Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5625,2,Mod(1,5625)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5625, 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("5625.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5625 = 3^{2} \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5625.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: | \(44.9158511370\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.6152203125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 625) |
| 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
\( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 2\nu^{4} + 17\nu^{3} - \nu^{2} - 15\nu + 1 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} - 2\nu^{5} + 17\nu^{4} - \nu^{3} - 15\nu^{2} + 4\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 4\nu^{5} + 5\nu^{4} - 26\nu^{3} - 5\nu^{2} + 36\nu - 4 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{7} + 8\nu^{6} + 7\nu^{5} - 43\nu^{4} - 4\nu^{3} + 51\nu^{2} - 8\nu ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} + 8\nu^{6} + 7\nu^{5} - 43\nu^{4} - 7\nu^{3} + 57\nu^{2} + \nu - 6 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + 2\beta_{2} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{7} + 3\beta_{6} + \beta_{5} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{7} + 12\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_{3} + 21\beta_{2} + 33\beta _1 + 44 \)
|
| \(\nu^{6}\) | \(=\) |
\( -33\beta_{7} + 37\beta_{6} + 14\beta_{5} + 8\beta_{4} + 17\beta_{3} + 67\beta_{2} + 83\beta _1 + 148 \)
|
| \(\nu^{7}\) | \(=\) |
\( -104\beta_{7} + 122\beta_{6} + 45\beta_{5} + 39\beta_{4} + 57\beta_{3} + 191\beta_{2} + 244\beta _1 + 406 \)
|
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 |
|
−2.66501 | 0 | 5.10229 | 0 | 0 | −2.04213 | −8.26764 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.47435 | 0 | 4.12242 | 0 | 0 | −0.973070 | −5.25163 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.32610 | 0 | 3.41075 | 0 | 0 | 3.59425 | −3.28154 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.05737 | 0 | −0.881958 | 0 | 0 | 1.01199 | 3.04731 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.500989 | 0 | −1.74901 | 0 | 0 | 0.0237879 | 1.87821 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.326751 | 0 | −1.89323 | 0 | 0 | 3.42409 | −1.27212 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.68341 | 0 | 0.833870 | 0 | 0 | 4.59110 | −1.96307 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.01367 | 0 | 2.05487 | 0 | 0 | 0.369971 | 0.110485 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 5625.2.a.s | 8 | |
| 3.b | odd | 2 | 1 | 625.2.a.g | yes | 8 | |
| 5.b | even | 2 | 1 | 5625.2.a.be | 8 | ||
| 12.b | even | 2 | 1 | 10000.2.a.be | 8 | ||
| 15.d | odd | 2 | 1 | 625.2.a.e | ✓ | 8 | |
| 15.e | even | 4 | 2 | 625.2.b.d | 16 | ||
| 60.h | even | 2 | 1 | 10000.2.a.bn | 8 | ||
| 75.h | odd | 10 | 2 | 625.2.d.p | 16 | ||
| 75.h | odd | 10 | 2 | 625.2.d.q | 16 | ||
| 75.j | odd | 10 | 2 | 625.2.d.m | 16 | ||
| 75.j | odd | 10 | 2 | 625.2.d.n | 16 | ||
| 75.l | even | 20 | 4 | 625.2.e.j | 32 | ||
| 75.l | even | 20 | 4 | 625.2.e.k | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 625.2.a.e | ✓ | 8 | 15.d | odd | 2 | 1 | |
| 625.2.a.g | yes | 8 | 3.b | odd | 2 | 1 | |
| 625.2.b.d | 16 | 15.e | even | 4 | 2 | ||
| 625.2.d.m | 16 | 75.j | odd | 10 | 2 | ||
| 625.2.d.n | 16 | 75.j | odd | 10 | 2 | ||
| 625.2.d.p | 16 | 75.h | odd | 10 | 2 | ||
| 625.2.d.q | 16 | 75.h | odd | 10 | 2 | ||
| 625.2.e.j | 32 | 75.l | even | 20 | 4 | ||
| 625.2.e.k | 32 | 75.l | even | 20 | 4 | ||
| 5625.2.a.s | 8 | 1.a | even | 1 | 1 | trivial | |
| 5625.2.a.be | 8 | 5.b | even | 2 | 1 | ||
| 10000.2.a.be | 8 | 12.b | even | 2 | 1 | ||
| 10000.2.a.bn | 8 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):
|
\( T_{2}^{8} + 5T_{2}^{7} - T_{2}^{6} - 35T_{2}^{5} - 29T_{2}^{4} + 60T_{2}^{3} + 69T_{2}^{2} - 9 \)
|
|
\( T_{7}^{8} - 10T_{7}^{7} + 24T_{7}^{6} + 35T_{7}^{5} - 154T_{7}^{4} + 25T_{7}^{3} + 124T_{7}^{2} - 45T_{7} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 5 T^{7} + \cdots - 9 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 10 T^{7} + \cdots + 1 \)
|
| $11$ |
\( T^{8} + T^{7} + \cdots + 2421 \)
|
| $13$ |
\( T^{8} - 10 T^{7} + \cdots + 361 \)
|
| $17$ |
\( T^{8} + 15 T^{7} + \cdots + 1611 \)
|
| $19$ |
\( T^{8} + 10 T^{7} + \cdots + 10525 \)
|
| $23$ |
\( T^{8} + 30 T^{7} + \cdots - 46089 \)
|
| $29$ |
\( T^{8} + 10 T^{7} + \cdots - 60975 \)
|
| $31$ |
\( T^{8} + 9 T^{7} + \cdots + 24001 \)
|
| $37$ |
\( T^{8} + 10 T^{7} + \cdots + 81 \)
|
| $41$ |
\( T^{8} - 4 T^{7} + \cdots - 487629 \)
|
| $43$ |
\( T^{8} - 99 T^{6} + \cdots - 1949 \)
|
| $47$ |
\( T^{8} + 30 T^{7} + \cdots + 56961 \)
|
| $53$ |
\( T^{8} + 10 T^{7} + \cdots - 1899 \)
|
| $59$ |
\( T^{8} - 5 T^{7} + \cdots - 225 \)
|
| $61$ |
\( T^{8} - 6 T^{7} + \cdots - 103529 \)
|
| $67$ |
\( T^{8} - 10 T^{7} + \cdots - 1746299 \)
|
| $71$ |
\( T^{8} - 9 T^{7} + \cdots - 16749 \)
|
| $73$ |
\( T^{8} - 314 T^{6} + \cdots + 237091 \)
|
| $79$ |
\( T^{8} + 20 T^{7} + \cdots + 249525 \)
|
| $83$ |
\( T^{8} + 40 T^{7} + \cdots + 12262851 \)
|
| $89$ |
\( T^{8} - 5 T^{7} + \cdots + 1849275 \)
|
| $97$ |
\( T^{8} - 321 T^{6} + \cdots + 972421 \)
|
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