Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7623,2,Mod(1,7623)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, 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("7623.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7623.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: | \(60.8699614608\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3829849.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 15x^{3} - 5x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 6x^{4} + 15x^{3} - 5x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} + 3\nu + 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 7\nu^{3} - 8\nu^{2} - 2\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 8\nu^{2} + 4\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{5} + 3\nu^{4} + 14\nu^{3} - 24\nu^{2} - 5\nu + 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( 3\nu^{5} - 4\nu^{4} - 19\nu^{3} + 32\nu^{2} - 3\nu - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - 2\beta_{3} + \beta _1 + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 5\beta_{3} + 6\beta_{2} + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -6\beta_{4} - 15\beta_{3} - 3\beta_{2} + 8\beta _1 + 25 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7\beta_{5} + 9\beta_{4} + 34\beta_{3} + 35\beta_{2} - 3 ) / 2 \)
|
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 |
|
−2.45784 | 0 | 4.04096 | 2.05098 | 0 | −1.00000 | −5.01636 | 0 | −5.04096 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.36768 | 0 | −0.129461 | 0.636509 | 0 | −1.00000 | 2.91241 | 0 | −0.870539 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.297484 | 0 | −1.91150 | −3.06404 | 0 | −1.00000 | 1.16361 | 0 | 0.911503 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.297484 | 0 | −1.91150 | 3.06404 | 0 | −1.00000 | −1.16361 | 0 | 0.911503 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.36768 | 0 | −0.129461 | −0.636509 | 0 | −1.00000 | −2.91241 | 0 | −0.870539 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.45784 | 0 | 4.04096 | −2.05098 | 0 | −1.00000 | 5.01636 | 0 | −5.04096 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7623.2.a.cq | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 7623.2.a.cq | ✓ | 6 |
| 11.b | odd | 2 | 1 | 7623.2.a.cr | yes | 6 | |
| 33.d | even | 2 | 1 | 7623.2.a.cr | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7623.2.a.cq | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 7623.2.a.cq | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 7623.2.a.cr | yes | 6 | 11.b | odd | 2 | 1 | |
| 7623.2.a.cr | yes | 6 | 33.d | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):
|
\( T_{2}^{6} - 8T_{2}^{4} + 12T_{2}^{2} - 1 \)
|
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\( T_{5}^{6} - 14T_{5}^{4} + 45T_{5}^{2} - 16 \)
|
|
\( T_{13}^{3} + 2T_{13}^{2} - 19T_{13} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 8 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 14 T^{4} + \cdots - 16 \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( (T^{3} + 2 T^{2} - 19 T - 4)^{2} \)
|
| $17$ |
\( T^{6} - 48 T^{4} + \cdots - 64 \)
|
| $19$ |
\( (T^{3} - 37 T + 16)^{2} \)
|
| $23$ |
\( T^{6} - 108 T^{4} + \cdots - 16384 \)
|
| $29$ |
\( T^{6} - 122 T^{4} + \cdots - 42436 \)
|
| $31$ |
\( (T^{3} - 2 T^{2} + \cdots + 304)^{2} \)
|
| $37$ |
\( (T^{3} - 4 T^{2} - 15 T + 34)^{2} \)
|
| $41$ |
\( T^{6} - 172 T^{4} + \cdots - 23104 \)
|
| $43$ |
\( (T^{3} + 10 T^{2} + \cdots - 32)^{2} \)
|
| $47$ |
\( T^{6} - 158 T^{4} + \cdots - 107584 \)
|
| $53$ |
\( T^{6} - 220 T^{4} + \cdots - 102400 \)
|
| $59$ |
\( T^{6} - 238 T^{4} + \cdots - 65536 \)
|
| $61$ |
\( (T^{3} + 8 T^{2} + \cdots - 272)^{2} \)
|
| $67$ |
\( (T^{3} - 12 T^{2} + \cdots + 68)^{2} \)
|
| $71$ |
\( T^{6} - 408 T^{4} + \cdots - 1638400 \)
|
| $73$ |
\( (T^{3} + 22 T^{2} + \cdots + 236)^{2} \)
|
| $79$ |
\( (T^{3} + 4 T^{2} + \cdots - 640)^{2} \)
|
| $83$ |
\( T^{6} - 236 T^{4} + \cdots - 6400 \)
|
| $89$ |
\( T^{6} - 104 T^{4} + \cdots - 30976 \)
|
| $97$ |
\( (T^{3} + 8 T^{2} + \cdots - 440)^{2} \)
|
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