Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,2,Mod(49,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 5, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.bx (of order \(10\), degree \(4\), 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: | \(6.58765816676\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\Q(\zeta_{20})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).
| \(n\) | \(376\) | \(551\) | \(727\) |
| \(\chi(n)\) | \(-\zeta_{20}^{6}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
−0.587785 | + | 0.190983i | 0.587785 | − | 0.809017i | −1.30902 | + | 0.951057i | 0 | −0.190983 | + | 0.587785i | 1.76336 | + | 2.42705i | 1.31433 | − | 1.80902i | −0.309017 | − | 0.951057i | 0 | ||||||||||||||||||||||||||||
| 49.2 | 0.587785 | − | 0.190983i | −0.587785 | + | 0.809017i | −1.30902 | + | 0.951057i | 0 | −0.190983 | + | 0.587785i | −1.76336 | − | 2.42705i | −1.31433 | + | 1.80902i | −0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||
| 124.1 | −0.951057 | − | 1.30902i | 0.951057 | − | 0.309017i | −0.190983 | + | 0.587785i | 0 | −1.30902 | − | 0.951057i | 2.85317 | + | 0.927051i | −2.12663 | + | 0.690983i | 0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||
| 124.2 | 0.951057 | + | 1.30902i | −0.951057 | + | 0.309017i | −0.190983 | + | 0.587785i | 0 | −1.30902 | − | 0.951057i | −2.85317 | − | 0.927051i | 2.12663 | − | 0.690983i | 0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||
| 499.1 | −0.951057 | + | 1.30902i | 0.951057 | + | 0.309017i | −0.190983 | − | 0.587785i | 0 | −1.30902 | + | 0.951057i | 2.85317 | − | 0.927051i | −2.12663 | − | 0.690983i | 0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||
| 499.2 | 0.951057 | − | 1.30902i | −0.951057 | − | 0.309017i | −0.190983 | − | 0.587785i | 0 | −1.30902 | + | 0.951057i | −2.85317 | + | 0.927051i | 2.12663 | + | 0.690983i | 0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||
| 724.1 | −0.587785 | − | 0.190983i | 0.587785 | + | 0.809017i | −1.30902 | − | 0.951057i | 0 | −0.190983 | − | 0.587785i | 1.76336 | − | 2.42705i | 1.31433 | + | 1.80902i | −0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||
| 724.2 | 0.587785 | + | 0.190983i | −0.587785 | − | 0.809017i | −1.30902 | − | 0.951057i | 0 | −0.190983 | − | 0.587785i | −1.76336 | + | 2.42705i | −1.31433 | − | 1.80902i | −0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 11.c | even | 5 | 1 | inner |
| 55.j | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 825.2.bx.a | 8 | |
| 5.b | even | 2 | 1 | inner | 825.2.bx.a | 8 | |
| 5.c | odd | 4 | 1 | 825.2.n.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 825.2.n.e | yes | 4 | |
| 11.c | even | 5 | 1 | inner | 825.2.bx.a | 8 | |
| 55.j | even | 10 | 1 | inner | 825.2.bx.a | 8 | |
| 55.k | odd | 20 | 1 | 825.2.n.a | ✓ | 4 | |
| 55.k | odd | 20 | 1 | 825.2.n.e | yes | 4 | |
| 55.k | odd | 20 | 1 | 9075.2.a.y | 2 | ||
| 55.k | odd | 20 | 1 | 9075.2.a.bz | 2 | ||
| 55.l | even | 20 | 1 | 9075.2.a.bb | 2 | ||
| 55.l | even | 20 | 1 | 9075.2.a.bu | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.2.n.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 825.2.n.a | ✓ | 4 | 55.k | odd | 20 | 1 | |
| 825.2.n.e | yes | 4 | 5.c | odd | 4 | 1 | |
| 825.2.n.e | yes | 4 | 55.k | odd | 20 | 1 | |
| 825.2.bx.a | 8 | 1.a | even | 1 | 1 | trivial | |
| 825.2.bx.a | 8 | 5.b | even | 2 | 1 | inner | |
| 825.2.bx.a | 8 | 11.c | even | 5 | 1 | inner | |
| 825.2.bx.a | 8 | 55.j | even | 10 | 1 | inner | |
| 9075.2.a.y | 2 | 55.k | odd | 20 | 1 | ||
| 9075.2.a.bb | 2 | 55.l | even | 20 | 1 | ||
| 9075.2.a.bu | 2 | 55.l | even | 20 | 1 | ||
| 9075.2.a.bz | 2 | 55.k | odd | 20 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{8} + T_{2}^{6} + 6T_{2}^{4} - 4T_{2}^{2} + 1 \)
|
|
\( T_{13}^{8} - 36T_{13}^{6} + 486T_{13}^{4} + 729T_{13}^{2} + 6561 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{8} - T^{6} + T^{4} + \cdots + 1 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 9 T^{6} + \cdots + 6561 \)
|
| $11$ |
\( (T^{4} + T^{3} + 21 T^{2} + \cdots + 121)^{2} \)
|
| $13$ |
\( T^{8} - 36 T^{6} + \cdots + 6561 \)
|
| $17$ |
\( T^{8} + 31 T^{6} + \cdots + 14641 \)
|
| $19$ |
\( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \)
|
| $23$ |
\( (T^{4} + 87 T^{2} + 1681)^{2} \)
|
| $29$ |
\( (T^{4} + 10 T^{2} + \cdots + 25)^{2} \)
|
| $31$ |
\( (T^{4} - 13 T^{3} + \cdots + 361)^{2} \)
|
| $37$ |
\( T^{8} - 124 T^{6} + \cdots + 130321 \)
|
| $41$ |
\( (T^{4} + 7 T^{3} + 19 T^{2} + \cdots + 1)^{2} \)
|
| $43$ |
\( (T^{4} + 42 T^{2} + 121)^{2} \)
|
| $47$ |
\( T^{8} - 59 T^{6} + \cdots + 141158161 \)
|
| $53$ |
\( T^{8} - 116 T^{6} + \cdots + 256 \)
|
| $59$ |
\( (T^{4} + 15 T^{3} + \cdots + 2025)^{2} \)
|
| $61$ |
\( (T^{4} - 3 T^{3} + \cdots + 3481)^{2} \)
|
| $67$ |
\( (T^{4} + 98 T^{2} + 1681)^{2} \)
|
| $71$ |
\( (T^{4} + 7 T^{3} + \cdots + 121)^{2} \)
|
| $73$ |
\( T^{8} - 76 T^{6} + \cdots + 1 \)
|
| $79$ |
\( (T^{4} + 20 T^{3} + \cdots + 9025)^{2} \)
|
| $83$ |
\( T^{8} - 191 T^{6} + \cdots + 294499921 \)
|
| $89$ |
\( (T^{2} - 15 T - 45)^{4} \)
|
| $97$ |
\( T^{8} - 369 T^{6} + \cdots + 855036081 \)
|
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