Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4275,2,Mod(1,4275)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4275.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4275.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: | \(34.1360468641\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - 3x^{6} - 8x^{5} + 26x^{4} + 11x^{3} - 51x^{2} + 12x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 285) |
| 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
\( x^{7} - 3x^{6} - 8x^{5} + 26x^{4} + 11x^{3} - 51x^{2} + 12x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 10\nu^{3} + 21\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 8\nu^{3} + 25\nu^{2} - 15\nu - 8 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 16\nu^{3} + 11\nu^{2} - 28\nu + 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{4} + 10\beta_{3} + 10\beta_{2} + 29\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{6} - 8\beta_{5} + 12\beta_{4} + 12\beta_{3} + 57\beta_{2} + 14\beta _1 + 132 \)
|
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 |
|
−2.73532 | 0 | 5.48197 | 0 | 0 | 2.95440 | −9.52431 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.47637 | 0 | 4.13242 | 0 | 0 | −3.36493 | −5.28066 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.57229 | 0 | 0.472094 | 0 | 0 | 1.87913 | 2.40231 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.498112 | 0 | −1.75188 | 0 | 0 | 4.62756 | 1.86886 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.184902 | 0 | −1.96581 | 0 | 0 | −1.90997 | −0.733287 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.70383 | 0 | 0.903045 | 0 | 0 | −0.338398 | −1.86903 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.39336 | 0 | 3.72816 | 0 | 0 | 4.15221 | 4.13612 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 4275.2.a.bv | 7 | |
| 3.b | odd | 2 | 1 | 1425.2.a.z | 7 | ||
| 5.b | even | 2 | 1 | 4275.2.a.bw | 7 | ||
| 5.c | odd | 4 | 2 | 855.2.c.g | 14 | ||
| 15.d | odd | 2 | 1 | 1425.2.a.y | 7 | ||
| 15.e | even | 4 | 2 | 285.2.c.b | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.2.c.b | ✓ | 14 | 15.e | even | 4 | 2 | |
| 855.2.c.g | 14 | 5.c | odd | 4 | 2 | ||
| 1425.2.a.y | 7 | 15.d | odd | 2 | 1 | ||
| 1425.2.a.z | 7 | 3.b | odd | 2 | 1 | ||
| 4275.2.a.bv | 7 | 1.a | even | 1 | 1 | trivial | |
| 4275.2.a.bw | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4275))\):
|
\( T_{2}^{7} + 3T_{2}^{6} - 8T_{2}^{5} - 26T_{2}^{4} + 11T_{2}^{3} + 51T_{2}^{2} + 12T_{2} - 4 \)
|
|
\( T_{7}^{7} - 8T_{7}^{6} - T_{7}^{5} + 126T_{7}^{4} - 166T_{7}^{3} - 418T_{7}^{2} + 568T_{7} + 232 \)
|
|
\( T_{11}^{7} - 4T_{11}^{6} - 43T_{11}^{5} + 112T_{11}^{4} + 680T_{11}^{3} - 546T_{11}^{2} - 4016T_{11} - 3232 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 3 T^{6} + \cdots - 4 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} - 8 T^{6} + \cdots + 232 \)
|
| $11$ |
\( T^{7} - 4 T^{6} + \cdots - 3232 \)
|
| $13$ |
\( T^{7} - 8 T^{6} + \cdots - 800 \)
|
| $17$ |
\( T^{7} + 4 T^{6} + \cdots - 40000 \)
|
| $19$ |
\( (T - 1)^{7} \)
|
| $23$ |
\( T^{7} + 10 T^{6} + \cdots + 14272 \)
|
| $29$ |
\( T^{7} - 6 T^{6} + \cdots - 20000 \)
|
| $31$ |
\( T^{7} - 4 T^{6} + \cdots - 13568 \)
|
| $37$ |
\( T^{7} - 14 T^{6} + \cdots - 21376 \)
|
| $41$ |
\( T^{7} + 2 T^{6} + \cdots - 128 \)
|
| $43$ |
\( T^{7} + 2 T^{6} + \cdots - 12776 \)
|
| $47$ |
\( T^{7} + 30 T^{6} + \cdots - 128944 \)
|
| $53$ |
\( T^{7} - 64 T^{5} + \cdots + 1856 \)
|
| $59$ |
\( T^{7} - 18 T^{6} + \cdots - 40960 \)
|
| $61$ |
\( T^{7} - 12 T^{6} + \cdots + 5641360 \)
|
| $67$ |
\( T^{7} - 18 T^{6} + \cdots + 10240 \)
|
| $71$ |
\( T^{7} - 18 T^{6} + \cdots + 10240 \)
|
| $73$ |
\( T^{7} - 10 T^{6} + \cdots - 136832 \)
|
| $79$ |
\( T^{7} + 4 T^{6} + \cdots - 81920 \)
|
| $83$ |
\( T^{7} + 18 T^{6} + \cdots - 145856 \)
|
| $89$ |
\( T^{7} - 8 T^{6} + \cdots + 652000 \)
|
| $97$ |
\( T^{7} - 20 T^{6} + \cdots + 18272 \)
|
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