Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7254,2,Mod(1,7254)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7254, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("7254.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7254 = 2 \cdot 3^{2} \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7254.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: | \(57.9234816262\) |
| 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: |
\( x^{7} - 3x^{6} - 14x^{5} + 23x^{4} + 76x^{3} - 3x^{2} - 59x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{7} - 3x^{6} - 14x^{5} + 23x^{4} + 76x^{3} - 3x^{2} - 59x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 6\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 3\nu^{2} + 4\nu - 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 5\nu^{5} - 4\nu^{4} + 32\nu^{3} + 11\nu^{2} - 34\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + 5\nu^{5} + 5\nu^{4} - 35\nu^{3} - 15\nu^{2} + 40\nu - 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( 4\nu^{6} - 19\nu^{5} - 22\nu^{4} + 129\nu^{3} + 72\nu^{2} - 134\nu + 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 3\beta_{2} + 10\beta _1 + 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 10\beta_{3} + 13\beta_{2} + 32\beta _1 + 50 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 6\beta_{5} + 2\beta_{4} + 30\beta_{3} + 47\beta_{2} + 124\beta _1 + 156 \)
|
| \(\nu^{6}\) | \(=\) |
\( 5\beta_{6} + 34\beta_{5} + 15\beta_{4} + 115\beta_{3} + 180\beta_{2} + 440\beta _1 + 634 \)
|
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 |
|
1.00000 | 0 | 1.00000 | −3.36122 | 0 | −0.817634 | 1.00000 | 0 | −3.36122 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −2.77783 | 0 | 3.45739 | 1.00000 | 0 | −2.77783 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | −2.20664 | 0 | −1.10429 | 1.00000 | 0 | −2.20664 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | −0.353010 | 0 | −2.13633 | 1.00000 | 0 | −0.353010 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 1.24414 | 0 | 2.39401 | 1.00000 | 0 | 1.24414 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0 | 1.00000 | 1.35474 | 0 | 1.57887 | 1.00000 | 0 | 1.35474 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0 | 1.00000 | 3.09981 | 0 | −3.37201 | 1.00000 | 0 | 3.09981 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
| \(31\) | \( -1 \) |
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 | 7254.2.a.bq | yes | 7 |
| 3.b | odd | 2 | 1 | 7254.2.a.bp | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7254.2.a.bp | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 7254.2.a.bq | yes | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7254))\):
|
\( T_{5}^{7} + 3T_{5}^{6} - 14T_{5}^{5} - 39T_{5}^{4} + 50T_{5}^{3} + 103T_{5}^{2} - 79T_{5} - 38 \)
|
|
\( T_{7}^{7} - 19T_{7}^{5} - 3T_{7}^{4} + 97T_{7}^{3} + 37T_{7}^{2} - 132T_{7} - 85 \)
|
|
\( T_{11}^{7} + 7T_{11}^{6} - 20T_{11}^{5} - 219T_{11}^{4} - 264T_{11}^{3} + 1043T_{11}^{2} + 2785T_{11} + 1822 \)
|
|
\( T_{17}^{7} + 12T_{17}^{6} - 11T_{17}^{5} - 523T_{17}^{4} - 1347T_{17}^{3} + 1171T_{17}^{2} + 3816T_{17} - 1989 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + 3 T^{6} + \cdots - 38 \)
|
| $7$ |
\( T^{7} - 19 T^{5} + \cdots - 85 \)
|
| $11$ |
\( T^{7} + 7 T^{6} + \cdots + 1822 \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( T^{7} + 12 T^{6} + \cdots - 1989 \)
|
| $19$ |
\( T^{7} + T^{6} + \cdots - 2092 \)
|
| $23$ |
\( T^{7} + 8 T^{6} + \cdots - 101088 \)
|
| $29$ |
\( T^{7} + 19 T^{6} + \cdots - 34940 \)
|
| $31$ |
\( (T - 1)^{7} \)
|
| $37$ |
\( T^{7} + 3 T^{6} + \cdots + 4252 \)
|
| $41$ |
\( T^{7} + 22 T^{6} + \cdots - 78079 \)
|
| $43$ |
\( T^{7} + 3 T^{6} + \cdots + 182394 \)
|
| $47$ |
\( T^{7} - 106 T^{5} + \cdots + 4352 \)
|
| $53$ |
\( T^{7} + 8 T^{6} + \cdots + 3904 \)
|
| $59$ |
\( T^{7} + 9 T^{6} + \cdots + 9748 \)
|
| $61$ |
\( T^{7} + 13 T^{6} + \cdots - 152956 \)
|
| $67$ |
\( T^{7} + 25 T^{6} + \cdots + 352994 \)
|
| $71$ |
\( T^{7} + 12 T^{6} + \cdots - 32904 \)
|
| $73$ |
\( T^{7} + 12 T^{6} + \cdots - 2139003 \)
|
| $79$ |
\( T^{7} + 4 T^{6} + \cdots + 485671 \)
|
| $83$ |
\( T^{7} + 13 T^{6} + \cdots - 340 \)
|
| $89$ |
\( T^{7} + 36 T^{6} + \cdots - 245744 \)
|
| $97$ |
\( T^{7} + 8 T^{6} + \cdots - 104 \)
|
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