Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [403,2,Mod(1,403)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, 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("403.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 403.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: | \(3.21797120146\) |
| 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: |
\( x^{7} - 2x^{6} - 9x^{5} + 12x^{4} + 22x^{3} - 18x^{2} - 13x - 1 \)
|
| 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
\( x^{7} - 2x^{6} - 9x^{5} + 12x^{4} + 22x^{3} - 18x^{2} - 13x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 9\nu^{3} + 2\nu^{2} + 15\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 9\nu^{3} + \nu^{2} + 16\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 9\nu^{4} - \nu^{3} - 17\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 10\nu^{4} - 4\nu^{3} - 21\nu^{2} + 7\nu + 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - \nu^{5} - 19\nu^{4} - 6\nu^{3} + 33\nu^{2} + 19\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 2\beta_{4} - 3\beta_{3} + 2\beta_{2} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + \beta_{5} + 5\beta_{4} - 13\beta_{3} + 10\beta_{2} + 14\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12\beta_{6} + \beta_{5} + 23\beta_{4} - 38\beta_{3} + 27\beta_{2} + 60\beta _1 + 17 \)
|
| \(\nu^{6}\) | \(=\) |
\( 38\beta_{6} + 10\beta_{5} + 65\beta_{4} - 135\beta_{3} + 98\beta_{2} + 158\beta _1 + 103 \)
|
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 |
|
−2.10645 | −2.17535 | 2.43713 | 2.39413 | 4.58226 | 1.34409 | −0.920801 | 1.73214 | −5.04311 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.07211 | 1.43412 | 2.29366 | 0.0880198 | −2.97167 | 0.184197 | −0.608494 | −0.943287 | −0.182387 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.299542 | 3.01774 | −1.91027 | 3.67569 | −0.903940 | −0.783156 | 1.17129 | 6.10678 | −1.10102 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.406064 | −0.564229 | −1.83511 | 2.08199 | −0.229113 | 4.31735 | −1.55730 | −2.68165 | 0.845422 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.39140 | 2.70914 | −0.0639949 | −0.456939 | 3.76951 | 1.81179 | −2.87185 | 4.33945 | −0.635787 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.09092 | −0.510463 | 2.37194 | 4.43975 | −1.06734 | −2.30124 | 0.777706 | −2.73943 | 9.28317 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.58972 | 1.08903 | 4.70665 | −1.22264 | 2.82028 | −0.573038 | 7.00945 | −1.81401 | −3.16629 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
| \(31\) | \(-1\) |
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 | 403.2.a.c | ✓ | 7 |
| 3.b | odd | 2 | 1 | 3627.2.a.n | 7 | ||
| 4.b | odd | 2 | 1 | 6448.2.a.ba | 7 | ||
| 13.b | even | 2 | 1 | 5239.2.a.h | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 403.2.a.c | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 3627.2.a.n | 7 | 3.b | odd | 2 | 1 | ||
| 5239.2.a.h | 7 | 13.b | even | 2 | 1 | ||
| 6448.2.a.ba | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 2T_{2}^{6} - 9T_{2}^{5} + 17T_{2}^{4} + 20T_{2}^{3} - 37T_{2}^{2} + T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 4 \)
|
| $3$ |
\( T^{7} - 5 T^{6} + \cdots + 8 \)
|
| $5$ |
\( T^{7} - 11 T^{6} + \cdots - 4 \)
|
| $7$ |
\( T^{7} - 4 T^{6} + \cdots + 2 \)
|
| $11$ |
\( T^{7} - 8 T^{6} + \cdots + 283 \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( T^{7} - 7 T^{6} + \cdots - 184 \)
|
| $19$ |
\( T^{7} - T^{6} + \cdots - 40 \)
|
| $23$ |
\( T^{7} - 6 T^{6} + \cdots - 3812 \)
|
| $29$ |
\( T^{7} + 2 T^{6} + \cdots - 3670 \)
|
| $31$ |
\( (T - 1)^{7} \)
|
| $37$ |
\( T^{7} - 28 T^{6} + \cdots + 77417 \)
|
| $41$ |
\( T^{7} - 3 T^{6} + \cdots - 4690 \)
|
| $43$ |
\( T^{7} + T^{6} + \cdots + 2536 \)
|
| $47$ |
\( T^{7} + T^{6} + \cdots + 39428 \)
|
| $53$ |
\( T^{7} - 29 T^{6} + \cdots - 270268 \)
|
| $59$ |
\( T^{7} - 3 T^{6} + \cdots - 250090 \)
|
| $61$ |
\( T^{7} - 5 T^{6} + \cdots - 54112 \)
|
| $67$ |
\( T^{7} + 32 T^{6} + \cdots - 1718 \)
|
| $71$ |
\( T^{7} - 5 T^{6} + \cdots - 662786 \)
|
| $73$ |
\( T^{7} - T^{6} + \cdots - 1339357 \)
|
| $79$ |
\( T^{7} + 15 T^{6} + \cdots - 158222 \)
|
| $83$ |
\( T^{7} - 17 T^{6} + \cdots + 24052 \)
|
| $89$ |
\( T^{7} - 26 T^{6} + \cdots + 1429 \)
|
| $97$ |
\( T^{7} - 11 T^{6} + \cdots + 357056 \)
|
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