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: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.13366265625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1875) |
| 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
\( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 10\nu^{4} + 41\nu^{3} - 20\nu^{2} - 40\nu + 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 34\nu^{4} - 21\nu^{3} + 94\nu^{2} + 8\nu - 56 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 10\nu^{5} + 30\nu^{4} + 21\nu^{3} - 62\nu^{2} - 12\nu + 20 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 30\nu^{4} - 21\nu^{3} + 66\nu^{2} + 12\nu - 32 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + 3\nu^{6} + 36\nu^{5} - 30\nu^{4} - 115\nu^{3} + 60\nu^{2} + 88\nu - 24 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 30\nu^{4} - 52\nu^{3} + 61\nu^{2} + 30\nu - 28 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 3\beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{5} + 7\beta_{4} - 2\beta_{3} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} + 10\beta_{6} - \beta_{4} + 24\beta_{2} + 22\beta _1 - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -2\beta_{7} + 59\beta_{5} + 46\beta_{4} - 20\beta_{3} - 2\beta_{2} - 14\beta _1 + 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( -26\beta_{7} + 79\beta_{6} - \beta_{5} - 16\beta_{4} + 171\beta_{2} + 136\beta _1 - 44 \)
|
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 |
|
−2.59716 | 0 | 4.74525 | 0 | 0 | 3.28414 | −7.12986 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.23365 | 0 | 2.98921 | 0 | 0 | −1.03143 | −2.20956 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.31354 | 0 | −0.274605 | 0 | 0 | 4.19091 | 2.98779 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.741379 | 0 | −1.45036 | 0 | 0 | 1.03586 | 2.55802 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.770071 | 0 | −1.40699 | 0 | 0 | 3.98808 | −2.62363 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.895394 | 0 | −1.19827 | 0 | 0 | 5.08992 | −2.86371 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.52260 | 0 | 0.318310 | 0 | 0 | −0.990985 | −2.56054 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.69767 | 0 | 5.27745 | 0 | 0 | −3.56649 | 8.84149 | 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.u | 8 | |
| 3.b | odd | 2 | 1 | 1875.2.a.o | yes | 8 | |
| 5.b | even | 2 | 1 | 5625.2.a.bc | 8 | ||
| 15.d | odd | 2 | 1 | 1875.2.a.n | ✓ | 8 | |
| 15.e | even | 4 | 2 | 1875.2.b.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1875.2.a.n | ✓ | 8 | 15.d | odd | 2 | 1 | |
| 1875.2.a.o | yes | 8 | 3.b | odd | 2 | 1 | |
| 1875.2.b.g | 16 | 15.e | even | 4 | 2 | ||
| 5625.2.a.u | 8 | 1.a | even | 1 | 1 | trivial | |
| 5625.2.a.bc | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):
|
\( T_{2}^{8} + T_{2}^{7} - 12T_{2}^{6} - 10T_{2}^{5} + 41T_{2}^{4} + 20T_{2}^{3} - 48T_{2}^{2} - 8T_{2} + 16 \)
|
|
\( T_{7}^{8} - 12T_{7}^{7} + 29T_{7}^{6} + 142T_{7}^{5} - 654T_{7}^{4} + 160T_{7}^{3} + 1650T_{7}^{2} - 320T_{7} - 1055 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{7} + \cdots + 16 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 12 T^{7} + \cdots - 1055 \)
|
| $11$ |
\( T^{8} + 12 T^{7} + \cdots + 16 \)
|
| $13$ |
\( T^{8} - 14 T^{7} + \cdots + 8731 \)
|
| $17$ |
\( T^{8} - T^{7} + \cdots + 8656 \)
|
| $19$ |
\( T^{8} - 16 T^{7} + \cdots - 14975 \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 28720 \)
|
| $29$ |
\( T^{8} + 2 T^{7} + \cdots + 57520 \)
|
| $31$ |
\( T^{8} - 13 T^{7} + \cdots + 801025 \)
|
| $37$ |
\( T^{8} + 8 T^{7} + \cdots + 25 \)
|
| $41$ |
\( T^{8} - 12 T^{7} + \cdots - 48080 \)
|
| $43$ |
\( T^{8} - 20 T^{7} + \cdots - 74369 \)
|
| $47$ |
\( T^{8} - 15 T^{7} + \cdots - 255824 \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots + 28720 \)
|
| $59$ |
\( T^{8} + 14 T^{7} + \cdots + 11920 \)
|
| $61$ |
\( T^{8} - 10 T^{7} + \cdots - 1093919 \)
|
| $67$ |
\( T^{8} - 19 T^{7} + \cdots + 20204221 \)
|
| $71$ |
\( T^{8} + 21 T^{7} + \cdots + 67696 \)
|
| $73$ |
\( T^{8} + 19 T^{7} + \cdots - 379655 \)
|
| $79$ |
\( T^{8} - 10 T^{7} + \cdots + 6951025 \)
|
| $83$ |
\( T^{8} - 27 T^{7} + \cdots - 113744 \)
|
| $89$ |
\( T^{8} - 9 T^{7} + \cdots - 12105680 \)
|
| $97$ |
\( T^{8} - 24 T^{7} + \cdots - 401939 \)
|
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