Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1225,2,Mod(99,1225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, 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("1225.99");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1225 = 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1225.b (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: | \(9.78167424761\) |
| Analytic rank: | \(0\) |
| 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: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 245) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{3} + \zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{8}^{3} + \zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(1177\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 |
|
− | 1.41421i | − | 2.41421i | 0 | 0 | −3.41421 | 0 | − | 2.82843i | −2.82843 | 0 | |||||||||||||||||||||||||||
| 99.2 | − | 1.41421i | − | 0.414214i | 0 | 0 | −0.585786 | 0 | − | 2.82843i | 2.82843 | 0 | ||||||||||||||||||||||||||||
| 99.3 | 1.41421i | 0.414214i | 0 | 0 | −0.585786 | 0 | 2.82843i | 2.82843 | 0 | |||||||||||||||||||||||||||||||
| 99.4 | 1.41421i | 2.41421i | 0 | 0 | −3.41421 | 0 | 2.82843i | −2.82843 | 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 | 1225.2.b.i | 4 | |
| 5.b | even | 2 | 1 | inner | 1225.2.b.i | 4 | |
| 5.c | odd | 4 | 1 | 245.2.a.e | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 1225.2.a.r | 2 | ||
| 7.b | odd | 2 | 1 | 1225.2.b.j | 4 | ||
| 15.e | even | 4 | 1 | 2205.2.a.v | 2 | ||
| 20.e | even | 4 | 1 | 3920.2.a.bw | 2 | ||
| 35.c | odd | 2 | 1 | 1225.2.b.j | 4 | ||
| 35.f | even | 4 | 1 | 245.2.a.f | yes | 2 | |
| 35.f | even | 4 | 1 | 1225.2.a.p | 2 | ||
| 35.k | even | 12 | 2 | 245.2.e.f | 4 | ||
| 35.l | odd | 12 | 2 | 245.2.e.g | 4 | ||
| 105.k | odd | 4 | 1 | 2205.2.a.t | 2 | ||
| 140.j | odd | 4 | 1 | 3920.2.a.br | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 245.2.a.e | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 245.2.a.f | yes | 2 | 35.f | even | 4 | 1 | |
| 245.2.e.f | 4 | 35.k | even | 12 | 2 | ||
| 245.2.e.g | 4 | 35.l | odd | 12 | 2 | ||
| 1225.2.a.p | 2 | 35.f | even | 4 | 1 | ||
| 1225.2.a.r | 2 | 5.c | odd | 4 | 1 | ||
| 1225.2.b.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 1225.2.b.i | 4 | 5.b | even | 2 | 1 | inner | |
| 1225.2.b.j | 4 | 7.b | odd | 2 | 1 | ||
| 1225.2.b.j | 4 | 35.c | odd | 2 | 1 | ||
| 2205.2.a.t | 2 | 105.k | odd | 4 | 1 | ||
| 2205.2.a.v | 2 | 15.e | even | 4 | 1 | ||
| 3920.2.a.br | 2 | 140.j | odd | 4 | 1 | ||
| 3920.2.a.bw | 2 | 20.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 \(S_{2}^{\mathrm{new}}(1225, [\chi])\):
|
\( T_{2}^{2} + 2 \)
|
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\( T_{19} - 6 \)
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\( T_{31}^{2} + 12T_{31} + 18 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
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| $3$ |
\( T^{4} + 6T^{2} + 1 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( (T^{2} + 6 T + 1)^{2} \)
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| $13$ |
\( T^{4} + 22T^{2} + 49 \)
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| $17$ |
\( T^{4} + 38T^{2} + 289 \)
|
| $19$ |
\( (T - 6)^{4} \)
|
| $23$ |
\( T^{4} + 76T^{2} + 1156 \)
|
| $29$ |
\( (T^{2} - 6 T - 23)^{2} \)
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| $31$ |
\( (T^{2} + 12 T + 18)^{2} \)
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| $37$ |
\( T^{4} + 44T^{2} + 196 \)
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| $41$ |
\( (T^{2} - 4 T - 14)^{2} \)
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| $43$ |
\( (T^{2} + 4)^{2} \)
|
| $47$ |
\( T^{4} + 54T^{2} + 81 \)
|
| $53$ |
\( (T^{2} + 18)^{2} \)
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| $59$ |
\( (T^{2} - 4 T - 14)^{2} \)
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| $61$ |
\( (T^{2} - 8)^{2} \)
|
| $67$ |
\( T^{4} + 68T^{2} + 4 \)
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| $71$ |
\( (T^{2} + 12 T + 28)^{2} \)
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| $73$ |
\( (T^{2} + 72)^{2} \)
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| $79$ |
\( (T^{2} - 14 T - 23)^{2} \)
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| $83$ |
\( T^{4} \)
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| $89$ |
\( (T + 8)^{4} \)
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| $97$ |
\( T^{4} + 198T^{2} + 3969 \)
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