Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,18,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, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| 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: | \(12.8255461141\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - 2x^{3} - 2290x^{2} - 4009x + 1104138 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2}\cdot 7 \) |
| 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
\( x^{4} - 2x^{3} - 2290x^{2} - 4009x + 1104138 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 6\nu^{3} - 212\nu^{2} - 6952\nu + 207760 ) / 11 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 20\nu^{3} - 670\nu^{2} - 22418\nu + 650154 ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 92\nu^{3} - 2950\nu^{2} - 109274\nu + 2842512 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( ( -5\beta_{3} + 41\beta_{2} - 60\beta _1 + 1986 ) / 4032 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 103\beta_{3} + 365\beta_{2} - 2796\beta _1 + 4619274 ) / 4032 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -359\beta_{3} + 10067\beta_{2} - 26820\beta _1 + 4316790 ) / 672 \)
|
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 |
|
−404.862 | −14458.1 | 32840.9 | 1.27737e6 | 5.85352e6 | −5.76480e6 | 3.97700e7 | 7.98958e7 | −5.17159e8 | ||||||||||||||||||||||||||||||
| 1.2 | −181.857 | 16535.0 | −98000.2 | −359290. | −3.00701e6 | −5.76480e6 | 4.16583e7 | 1.44267e8 | 6.53393e7 | |||||||||||||||||||||||||||||||
| 1.3 | 307.439 | −5612.83 | −36553.2 | 904274. | −1.72560e6 | −5.76480e6 | −5.15345e7 | −9.76363e7 | 2.78009e8 | |||||||||||||||||||||||||||||||
| 1.4 | 465.279 | 749.859 | 85412.5 | −1.54763e6 | 348894. | −5.76480e6 | −2.12444e7 | −1.28578e8 | −7.20081e8 | |||||||||||||||||||||||||||||||
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 | 7.18.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 63.18.a.b | 4 | ||
| 4.b | odd | 2 | 1 | 112.18.a.f | 4 | ||
| 7.b | odd | 2 | 1 | 49.18.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.18.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 49.18.a.c | 4 | 7.b | odd | 2 | 1 | ||
| 63.18.a.b | 4 | 3.b | odd | 2 | 1 | ||
| 112.18.a.f | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 186T_{2}^{3} - 236696T_{2}^{2} + 27034368T_{2} + 10531932160 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 10531932160 \)
|
| $3$ |
\( T^{4} + \cdots + 10\!\cdots\!04 \)
|
| $5$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
|
| $7$ |
\( (T + 5764801)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 75\!\cdots\!24 \)
|
| $13$ |
\( T^{4} + \cdots - 45\!\cdots\!40 \)
|
| $17$ |
\( T^{4} + \cdots + 17\!\cdots\!08 \)
|
| $19$ |
\( T^{4} + \cdots + 54\!\cdots\!20 \)
|
| $23$ |
\( T^{4} + \cdots + 95\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 50\!\cdots\!40 \)
|
| $31$ |
\( T^{4} + \cdots + 60\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 30\!\cdots\!96 \)
|
| $41$ |
\( T^{4} + \cdots - 19\!\cdots\!96 \)
|
| $43$ |
\( T^{4} + \cdots + 12\!\cdots\!20 \)
|
| $47$ |
\( T^{4} + \cdots - 81\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 70\!\cdots\!12 \)
|
| $59$ |
\( T^{4} + \cdots - 29\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 16\!\cdots\!96 \)
|
| $67$ |
\( T^{4} + \cdots - 15\!\cdots\!72 \)
|
| $71$ |
\( T^{4} + \cdots - 16\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots - 58\!\cdots\!76 \)
|
| $79$ |
\( T^{4} + \cdots + 13\!\cdots\!60 \)
|
| $83$ |
\( T^{4} + \cdots - 91\!\cdots\!92 \)
|
| $89$ |
\( T^{4} + \cdots + 70\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots - 51\!\cdots\!52 \)
|
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