Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [575,4,Mod(1,575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(575, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("575.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 575 = 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 575.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: | \(33.9260982533\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 37x^{5} + 123x^{4} + 304x^{3} - 1196x^{2} + 264x + 864 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 37x^{5} + 123x^{4} + 304x^{3} - 1196x^{2} + 264x + 864 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 31\nu^{4} + 18\nu^{3} + 208\nu^{2} - 236\nu + 48 ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 37\nu^{4} - 6\nu^{3} - 346\nu^{2} + 68\nu + 432 ) / 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 37\nu^{4} + 18\nu^{3} + 346\nu^{2} - 296\nu - 312 ) / 24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 3\nu^{5} - 31\nu^{4} - 75\nu^{3} + 250\nu^{2} + 388\nu - 384 ) / 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{4} + 19\beta _1 - 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{5} + 4\beta_{3} + 23\beta_{2} - 33\beta _1 + 216 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{6} + 62\beta_{5} + 62\beta_{4} - 8\beta_{3} - 14\beta_{2} + 395\beta _1 - 334 \)
|
| \(\nu^{6}\) | \(=\) |
\( -160\beta_{5} - 36\beta_{4} + 148\beta_{3} + 505\beta_{2} - 921\beta _1 + 4332 \)
|
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 |
|
−4.39200 | 3.27927 | 11.2896 | 0 | −14.4025 | −13.7978 | −14.4481 | −16.2464 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.84004 | −4.08871 | 6.74592 | 0 | 15.7008 | 27.0878 | 4.81574 | −10.2825 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.80947 | 9.02517 | −0.106856 | 0 | −25.3560 | −11.4533 | 22.7760 | 54.4537 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.37904 | −5.53432 | −6.09824 | 0 | 7.63206 | −8.23397 | 19.4421 | 3.62871 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.711047 | 0.689108 | −7.49441 | 0 | 0.489988 | 19.0526 | −11.0173 | −26.5251 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.74805 | 2.78570 | 6.04789 | 0 | 10.4409 | −9.96305 | −7.31660 | −19.2399 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.96146 | −7.15622 | 16.6161 | 0 | −35.5053 | −3.69237 | 42.7482 | 24.2114 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 575.4.a.l | ✓ | 7 |
| 5.b | even | 2 | 1 | 575.4.a.m | yes | 7 | |
| 5.c | odd | 4 | 2 | 575.4.b.j | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 575.4.a.l | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 575.4.a.m | yes | 7 | 5.b | even | 2 | 1 | |
| 575.4.b.j | 14 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(575))\):
|
\( T_{2}^{7} + 3T_{2}^{6} - 37T_{2}^{5} - 123T_{2}^{4} + 304T_{2}^{3} + 1196T_{2}^{2} + 264T_{2} - 864 \)
|
|
\( T_{3}^{7} + T_{3}^{6} - 99T_{3}^{5} - 162T_{3}^{4} + 2175T_{3}^{3} + 1410T_{3}^{2} - 15280T_{3} + 9200 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 3 T^{6} + \cdots - 864 \)
|
| $3$ |
\( T^{7} + T^{6} + \cdots + 9200 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} + T^{6} + \cdots + 24704352 \)
|
| $11$ |
\( T^{7} + 52 T^{6} + \cdots - 935052376 \)
|
| $13$ |
\( T^{7} + \cdots + 14970390426 \)
|
| $17$ |
\( T^{7} + \cdots + 864344916480 \)
|
| $19$ |
\( T^{7} + \cdots + 451105530840 \)
|
| $23$ |
\( (T - 23)^{7} \)
|
| $29$ |
\( T^{7} + \cdots + 323393919122280 \)
|
| $31$ |
\( T^{7} + \cdots + 38841367049280 \)
|
| $37$ |
\( T^{7} + \cdots - 774288681572352 \)
|
| $41$ |
\( T^{7} + \cdots - 15\!\cdots\!85 \)
|
| $43$ |
\( T^{7} + \cdots - 56\!\cdots\!00 \)
|
| $47$ |
\( T^{7} + \cdots - 13\!\cdots\!20 \)
|
| $53$ |
\( T^{7} + \cdots + 54\!\cdots\!96 \)
|
| $59$ |
\( T^{7} + \cdots + 771684969846528 \)
|
| $61$ |
\( T^{7} + \cdots - 67\!\cdots\!12 \)
|
| $67$ |
\( T^{7} + \cdots + 43\!\cdots\!60 \)
|
| $71$ |
\( T^{7} + \cdots + 30\!\cdots\!00 \)
|
| $73$ |
\( T^{7} + \cdots - 11\!\cdots\!37 \)
|
| $79$ |
\( T^{7} + \cdots + 30\!\cdots\!36 \)
|
| $83$ |
\( T^{7} + \cdots - 11\!\cdots\!04 \)
|
| $89$ |
\( T^{7} + \cdots + 28\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 19\!\cdots\!68 \)
|
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