Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6525,2,Mod(1,6525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6525, 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("6525.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6525.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: | \(52.1023873189\) |
| 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: |
\( x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2175) |
| 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
\( x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{6} + \nu^{5} + 19\nu^{4} - 7\nu^{3} - 41\nu^{2} + 9\nu + 7 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{6} + \nu^{5} + 24\nu^{4} - 7\nu^{3} - 76\nu^{2} + 9\nu + 37 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{6} - 4\nu^{5} - 31\nu^{4} + 33\nu^{3} + 84\nu^{2} - 56\nu - 38 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{6} - \beta_{5} - 2\beta_{4} + 9\beta_{3} + 2\beta_{2} + 28\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{6} + 9\beta_{5} - 13\beta_{4} + \beta_{3} + 47\beta_{2} + \beta _1 + 85 \)
|
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 |
|
−2.66356 | 0 | 5.09453 | 0 | 0 | −0.0170416 | −8.24244 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.26695 | 0 | 3.13907 | 0 | 0 | 0.159887 | −2.58223 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.14889 | 0 | −0.680059 | 0 | 0 | 3.52165 | 3.07908 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.792771 | 0 | −1.37151 | 0 | 0 | −2.01830 | 2.67284 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.560139 | 0 | −1.68624 | 0 | 0 | 4.36313 | −2.06481 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.83397 | 0 | 1.36343 | 0 | 0 | −4.17416 | −1.16745 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.47806 | 0 | 4.14079 | 0 | 0 | −2.83516 | 5.30500 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( -1 \) |
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 | 6525.2.a.bu | 7 | |
| 3.b | odd | 2 | 1 | 2175.2.a.bb | yes | 7 | |
| 5.b | even | 2 | 1 | 6525.2.a.bx | 7 | ||
| 15.d | odd | 2 | 1 | 2175.2.a.ba | ✓ | 7 | |
| 15.e | even | 4 | 2 | 2175.2.c.o | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2175.2.a.ba | ✓ | 7 | 15.d | odd | 2 | 1 | |
| 2175.2.a.bb | yes | 7 | 3.b | odd | 2 | 1 | |
| 2175.2.c.o | 14 | 15.e | even | 4 | 2 | ||
| 6525.2.a.bu | 7 | 1.a | even | 1 | 1 | trivial | |
| 6525.2.a.bx | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):
|
\( T_{2}^{7} + 2T_{2}^{6} - 10T_{2}^{5} - 19T_{2}^{4} + 24T_{2}^{3} + 44T_{2}^{2} - 3T_{2} - 14 \)
|
|
\( T_{7}^{7} + T_{7}^{6} - 30T_{7}^{5} - 38T_{7}^{4} + 217T_{7}^{3} + 337T_{7}^{2} - 53T_{7} - 1 \)
|
|
\( T_{11}^{7} + 4T_{11}^{6} - 60T_{11}^{5} - 190T_{11}^{4} + 1105T_{11}^{3} + 2448T_{11}^{2} - 5733T_{11} - 10746 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots - 14 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} + T^{6} - 30 T^{5} + \cdots - 1 \)
|
| $11$ |
\( T^{7} + 4 T^{6} + \cdots - 10746 \)
|
| $13$ |
\( T^{7} + T^{6} + \cdots - 6803 \)
|
| $17$ |
\( T^{7} + 8 T^{6} + \cdots - 86134 \)
|
| $19$ |
\( T^{7} - 15 T^{6} + \cdots - 15680 \)
|
| $23$ |
\( T^{7} + 14 T^{6} + \cdots + 27008 \)
|
| $29$ |
\( (T - 1)^{7} \)
|
| $31$ |
\( T^{7} - 5 T^{6} + \cdots - 12096 \)
|
| $37$ |
\( T^{7} + 6 T^{6} + \cdots - 13696 \)
|
| $41$ |
\( T^{7} + 22 T^{6} + \cdots + 626632 \)
|
| $43$ |
\( T^{7} + 19 T^{6} + \cdots - 42304 \)
|
| $47$ |
\( T^{7} + 22 T^{6} + \cdots + 2842 \)
|
| $53$ |
\( T^{7} + 10 T^{6} + \cdots + 1229824 \)
|
| $59$ |
\( T^{7} + 6 T^{6} + \cdots + 87680 \)
|
| $61$ |
\( T^{7} - 23 T^{6} + \cdots + 51776 \)
|
| $67$ |
\( T^{7} + 13 T^{6} + \cdots + 37259 \)
|
| $71$ |
\( T^{7} + 26 T^{6} + \cdots + 34048 \)
|
| $73$ |
\( T^{7} + 24 T^{6} + \cdots + 1658752 \)
|
| $79$ |
\( T^{7} - 14 T^{6} + \cdots + 2560 \)
|
| $83$ |
\( T^{7} + 10 T^{6} + \cdots + 5248 \)
|
| $89$ |
\( T^{7} + 14 T^{6} + \cdots - 231800 \)
|
| $97$ |
\( T^{7} + 31 T^{6} + \cdots + 6565312 \)
|
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