Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,8,Mod(1,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.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: | \(113.395764251\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.115512.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 70x - 194 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 33) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 70x - 194 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 6\nu - 45 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 6\nu - 92 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 47 \)
|
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 |
|
−17.9120 | −27.0000 | 192.840 | 84.6717 | 483.624 | −1679.06 | −1161.42 | 729.000 | −1516.64 | |||||||||||||||||||||||||||
| 1.2 | 2.40077 | −27.0000 | −122.236 | −505.769 | −64.8207 | −1401.07 | −600.759 | 729.000 | −1214.23 | ||||||||||||||||||||||||||||
| 1.3 | 6.51124 | −27.0000 | −85.6037 | −22.9029 | −175.804 | 1466.13 | −1390.83 | 729.000 | −149.126 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
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 | 363.8.a.e | 3 | |
| 11.b | odd | 2 | 1 | 33.8.a.d | ✓ | 3 | |
| 33.d | even | 2 | 1 | 99.8.a.e | 3 | ||
| 44.c | even | 2 | 1 | 528.8.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.8.a.d | ✓ | 3 | 11.b | odd | 2 | 1 | |
| 99.8.a.e | 3 | 33.d | even | 2 | 1 | ||
| 363.8.a.e | 3 | 1.a | even | 1 | 1 | trivial | |
| 528.8.a.o | 3 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 9T_{2}^{2} - 144T_{2} + 280 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(363))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 9 T^{2} + \cdots + 280 \)
|
| $3$ |
\( (T + 27)^{3} \)
|
| $5$ |
\( T^{3} + 444 T^{2} + \cdots - 980800 \)
|
| $7$ |
\( T^{3} + \cdots - 3449053112 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 665759180384 \)
|
| $17$ |
\( T^{3} + \cdots + 11730861043168 \)
|
| $19$ |
\( T^{3} + \cdots + 5608943166816 \)
|
| $23$ |
\( T^{3} + \cdots + 200462606267008 \)
|
| $29$ |
\( T^{3} + \cdots - 61407931779072 \)
|
| $31$ |
\( T^{3} + \cdots + 19\!\cdots\!96 \)
|
| $37$ |
\( T^{3} + \cdots + 11\!\cdots\!64 \)
|
| $41$ |
\( T^{3} + \cdots + 12\!\cdots\!52 \)
|
| $43$ |
\( T^{3} + \cdots + 20\!\cdots\!12 \)
|
| $47$ |
\( T^{3} + \cdots + 94\!\cdots\!80 \)
|
| $53$ |
\( T^{3} + \cdots + 29\!\cdots\!44 \)
|
| $59$ |
\( T^{3} + \cdots + 38\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 45\!\cdots\!60 \)
|
| $67$ |
\( T^{3} + \cdots - 13\!\cdots\!00 \)
|
| $71$ |
\( T^{3} + \cdots + 13\!\cdots\!84 \)
|
| $73$ |
\( T^{3} + \cdots - 26\!\cdots\!96 \)
|
| $79$ |
\( T^{3} + \cdots + 11\!\cdots\!12 \)
|
| $83$ |
\( T^{3} + \cdots - 81\!\cdots\!28 \)
|
| $89$ |
\( T^{3} + \cdots - 26\!\cdots\!48 \)
|
| $97$ |
\( T^{3} + \cdots + 24\!\cdots\!84 \)
|
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