Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2415,2,Mod(1,2415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, 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("2415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2415.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: | \(19.2838720881\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 87x^{6} - 143x^{5} - 196x^{4} + 244x^{3} + 160x^{2} - 89x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 87x^{6} - 143x^{5} - 196x^{4} + 244x^{3} + 160x^{2} - 89x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 15\nu^{7} + 11\nu^{6} + 72\nu^{5} - 25\nu^{4} - 123\nu^{3} - 5\nu^{2} + 37\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 15\nu^{7} + 11\nu^{6} + 72\nu^{5} - 25\nu^{4} - 125\nu^{3} - 5\nu^{2} + 47\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{9} - 16\nu^{7} - 2\nu^{6} + 84\nu^{5} + 24\nu^{4} - 158\nu^{3} - 62\nu^{2} + 57\nu + 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{9} + 16\nu^{7} + 3\nu^{6} - 83\nu^{5} - 35\nu^{4} + 149\nu^{3} + 91\nu^{2} - 39\nu - 14 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{9} - \nu^{8} - 47\nu^{7} + 7\nu^{6} + 238\nu^{5} + 23\nu^{4} - 421\nu^{3} - 131\nu^{2} + 119\nu + 16 ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( -2\nu^{9} + 33\nu^{7} + 4\nu^{6} - 178\nu^{5} - 47\nu^{4} + 335\nu^{3} + 121\nu^{2} - 103\nu - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 7\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{8} + \beta_{6} - 9\beta_{5} + 10\beta_{4} - \beta_{2} + 29\beta _1 + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{4} + 11\beta_{3} + 49\beta_{2} - 2\beta _1 + 135 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{9} - 10\beta_{8} + 12\beta_{6} - 71\beta_{5} + 81\beta_{4} - \beta_{3} - 14\beta_{2} + 184\beta _1 + 38 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{9} + 15 \beta_{8} + 13 \beta_{7} + \beta_{6} + 2 \beta_{5} - 19 \beta_{4} + 93 \beta_{3} + \cdots + 885 \)
|
| \(\nu^{9}\) | \(=\) |
\( 16 \beta_{9} - 74 \beta_{8} + 2 \beta_{7} + 109 \beta_{6} - 538 \beta_{5} + 612 \beta_{4} - 18 \beta_{3} + \cdots + 161 \)
|
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 |
|
−2.75158 | 1.00000 | 5.57120 | 1.00000 | −2.75158 | 1.00000 | −9.82644 | 1.00000 | −2.75158 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.90103 | 1.00000 | 1.61393 | 1.00000 | −1.90103 | 1.00000 | 0.733935 | 1.00000 | −1.90103 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.78717 | 1.00000 | 1.19399 | 1.00000 | −1.78717 | 1.00000 | 1.44048 | 1.00000 | −1.78717 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.910389 | 1.00000 | −1.17119 | 1.00000 | −0.910389 | 1.00000 | 2.88702 | 1.00000 | −0.910389 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.115446 | 1.00000 | −1.98667 | 1.00000 | −0.115446 | 1.00000 | 0.460244 | 1.00000 | −0.115446 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.509075 | 1.00000 | −1.74084 | 1.00000 | 0.509075 | 1.00000 | −1.90437 | 1.00000 | 0.509075 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.88703 | 1.00000 | 1.56089 | 1.00000 | 1.88703 | 1.00000 | −0.828610 | 1.00000 | 1.88703 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.91628 | 1.00000 | 1.67212 | 1.00000 | 1.91628 | 1.00000 | −0.628312 | 1.00000 | 1.91628 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.51086 | 1.00000 | 4.30443 | 1.00000 | 2.51086 | 1.00000 | 5.78609 | 1.00000 | 2.51086 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.64238 | 1.00000 | 4.98215 | 1.00000 | 2.64238 | 1.00000 | 7.87996 | 1.00000 | 2.64238 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 2415.2.a.w | ✓ | 10 |
| 3.b | odd | 2 | 1 | 7245.2.a.bv | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2415.2.a.w | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 7245.2.a.bv | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):
|
\( T_{2}^{10} - 2 T_{2}^{9} - 16 T_{2}^{8} + 30 T_{2}^{7} + 87 T_{2}^{6} - 143 T_{2}^{5} - 196 T_{2}^{4} + \cdots - 12 \)
|
|
\( T_{11}^{10} - 9 T_{11}^{9} - 31 T_{11}^{8} + 415 T_{11}^{7} - 223 T_{11}^{6} - 4185 T_{11}^{5} + \cdots + 13056 \)
|
|
\( T_{13}^{10} - 14 T_{13}^{9} + 23 T_{13}^{8} + 448 T_{13}^{7} - 1811 T_{13}^{6} - 3002 T_{13}^{5} + \cdots + 25024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots - 12 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( (T - 1)^{10} \)
|
| $11$ |
\( T^{10} - 9 T^{9} + \cdots + 13056 \)
|
| $13$ |
\( T^{10} - 14 T^{9} + \cdots + 25024 \)
|
| $17$ |
\( T^{10} - 8 T^{9} + \cdots - 214464 \)
|
| $19$ |
\( T^{10} - 13 T^{9} + \cdots - 7808 \)
|
| $23$ |
\( (T + 1)^{10} \)
|
| $29$ |
\( T^{10} - 10 T^{9} + \cdots + 24611328 \)
|
| $31$ |
\( T^{10} - 8 T^{9} + \cdots + 188416 \)
|
| $37$ |
\( T^{10} - 8 T^{9} + \cdots + 4336304 \)
|
| $41$ |
\( T^{10} + 5 T^{9} + \cdots + 2321376 \)
|
| $43$ |
\( T^{10} - 4 T^{9} + \cdots - 55440128 \)
|
| $47$ |
\( T^{10} - T^{9} + \cdots - 16662528 \)
|
| $53$ |
\( T^{10} - 9 T^{9} + \cdots + 1388928 \)
|
| $59$ |
\( T^{10} + 17 T^{9} + \cdots + 15264 \)
|
| $61$ |
\( T^{10} - 19 T^{9} + \cdots - 41801456 \)
|
| $67$ |
\( T^{10} - 239 T^{8} + \cdots + 2011136 \)
|
| $71$ |
\( T^{10} - 232 T^{8} + \cdots - 49152 \)
|
| $73$ |
\( T^{10} - 6 T^{9} + \cdots + 3179632 \)
|
| $79$ |
\( T^{10} + \cdots + 1067433984 \)
|
| $83$ |
\( T^{10} + \cdots - 1263122688 \)
|
| $89$ |
\( T^{10} - 10 T^{9} + \cdots + 771072 \)
|
| $97$ |
\( T^{10} + \cdots + 329532928 \)
|
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