Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,19,Mod(17,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 19, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.17");
S:= CuspForms(chi, 19);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 19 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.e (of order \(2\), degree \(1\), 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: | \(98.5853461007\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8797x^{4} + 1809980x^{3} + 107861490x^{2} - 7180095600x + 788142376800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{36}\cdot 3^{16} \) |
| Twist minimal: | no (minimal twist has level 6) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 8797x^{4} + 1809980x^{3} + 107861490x^{2} - 7180095600x + 788142376800 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 534036469 \nu^{5} - 30088714589 \nu^{4} - 8686261738486 \nu^{3} + 958886787696422 \nu^{2} + \cdots - 81\!\cdots\!20 ) / 465336994768800 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 12196612195 \nu^{5} - 177322662981 \nu^{4} + 68787906245722 \nu^{3} + \cdots - 14\!\cdots\!20 ) / 41\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1820471647 \nu^{5} - 519915315321 \nu^{4} + 59341232079394 \nu^{3} + \cdots + 14\!\cdots\!00 ) / 130876029778725 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13221659621 \nu^{5} + 3982525238835 \nu^{4} + 200176939060714 \nu^{3} + \cdots + 49\!\cdots\!40 ) / 380730268447200 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 160347441791 \nu^{5} + 7971468883017 \nu^{4} - 898261134207650 \nu^{3} + \cdots - 81\!\cdots\!20 ) / 13\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{5} - 8\beta_{4} - 4\beta_{3} - 153\beta_{2} - 595\beta _1 + 55296 ) / 165888 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3544\beta_{5} - 1184\beta_{4} + 1296\beta_{3} + 108061\beta_{2} - 28949\beta _1 + 1459648512 ) / 497664 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 137024\beta_{5} + 57488\beta_{4} + 664104\beta_{3} - 3757675\beta_{2} - 16939909\beta _1 - 446002329600 ) / 497664 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5920936 \beta_{5} + 38030368 \beta_{4} - 44387280 \beta_{3} + 1248885187 \beta_{2} + \cdots - 24137584367616 ) / 497664 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 3366247720 \beta_{5} + 4334030912 \beta_{4} + 5539119168 \beta_{3} - 201415696981 \beta_{2} + \cdots - 36\!\cdots\!12 ) / 497664 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(31\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −19502.4 | − | 2660.56i | 0 | − | 1.30525e6i | 0 | −3.15845e7 | 0 | 3.73263e8 | + | 1.03774e8i | 0 | |||||||||||||||||||||||||||||||
| 17.2 | 0 | −19502.4 | + | 2660.56i | 0 | 1.30525e6i | 0 | −3.15845e7 | 0 | 3.73263e8 | − | 1.03774e8i | 0 | |||||||||||||||||||||||||||||||||
| 17.3 | 0 | 3612.02 | − | 19348.7i | 0 | − | 3.69488e6i | 0 | −3.18077e7 | 0 | −3.61327e8 | − | 1.39776e8i | 0 | ||||||||||||||||||||||||||||||||
| 17.4 | 0 | 3612.02 | + | 19348.7i | 0 | 3.69488e6i | 0 | −3.18077e7 | 0 | −3.61327e8 | + | 1.39776e8i | 0 | |||||||||||||||||||||||||||||||||
| 17.5 | 0 | 19019.3 | − | 5068.08i | 0 | 2.98937e6i | 0 | 4.92752e7 | 0 | 3.36050e8 | − | 1.92783e8i | 0 | |||||||||||||||||||||||||||||||||
| 17.6 | 0 | 19019.3 | + | 5068.08i | 0 | − | 2.98937e6i | 0 | 4.92752e7 | 0 | 3.36050e8 | + | 1.92783e8i | 0 | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.19.e.d | 6 | |
| 3.b | odd | 2 | 1 | inner | 48.19.e.d | 6 | |
| 4.b | odd | 2 | 1 | 6.19.b.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 6.19.b.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.19.b.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 6.19.b.a | ✓ | 6 | 12.b | even | 2 | 1 | |
| 48.19.e.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 48.19.e.d | 6 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 24292116155520 T_{5}^{4} + \cdots + 20\!\cdots\!00 \)
acting on \(S_{19}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 58\!\cdots\!69 \)
|
| $5$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots - 49\!\cdots\!88)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 24\!\cdots\!88 \)
|
| $13$ |
\( (T^{3} + \cdots + 54\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 28\!\cdots\!52 \)
|
| $19$ |
\( (T^{3} + \cdots + 12\!\cdots\!32)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 11\!\cdots\!12 \)
|
| $29$ |
\( T^{6} + \cdots + 67\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 23\!\cdots\!88)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 28\!\cdots\!88)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 17\!\cdots\!48 \)
|
| $43$ |
\( (T^{3} + \cdots - 10\!\cdots\!64)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 35\!\cdots\!72 \)
|
| $59$ |
\( T^{6} + \cdots + 16\!\cdots\!68 \)
|
| $61$ |
\( (T^{3} + \cdots + 37\!\cdots\!36)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots - 28\!\cdots\!32)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 25\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 17\!\cdots\!88)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 45\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots + 52\!\cdots\!88 \)
|
| $97$ |
\( (T^{3} + \cdots - 53\!\cdots\!16)^{2} \)
|
| show more | |
| show less |