Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3775,2,Mod(1,3775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3775, 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("3775.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3775 = 5^{2} \cdot 151 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3775.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: | \(30.1435267630\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.4838537.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 3x^{3} + 13x^{2} + 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 151) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 7x^{4} + 3x^{3} + 13x^{2} + 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 6\nu^{3} + 3\nu^{2} + 9\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 6\nu^{3} - 8\nu^{2} - 10\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 5\beta_{2} + 6\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 6\beta_{3} + 2\beta_{2} + 18\beta _1 + 7 \)
|
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 |
|
−2.34838 | 1.59233 | 3.51488 | 0 | −3.73938 | 3.82411 | −3.55751 | −0.464497 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.05089 | 0.642853 | 2.20615 | 0 | −1.31842 | −4.07943 | −0.422789 | −2.58674 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.183668 | 3.29466 | −1.96627 | 0 | −0.605125 | −3.65107 | 0.728478 | 7.85477 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.503289 | −3.23034 | −1.74670 | 0 | −1.62580 | −2.18587 | −1.88567 | 7.43510 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.18639 | −0.248968 | −0.592471 | 0 | −0.295374 | −1.68227 | −3.07569 | −2.93801 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.89325 | 2.94947 | 1.58441 | 0 | 5.58410 | 4.77452 | −0.786819 | 5.69938 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(151\) | \( -1 \) |
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 | 3775.2.a.p | 6 | |
| 5.b | even | 2 | 1 | 151.2.a.c | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 1359.2.a.i | 6 | ||
| 20.d | odd | 2 | 1 | 2416.2.a.o | 6 | ||
| 35.c | odd | 2 | 1 | 7399.2.a.e | 6 | ||
| 40.e | odd | 2 | 1 | 9664.2.a.bc | 6 | ||
| 40.f | even | 2 | 1 | 9664.2.a.bh | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 151.2.a.c | ✓ | 6 | 5.b | even | 2 | 1 | |
| 1359.2.a.i | 6 | 15.d | odd | 2 | 1 | ||
| 2416.2.a.o | 6 | 20.d | odd | 2 | 1 | ||
| 3775.2.a.p | 6 | 1.a | even | 1 | 1 | trivial | |
| 7399.2.a.e | 6 | 35.c | odd | 2 | 1 | ||
| 9664.2.a.bc | 6 | 40.e | odd | 2 | 1 | ||
| 9664.2.a.bh | 6 | 40.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3775))\):
|
\( T_{2}^{6} + T_{2}^{5} - 7T_{2}^{4} - 3T_{2}^{3} + 13T_{2}^{2} - 3T_{2} - 1 \)
|
|
\( T_{3}^{6} - 5T_{3}^{5} - 4T_{3}^{4} + 51T_{3}^{3} - 68T_{3}^{2} + 12T_{3} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 7 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} - 5 T^{5} + \cdots + 8 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots + 1000 \)
|
| $11$ |
\( T^{6} - 8 T^{5} + \cdots + 49 \)
|
| $13$ |
\( T^{6} - T^{5} + \cdots - 328 \)
|
| $17$ |
\( T^{6} + 9 T^{5} + \cdots + 253 \)
|
| $19$ |
\( T^{6} + 6 T^{5} + \cdots + 115 \)
|
| $23$ |
\( T^{6} - 4 T^{5} + \cdots - 64 \)
|
| $29$ |
\( T^{6} + 2 T^{5} + \cdots - 5 \)
|
| $31$ |
\( T^{6} + 8 T^{5} + \cdots + 271 \)
|
| $37$ |
\( T^{6} - 12 T^{5} + \cdots + 56789 \)
|
| $41$ |
\( T^{6} - 41 T^{5} + \cdots + 73432 \)
|
| $43$ |
\( T^{6} + T^{5} + \cdots - 11425 \)
|
| $47$ |
\( T^{6} + 28 T^{5} + \cdots - 65843 \)
|
| $53$ |
\( T^{6} + 14 T^{5} + \cdots + 24664 \)
|
| $59$ |
\( T^{6} - 12 T^{5} + \cdots - 5 \)
|
| $61$ |
\( T^{6} - 5 T^{5} + \cdots - 16984 \)
|
| $67$ |
\( T^{6} - 15 T^{5} + \cdots + 14696 \)
|
| $71$ |
\( T^{6} + 2 T^{5} + \cdots - 4024 \)
|
| $73$ |
\( T^{6} - 7 T^{5} + \cdots - 135872 \)
|
| $79$ |
\( T^{6} + 9 T^{5} + \cdots - 195080 \)
|
| $83$ |
\( T^{6} - 11 T^{5} + \cdots - 260696 \)
|
| $89$ |
\( T^{6} - 36 T^{5} + \cdots + 64000 \)
|
| $97$ |
\( T^{6} + 11 T^{5} + \cdots - 193 \)
|
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