Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3283,2,Mod(1,3283)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3283, 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("3283.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3283 = 7^{2} \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3283.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: | \(26.2148869836\) |
| 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: |
\( x^{7} - x^{6} - 12x^{5} + 9x^{4} + 43x^{3} - 17x^{2} - 44x - 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 469) |
| 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
\( x^{7} - x^{6} - 12x^{5} + 9x^{4} + 43x^{3} - 17x^{2} - 44x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 12\nu^{4} + \nu^{3} + 36\nu^{2} - 5\nu - 13 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 10\nu^{4} - 15\nu^{3} - 28\nu^{2} + 21\nu + 19 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 10\nu^{4} - 21\nu^{3} + 20\nu^{2} + 47\nu + 13 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 10\nu^{4} + 19\nu^{3} + 28\nu^{2} - 41\nu - 23 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 6\beta_{2} + \beta _1 + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{6} + \beta_{5} + 10\beta_{4} + 2\beta_{2} + 28\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 23\beta_{6} + 12\beta_{5} + 11\beta_{4} - 20\beta_{3} + 36\beta_{2} + 12\beta _1 + 120 \)
|
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 |
|
−2.44888 | 2.86037 | 3.99700 | −1.91383 | −7.00469 | 0 | −4.89041 | 5.18171 | 4.68672 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.98532 | −2.45084 | 1.94149 | −1.22508 | 4.86570 | 0 | 0.116156 | 3.00662 | 2.43217 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.735445 | −2.59968 | −1.45912 | −3.13354 | 1.91192 | 0 | 2.54399 | 3.75833 | 2.30455 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.317096 | 0.986726 | −1.89945 | −1.14946 | −0.312887 | 0 | 1.23650 | −2.02637 | 0.364490 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.77723 | −3.30721 | 1.15856 | 3.03353 | −5.87769 | 0 | −1.49543 | 7.93764 | 5.39129 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.06166 | −0.616280 | 2.25044 | −3.56218 | −1.27056 | 0 | 0.516316 | −2.62020 | −7.34400 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.64784 | −0.873086 | 5.01108 | 1.95056 | −2.31180 | 0 | 7.97287 | −2.23772 | 5.16478 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(67\) | \( +1 \) |
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 | 3283.2.a.t | 7 | |
| 7.b | odd | 2 | 1 | 469.2.a.h | ✓ | 7 | |
| 21.c | even | 2 | 1 | 4221.2.a.v | 7 | ||
| 28.d | even | 2 | 1 | 7504.2.a.bp | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 469.2.a.h | ✓ | 7 | 7.b | odd | 2 | 1 | |
| 3283.2.a.t | 7 | 1.a | even | 1 | 1 | trivial | |
| 4221.2.a.v | 7 | 21.c | even | 2 | 1 | ||
| 7504.2.a.bp | 7 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3283))\):
|
\( T_{2}^{7} - T_{2}^{6} - 12T_{2}^{5} + 9T_{2}^{4} + 43T_{2}^{3} - 17T_{2}^{2} - 44T_{2} - 11 \)
|
|
\( T_{3}^{7} + 6T_{3}^{6} + T_{3}^{5} - 51T_{3}^{4} - 85T_{3}^{3} + 12T_{3}^{2} + 80T_{3} + 32 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} + \cdots - 11 \)
|
| $3$ |
\( T^{7} + 6 T^{6} + \cdots + 32 \)
|
| $5$ |
\( T^{7} + 6 T^{6} + \cdots + 178 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} - 8 T^{6} + \cdots - 16 \)
|
| $13$ |
\( T^{7} - 8 T^{6} + \cdots + 394 \)
|
| $17$ |
\( T^{7} - 7 T^{6} + \cdots - 7232 \)
|
| $19$ |
\( T^{7} - 11 T^{6} + \cdots - 400 \)
|
| $23$ |
\( T^{7} - 19 T^{6} + \cdots - 27008 \)
|
| $29$ |
\( T^{7} - 3 T^{6} + \cdots - 116 \)
|
| $31$ |
\( T^{7} + 18 T^{6} + \cdots - 49424 \)
|
| $37$ |
\( T^{7} - 20 T^{6} + \cdots - 69416 \)
|
| $41$ |
\( T^{7} - 20 T^{6} + \cdots + 3222 \)
|
| $43$ |
\( T^{7} + 11 T^{6} + \cdots - 619568 \)
|
| $47$ |
\( T^{7} + 24 T^{6} + \cdots + 35244 \)
|
| $53$ |
\( T^{7} - 36 T^{6} + \cdots - 68144 \)
|
| $59$ |
\( T^{7} + 26 T^{6} + \cdots - 381116 \)
|
| $61$ |
\( T^{7} + 8 T^{6} + \cdots - 716 \)
|
| $67$ |
\( (T + 1)^{7} \)
|
| $71$ |
\( T^{7} - 16 T^{6} + \cdots + 320624 \)
|
| $73$ |
\( T^{7} - 37 T^{6} + \cdots - 172304 \)
|
| $79$ |
\( T^{7} + 30 T^{6} + \cdots + 150336 \)
|
| $83$ |
\( T^{7} + 6 T^{6} + \cdots + 1504 \)
|
| $89$ |
\( T^{7} + 18 T^{6} + \cdots - 29830208 \)
|
| $97$ |
\( T^{7} + T^{6} + \cdots - 2564 \)
|
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