 | 1.60580 | 1.88935 | −0.656310 | −0.813063 | 1.00000 | 3.03393 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.73260 | 1.00000 | 5.46713 | −3.35117 | 2.73260 | −4.68950 | 9.47430 | 1.00000 | −9.15742 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(79\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 237.2.a.c | ✓ | 7 |
| 3.b | odd | 2 | 1 | 711.2.a.g | 7 | ||
| 4.b | odd | 2 | 1 | 3792.2.a.y | 7 | ||
| 5.b | even | 2 | 1 | 5925.2.a.t | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 237.2.a.c | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 711.2.a.g | 7 | 3.b | odd | 2 | 1 | ||
| 3792.2.a.y | 7 | 4.b | odd | 2 | 1 | ||
| 5925.2.a.t | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 2T_{2}^{6} - 11T_{2}^{5} + 22T_{2}^{4} + 30T_{2}^{3} - 65T_{2}^{2} - 2T_{2} + 23 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(237))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 23 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( T^{7} + 2 T^{6} + \cdots + 32 \)
|
| $7$ |
\( T^{7} - 4 T^{6} + \cdots + 128 \)
|
| $11$ |
\( T^{7} - 2 T^{6} + \cdots + 116 \)
|
| $13$ |
\( T^{7} - 6 T^{6} + \cdots + 58 \)
|
| $17$ |
\( T^{7} + 8 T^{6} + \cdots - 54176 \)
|
| $19$ |
\( T^{7} - 4 T^{6} + \cdots - 80 \)
|
| $23$ |
\( T^{7} - 8 T^{6} + \cdots - 1928 \)
|
| $29$ |
\( T^{7} + 10 T^{6} + \cdots - 233440 \)
|
| $31$ |
\( T^{7} - 4 T^{6} + \cdots + 2560 \)
|
| $37$ |
\( T^{7} - 10 T^{6} + \cdots + 133888 \)
|
| $41$ |
\( T^{7} + 20 T^{6} + \cdots - 344320 \)
|
| $43$ |
\( T^{7} - 22 T^{6} + \cdots - 519616 \)
|
| $47$ |
\( T^{7} + 10 T^{6} + \cdots - 1024 \)
|
| $53$ |
\( T^{7} - 236 T^{5} + \cdots + 663808 \)
|
| $59$ |
\( T^{7} + 2 T^{6} + \cdots + 6649600 \)
|
| $61$ |
\( T^{7} - 4 T^{6} + \cdots - 3712 \)
|
| $67$ |
\( T^{7} - 20 T^{6} + \cdots - 6368 \)
|
| $71$ |
\( T^{7} + 20 T^{6} + \cdots + 5120 \)
|
| $73$ |
\( T^{7} - 2 T^{6} + \cdots - 8434 \)
|
| $79$ |
\( (T + 1)^{7} \)
|
| $83$ |
\( T^{7} - 10 T^{6} + \cdots + 16384 \)
|
| $89$ |
\( T^{7} + 20 T^{6} + \cdots - 55040 \)
|
| $97$ |
\( T^{7} - 14 T^{6} + \cdots + 372254 \)
|
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