Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7448,2,Mod(1,7448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("7448.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7448.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: | \(59.4725794254\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 15x^{6} - x^{5} + 66x^{4} + 4x^{3} - 76x^{2} + 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1064) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 15x^{6} - x^{5} + 66x^{4} + 4x^{3} - 76x^{2} + 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 11\nu^{4} + 17\nu^{3} + 32\nu^{2} - 24\nu - 16 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 13\nu^{5} - 16\nu^{4} + 43\nu^{3} + 64\nu^{2} - 12\nu - 32 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 13\nu^{5} + 5\nu^{4} - 46\nu^{3} - 34\nu^{2} + 28\nu + 18 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{7} + \nu^{6} - 28\nu^{5} - 17\nu^{4} + 109\nu^{3} + 68\nu^{2} - 84\nu - 24 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 15\nu^{5} - 14\nu^{4} + 61\nu^{3} + 50\nu^{2} - 42\nu - 20 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + 8\beta_{2} + \beta _1 + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{7} + 10\beta_{6} + \beta_{5} + 2\beta_{4} + 8\beta_{3} + \beta_{2} + 40\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -3\beta_{7} + 14\beta_{6} + 13\beta_{5} + 4\beta_{4} - 8\beta_{3} + 58\beta_{2} + 13\beta _1 + 169 \)
|
| \(\nu^{7}\) | \(=\) |
\( -84\beta_{7} + 89\beta_{6} + 16\beta_{5} + 26\beta_{4} + 53\beta_{3} + 19\beta_{2} + 277\beta _1 + 46 \)
|
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 | −2.59528 | 0 | 1.66621 | 0 | 0 | 0 | 3.73546 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.46993 | 0 | −4.03871 | 0 | 0 | 0 | 3.10055 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.34151 | 0 | 2.90869 | 0 | 0 | 0 | −1.20034 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.436576 | 0 | −0.299640 | 0 | 0 | 0 | −2.80940 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.684189 | 0 | −3.65096 | 0 | 0 | 0 | −2.53189 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.912296 | 0 | −1.66528 | 0 | 0 | 0 | −2.16772 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.39016 | 0 | 3.25179 | 0 | 0 | 0 | 2.71288 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.85665 | 0 | 0.827910 | 0 | 0 | 0 | 5.16044 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(1\) |
| \(19\) | \(-1\) |
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 | 7448.2.a.bq | 8 | |
| 7.b | odd | 2 | 1 | 7448.2.a.br | 8 | ||
| 7.c | even | 3 | 2 | 1064.2.q.n | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1064.2.q.n | ✓ | 16 | 7.c | even | 3 | 2 | |
| 7448.2.a.bq | 8 | 1.a | even | 1 | 1 | trivial | |
| 7448.2.a.br | 8 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):
|
\( T_{3}^{8} - 15T_{3}^{6} - T_{3}^{5} + 66T_{3}^{4} + 4T_{3}^{3} - 76T_{3}^{2} + 8T_{3} + 16 \)
|
|
\( T_{5}^{8} + T_{5}^{7} - 27T_{5}^{6} - 9T_{5}^{5} + 222T_{5}^{4} - 52T_{5}^{3} - 464T_{5}^{2} + 192T_{5} + 96 \)
|
|
\( T_{11}^{8} - 9T_{11}^{7} - 6T_{11}^{6} + 234T_{11}^{5} - 575T_{11}^{4} + 219T_{11}^{3} + 276T_{11}^{2} + 12T_{11} - 8 \)
|
|
\( T_{13}^{8} - 41T_{13}^{6} + T_{13}^{5} + 310T_{13}^{4} + 200T_{13}^{3} - 516T_{13}^{2} - 488T_{13} - 16 \)
|
|
\( T_{17}^{8} - 4T_{17}^{7} - 78T_{17}^{6} + 291T_{17}^{5} + 1921T_{17}^{4} - 6693T_{17}^{3} - 13866T_{17}^{2} + 49060T_{17} - 24632 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 15 T^{6} + \cdots + 16 \)
|
| $5$ |
\( T^{8} + T^{7} + \cdots + 96 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} - 9 T^{7} + \cdots - 8 \)
|
| $13$ |
\( T^{8} - 41 T^{6} + \cdots - 16 \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots - 24632 \)
|
| $19$ |
\( (T - 1)^{8} \)
|
| $23$ |
\( T^{8} - 25 T^{7} + \cdots + 8831 \)
|
| $29$ |
\( T^{8} - 6 T^{7} + \cdots + 6672 \)
|
| $31$ |
\( T^{8} - 100 T^{6} + \cdots + 9344 \)
|
| $37$ |
\( T^{8} - 13 T^{7} + \cdots - 536000 \)
|
| $41$ |
\( (T + 2)^{8} \)
|
| $43$ |
\( T^{8} - 17 T^{7} + \cdots - 51712 \)
|
| $47$ |
\( T^{8} + 24 T^{7} + \cdots + 13177 \)
|
| $53$ |
\( T^{8} - 2 T^{7} + \cdots - 1827792 \)
|
| $59$ |
\( T^{8} - 2 T^{7} + \cdots + 952576 \)
|
| $61$ |
\( T^{8} + 13 T^{7} + \cdots + 124920 \)
|
| $67$ |
\( T^{8} - 2 T^{7} + \cdots + 14848 \)
|
| $71$ |
\( T^{8} - 10 T^{7} + \cdots - 712576 \)
|
| $73$ |
\( T^{8} - 5 T^{7} + \cdots - 282879 \)
|
| $79$ |
\( T^{8} - 16 T^{7} + \cdots - 2067840 \)
|
| $83$ |
\( T^{8} + 43 T^{7} + \cdots + 95768 \)
|
| $89$ |
\( T^{8} - 8 T^{7} + \cdots - 512640 \)
|
| $97$ |
\( T^{8} + 12 T^{7} + \cdots + 580864 \)
|
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