Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2015,2,Mod(1,2015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, 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("2015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2015 = 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2015.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: | \(16.0898560073\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.1299600812.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 6x^{6} + 10x^{5} + 12x^{4} - 13x^{3} - 8x^{2} + 3x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 6x^{6} + 10x^{5} + 12x^{4} - 13x^{3} - 8x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 4\nu^{4} + 7\nu^{3} + 4\nu^{2} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 5\nu^{5} - 8\nu^{4} - 8\nu^{3} + 6\nu^{2} + 4\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 5\nu^{5} + 9\nu^{4} + 7\nu^{3} - 11\nu^{2} - \nu + 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 10\nu^{4} + 11\nu^{3} - 12\nu^{2} - 4\nu + 1 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 11\nu^{4} + 10\nu^{3} - 16\nu^{2} - 3\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{7} + 3\beta_{6} + 3\beta_{5} + 3\beta_{4} + 2\beta_{2} + 5\beta _1 + 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{2} + 6\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -9\beta_{7} + 7\beta_{6} + 13\beta_{5} + 11\beta_{4} + 10\beta_{2} + 27\beta _1 + 23 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + 9\beta_{5} + 7\beta_{4} + \beta_{3} + 7\beta_{2} + 32\beta _1 + 29 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -29\beta_{7} + 11\beta_{6} + 65\beta_{5} + 45\beta_{4} + 4\beta_{3} + 46\beta_{2} + 143\beta _1 + 109 ) / 2 \)
|
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 |
|
−2.34510 | 1.76768 | 3.49951 | 1.00000 | −4.14539 | −0.300948 | −3.51650 | 0.124690 | −2.34510 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.32645 | −0.727407 | −0.240519 | 1.00000 | 0.964872 | −3.58712 | 2.97195 | −2.47088 | −1.32645 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.19127 | 1.64128 | −0.580874 | 1.00000 | −1.95521 | 1.27906 | 3.07452 | −0.306188 | −1.19127 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.544536 | −2.08393 | −1.70348 | 1.00000 | 1.13478 | 2.03459 | 2.01668 | 1.34277 | −0.544536 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.592215 | −2.25183 | −1.64928 | 1.00000 | −1.33357 | −1.51375 | −2.16116 | 2.07074 | 0.592215 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.889024 | −0.0365626 | −1.20964 | 1.00000 | −0.0325050 | 3.61512 | −2.85344 | −2.99866 | 0.889024 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.66553 | 1.13151 | 0.773988 | 1.00000 | 1.88456 | −2.62603 | −2.04196 | −1.71969 | 1.66553 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.26060 | −2.44074 | 3.11029 | 1.00000 | −5.51753 | 0.0990778 | 2.50992 | 2.95722 | 2.26060 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
| \(31\) | \(-1\) |
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 | 2015.2.a.h | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2015.2.a.h | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\):
|
\( T_{2}^{8} - 9T_{2}^{6} + 23T_{2}^{4} + T_{2}^{3} - 19T_{2}^{2} + 4 \)
|
|
\( T_{3}^{8} + 3T_{3}^{7} - 7T_{3}^{6} - 22T_{3}^{5} + 17T_{3}^{4} + 49T_{3}^{3} - 16T_{3}^{2} - 28T_{3} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 9 T^{6} + \cdots + 4 \)
|
| $3$ |
\( T^{8} + 3 T^{7} + \cdots - 1 \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} + T^{7} - 20 T^{6} + \cdots + 4 \)
|
| $11$ |
\( T^{8} + 13 T^{7} + \cdots + 4 \)
|
| $13$ |
\( (T + 1)^{8} \)
|
| $17$ |
\( T^{8} + T^{7} + \cdots - 1009 \)
|
| $19$ |
\( T^{8} + 3 T^{7} + \cdots - 251 \)
|
| $23$ |
\( T^{8} + 7 T^{7} + \cdots + 16 \)
|
| $29$ |
\( T^{8} + 24 T^{7} + \cdots + 95488 \)
|
| $31$ |
\( (T - 1)^{8} \)
|
| $37$ |
\( T^{8} + 18 T^{7} + \cdots + 3721 \)
|
| $41$ |
\( T^{8} + 8 T^{7} + \cdots - 54361 \)
|
| $43$ |
\( T^{8} + 13 T^{7} + \cdots - 17957 \)
|
| $47$ |
\( T^{8} - 9 T^{7} + \cdots + 162164 \)
|
| $53$ |
\( T^{8} + 5 T^{7} + \cdots - 390227 \)
|
| $59$ |
\( T^{8} + 9 T^{7} + \cdots + 571 \)
|
| $61$ |
\( T^{8} + 23 T^{7} + \cdots + 63968 \)
|
| $67$ |
\( T^{8} + 11 T^{7} + \cdots + 194528 \)
|
| $71$ |
\( T^{8} + 3 T^{7} + \cdots + 59767 \)
|
| $73$ |
\( T^{8} + 9 T^{7} + \cdots + 103249 \)
|
| $79$ |
\( T^{8} + 16 T^{7} + \cdots - 17536 \)
|
| $83$ |
\( T^{8} - 33 T^{7} + \cdots + 70421399 \)
|
| $89$ |
\( T^{8} + 44 T^{7} + \cdots + 2456876 \)
|
| $97$ |
\( T^{8} + 39 T^{7} + \cdots - 2554228 \)
|
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