Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,16,Mod(1,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.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: | \(35.6733762750\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 29397x^{4} + 153469728x^{2} - 65015354624 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{4}\cdot 5^{6} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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
\( x^{6} - 29397x^{4} + 153469728x^{2} - 65015354624 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13\nu^{5} - 368353\nu^{3} + 1648953712\nu ) / 4805184 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19\nu^{5} + 200897\nu^{3} - 11605603568\nu ) / 2402592 \)
|
| \(\beta_{4}\) | \(=\) |
\( 4\nu^{2} - 39196 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 24373\nu^{2} + 53082976 ) / 174 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 39196 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 13\beta_{3} - 38\beta_{2} + 37918\beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 696\beta_{5} + 24373\beta_{4} + 742992204 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 368353\beta_{3} + 401794\beta_{2} + 820715510\beta_1 ) / 4 \)
|
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 |
|
−301.910 | 779.290 | 58381.4 | 0 | −235275. | 2.08791e6 | −7.73292e6 | −1.37416e7 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −156.785 | −1705.52 | −8186.44 | 0 | 267401. | −1.54725e6 | 6.42105e6 | −1.14401e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −43.0940 | 6725.99 | −30910.9 | 0 | −289850. | −1.29717e6 | 2.74418e6 | 3.08900e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 43.0940 | −6725.99 | −30910.9 | 0 | −289850. | 1.29717e6 | −2.74418e6 | 3.08900e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 156.785 | 1705.52 | −8186.44 | 0 | 267401. | 1.54725e6 | −6.42105e6 | −1.14401e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 301.910 | −779.290 | 58381.4 | 0 | −235275. | −2.08791e6 | 7.73292e6 | −1.37416e7 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.16.a.f | 6 | |
| 5.b | even | 2 | 1 | inner | 25.16.a.f | 6 | |
| 5.c | odd | 4 | 2 | 5.16.b.a | ✓ | 6 | |
| 15.e | even | 4 | 2 | 45.16.b.b | 6 | ||
| 20.e | even | 4 | 2 | 80.16.c.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.16.b.a | ✓ | 6 | 5.c | odd | 4 | 2 | |
| 25.16.a.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 25.16.a.f | 6 | 5.b | even | 2 | 1 | inner | |
| 45.16.b.b | 6 | 15.e | even | 4 | 2 | ||
| 80.16.c.a | 6 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 117588T_{2}^{4} + 2455515648T_{2}^{2} - 4160982695936 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 4160982695936 \)
|
| $3$ |
\( T^{6} + \cdots - 79\!\cdots\!04 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 17\!\cdots\!56 \)
|
| $11$ |
\( (T^{3} + \cdots - 91\!\cdots\!08)^{2} \)
|
| $13$ |
\( T^{6} + \cdots - 12\!\cdots\!64 \)
|
| $17$ |
\( T^{6} + \cdots - 43\!\cdots\!96 \)
|
| $19$ |
\( (T^{3} + \cdots - 20\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 61\!\cdots\!24 \)
|
| $29$ |
\( (T^{3} + \cdots - 81\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots + 49\!\cdots\!92)^{2} \)
|
| $37$ |
\( T^{6} + \cdots - 30\!\cdots\!76 \)
|
| $41$ |
\( (T^{3} + \cdots - 12\!\cdots\!08)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 19\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots - 11\!\cdots\!16 \)
|
| $53$ |
\( T^{6} + \cdots - 40\!\cdots\!04 \)
|
| $59$ |
\( (T^{3} + \cdots + 25\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 78\!\cdots\!92)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 52\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots + 14\!\cdots\!92)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 24\!\cdots\!24 \)
|
| $79$ |
\( (T^{3} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 13\!\cdots\!84 \)
|
| $89$ |
\( (T^{3} + \cdots + 47\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots - 17\!\cdots\!16 \)
|
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