Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,2,Mod(49,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.n (of order \(8\), 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: | \(3.39364208590\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 17) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(326\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{8}\) |
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 |
|
−1.70711 | + | 1.70711i | −1.00000 | + | 0.414214i | − | 3.82843i | 0 | 1.00000 | − | 2.41421i | −1.00000 | + | 2.41421i | 3.12132 | + | 3.12132i | −1.29289 | + | 1.29289i | 0 | |||||||||||||||||
| 274.1 | −0.292893 | + | 0.292893i | −1.00000 | − | 2.41421i | 1.82843i | 0 | 1.00000 | + | 0.414214i | −1.00000 | − | 0.414214i | −1.12132 | − | 1.12132i | −2.70711 | + | 2.70711i | 0 | |||||||||||||||||||
| 349.1 | −0.292893 | − | 0.292893i | −1.00000 | + | 2.41421i | − | 1.82843i | 0 | 1.00000 | − | 0.414214i | −1.00000 | + | 0.414214i | −1.12132 | + | 1.12132i | −2.70711 | − | 2.70711i | 0 | ||||||||||||||||||
| 399.1 | −1.70711 | − | 1.70711i | −1.00000 | − | 0.414214i | 3.82843i | 0 | 1.00000 | + | 2.41421i | −1.00000 | − | 2.41421i | 3.12132 | − | 3.12132i | −1.29289 | − | 1.29289i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 85.m | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 425.2.n.a | 4 | |
| 5.b | even | 2 | 1 | 425.2.n.b | 4 | ||
| 5.c | odd | 4 | 1 | 17.2.d.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 425.2.m.a | 4 | ||
| 15.e | even | 4 | 1 | 153.2.l.c | 4 | ||
| 17.d | even | 8 | 1 | 425.2.n.b | 4 | ||
| 20.e | even | 4 | 1 | 272.2.v.d | 4 | ||
| 35.f | even | 4 | 1 | 833.2.l.a | 4 | ||
| 35.k | even | 12 | 2 | 833.2.v.a | 8 | ||
| 35.l | odd | 12 | 2 | 833.2.v.b | 8 | ||
| 85.f | odd | 4 | 1 | 289.2.d.c | 4 | ||
| 85.g | odd | 4 | 1 | 289.2.d.a | 4 | ||
| 85.i | odd | 4 | 1 | 289.2.d.b | 4 | ||
| 85.k | odd | 8 | 1 | 289.2.d.b | 4 | ||
| 85.k | odd | 8 | 1 | 289.2.d.c | 4 | ||
| 85.k | odd | 8 | 1 | 425.2.m.a | 4 | ||
| 85.m | even | 8 | 1 | inner | 425.2.n.a | 4 | |
| 85.n | odd | 8 | 1 | 17.2.d.a | ✓ | 4 | |
| 85.n | odd | 8 | 1 | 289.2.d.a | 4 | ||
| 85.o | even | 16 | 4 | 289.2.c.c | 8 | ||
| 85.o | even | 16 | 2 | 7225.2.a.u | 4 | ||
| 85.r | even | 16 | 2 | 289.2.a.f | 4 | ||
| 85.r | even | 16 | 2 | 289.2.b.b | 4 | ||
| 255.v | even | 8 | 1 | 153.2.l.c | 4 | ||
| 255.bj | odd | 16 | 2 | 2601.2.a.bb | 4 | ||
| 340.w | even | 8 | 1 | 272.2.v.d | 4 | ||
| 340.bj | odd | 16 | 2 | 4624.2.a.bp | 4 | ||
| 595.bj | even | 8 | 1 | 833.2.l.a | 4 | ||
| 595.ce | odd | 24 | 2 | 833.2.v.b | 8 | ||
| 595.cf | even | 24 | 2 | 833.2.v.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.2.d.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 17.2.d.a | ✓ | 4 | 85.n | odd | 8 | 1 | |
| 153.2.l.c | 4 | 15.e | even | 4 | 1 | ||
| 153.2.l.c | 4 | 255.v | even | 8 | 1 | ||
| 272.2.v.d | 4 | 20.e | even | 4 | 1 | ||
| 272.2.v.d | 4 | 340.w | even | 8 | 1 | ||
| 289.2.a.f | 4 | 85.r | even | 16 | 2 | ||
| 289.2.b.b | 4 | 85.r | even | 16 | 2 | ||
| 289.2.c.c | 8 | 85.o | even | 16 | 4 | ||
| 289.2.d.a | 4 | 85.g | odd | 4 | 1 | ||
| 289.2.d.a | 4 | 85.n | odd | 8 | 1 | ||
| 289.2.d.b | 4 | 85.i | odd | 4 | 1 | ||
| 289.2.d.b | 4 | 85.k | odd | 8 | 1 | ||
| 289.2.d.c | 4 | 85.f | odd | 4 | 1 | ||
| 289.2.d.c | 4 | 85.k | odd | 8 | 1 | ||
| 425.2.m.a | 4 | 5.c | odd | 4 | 1 | ||
| 425.2.m.a | 4 | 85.k | odd | 8 | 1 | ||
| 425.2.n.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 425.2.n.a | 4 | 85.m | even | 8 | 1 | inner | |
| 425.2.n.b | 4 | 5.b | even | 2 | 1 | ||
| 425.2.n.b | 4 | 17.d | even | 8 | 1 | ||
| 833.2.l.a | 4 | 35.f | even | 4 | 1 | ||
| 833.2.l.a | 4 | 595.bj | even | 8 | 1 | ||
| 833.2.v.a | 8 | 35.k | even | 12 | 2 | ||
| 833.2.v.a | 8 | 595.cf | even | 24 | 2 | ||
| 833.2.v.b | 8 | 35.l | odd | 12 | 2 | ||
| 833.2.v.b | 8 | 595.ce | odd | 24 | 2 | ||
| 2601.2.a.bb | 4 | 255.bj | odd | 16 | 2 | ||
| 4624.2.a.bp | 4 | 340.bj | odd | 16 | 2 | ||
| 7225.2.a.u | 4 | 85.o | even | 16 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} + 8T_{2}^{2} + 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 1 \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 8 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 8 \)
|
| $11$ |
\( T^{4} + 4 T^{3} + \cdots + 8 \)
|
| $13$ |
\( (T^{2} - 2)^{2} \)
|
| $17$ |
\( (T^{2} + 6 T + 17)^{2} \)
|
| $19$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
|
| $23$ |
\( T^{4} + 12 T^{3} + \cdots + 392 \)
|
| $29$ |
\( T^{4} - 4 T^{3} + \cdots + 2 \)
|
| $31$ |
\( T^{4} + 12 T^{3} + \cdots + 648 \)
|
| $37$ |
\( T^{4} - 20 T^{3} + \cdots + 1250 \)
|
| $41$ |
\( T^{4} + 4 T^{3} + \cdots + 98 \)
|
| $43$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
|
| $47$ |
\( (T^{2} + 16 T + 56)^{2} \)
|
| $53$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $59$ |
\( T^{4} + 1296 \)
|
| $61$ |
\( T^{4} + 50 T^{2} + \cdots + 1250 \)
|
| $67$ |
\( T^{4} + 48T^{2} + 64 \)
|
| $71$ |
\( T^{4} - 20 T^{3} + \cdots + 5000 \)
|
| $73$ |
\( T^{4} + 98 T^{2} + \cdots + 4802 \)
|
| $79$ |
\( T^{4} - 4 T^{3} + \cdots + 392 \)
|
| $83$ |
\( T^{4} + 16 T^{3} + \cdots + 16 \)
|
| $89$ |
\( T^{4} + 132T^{2} + 3844 \)
|
| $97$ |
\( T^{4} - 4 T^{3} + \cdots + 4418 \)
|
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