Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,16,Mod(1,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.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: | \(9.98854535699\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 7385x^{2} - 167250x - 586980 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3\cdot 5\cdot 7 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 7385x^{2} - 167250x - 586980 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 7\nu^{3} - 101\nu^{2} - 51090\nu - 556746 ) / 1138 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 113\nu^{3} - 655\nu^{2} - 645910\nu - 12683370 ) / 3414 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -43\nu^{3} + 6473\nu^{2} + 112250\nu - 18093066 ) / 3414 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{3} + 18\beta_{2} - 103\beta _1 + 582 ) / 1120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 55\beta_{3} + 62\beta_{2} - 221\beta _1 + 413698 ) / 112 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -349\beta_{3} + 3508\beta_{2} - 15039\beta _1 + 3825446 ) / 28 \)
|
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 |
|
−273.692 | 4148.79 | 42139.3 | −142812. | −1.13549e6 | −823543. | −2.56484e6 | 2.86353e6 | 3.90866e7 | ||||||||||||||||||||||||||||||
| 1.2 | −64.9617 | −4317.07 | −28548.0 | −78958.0 | 280444. | −823543. | 3.98319e6 | 4.28816e6 | 5.12925e6 | |||||||||||||||||||||||||||||||
| 1.3 | 92.0462 | 5331.04 | −24295.5 | 305120. | 490702. | −823543. | −5.25248e6 | 1.40711e7 | 2.80852e7 | |||||||||||||||||||||||||||||||
| 1.4 | 339.607 | 3391.24 | 82565.2 | −75579.8 | 1.15169e6 | −823543. | 1.69115e7 | −2.84842e6 | −2.56674e7 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
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 | 7.16.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 63.16.a.e | 4 | ||
| 4.b | odd | 2 | 1 | 112.16.a.f | 4 | ||
| 7.b | odd | 2 | 1 | 49.16.a.d | 4 | ||
| 7.c | even | 3 | 2 | 49.16.c.f | 8 | ||
| 7.d | odd | 6 | 2 | 49.16.c.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.16.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 49.16.a.d | 4 | 7.b | odd | 2 | 1 | ||
| 49.16.c.f | 8 | 7.c | even | 3 | 2 | ||
| 49.16.c.g | 8 | 7.d | odd | 6 | 2 | ||
| 63.16.a.e | 4 | 3.b | odd | 2 | 1 | ||
| 112.16.a.f | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 93T_{2}^{3} - 97142T_{2}^{2} + 2911584T_{2} + 555779200 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 93 T^{3} + \cdots + 555779200 \)
|
| $3$ |
\( T^{4} + \cdots - 323802472565376 \)
|
| $5$ |
\( T^{4} + \cdots - 26\!\cdots\!00 \)
|
| $7$ |
\( (T + 823543)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 15\!\cdots\!44 \)
|
| $13$ |
\( T^{4} + \cdots + 30\!\cdots\!20 \)
|
| $17$ |
\( T^{4} + \cdots + 13\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 46\!\cdots\!56 \)
|
| $29$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 40\!\cdots\!92 \)
|
| $37$ |
\( T^{4} + \cdots - 86\!\cdots\!76 \)
|
| $41$ |
\( T^{4} + \cdots - 75\!\cdots\!36 \)
|
| $43$ |
\( T^{4} + \cdots + 59\!\cdots\!60 \)
|
| $47$ |
\( T^{4} + \cdots + 32\!\cdots\!08 \)
|
| $53$ |
\( T^{4} + \cdots - 29\!\cdots\!32 \)
|
| $59$ |
\( T^{4} + \cdots - 68\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 53\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots - 21\!\cdots\!68 \)
|
| $71$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 16\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots + 92\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 15\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots - 24\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 10\!\cdots\!68 \)
|
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