Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2450,2,Mod(99,2450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2450.99");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2450.c (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: | \(19.5633484952\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.959512576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 7x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{8} + 7x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 16\nu^{2} ) / 45 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{4} + 7 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} + 9\nu^{5} + 13\nu^{3} + 9\nu ) / 135 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} - 9\nu^{5} + 13\nu^{3} - 9\nu ) / 135 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{2} ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{5} - 16\nu^{3} + 48\nu ) / 45 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{5} + 16\nu^{3} + 48\nu ) / 45 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 5\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} - 2\beta_{6} + 3\beta_{4} + 3\beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{2} - 7 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - 16\beta_{4} + 16\beta_{3} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{5} + 5\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 13\beta_{7} - 13\beta_{6} - 48\beta_{4} - 48\beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(1177\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 1.00000i | − | 3.05231i | −1.00000 | 0 | −3.05231 | 0 | 1.00000i | −6.31662 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 99.2 | − | 1.00000i | − | 1.63810i | −1.00000 | 0 | −1.63810 | 0 | 1.00000i | 0.316625 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 99.3 | − | 1.00000i | 1.63810i | −1.00000 | 0 | 1.63810 | 0 | 1.00000i | 0.316625 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 99.4 | − | 1.00000i | 3.05231i | −1.00000 | 0 | 3.05231 | 0 | 1.00000i | −6.31662 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 99.5 | 1.00000i | − | 3.05231i | −1.00000 | 0 | 3.05231 | 0 | − | 1.00000i | −6.31662 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 99.6 | 1.00000i | − | 1.63810i | −1.00000 | 0 | 1.63810 | 0 | − | 1.00000i | 0.316625 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 99.7 | 1.00000i | 1.63810i | −1.00000 | 0 | −1.63810 | 0 | − | 1.00000i | 0.316625 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 99.8 | 1.00000i | 3.05231i | −1.00000 | 0 | −3.05231 | 0 | − | 1.00000i | −6.31662 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2450.2.c.x | 8 | |
| 5.b | even | 2 | 1 | inner | 2450.2.c.x | 8 | |
| 5.c | odd | 4 | 1 | 2450.2.a.bt | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 2450.2.a.bu | yes | 4 | |
| 7.b | odd | 2 | 1 | inner | 2450.2.c.x | 8 | |
| 35.c | odd | 2 | 1 | inner | 2450.2.c.x | 8 | |
| 35.f | even | 4 | 1 | 2450.2.a.bt | ✓ | 4 | |
| 35.f | even | 4 | 1 | 2450.2.a.bu | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2450.2.a.bt | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 2450.2.a.bt | ✓ | 4 | 35.f | even | 4 | 1 | |
| 2450.2.a.bu | yes | 4 | 5.c | odd | 4 | 1 | |
| 2450.2.a.bu | yes | 4 | 35.f | even | 4 | 1 | |
| 2450.2.c.x | 8 | 1.a | even | 1 | 1 | trivial | |
| 2450.2.c.x | 8 | 5.b | even | 2 | 1 | inner | |
| 2450.2.c.x | 8 | 7.b | odd | 2 | 1 | inner | |
| 2450.2.c.x | 8 | 35.c | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2450, [\chi])\):
|
\( T_{3}^{4} + 12T_{3}^{2} + 25 \)
|
|
\( T_{11}^{2} - 4T_{11} - 7 \)
|
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\( T_{13}^{4} + 48T_{13}^{2} + 400 \)
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\( T_{19}^{4} - 12T_{19}^{2} + 25 \)
|
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\( T_{31}^{2} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{4} + 12 T^{2} + 25)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{2} - 4 T - 7)^{4} \)
|
| $13$ |
\( (T^{4} + 48 T^{2} + 400)^{2} \)
|
| $17$ |
\( (T^{4} + 60 T^{2} + 361)^{2} \)
|
| $19$ |
\( (T^{4} - 12 T^{2} + 25)^{2} \)
|
| $23$ |
\( (T^{4} + 96 T^{2} + 1600)^{2} \)
|
| $29$ |
\( (T^{2} + 8 T - 28)^{4} \)
|
| $31$ |
\( (T^{2} - 32)^{4} \)
|
| $37$ |
\( (T^{4} + 96 T^{2} + 1600)^{2} \)
|
| $41$ |
\( (T^{4} - 60 T^{2} + 361)^{2} \)
|
| $43$ |
\( (T^{2} + 44)^{4} \)
|
| $47$ |
\( (T^{4} + 48 T^{2} + 400)^{2} \)
|
| $53$ |
\( (T^{2} + 64)^{4} \)
|
| $59$ |
\( (T^{4} - 60 T^{2} + 196)^{2} \)
|
| $61$ |
\( (T^{4} - 48 T^{2} + 400)^{2} \)
|
| $67$ |
\( (T^{4} + 138 T^{2} + 361)^{2} \)
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| $71$ |
\( (T + 2)^{8} \)
|
| $73$ |
\( (T^{4} + 60 T^{2} + 361)^{2} \)
|
| $79$ |
\( (T^{2} + 16 T + 20)^{4} \)
|
| $83$ |
\( (T^{4} + 236 T^{2} + 11449)^{2} \)
|
| $89$ |
\( (T^{4} - 108 T^{2} + 2025)^{2} \)
|
| $97$ |
\( (T^{2} + 18)^{4} \)
|
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