Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7600,2,Mod(1,7600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("7600.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7600.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: | \(60.6863055362\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.11344.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 95) |
| 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
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 2\nu - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 11 ) / 2 \)
|
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 |
|
0 | −2.87834 | 0 | 0 | 0 | 3.10482 | 0 | 5.28487 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0.296842 | 0 | 0 | 0 | −3.56331 | 0 | −2.91188 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.53844 | 0 | 0 | 0 | 5.03316 | 0 | −0.633188 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.04306 | 0 | 0 | 0 | −0.574672 | 0 | 6.26020 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(19\) | \(1\) |
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 | 7600.2.a.cf | 4 | |
| 4.b | odd | 2 | 1 | 475.2.a.i | 4 | ||
| 5.b | even | 2 | 1 | 1520.2.a.t | 4 | ||
| 12.b | even | 2 | 1 | 4275.2.a.bo | 4 | ||
| 20.d | odd | 2 | 1 | 95.2.a.b | ✓ | 4 | |
| 20.e | even | 4 | 2 | 475.2.b.e | 8 | ||
| 40.e | odd | 2 | 1 | 6080.2.a.cc | 4 | ||
| 40.f | even | 2 | 1 | 6080.2.a.ch | 4 | ||
| 60.h | even | 2 | 1 | 855.2.a.m | 4 | ||
| 76.d | even | 2 | 1 | 9025.2.a.bf | 4 | ||
| 140.c | even | 2 | 1 | 4655.2.a.y | 4 | ||
| 380.d | even | 2 | 1 | 1805.2.a.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.a.b | ✓ | 4 | 20.d | odd | 2 | 1 | |
| 475.2.a.i | 4 | 4.b | odd | 2 | 1 | ||
| 475.2.b.e | 8 | 20.e | even | 4 | 2 | ||
| 855.2.a.m | 4 | 60.h | even | 2 | 1 | ||
| 1520.2.a.t | 4 | 5.b | even | 2 | 1 | ||
| 1805.2.a.p | 4 | 380.d | even | 2 | 1 | ||
| 4275.2.a.bo | 4 | 12.b | even | 2 | 1 | ||
| 4655.2.a.y | 4 | 140.c | even | 2 | 1 | ||
| 6080.2.a.cc | 4 | 40.e | odd | 2 | 1 | ||
| 6080.2.a.ch | 4 | 40.f | even | 2 | 1 | ||
| 7600.2.a.cf | 4 | 1.a | even | 1 | 1 | trivial | |
| 9025.2.a.bf | 4 | 76.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 8T_{3}^{2} + 16T_{3} - 4 \)
|
|
\( T_{7}^{4} - 4T_{7}^{3} - 16T_{7}^{2} + 48T_{7} + 32 \)
|
|
\( T_{11}^{4} + 4T_{11}^{3} - 16T_{11}^{2} - 32T_{11} + 48 \)
|
|
\( T_{13}^{4} + 2T_{13}^{3} - 24T_{13}^{2} - 32T_{13} + 20 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots - 4 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 32 \)
|
| $11$ |
\( T^{4} + 4 T^{3} + \cdots + 48 \)
|
| $13$ |
\( T^{4} + 2 T^{3} + \cdots + 20 \)
|
| $17$ |
\( T^{4} + 4 T^{3} + \cdots + 48 \)
|
| $19$ |
\( (T + 1)^{4} \)
|
| $23$ |
\( T^{4} + 8 T^{3} + \cdots + 288 \)
|
| $29$ |
\( T^{4} - 4 T^{3} + \cdots + 48 \)
|
| $31$ |
\( T^{4} + 4 T^{3} + \cdots - 640 \)
|
| $37$ |
\( T^{4} - 6 T^{3} + \cdots + 4 \)
|
| $41$ |
\( T^{4} - 16 T^{3} + \cdots - 240 \)
|
| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 32 \)
|
| $47$ |
\( T^{4} + 12 T^{3} + \cdots + 1056 \)
|
| $53$ |
\( T^{4} - 10 T^{3} + \cdots - 348 \)
|
| $59$ |
\( T^{4} - 64 T^{2} + \cdots - 192 \)
|
| $61$ |
\( T^{4} - 20 T^{3} + \cdots - 2656 \)
|
| $67$ |
\( T^{4} + 18 T^{3} + \cdots - 1076 \)
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| $71$ |
\( T^{4} - 20 T^{3} + \cdots - 4224 \)
|
| $73$ |
\( T^{4} + 28 T^{3} + \cdots + 176 \)
|
| $79$ |
\( T^{4} - 16 T^{3} + \cdots - 1856 \)
|
| $83$ |
\( T^{4} - 72 T^{2} + \cdots + 480 \)
|
| $89$ |
\( T^{4} - 4 T^{3} + \cdots + 240 \)
|
| $97$ |
\( T^{4} + 30 T^{3} + \cdots - 1388 \)
|
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