Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,4,Mod(274,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.274");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.d (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(30.9760027530\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 48x^{6} + 668x^{4} + 2217x^{2} + 2116 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 48x^{6} + 668x^{4} + 2217x^{2} + 2116 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 48\nu^{5} + 622\nu^{3} + 1113\nu ) / 138 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 24\nu^{2} + 46 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 48\nu^{5} + 691\nu^{3} + 2493\nu ) / 69 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 6\nu^{7} + 265\nu^{5} + 3180\nu^{3} + 5620\nu ) / 69 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} + 44\nu^{4} + 526\nu^{2} + 926 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{3} - 20\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{4} - 24\beta_{2} + 242 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{6} - 24\beta_{5} + 84\beta_{3} + 434\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{7} - 132\beta_{4} + 530\beta_{2} - 5262 \)
|
| \(\nu^{7}\) | \(=\) |
\( 144\beta_{6} + 530\beta_{5} - 2650\beta_{3} - 9505\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(176\) | \(451\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 274.1 |
|
− | 4.75345i | 3.00000i | −14.5953 | 0 | 14.2604 | − | 7.00000i | 31.3504i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 274.2 | − | 4.60171i | − | 3.00000i | −13.1758 | 0 | −13.8051 | 7.00000i | 23.8174i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 274.3 | − | 1.52801i | − | 3.00000i | 5.66519 | 0 | −4.58402 | 7.00000i | − | 20.8805i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 274.4 | − | 1.37627i | 3.00000i | 6.10588 | 0 | 4.12881 | − | 7.00000i | − | 19.4135i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 274.5 | 1.37627i | − | 3.00000i | 6.10588 | 0 | 4.12881 | 7.00000i | 19.4135i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 274.6 | 1.52801i | 3.00000i | 5.66519 | 0 | −4.58402 | − | 7.00000i | 20.8805i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 274.7 | 4.60171i | 3.00000i | −13.1758 | 0 | −13.8051 | − | 7.00000i | − | 23.8174i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 274.8 | 4.75345i | − | 3.00000i | −14.5953 | 0 | 14.2604 | 7.00000i | − | 31.3504i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
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 | 525.4.d.n | 8 | |
| 5.b | even | 2 | 1 | inner | 525.4.d.n | 8 | |
| 5.c | odd | 4 | 1 | 525.4.a.t | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 525.4.a.u | yes | 4 | |
| 15.e | even | 4 | 1 | 1575.4.a.bj | 4 | ||
| 15.e | even | 4 | 1 | 1575.4.a.bk | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.4.a.t | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 525.4.a.u | yes | 4 | 5.c | odd | 4 | 1 | |
| 525.4.d.n | 8 | 1.a | even | 1 | 1 | trivial | |
| 525.4.d.n | 8 | 5.b | even | 2 | 1 | inner | |
| 1575.4.a.bj | 4 | 15.e | even | 4 | 1 | ||
| 1575.4.a.bk | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(525, [\chi])\):
|
\( T_{2}^{8} + 48T_{2}^{6} + 668T_{2}^{4} + 2217T_{2}^{2} + 2116 \)
|
|
\( T_{11}^{4} - 21T_{11}^{3} - 2643T_{11}^{2} + 61617T_{11} - 304010 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 48 T^{6} + \cdots + 2116 \)
|
| $3$ |
\( (T^{2} + 9)^{4} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{2} + 49)^{4} \)
|
| $11$ |
\( (T^{4} - 21 T^{3} + \cdots - 304010)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 297426255424 \)
|
| $17$ |
\( T^{8} + \cdots + 659698266206464 \)
|
| $19$ |
\( (T^{4} + 72 T^{3} + \cdots + 1017760)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 152816100889 \)
|
| $29$ |
\( (T^{4} - 240 T^{3} + \cdots + 130344085)^{2} \)
|
| $31$ |
\( (T^{4} - 351 T^{3} + \cdots + 137611224)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 22\!\cdots\!24 \)
|
| $41$ |
\( (T^{4} - 381 T^{3} + \cdots - 91750400)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 13\!\cdots\!49 \)
|
| $47$ |
\( T^{8} + \cdots + 12\!\cdots\!16 \)
|
| $53$ |
\( T^{8} + \cdots + 72\!\cdots\!00 \)
|
| $59$ |
\( (T^{4} - 855 T^{3} + \cdots - 35121553400)^{2} \)
|
| $61$ |
\( (T^{4} - 687 T^{3} + \cdots + 13537528704)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 14\!\cdots\!44 \)
|
| $71$ |
\( (T^{4} + 681 T^{3} + \cdots - 3046275956)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 24\!\cdots\!44 \)
|
| $79$ |
\( (T^{4} + 345 T^{3} + \cdots - 14999641370)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} - 198 T^{3} + \cdots - 25670269520)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 36\!\cdots\!24 \)
|
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