Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,4,Mod(7,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 38.c (of order \(3\), 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: | \(2.24207258022\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 64\nu^{4} + 4096\nu^{3} - 3984\nu^{2} + 945\nu - 60480 ) / 254031 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21\nu^{5} - 1344\nu^{4} + 1339\nu^{3} - 83664\nu^{2} + 19845\nu - 3641036 ) / 169354 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1344\nu^{5} - 1339\nu^{4} + 85696\nu^{3} + 64832\nu^{2} + 5334576\nu + 4800 ) / 1270155 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 57792\nu^{5} - 57577\nu^{4} + 3684928\nu^{3} + 1517621\nu^{2} + 229386768\nu - 54410265 ) / 2540310 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{5} + 43\beta_{4} - 43 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + 63\beta_{2} - 28 \)
|
| \(\nu^{4}\) | \(=\) |
\( 128\beta_{5} - 2737\beta_{4} - 128\beta_{3} - 48\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 224\beta_{5} - 3856\beta_{4} - 4017\beta_{2} - 4017\beta _1 + 3856 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) |
| \(\chi(n)\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
1.00000 | + | 1.73205i | −4.78825 | − | 8.29349i | −2.00000 | + | 3.46410i | −7.88908 | − | 13.6643i | 9.57650 | − | 16.5870i | 16.5765 | −8.00000 | −32.3546 | + | 56.0399i | 15.7782 | − | 27.3286i | ||||||||||||||||||||||
| 7.2 | 1.00000 | + | 1.73205i | −0.881294 | − | 1.52645i | −2.00000 | + | 3.46410i | 10.3546 | + | 17.9347i | 1.76259 | − | 3.05289i | 8.76259 | −8.00000 | 11.9466 | − | 20.6922i | −20.7092 | + | 35.8694i | |||||||||||||||||||||||
| 7.3 | 1.00000 | + | 1.73205i | 3.16954 | + | 5.48981i | −2.00000 | + | 3.46410i | −2.96554 | − | 5.13646i | −6.33908 | + | 10.9796i | 0.660916 | −8.00000 | −6.59199 | + | 11.4177i | 5.93108 | − | 10.2729i | |||||||||||||||||||||||
| 11.1 | 1.00000 | − | 1.73205i | −4.78825 | + | 8.29349i | −2.00000 | − | 3.46410i | −7.88908 | + | 13.6643i | 9.57650 | + | 16.5870i | 16.5765 | −8.00000 | −32.3546 | − | 56.0399i | 15.7782 | + | 27.3286i | |||||||||||||||||||||||
| 11.2 | 1.00000 | − | 1.73205i | −0.881294 | + | 1.52645i | −2.00000 | − | 3.46410i | 10.3546 | − | 17.9347i | 1.76259 | + | 3.05289i | 8.76259 | −8.00000 | 11.9466 | + | 20.6922i | −20.7092 | − | 35.8694i | |||||||||||||||||||||||
| 11.3 | 1.00000 | − | 1.73205i | 3.16954 | − | 5.48981i | −2.00000 | − | 3.46410i | −2.96554 | + | 5.13646i | −6.33908 | − | 10.9796i | 0.660916 | −8.00000 | −6.59199 | − | 11.4177i | 5.93108 | + | 10.2729i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 38.4.c.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.4.g.f | 6 | ||
| 4.b | odd | 2 | 1 | 304.4.i.e | 6 | ||
| 19.c | even | 3 | 1 | inner | 38.4.c.c | ✓ | 6 |
| 19.c | even | 3 | 1 | 722.4.a.j | 3 | ||
| 19.d | odd | 6 | 1 | 722.4.a.k | 3 | ||
| 57.h | odd | 6 | 1 | 342.4.g.f | 6 | ||
| 76.g | odd | 6 | 1 | 304.4.i.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.4.c.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 38.4.c.c | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 304.4.i.e | 6 | 4.b | odd | 2 | 1 | ||
| 304.4.i.e | 6 | 76.g | odd | 6 | 1 | ||
| 342.4.g.f | 6 | 3.b | odd | 2 | 1 | ||
| 342.4.g.f | 6 | 57.h | odd | 6 | 1 | ||
| 722.4.a.j | 3 | 19.c | even | 3 | 1 | ||
| 722.4.a.k | 3 | 19.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 5T_{3}^{5} + 80T_{3}^{4} - 61T_{3}^{3} + 3560T_{3}^{2} + 5885T_{3} + 11449 \)
acting on \(S_{4}^{\mathrm{new}}(38, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{3} \)
|
| $3$ |
\( T^{6} + 5 T^{5} + \cdots + 11449 \)
|
| $5$ |
\( T^{6} + T^{5} + \cdots + 3755844 \)
|
| $7$ |
\( (T^{3} - 26 T^{2} + \cdots - 96)^{2} \)
|
| $11$ |
\( (T^{3} - 4 T^{2} + \cdots + 49980)^{2} \)
|
| $13$ |
\( T^{6} - 129 T^{5} + \cdots + 129322384 \)
|
| $17$ |
\( T^{6} + \cdots + 11293737984 \)
|
| $19$ |
\( T^{6} + \cdots + 322687697779 \)
|
| $23$ |
\( T^{6} + \cdots + 4555440036 \)
|
| $29$ |
\( T^{6} + \cdots + 5937750562500 \)
|
| $31$ |
\( (T^{3} + 50 T^{2} + \cdots + 3809848)^{2} \)
|
| $37$ |
\( (T^{3} + 188 T^{2} + \cdots - 88004)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 81183541856481 \)
|
| $43$ |
\( T^{6} + \cdots + 259289089231936 \)
|
| $47$ |
\( T^{6} + \cdots + 25671752892900 \)
|
| $53$ |
\( T^{6} + \cdots + 10857156800400 \)
|
| $59$ |
\( T^{6} + \cdots + 12\!\cdots\!21 \)
|
| $61$ |
\( T^{6} + \cdots + 356206057990084 \)
|
| $67$ |
\( T^{6} + \cdots + 964239549849 \)
|
| $71$ |
\( T^{6} + \cdots + 68\!\cdots\!16 \)
|
| $73$ |
\( T^{6} + \cdots + 713681971209 \)
|
| $79$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( (T^{3} - 588 T^{2} + \cdots - 162474984)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 27\!\cdots\!04 \)
|
| $97$ |
\( T^{6} + \cdots + 68\!\cdots\!25 \)
|
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