Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,4,Mod(1,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.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: | \(25.0758117524\) |
| 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: |
\( x^{6} - x^{5} - 30x^{4} + 30x^{3} + 213x^{2} - 109x - 276 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} - 30x^{4} + 30x^{3} + 213x^{2} - 109x - 276 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 23\nu^{3} - 11\nu^{2} + 96\nu + 28 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} + 7\nu^{4} - 31\nu^{3} - 107\nu^{2} - 3\nu + 156 ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 6\nu^{4} - 18\nu^{3} - 106\nu^{2} + 51\nu + 208 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 15\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} - \beta_{4} - \beta_{3} + 20\beta_{2} - 6\beta _1 + 156 \)
|
| \(\nu^{5}\) | \(=\) |
\( -26\beta_{5} + 24\beta_{4} - 12\beta_{3} - 32\beta_{2} + 255\beta _1 - 120 \)
|
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 |
|
−3.95575 | 3.53947 | 7.64793 | 0 | −14.0012 | −13.8166 | 1.39269 | −14.4722 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.74408 | −3.16721 | 6.01816 | 0 | 11.8583 | 26.6627 | 7.42016 | −16.9688 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.52037 | 9.11646 | −5.68846 | 0 | −13.8604 | −16.3642 | 20.8116 | 56.1099 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.03585 | −4.77225 | −6.92701 | 0 | −4.94335 | 16.4564 | −15.4622 | −4.22566 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.55730 | 3.86147 | −1.46020 | 0 | 9.87494 | 5.49760 | −24.1926 | −12.0891 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.62705 | −2.57794 | 13.4096 | 0 | −11.9282 | −22.4360 | 25.0303 | −20.3542 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(17\) | \(1\) |
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 | 425.4.a.j | ✓ | 6 |
| 5.b | even | 2 | 1 | 425.4.a.k | yes | 6 | |
| 5.c | odd | 4 | 2 | 425.4.b.j | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 425.4.a.j | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 425.4.a.k | yes | 6 | 5.b | even | 2 | 1 | |
| 425.4.b.j | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(425))\):
|
\( T_{2}^{6} + T_{2}^{5} - 30T_{2}^{4} - 30T_{2}^{3} + 213T_{2}^{2} + 109T_{2} - 276 \)
|
|
\( T_{3}^{6} - 6T_{3}^{5} - 57T_{3}^{4} + 180T_{3}^{3} + 933T_{3}^{2} - 1272T_{3} - 4855 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots - 276 \)
|
| $3$ |
\( T^{6} - 6 T^{5} + \cdots - 4855 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 4 T^{5} + \cdots - 12236443 \)
|
| $11$ |
\( T^{6} + 20 T^{5} + \cdots - 98280800 \)
|
| $13$ |
\( T^{6} + 42 T^{5} + \cdots + 819022489 \)
|
| $17$ |
\( (T + 17)^{6} \)
|
| $19$ |
\( T^{6} + 70 T^{5} + \cdots - 977808960 \)
|
| $23$ |
\( T^{6} + \cdots - 14481037920 \)
|
| $29$ |
\( T^{6} + \cdots + 2867744451360 \)
|
| $31$ |
\( T^{6} + \cdots - 4319982391195 \)
|
| $37$ |
\( T^{6} + \cdots - 42410952261024 \)
|
| $41$ |
\( T^{6} + \cdots - 15523734022400 \)
|
| $43$ |
\( T^{6} + \cdots + 77226368798496 \)
|
| $47$ |
\( T^{6} + \cdots + 2459602413120 \)
|
| $53$ |
\( T^{6} + \cdots + 16763419208545 \)
|
| $59$ |
\( T^{6} + \cdots - 132455594307936 \)
|
| $61$ |
\( T^{6} + \cdots + 5378930561600 \)
|
| $67$ |
\( T^{6} + \cdots + 57\!\cdots\!04 \)
|
| $71$ |
\( T^{6} + \cdots + 61\!\cdots\!61 \)
|
| $73$ |
\( T^{6} + \cdots + 45\!\cdots\!96 \)
|
| $79$ |
\( T^{6} + \cdots - 61\!\cdots\!07 \)
|
| $83$ |
\( T^{6} + \cdots - 11\!\cdots\!36 \)
|
| $89$ |
\( T^{6} + \cdots - 13\!\cdots\!20 \)
|
| $97$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
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