Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1004,2,Mod(1,1004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1004, 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("1004.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1004 = 2^{2} \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1004.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: | \(8.01698036294\) |
| 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: |
\( x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 13\nu^{3} + 15\nu^{2} - 17\nu - 5 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} + 6\nu^{4} - 18\nu^{3} - 7\nu^{2} + 17\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 6\nu^{4} + 18\nu^{3} + 8\nu^{2} - 18\nu - 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{6} - 16\nu^{5} - 25\nu^{4} + 92\nu^{3} + 9\nu^{2} - 73\nu - 16 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{6} - 23\nu^{5} - 35\nu^{4} + 136\nu^{3} + 15\nu^{2} - 125\nu - 29 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 6\beta_{4} + 9\beta_{3} + 2\beta_{2} + 8\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{6} - 5\beta_{5} + 4\beta_{4} + 16\beta_{3} + 12\beta_{2} + 38\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} + 3\beta_{5} + 41\beta_{4} + 76\beta_{3} + 30\beta_{2} + 64\beta _1 + 60 \)
|
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 |
|
0 | −2.85375 | 0 | 0.454452 | 0 | −2.18653 | 0 | 5.14389 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.29157 | 0 | −1.78468 | 0 | 0.671050 | 0 | 2.25130 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.40474 | 0 | 2.59588 | 0 | 1.10443 | 0 | −1.02671 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.164919 | 0 | −1.38175 | 0 | 1.87004 | 0 | −2.97280 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.358013 | 0 | 2.25358 | 0 | −4.66949 | 0 | −2.87183 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.844838 | 0 | −2.47034 | 0 | −0.972158 | 0 | −2.28625 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.18229 | 0 | −1.66714 | 0 | −1.81734 | 0 | 1.76240 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(251\) | \(-1\) |
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 | 1004.2.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 9036.2.a.i | 7 | ||
| 4.b | odd | 2 | 1 | 4016.2.a.g | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1004.2.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 4016.2.a.g | 7 | 4.b | odd | 2 | 1 | ||
| 9036.2.a.i | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + 3T_{3}^{6} - 6T_{3}^{5} - 18T_{3}^{4} + 8T_{3}^{3} + 17T_{3}^{2} - 9T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1004))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} + 3 T^{6} + \cdots + 1 \)
|
| $5$ |
\( T^{7} + 2 T^{6} + \cdots - 27 \)
|
| $7$ |
\( T^{7} + 6 T^{6} + \cdots - 25 \)
|
| $11$ |
\( T^{7} + 5 T^{6} + \cdots - 72 \)
|
| $13$ |
\( T^{7} + T^{6} + \cdots + 15 \)
|
| $17$ |
\( T^{7} + 8 T^{6} + \cdots + 21 \)
|
| $19$ |
\( T^{7} + 15 T^{6} + \cdots + 1896 \)
|
| $23$ |
\( T^{7} + 5 T^{6} + \cdots + 3517 \)
|
| $29$ |
\( T^{7} - 71 T^{5} + \cdots + 5736 \)
|
| $31$ |
\( T^{7} + 21 T^{6} + \cdots - 433 \)
|
| $37$ |
\( T^{7} + T^{6} + \cdots - 82184 \)
|
| $41$ |
\( T^{7} + 10 T^{6} + \cdots + 183 \)
|
| $43$ |
\( T^{7} + 23 T^{6} + \cdots - 2232 \)
|
| $47$ |
\( T^{7} + 10 T^{6} + \cdots + 45376 \)
|
| $53$ |
\( T^{7} + T^{6} + \cdots + 24 \)
|
| $59$ |
\( T^{7} + 4 T^{6} + \cdots - 9864 \)
|
| $61$ |
\( T^{7} - 3 T^{6} + \cdots + 2504 \)
|
| $67$ |
\( T^{7} + 28 T^{6} + \cdots + 289511 \)
|
| $71$ |
\( T^{7} + 18 T^{6} + \cdots + 24056 \)
|
| $73$ |
\( T^{7} + 7 T^{6} + \cdots - 39339 \)
|
| $79$ |
\( T^{7} + 30 T^{6} + \cdots + 26831 \)
|
| $83$ |
\( T^{7} - 13 T^{6} + \cdots - 93888 \)
|
| $89$ |
\( T^{7} - 514 T^{5} + \cdots - 10211349 \)
|
| $97$ |
\( T^{7} + 2 T^{6} + \cdots + 3352 \)
|
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