Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [404,2,Mod(1,404)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(404, 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("404.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 404 = 2^{2} \cdot 101 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 404.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: | \(3.22595624166\) |
| 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: |
\( x^{7} - 2x^{6} - 17x^{5} + 36x^{4} + 64x^{3} - 148x^{2} + 11x + 58 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{7} - 2x^{6} - 17x^{5} + 36x^{4} + 64x^{3} - 148x^{2} + 11x + 58 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\nu^{6} - 2\nu^{5} + 28\nu^{4} + 13\nu^{3} - 89\nu^{2} + 18\nu + 34 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{6} + 3\nu^{5} - 42\nu^{4} - 19\nu^{3} + 134\nu^{2} - 31\nu - 52 \)
|
| \(\beta_{4}\) | \(=\) |
\( -10\nu^{6} - 9\nu^{5} + 143\nu^{4} + 53\nu^{3} - 475\nu^{2} + 116\nu + 201 \)
|
| \(\beta_{5}\) | \(=\) |
\( 11\nu^{6} + 11\nu^{5} - 155\nu^{4} - 71\nu^{3} + 502\nu^{2} - 103\nu - 206 \)
|
| \(\beta_{6}\) | \(=\) |
\( -20\nu^{6} - 19\nu^{5} + 284\nu^{4} + 118\nu^{3} - 933\nu^{2} + 206\nu + 394 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} + 8\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -12\beta_{6} - 13\beta_{5} + 12\beta_{4} + 13\beta_{3} + 8\beta_{2} + 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( 18\beta_{6} + 21\beta_{5} - 17\beta_{4} + 3\beta_{3} + 25\beta_{2} + 70\beta _1 - 43 \)
|
| \(\nu^{6}\) | \(=\) |
\( -135\beta_{6} - 152\beta_{5} + 134\beta_{4} + 141\beta_{3} + 55\beta_{2} - 9\beta _1 + 406 \)
|
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 | −3.39477 | 0 | −4.22882 | 0 | −2.00794 | 0 | 8.52447 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.39772 | 0 | 2.38721 | 0 | −4.58715 | 0 | 2.74907 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.553034 | 0 | −3.58964 | 0 | 3.89292 | 0 | −2.69415 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.09724 | 0 | 4.24890 | 0 | 2.57755 | 0 | −1.79608 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.35023 | 0 | 0.480650 | 0 | 0.469556 | 0 | −1.17687 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.94838 | 0 | −1.52030 | 0 | 4.92250 | 0 | 5.69295 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.94968 | 0 | 2.22200 | 0 | −3.26744 | 0 | 5.70061 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(101\) | \(1\) |
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 | 404.2.a.c | ✓ | 7 |
| 3.b | odd | 2 | 1 | 3636.2.a.j | 7 | ||
| 4.b | odd | 2 | 1 | 1616.2.a.l | 7 | ||
| 8.b | even | 2 | 1 | 6464.2.a.bc | 7 | ||
| 8.d | odd | 2 | 1 | 6464.2.a.be | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 404.2.a.c | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1616.2.a.l | 7 | 4.b | odd | 2 | 1 | ||
| 3636.2.a.j | 7 | 3.b | odd | 2 | 1 | ||
| 6464.2.a.bc | 7 | 8.b | even | 2 | 1 | ||
| 6464.2.a.be | 7 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 2T_{3}^{6} - 17T_{3}^{5} + 36T_{3}^{4} + 64T_{3}^{3} - 148T_{3}^{2} + 11T_{3} + 58 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(404))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 2 T^{6} + \cdots + 58 \)
|
| $5$ |
\( T^{7} - 31 T^{5} + \cdots + 250 \)
|
| $7$ |
\( T^{7} - 2 T^{6} + \cdots + 698 \)
|
| $11$ |
\( T^{7} - 4 T^{6} + \cdots + 3482 \)
|
| $13$ |
\( T^{7} - 4 T^{6} + \cdots + 58 \)
|
| $17$ |
\( T^{7} - 4 T^{6} + \cdots + 522 \)
|
| $19$ |
\( T^{7} + 4 T^{6} + \cdots - 48640 \)
|
| $23$ |
\( T^{7} - 8 T^{6} + \cdots + 19456 \)
|
| $29$ |
\( T^{7} - 6 T^{6} + \cdots + 67456 \)
|
| $31$ |
\( T^{7} - 4 T^{6} + \cdots - 3840 \)
|
| $37$ |
\( T^{7} - 32 T^{6} + \cdots + 84646 \)
|
| $41$ |
\( T^{7} + 14 T^{6} + \cdots - 1024 \)
|
| $43$ |
\( T^{7} + 24 T^{6} + \cdots + 5888 \)
|
| $47$ |
\( T^{7} + 4 T^{6} + \cdots + 11776 \)
|
| $53$ |
\( T^{7} - 6 T^{6} + \cdots + 70016 \)
|
| $59$ |
\( T^{7} + 30 T^{6} + \cdots - 1774 \)
|
| $61$ |
\( T^{7} - 10 T^{6} + \cdots + 161152 \)
|
| $67$ |
\( T^{7} + 2 T^{6} + \cdots - 61150 \)
|
| $71$ |
\( T^{7} - 396 T^{5} + \cdots + 1203968 \)
|
| $73$ |
\( T^{7} + 14 T^{6} + \cdots - 1811456 \)
|
| $79$ |
\( T^{7} - 24 T^{6} + \cdots + 148224 \)
|
| $83$ |
\( T^{7} + 24 T^{6} + \cdots + 9026 \)
|
| $89$ |
\( T^{7} + 6 T^{6} + \cdots + 5248 \)
|
| $97$ |
\( T^{7} + 8 T^{6} + \cdots - 72770 \)
|
| show more | |
| show less |