Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7569,2,Mod(1,7569)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7569, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("7569.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7569 = 3^{2} \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7569.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: | \(60.4387692899\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.8902000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 10x^{3} + 13x^{2} - 9x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2523) |
| 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
\( x^{6} - x^{5} - 9x^{4} + 10x^{3} + 13x^{2} - 9x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 6\nu^{2} - 3\nu + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 6\nu^{3} - 3\nu^{2} + \nu + 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + 5\nu^{2} + 4\nu - 7 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} + 15\nu^{3} - 17\nu + 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} - \beta_{2} + 6\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + 6\beta_{4} - 8\beta_{3} + 9\beta_{2} - 3\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{5} - 3\beta_{4} + 19\beta_{3} - 12\beta_{2} + 38\beta _1 - 34 \)
|
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 |
|
−2.78072 | 0 | 5.73239 | 0.924777 | 0 | 3.48365 | −10.3787 | 0 | −2.57154 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.14760 | 0 | −0.683006 | 0.723161 | 0 | −4.16166 | 3.07903 | 0 | −0.829902 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.0983010 | 0 | −1.99034 | −3.84813 | 0 | 3.11782 | 0.392254 | 0 | 0.378275 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.674113 | 0 | −1.54557 | 2.11882 | 0 | 3.99177 | −2.39012 | 0 | 1.42833 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.09152 | 0 | 2.37448 | −4.22395 | 0 | −1.68421 | 0.783225 | 0 | −8.83449 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.26098 | 0 | 3.11205 | −0.694681 | 0 | −1.74736 | 2.51433 | 0 | −1.57066 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 7569.2.a.bb | 6 | |
| 3.b | odd | 2 | 1 | 2523.2.a.k | ✓ | 6 | |
| 29.b | even | 2 | 1 | 7569.2.a.z | 6 | ||
| 87.d | odd | 2 | 1 | 2523.2.a.l | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2523.2.a.k | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 2523.2.a.l | yes | 6 | 87.d | odd | 2 | 1 | |
| 7569.2.a.z | 6 | 29.b | even | 2 | 1 | ||
| 7569.2.a.bb | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):
|
\( T_{2}^{6} - T_{2}^{5} - 9T_{2}^{4} + 10T_{2}^{3} + 13T_{2}^{2} - 9T_{2} - 1 \)
|
|
\( T_{5}^{6} + 5T_{5}^{5} - 7T_{5}^{4} - 36T_{5}^{3} + 36T_{5}^{2} + 16T_{5} - 16 \)
|
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\( T_{7}^{6} - 3T_{7}^{5} - 26T_{7}^{4} + 69T_{7}^{3} + 182T_{7}^{2} - 291T_{7} - 531 \)
|
|
\( T_{19}^{6} - T_{19}^{5} - 56T_{19}^{4} + 107T_{19}^{3} + 384T_{19}^{2} - 705T_{19} - 269 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} - 9 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 5 T^{5} + \cdots - 16 \)
|
| $7$ |
\( T^{6} - 3 T^{5} + \cdots - 531 \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots - 64 \)
|
| $13$ |
\( T^{6} - 8 T^{5} + \cdots + 261 \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 2304 \)
|
| $19$ |
\( T^{6} - T^{5} + \cdots - 269 \)
|
| $23$ |
\( T^{6} + 7 T^{5} + \cdots - 576 \)
|
| $29$ |
\( T^{6} \)
|
| $31$ |
\( T^{6} + 28 T^{5} + \cdots + 649 \)
|
| $37$ |
\( T^{6} + 4 T^{5} + \cdots - 4756 \)
|
| $41$ |
\( T^{6} + 36 T^{5} + \cdots + 12176 \)
|
| $43$ |
\( T^{6} + 3 T^{5} + \cdots - 42849 \)
|
| $47$ |
\( T^{6} + T^{5} + \cdots - 132304 \)
|
| $53$ |
\( T^{6} + 16 T^{5} + \cdots + 1024 \)
|
| $59$ |
\( T^{6} + 25 T^{5} + \cdots + 86544 \)
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| $61$ |
\( T^{6} + 2 T^{5} + \cdots - 5949 \)
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| $67$ |
\( T^{6} - 184 T^{4} + \cdots - 19111 \)
|
| $71$ |
\( T^{6} - 3 T^{5} + \cdots + 12176 \)
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| $73$ |
\( T^{6} + 21 T^{5} + \cdots + 11959 \)
|
| $79$ |
\( T^{6} - 6 T^{5} + \cdots - 2169 \)
|
| $83$ |
\( T^{6} + 14 T^{5} + \cdots - 150336 \)
|
| $89$ |
\( T^{6} - 6 T^{5} + \cdots + 16 \)
|
| $97$ |
\( T^{6} - 8 T^{5} + \cdots + 1996 \)
|
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