Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [361,6,Mod(1,361)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("361.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 361.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: | \(57.8985589525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{177}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 19) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−3.15207 | 23.4562 | −22.0645 | −33.2397 | −73.9355 | 129.129 | 170.415 | 307.193 | 104.774 | ||||||||||||||||||||||||
| 1.2 | 10.1521 | −16.4562 | 71.0645 | −99.7603 | −167.064 | −57.1289 | 396.585 | 27.8066 | −1012.77 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \(-1\) |
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 | 361.6.a.d | 2 | |
| 19.b | odd | 2 | 1 | 19.6.a.c | ✓ | 2 | |
| 57.d | even | 2 | 1 | 171.6.a.f | 2 | ||
| 76.d | even | 2 | 1 | 304.6.a.g | 2 | ||
| 95.d | odd | 2 | 1 | 475.6.a.d | 2 | ||
| 133.c | even | 2 | 1 | 931.6.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.6.a.c | ✓ | 2 | 19.b | odd | 2 | 1 | |
| 171.6.a.f | 2 | 57.d | even | 2 | 1 | ||
| 304.6.a.g | 2 | 76.d | even | 2 | 1 | ||
| 361.6.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 475.6.a.d | 2 | 95.d | odd | 2 | 1 | ||
| 931.6.a.c | 2 | 133.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 7T_{2} - 32 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(361))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 7T - 32 \)
|
| $3$ |
\( T^{2} - 7T - 386 \)
|
| $5$ |
\( T^{2} + 133T + 3316 \)
|
| $7$ |
\( T^{2} - 72T - 7377 \)
|
| $11$ |
\( T^{2} + 705T + 116778 \)
|
| $13$ |
\( T^{2} - 1341 T + 412356 \)
|
| $17$ |
\( T^{2} - 2784 T + 1859607 \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( T^{2} + 2713 T - 1457816 \)
|
| $29$ |
\( T^{2} - 7775 T + 13694842 \)
|
| $31$ |
\( T^{2} + 7132 T - 21863072 \)
|
| $37$ |
\( T^{2} - 6248 T - 28020212 \)
|
| $41$ |
\( T^{2} - 4174 T - 52950128 \)
|
| $43$ |
\( T^{2} - 25357 T + 103337908 \)
|
| $47$ |
\( T^{2} - 11727 T + 34377048 \)
|
| $53$ |
\( T^{2} - 29133 T + 137515428 \)
|
| $59$ |
\( T^{2} + \cdots + 1040195802 \)
|
| $61$ |
\( T^{2} + 40939 T - 280177226 \)
|
| $67$ |
\( T^{2} + 19039 T - 80586308 \)
|
| $71$ |
\( T^{2} - 70236 T - 170310588 \)
|
| $73$ |
\( T^{2} - 67058 T + 555800113 \)
|
| $79$ |
\( T^{2} - 32850 T + 93134448 \)
|
| $83$ |
\( T^{2} - 71534 T + 666270472 \)
|
| $89$ |
\( T^{2} + \cdots - 2036009792 \)
|
| $97$ |
\( T^{2} - 62458 T + 832198864 \)
|
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