Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3325,2,Mod(1,3325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3325, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3325.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3325 = 5^{2} \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3325.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: | \(26.5502586721\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 12x^{6} + 24x^{5} + 36x^{4} - 70x^{3} - 20x^{2} + 32x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 665) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 12x^{6} + 24x^{5} + 36x^{4} - 70x^{3} - 20x^{2} + 32x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{4} + 27\nu^{2} + 4\nu - 5 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{4} + 31\nu^{2} + 4\nu - 17 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{4} + 4\nu^{3} + 27\nu^{2} - 20\nu - 1 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 11\nu^{5} + 11\nu^{4} + 27\nu^{3} - 23\nu^{2} - 5\nu - 3 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 13\nu^{5} + 2\nu^{4} - 47\nu^{3} - 16\nu^{2} + 43\nu + 10 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 13\nu^{5} + 13\nu^{4} + 47\nu^{3} - 39\nu^{2} - 43\nu + 7 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{2} + 6\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 7\beta_{3} - 6\beta_{2} - \beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + 2\beta_{5} + 10\beta_{4} - \beta_{3} - 8\beta_{2} + 40\beta _1 - 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{7} + 11\beta_{6} + 50\beta_{3} - 35\beta_{2} - 15\beta _1 + 122 \)
|
| \(\nu^{7}\) | \(=\) |
\( -11\beta_{7} + 11\beta_{6} + 26\beta_{5} + 83\beta_{4} - 15\beta_{3} - 53\beta_{2} + 279\beta _1 - 98 \)
|
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 |
|
−2.66888 | −0.321744 | 5.12292 | 0 | 0.858696 | −1.00000 | −8.33471 | −2.89648 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.22346 | 2.66100 | 2.94379 | 0 | −5.91664 | −1.00000 | −2.09848 | 4.08093 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.84638 | −1.55242 | 1.40913 | 0 | 2.86637 | −1.00000 | 1.09097 | −0.589984 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.777584 | −2.65879 | −1.39536 | 0 | 2.06743 | −1.00000 | 2.64018 | 4.06915 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0903326 | 1.28544 | −1.99184 | 0 | 0.116117 | −1.00000 | −0.360593 | −1.34766 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.730875 | −0.878233 | −1.46582 | 0 | −0.641878 | −1.00000 | −2.53308 | −2.22871 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.92614 | 3.21872 | 1.71000 | 0 | 6.19970 | −1.00000 | −0.558573 | 7.36017 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.76897 | 1.24603 | 5.66718 | 0 | 3.45021 | −1.00000 | 10.1543 | −1.44742 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 3325.2.a.y | 8 | |
| 5.b | even | 2 | 1 | 665.2.a.k | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 5985.2.a.bq | 8 | ||
| 35.c | odd | 2 | 1 | 4655.2.a.bi | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 665.2.a.k | ✓ | 8 | 5.b | even | 2 | 1 | |
| 3325.2.a.y | 8 | 1.a | even | 1 | 1 | trivial | |
| 4655.2.a.bi | 8 | 35.c | odd | 2 | 1 | ||
| 5985.2.a.bq | 8 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3325))\):
|
\( T_{2}^{8} + 2T_{2}^{7} - 12T_{2}^{6} - 24T_{2}^{5} + 36T_{2}^{4} + 70T_{2}^{3} - 20T_{2}^{2} - 32T_{2} + 3 \)
|
|
\( T_{3}^{8} - 3T_{3}^{7} - 11T_{3}^{6} + 31T_{3}^{5} + 32T_{3}^{4} - 76T_{3}^{3} - 32T_{3}^{2} + 48T_{3} + 16 \)
|
|
\( T_{11}^{8} - 16T_{11}^{7} + 64T_{11}^{6} + 156T_{11}^{5} - 1392T_{11}^{4} + 1408T_{11}^{3} + 3456T_{11}^{2} - 3584T_{11} - 3072 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 3 \)
|
| $3$ |
\( T^{8} - 3 T^{7} + \cdots + 16 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T + 1)^{8} \)
|
| $11$ |
\( T^{8} - 16 T^{7} + \cdots - 3072 \)
|
| $13$ |
\( T^{8} + 2 T^{7} + \cdots - 6912 \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots + 171936 \)
|
| $19$ |
\( (T - 1)^{8} \)
|
| $23$ |
\( T^{8} + 7 T^{7} + \cdots - 1294848 \)
|
| $29$ |
\( T^{8} - 6 T^{7} + \cdots + 35712 \)
|
| $31$ |
\( T^{8} - 6 T^{7} + \cdots - 49152 \)
|
| $37$ |
\( T^{8} + 3 T^{7} + \cdots + 5554096 \)
|
| $41$ |
\( T^{8} + 7 T^{7} + \cdots - 207456 \)
|
| $43$ |
\( T^{8} + 23 T^{7} + \cdots - 72704 \)
|
| $47$ |
\( T^{8} - 144 T^{6} + \cdots - 98304 \)
|
| $53$ |
\( T^{8} - 11 T^{7} + \cdots - 92016 \)
|
| $59$ |
\( T^{8} - 4 T^{7} + \cdots + 13154304 \)
|
| $61$ |
\( T^{8} - 18 T^{7} + \cdots + 445312 \)
|
| $67$ |
\( T^{8} + 4 T^{7} + \cdots - 5514688 \)
|
| $71$ |
\( T^{8} - 13 T^{7} + \cdots - 33408 \)
|
| $73$ |
\( T^{8} - T^{7} + \cdots - 32 \)
|
| $79$ |
\( T^{8} - 10 T^{7} + \cdots - 1986048 \)
|
| $83$ |
\( T^{8} - 20 T^{7} + \cdots - 13879296 \)
|
| $89$ |
\( T^{8} + 2 T^{7} + \cdots + 17318016 \)
|
| $97$ |
\( T^{8} - 42 T^{7} + \cdots + 51008 \)
|
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