Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [335,2,Mod(1,335)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(335, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("335.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 335 = 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 335.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: | \(2.67498846771\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - x^{6} - 8x^{5} + 4x^{4} + 16x^{3} - 3x^{2} - 4x + 1 \)
|
| 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
\( x^{7} - x^{6} - 8x^{5} + 4x^{4} + 16x^{3} - 3x^{2} - 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 8\nu^{4} - 4\nu^{3} - 15\nu^{2} + 2\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 8\nu^{4} + 4\nu^{3} + 16\nu^{2} - 3\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{6} + 2\nu^{5} + 15\nu^{4} - 6\nu^{3} - 27\nu^{2} - \nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} - \nu^{5} - 17\nu^{4} + \nu^{3} + 34\nu^{2} + 6\nu - 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\nu^{6} + \nu^{5} + 17\nu^{4} - 35\nu^{2} - 10\nu + 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} - \beta_{4} + 5\beta_{3} + 7\beta_{2} + 8\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{6} + 10\beta_{5} - \beta_{4} + 8\beta_{3} + 12\beta_{2} + 29\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 21\beta_{6} + 22\beta_{5} - 9\beta_{4} + 29\beta_{3} + 48\beta_{2} + 60\beta _1 + 38 \)
|
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 |
|
−2.51537 | 3.01721 | 4.32706 | −1.00000 | −7.58939 | 0.440983 | −5.85342 | 6.10357 | 2.51537 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.06332 | −0.323126 | 2.25729 | −1.00000 | 0.666713 | 3.61906 | −0.530878 | −2.89559 | 2.06332 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.328251 | −0.506742 | −1.89225 | −1.00000 | 0.166338 | 0.631798 | 1.27763 | −2.74321 | 0.328251 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.260755 | 2.62971 | −1.93201 | −1.00000 | −0.685711 | 3.37927 | 1.02529 | 3.91538 | 0.260755 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.21690 | −3.25732 | 2.91467 | −1.00000 | −7.22118 | 5.05532 | 2.02773 | 7.61016 | −2.21690 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.28820 | 2.22816 | 3.23584 | −1.00000 | 5.09848 | −4.76054 | 2.82784 | 1.96472 | −2.28820 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.66259 | 0.212104 | 5.08939 | −1.00000 | 0.564747 | 1.63411 | 8.22580 | −2.95501 | −2.66259 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(67\) | \(-1\) |
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 | 335.2.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 3015.2.a.m | 7 | ||
| 4.b | odd | 2 | 1 | 5360.2.a.bj | 7 | ||
| 5.b | even | 2 | 1 | 1675.2.a.i | 7 | ||
| 5.c | odd | 4 | 2 | 1675.2.c.h | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 335.2.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1675.2.a.i | 7 | 5.b | even | 2 | 1 | ||
| 1675.2.c.h | 14 | 5.c | odd | 4 | 2 | ||
| 3015.2.a.m | 7 | 3.b | odd | 2 | 1 | ||
| 5360.2.a.bj | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 2T_{2}^{6} - 12T_{2}^{5} + 21T_{2}^{4} + 42T_{2}^{3} - 52T_{2}^{2} - 39T_{2} - 6 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(335))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots - 6 \)
|
| $3$ |
\( T^{7} - 4 T^{6} + \cdots + 2 \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( T^{7} - 10 T^{6} + \cdots + 134 \)
|
| $11$ |
\( T^{7} - 6 T^{6} + \cdots + 3072 \)
|
| $13$ |
\( T^{7} - 4 T^{6} + \cdots - 778 \)
|
| $17$ |
\( T^{7} - 17 T^{6} + \cdots - 5952 \)
|
| $19$ |
\( T^{7} - 3 T^{6} + \cdots + 12343 \)
|
| $23$ |
\( T^{7} - 5 T^{6} + \cdots - 8256 \)
|
| $29$ |
\( T^{7} + T^{6} + \cdots - 32259 \)
|
| $31$ |
\( T^{7} - 184 T^{5} + \cdots + 33664 \)
|
| $37$ |
\( T^{7} - 15 T^{6} + \cdots + 295616 \)
|
| $41$ |
\( T^{7} + 20 T^{6} + \cdots - 581376 \)
|
| $43$ |
\( T^{7} - 20 T^{6} + \cdots - 126946 \)
|
| $47$ |
\( T^{7} - 19 T^{6} + \cdots - 856896 \)
|
| $53$ |
\( T^{7} - 18 T^{6} + \cdots - 17478 \)
|
| $59$ |
\( T^{7} + 39 T^{6} + \cdots - 61731 \)
|
| $61$ |
\( T^{7} + 6 T^{6} + \cdots + 526976 \)
|
| $67$ |
\( (T - 1)^{7} \)
|
| $71$ |
\( T^{7} + 28 T^{6} + \cdots + 48 \)
|
| $73$ |
\( T^{7} - 41 T^{6} + \cdots + 1210304 \)
|
| $79$ |
\( T^{7} - 24 T^{6} + \cdots - 88064 \)
|
| $83$ |
\( T^{7} + 38 T^{6} + \cdots + 97536 \)
|
| $89$ |
\( T^{7} + T^{6} + \cdots + 56673 \)
|
| $97$ |
\( T^{7} - 18 T^{6} + \cdots + 16948 \)
|
| show more | |
| show less |