Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [114,4,Mod(7,114)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(114, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("114.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 114 = 2 \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 114.e (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: | \(6.72621774065\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.627014547.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 26x^{4} + 169x^{2} + 147 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{3} \) |
| 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
\( x^{6} + 26x^{4} + 169x^{2} + 147 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 13\nu + 7 ) / 14 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{4} + 40\nu^{2} + 119 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10\nu^{4} + 116\nu^{2} - 119 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} - 10\nu^{4} + 43\nu^{3} - 116\nu^{2} + 431\nu + 119 ) / 14 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 2\nu^{4} + 43\nu^{3} - 40\nu^{2} + 179\nu - 119 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 5\beta_{2} - 102 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 26\beta_{5} - 26\beta_{4} - 13\beta_{3} + 13\beta_{2} + 504\beta _1 - 252 ) / 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 10\beta_{3} - 29\beta_{2} + 663 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -64\beta_{5} + 190\beta_{4} + 95\beta_{3} - 32\beta_{2} - 5418\beta _1 + 2709 ) / 18 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −7.12716 | − | 12.3446i | 3.00000 | − | 5.19615i | 13.2543 | 8.00000 | −4.50000 | + | 7.79423i | −14.2543 | + | 24.6892i | ||||||||||||||||||||||
| 7.2 | −1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −2.09405 | − | 3.62701i | 3.00000 | − | 5.19615i | 3.18810 | 8.00000 | −4.50000 | + | 7.79423i | −4.18810 | + | 7.25401i | |||||||||||||||||||||||
| 7.3 | −1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 4.22121 | + | 7.31135i | 3.00000 | − | 5.19615i | −9.44242 | 8.00000 | −4.50000 | + | 7.79423i | 8.44242 | − | 14.6227i | |||||||||||||||||||||||
| 49.1 | −1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −7.12716 | + | 12.3446i | 3.00000 | + | 5.19615i | 13.2543 | 8.00000 | −4.50000 | − | 7.79423i | −14.2543 | − | 24.6892i | |||||||||||||||||||||||
| 49.2 | −1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −2.09405 | + | 3.62701i | 3.00000 | + | 5.19615i | 3.18810 | 8.00000 | −4.50000 | − | 7.79423i | −4.18810 | − | 7.25401i | |||||||||||||||||||||||
| 49.3 | −1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 4.22121 | − | 7.31135i | 3.00000 | + | 5.19615i | −9.44242 | 8.00000 | −4.50000 | − | 7.79423i | 8.44242 | + | 14.6227i | |||||||||||||||||||||||
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 | 114.4.e.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.4.g.h | 6 | ||
| 19.c | even | 3 | 1 | inner | 114.4.e.d | ✓ | 6 |
| 19.c | even | 3 | 1 | 2166.4.a.u | 3 | ||
| 19.d | odd | 6 | 1 | 2166.4.a.t | 3 | ||
| 57.h | odd | 6 | 1 | 342.4.g.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.4.e.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 114.4.e.d | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 342.4.g.h | 6 | 3.b | odd | 2 | 1 | ||
| 342.4.g.h | 6 | 57.h | odd | 6 | 1 | ||
| 2166.4.a.t | 3 | 19.d | odd | 6 | 1 | ||
| 2166.4.a.u | 3 | 19.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 10T_{5}^{5} + 196T_{5}^{4} + 48T_{5}^{3} + 14256T_{5}^{2} + 48384T_{5} + 254016 \)
acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{3} \)
|
| $5$ |
\( T^{6} + 10 T^{5} + \cdots + 254016 \)
|
| $7$ |
\( (T^{3} - 7 T^{2} + \cdots + 399)^{2} \)
|
| $11$ |
\( (T^{3} + 44 T^{2} + \cdots - 217224)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 1420159225 \)
|
| $17$ |
\( T^{6} + \cdots + 673712640000 \)
|
| $19$ |
\( T^{6} + \cdots + 322687697779 \)
|
| $23$ |
\( T^{6} + \cdots + 11401114176 \)
|
| $29$ |
\( T^{6} + \cdots + 1696079056896 \)
|
| $31$ |
\( (T^{3} + 109 T^{2} + \cdots - 9721957)^{2} \)
|
| $37$ |
\( (T^{3} - 245 T^{2} + \cdots - 1317799)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 105675768336384 \)
|
| $43$ |
\( T^{6} + \cdots + 39054112950241 \)
|
| $47$ |
\( T^{6} + \cdots + 136793983242816 \)
|
| $53$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 134994517056 \)
|
| $61$ |
\( T^{6} + \cdots + 5456924641 \)
|
| $67$ |
\( T^{6} + \cdots + 92437321911489 \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!96 \)
|
| $73$ |
\( T^{6} + \cdots + 49\!\cdots\!25 \)
|
| $79$ |
\( T^{6} + \cdots + 26\!\cdots\!29 \)
|
| $83$ |
\( (T^{3} - 2496 T^{2} + \cdots - 372278592)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 585649680040000 \)
|
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