Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5175,2,Mod(1,5175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5175, 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("5175.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5175.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: | \(41.3225830460\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.98838128.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 10x^{4} - x^{3} + 16x^{2} + 5x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1035) |
| 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
\( x^{6} - x^{5} - 10x^{4} - x^{3} + 16x^{2} + 5x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 6\nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 4\nu + 5 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 3\nu^{4} - 18\nu^{3} + 7\nu^{2} + 24\nu - 3 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} - 3\nu^{4} - 18\nu^{3} + 5\nu^{2} + 26\nu + 3 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 3\nu^{4} - 18\nu^{3} + 5\nu^{2} + 30\nu + 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 3\beta_{4} + 2\beta_{3} + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} - 6\beta_{4} + 2\beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 19\beta_{5} - 41\beta_{4} + 22\beta_{3} + 4\beta_{2} + 4\beta _1 + 48 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 85\beta_{5} - 147\beta_{4} + 64\beta_{3} + 6\beta_{2} + 24\beta _1 + 126 ) / 2 \)
|
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 |
|
−2.50089 | 0 | 4.25445 | 0 | 0 | −3.78907 | −5.63815 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.11684 | 0 | 2.48100 | 0 | 0 | 1.01690 | −1.01820 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.605771 | 0 | −1.63304 | 0 | 0 | −3.25359 | 2.20079 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.810417 | 0 | −1.34322 | 0 | 0 | 4.60617 | −2.70941 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.70496 | 0 | 0.906890 | 0 | 0 | −3.36634 | −1.86371 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.70812 | 0 | 5.33392 | 0 | 0 | −1.21406 | 9.02867 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
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 | 5175.2.a.by | 6 | |
| 3.b | odd | 2 | 1 | 5175.2.a.bz | 6 | ||
| 5.b | even | 2 | 1 | 1035.2.a.q | yes | 6 | |
| 15.d | odd | 2 | 1 | 1035.2.a.p | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1035.2.a.p | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 1035.2.a.q | yes | 6 | 5.b | even | 2 | 1 | |
| 5175.2.a.by | 6 | 1.a | even | 1 | 1 | trivial | |
| 5175.2.a.bz | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5175))\):
|
\( T_{2}^{6} - 11T_{2}^{4} + 30T_{2}^{2} - 4T_{2} - 12 \)
|
|
\( T_{7}^{6} + 6T_{7}^{5} - 12T_{7}^{4} - 134T_{7}^{3} - 201T_{7}^{2} + 116T_{7} + 236 \)
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\( T_{11}^{6} + 4T_{11}^{5} - 42T_{11}^{4} - 196T_{11}^{3} + 100T_{11}^{2} + 928T_{11} + 576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 11 T^{4} + \cdots - 12 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 236 \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots + 576 \)
|
| $13$ |
\( T^{6} + 12 T^{5} + \cdots + 496 \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots + 108 \)
|
| $19$ |
\( T^{6} - 8 T^{5} + \cdots + 64 \)
|
| $23$ |
\( (T - 1)^{6} \)
|
| $29$ |
\( T^{6} - 6 T^{5} + \cdots + 3408 \)
|
| $31$ |
\( T^{6} - 8 T^{5} + \cdots + 64 \)
|
| $37$ |
\( T^{6} + 22 T^{5} + \cdots - 63824 \)
|
| $41$ |
\( T^{6} - 18 T^{5} + \cdots - 7056 \)
|
| $43$ |
\( T^{6} + 8 T^{5} + \cdots - 12096 \)
|
| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 6912 \)
|
| $53$ |
\( T^{6} - 4 T^{5} + \cdots + 1212 \)
|
| $59$ |
\( T^{6} - 6 T^{5} + \cdots + 348 \)
|
| $61$ |
\( T^{6} - 126 T^{4} + \cdots + 16784 \)
|
| $67$ |
\( T^{6} + 10 T^{5} + \cdots - 19892 \)
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| $71$ |
\( T^{6} - 22 T^{5} + \cdots + 308076 \)
|
| $73$ |
\( T^{6} + 12 T^{5} + \cdots - 336 \)
|
| $79$ |
\( T^{6} - 4 T^{5} + \cdots - 12288 \)
|
| $83$ |
\( T^{6} - 24 T^{5} + \cdots + 31536 \)
|
| $89$ |
\( T^{6} - 8 T^{5} + \cdots - 575232 \)
|
| $97$ |
\( T^{6} + 12 T^{5} + \cdots + 93952 \)
|
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