Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4225,2,Mod(1,4225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4225 = 5^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4225.a (trivial) |
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: | yes |
| Analytic conductor: | \(33.7367948540\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 65) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.41421 | 1.41421 | 3.82843 | 0 | −3.41421 | 4.82843 | −4.41421 | −1.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0.414214 | −1.41421 | −1.82843 | 0 | −0.585786 | −0.828427 | −1.58579 | −1.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(13\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4225.2.a.r | 2 | |
| 5.b | even | 2 | 1 | 845.2.a.g | 2 | ||
| 13.b | even | 2 | 1 | 325.2.a.i | 2 | ||
| 15.d | odd | 2 | 1 | 7605.2.a.x | 2 | ||
| 39.d | odd | 2 | 1 | 2925.2.a.u | 2 | ||
| 52.b | odd | 2 | 1 | 5200.2.a.bu | 2 | ||
| 65.d | even | 2 | 1 | 65.2.a.b | ✓ | 2 | |
| 65.g | odd | 4 | 2 | 845.2.c.b | 4 | ||
| 65.h | odd | 4 | 2 | 325.2.b.f | 4 | ||
| 65.l | even | 6 | 2 | 845.2.e.h | 4 | ||
| 65.n | even | 6 | 2 | 845.2.e.c | 4 | ||
| 65.s | odd | 12 | 4 | 845.2.m.f | 8 | ||
| 195.e | odd | 2 | 1 | 585.2.a.m | 2 | ||
| 195.s | even | 4 | 2 | 2925.2.c.r | 4 | ||
| 260.g | odd | 2 | 1 | 1040.2.a.j | 2 | ||
| 455.h | odd | 2 | 1 | 3185.2.a.j | 2 | ||
| 520.b | odd | 2 | 1 | 4160.2.a.z | 2 | ||
| 520.p | even | 2 | 1 | 4160.2.a.bf | 2 | ||
| 715.c | odd | 2 | 1 | 7865.2.a.j | 2 | ||
| 780.d | even | 2 | 1 | 9360.2.a.cd | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.a.b | ✓ | 2 | 65.d | even | 2 | 1 | |
| 325.2.a.i | 2 | 13.b | even | 2 | 1 | ||
| 325.2.b.f | 4 | 65.h | odd | 4 | 2 | ||
| 585.2.a.m | 2 | 195.e | odd | 2 | 1 | ||
| 845.2.a.g | 2 | 5.b | even | 2 | 1 | ||
| 845.2.c.b | 4 | 65.g | odd | 4 | 2 | ||
| 845.2.e.c | 4 | 65.n | even | 6 | 2 | ||
| 845.2.e.h | 4 | 65.l | even | 6 | 2 | ||
| 845.2.m.f | 8 | 65.s | odd | 12 | 4 | ||
| 1040.2.a.j | 2 | 260.g | odd | 2 | 1 | ||
| 2925.2.a.u | 2 | 39.d | odd | 2 | 1 | ||
| 2925.2.c.r | 4 | 195.s | even | 4 | 2 | ||
| 3185.2.a.j | 2 | 455.h | odd | 2 | 1 | ||
| 4160.2.a.z | 2 | 520.b | odd | 2 | 1 | ||
| 4160.2.a.bf | 2 | 520.p | even | 2 | 1 | ||
| 4225.2.a.r | 2 | 1.a | even | 1 | 1 | trivial | |
| 5200.2.a.bu | 2 | 52.b | odd | 2 | 1 | ||
| 7605.2.a.x | 2 | 15.d | odd | 2 | 1 | ||
| 7865.2.a.j | 2 | 715.c | odd | 2 | 1 | ||
| 9360.2.a.cd | 2 | 780.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):
|
\( T_{2}^{2} + 2T_{2} - 1 \)
|
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\( T_{3}^{2} - 2 \)
|
|
\( T_{7}^{2} - 4T_{7} - 4 \)
|
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\( T_{11}^{2} + 4T_{11} + 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T - 1 \)
|
| $3$ |
\( T^{2} - 2 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 4T - 4 \)
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| $11$ |
\( T^{2} + 4T + 2 \)
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| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} - 4T - 4 \)
|
| $19$ |
\( T^{2} + 4T + 2 \)
|
| $23$ |
\( T^{2} - 2 \)
|
| $29$ |
\( T^{2} - 32 \)
|
| $31$ |
\( T^{2} + 12T + 18 \)
|
| $37$ |
\( T^{2} - 72 \)
|
| $41$ |
\( T^{2} - 12T + 28 \)
|
| $43$ |
\( T^{2} - 8T - 34 \)
|
| $47$ |
\( T^{2} + 4T - 4 \)
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| $53$ |
\( T^{2} - 12T - 36 \)
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| $59$ |
\( T^{2} + 12T + 18 \)
|
| $61$ |
\( (T + 8)^{2} \)
|
| $67$ |
\( (T + 2)^{2} \)
|
| $71$ |
\( T^{2} + 4T - 94 \)
|
| $73$ |
\( T^{2} - 72 \)
|
| $79$ |
\( T^{2} - 72 \)
|
| $83$ |
\( T^{2} + 12T + 28 \)
|
| $89$ |
\( (T + 6)^{2} \)
|
| $97$ |
\( T^{2} + 4T - 28 \)
|
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