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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 11\nu^{4} + 2\nu^{3} + 28\nu^{2} - 10\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} + \nu^{5} - 10\nu^{4} - 8\nu^{3} + 22\nu^{2} + 11\nu - 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - 11\nu^{5} + 2\nu^{4} + 29\nu^{3} - 11\nu^{2} - 8\nu + 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 11\nu^{5} + 2\nu^{4} + 30\nu^{3} - 11\nu^{2} - 14\nu + 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + \nu^{6} + 11\nu^{5} - 12\nu^{4} - 28\nu^{3} + 32\nu^{2} + 4\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{3} + 7\beta_{2} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + 9\beta_{6} - 10\beta_{5} + \beta_{4} - \beta_{2} + 39\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{7} + 9\beta_{6} + 2\beta_{5} - 10\beta_{3} + 49\beta_{2} - 2\beta _1 + 156 \)
|
| \(\nu^{7}\) | \(=\) |
\( -13\beta_{7} + 68\beta_{6} - 80\beta_{5} + 11\beta_{4} + 2\beta_{3} - 14\beta_{2} + 263\beta _1 - 17 \)
|
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 |
|
−2.59587 | 0 | 4.73855 | 0 | 0 | −3.93826 | −7.10893 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.57789 | 0 | 4.64553 | 0 | 0 | 4.69867 | −6.81989 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.64893 | 0 | 0.718971 | 0 | 0 | 1.44112 | 2.11233 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.510732 | 0 | −1.73915 | 0 | 0 | −4.82343 | 1.90970 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.135002 | 0 | −1.98177 | 0 | 0 | 1.12340 | −0.537546 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.485464 | 0 | −1.76432 | 0 | 0 | 1.94912 | −1.82745 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.98486 | 0 | 1.93969 | 0 | 0 | 2.16633 | −0.119715 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.72810 | 0 | 5.44251 | 0 | 0 | −4.61695 | 9.39150 | 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.by | 8 | |
| 3.b | odd | 2 | 1 | 2175.2.a.bd | yes | 8 | |
| 5.b | even | 2 | 1 | 6525.2.a.bz | 8 | ||
| 15.d | odd | 2 | 1 | 2175.2.a.bc | ✓ | 8 | |
| 15.e | even | 4 | 2 | 2175.2.c.p | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2175.2.a.bc | ✓ | 8 | 15.d | odd | 2 | 1 | |
| 2175.2.a.bd | yes | 8 | 3.b | odd | 2 | 1 | |
| 2175.2.c.p | 16 | 15.e | even | 4 | 2 | ||
| 6525.2.a.by | 8 | 1.a | even | 1 | 1 | trivial | |
| 6525.2.a.bz | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):
|
\( T_{2}^{8} + 2T_{2}^{7} - 12T_{2}^{6} - 23T_{2}^{5} + 36T_{2}^{4} + 62T_{2}^{3} - 15T_{2}^{2} - 14T_{2} + 2 \)
|
|
\( T_{7}^{8} + 2T_{7}^{7} - 45T_{7}^{6} - 44T_{7}^{5} + 667T_{7}^{4} - 270T_{7}^{3} - 3428T_{7}^{2} + 5898T_{7} - 2817 \)
|
|
\( T_{11}^{8} + 6T_{11}^{7} - 36T_{11}^{6} - 238T_{11}^{5} + 265T_{11}^{4} + 2382T_{11}^{3} - 21T_{11}^{2} - 6052T_{11} - 1140 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 2 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots - 2817 \)
|
| $11$ |
\( T^{8} + 6 T^{7} + \cdots - 1140 \)
|
| $13$ |
\( T^{8} + 6 T^{7} + \cdots - 10503 \)
|
| $17$ |
\( T^{8} + 12 T^{7} + \cdots + 18954 \)
|
| $19$ |
\( T^{8} - 99 T^{6} + \cdots + 205760 \)
|
| $23$ |
\( T^{8} + 14 T^{7} + \cdots - 14976 \)
|
| $29$ |
\( (T + 1)^{8} \)
|
| $31$ |
\( T^{8} - 8 T^{7} + \cdots + 1225792 \)
|
| $37$ |
\( T^{8} + 4 T^{7} + \cdots + 20929536 \)
|
| $41$ |
\( T^{8} + 2 T^{7} + \cdots - 2160 \)
|
| $43$ |
\( T^{8} + 2 T^{7} + \cdots + 486720 \)
|
| $47$ |
\( T^{8} + 12 T^{7} + \cdots + 2390796 \)
|
| $53$ |
\( T^{8} + 4 T^{7} + \cdots + 10368 \)
|
| $59$ |
\( T^{8} + 18 T^{7} + \cdots + 10368 \)
|
| $61$ |
\( T^{8} - 12 T^{7} + \cdots + 386112 \)
|
| $67$ |
\( T^{8} + 26 T^{7} + \cdots - 48194129 \)
|
| $71$ |
\( T^{8} + 24 T^{7} + \cdots - 16256 \)
|
| $73$ |
\( T^{8} - 14 T^{7} + \cdots + 19650816 \)
|
| $79$ |
\( T^{8} - 10 T^{7} + \cdots + 2032128 \)
|
| $83$ |
\( T^{8} + 40 T^{7} + \cdots - 144320256 \)
|
| $89$ |
\( T^{8} + 34 T^{7} + \cdots - 12296640 \)
|
| $97$ |
\( T^{8} + 30 T^{7} + \cdots - 9904960 \)
|
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