Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1575,4,Mod(1,1575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1575.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1575.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: | \(92.9280082590\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 32x^{2} - 35x + 120 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 175) |
| 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 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 in terms of a root \(\nu\) of
\( x^{4} - 32x^{2} - 35x + 120 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 20\nu + 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu^{2} - 18\nu + 38 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 2\beta _1 + 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{3} + 4\beta_{2} + 26\beta _1 + 26 \)
|
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 |
|
−4.87199 | 0 | 15.7363 | 0 | 0 | −7.00000 | −37.6910 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.504784 | 0 | −7.74519 | 0 | 0 | −7.00000 | 7.94792 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 4.53510 | 0 | 12.5671 | 0 | 0 | −7.00000 | 20.7124 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 4.84167 | 0 | 15.4418 | 0 | 0 | −7.00000 | 36.0308 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(1\) |
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 | 1575.4.a.bl | 4 | |
| 3.b | odd | 2 | 1 | 175.4.a.g | ✓ | 4 | |
| 5.b | even | 2 | 1 | 1575.4.a.bg | 4 | ||
| 15.d | odd | 2 | 1 | 175.4.a.h | yes | 4 | |
| 15.e | even | 4 | 2 | 175.4.b.f | 8 | ||
| 21.c | even | 2 | 1 | 1225.4.a.z | 4 | ||
| 105.g | even | 2 | 1 | 1225.4.a.bd | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 175.4.a.g | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 175.4.a.h | yes | 4 | 15.d | odd | 2 | 1 | |
| 175.4.b.f | 8 | 15.e | even | 4 | 2 | ||
| 1225.4.a.z | 4 | 21.c | even | 2 | 1 | ||
| 1225.4.a.bd | 4 | 105.g | even | 2 | 1 | ||
| 1575.4.a.bg | 4 | 5.b | even | 2 | 1 | ||
| 1575.4.a.bl | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):
|
\( T_{2}^{4} - 4T_{2}^{3} - 26T_{2}^{2} + 95T_{2} + 54 \)
|
|
\( T_{11}^{4} + 100T_{11}^{3} - 586T_{11}^{2} - 292452T_{11} - 6827031 \)
|
|
\( T_{13}^{4} + 44T_{13}^{3} - 3968T_{13}^{2} - 135688T_{13} - 1040448 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4 T^{3} + \cdots + 54 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} + 100 T^{3} + \cdots - 6827031 \)
|
| $13$ |
\( T^{4} + 44 T^{3} + \cdots - 1040448 \)
|
| $17$ |
\( T^{4} + 53 T^{3} + \cdots - 85632 \)
|
| $19$ |
\( T^{4} + 29 T^{3} + \cdots + 56837320 \)
|
| $23$ |
\( T^{4} - 295 T^{3} + \cdots - 138903216 \)
|
| $29$ |
\( T^{4} + 129 T^{3} + \cdots + 18445050 \)
|
| $31$ |
\( T^{4} - 114 T^{3} + \cdots + 44772800 \)
|
| $37$ |
\( T^{4} + \cdots - 1237182776 \)
|
| $41$ |
\( T^{4} + \cdots - 1082974824 \)
|
| $43$ |
\( T^{4} + \cdots + 2143095488 \)
|
| $47$ |
\( T^{4} + 8 T^{3} + \cdots + 466065552 \)
|
| $53$ |
\( T^{4} + \cdots + 24137178144 \)
|
| $59$ |
\( T^{4} + \cdots + 3039063360 \)
|
| $61$ |
\( T^{4} + \cdots + 1189016144 \)
|
| $67$ |
\( T^{4} + \cdots - 1287279571 \)
|
| $71$ |
\( T^{4} + \cdots + 2950374906 \)
|
| $73$ |
\( T^{4} + \cdots + 20054062888 \)
|
| $79$ |
\( T^{4} + \cdots - 178640145850 \)
|
| $83$ |
\( T^{4} + \cdots + 194459351136 \)
|
| $89$ |
\( T^{4} + \cdots + 151094328480 \)
|
| $97$ |
\( T^{4} + \cdots - 802513451424 \)
|
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