Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3315,2,Mod(1,3315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("3315.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3315 = 3 \cdot 5 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3315.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: | \(26.4704082701\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 9x^{5} + 7x^{4} + 20x^{3} - 11x^{2} - 8x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 9x^{5} + 7x^{4} + 20x^{3} - 11x^{2} - 8x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 9\nu^{4} + 7\nu^{3} + 19\nu^{2} - 11\nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 9\nu^{4} + 7\nu^{3} + 20\nu^{2} - 11\nu - 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{6} - 3\nu^{5} - 17\nu^{4} + 22\nu^{3} + 34\nu^{2} - 36\nu - 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( 3\nu^{6} - 3\nu^{5} - 26\nu^{4} + 21\nu^{3} + 52\nu^{2} - 35\nu - 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( 3\nu^{6} - 4\nu^{5} - 26\nu^{4} + 30\nu^{3} + 53\nu^{2} - 52\nu - 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{4} - \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} - 8\beta_{2} + 2\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{6} + \beta_{5} - 9\beta_{4} + \beta_{3} - 10\beta_{2} + 28\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 10\beta_{5} - 2\beta_{4} + 27\beta_{3} - 55\beta_{2} + 22\beta _1 + 86 \)
|
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 |
|
−2.59918 | −1.00000 | 4.75574 | 1.00000 | 2.59918 | 3.12665 | −7.16266 | 1.00000 | −2.59918 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.67798 | −1.00000 | 0.815617 | 1.00000 | 1.67798 | 0.0210250 | 1.98737 | 1.00000 | −1.67798 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.13883 | −1.00000 | −0.703059 | 1.00000 | 1.13883 | −2.88384 | 3.07833 | 1.00000 | −1.13883 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.191596 | −1.00000 | −1.96329 | 1.00000 | −0.191596 | −1.52789 | −0.759350 | 1.00000 | 0.191596 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.278179 | −1.00000 | −1.92262 | 1.00000 | −0.278179 | 4.49505 | −1.09119 | 1.00000 | 0.278179 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.63304 | −1.00000 | 0.666817 | 1.00000 | −1.63304 | −0.579281 | −2.17714 | 1.00000 | 1.63304 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.31318 | −1.00000 | 3.35080 | 1.00000 | −2.31318 | −2.65172 | 3.12463 | 1.00000 | 2.31318 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
| \(17\) | \(-1\) |
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 | 3315.2.a.r | ✓ | 7 |
| 3.b | odd | 2 | 1 | 9945.2.a.bc | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3315.2.a.r | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 9945.2.a.bc | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3315))\):
|
\( T_{2}^{7} + T_{2}^{6} - 9T_{2}^{5} - 7T_{2}^{4} + 20T_{2}^{3} + 11T_{2}^{2} - 8T_{2} + 1 \)
|
|
\( T_{7}^{7} - 24T_{7}^{5} - 25T_{7}^{4} + 131T_{7}^{3} + 241T_{7}^{2} + 90T_{7} - 2 \)
|
|
\( T_{23}^{7} - 113T_{23}^{5} - 91T_{23}^{4} + 3608T_{23}^{3} + 4548T_{23}^{2} - 22112T_{23} + 5440 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} - 9 T^{5} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( T^{7} - 24 T^{5} + \cdots - 2 \)
|
| $11$ |
\( T^{7} + 10 T^{6} + \cdots + 68 \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( (T - 1)^{7} \)
|
| $19$ |
\( T^{7} + 2 T^{6} + \cdots - 2762 \)
|
| $23$ |
\( T^{7} - 113 T^{5} + \cdots + 5440 \)
|
| $29$ |
\( T^{7} + 25 T^{6} + \cdots + 37748 \)
|
| $31$ |
\( T^{7} - 5 T^{6} + \cdots - 78160 \)
|
| $37$ |
\( T^{7} + 2 T^{6} + \cdots - 208268 \)
|
| $41$ |
\( T^{7} + 38 T^{6} + \cdots + 25400 \)
|
| $43$ |
\( T^{7} + T^{6} + \cdots - 13976 \)
|
| $47$ |
\( T^{7} - 8 T^{6} + \cdots + 42016 \)
|
| $53$ |
\( T^{7} + 13 T^{6} + \cdots - 112268 \)
|
| $59$ |
\( T^{7} + 13 T^{6} + \cdots - 448904 \)
|
| $61$ |
\( T^{7} + 25 T^{6} + \cdots - 189040 \)
|
| $67$ |
\( T^{7} - T^{6} + \cdots - 659632 \)
|
| $71$ |
\( T^{7} + 31 T^{6} + \cdots + 120128 \)
|
| $73$ |
\( T^{7} + 21 T^{6} + \cdots + 1040308 \)
|
| $79$ |
\( T^{7} - 23 T^{6} + \cdots - 1451848 \)
|
| $83$ |
\( T^{7} - 22 T^{6} + \cdots - 2162624 \)
|
| $89$ |
\( T^{7} + 15 T^{6} + \cdots - 18112 \)
|
| $97$ |
\( T^{7} + 14 T^{6} + \cdots + 699376 \)
|
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