Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(79,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.79");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 637.e (of order \(3\), degree \(2\), not 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: | \(5.08647060876\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.2696112.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 91) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 18\nu^{2} + 8\nu - 40 ) / 82 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 10\nu^{4} + 9\nu^{3} - 36\nu^{2} + 16\nu - 121 ) / 41 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -10\nu^{5} + 9\nu^{4} - 45\nu^{3} - 25\nu^{2} - 162\nu + 72 ) / 82 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -26\nu^{5} + 7\nu^{4} - 117\nu^{3} - 65\nu^{2} - 454\nu - 26 ) / 82 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} - \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 4\beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{5} + 13\beta_{4} - 2\beta _1 - 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} + 11\beta_{4} + 7\beta_{3} - 18\beta_{2} - 18\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(248\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
−1.17146 | + | 2.02903i | −0.573183 | − | 0.992782i | −1.74464 | − | 3.02181i | −0.671462 | + | 1.16301i | 2.68585 | 0 | 3.48929 | 0.842923 | − | 1.45999i | −1.57318 | − | 2.72483i | ||||||||||||||||||||||||
| 79.2 | −0.235342 | + | 0.407624i | 1.12457 | + | 1.94781i | 0.889229 | + | 1.54019i | 0.264658 | − | 0.458402i | −1.05863 | 0 | −1.77846 | −1.02932 | + | 1.78283i | 0.124570 | + | 0.215762i | |||||||||||||||||||||||||
| 79.3 | 0.906803 | − | 1.57063i | −1.55139 | − | 2.68708i | −0.644584 | − | 1.11645i | 1.40680 | − | 2.43665i | −5.62721 | 0 | 1.28917 | −3.31361 | + | 5.73933i | −2.55139 | − | 4.41913i | |||||||||||||||||||||||||
| 508.1 | −1.17146 | − | 2.02903i | −0.573183 | + | 0.992782i | −1.74464 | + | 3.02181i | −0.671462 | − | 1.16301i | 2.68585 | 0 | 3.48929 | 0.842923 | + | 1.45999i | −1.57318 | + | 2.72483i | |||||||||||||||||||||||||
| 508.2 | −0.235342 | − | 0.407624i | 1.12457 | − | 1.94781i | 0.889229 | − | 1.54019i | 0.264658 | + | 0.458402i | −1.05863 | 0 | −1.77846 | −1.02932 | − | 1.78283i | 0.124570 | − | 0.215762i | |||||||||||||||||||||||||
| 508.3 | 0.906803 | + | 1.57063i | −1.55139 | + | 2.68708i | −0.644584 | + | 1.11645i | 1.40680 | + | 2.43665i | −5.62721 | 0 | 1.28917 | −3.31361 | − | 5.73933i | −2.55139 | + | 4.41913i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 637.2.e.i | 6 | |
| 7.b | odd | 2 | 1 | 637.2.e.j | 6 | ||
| 7.c | even | 3 | 1 | 637.2.a.j | 3 | ||
| 7.c | even | 3 | 1 | inner | 637.2.e.i | 6 | |
| 7.d | odd | 6 | 1 | 91.2.a.d | ✓ | 3 | |
| 7.d | odd | 6 | 1 | 637.2.e.j | 6 | ||
| 21.g | even | 6 | 1 | 819.2.a.i | 3 | ||
| 21.h | odd | 6 | 1 | 5733.2.a.x | 3 | ||
| 28.f | even | 6 | 1 | 1456.2.a.t | 3 | ||
| 35.i | odd | 6 | 1 | 2275.2.a.m | 3 | ||
| 56.j | odd | 6 | 1 | 5824.2.a.by | 3 | ||
| 56.m | even | 6 | 1 | 5824.2.a.bs | 3 | ||
| 91.r | even | 6 | 1 | 8281.2.a.bg | 3 | ||
| 91.s | odd | 6 | 1 | 1183.2.a.i | 3 | ||
| 91.bb | even | 12 | 2 | 1183.2.c.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.2.a.d | ✓ | 3 | 7.d | odd | 6 | 1 | |
| 637.2.a.j | 3 | 7.c | even | 3 | 1 | ||
| 637.2.e.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 637.2.e.i | 6 | 7.c | even | 3 | 1 | inner | |
| 637.2.e.j | 6 | 7.b | odd | 2 | 1 | ||
| 637.2.e.j | 6 | 7.d | odd | 6 | 1 | ||
| 819.2.a.i | 3 | 21.g | even | 6 | 1 | ||
| 1183.2.a.i | 3 | 91.s | odd | 6 | 1 | ||
| 1183.2.c.f | 6 | 91.bb | even | 12 | 2 | ||
| 1456.2.a.t | 3 | 28.f | even | 6 | 1 | ||
| 2275.2.a.m | 3 | 35.i | odd | 6 | 1 | ||
| 5733.2.a.x | 3 | 21.h | odd | 6 | 1 | ||
| 5824.2.a.bs | 3 | 56.m | even | 6 | 1 | ||
| 5824.2.a.by | 3 | 56.j | odd | 6 | 1 | ||
| 8281.2.a.bg | 3 | 91.r | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):
|
\( T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} + 18T_{2}^{2} + 8T_{2} + 4 \)
|
|
\( T_{3}^{6} + 2T_{3}^{5} + 10T_{3}^{4} + 4T_{3}^{3} + 52T_{3}^{2} + 48T_{3} + 64 \)
|
|
\( T_{5}^{6} - 2T_{5}^{5} + 7T_{5}^{4} + 2T_{5}^{3} + 13T_{5}^{2} - 6T_{5} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + 5 T^{4} + \cdots + 4 \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots + 64 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 4 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 2 T^{5} + \cdots + 64 \)
|
| $13$ |
\( (T + 1)^{6} \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots + 16 \)
|
| $19$ |
\( T^{6} + 4 T^{5} + \cdots + 16 \)
|
| $23$ |
\( T^{6} + 10 T^{5} + \cdots + 18496 \)
|
| $29$ |
\( (T^{3} - 24 T^{2} + \cdots - 454)^{2} \)
|
| $31$ |
\( T^{6} + 4 T^{5} + \cdots + 256 \)
|
| $37$ |
\( T^{6} + 58 T^{4} + \cdots + 15376 \)
|
| $41$ |
\( (T^{3} + 2 T^{2} - 28 T + 8)^{2} \)
|
| $43$ |
\( (T^{3} - 10 T^{2} + \cdots + 628)^{2} \)
|
| $47$ |
\( T^{6} + 8 T^{5} + \cdots + 295936 \)
|
| $53$ |
\( T^{6} + 8 T^{5} + \cdots + 484 \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots + 473344 \)
|
| $61$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $67$ |
\( T^{6} - 12 T^{5} + \cdots + 952576 \)
|
| $71$ |
\( (T^{3} + 6 T^{2} - 22 T + 16)^{2} \)
|
| $73$ |
\( T^{6} + 10 T^{5} + \cdots + 75076 \)
|
| $79$ |
\( T^{6} - 14 T^{5} + \cdots + 256 \)
|
| $83$ |
\( (T^{3} - 12 T^{2} + \cdots + 3268)^{2} \)
|
| $89$ |
\( T^{6} - 2 T^{5} + \cdots + 178084 \)
|
| $97$ |
\( (T^{3} - 10 T^{2} + \cdots - 22)^{2} \)
|
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